formulating moderation hypothesis

The Three Most Common Types of Hypotheses

In this post, I discuss three of the most common hypotheses in psychology research, and what statistics are often used to test them.

Post author By sean
Post date September 28, 2013
37 Comments on The Three Most Common Types of Hypotheses

Simple main effects (i.e., X leads to Y) are usually not going to get you published. Main effects can be exciting in the early stages of research to show the existence of a new effect, but as a field matures the types of questions that scientists are trying to answer tend to become more nuanced and specific. In this post, I’ll briefly describe the three most common kinds of hypotheses that expand upon simple main effects – at least, the most common ones I’ve seen in my research career in psychology – as well as providing some resources to help you learn about how to test these hypotheses using statistics.

Incremental Validity

“Can X predict Y over and above other important predictors?”

This is probably the simplest of the three hypotheses I propose. Basically, you attempt to rule out potential confounding variables by controlling for them in your analysis. We do this because (in many cases) our predictor variables are correlated with each other. This is undesirable from a statistical perspective, but is common with real data. The idea is that we want to see if X can predict unique variance in Y over and above the other variables you include.

In terms of analysis, you are probably going to use some variation of multiple regression or partial correlations. For example, in my own work I’ve shown in the past that friendship intimacy as coded from autobiographical narratives can predict concern for the next generation over and above numerous other variables, such as optimism, depression, and relationship status ( Mackinnon et al., 2011 ).

“Under what conditions does X lead to Y?”

Of the three techniques I describe, moderation is probably the most tricky to understand. Essentially, it proposes that the size of a relationship between two variables changes depending upon the value of a third variable, known as a “moderator.” For example, in the diagram below you might find a simple main effect that is moderated by sex. That is, the relationship is stronger for women than for men:

With moderation, it is important to note that the moderating variable can be a category (e.g., sex) or it can be a continuous variable (e.g., scores on a personality questionnaire). When a moderator is continuous, usually you’re making statements like: “As the value of the moderator increases, the relationship between X and Y also increases.”

“Does X predict M, which in turn predicts Y?”

We might know that X leads to Y, but a mediation hypothesis proposes a mediating, or intervening variable. That is, X leads to M, which in turn leads to Y. In the diagram below I use a different way of visually representing things consistent with how people typically report things when using path analysis.

I use mediation a lot in my own research. For example, I’ve published data suggesting the relationship between perfectionism and depression is mediated by relationship conflict ( Mackinnon et al., 2012 ). That is, perfectionism leads to increased conflict, which in turn leads to heightened depression. Another way of saying this is that perfectionism has an indirect effect on depression through conflict.

Helpful links to get you started testing these hypotheses

Depending on the nature of your data, there are multiple ways to address each of these hypotheses using statistics. They can also be combined together (e.g., mediated moderation). Nonetheless, a core understanding of these three hypotheses and how to analyze them using statistics is essential for any researcher in the social or health sciences. Below are a few links that might help you get started:

Are you a little rusty with multiple regression? The basics of this technique are required for most common tests of these hypotheses. You might check out this guide as a helpful resource:

https://statistics.laerd.com/spss-tutorials/multiple-regression-using-spss-statistics.php

David Kenny’s Mediation Website provides an excellent overview of mediation and moderation for the beginner.

http://davidakenny.net/cm/mediate.htm

http://davidakenny.net/cm/moderation.htm

Preacher and Haye’s INDIRECT Macro is a great, easy way to implement mediation in SPSS software, and their MODPROBE macro is a useful tool for testing moderation.

http://afhayes.com/spss-sas-and-mplus-macros-and-code.html

If you want to graph the results of your moderation analyses, the excel calculators provided on Jeremy Dawson’s webpage are fantastic, easy-to-use tools:

http://www.jeremydawson.co.uk/slopes.htm

Tags mediation , moderation , regression , tutorial

37 replies on “The Three Most Common Types of Hypotheses”

I want to see clearly the three types of hypothesis

Thanks for your information. I really like this

Thank you so much, writing up my masters project now and wasn’t sure whether one of my variables was mediating or moderating….Much clearer now.

Thank you for simplified presentation. It is clearer to me now than ever before.

Thank you. Concise and clear

hello there

I would like to ask about mediation relationship: If I have three variables( X-M-Y)how many hypotheses should I write down? Should I have 2 or 3? In other words, should I have hypotheses for the mediating relationship? What about questions and objectives? Should be 3? Thank you.

Hi Osama. It’s really a stylistic thing. You could write it out as 3 separate hypotheses (X -> Y; X -> M; M -> Y) or you could just write out one mediation hypotheses “X will have an indirect effect on Y through M.” Usually, I’d write just the 1 because it conserves space, but either would be appropriate.

Hi Sean, according to the three steps model (Dudley, Benuzillo and Carrico, 2004; Pardo and Román, 2013)., we can test hypothesis of mediator variable in three steps: (X -> Y; X -> M; X and M -> Y). Then, we must use the Sobel test to make sure that the effect is significant after using the mediator variable.

Yes, but this is older advice. Best practice now is to calculate an indirect effect and use bootstrapping, rather than the causal steps approach and the more out-dated Sobel test. I’d recommend reading Hayes (2018) book for more info:

Hayes, A. F. (2018). Introduction to mediation, moderation, and conditional process analysis: A regression-based approach (2nd ed). Guilford Publications.

Hi! It’s been really helpful but I still don’t know how to formulate the hypothesis with my mediating variable.

I have one dependent variable DV which is formed by DV1 and DV2, then I have MV (mediating variable), and then 2 independent variables IV1, and IV2.

How many hypothesis should I write? I hope you can help me 🙂

Thank you so much!!

If I’m understanding you correctly, I guess 2 mediation hypotheses:

IV1 –> Med –> DV1&2 IV2 –> Med –> DV1&2

Thank you so much for your quick answer! ^^

Could you help me formulate my research question? English is not my mother language and I have trouble choosing the right words. My x = psychopathy y = aggression m = deficis in emotion recognition

thank you in advance

I have mediator and moderator how should I make my hypothesis

Can you have a negative partial effect? IV – M – DV. That is my M will have negative effect on the DV – e.g Social media usage (M) will partial negative mediate the relationship between father status (IV) and social connectedness (DV)?

Thanks in advance

Hi Ashley. Yes, this is possible, but often it means you have a condition known as “inconsistent mediation” which isn’t usually desirable. See this entry on David Kenny’s page:

Or look up “inconsistent mediation” in this reference:

MacKinnon, D. P., Fairchild, A. J., & Fritz, M. S. (2007). Mediation analysis. Annual Review of Psychology, 58, 593-614.

This is very interesting presentation. i love it.

This is very interesting and educative. I love it.

Hello, you mentioned that for the moderator, it changes the relationship between iv and dv depending on its strength. How would one describe a situation where if the iv is high iv and dv relationship is opposite from when iv is low. And then a 3rd variable maybe the moderator increases dv when iv is low and decreases dv when iv is high.

This isn’t problematic for moderation. Moderation just proposes that the magnitude of the relationship changes as levels of the moderator changes. If the sign flips, probably the original relationship was small. Sometimes people call this a “cross-over” effect, but really, it’s nothing special and can happen in any moderation analysis.

i want to use an independent variable as moderator after this i will have 3 independent variable and 1 dependent variable…. my confusion is do i need to have some past evidence of the X variable moderate the relationship of Y independent variable and Z dependent variable.

Dear Sean It is really helpful as my research model will use mediation. Because I still face difficulty in developing hyphothesis, can you give examples ? Thank you

Hi! is it possible to have all three pathways negative? My regression analysis showed significant negative relationships between x to y, x to m and m to y.

Hi, I have 1 independent variable, 1 dependent variable and 4 mediating variable May I know how many hypothesis should I develop?

Hello I have 4 IV , 1 mediating Variable and 1 DV

My model says that 4 IVs when mediated by 1MV leads to 1 Dv

Pls tell me how to set the hypothesis for mediation

Hi I have 4 IVs ,2 Mediating Variables , 1DV and 3 Outcomes (criterion variables).

Pls can u tell me how many hypotheses to set.

Thankyou in advance

I am in fact happy to read this webpage posts which carries tons of useful information, thanks for providing such data.

I see you don’t monetize savvystatistics.com, don’t waste your traffic, you can earn additional bucks every month with new monetization method. This is the best adsense alternative for any type of website (they approve all websites), for more info simply search in gooogle: murgrabia’s tools

what if the hypothesis and moderator significant in regrestion and insgificant in moderation?

Thank you so much!! Your slide on the mediator variable let me understand!

Very informative material. The author has used very clear language and I would recommend this for any student of research/

Hi Sean, thanks for the nice material. I have a question: for the second type of hypothesis, you state “That is, the relationship is stronger for men than for women”. Based on the illustration, wouldn’t the opposite be true?

Yes, your right! I updated the post to fix the typo, thank you!

I have 3 independent variable one mediator and 2 dependant variable how many hypothesis I have 2 write?

Sounds like 6 mediation hypotheses total:

X1 -> M -> Y1 X2 -> M -> Y1 X3 -> M -> Y1 X1 -> M -> Y2 X2 -> M -> Y2 X3 -> M -> Y2

Clear explanation! Thanks!

Section 7.3: Moderation Models, Assumptions, Interpretation, and Write Up

Last updated
Save as PDF
Page ID 31601

Learning Objectives

At the end of this section you should be able to answer the following questions:

What are some basic assumptions behind moderation?
What are the key components of a write up of moderation analysis?

Moderation Models

Difference between mediation & moderation.

The main difference between a simple interaction, like in ANOVA models or in moderation models, is that mediation implies that there is a causal sequence. In this case, we know that stress causes ill effects on health, so that would be the causal factor.

Some predictor variables interact in a sequence, rather than impacting the outcome variable singly or as a group (like regression).

Moderation and mediation is a form of regression that allows researchers to analyse how a third variable effects the relationship of the predictor and outcome variable.

Moderation analyses imply an interaction on the different levels of M

PowerPoint: Basic Moderation Model

Consider the below model:

Chapter Seven – Basic Moderation Model

Would the muscle percentage be the same for young, middle-aged, and older participants after training? We know that it is harder to build muscle as we age, so would training have a lower effect on muscle growth in older people?

Example Research Question:

Does cyberbullying moderate the relationship between perceived stress and mental distress?

Moderation Assumptions

The dependent and independent variables should be measured on a continuous scale.
There should be a moderator variable that is a nominal variable with at least two groups.
The variables of interest (the dependent variable and the independent and moderator variables) should have a linear relationship, which you can check with a scatterplot.
The data must not show multicollinearity (see Multiple Regression).
There should be no significant outliers, and the distribution of the variables should be approximately normal.

Moderation Interpretation

PowerPoint: Moderation menu, results and output

Please have a look at the following link for the Moderation Menu and Output:

Chapter Seven – Moderation Output

Interpretation

The effects of cyberbullying can be seen in blue, with the perceived stress in green. These are the main effects of the X and M variable on the outcome variable (Y). The interaction effect can be seen in purple. This will tell us if perceived stress is effecting mental distress equally for average, lower than average or higher than average levels of cyberbullying. If this is significant, then there is a difference in that effect. As can be seen in yellow and grey, cyberbullying has an effect on mental distress, but the effect is stronger for those who report higher levels of cyberbullying (see graph).

Moderation Write Up

The following text represents a moderation write up:

A moderation test was run, with perceived stress as the predictor, mental distress as the dependant, and cyberbullying as a moderator. There was a significant main effect found between perceived stress and mental distress, b = -1.23, BCa CI [1.11, 1.34], z =21.38 , p <.001, and nonsignificant main effect of cyberbullying on mental distress b = 1.05, BCa CI [0.72, 1.38], z=6.28, p < .001. There was a significant interaction found by cyberbullying on perceived stress and mental distress, b = -0.05, BCa CI [0.01, 0.09], z=2.16, p =.031. It was found that participants who reported higher than average levels of cyberbullying experienced a greater effect of perceived stress on mental distress ( b = 1.35, BCa CI [1.19, 1.50], z=17.1, p < .001), when compared to average or lower than average levels of cyberbullying ( b = 1.23, BCa CI [1.11, 1.34], z=21.3, p < .001, b = 1.11, BCa CI [0.95, 1.27], z=13.8, p < .001, respectively). From these results, it can be concluded that the effect of perceived stress on mental distress is partially moderated by cyberbullying.

Have a language expert improve your writing

Run a free plagiarism check in 10 minutes, generate accurate citations for free.

Knowledge Base

Methodology

How to Write a Strong Hypothesis | Steps & Examples

How to Write a Strong Hypothesis | Steps & Examples

Published on May 6, 2022 by Shona McCombes . Revised on November 20, 2023.

A hypothesis is a statement that can be tested by scientific research. If you want to test a relationship between two or more variables, you need to write hypotheses before you start your experiment or data collection .

Example: Hypothesis

Daily apple consumption leads to fewer doctor’s visits.

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, other interesting articles, frequently asked questions about writing hypotheses.

A hypothesis states your predictions about what your research will find. It is a tentative answer to your research question that has not yet been tested. For some research projects, you might have to write several hypotheses that address different aspects of your research question.

A hypothesis is not just a guess – it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Variables in hypotheses

Hypotheses propose a relationship between two or more types of variables .

An independent variable is something the researcher changes or controls.
A dependent variable is something the researcher observes and measures.

If there are any control variables , extraneous variables , or confounding variables , be sure to jot those down as you go to minimize the chances that research bias will affect your results.

In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .

Here's why students love Scribbr's proofreading services

Discover proofreading & editing

Step 1. Ask a question

Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.

Step 2. Do some preliminary research

Your initial answer to the question should be based on what is already known about the topic. Look for theories and previous studies to help you form educated assumptions about what your research will find.

At this stage, you might construct a conceptual framework to ensure that you’re embarking on a relevant topic . This can also help you identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalize more complex constructs.

Step 3. Formulate your hypothesis

Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.

4. Refine your hypothesis

You need to make sure your hypothesis is specific and testable. There are various ways of phrasing a hypothesis, but all the terms you use should have clear definitions, and the hypothesis should contain:

The relevant variables
The specific group being studied
The predicted outcome of the experiment or analysis

5. Phrase your hypothesis in three ways

To identify the variables, you can write a simple prediction in if…then form. The first part of the sentence states the independent variable and the second part states the dependent variable.

In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.

If you are comparing two groups, the hypothesis can state what difference you expect to find between them.

6. Write a null hypothesis

If your research involves statistical hypothesis testing , you will also have to write a null hypothesis . The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0 , while the alternative hypothesis is H 1 or H a .

H 0 : The number of lectures attended by first-year students has no effect on their final exam scores.
H 1 : The number of lectures attended by first-year students has a positive effect on their final exam scores.

If you want to know more about the research process , methodology , research bias , or statistics , make sure to check out some of our other articles with explanations and examples.

Sampling methods
Simple random sampling
Stratified sampling
Cluster sampling
Likert scales
Reproducibility

Statistics

Null hypothesis
Statistical power
Probability distribution
Effect size
Poisson distribution

Research bias

Optimism bias
Cognitive bias
Implicit bias
Hawthorne effect
Anchoring bias
Explicit bias

Receive feedback on language, structure, and formatting

Professional editors proofread and edit your paper by focusing on:

Academic style
Vague sentences
Style consistency

See an example

A hypothesis is not just a guess — it should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations and statistical analysis of data).

Null and alternative hypotheses are used in statistical hypothesis testing . The null hypothesis of a test always predicts no effect or no relationship between variables, while the alternative hypothesis states your research prediction of an effect or relationship.

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

Cite this Scribbr article

If you want to cite this source, you can copy and paste the citation or click the “Cite this Scribbr article” button to automatically add the citation to our free Citation Generator.

McCombes, S. (2023, November 20). How to Write a Strong Hypothesis | Steps & Examples. Scribbr. Retrieved March 29, 2024, from https://www.scribbr.com/methodology/hypothesis/

Is this article helpful?

Shona McCombes

Other students also liked, construct validity | definition, types, & examples, what is a conceptual framework | tips & examples, operationalization | a guide with examples, pros & cons, what is your plagiarism score.

Have a language expert improve your writing

Run a free plagiarism check in 10 minutes, automatically generate references for free.

Knowledge Base
Methodology
How to Write a Strong Hypothesis | Guide & Examples

How to Write a Strong Hypothesis | Guide & Examples

Published on 6 May 2022 by Shona McCombes .

What is a hypothesis, developing a hypothesis (with example), hypothesis examples, frequently asked questions about writing hypotheses.

Variables in hypotheses

Hypotheses propose a relationship between two or more variables . An independent variable is something the researcher changes or controls. A dependent variable is something the researcher observes and measures.

In this example, the independent variable is exposure to the sun – the assumed cause . The dependent variable is the level of happiness – the assumed effect .

Prevent plagiarism, run a free check.

Step 1: ask a question.

Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project.

Step 2: Do some preliminary research

At this stage, you might construct a conceptual framework to identify which variables you will study and what you think the relationships are between them. Sometimes, you’ll have to operationalise more complex constructs.

Step 3: Formulate your hypothesis

Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence.

Step 4: Refine your hypothesis

The relevant variables
The specific group being studied
The predicted outcome of the experiment or analysis

Step 5: Phrase your hypothesis in three ways

To identify the variables, you can write a simple prediction in if … then form. The first part of the sentence states the independent variable and the second part states the dependent variable.

In academic research, hypotheses are more commonly phrased in terms of correlations or effects, where you directly state the predicted relationship between variables.

If you are comparing two groups, the hypothesis can state what difference you expect to find between them.

Step 6. Write a null hypothesis

If your research involves statistical hypothesis testing , you will also have to write a null hypothesis. The null hypothesis is the default position that there is no association between the variables. The null hypothesis is written as H 0 , while the alternative hypothesis is H 1 or H a .

A hypothesis is not just a guess. It should be based on existing theories and knowledge. It also has to be testable, which means you can support or refute it through scientific research methods (such as experiments, observations, and statistical analysis of data).

A research hypothesis is your proposed answer to your research question. The research hypothesis usually includes an explanation (‘ x affects y because …’).

A statistical hypothesis, on the other hand, is a mathematical statement about a population parameter. Statistical hypotheses always come in pairs: the null and alternative hypotheses. In a well-designed study , the statistical hypotheses correspond logically to the research hypothesis.

Cite this Scribbr article

If you want to cite this source, you can copy and paste the citation or click the ‘Cite this Scribbr article’ button to automatically add the citation to our free Reference Generator.

McCombes, S. (2022, May 06). How to Write a Strong Hypothesis | Guide & Examples. Scribbr. Retrieved 25 March 2024, from https://www.scribbr.co.uk/research-methods/hypothesis-writing/

Is this article helpful?

Shona McCombes

Other students also liked, operationalisation | a guide with examples, pros & cons, what is a conceptual framework | tips & examples, a quick guide to experimental design | 5 steps & examples.

User Preferences

Content preview.

Arcu felis bibendum ut tristique et egestas quis:

Ut enim ad minim veniam, quis nostrud exercitation ullamco laboris
Duis aute irure dolor in reprehenderit in voluptate
Excepteur sint occaecat cupidatat non proident

Keyboard Shortcuts

5.2 - writing hypotheses.

The first step in conducting a hypothesis test is to write the hypothesis statements that are going to be tested. For each test you will have a null hypothesis ($H_0$) and an alternative hypothesis ($H_a$).

When writing hypotheses there are three things that we need to know: (1) the parameter that we are testing (2) the direction of the test (non-directional, right-tailed or left-tailed), and (3) the value of the hypothesized parameter.

At this point we can write hypotheses for a single mean ($\mu$), paired means($\mu_d$), a single proportion ($p$), the difference between two independent means ($\mu_1-\mu_2$), the difference between two proportions ($p_1-p_2$), a simple linear regression slope ($\beta$), and a correlation ($\rho$).
The research question will give us the information necessary to determine if the test is two-tailed (e.g., "different from," "not equal to"), right-tailed (e.g., "greater than," "more than"), or left-tailed (e.g., "less than," "fewer than").
The research question will also give us the hypothesized parameter value. This is the number that goes in the hypothesis statements (i.e., $\mu_0$ and $p_0$). For the difference between two groups, regression, and correlation, this value is typically 0.

Hypotheses are always written in terms of population parameters (e.g., $p$ and $\mu$). The tables below display all of the possible hypotheses for the parameters that we have learned thus far. Note that the null hypothesis always includes the equality (i.e., =).

Chapter 1: Introduction
Chapter 2: Indexing
Chapter 3: Loops & Logicals
Chapter 4: Apply Family
Chapter 5: Plyr Package
Chapter 6: Vectorizing
Chapter 7: Sample & Replicate
Chapter 8: Melting & Casting
Chapter 9: Tidyr Package
Chapter 10: GGPlot1: Basics
Chapter 11: GGPlot2: Bars & Boxes
Chapter 12: Linear & Multiple
Chapter 13: Ploting Interactions
Chapter 14: Moderation/Mediation
Chapter 15: Moderated-Mediation
Chapter 16: MultiLevel Models
Chapter 17: Mixed Models
Chapter 18: Mixed Assumptions Testing
Chapter 19: Logistic & Poisson
Chapter 20: Between-Subjects
Chapter 21: Within- & Mixed-Subjects
Chapter 22: Correlations
Chapter 23: ARIMA
Chapter 24: Decision Trees
Chapter 25: Signal Detection
Chapter 26: Intro to Shiny
Chapter 27: ANOVA Variance
Download Rmd

Chapter 14: Mediation and Moderation

Alyssa blair, 1 what are mediation and moderation.

Mediation analysis tests a hypothetical causal chain where one variable X affects a second variable M and, in turn, that variable affects a third variable Y. Mediators describe the how or why of a (typically well-established) relationship between two other variables and are sometimes called intermediary variables since they often describe the process through which an effect occurs. This is also sometimes called an indirect effect. For instance, people with higher incomes tend to live longer but this effect is explained by the mediating influence of having access to better health care.

In R, this kind of analysis may be conducted in two ways: Baron & Kenny’s (1986) 4-step indirect effect method and the more recent mediation package (Tingley, Yamamoto, Hirose, Keele, & Imai, 2014). The Baron & Kelly method is among the original methods for testing for mediation but tends to have low statistical power. It is covered in this chapter because it provides a very clear approach to establishing relationships between variables and is still occassionally requested by reviewers. However, the mediation package method is highly recommended as a more flexible and statistically powerful approach.

Moderation analysis also allows you to test for the influence of a third variable, Z, on the relationship between variables X and Y. Rather than testing a causal link between these other variables, moderation tests for when or under what conditions an effect occurs. Moderators can stength, weaken, or reverse the nature of a relationship. For example, academic self-efficacy (confidence in own’s ability to do well in school) moderates the relationship between task importance and the amount of test anxiety a student feels (Nie, Lau, & Liau, 2011). Specifically, students with high self-efficacy experience less anxiety on important tests than students with low self-efficacy while all students feel relatively low anxiety for less important tests. Self-efficacy is considered a moderator in this case because it interacts with task importance, creating a different effect on test anxiety at different levels of task importance.

In general (and thus in R), moderation can be tested by interacting variables of interest (moderator with IV) and plotting the simple slopes of the interaction, if present. A variety of packages also include functions for testing moderation but as the underlying statistical approaches are the same, only the “by hand” approach is covered in detail in here.

Finally, this chapter will cover these basic mediation and moderation techniques only. For more complicated techniques, such as multiple mediation, moderated mediation, or mediated moderation please see the mediation package’s full documentation.

1.1 Getting Started

If necessary, review the Chapter on regression. Regression test assumptions may be tested with gvlma . You may load all the libraries below or load them as you go along. Review the help section of any packages you may be unfamiliar with ?(packagename).

2 Mediation Analyses

Mediation tests whether the effects of X (the independent variable) on Y (the dependent variable) operate through a third variable, M (the mediator). In this way, mediators explain the causal relationship between two variables or “how” the relationship works, making it a very popular method in psychological research.

Both mediation and moderation assume that there is little to no measurement error in the mediator/moderator variable and that the DV did not CAUSE the mediator/moderator. If mediator error is likely to be high, researchers should collect multiple indicators of the construct and use SEM to estimate latent variables. The safest ways to make sure your mediator is not caused by your DV are to experimentally manipulate the variable or collect the measurement of your mediator before you introduce your IV.

Total Effect Model.

Basic Mediation Model.

c = the total effect of X on Y c = c’ + ab c’= the direct effect of X on Y after controlling for M; c’=c-ab ab= indirect effect of X on Y

The above shows the standard mediation model. Perfect mediation occurs when the effect of X on Y decreases to 0 with M in the model. Partial mediation occurs when the effect of X on Y decreases by a nontrivial amount (the actual amount is up for debate) with M in the model.

2.1 Example Mediation Data

Set an appropriate working directory and generate the following data set.

In this example we’ll say we are interested in whether the number of hours since dawn (X) affect the subjective ratings of wakefulness (Y) 100 graduate students through the consumption of coffee (M).

Note that we are intentionally creating a mediation effect here (because statistics is always more fun if we have something to find) and we do so below by creating M so that it is related to X and Y so that it is related to M. This creates the causal chain for our analysis to parse.

2.2 Method 1: Baron & Kenny

This is the original 4-step method used to describe a mediation effect. Steps 1 and 2 use basic linear regression while steps 3 and 4 use multiple regression. For help with regression, see Chapter 10.

The Steps: 1. Estimate the relationship between X on Y (hours since dawn on degree of wakefulness) -Path “c” must be significantly different from 0; must have a total effect between the IV & DV

Estimate the relationship between X on M (hours since dawn on coffee consumption) -Path “a” must be significantly different from 0; IV and mediator must be related.

Estimate the relationship between M on Y controlling for X (coffee consumption on wakefulness, controlling for hours since dawn) -Path “b” must be significantly different from 0; mediator and DV must be related. -The effect of X on Y decreases with the inclusion of M in the model

Estimate the relationship between Y on X controlling for M (wakefulness on hours since dawn, controlling for coffee consumption) -Should be non-significant and nearly 0.

2.3 Interpreting Barron & Kenny Results

Here we find that our total effect model shows a significant positive relationship between hours since dawn (X) and wakefulness (Y). Our Path A model shows that hours since down (X) is also positively related to coffee consumption (M). Our Path B model then shows that coffee consumption (M) positively predicts wakefulness (Y) when controlling for hours since dawn (X). Finally, wakefulness (Y) does not predict hours since dawn (X) when controlling for coffee consumption (M).

Since the relationship between hours since dawn and wakefulness is no longer significant when controlling for coffee consumption, this suggests that coffee consumption does in fact mediate this relationship. However, this method alone does not allow for a formal test of the indirect effect so we don’t know if the change in this relationship is truly meaningful.

There are two primary methods for formally testing the significance of the indirect test: the Sobel test & bootstrapping (covered under the mediatation method).

The Sobel Test uses a specialized t-test to determine if there is a significant reduction in the effect of X on Y when M is present. Using the sobel function of the multilevel package will show provide you with three of the basic models we ran before (Mod1 = Total Effect; Mod2 = Path B; and Mod3 = Path A) as well as an estimate of the indirect effect, the standard error of that effect, and the z-value for that effect. You can either use this value to calculate your p-value or run the mediation.test function from the bda package to receive a p-value for this estimate.

In this case, we can now confirm that the relationship between hours since dawn and feelings of wakefulness are significantly mediated by the consumption of coffee (z’ = 3.84, p < .001).

However, the Sobel Test is largely considered an outdated method since it assumes that the indirect effect (ab) is normally distributed and tends to only have adequate power with large sample sizes. Thus, again, it is highly recommended to use the mediation bootstrapping method instead.

2.4 Method 2: The Mediation Pacakge Method

This package uses the more recent bootstrapping method of Preacher & Hayes (2004) to address the power limitations of the Sobel Test. This method computes the point estimate of the indirect effect (ab) over a large number of random sample (typically 1000) so it does not assume that the data are normally distributed and is especially more suitable for small sample sizes than the Barron & Kenny method.

To run the mediate function, we will again need a model of our IV (hours since dawn), predicting our mediator (coffee consumption) like our Path A model above. We will also need a model of the direct effect of our IV (hours since dawn) on our DV (wakefulness), when controlling for our mediator (coffee consumption). When can then use mediate to repeatedly simulate a comparsion between these models and to test the signifcance of the indirect effect of coffee consumption.

2.5 Interpreting Mediation Results

The mediate function gives us our Average Causal Mediation Effects (ACME), our Average Direct Effects (ADE), our combined indirect and direct effects (Total Effect), and the ratio of these estimates (Prop. Mediated). The ACME here is the indirect effect of M (total effect - direct effect) and thus this value tells us if our mediation effect is significant.

In this case, our fitMed model again shows a signifcant affect of coffee consumption on the relationship between hours since dawn and feelings of wakefulness, (ACME = .28, p < .001) with no direct effect of hours since dawn (ADE = -0.11, p = .27) and significant total effect ( p < .05).

We can then bootstrap this comparison to verify this result in fitMedBoot and again find a significant mediation effect (ACME = .28, p < .001) and no direct effect of hours since dawn (ADE = -0.11, p = .27). However, with increased power, this analysis no longer shows a significant total effect ( p = .08).

3 Moderation Analyses

Moderation tests whether a variable (Z) affects the direction and/or strength of the relation between an IV (X) and a DV (Y). In other words, moderation tests for interactions that affect WHEN relationships between variables occur. Moderators are conceptually different from mediators (when versus how/why) but some variables may be a moderator or a mediator depending on your question. See the mediation package documentation for ways of testing more complicated mediated moderation/moderated mediation relationships.

Like mediation, moderation assumes that there is little to no measurement error in the moderator variable and that the DV did not CAUSE the moderator. If moderator error is likely to be high, researchers should collect multiple indicators of the construct and use SEM to estimate latent variables. The safest ways to make sure your moderator is not caused by your DV are to experimentally manipulate the variable or collect the measurement of your moderator before you introduce your IV.

Basic Moderation Model.

3.1 Example Moderation Data

In this example we’ll say we are interested in whether the relationship between the number of hours of sleep (X) a graduate student receives and the attention that they pay to this tutorial (Y) is influenced by their consumption of coffee (Z). Here we create the moderation effect by making our DV (Y) the product of levels of the IV (X) and our moderator (Z).

3.2 Moderation Analysis

Moderation can be tested by looking for significant interactions between the moderating variable (Z) and the IV (X). Notably, it is important to mean center both your moderator and your IV to reduce multicolinearity and make interpretation easier. Centering can be done using the scale function, which subtracts the mean of a variable from each value in that variable. For more information on the use of centering, see ?scale and any number of statistical textbooks that cover regression (we recommend Cohen, 2008).

A number of packages in R can also be used to conduct and plot moderation analyses, including the moderate.lm function of the QuantPsyc package and the pequod package. However, it is simple to do this “by hand” using traditional multiple regression, as shown here, and the underlying analysis (interacting the moderator and the IV) in these packages is identical to this approach. The rockchalk package used here is one of many graphing and plotting packages available in R and was chosen because it was especially designed for use with regression analyses (unlike the more general graphing options described in Chapters 8 & 9).

3.3 Interpreting Moderation Results

Results are presented similar to regular multiple regression results (see Chapter 10). Since we have significant interactions in this model, there is no need to interpret the separate main effects of either our IV or our moderator.

Our by hand model shows a significant interaction between hours slept and coffee consumption on attention paid to this tutorial (b = .23, SE = .04, p < .001). However, we’ll need to unpack this interaction visually to get a better idea of what this means.

The rockchalk function will automatically plot the simple slopes (1 SD above and 1 SD below the mean) of the moderating effect. This figure shows that those who drank less coffee (the black line) paid more attention with the more sleep that they got last night but paid less attention overall that average (the red line). Those who drank more coffee (the green line) paid more when they slept more as well and paid more attention than average. The difference in the slopes for those who drank more or less coffee shows that coffee consumption moderates the relationship between hours of sleep and attention paid.

4 References and Further Reading

Baron, R., & Kenny, D. (1986). The moderator-mediator variable distinction in social psychological research: Conceptual, strategic, and statistical considerations. Journal of Personality and Social Psychology, 51, 1173-1182.

Cohen, B. H. (2008). Explaining psychological statistics. John Wiley & Sons.

Imai, K., Keele, L., & Tingley, D. (2010). A general approach to causal mediation analysis. Psychological methods, 15(4), 309.

MacKinnon, D. P., Lockwood, C. M., Hoffman, J. M., West, S. G., & Sheets, V. (2002). A comparison of methods to test mediation and other intervening variable effects. Psychological methods, 7(1), 83.

Nie, Y., Lau, S., & Liau, A. K. (2011). Role of academic self-efficacy in moderating the relation between task importance and test anxiety. Learning and Individual Differences, 21(6), 736-741.

Tingley, D., Yamamoto, T., Hirose, K., Keele, L., & Imai, K. (2014). Mediation: R package for causal mediation analysis.

Want to create or adapt books like this? Learn more about how Pressbooks supports open publishing practices.

Section 7.1: Mediation and Moderation Models

Learning Objectives

At the end of this section you should be able to answer the following questions:

Define the concept of a moderator variable.
Define the concept of a mediator variable.

As we discussed in the lesson on correlations and regressions, understanding associations between psychological constructs can tell researchers a great deal about how certain mental health concerns and behaviours affects us on an emotional level. Correlation analyses focus on the relationship between two variables, and regression is the association of multiple independent variables with a single dependant variable.

Some predictor variables interact in a sequence, rather than impacting the outcome variable singly or as a group (like regression).

Moderation and mediation is a form of regression that allows researchers to analyse how a third variable effects the relationship of the predictor and outcome variable.

PowerPoint: Basic Mediation Model

Consider the Basic Mediation Model in this slide:

Chapter Seven – Basic Mediation Model

We know that high levels of stress can negatively impact health, we also know that a high level of social support can be beneficial to health. With these two points of knowledge, could it be that social support might provide a protective factor from the effects of stress on health? Thinking about a sequence of effects, perhaps social support can mediate the effect of stress on health.

Mediation is a more complicated extension of multiple regression procedures. Mediation examines the pattern of relationships among three variables (Simple Mediation Model), and can be used on four or more variables.

Examples of Research Questions

Here are some examples of research questions that could use a mediation analysis.

If an intervention increases secure attachment among young children, do behavioural problems decrease when the children enter school?
Does physical abuse in early childhood lead to deviant processing of social information that leads to aggressive behaviour?
Do performance expectations start a self-fulfilling prophecy that affects behaviour?
Can changes in cognitive attributions reduce depression?

PowerPoint: Three Mediation Figures

Consider the Three Figures Illustrating Mediation from the following slides:

Chapter Seven – Three Mediation Figures

Looking at this conceptual model, you can see the direct effect of X on Y. You can also see the effect of M on Y. What we are interested in is the effects of X on Y, accounting for the effects of M.

An example mediation model is that of the mediating effect of health-related behaviours on conscientiousness and overall physical health. Conscientiousness, or the personality trait associated with hardworking has relationship with overall physical health, but if an individual is hardworking, but does not perform health-related behaviours like exercise or diet control, then they are likely to be less healthy. From this, we can assume that health-related behaviours mediates the relationship between conscientiousness and physical health.

Share This Book

Statistics: Data analysis and modelling

Chapter 6 moderation and mediation.

In this chapter, we will focus on two ways in which one predictor variable may affect the relation between another predictor variable and the dependent variable. Moderation means the strength of the relation (in terms of the slope) of a predictor variable is determined by the value of another predictor variable. For instance, while physical attractiveness is generally positively related to mating success, for very rich people, physical attractiveness may not be so important. This is also called an interaction between the two predictor variables. Mediation is a different way in which two predictors affect a dependent variable. It is best thought of as a causal chain , where one predictor variable determines the value of another predictor variable, which then in turn determines the value of the dependent variable. the difference between moderation and mediation is illustrated in Figure 6.1 .

Figure 6.1: Graphical depiction of the difference between moderation and mediation. Moderation means that the effect of a predictor ( $X_1$ ) on the dependent variable ( $Y$ ) depends on the value of another predictor ( $X_2$ ). Mediation means that a predictor ( $X_1$ ) affects the dependent variable ( $Y$ ) indirectly, through its relation to another predictor ( $X_2$ ) which is directly related to the dependent variable.

6.1 Moderation

6.1.1 physical attractiveness and intelligence in speed dating.

Fisman, Iyengar, Kamenica, & Simonson ( 2006 ) conducted a large scale experiment 15 on dating behaviour. They placed their participants in a speed dating context, where they were randomly matched with a number of potential partners (between 5 and 20) and could converse for four minutes. As part of the study, after each meeting, participants rated how much they liked their speed dating partners, as well as more specifically on their attractiveness, sincerity, intelligence, fun, and ambition. We will focus in particular on ratings of physical attractiveness, fun, and intelligence, and how these are related to the general liking of a person. Ratings were given on a 10-point scale, from 1 (“awful”) to 10 (“great”). A multiple regression analysis predicting general liking from attractiveness, fun, and intelligence (Table 6.1 ) shows that all three predictors have a significant and positive relation with general liking.

6.1.2 Conditional slopes

If we were to model the relation between overall liking and physical attractiveness and intelligence, we might use a multiple regression model such as: 16 \[\texttt{like}_i = \beta_0 + \beta_{\texttt{attr}} \times \texttt{attr}_i + \beta_\texttt{intel} \times \texttt{intel}_i + \epsilon_i \quad \quad \epsilon_i \sim \mathbf{Normal}(0,\sigma_\epsilon)\] which is estimated as \[\texttt{like}_i = -0.0733 + 0.527 \times \texttt{attr}_i + 0.392 \times \texttt{intel}_i + \hat{\epsilon}_i \quad \quad \hat{\epsilon}_i \sim \mathbf{Normal}(0, 1.25)\] The estimates indicate a positive relation to liking of both attractiveness and intelligence. Note that the values of the slopes are different from those in Table 6.1 . The reason for this is that the model in the Table also includes fun as a predictor. Because the slopes reflect unique effects , these depend on all predictors included in the model. When there is dependence between the predictors (i.e. there is multicollinearity) both the estimates of the slopes and the corresponding significance tests will vary when you add or remove predictors from the model.

In the model above, a relative lack in physical attractiveness can be overcome by high intelligence, because in the end, the general liking of someone depends on the sum of both attractiveness and intelligence (each “scaled” by their corresponding slope). For example, someone with an attractiveness rating of $\texttt{attr}_i = 8$ and an intelligence rating of $\texttt{intel}_i = 2$ would be expected to be liked as much as a partner as someone with an attractiveness rating of $\texttt{attr}_i = 3.538$ and an intelligence rating of $\texttt{intel}_i = 8$ : \[\begin{aligned} \texttt{like}_i &= -0.073 + 0.527 \times 8 + 0.392 \times 2 = 4.924 \\ \texttt{like}_i &= -0.073 + 0.527 \times 3.538 + 0.392 \times 8 = 4.924 \end{aligned}\]

But what if for those lucky people who are very physically attractive, their intelligence doesn’t matter that much , or even at all ? And what if, for those lucky people who are very intelligent, their physical attractiveness doesn’t really matter much or at all? In other words, what if the more attractive people are, the less intelligence determines how much other people like them as a potential partner, and conversely, the more intelligent people are, the less attractiveness determines how much others like them as a potential partner? This implies that the effect of attractiveness on liking depends on intelligence, and that the effect of intelligence on liking depends on attractiveness. Such dependence is not captured by the multiple regression model above. While a relative lack of intelligence might be overcome by a relative abundance of attractiveness, for any level of intelligence, the additional effect of attractiveness is the same (i.e., an increase in attractiveness by one unit will always result in an increase of the predicted liking of 0.527).

Let’s define $\beta_{\texttt{attr}|\texttt{intel}_i}$ as the slope of $\texttt{attr}$ conditional on the value of $\texttt{intel}_i$ . That is, we allow the slope of $\texttt{attr}$ to vary as a function of $\texttt{intel}$ . Similarly, we can define $\beta_{\texttt{intel}|\texttt{attr}_i}$ as the slope of $\texttt{intel}$ conditional on the value of $\texttt{attr}$ . Our regression model can then be written as: \[\begin{equation} \texttt{like}_i = \beta_0 + \beta_{\texttt{attr}|\texttt{intel}_i} \times \texttt{attr}_i + \beta_{\texttt{intel} | \texttt{attr}_i} \times \texttt{intel}_i + \epsilon_i \tag{6.1} \end{equation}\] That’s a good start, but what would the value of $\beta_{\texttt{attr}|\texttt{intel}_i}$ be? Estimating the slope of $\texttt{attr}$ for each value of $\texttt{intel}$ by fitting regression models to each subset of data with a particular value of $\texttt{intel}$ is not really doable. We’d need lots and lots of data, and furthermore, we wouldn’t also be able to simultaneously estimate the value of $\beta_{\texttt{intel} | \texttt{attr}_i}$ . We need to supply some structure to $\beta_{\texttt{attr}|\texttt{intel}_i}$ to allow us to estimate its value without overcomplicating things.

6.1.3 Modeling slopes with linear models

One idea is to define $\beta_{\texttt{attr}|\texttt{intel}_i}$ with a linear model: \[\beta_{\texttt{attr}|\texttt{intel}_i} = \beta_{\texttt{attr},0} + \beta_{\texttt{attr},1} \times \texttt{intel}_i\] This is just like a simple linear regression model, but now the “dependent variable” is the slope of $\texttt{attr}$ . Defined in this way, the slope of $\texttt{attr}$ is $\beta_{\texttt{attr},0}$ when $\texttt{intel}_i = 0$ , and for every one-unit increase in $\texttt{intel}_i$ , the slope of $\texttt{attr}$ increases (or decreases) by $\beta_{\texttt{attr},1}$ . For example, let’s assume $\beta_{\texttt{attr},0} = 1$ and $\beta_{\texttt{attr},1} = 0.5$ . For someone with an intelligence rating of $\texttt{intel}_i = 0$ , the slope of $\texttt{attr}$ is \[\beta_{\texttt{attr}|\texttt{intel}_i} = 1 + 0.5 \times 0 = 1\] For someone with an intelligence rating of $\texttt{intel}_i = 1$ , the slope of $\texttt{attr}$ is \[\beta_{\texttt{attr}|\texttt{intel}_i} = 1 + 0.5 \times 1 = 1.5\] For someone with an intelligence rating of $\texttt{intel}_i = 2$ , the slope of $\texttt{attr}$ is \[\beta_{\texttt{attr}|\texttt{intel}_i} = 1 + 0.5 \times 2 = 2\] As you can see, for every increase in intelligence rating by 1 point, the slope of $\texttt{attr}$ increases by 0.5. In such a model, there will be values of $\texttt{intel}$ which result in a negative slope of $\texttt{attr}$ . For instance, for $\texttt{intel}_i = -4$ , the slope of $\texttt{attr}$ is \[\beta_{\texttt{attr}|\texttt{intel}_i} = 1 + 0.5 \times (-4) = - 1\]

We can define the slope of $\texttt{intel}$ in a similar manner as \[\beta_{\texttt{intel}|\texttt{attr}_i} = \beta_{\texttt{intel},0} + \beta_{\texttt{intel},1} \times \texttt{attr}_i\] When we plug these definitions into Equation (6.1) , we get \[\begin{aligned} \texttt{like}_i &= \beta_0 + (\beta_{\texttt{attr},0} + \beta_{\texttt{attr},1} \times \texttt{intel}_i) \times \texttt{attr}_i + (\beta_{\texttt{intel},0} + \beta_{\texttt{intel},1} \times \texttt{attr}_i) \times \texttt{intel}_i + \epsilon_i \\ &= \beta_0 + \beta_{\texttt{attr},0} \times \texttt{attr}_i + \beta_{\texttt{intel},0} \times \texttt{intel}_i + (\beta_{\texttt{attr},1} + \beta_{\texttt{intel},1}) \times (\texttt{attr}_i \times \texttt{intel}_i) + \epsilon_i \end{aligned}\]

Looking carefully at this formula, you can recognize a multiple regression model with three predictors: $\texttt{attr}$ , $\texttt{intel}$ , and a new predictor $\texttt{attr}_i \times \texttt{intel}_i$ , which is computed as the product of these two variables. While it is thus related to both variables, we can treat this product as just another predictor in the model. The slope of this new predictor is the sum of two terms, $\beta_{\texttt{attr},1} + \beta_{\texttt{intel},1}$ . Although we have defined these as different things (i.e. as the effect of $\texttt{intel}$ on the slope of $\texttt{attr}$ , and the effect of $\texttt{attr}$ on the slope of $\texttt{intel}$ , respectively), their value can not be estimated uniquely. We can only estimate their summed value. That means that moderation in regression is “symmetric”, in the sense that each predictor determines the slope of the other one. We can not say that it is just intelligence that determines the effect of attraction on liking, nor can we say that it is just attraction that determines the effect of intelligence on liking. The two variables interact and each determine the other’s effect on the dependent variable.

With that in mind, we can simplify the notation of the resulting model somewhat, by renaming the slopes of the two predictors to $\beta_{\texttt{attr}} = \beta_{\texttt{attr},0}$ and $\beta_{\texttt{intel}} = \beta_{\texttt{intel},0}$ , and using a single parameter for the sum $\beta_{\texttt{attr} \times \texttt{intel}} = \beta_{\texttt{attr},1} + \beta_{\texttt{intel},1}$ :

\[\begin{equation} \texttt{like}_i = \beta_0 + \beta_{\texttt{attr}} \times \texttt{attr}_i + \beta_{\texttt{intel}} \times \texttt{intel}_i + \beta_{\texttt{attr} \times \texttt{intel}} \times (\texttt{attr} \times \texttt{intel})_i + \epsilon_i \end{equation}\]

Estimating this model gives \[\texttt{like}_i = -0.791 + 0.657 \times \texttt{attr}_i + 0.488 \times \texttt{intel}_i - 0.0171 \times \texttt{(attr}\times\texttt{intel)}_i + \hat{\epsilon}_i \] The estimate of the slope of the interaction, $\hat{\beta}_{\texttt{attr} \times \texttt{intel}} = -0.017$ , is negative. That means that the higher the value of $\texttt{intel}$ , the less steep the regression line relating $\texttt{attr}$ to $\texttt{like}$ . At the same time, the higher the value of $\texttt{attr}$ , the less steep the regression line relating $\texttt{intel}$ to $\texttt{like}$ . You can interpret this as meaning that for more intelligent people, physical attractiveness is less of a defining factor in their liking by a potential partner. And for more attractive people, intelligence is less important.

A graphical view of this model, and the earlier one without moderation, is provided in Figure 6.2 . The plot on the left represents the model which does not allow for interaction. You can see that, for different values of intelligence, the model predicts parallel regression lines for the relation between attractiveness and liking. While intelligence affects the intercept of these regression lines, it does not affect the slope. In the plot on the right – although subtle – you can see that the regression lines are not parallel. This is a model with an interaction between intelligence and attractiveness. For different values of intelligence, the model predicts a linear relation between attractiveness and liking, but crucially, intelligence determines both the intercept and slope of these lines.

Figure 6.2: Liking as a function of attractiveness (intelligence) for different levels of intelligence (attractiveness), either without moderation or with moderation of the slope of attraciveness by intelligence. Note that the actual values of liking, attractiveness, and intelligence, are whole numbers (ratings on a scale between 1 and 10). For visualization purposes, the values have been randomly jittered by adding a Normal-distributed displacement term.

Note that we have constructed this model by simply including a new predictor in the model, which is computed by multiplying the values of $\texttt{attr}$ and $\texttt{intel}$ . While including such an “interaction predictor” has important implications for the resulting relations between $\texttt{attr}$ and $\texttt{like}$ for different values of $\texttt{intel}$ , as well as the relations between $\texttt{intel}$ and $\texttt{like}$ for different values of $\texttt{attr}$ , the model itself is just like any other regression model. Thus, parameter estimation and inference are exactly the same as before. Table 6.2 shows the results of comparing the full MODEL G (with three predictors) to different versions of MODEL R, where in each we fix one of the parameters to 0. As you can see, these comparisons indicate that we can reject the null hypothesis $H_0$ : $\beta_0 = 0$ , as well as $H_0$ : $\beta_{\texttt{attr}} = 0$ and $H_0$ : $\beta_{\texttt{intel}} = 0$ . However, as the p-value is above the conventional significance level of $\alpha=.05$ , we would not reject the null hypothesis $H_0$ : $\beta_{\texttt{attr} \times \texttt{intel}} = 0$ . That implies that, in the context of this model, there is not sufficient evidence that there is an interaction. That may seem a little disappointing. We’ve done a lot of work to construct a model where we allow the effect of attractiveness to depend on intelligence, and vice versa. And now the hypothesis test indicates that there is no evidence that this moderation is present. As we will see later, there is evidence of this moderation when we also include $\texttt{fun}$ in the model. I have left this predictor out of the model for now to keep things as simple as possible.

6.1.4 Simple slopes and centering

It is very important to realise that in a model with interactions, there is no single slope for any of the predictors involved in an interaction, that is particularly meaningful in principle. An interaction means that the slope of one predictor varies as a function of another predictor. Depending on which value of that other predictor you focus on, the slope of the predictor can be positive, negative, or zero. Let’s consider the model we estimated again: \[\texttt{like}_i = -0.791 + 0.657 \times \texttt{attr}_i + 0.488 \times \texttt{intel}_i - 0.0171 \times \texttt{(attr}\times\texttt{intel)}_i + \hat{\epsilon}_i \] If we fill in a particular value for intelligence, say $\texttt{intel} = 1$ , we can write this as

\[\begin{aligned} \texttt{intel}_i &= -0.791 + 0.657 \times \texttt{attr}_i + 0.488 \times 1 -0.017 \times (\texttt{attr} \times 1)_i + \epsilon_i \\ &= (-0.791 + 0.488) + (0.657 -0.017) \times \texttt{attr}_i + \epsilon_i \\ &= -0.303 + 0.64 \times \texttt{attr}_i + \epsilon_i \end{aligned}\]

If we pick a different value, say $\texttt{intel} = 10$ , the the model becomes \[\begin{aligned} \texttt{intel}_i &= -0.791 + 0.657 \times \texttt{attr}_i + 0.488 \times 10 -0.017 \times (\texttt{attr} \times 10)_i + \epsilon_i \\ &= (-0.791 + 0.488 \times 10) + (0.657 -0.017\times 10) \times \texttt{attr}_i + \epsilon_i \\ &= 4.09 + 0.486 \times \texttt{attr}_i + \epsilon_i \end{aligned}\] This shows that the higher the value of intelligence, the lower the slope of $\texttt{attr}$ becomes. If you’d pick $\texttt{intel} = 38.337$ , the slope would be exactly equal to 0. 17 Because there is not just a single value of the slope, testing whether “the” slope of $\texttt{attr}$ is equal to 0 doesn’t really make sense, because there is no single value to represent “the” slope. What, then, does $\hat{\beta}_\texttt{attr} = 0.657$ represent? Well, it is the (estimated) slope of $\texttt{attr}$ when $\texttt{intel}_i = 0$ . Similarly, $\hat{\beta}_\texttt{intel} = 0.488$ is the estimated slope of $\texttt{intel}$ when $\texttt{attr}_i = 0$

A significance test of the null hypothesis $H_0$ : $\beta_\texttt{attr} = 0$ is thus a test whether, when $\texttt{intel} = 0$ , the slope of $\texttt{attr}$ is 0. This test is easy enough to perform, but is it interesting to know whether liking is related to attractiveness for people who’s intelligence was rated as 0? Perhaps not. For one thing, the ratings were on a scale from 1 to 10, so no one could actually receive a rating of 0. Because the slope depends on $\texttt{intel}$ and we know that for some value of $\texttt{intel}$ , the slope of $\texttt{attr}$ will equal 0, the hypothesis test will not be significant for some values of $\texttt{intel}$ , and will be significant for others. At which value of $\texttt{intel}$ we might want to perform such a test is up to us, but the result seems somewhat arbitrary.

That said, we might be interested in assessing whether there is an effect of $\texttt{attr}$ for particular values of $\texttt{intel}$ . For instance, whether, for someone with an average intelligence rating, their physical attractiveness matters for how much someone likes them as a potential partner. We can obtain this test by centering the predictors. Centering is basically just subtracting the sample mean of each value of a variable. So for example, we can center $\texttt{attr}$ as follows: \[\texttt{attr_cent}_i = \texttt{attr}_i - \overline{\texttt{attr}}\] Centering does not affect the relation between variables. You can view it as a simple relabelling of the values, where the value which was the sample mean is now $\texttt{attr_cent}_i = \overline{\texttt{attr}} - \overline{\texttt{attr}} = 0$ , all values below the mean are now negative, and values above the mean are now positive. The important part of this is that the centered predictor is 0 where the original predictor was at the sample mean. In a model with centered predictors \[\begin{align} \texttt{like}_i =& \beta_0 + \beta_{\texttt{attr_cent}} \times \texttt{attr_cent}_i + \beta_{\texttt{intel_cent}} \times \texttt{intel_cent}_i \\ &+ \beta_{\texttt{attr_cent} \times \texttt{intel_cent}} \times (\texttt{attr_cent} \times \texttt{intel_cent})_i + \epsilon_i \end{align}\] the slope $\beta_{\texttt{attr_cent}}$ is, as usual, the slope of $\texttt{attr_cent}$ whenever $\texttt{intel_cent}_i = 0$ . We know that $\texttt{intel_cent}_i = 0$ when $\texttt{intel}_i = \overline{\texttt{intel}}$ . Hence, $\beta_{\texttt{attr_cent}}$ is the slope of $\texttt{attr}$ when $\texttt{intel} = \overline{\texttt{intel}}$ , i.e. it represents the effect of $\texttt{attr}$ for those with an average intelligence ratings.

Figure 6.3 shows the resulting model after centering both attractiveness and intelligence. When you compare this to the corresponding plot in Figure 6.2 , you can see that the only real difference is in the labels for the x-axis and the scale for intelligence. In all other respects, the uncentered and centered models predict the same relations between attractiveness and liking, and the models provide an equally good account, providing the same prediction errors.

Figure 6.3: Liking as a function of centered attractiveness for different levels of (centered) intelligence in a model including an interaction between attractiveness and intelligence. Note that the actual values of liking, attractiveness, and intelligence, are whole numbers (ratings on a scale between 1 and 10). For visualization purposes, the values have been randomly jittered by adding a Normal-distributed displacement term.

The results of all model comparisons after centering are given in Table 6.3 . A first important thing to notice is that centering does not affect the estimate and test of the interaction term . The slope of the interaction predictor reflects the increase in the slope relating $\texttt{attr}$ to $\texttt{like}$ for every one-unit increase in $\texttt{intel}$ . Such changes to the steepness of the relation between $\texttt{attr}$ and $\texttt{like}$ should not – and are not – affected by changing the 0-point of the predictors through centering. A second thing to notice is that centering changes the estimates and test of the “simple slopes” and intercept . In the centered model, the simple slope $\hat{\beta}_\texttt{attr_cent}$ reflects the effect of $\texttt{attr}$ on $\texttt{like}$ for cases with an average rating on $\texttt{intel}$ . In Figure 6.3 , this is (approximately) the regression line in the middle. In the uncentered model, the simple slope $\hat{\beta}_\texttt{attr}$ reflects the effect of $\texttt{attr}$ on $\texttt{like}$ for cases with $\texttt{intel} = 0$ . In the top right plot in Figure 6.2 , this is (approximately) the lower regression line. This latter regression line is quite far removed from most of the data, because there are no cases with an intelligence rating of 0. The regression line for people with an average intelligence rating lies much more “within the cloud of data points”, and reflects the model predictions for many more cases in the data. As a result, the reduction in the SSE that can be attributed to the simple slope is much higher in the centered model (Table 6.3 ) than the uncentered one (Table 6.2 ). This results in a much higher $F$ statistic. You can also think of this as follows: because there are hardly any cases with an intelligence rating close to 0, estimating the effect of attractiveness on liking for these cases is rather difficult and unreliable. Estimating the effect of attractiveness on liking for cases with an average intelligence rating is much more reliable, because there are many more cases with a close-to-average intelligence rating.

6.1.5 Don’t forget about fun! A model with multiple interactions

Up to now, we have looked at a model with two predictors, attractiveness and intelligence, and have allowed for an interaction between these. To simplify the discussion a little, we have not included $\texttt{fun}$ in the model. It is relatively straightforward to extend this idea to multiple predictors. For instance, it might also be the case that the effect of $\texttt{fun}$ is moderated by $\texttt{intel}$ . To investigate this, we can estimate the following regression model:

\[\begin{aligned} \texttt{like}_i =& \beta_0 + \beta_{\texttt{attr}} \times \texttt{attr}_i + \beta_{\texttt{intel}} \times \texttt{intel}_i + \beta_{\texttt{fun}} \times \texttt{fun}_i \\ &+ \beta_{\texttt{attr} \times \texttt{intel}} \times (\texttt{attr} \times \texttt{intel})_i + \beta_{\texttt{fun} \times \texttt{intel}} \times (\texttt{fun} \times \texttt{intel})_i + \epsilon_i \end{aligned}\]

The results, having centered all predictors, are given in Table 6.4 . As you can see there, the simple slopes of $\texttt{attr}$ , $\texttt{intel}$ , and $\texttt{fun}$ are all positive. Each of these represents the effect of that predictor when the other predictors have the value 0. Because the predictors are centered, that means that e.g. the slope of $\texttt{attr}$ reflects the effect of attractiveness for people with an average rating on intelligence and fun. As before, the estimated interaction between $\texttt{attr}$ and $\texttt{intel}$ is negative, indicating that attractiveness has less of an effect on liking for those seen as more intelligent, and that intelligence has less of an effect for those seen as more attractive. The hypothesis test of this effect is now also significant, indicating that we have reliable evidence for this moderation. This shows that by including more predictors in a model, it is possible to increase the reliability of the estimates for other predictors. There is also a significant interaction between $\texttt{fun}$ and $\texttt{intel}.$ The estimated interaction is positive here. This indicates that fun has more of an effect on liking for those seen as more intelligent, and that intelligence has more of an effect for those seen as more fun. Perhaps you can think of a reason why intelligence appears to lessen the effect of attractiveness, but appears to strengthen the effect of fun…

6.2 Mediation

6.2.1 legacy motives and pro-environmental behaviours.

Zaval, Markowitz, & Weber ( 2015 ) investigated whether there is a relation between individuals’ motivation to leave a positive legacy in the world, and their pro-environmental behaviours and intentions. The authors reasoned that long time horizons and social distance are key psychological barriers to pro-environmental action, particularly regarding climate change. But if people with a legacy motivation put more emphasis on future others than those without such motivation, they may also be motivated to behave more pro-environmentally in order to benefit those future others. In a pilot study, they recruited a diverse sample of 245 U.S. participants through Amazon’s Mechanical Turk. Participants answered three sets of questions: one assessing individual differences in legacy motives, one assessing their beliefs about climate change, and one assessing their willingness to take pro-environmental action. Following these sets of questions, participants were told they would be entered into a lottery to win a $10 bonus. They were then given the option to donate part (between $0 and $10) of their bonus to an environmental cause (Trees for the Future). This last measure was meant to test whether people actually act on any intention to act pro-environmentally.

For ease of analysis, the three sets of questions measuring legacy motive, belief about the reality of climate change, and intention to take pro-environmental action, were transformed into three overall scores by computing the average over the items in each set. After eliminating participants who did not answer all questions, we have data from $n = 237$ participants. Figure 6.4 depicts the pairwise relations between the four variables. As can be seen, all variables are significantly correlated. The relation is most obvious for $\texttt{belief}$ and $\texttt{intention}$ . Looking at the histogram of $\texttt{donation}$ , you can see that although all whole amounts between $0 and $10 have been chosen at least once, it looks like three values were particularly popular, namely $0, $5, and to a lesser extent $10. This results in what looks like a tri-modal distribution. This is not necessarily an issue when modelling $\texttt{donation}$ with a regression model, as the assumptions in a regression model concern the prediction errors , and not the dependent variable itself.

Figure 6.4: Pairwise plots for legacy motives, climate change belief, intention for pro-environmental action, and donations.

According to the Theory of Planned Behavior ( Ajzen, 1991 ) , attitudes and norms shape a person’s behavioural intentions, which in turn result in behaviour itself. In the context of the present example, that could mean that legacy motive and climate change beliefs do not directly determine whether someone behaves in a pro-environmental way. Rather, these factors shape a person’s intentions towards pro-environmental behaviour, which in turn may actually lead to said pro-environmental behaviour. This is an example of an assumed causal chain , where legacy motive (partly) determines behavioural intention, and intention determines behaviour. Mediation analysis is aimed at detecting an indirect effect of a predictor (e.g. $\texttt{legacy}$ ) on the dependent variable (e.g. $\texttt{donation}$ ), via another variable called the mediator (e.g. $\texttt{intention}$ ), which is the middle variable in the causal chain.

6.2.2 Causal steps

A traditional method to assess mediation is the so-called causal steps approach ( Baron & Kenny, 1986 ) . The basic idea behind the causal steps approach is as follows: if there is a causal chain from predictor ( $X$ ) to mediator ( $M$ ) to dependent variable ( $Y$ ), then, ignoring the mediator for the moment, we should be able to see a relation between the predictor and dependent variable. This relation reflects the indirect effect of the predictor on the dependent variable. We should also be able to detect an effect of the predictor on the mediator, as well as an effect of the mediator on the dependent variable. Crucially, if there is a true causal chain, then the predictor should not offer any additional predictive power over the mediator. Because the effect of the predictor is assumed to go only “through” the mediator, once we know the value of the mediator, this should be all we need to predict the dependent variable. In more fancy statistical terms, this means that conditional on the mediator, the dependent variable is independent of the predictor, i.e. $p(Y \mid M, X) = p(Y \mid M)$ . In the context of a multiple regression model, we could say that in a model where we predict $Y$ from $M$ , the predictor $X$ would not have a unique effect on $Y$ (i.e. its slope would equal $\beta_X = 0$ ).

The causal steps (Figure 6.5 ) approach involves assessing a pattern of significant relations in three different regression models. The first model is a simple regression model where we predict $Y$ from $X$ . In this model, we should find evidence for a relation between $X$ and $Y$ , meaning that we can reject the null hypothesis that the slope of $X$ on $Y$ (referred to here as $\beta_X = c$ ) equals 0. The second model is a simple regression model where we predict $M$ from $X$ . In this model, we should find evidence for a relation between $X$ and $M$ , meaning that we can reject the null hypothesis that the slope of $X$ on $M$ (referred to here as $\beta_X = a$ here) equals 0. The third model is a multiple regression model where we predict $Y$ from both $M$ and $X$ . In this model, we should find evidence for a unique relation between $M$ and $Y$ , meaning that we can reject the null hypothesis that the slope of $M$ on $Y$ (referred to here as $\beta_M = b$ here) equals 0. Controlling for the effect of $M$ on $Y$ , in a true causal chain, there should no longer be evidence for a relation between $X$ and $Y$ (as any relation between $X$ and $Y$ is captured through $M$ ). Hence, we should not be able to reject the null hypothesis that the slope of $X$ on $Y$ in this model (referred to here as $\beta_X = c$ ’, to distinguish it from the relation between $X$ and $Y$ in the first model, which was labelled as $c$ ) equals 0. If this is so, then we speak of full mediation . When there is still evidence of a unique relation between $X$ and $Y$ in the model that includes $M$ , but the relation is reduced (i.e. $|c'| < |c|$ ), we speak of partial mediation .

Figure 6.5: Assessing mediation with the causal steps approach involves testing parameters of three models. MODEL 1 is a simple regression model predicting $Y$ from $X$ and the slope of $X$ ( $c$ ) should be significant MODEL 2 is a simple regression model predicting $M$ from $X$ and the slope of $X$ ( $a$ ) should be significant. MODEL 3 is a multiple regression model predicting $Y$ from both $X$ and $M$ . The slope of $M$ ( $b$ ) should be significant. The slope of $X$ ( $c$ ’) should not be significant (“full” mediation) or be substantially smaller in absolute value (“partial” mediation).

6.2.2.1 Testing mediation of legacy motive by intention with the causal steps approach

Let’s see how the causal steps approach works in practice by assessing whether the relation between $\texttt{legacy}$ on $\texttt{donation}$ is mediated by $\texttt{intention}$ .

In MODEL 1 (Table 6.5 ), we assess the relation between $\texttt{legacy}$ and $\texttt{donation}$ . In this model, we find a significant and positive relation between legacy motives and donations, such that people with stronger legacy motives donate more of their potential bonus to a pro-environmental cause. The question is now whether this is a direct effect of legacy motive, or an indirect effect “via” behavioural intent.

In MODEL 2 (Table 6.6 ), we assess the relation between $\texttt{legacy}$ and $\texttt{intention}$ . In this model, we find a significant and positive relation between legacy motives and intention to act pro-environmentally, such that people with stronger legacy motives have a stronger intention to act pro-environmentally.

In MODEL 3 (Table 6.7 ), we assess the relation between $\texttt{legacy}$ , $\texttt{intention}$ , and $\texttt{donation}$ . In this model, we find a significant and positive relation between intention to act pro-environmentally and donation to a pro-environmental cause, such that people with stronger intentions donate more. We also find evidence of a unique and positive effect of legacy motive on donation, such that people with stronger legacy motives donate more. Because there is still evidence of an effect of legacy motive on donations, after controlling for the effect of behavioural intent, we would not conclude that the effect of legacy motive is fully mediated by intent. When you compare the slope of $\texttt{legacy}$ in MODEL 3 to that in MODEL 1, you can however see that the (absolute) value is smaller. Hence, when controlling for the effect of behavioural intent, a one-unit increase in $\texttt{legacy}$ is estimated to increase the amount of donation less then in a model where $\texttt{intention}$ is not taken into account.

In conclusion, the causal steps approach indicates that the effect of legacy motive of pro-environmental action (donations) is partially mediated by pro-environmental behavioural intentions. There is a residual direct effect of legacy motive on donations that is not captured by behavioural intentions.

6.2.3 Estimating the mediated effect

One potential problem with the causal steps approach is that it is based on a pattern of significance in four hypothesis tests (one for each parameter $a$ , $b$ , $c$ , and $c'$ ). This can result in a rather low power of the procedure ( MacKinnon, Fairchild, & Fritz, 2007 ) , which seems to be particularly related to the requirement of a significant $c$ (the direct effect of $X$ on $Y$ in the model without the mediator).

An alternative to the causal steps approach is to estimate the mediated (indirect) effect of the predictor on the dependent variable directly. Algebraically, this mediated effect can be worked out as ( MacKinnon et al., 2007 ) :

\[\begin{equation} \text{mediated effect} = a \times b \end{equation}\]

The rationale behind this is reasonably straightforward. The slope $a$ reflects the increase in the mediator $M$ for every one-unit increase in the predictor $X$ . The slope $b$ reflects the increase in the dependent variable $Y$ for every one unit increase in the mediator. So a one-unit increase in $X$ implies an increase in $M$ by $a$ units, which in turn implies an increase in $Y$ of $a \times b$ units. Hence, the mediated effect can be expressed as $a \times b$ .

In a single mediator model such as the one looked at here, the mediated effect $a \times b$ turns out to be equal to $c - c'$ , i.e. the difference between the direct effect of $X$ on $Y$ in a model without the mediator, and the unique direct effect of $X$ on $Y$ in a model which includes the mediator.

To test whether the mediated effect differs from 0, we can try to work out the sampling distribution of the estimated effect $\hat{a} \times \hat{b}$ , under the null-hypothesis that in reality, $a \times b = 0$ . Note that this null hypothesis can be true when $a = 0$ , $b = 0$ , or both $a = b = 0$ . In the so-called Sobel-Aroian test, this sampling distribution is assumed to be Normal. However, it has been found that this assumption is often inaccurate. As there is no method to derive an accurate sampling distribution analytically, modern procedures rely on simulation. There are different ways to do this, but we’ll focus on one, namely the nonparametric bootstrap approach ( Preacher & Hayes, 2008 ) . This involves generating a large number (e.g. $>1000$ ) of simulated datasets by randomly sampling $n$ cases with replacement from the original dataset. This means that any given case (i.e. a row in the dataset) can occur 0, 1, 2, times in a simulated dataset. For each simulated dataset, we can estimate $\hat{a} \times \hat{b}$ by fitting the two corresponding regression models. The variance in these estimates over the different datasets forms an estimate of the variance of the sampling distribution. A 95% confidence interval can then also be computed through by determining the 2.5 and 97.5 percentiles. Because just the original data is used, there is no direct assumption made about the distribution of the variables, apart from that the original data is a representative sample from the Data Generating Process. Applying this procedure (with 1000 simulated datasets) provides a 95% confidence interval for $a \times b$ of $[0.104, 0.446]$ . As this interval does not contain the value 0, we reject the null hypothesis that the mediated effect of $\texttt{legacy}$ on $\texttt{donation}$ “via” $\texttt{intention}$ equals 0.

Note that in solely focusing on the mediated effect, we do not address the issue of total vs partial mediation. Using our simulated datasets, we can however also compute a bootstrap confidence interval for $c'$ . For the present set of simulations, the 95% confidence interval for $c'$ is $[0.189, 0.807]$ . As this interval does not contain the value 0, we reject the null hypothesis that the unique direct effect of $\texttt{legacy}$ on $\texttt{donation}$ equals 0. This thus provides a similar conclusion to the causal steps approach.

Here, we analyse only a subset of their data. ↩︎

Note that I’m using more descriptive labels here. If you prefer the more abstract version, then you can replace $Y_i = \texttt{like}_i$ , $\beta_1 = \beta_{\texttt{attr}}$ , $X_{1,i} = \texttt{attr}_i$ . $\beta_2 = \beta_{\texttt{intel}}$ , $X_{2,i} = \texttt{intel}_i$ . ↩︎

The value for which the slope is 0 is easily worked out as $\frac{\hat{\beta}_\texttt{attr}}{- \hat{\beta}_{\texttt{attr} \times \texttt{intel}}}$ . ↩︎

How Do You Formulate (Important) Hypotheses?

Open Access
First Online: 03 December 2022

Cite this chapter

You have full access to this open access chapter

James Hiebert 6 ,
Jinfa Cai 7 ,
Stephen Hwang 7 ,
Anne K Morris 6 &
Charles Hohensee 6

Part of the book series: Research in Mathematics Education ((RME))

10k Accesses

Building on the ideas in Chap. 1, we describe formulating, testing, and revising hypotheses as a continuing cycle of clarifying what you want to study, making predictions about what you might find together with developing your reasons for these predictions, imagining tests of these predictions, revising your predictions and rationales, and so on. Many resources feed this process, including reading what others have found about similar phenomena, talking with colleagues, conducting pilot studies, and writing drafts as you revise your thinking. Although you might think you cannot predict what you will find, it is always possible—with enough reading and conversations and pilot studies—to make some good guesses. And, once you guess what you will find and write out the reasons for these guesses you are on your way to scientific inquiry. As you refine your hypotheses, you can assess their research importance by asking how connected they are to problems your research community really wants to solve.

Download chapter PDF

Part I. Getting Started

We want to begin by addressing a question you might have had as you read the title of this chapter. You are likely to hear, or read in other sources, that the research process begins by asking research questions . For reasons we gave in Chap. 1 , and more we will describe in this and later chapters, we emphasize formulating, testing, and revising hypotheses. However, it is important to know that asking and answering research questions involve many of the same activities, so we are not describing a completely different process.

We acknowledge that many researchers do not actually begin by formulating hypotheses. In other words, researchers rarely get a researchable idea by writing out a well-formulated hypothesis. Instead, their initial ideas for what they study come from a variety of sources. Then, after they have the idea for a study, they do lots of background reading and thinking and talking before they are ready to formulate a hypothesis. So, for readers who are at the very beginning and do not yet have an idea for a study, let’s back up. Where do research ideas come from?

There are no formulas or algorithms that spawn a researchable idea. But as you begin the process, you can ask yourself some questions. Your answers to these questions can help you move forward.

What are you curious about? What are you passionate about? What have you wondered about as an educator? These are questions that look inward, questions about yourself.

What do you think are the most pressing educational problems? Which problems are you in the best position to address? What change(s) do you think would help all students learn more productively? These are questions that look outward, questions about phenomena you have observed.

What are the main areas of research in the field? What are the big questions that are being asked? These are questions about the general landscape of the field.

What have you read about in the research literature that caught your attention? What have you read that prompted you to think about extending the profession’s knowledge about this? What have you read that made you ask, “I wonder why this is true?” These are questions about how you can build on what is known in the field.

What are some research questions or testable hypotheses that have been identified by other researchers for future research? This, too, is a question about how you can build on what is known in the field. Taking up such questions or hypotheses can help by providing some existing scaffolding that others have constructed.

What research is being done by your immediate colleagues or your advisor that is of interest to you? These are questions about topics for which you will likely receive local support.

Exercise 2.1

Brainstorm some answers for each set of questions. Record them. Then step back and look at the places of intersection. Did you have similar answers across several questions? Write out, as clearly as you can, the topic that captures your primary interest, at least at this point. We will give you a chance to update your responses as you study this book.

Part II. Paths from a General Interest to an Informed Hypothesis

There are many different paths you might take from conceiving an idea for a study, maybe even a vague idea, to formulating a prediction that leads to an informed hypothesis that can be tested. We will explore some of the paths we recommend.

We will assume you have completed Exercise 2.1 in Part I and have some written answers to the six questions that preceded it as well as a statement that describes your topic of interest. This very first statement could take several different forms: a description of a problem you want to study, a question you want to address, or a hypothesis you want to test. We recommend that you begin with one of these three forms, the one that makes most sense to you. There is an advantage to using all three and flexibly choosing the one that is most meaningful at the time and for a particular study. You can then move from one to the other as you think more about your research study and you develop your initial idea. To get a sense of how the process might unfold, consider the following alternative paths.

Beginning with a Prediction If You Have One

Sometimes, when you notice an educational problem or have a question about an educational situation or phenomenon, you quickly have an idea that might help solve the problem or answer the question. Here are three examples.

You are a teacher, and you noticed a problem with the way the textbook presented two related concepts in two consecutive lessons. Almost as soon as you noticed the problem, it occurred to you that the two lessons could be taught more effectively in the reverse order. You predicted better outcomes if the order was reversed, and you even had a preliminary rationale for why this would be true.

You are a graduate student and you read that students often misunderstand a particular aspect of graphing linear functions. You predicted that, by listening to small groups of students working together, you could hear new details that would help you understand this misconception.

You are a curriculum supervisor and you observed sixth-grade classrooms where students were learning about decimal fractions. After talking with several experienced teachers, you predicted that beginning with percentages might be a good way to introduce students to decimal fractions.

We begin with the path of making predictions because we see the other two paths as leading into this one at some point in the process (see Fig. 2.1 ). Starting with this path does not mean you did not sense a problem you wanted to solve or a question you wanted to answer.

The process flow diagram of initiation of hypothesis. It starts with a problem situation and leads to a prediction following the question to the hypothesis.

Three Pathways to Formulating Informed Hypotheses

Notice that your predictions can come from a variety of sources—your own experience, reading, and talking with colleagues. Most likely, as you write out your predictions you also think about the educational problem for which your prediction is a potential solution. Writing a clear description of the problem will be useful as you proceed. Notice also that it is easy to change each of your predictions into a question. When you formulate a prediction, you are actually answering a question, even though the question might be implicit. Making that implicit question explicit can generate a first draft of the research question that accompanies your prediction. For example, suppose you are the curriculum supervisor who predicts that teaching percentages first would be a good way to introduce decimal fractions. In an obvious shift in form, you could ask, “In what ways would teaching percentages benefit students’ initial learning of decimal fractions?”

The picture has a difference between a question and a prediction: a question simply asks what you will find whereas a prediction also says what you expect to find; written.

There are advantages to starting with the prediction form if you can make an educated guess about what you will find. Making a prediction forces you to think now about several things you will need to think about at some point anyway. It is better to think about them earlier rather than later. If you state your prediction clearly and explicitly, you can begin to ask yourself three questions about your prediction: Why do I expect to observe what I am predicting? Why did I make that prediction? (These two questions essentially ask what your rationale is for your prediction.) And, how can I test to see if it’s right? This is where the benefits of making predictions begin.

Asking yourself why you predicted what you did, and then asking yourself why you answered the first “why” question as you did, can be a powerful chain of thought that lays the groundwork for an increasingly accurate prediction and an increasingly well-reasoned rationale. For example, suppose you are the curriculum supervisor above who predicted that beginning by teaching percentages would be a good way to introduce students to decimal fractions. Why did you make this prediction? Maybe because students are familiar with percentages in everyday life so they could use what they know to anchor their thinking about hundredths. Why would that be helpful? Because if students could connect hundredths in percentage form with hundredths in decimal fraction form, they could bring their meaning of percentages into decimal fractions. But how would that help? If students understood that a decimal fraction like 0.35 meant 35 of 100, then they could use their understanding of hundredths to explore the meaning of tenths, thousandths, and so on. Why would that be useful? By continuing to ask yourself why you gave the previous answer, you can begin building your rationale and, as you build your rationale, you will find yourself revisiting your prediction, often making it more precise and explicit. If you were the curriculum supervisor and continued the reasoning in the previous sentences, you might elaborate your prediction by specifying the way in which percentages should be taught in order to have a positive effect on particular aspects of students’ understanding of decimal fractions.

Developing a Rationale for Your Predictions

Keeping your initial predictions in mind, you can read what others already know about the phenomenon. Your reading can now become targeted with a clear purpose.

By reading and talking with colleagues, you can develop more complete reasons for your predictions. It is likely that you will also decide to revise your predictions based on what you learn from your reading. As you develop sound reasons for your predictions, you are creating your rationales, and your predictions together with your rationales become your hypotheses. The more you learn about what is already known about your research topic, the more refined will be your predictions and the clearer and more complete your rationales. We will use the term more informed hypotheses to describe this evolution of your hypotheses.

The picture says you develop sound reasons for your predictions, you are creating your rationales, and your predictions together with your rationales become your hypotheses.

Developing more informed hypotheses is a good thing because it means: (1) you understand the reasons for your predictions; (2) you will be able to imagine how you can test your hypotheses; (3) you can more easily convince your colleagues that they are important hypotheses—they are hypotheses worth testing; and (4) at the end of your study, you will be able to more easily interpret the results of your test and to revise your hypotheses to demonstrate what you have learned by conducting the study.

Imagining Testing Your Hypotheses

Because we have tied together predictions and rationales to constitute hypotheses, testing hypotheses means testing predictions and rationales. Testing predictions means comparing empirical observations, or findings, with the predictions. Testing rationales means using these comparisons to evaluate the adequacy or soundness of the rationales.

Imagining how you might test your hypotheses does not mean working out the details for exactly how you would test them. Rather, it means thinking ahead about how you could do this. Recall the descriptor of scientific inquiry: “experience carefully planned in advance” (Fisher, 1935). Asking whether predictions are testable and whether rationales can be evaluated is simply planning in advance.

You might read that testing hypotheses means simply assessing whether predictions are correct or incorrect. In our view, it is more useful to think of testing as a means of gathering enough information to compare your findings with your predictions, revise your rationales, and propose more accurate predictions. So, asking yourself whether hypotheses can be tested means asking whether information could be collected to assess the accuracy of your predictions and whether the information will show you how to revise your rationales to sharpen your predictions.

Cycles of Building Rationales and Planning to Test Your Predictions

Scientific reasoning is a dialogue between the possible and the actual, an interplay between hypotheses and the logical expectations they give rise to: there is a restless to-and-fro motion of thought, the formulation and rectification of hypotheses (Medawar, 1982 , p.72).

As you ask yourself about how you could test your predictions, you will inevitably revise your rationales and sharpen your predictions. Your hypotheses will become more informed, more targeted, and more explicit. They will make clearer to you and others what, exactly, you plan to study.

When will you know that your hypotheses are clear and precise enough? Because of the way we define hypotheses, this question asks about both rationales and predictions. If a rationale you are building lets you make a number of quite different predictions that are equally plausible rather than a single, primary prediction, then your hypothesis needs further refinement by building a more complete and precise rationale. Also, if you cannot briefly describe to your colleagues a believable way to test your prediction, then you need to phrase it more clearly and precisely.

Each time you strengthen your rationales, you might need to adjust your predictions. And, each time you clarify your predictions, you might need to adjust your rationales. The cycle of going back and forth to keep your predictions and rationales tightly aligned has many payoffs down the road. Every decision you make from this point on will be in the interests of providing a transparent and convincing test of your hypotheses and explaining how the results of your test dictate specific revisions to your hypotheses. As you make these decisions (described in the succeeding chapters), you will probably return to clarify your hypotheses even further. But, you will be in a much better position, at each point, if you begin with well-informed hypotheses.

Beginning by Asking Questions to Clarify Your Interests

Instead of starting with predictions, a second path you might take devotes more time at the beginning to asking questions as you zero in on what you want to study. Some researchers suggest you start this way (e.g., Gournelos et al., 2019 ). Specifically, with this second path, the first statement you write to express your research interest would be a question. For example, you might ask, “Why do ninth-grade students change the way they think about linear equations after studying quadratic equations?” or “How do first graders solve simple arithmetic problems before they have been taught to add and subtract?”

The first phrasing of your question might be quite general or vague. As you think about your question and what you really want to know, you are likely to ask follow-up questions. These questions will almost always be more specific than your first question. The questions will also express more clearly what you want to know. So, the question “How do first graders solve simple arithmetic problems before they have been taught to add and subtract” might evolve into “Before first graders have been taught to solve arithmetic problems, what strategies do they use to solve arithmetic problems with sums and products below 20?” As you read and learn about what others already know about your questions, you will continually revise your questions toward clearer and more explicit and more precise versions that zero in on what you really want to know. The question above might become, “Before they are taught to solve arithmetic problems, what strategies do beginning first graders use to solve arithmetic problems with sums and products below 20 if they are read story problems and given physical counters to help them keep track of the quantities?”

Imagining Answers to Your Questions

If you monitor your own thinking as you ask questions, you are likely to begin forming some guesses about answers, even to the early versions of the questions. What do students learn about quadratic functions that influences changes in their proportional reasoning when dealing with linear functions? It could be that if you analyze the moments during instruction on quadratic equations that are extensions of the proportional reasoning involved in solving linear equations, there are times when students receive further experience reasoning proportionally. You might predict that these are the experiences that have a “backward transfer” effect (Hohensee, 2014 ).

These initial guesses about answers to your questions are your first predictions. The first predicted answers are likely to be hunches or fuzzy, vague guesses. This simply means you do not know very much yet about the question you are asking. Your first predictions, no matter how unfocused or tentative, represent the most you know at the time about the question you are asking. They help you gauge where you are in your thinking.

Shifting to the Hypothesis Formulation and Testing Path

Research questions can play an important role in the research process. They provide a succinct way of capturing your research interests and communicating them to others. When colleagues want to know about your work, they will often ask “What are your research questions?” It is good to have a ready answer.

However, research questions have limitations. They do not capture the three images of scientific inquiry presented in Chap. 1 . Due, in part, to this less expansive depiction of the process, research questions do not take you very far. They do not provide a guide that leads you through the phases of conducting a study.

Consequently, when you can imagine an answer to your research question, we recommend that you move onto the hypothesis formulation and testing path. Imagining an answer to your question means you can make plausible predictions. You can now begin clarifying the reasons for your predictions and transform your early predictions into hypotheses (predictions along with rationales). We recommend you do this as soon as you have guesses about the answers to your questions because formulating, testing, and revising hypotheses offers a tool that puts you squarely on the path of scientific inquiry. It is a tool that can guide you through the entire process of conducting a research study.

This does not mean you are finished asking questions. Predictions are often created as answers to questions. So, we encourage you to continue asking questions to clarify what you want to know. But your target shifts from only asking questions to also proposing predictions for the answers and developing reasons the answers will be accurate predictions. It is by predicting answers, and explaining why you made those predictions, that you become engaged in scientific inquiry.

Cycles of Refining Questions and Predicting Answers

An example might provide a sense of how this process plays out. Suppose you are reading about Vygotsky’s ( 1987 ) zone of proximal development (ZPD), and you realize this concept might help you understand why your high school students had trouble learning exponential functions. Maybe they were outside this zone when you tried to teach exponential functions. In order to recognize students who would benefit from instruction, you might ask, “How can I identify students who are within the ZPD around exponential functions?” What would you predict? Maybe students in this ZPD are those who already had knowledge of related functions. You could write out some reasons for this prediction, like “students who understand linear and quadratic functions are more likely to extend their knowledge to exponential functions.” But what kind of data would you need to test this? What would count as “understanding”? Are linear and quadratic the functions you should assess? Even if they are, how could you tell whether students who scored well on tests of linear and quadratic functions were within the ZPD of exponential functions? How, in the end, would you measure what it means to be in this ZPD? So, asking a series of reasonable questions raised some red flags about the way your initial question was phrased, and you decide to revise it.

You set the stage for revising your question by defining ZPD as the zone within which students can solve an exponential function problem by making only one additional conceptual connection between what they already know and exponential functions. Your revised question is, “Based on students’ knowledge of linear and quadratic functions, which students are within the ZPD of exponential functions?” This time you know what kind of data you need: the number of conceptual connections students need to bridge from their knowledge of related functions to exponential functions. How can you collect these data? Would you need to see into the minds of the students? Or, are there ways to test the number of conceptual connections someone makes to move from one topic to another? Do methods exist for gathering these data? You decide this is not realistic, so you now have a choice: revise the question further or move your research in a different direction.

Notice that we do not use the term research question for all these early versions of questions that begin clarifying for yourself what you want to study. These early versions are too vague and general to be called research questions. In this book, we save the term research question for a question that comes near the end of the work and captures exactly what you want to study . By the time you are ready to specify a research question, you will be thinking about your study in terms of hypotheses and tests. When your hypotheses are in final form and include clear predictions about what you will find, it will be easy to state the research questions that accompany your predictions.

To reiterate one of the key points of this chapter: hypotheses carry much more information than research questions. Using our definition, hypotheses include predictions about what the answer might be to the question plus reasons for why you think so. Unlike research questions, hypotheses capture all three images of scientific inquiry presented in Chap. 1 (planning, observing and explaining, and revising one’s thinking). Your hypotheses represent the most you know, at the moment, about your research topic. The same cannot be said for research questions.

Beginning with a Research Problem

When you wrote answers to the six questions at the end of Part I of this chapter, you might have identified a research interest by stating it as a problem. This is the third path you might take to begin your research. Perhaps your description of your problem might look something like this: “When I tried to teach my middle school students by presenting them with a challenging problem without showing them how to solve similar problems, they didn’t exert much effort trying to find a solution but instead waited for me to show them how to solve the problem.” You do not have a specific question in mind, and you do not have an idea for why the problem exists, so you do not have a prediction about how to solve it. Writing a statement of this problem as clearly as possible could be the first step in your research journey.

As you think more about this problem, it will feel natural to ask questions about it. For example, why did some students show more initiative than others? What could I have done to get them started? How could I have encouraged the students to keep trying without giving away the solution? You are now on the path of asking questions—not research questions yet, but questions that are helping you focus your interest.

As you continue to think about these questions, reflect on your own experience, and read what others know about this problem, you will likely develop some guesses about the answers to the questions. They might be somewhat vague answers, and you might not have lots of confidence they are correct, but they are guesses that you can turn into predictions. Now you are on the hypothesis-formulation-and-testing path. This means you are on the path of asking yourself why you believe the predictions are correct, developing rationales for the predictions, asking what kinds of empirical observations would test your predictions, and refining your rationales and predictions as you read the literature and talk with colleagues.

A simple diagram that summarizes the three paths we have described is shown in Fig. 2.1 . Each row of arrows represents one pathway for formulating an informed hypothesis. The dotted arrows in the first two rows represent parts of the pathways that a researcher may have implicitly travelled through already (without an intent to form a prediction) but that ultimately inform the researcher’s development of a question or prediction.

Part III. One Researcher’s Experience Launching a Scientific Inquiry

Martha was in her third year of her doctoral program and beginning to identify a topic for her dissertation. Based on (a) her experience as a high school mathematics teacher and a curriculum supervisor, (b) the reading she has done to this point, and (c) her conversations with her colleagues, she has developed an interest in what kinds of professional development experiences (let’s call them learning opportunities [LOs] for teachers) are most effective. Where does she go from here?

Exercise 2.2

Before you continue reading, please write down some suggestions for Martha about where she should start.

A natural thing for Martha to do at this point is to ask herself some additional questions, questions that specify further what she wants to learn: What kinds of LOs do most teachers experience? How do these experiences change teachers’ practices and beliefs? Are some LOs more effective than others? What makes them more effective?

To focus her questions and decide what she really wants to know, she continues reading but now targets her reading toward everything she can find that suggests possible answers to these questions. She also talks with her colleagues to get more ideas about possible answers to these or related questions. Over several weeks or months, she finds herself being drawn to questions about what makes LOs effective, especially for helping teachers teach more conceptually. She zeroes in on the question, “What makes LOs for teachers effective for improving their teaching for conceptual understanding?”

This question is more focused than her first questions, but it is still too general for Martha to define a research study. How does she know it is too general? She uses two criteria. First, she notices that the predictions she makes about the answers to the question are all over the place; they are not constrained by the reasons she has assembled for her predictions. One prediction is that LOs are more effective when they help teachers learn content. Martha makes this guess because previous research suggests that effective LOs for teachers include attention to content. But this rationale allows lots of different predictions. For example, LOs are more effective when they focus on the content teachers will teach; LOs are more effective when they focus on content beyond what teachers will teach so teachers see how their instruction fits with what their students will encounter later; and LOs are more effective when they are tailored to the level of content knowledge participants have when they begin the LOs. The rationale she can provide at this point does not point to a particular prediction.

A second measure Martha uses to decide her question is too general is that the predictions she can make regarding the answers seem very difficult to test. How could she test, for example, whether LOs should focus on content beyond what teachers will teach? What does “content beyond what teachers teach” mean? How could you tell whether teachers use their new knowledge of later content to inform their teaching?

Before anticipating what Martha’s next question might be, it is important to pause and recognize how predicting the answers to her questions moved Martha into a new phase in the research process. As she makes predictions, works out the reasons for them, and imagines how she might test them, she is immersed in scientific inquiry. This intellectual work is the main engine that drives the research process. Also notice that revisions in the questions asked, the predictions made, and the rationales built represent the updated thinking (Chap. 1 ) that occurs as Martha continues to define her study.

Based on all these considerations and her continued reading, Martha revises the question again. The question now reads, “Do LOs that engage middle school mathematics teachers in studying mathematics content help teachers teach this same content with more of a conceptual emphasis?” Although she feels like the question is more specific, she realizes that the answer to the question is either “yes” or “no.” This, by itself, is a red flag. Answers of “yes” or “no” would not contribute much to understanding the relationships between these LOs for teachers and changes in their teaching. Recall from Chap. 1 that understanding how things work, explaining why things work, is the goal of scientific inquiry.

Martha continues by trying to understand why she believes the answer is “yes.” When she tries to write out reasons for predicting “yes,” she realizes that her prediction depends on a variety of factors. If teachers already have deep knowledge of the content, the LOs might not affect them as much as other teachers. If the LOs do not help teachers develop their own conceptual understanding, they are not likely to change their teaching. By trying to build the rationale for her prediction—thus formulating a hypothesis—Martha realizes that the question still is not precise and clear enough.

Martha uses what she learned when developing the rationale and rephrases the question as follows: “ Under what conditions do LOs that engage middle school mathematics teachers in studying mathematics content help teachers teach this same content with more of a conceptual emphasis?” Through several additional cycles of thinking through the rationale for her predictions and how she might test them, Martha specifies her question even further: “Under what conditions do middle school teachers who lack conceptual knowledge of linear functions benefit from LOs that engage them in conceptual learning of linear functions as assessed by changes in their teaching toward a more conceptual emphasis on linear functions?”

Each version of Martha’s question has become more specific. This has occurred as she has (a) identified a starting condition for the teachers—they lack conceptual knowledge of linear functions, (b) specified the mathematics content as linear functions, and (c) included a condition or purpose of the LO—it is aimed at conceptual learning.

Because of the way Martha’s question is now phrased, her predictions will require thinking about the conditions that could influence what teachers learn from the LOs and how this learning could affect their teaching. She might predict that if teachers engaged in LOs that extended over multiple sessions, they would develop deeper understanding which would, in turn, prompt changes in their teaching. Or she might predict that if the LOs included examples of how their conceptual learning could translate into different instructional activities for their students, teachers would be more likely to change their teaching. Reasons for these predictions would likely come from research about the effects of professional development on teachers’ practice.

As Martha thinks about testing her predictions, she realizes it will probably be easier to measure the conditions under which teachers are learning than the changes in the conceptual emphasis in their instruction. She makes a note to continue searching the literature for ways to measure the “conceptualness” of teaching.

As she refines her predictions and expresses her reasons for the predictions, she formulates a hypothesis (in this case several hypotheses) that will guide her research. As she makes predictions and develops the rationales for these predictions, she will probably continue revising her question. She might decide, for example, that she is not interested in studying the condition of different numbers of LO sessions and so decides to remove this condition from consideration by including in her question something like “. . . over five 2-hour sessions . . .”

At this point, Martha has developed a research question, articulated a number of predictions, and developed rationales for them. Her current question is: “Under what conditions do middle school teachers who lack conceptual knowledge of linear functions benefit from five 2-hour LO sessions that engage them in conceptual learning of linear functions as assessed by changes in their teaching toward a more conceptual emphasis on linear functions?” Her hypothesis is:

Prediction: Participating teachers will show changes in their teaching with a greater emphasis on conceptual understanding, with larger changes on linear function topics directly addressed in the LOs than on other topics.

Brief Description of Rationale: (1) Past research has shown correlations between teachers’ specific mathematics knowledge of a topic and the quality of their teaching of that topic. This does not mean an increase in knowledge causes higher quality teaching but it allows for that possibility. (2) Transfer is usually difficult for teachers, but the examples developed during the LO sessions will help them use what they learned to teach for conceptual understanding. This is because the examples developed during the LO sessions are much like those that will be used by the teachers. So larger changes will be found when teachers are teaching the linear function topics addressed in the LOs.

Notice it is more straightforward to imagine how Martha could test this prediction because it is more precise than previous predictions. Notice also that by asking how to test a particular prediction, Martha will be faced with a decision about whether testing this prediction will tell her something she wants to learn. If not, she can return to the research question and consider how to specify it further and, perhaps, constrain further the conditions that could affect the data.

As Martha formulates her hypotheses and goes through multiple cycles of refining her question(s), articulating her predictions, and developing her rationales, she is constantly building the theoretical framework for her study. Because the theoretical framework is the topic for Chap. 3 , we will pause here and pick up Martha’s story in the next chapter. Spoiler alert: Martha’s experience contains some surprising twists and turns.

Before leaving Martha, however, we point out two aspects of the process in which she has been engaged. First, it can be useful to think about the process as identifying (1) the variables targeted in her predictions, (2) the mechanisms she believes explain the relationships among the variables, and (3) the definitions of all the terms that are special to her educational problem. By variables, we mean things that can be measured and, when measured, can take on different values. In Martha’s case, the variables are the conceptualness of teaching and the content topics addressed in the LOs. The mechanisms are cognitive processes that enable teachers to see the relevance of what they learn in PD to their own teaching and that enable the transfer of learning from one setting to another. Definitions are the precise descriptions of how the important ideas relevant to the research are conceptualized. In Martha’s case, definitions must be provided for terms like conceptual understanding, linear functions, LOs, each of the topics related to linear functions, instructional setting, and knowledge transfer.

A second aspect of the process is a practice that Martha acquired as part of her graduate program, a practice that can go unnoticed. Martha writes out, in full sentences, her thinking as she wrestles with her research question, her predictions of the answers, and the rationales for her predictions. Writing is a tool for organizing thinking and we recommend you use it throughout the scientific inquiry process. We say more about this at the end of the chapter.

Here are the questions Martha wrote as she developed a clearer sense of what question she wanted to answer and what answer she predicted. The list shows the increasing refinement that occurred as she continued to read, think, talk, and write.

Early questions: What kinds of LOs do most teachers experience? How do these experiences change teachers’ practices and beliefs? Are some LOs more effective than others? What makes them more effective?

First focused question: What makes LOs for teachers effective for improving their teaching for conceptual understanding?

Question after trying to predict the answer and imagining how to test the prediction: Do LOs that engage middle school mathematics teachers in studying mathematics content help teachers teach this same content with more of a conceptual emphasis?

Question after developing an initial rationale for her prediction: Under what conditions do LOs that engage middle school mathematics teachers in studying mathematics content help teachers teach this same content with more of a conceptual emphasis?

Question after developing a more precise prediction and richer rationale: Under what conditions do middle school teachers who lack conceptual knowledge of linear functions benefit from five 2-hour LO sessions that engage them in conceptual learning of linear functions as assessed by changes in their teaching toward a more conceptual emphasis on linear functions?

Part IV. An Illustrative Dialogue

The story of Martha described the major steps she took to refine her thinking. However, there is a lot of work that went on behind the scenes that wasn’t part of the story. For example, Martha had conversations with fellow students and professors that sharpened her thinking. What do these conversations look like? Because they are such an important part of the inquiry process, it will be helpful to “listen in” on the kinds of conversations that students might have with their advisors.

Here is a dialogue between a beginning student, Sam (S), and their advisor, Dr. Avery (A). They are meeting to discuss data Sam collected for a course project. The dialogue below is happening very early on in Sam’s conceptualization of the study, prior even to systematic reading of the literature.

Thanks for meeting with me today. As you know, I was able to collect some data for a course project a few weeks ago, but I’m having trouble analyzing the data, so I need your help. Let me try to explain the problem. As you know, I wanted to understand what middle-school teachers do to promote girls’ achievement in a mathematics class. I conducted four observations in each of three teachers’ classrooms. I also interviewed each teacher once about the four lessons I observed, and I interviewed two girls from each of the teachers’ classes. Obviously, I have a ton of data. But when I look at all these data, I don’t really know what I learned about my topic. When I was observing the teachers, I thought I might have observed some ways the teachers were promoting girls’ achievement, but then I wasn’t sure how to interpret my data. I didn’t know if the things I was observing were actually promoting girls’ achievement.

What were some of your observations?

Well, in a couple of my classroom observations, teachers called on girls to give an answer, even when the girls didn’t have their hands up. I thought that this might be a way that teachers were promoting the girls’ achievement. But then the girls didn’t say anything about that when I interviewed them and also the teachers didn’t do it in every class. So, it’s hard to know what effect, if any, this might have had on their learning or their motivation to learn. I didn’t want to ask the girls during the interview specifically about the teacher calling on them, and without the girls bringing it up themselves, I didn’t know if it had any effect.

Well, why didn’t you want to ask the girls about being called on?

Because I wanted to leave it as open as possible; I didn’t want to influence what they were going to say. I didn’t want to put words in their mouths. I wanted to know what they thought the teacher was doing that promoted their mathematical achievement and so I only asked the girls general questions, like “Do you think the teacher does things to promote girls’ mathematical achievement?” and “Can you describe specific experiences you have had that you believe do and do not promote your mathematical achievement?”

So then, how did they answer those general questions?

Well, with very general answers, such as that the teacher knows their names, offers review sessions, grades their homework fairly, gives them opportunities to earn extra credit, lets them ask questions, and always answers their questions. Nothing specific that helps me know what teaching actions specifically target girls’ mathematics achievement.

OK. Any ideas about what you might do next?

Well, I remember that when I was planning this data collection for my course, you suggested I might want to be more targeted and specific about what I was looking for. I can see now that more targeted questions would have made my data more interpretable in terms of connecting teaching actions to the mathematical achievement of girls. But I just didn’t want to influence what the girls would say.

Yes, I remember when you were planning your course project, you wanted to keep it open. You didn’t want to miss out on discovering something new and interesting. What do you think now about this issue?

Well, I still don’t want to put words in their mouths. I want to know what they think. But I see that if I ask really open questions, I have no guarantee they will talk about what I want them to talk about. I guess I still like the idea of an open study, but I see that it’s a risky approach. Leaving the questions too open meant I didn’t constrain their responses and there were too many ways they could interpret and answer the questions. And there are too many ways I could interpret their responses.

By this point in the dialogue, Sam has realized that open data (i.e., data not testing a specific prediction) is difficult to interpret. In the next part, Dr. Avery explains why collecting open data was not helping Sam achieve goals for her study that had motivated collecting open data in the first place.

Yes, I totally agree. Even for an experienced researcher, it can be difficult to make sense of this kind of open, messy data. However, if you design a study with a more specific focus, you can create questions for participants that are more targeted because you will be interested in their answers to these specific questions. Let’s reflect back on your data collection. What can you learn from it for the future?

When I think about it now, I realize that I didn’t think about the distinction between all the different constructs at play in my study, and I didn’t choose which one I was focusing on. One construct was the teaching moves that teachers think could be promoting achievement. Another is what teachers deliberately do to promote girls’ mathematics achievement, if anything. Another was the teaching moves that actually do support girls’ mathematics achievement. Another was what teachers were doing that supported girls’ mathematics achievement versus the mathematics achievement of all students. Another was students’ perception of what their teacher was doing to promote girls’ mathematics achievement. I now see that any one of these constructs could have been the focus of a study and that I didn’t really decide which of these was the focus of my course project prior to collecting data.

So, since you told me that the topic of this course project is probably what you’ll eventually want to study for your dissertation, which of these constructs are you most interested in?

I think I’m more interested in the teacher moves that teachers deliberately do to promote girls’ achievement. But I’m still worried about asking teachers directly and getting too specific about what they do because I don’t want to bias what they will say. And I chose qualitative methods and an exploratory design because I thought it would allow for a more open approach, an approach that helps me see what’s going on and that doesn’t bias or predetermine the results.

Well, it seems to me you are conflating three issues. One issue is how to conduct an unbiased study. Another issue is how specific to make your study. And the third issue is whether or not to choose an exploratory or qualitative study design. Those three issues are not the same. For example, designing a study that’s more open or more exploratory is not how researchers make studies fair and unbiased. In fact, it would be quite easy to create an open study that is biased. For example, you could ask very open questions and then interpret the responses in a way that unintentionally, and even unknowingly, aligns with what you were hoping the findings would say. Actually, you could argue that by adding more specificity and narrowing your focus, you’re creating constraints that prevent bias. The same goes for an exploratory or qualitative study; they can be biased or unbiased. So, let’s talk about what is meant by getting more specific. Within your new focus on what teachers deliberately do, there are many things that would be interesting to look at, such as teacher moves that address math anxiety, moves that allow girls to answer questions more frequently, moves that are specifically fitted to student thinking about specific mathematical content, and so on. What are one or two things that are most interesting to you? One way to answer this question is by thinking back to where your interest in this topic began.

In the preceding part of the dialogue, Dr. Avery explained how the goals Sam had for their study were not being met with open data. In the next part, Sam begins to articulate a prediction, which Sam and Dr. Avery then sharpen.

Actually, I became interested in this topic because of an experience I had in college when I was in a class of mostly girls. During whole class discussions, we were supposed to critically evaluate each other’s mathematical thinking, but we were too polite to do that. Instead, we just praised each other’s work. But it was so different in our small groups. It seemed easier to critique each other’s thinking and to push each other to better solutions in small groups. I began wondering how to get girls to be more critical of each other’s thinking in a whole class discussion in order to push everyone’s thinking.

Okay, this is great information. Why not use this idea to zoom-in on a more manageable and interpretable study? You could look specifically at how teachers support girls in critically evaluating each other’s thinking during whole class discussions. That would be a much more targeted and specific topic. Do you have predictions about what teachers could do in that situation, keeping in mind that you are looking specifically at girls’ mathematical achievement, not students in general?

Well, what I noticed was that small groups provided more social and emotional support for girls, whereas the whole class discussion did not provide that same support. The girls felt more comfortable critiquing each other’s thinking in small groups. So, I guess I predict that when the social and emotional supports that are present in small groups are extended to the whole class discussion, girls would be more willing to evaluate each other’s mathematical thinking critically during whole class discussion . I guess ultimately, I’d like to know how the whole class discussion could be used to enhance, rather than undermine, the social and emotional support that is present in the small groups.

Okay, then where would you start? Would you start with a study of what the teachers say they will do during whole class discussion and then observe if that happens during whole class discussion?

But part of my prediction also involves the small groups. So, I’d also like to include small groups in my study if possible. If I focus on whole groups, I won’t be exploring what I am interested in. My interest is broader than just the whole class discussion.

That makes sense, but there are many different things you could look at as part of your prediction, more than you can do in one study. For instance, if your prediction is that when the social and emotional supports that are present in small groups are extended to whole class discussions, girls would be more willing to evaluate each other’s mathematical thinking critically during whole class discussions , then you could ask the following questions: What are the social and emotional supports that are present in small groups?; In which small groups do they exist?; Is it groups that are made up only of girls?; Does every small group do this, and for groups that do this, when do these supports get created?; What kinds of small group activities that teachers ask them to work on are associated with these supports?; Do the same social and emotional supports that apply to small groups even apply to whole group discussion?

All your questions make me realize that my prediction about extending social and emotional supports to whole class discussions first requires me to have a better understanding of the social and emotional supports that exist in small groups. In fact, I first need to find out whether those supports commonly exist in small groups or is that just my experience working in small groups. So, I think I will first have to figure out what small groups do to support each other and then, in a later study, I could ask a teacher to implement those supports during whole class discussions and find out how you can do that. Yeah, now I’m seeing that.

The previous part of the dialogue illustrates how continuing to ask questions about one’s initial prediction is a good way to make it more and more precise (and researchable). In the next part, we see how developing a precise prediction has the added benefit of setting the researcher up for future studies.

Yes, I agree that for your first study, you should probably look at small groups. In other words, you should focus on only a part of your prediction for now, namely the part that says there are social and emotional supports in small groups that support girls in critiquing each other’s thinking . That begins to sharpen the focus of your prediction, but you’ll want to continue to refine it. For example, right now, the question that this prediction leads to is a question with a yes or no answer, but what you’ve said so far suggests to me that you are looking for more than that.

Yes, I want to know more than just whether there are supports. I’d like to know what kinds. That’s why I wanted to do a qualitative study.

Okay, this aligns more with my thinking about research as being prediction driven. It’s about collecting data that would help you revise your existing predictions into better ones. What I mean is that you would focus on collecting data that would allow you to refine your prediction, make it more nuanced, and go beyond what is already known. Does that make sense, and if so, what would that look like for your prediction?

Oh yes, I like that. I guess that would mean that, based on the data I collect for this next study, I could develop a more refined prediction that, for example, more specifically identifies and differentiates between different kinds of social and emotional supports that are present in small groups, or maybe that identifies the kinds of small groups that they occur in, or that predicts when and how frequently or infrequently they occur, or about the features of the small group tasks in which they occur, etc. I now realize that, although I chose qualitative research to make my study be more open, really the reason qualitative research fits my purposes is because it will allow me to explore fine-grained aspects of social and emotional supports that may exist for girls in small groups.

Yes, exactly! And then, based on the data you collect, you can include in your revised prediction those new fine-grained aspects. Furthermore, you will have a story to tell about your study in your written report, namely the story about your evolving prediction. In other words, your written report can largely tell how you filled out and refined your prediction as you learned more from carrying out the study. And even though you might not use them right away, you are also going to be able to develop new predictions that you would not have even thought of about social and emotional supports in small groups and your aim of extending them to whole-class discussions, had you not done this study. That will set you up to follow up on those new predictions in future studies. For example, you might have more refined ideas after you collect the data about the goals for critiquing student thinking in small groups versus the goals for critiquing student thinking during whole class discussion. You might even begin to think that some of the social and emotional supports you observe are not even replicable or even applicable to or appropriate for whole-class discussions, because the supports play different roles in different contexts. So, to summarize what I’m saying, what you look at in this study, even though it will be very focused, sets you up for a research program that will allow you to more fully investigate your broader interest in this topic, where each new study builds on your prior body of work. That’s why it is so important to be explicit about the best place to start this research, so that you can build on it.

I see what you are saying. We started this conversation talking about my course project data. What I think I should have done was figure out explicitly what I needed to learn with that study with the intention of then taking what I learned and using it as the basis for the next study. I didn’t do that, and so I didn’t collect data that pushed forward my thinking in ways that would guide my next study. It would be as if I was starting over with my next study.

Sam and Dr. Avery have just explored how specifying a prediction reveals additional complexities that could become fodder for developing a systematic research program. Next, we watch Sam beginning to recognize the level of specificity required for a prediction to be testable.

One thing that would have really helped would have been if you had had a specific prediction going into your data collection for your course project.

Well, I didn’t really have much of an explicit prediction in mind when I designed my methods.

Think back, you must have had some kind of prediction, even if it was implicit.

Well, yes, I guess I was predicting that teachers would enact moves that supported girls’ mathematical achievement. And I observed classrooms to identify those teacher moves, I interviewed teachers to ask them about the moves I observed, and I interviewed students to see if they mentioned those moves as promoting their mathematical achievement. The goal of my course project was to identify teacher moves that support girls’ mathematical achievement. And my specific research question was: What teacher moves support girls’ mathematical achievement?

So, really you were asking the teacher and students to show and tell you what those moves are and the effects of those moves, as a result putting the onus on your participants to provide the answers to your research question for you. I have an idea, let’s try a thought experiment. You come up with data collection methods for testing the prediction that there are social and emotional supports in small groups that support girls in critiquing each other’s thinking that still puts the onus on the participants. And then I’ll see if I can think of data collection methods that would not put the onus on the participants.

Hmm, well. .. I guess I could simply interview girls who participated in small groups and ask them “are there social and emotional supports that you use in small groups that support your group in critiquing each other’s thinking and if so, what are they?” In that case, I would be putting the onus on them to be aware of the social dynamics of small groups and to have thought about these constructs as much as I have. Okay now can you continue the thought experiment? What might the data collection methods look like if I didn’t put the onus on the participants?

First, I would pick a setting in which it was only girls at this point to reduce the number of variables. Then, personally I would want to observe a lot of groups of girls interacting in groups around tasks. I would be looking for instances when the conversation about students’ ideas was shut down and instances when the conversation about students’ ideas involved critiquing of ideas and building on each other’s thinking. I would also look at what happened just before and during those instances, such as: did the student continue to talk after their thinking was critiqued, did other students do anything to encourage the student to build on their own thinking (i.e., constructive criticism) or how did they support or shut down continued participation. In fact, now that I think about it, “critiquing each other’s thinking” can be defined in a number of different ways. I could mean just commenting on someone’s thinking, judging correctness and incorrectness, constructive criticism that moves the thinking forward, etc. If you put the onus on the participants to answer your research question, you are stuck with their definition, and they won’t have thought about this very much, if at all.

I think that what you are also saying is that my definitions would affect my data collection. If I think that critiquing each other’s thinking means that the group moves their thinking forward toward more valid and complete mathematical solutions, then I’m going to focus on different moves than if I define it another way, such as just making a comment on each other’s thinking and making each other feel comfortable enough to keep participating. In fact, am I going to look at individual instances of critiquing or look at entire sequences in which the critiquing leads to a goal? This seems like a unit of analysis question, and I would need to develop a more nuanced prediction that would make explicit what that unit of analysis is.

I agree, your definition of “critiquing each other’s thinking” could entirely change what you are predicting. One prediction could be based on defining critiquing as a one-shot event in which someone makes one comment on another person’s thinking. In this case the prediction would be that there are social and emotional supports in small groups that support girls in making an evaluative comment on another student’s thinking. Another prediction could be based on defining critiquing as a back-and-forth process in which the thinking gets built on and refined. In that case, the prediction would be something like that there are social and emotional supports in small groups that support girls in critiquing each other’s thinking in ways that do not shut down the conversation but that lead to sustained conversations that move each other toward more valid and complete solutions.

Well, I think I am more interested in the second prediction because it is more compatible with my long-term interests, which are that I’m interested in extending small group supports to whole class discussions. The second prediction is more appropriate for eventually looking at girls in whole class discussion. During whole class discussion, the teacher tries to get a sustained conversation going that moves the students’ thinking forward. So, if I learn about small group supports that lead to sustained conversations that move each other toward more valid and complete solutions , those supports might transfer to whole class discussions.

In the previous part of the dialogue, Dr. Avery and Sam showed how narrowing down a prediction to one that is testable requires making numerous important decisions, including how to define the constructs referred to in the prediction. In the final part of the dialogue, Dr. Avery and Sam begin to outline the reading Sam will have to do to develop a rationale for the specific prediction.

Do you see how your prediction and definitions are getting more and more specific? You now need to read extensively to further refine your prediction.

Well, I should probably read about micro dynamics of small group interactions, anything about interactions in small groups, and what is already known about small group interactions that support sustained conversations that move students’ thinking toward more valid and complete solutions. I guess I could also look at research on whole-class discussion methods that support sustained conversations that move the class to more mathematically valid and complete solutions, because it might give me ideas for what to look for in the small groups. I might also need to focus on research about how learners develop understandings about a particular subject matter so that I know what “more valid and complete solutions” look like. I also need to read about social and emotional supports but focus on how they support students cognitively, rather than in other ways.

Sounds good, let’s get together after you have processed some of this literature and we can talk about refining your prediction based on what you read and also the methods that will best suit testing that prediction.

Great! Thanks for meeting with me. I feel like I have a much better set of tools that push my own thinking forward and allow me to target something specific that will lead to more interpretable data.

Part V. Is It Always Possible to Formulate Hypotheses?

In Chap. 1 , we noted you are likely to read that research does not require formulating hypotheses. Some sources describe doing research without making predictions and developing rationales for these predictions. Some researchers say you cannot always make predictions—you do not know enough about the situation. In fact, some argue for the value of not making predictions (e.g., Glaser & Holton, 2004 ; Merton, 1968 ; Nemirovsky, 2011 ). These are important points of view, so we will devote this section to discussing them.

Can You Always Predict What You Will Find?

One reason some researchers say you do not need to make predictions is that it can be difficult to imagine what you will find. This argument comes up most often for descriptive studies. Suppose you want to describe the nature of a situation you do not know much about. Can you still make a prediction about what you will find? We believe that, although you do not know exactly what you will find, you probably have a hunch or, at a minimum, a very fuzzy idea. It would be unusual to ask a question about a situation you want to know about without at least a fuzzy inkling of what you might find. The original question just would not occur to you. We acknowledge you might have only a vague idea of what you will find and you might not have much confidence in your prediction. However, we expect if you monitor your own thinking you will discover you have developed a suspicion along the way, regardless how vague the suspicion might be. Through the cyclic process we discussed above, that suspicion or hunch gradually evolves and turns into a prediction.

The Benefits of Making Predictions Even When They Are Wrong: An Example from the 1970s

One of us was a graduate student at the University of Wisconsin in the late 1970s, assigned as a research assistant to a project that was investigating young children’s thinking about simple arithmetic. A new curriculum was being written, and the developers wanted to know how to introduce the earliest concepts and skills to kindergarten and first-grade children. The directors of the project did not know what to expect because, at the time, there was little research on five- and six-year-olds’ pre-instruction strategies for adding and subtracting.

After consulting what literature was available, talking with teachers, analyzing the nature of different types of addition and subtraction problems, and debating with each other, the research team formulated some hypotheses about children’s performance. Following the usual assumptions at the time and recognizing the new curriculum would introduce the concepts, the researchers predicted that, before instruction, most children would not be able to solve the problems. Based on the rationale that some young children did not yet recognize the simple form for written problems (e.g., 5 + 3 = ___), the researchers predicted that the best chance for success would be to read problems as stories (e.g., Jesse had 5 apples and then found 3 more. How many does she have now?). They reasoned that, even though children would have difficulty on all the problems, some story problems would be easier because the semantic structure is easier to follow. For example, they predicted the above story about adding 3 apples to 5 would be easier than a problem like, “Jesse had some apples in the refrigerator. She put in 2 more and now has 6. How many were in the refrigerator at the beginning?” Based on the rationale that children would need to count to solve the problems and that it can be difficult to keep track of the numbers, they predicted children would be more successful if they were given counters. Finally, accepting the common reasoning that larger numbers are more difficult than smaller numbers, they predicted children would be more successful if all the numbers in a problem were below 10.

Although these predictions were not very precise and the rationales were not strongly convincing, these hypotheses prompted the researchers to design the study to test their predictions. This meant they would collect data by presenting a variety of problems under a variety of conditions. Because the goal was to describe children’s thinking, problems were presented to students in individual interviews. Problems with different semantic structures were included, counters were available for some problems but not others, and some problems had sums to 9 whereas others had sums to 20 or more.

The punchline of this story is that gathering data under these conditions, prompted by the predictions, made all the difference in what the researchers learned. Contrary to predictions, children could solve addition and subtraction problems before instruction. Counters were important because almost all the solution strategies were based on counting which meant that memory was an issue because many strategies require counting in two ways simultaneously. For example, subtracting 4 from 7 was usually solved by counting down from 7 while counting up from 1 to 4 to keep track of counting down. Because children acted out the stories with their counters, the semantic structure of the story was also important. Stories that were easier to read and write were also easier to solve.

To make a very long story very short, other researchers were, at about the same time, reporting similar results about children’s pre-instruction arithmetic capabilities. A clear pattern emerged regarding the relative difficulty of different problem types (semantic structures) and the strategies children used to solve each type. As the data were replicated, the researchers recognized that kindergarten and first-grade teachers could make good use of this information when they introduced simple arithmetic. This is how Cognitively Guided Instruction (CGI) was born (Carpenter et al., 1989 ; Fennema et al., 1996 ).

To reiterate, the point of this example is that the study conducted to describe children’s thinking would have looked quite different if the researchers had made no predictions. They would have had no reason to choose the particular problems and present them under different conditions. The fact that some of the predictions were completely wrong is not the point. The predictions created the conditions under which the predictions were tested which, in turn, created learning opportunities for the researchers that would not have existed without the predictions. The lesson is that even research that aims to simply describe a phenomenon can benefit from hypotheses. As signaled in Chap. 1 , this also serves as another example of “failing productively.”

Suggestions for What to Do When You Do Not Have Predictions

There likely are exceptions to our claim about being able to make a prediction about what you will find. For example, there could be rare cases where researchers truly have no idea what they will find and can come up with no predictions and even no hunches. And, no research has been reported on related phenomena that would offer some guidance. If you find yourself in this position, we suggest one of three approaches: revise your question, conduct a pilot study, or choose another question.

Because there are many advantages to making predictions explicit and then writing out the reasons for these predictions, one approach is to adjust your question just enough to allow you to make a prediction. Perhaps you can build on descriptions that other researchers have provided for related situations and consider how you can extend this work. Building on previous descriptions will enable you to make predictions about the situation you want to describe.

A second approach is to conduct a small pilot study or, better, a series of small pilot studies to develop some preliminary ideas of what you might find. If you can identify a small sample of participants who are similar to those in your study, you can try out at least some of your research plans to help make and refine your predictions. As we detail later, you can also use pilot studies to check whether key aspects of your methods (e.g., tasks, interview questions, data collection methods) work as you expect.

A third approach is to return to your list of interests and choose one that has been studied previously. Sometimes this is the wisest choice. It is very difficult for beginning researchers to conduct research in brand-new areas where no hunches or predictions are possible. In addition, the contributions of this research can be limited. Recall the earlier story about one of us “failing productively” by completing a dissertation in a somewhat new area. If, after an exhaustive search, you find that no one has investigated the phenomenon in which you are interested or even related phenomena, it can be best to move in a different direction. You will read recommendations in other sources to find a “gap” in the research and develop a study to “fill the gap.” This can be helpful advice if the gap is very small. However, if the gap is large, too large to predict what you might find, the study will present severe challenges. It will be more productive to extend work that has already been done than to launch into an entirely new area.

Should You Always Try to Predict What You Will Find?

In short, our answer to the question in the heading is “yes.” But this calls for further explanation.

Suppose you want to observe a second-grade classroom in order to investigate how students talk about adding and subtracting whole numbers. You might think, “I don’t want to bias my thinking; I want to be completely open to what I see in the classroom.” Sam shared a similar point of view at the beginning of the dialogue: “I wanted to leave it as open as possible; I didn’t want to influence what they were going to say.” Some researchers say that beginning your research study by making predictions is inappropriate precisely because it will bias your observations and results. The argument is that by bringing a set of preconceptions, you will confirm what you expected to find and be blind to other observations and outcomes. The following quote illustrates this view: “The first step in gaining theoretical sensitivity is to enter the research setting with as few predetermined ideas as possible—especially logically deducted, a priori hypotheses. In this posture, the analyst is able to remain sensitive to the data by being able to record events and detect happenings without first having them filtered through and squared with pre-existing hypotheses and biases” (Glaser, 1978, pp. 2–3).

We take a different point of view. In fact, we believe there are several compelling reasons for making your predictions explicit.

Making Your Predictions Explicit Increases Your Chances of Productive Observations

Because your predictions are an extension of what is already known, they prepare you to identify more nuanced relationships that can advance our understanding of a phenomenon. For example, rather than simply noticing, in a general sense, that students talking about addition and subtraction leads them to better understandings, you might, based on your prediction, make the specific observation that talking about addition and subtraction in a particular way helps students to think more deeply about a particular concept related to addition and subtraction. Going into a study without predictions can bring less sensitivity rather than more to the study of a phenomenon. Drawing on knowledge about related phenomena by reading the literature and conducting pilot studies allows you to be much more sensitive and your observations to be more productive.

Making Your Predictions Explicit Allows You to Guard Against Biases

Some genres and methods of educational research are, in fact, rooted in philosophical traditions (e.g., Husserl, 1929/ 1973 ) that explicitly call for researchers to temporarily “bracket” or set aside existing theory as well as their prior knowledge and experience to better enter into the experience of the participants in the research. However, this does not mean ignoring one’s own knowledge and experience or turning a blind eye to what has been learned by others. Much more than the simplistic image of emptying one’s mind of preconceptions and implicit biases (arguably an impossible feat to begin with), the goal is to be as reflective as possible about one’s prior knowledge and conceptions and as transparent as possible about how they may guide observations and shape interpretations (Levitt et al., 2018 ).

We believe it is better to be honest about the predictions you are almost sure to have because then you can deliberately plan to minimize the chances they will influence what you find and how you interpret your results. For starters, it is important to recognize that acknowledging you have some guesses about what you will find does not make them more influential. Because you are likely to have them anyway, we recommend being explicit about what they are. It is easier to deal with biases that are explicit than those that lurk in the background and are not acknowledged.

What do we mean by “deal with biases”? Some journals require you to include a statement about your “positionality” with respect to the participants in your study and the observations you are making to gather data. Formulating clear hypotheses is, in our view, a direct response to this request. The reasons for your predictions are your explicit statements about your positionality. Often there are methodological strategies you can use to protect the study from undue influences of bias. In other words, making your vague predictions explicit can help you design your study so you minimize the bias of your findings.

Making Your Predictions Explicit Can Help You See What You Did Not Predict

Making your predictions explicit does not need to blind you to what is different than expected. It does not need to force you to see only what you want to see. Instead, it can actually increase your sensitivity to noticing features of the situation that are surprising, features you did not predict. Results can stand out when you did not expect to see them.

In contrast, not bringing your biases to consciousness might subtly shift your attention away from these unexpected results in ways that you are not aware of. This path can lead to claiming no biases and no unexpected findings without being conscious of them. You cannot observe everything, and some things inevitably will be overlooked. If you have predicted what you will see, you can design your study so that the unexpected results become more salient rather than less.

Returning to the example of observing a second-grade classroom, we note that the field already knows a great deal about how students talk about addition and subtraction. Being cognizant of what others have observed allows you to enter the classroom with some clear predictions about what will happen. The rationales for these predictions are based on all the related knowledge you have before stepping into the classroom, and the predictions and rationales help you to better deal with what you see. This is partly because you are likely to be surprised by the things you did not anticipate. There is almost always something that will surprise you because your predictions will almost always be incomplete or too general. This sensitivity to the unanticipated—the sense of surprise that sparks your curiosity—is an indication of your openness to the phenomenon you are studying.

Making Your Predictions Explicit Allows You to Plan in Advance

Recall from Chap. 1 the descriptor of scientific inquiry: “Experience carefully planned in advance.” If you make no predictions about what might happen, it is very difficult, if not impossible, to plan your study in advance. Again, you cannot observe everything, so you must make decisions about what you will observe. What kind of data will you plan to collect? Why would you collect these data instead of others? If you have no idea what to expect, on what basis will you make these consequential decisions? Even if your predictions are vague and your rationales for the predictions are a bit shaky, at least they provide a direction for your plan. They allow you to explain why you are planning this study and collecting these data. They allow you to “carefully plan in advance.”

Making Your Predictions Explicit Allows You to Put Your Rationales in Harm’s Way

Rationales are developed to justify the predictions. Rationales represent your best reasoning about the research problem you are studying. How can you tell whether your reasoning is sound? You can try it out with colleagues. However, the best way to test it is to put it in “harm’s way” (Cobb, Confrey, diSessa, Lehrer, & Schauble, 2003 p. 10). And the best approach to putting your reasoning in harm’s way is to test the predictions it generates. Regardless if you are conducting a qualitative or quantitative study, rationales can be improved only if they generate testable predictions. This is possible only if predictions are explicit and precise. As we described earlier, rationales are evaluated for their soundness and refined in light of the specific differences between predictions and empirical observations.

Making Your Predictions Explicit Forces You to Organize and Extend Your (and the Field’s) Thinking

By writing out your predictions (even hunches or fuzzy guesses) and by reflecting on why you have these predictions and making these reasons explicit for yourself, you are advancing your thinking about the questions you really want to answer. This means you are making progress toward formulating your research questions and your final hypotheses. Making more progress in your own thinking before you conduct your study increases the chances your study will be of higher quality and will be exactly the study you intended. Making predictions, developing rationales, and imagining tests are tools you can use to push your thinking forward before you even collect data.

Suppose you wonder how preservice teachers in your university’s teacher preparation program will solve particular kinds of math problems. You are interested in this question because you have noticed several PSTs solve them in unexpected ways. As you ask the question you want to answer, you make predictions about what you expect to see. When you reflect on why you made these predictions, you realize that some PSTs might use particular solution strategies because they were taught to use some of them in an earlier course, and they might believe you expect them to solve the problems in these ways. By being explicit about why you are making particular predictions, you realize that you might be answering a different question than you intend (“How much do PSTs remember from previous courses?” or even “To what extent do PSTs believe different instructors have similar expectations?”). Now you can either change your question or change the design of your study (i.e., the sample of students you will use) or both. You are advancing your thinking by being explicit about your predictions and why you are making them.

The Costs of Not Making Predictions

Avoiding making predictions, for whatever reason, comes with significant costs. It prevents you from learning very much about your research topic. It would require not reading related research, not talking with your colleagues, and not conducting pilot studies because, if you do, you are likely to find a prediction creeping into your thinking. Not doing these things would forego the benefits of advancing your thinking before you collect data. It would amount to conducting the study with as little forethought as possible.

Part VI. How Do You Formulate Important Hypotheses?

We provided a partial answer in Chap. 1 to the question of a hypothesis’ importance when we encouraged considering the ultimate goal to which a study’s findings might contribute. You might want to reread Part III of Chap. 1 where we offered our opinions about the purposes of doing research. We also recommend reading the March 2019 editorial in the Journal for Research in Mathematics Education (Cai et al., 2019b ) in which we address what constitutes important educational research.

As we argued in Chap. 1 and in the March 2019 editorial, a worthy ultimate goal for educational research is to improve the learning opportunities for all students. However, arguments can be made for other ultimate goals as well. To gauge the importance of your hypotheses, think about how clearly you can connect them to a goal the educational community considers important. In addition, given the descriptors of scientific inquiry proposed in Chap. 1 , think about how testing your hypotheses will help you (and the community) understand what you are studying. Will you have a better explanation for the phenomenon after your study than before?

Although we address the question of importance again, and in more detail, in Chap. 5 , it is useful to know here that you can determine the significance or importance of your hypotheses when you formulate them. The importance need not depend on the data you collect or the results you report. The importance can come from the fact that, based on the results of your study, you will be able to offer revised hypotheses that help the field better understand an important issue. In large part, it is these revised hypotheses rather than the data that determine a study’s importance.

A critical caveat to this discussion is that few hypotheses are self-evidently important. They are important only if you make the case for their importance. Even if you follow closely the guidelines we suggest for formulating an important hypothesis, you must develop an argument that convinces others. This argument will be presented in the research paper you write.

The picture has a few hypotheses that are self-evidently important. They are important only if you make the case for their importance; written.

Consider Martha’s hypothesis presented earlier. When we left Martha, she predicted that “Participating teachers will show changes in their teaching with a greater emphasis on conceptual understanding with larger changes on linear function topics directly addressed in the LOs than on other topics.” For researchers and educators not intimately familiar with this area of research, it is not apparent why someone should spend a year or more conducting a dissertation to test this prediction. Her rationale, summarized earlier, begins to describe why this could be an important hypothesis. But it is by writing a clear argument that explains her rationale to readers that she will convince them of its importance.

How Martha fills in her rationale so she can create a clear written argument for its importance is taken up in Chap. 3 . As we indicated, Martha’s work in this regard led her to make some interesting decisions, in part due to her own assessment of what was important.

Part VII. Beginning to Write the Research Paper for Your Study

It is common to think that researchers conduct a study and then, after the data are collected and analyzed, begin writing the paper about the study. We recommend an alternative, especially for beginning researchers. We believe it is better to write drafts of the paper at the same time you are planning and conducting your study. The paper will gradually evolve as you work through successive phases of the scientific inquiry process. Consequently, we will call this paper your evolving research paper .

The picture has, we believe it is better to write drafts of the paper at the same time you are planning and conducting your study; written.

You will use your evolving research paper to communicate your study, but you can also use writing as a tool for thinking and organizing your thinking while planning and conducting the study. Used as a tool for thinking, you can write drafts of your ideas to check on the clarity of your thinking, and then you can step back and reflect on how to clarify it further. Be sure to avoid jargon and general terms that are not well defined. Ask yourself whether someone not in your field, maybe a sibling, a parent, or a friend, would be able to understand what you mean. You are likely to write multiple drafts with lots of scribbling, crossing out, and revising.

Used as a tool for communicating, writing the best version of what you know before moving to the next phase will help you record your decisions and the reasons for them before you forget important details. This best-version-for-now paper also provides the basis for your thinking about the next phase of your scientific inquiry.

At this point in the process, you will be writing your (research) questions, the answers you predict, and the rationales for your predictions. The predictions you make should be direct answers to your research questions and should flow logically from (or be directly supported by) the rationales you present. In addition, you will have a written statement of the study’s purpose or, said another way, an argument for the importance of the hypotheses you will be testing. It is in the early sections of your paper that you will convince your audience about the importance of your hypotheses.

In our experience, presenting research questions is a more common form of stating the goal of a research study than presenting well-formulated hypotheses. Authors sometimes present a hypothesis, often as a simple prediction of what they might find. The hypothesis is then forgotten and not used to guide the analysis or interpretations of the findings. In other words, authors seldom use hypotheses to do the kind of work we describe. This means that many research articles you read will not treat hypotheses as we suggest. We believe these are missed opportunities to present research in a more compelling and informative way. We intend to provide enough guidance in the remaining chapters for you to feel comfortable organizing your evolving research paper around formulating, testing, and revising hypotheses.

While we were editing one of the leading research journals in mathematics education ( JRME ), we conducted a study of reviewers’ critiques of papers submitted to the journal. Two of the five most common concerns were: (1) the research questions were unclear, and (2) the answers to the questions did not make a substantial contribution to the field. These are likely to be major concerns for the reviewers of all research journals. We hope the knowledge and skills you have acquired working through this chapter will allow you to write the opening to your evolving research paper in a way that addresses these concerns. Much of the chapter should help make your research questions clear, and the prior section on formulating “important hypotheses” will help you convey the contribution of your study.

Exercise 2.3

Look back at your answers to the sets of questions before part II of this chapter.

Think about how you would argue for the importance of your current interest.

Write your interest in the form of (1) a research problem, (2) a research question, and (3) a prediction with the beginnings of a rationale. You will update these as you read the remaining chapters.

Part VIII. The Heart of Scientific Inquiry

In this chapter, we have described the process of formulating hypotheses. This process is at the heart of scientific inquiry. It is where doing research begins. Conducting research always involves formulating, testing, and revising hypotheses. This is true regardless of your research questions and whether you are using qualitative, quantitative, or mixed methods. Without engaging in this process in a deliberate, intense, relentless way, your study will reveal less than it could. By engaging in this process, you are maximizing what you, and others, can learn from conducting your study.

In the next chapter, we build on the ideas we have developed in the first two chapters to describe the purpose and nature of theoretical frameworks . The term theoretical framework, along with closely related terms like conceptual framework, can be somewhat mysterious for beginning researchers and can seem like a requirement for writing a paper rather than an aid for conducting research. We will show how theoretical frameworks grow from formulating hypotheses—from developing rationales for the predicted answers to your research questions. We will propose some practical suggestions for building theoretical frameworks and show how useful they can be. In addition, we will continue Martha’s story from the point at which we paused earlier—developing her theoretical framework.

Cai, J., Morris, A., Hohensee, C., Hwang, S., Robison, V., Cirillo, M., Kramer, S. L., & Hiebert, J. (2019b). Posing significant research questions. Journal for Research in Mathematics Education, 50 (2), 114–120. https://doi.org/10.5951/jresematheduc.50.2.0114

Article Google Scholar

Carpenter, T. P., Fennema, E., Peterson, P. L., Chiang, C. P., & Loef, M. (1989). Using knowledge of children’s mathematics thinking in classroom teaching: An experimental study. American Educational Research Journal, 26 (4), 385–531.

Fennema, E., Carpenter, T. P., Franke, M. L., Levi, L., Jacobs, V. R., & Empson, S. B. (1996). A longitudinal study of learning to use children’s thinking in mathematics instruction. Journal for Research in Mathematics Education, 27 (4), 403–434.

Glaser, B. G., & Holton, J. (2004). Remodeling grounded theory. Forum: Qualitative Social Research, 5(2). https://www.qualitative-research.net/index.php/fqs/article/view/607/1316

Gournelos, T., Hammonds, J. R., & Wilson, M. A. (2019). Doing academic research: A practical guide to research methods and analysis . Routledge.

Book Google Scholar

Hohensee, C. (2014). Backward transfer: An investigation of the influence of quadratic functions instruction on students’ prior ways of reasoning about linear functions. Mathematical Thinking and Learning, 16 (2), 135–174.

Husserl, E. (1973). Cartesian meditations: An introduction to phenomenology (D. Cairns, Trans.). Martinus Nijhoff. (Original work published 1929).

Google Scholar

Levitt, H. M., Bamberg, M., Creswell, J. W., Frost, D. M., Josselson, R., & Suárez-Orozco, C. (2018). Journal article reporting standards for qualitative primary, qualitative meta-analytic, and mixed methods research in psychology: The APA Publications and Communications Board Task Force report. American Psychologist, 73 (1), 26–46.

Medawar, P. (1982). Pluto’s republic [no typo]. Oxford University Press.

Merton, R. K. (1968). Social theory and social structure (Enlarged edition). Free Press.

Nemirovsky, R. (2011). Episodic feelings and transfer of learning. Journal of the Learning Sciences, 20 (2), 308–337. https://doi.org/10.1080/10508406.2011.528316

Vygotsky, L. (1987). The development of scientific concepts in childhood: The design of a working hypothesis. In A. Kozulin (Ed.), Thought and language (pp. 146–209). The MIT Press.

Download references

Author information

Authors and affiliations.

School of Education, University of Delaware, Newark, DE, USA

James Hiebert, Anne K Morris & Charles Hohensee

Department of Mathematical Sciences, University of Delaware, Newark, DE, USA

Jinfa Cai & Stephen Hwang

You can also search for this author in PubMed Google Scholar

Rights and permissions

Open Access This chapter is licensed under the terms of the Creative Commons Attribution 4.0 International License ( http://creativecommons.org/licenses/by/4.0/ ), which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license and indicate if changes were made.

The images or other third party material in this chapter are included in the chapter's Creative Commons license, unless indicated otherwise in a credit line to the material. If material is not included in the chapter's Creative Commons license and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder.

Reprints and permissions

Copyright information

About this chapter

Hiebert, J., Cai, J., Hwang, S., Morris, A.K., Hohensee, C. (2023). How Do You Formulate (Important) Hypotheses?. In: Doing Research: A New Researcher’s Guide. Research in Mathematics Education. Springer, Cham. https://doi.org/10.1007/978-3-031-19078-0_2

Download citation

DOI : https://doi.org/10.1007/978-3-031-19078-0_2

Published : 03 December 2022

Publisher Name : Springer, Cham

Print ISBN : 978-3-031-19077-3

Online ISBN : 978-3-031-19078-0

eBook Packages : Education Education (R0)

Share this chapter

Anyone you share the following link with will be able to read this content:

Sorry, a shareable link is not currently available for this article.

Provided by the Springer Nature SharedIt content-sharing initiative

Publish with us

Policies and ethics

Find a journal
Track your research

Recoding Introduction to Mediation, Moderation, and Conditional Process Analysis

8 extending the fundamental principles of moderation analysis.

As Hayes opened, “in this chapter, [we’ll see] how [the] principles of moderation analysis are applied when the moderator is dichotomous (rather than a continuum, as in the previous chapter) as well as when both focal antecedent and moderator are continuous (p. 267).”

8.1 Moderation with a dichotomous moderator

Here we load a couple necessary packages, load the data, and take a glimpse() .

Regardless of whether the antecedent variables are continuous or binary, the equation for the simple moderation is still

\[Y = i_Y + b_1 X + b_2 W + b_3 XW + e_Y.\]

We can use that equation to fit our first moderation model with a binary $W$ (i.e., frame ) like so.

Check the summary.

We’ll compute our Bayeisan $R^2$ in the typical way.

This model should look familiar to you because it is exactly the same model estimated in the analysis presented in Chapter 7 (see Table 7.4, model 3). The only differences between these two analyses are how the corresponding question is framed, meaning which variable is deemed the focal antecedent and which is the moderator, and how these variables are symbolically labeled as $X$ and $W$ . In the analysis in Chapter 7, the focal antecedent variable was a dichotomous variable coding the framing of the cause of the disaster (labeled $X$ then, but $W$ now), whereas in this analysis, the focal antecedent is a continuous variable placing each person on a continuum of climate change skepticism (labeled $W$ then, but $X$ now), with the moderator being a dichotomous variable coding experimental condition. So this example illustrates the symmetry property of interactions introduced in section 7.1. (p. 272)

8.1.1 Visualizing and probing the interaction.

For the plots in this chapter, we’ll take our color palette from the ochRe package , which provides Australia-inspired colors. We’ll also use a few theme settings from good-old ggthemes . As in the last chapter, we’ll save our adjusted theme settings as an object, theme_08 .

Happily, the ochRe package has a handy convenience function, viz_palette() , that makes it easy to preview the colors available in a given palette. We’ll be using “olsen_qual” and “olsen_seq”.

Behold our Figure 8.3.

In addition to our fancy Australia-inspired colors, we’ll also play around a bit with spaghetti plots in this chapter. To my knowledge, this use of spaghetti plots is uniquely Bayesian. If you’re trying to wrap your head around what on earth we just did, take a look at the first few rows from posterior_samples() object, post .

The head() function returned six rows, each one corresponding to the credible parameter values from a given posterior draw. The lp__ is uniquely Bayesian and beyond the scope of this project. You might think of sigma as the Bayesian analogue to what the OLS folks often refer to as error or the residual variance. Hayes doesn’t tend to emphasize it in this text, but it’s something you’ll want to pay increasing attention to as you move along in your Bayesian career. All the columns starting with b_ are the regression parameters, the model coefficients or the fixed effects. But anyways, notice that those b_ columns correspond to the four parameter values in Formula 8.2 on page 270. Here they are, but reformatted to more closely mimic the text:

$\hat{Y}$ = 2.171 + 0.181 $X$ + -0.621 $W$ + 0.181 XW
$\hat{Y}$ = 2.113 + 0.193 $X$ + -0.535 $W$ + 0.175 XW
$\hat{Y}$ = 2.58 + 0.047 $X$ + -0.339 $W$ + 0.166 XW
$\hat{Y}$ = 2.609 + 0.046 $X$ + -0.394 $W$ + 0.159 XW
$\hat{Y}$ = 2.408 + 0.127 $X$ + -0.591 $W$ + 0.196 XW
$\hat{Y}$ = 2.384 + 0.115 $X$ + -0.596 $W$ + 0.23 XW

Each row of post , each iteration or posterior draw, yields a full model equation that is a credible description of the data–or at least as credible as we can get within the limits of the model we have specified, our priors (which we typically cop out on and just use defaults in this project), and how well those fit when applied to the data at hand. So when we use brms convenience functions like fitted() , we pass specific predictor values through those 4000 unique model equations, which returns 4000 similar but distinct expected $Y$ -values. So although a nice way to summarize those 4000 values is with summaries such as the posterior mean/median and 95% intervals, another way is to just plot an individual regression line for each of the iterations. That is what’s going on when we depict out models with a spaghetti plot.

The thing I like about spaghetti plots is that they give a three-dimensional sense of the posterior. Note that each individual line is very skinny and semitransparent. When you pile a whole bunch of them atop each other, the peaked or most credible regions of the posterior are the most saturated in color. Less credible posterior regions almost seamlessly merge into the background. Also, note how the combination of many similar but distinct straight lines results in a bowtie shape. Hopefully this clarifies where that shape’s been coming from when we use geom_ribbon() to plot the 95% intervals.

Back to the text, on the bottom of page 274, Hayes pointed out the conditional effect of skeptic when frame == 1 is $b_1 + b_3 = 0.306$ . We can show that with a little arithmetic followed up with tidybayes::mean_qi() .

But anyways, you could recode frame in a number of ways, including if_else() or, in this case, by simple arithmetic.

With frame_ep in hand, we’re ready to refit the model.

Our results match nicely with the formula on page 275.

If you want to follow along with Hayes on page 276 and isolate the 95% credible intervals for the skeptic parameter, you can use the posterior_interval() function.

8.2 Interaction between two quantitative variables

Here’s the glbwarm data.

In this section we add three covariates (i.e., $C$ variables) to the basic moderation model. Although Hayes made a distinction between the $X$ , $M$ , and $C$ variables in the text, that distinction is conceptual and doesn’t impact the way we enter them into brm() . Rather, the brm() formula clarifies they’re all just predictors.

Our results cohere nicely with the Hayes’s formula in the middle of page 278 or with the results he displayed in Table 8.2.

Here’s the $R^2$ summary.

As the $R^2$ is a good bit away from the boundaries, it’s nicely Gaussian.

8.2.1 Visualizing and probing the interaction.

For our version of Figure 8.5, we’ll need to adjust our nd data for fitted() .

Our fitted() and ggplot2 code will be quite similar to the last spaghetti plot. Only this time we’ll use filter() to reduce the number of posterior draws we show in the plot.

When we reduce the number of lines depicted in the plot, we lose some of the three-dimensional illusion. It’s nice, however, to get a closer look to each individual line. To each their own.

We’ll continue with our spaghetti plot approach for Figure 8.7. Again, when we made the JN technique plot for Chapter 7, we computed values for the posterior mean and the 95% intervals. Because the intervals follow a bowtie shape, we had to compute the $Y$ -values for many values across the x-axis in order to make the curve look smooth. But as long as we stick with the spaghetti plot approach, all we need are the values at the endpoints of each iteration. Although each line is straight, the combination of many lines is what produces the bowtie effect.

In other words, each of those orange lines is a credible expression of $\theta_{X \rightarrow Y}$ (i.e., $b_1 + b_3 W$ ) across a continuous range of $W$ values.

8.3 Hierarchical versus simultaneous entry

Many investigators test a moderation hypothesis in regression analysis using a method that on the surface seems different than the procedure described thus far. This alternative approach is to build a regression model by adding the product of $X$ and $W$ to a model already containing $X$ and $W$ . This procedure is sometimes called hierarchical regression or hierarchical variable entry (and easily confused by name with hierarchical linear modeling , which is an entirely different thing). The goal using this method is to determine whether allowing $X$ ’s effect to be contingent on $W$ produces a better fitting model than one in which the effect of $X$ is constrained to be unconditional on $W$ . According to the logic of hierarchical entry, if the contingent model accounts for more of the variation in Y than the model that forces $X$ ’s effect to be independent of $W$ , then the better model is one in which $W$ is allowed to moderate $X$ ’s effect. Although this approach works, it is a widely believed myth that it is necessary to use this approach in order to test a moderation hypothesis. (p. 289, emphasis in the original)

Although this method is not necessary, it can be handy to slowly build your model. This method can also serve nice rhetorical purposes in a paper. Anyway, here’s our multivariable but non-moderation model, model8.4 .

Here we’ll compute the corresponding $R^2$ and compare it with the one for the original interaction model with a difference score.

Note that the Bayesian $R^2$ performed differently than the $F$ -test in the text.

We can also compare these with the LOO, which, as is typical of information criteria, corrects for model complexity. First, we compute them and attach the results to the model fit objects.

Now use the loo_compare() function to compare them directly.

As a reminder, we generally prefer models with lower information criteria, which in this case is clearly the moderation model (i.e., model8.1 ). However, the standard error value (i.e., se_diff ) for the difference (i.e., elpd_diff ) is quite large, which suggests that the model with the lowest value isn’t the clear winner. Happily, these results match nicely with the Bayesian $R^2$ difference score. The moderation model appears somewhat better than the multivariable model, but its superiority is hardly decisive.

8.4 The equivalence between moderated regression analysis and a 2 X 2 factorial analysis of variance

I’m just not going to encourage ANOVA $F$ -testing methodology. However, I will show the Bayesian regression model. First, here are the data.

Fit the moderation model.

Those results don’t look anything like what Hayes reported in Tables 8.3 or 8.4. However, a little deft manipulation of the posterior samples can yield equivalent results to Hayes’s Table 8.3.

Here are the cell-specific means in Table 8.3.

And here are the marginal means from Table 8.3.

For kicks and giggles, here are what the cell-specific means look like in box plots.

And here are the same for the marginal means. This time we’ll show the shapes of the posteriors with violin plots with horizontal lines depicting the median and interquartile ranges.

On page 294, Hayes used point estimates to compute the simple effect of policy information among Kerry supporters and then the same thing among Bush supporters. Here’s how we’d do that when working with the full vector of posterior draws.

So then computing the main effect for policy information using the simple effects is little more than an extension of those steps.

And we get the same results by strategically subtracting the marginal means.

The main effect of for candidate is similarly computed using either approach.

We don’t have an $F$ -test for our Bayesian moderation model. But we do have an interaction term. Here’s its distribution.

Following Hayes’s work on the bottom of page 295, here’s how you’d reproduce that by manipulating our $\overline Y$ vectors.

Extending that logic, we also get the answer this way.

8.4.1 Simple effects parameterization.

We might reacquaint ourselves with the formula from model8.5 .

The results cohere nicely with the “Model 1” results at the top of Table 8.5.

The Bayesian $R^2$ portion looks on point, too.

Our various Y_bar transformations from before continue to cohere with the coefficients, above, just like in the text. E.g., the policy coefficient may be returned like so.

We can continue to use Hayes’s Y_bar transformations to return the kerry coefficient, too.

Here we compute $b_3$ with the difference between the simple effects of $X$ at levels of $W$ .

And now $b_{3}$ with the difference between the simple effects of $W$ at levels of $X$ .

8.4.2 Main effects parameterization.

A nice feature of brms is you can transform your data right within the brm() or update() functions. Here we’ll make our two new main-effects-coded variables, policy_me and kerry_me , with the mutate() function right within update() .

Transforming your data within the brms functions won’t change the original data structure. However, brms will save the data used to fit the model within the brm() object. You can access that data like so.

But we digress. Here’s our analogue to the “Model 2” portion of Table 8.5.

Like with model8.6 , above, we’ll need a bit of algebra to compute our $\overline Y_i$ vectors.

With our post for fit5 in hand, we’ll follow the formulas at the top of page 298 to compute our $b_1$ and $b_2$ distributions.

Hayes pointed out that the interaction effect, $b_3$ , is the same across models his OLS Models 1 and 2. This is largely true for our Bayesian HMC model8.5 and model8.6 models.

However, the results aren’t exactly the same because of simulation error. If you were working on a project requiring high precision, increase the number of posterior iterations. To demonstrate, here we’ll increase each chain’s post-warmup iteration count by an order of magnitude, resulting in 80,000 post-warmup iterations rather than the default 4,000.

Now they’re quite a bit closer.

And before you get fixate on how there are still differences after 80,000 iterations, each, consider comparing the two density plots.

8.4.3 Conducting a $2 \times 2$ between-participants factorial ANOVA using PROCESS another regression model with brms.

Since we’re square in single-level regression land with our brms approach, there’s no direct analogue for us, here. However, notice the post-ANOVA $t$ -tests Hayes presented on page 300. If we just want to consider the $2 \times 2$ structure of our two dummy variables as indicative of four groups, we have one more coding system available for the job. With the handy str_c() function, we’ll concatenate the policy and kerry values into a nominal variable, policy_kerry . Here’s what that looks like:

Now check out what happens if we reformat our formula to interest ~ 0 + policy_kerry .

The brm() function recnognized policy_kerry was a character vector and treated it as a nominal variable. The 0 + part of the function removed the model intercept. Here’s how that effects the output.

Without the typical intercept, brm() estimated the means for each of the four policy_kerry groups. It’s kinda like an intercept-only model, but with four intercepts. Here’s what their densities look like:

Since each of the four primary vectors in our post object is of a group mean, it’s trivial to compute difference scores. To compute the difference score analogous to Hayes’s two $t$ -tests, we’d do the following.

Hayes, A. F. (2018). Introduction to mediation, moderation, and conditional process analysis: A regression-based approach. (2nd ed.). New York, NY, US: The Guilford Press.

Session info

An official website of the United States government

The .gov means it’s official. Federal government websites often end in .gov or .mil. Before sharing sensitive information, make sure you’re on a federal government site.

The site is secure. The https:// ensures that you are connecting to the official website and that any information you provide is encrypted and transmitted securely.

Publications
Account settings

Preview improvements coming to the PMC website in October 2024. Learn More or Try it out now .

Advanced Search
Journal List
J Korean Med Sci
v.36(50); 2021 Dec 27

Formulating Hypotheses for Different Study Designs

Durga prasanna misra.

1 Department of Clinical Immunology and Rheumatology, Sanjay Gandhi Postgraduate Institute of Medical Sciences, Lucknow, India.

Armen Yuri Gasparyan

2 Departments of Rheumatology and Research and Development, Dudley Group NHS Foundation Trust (Teaching Trust of the University of Birmingham, UK), Russells Hall Hospital, Dudley, UK.

Olena Zimba

3 Department of Internal Medicine #2, Danylo Halytsky Lviv National Medical University, Lviv, Ukraine.

Marlen Yessirkepov

4 Department of Biology and Biochemistry, South Kazakhstan Medical Academy, Shymkent, Kazakhstan.

Vikas Agarwal

George d. kitas.

5 Centre for Epidemiology versus Arthritis, University of Manchester, Manchester, UK.

Generating a testable working hypothesis is the first step towards conducting original research. Such research may prove or disprove the proposed hypothesis. Case reports, case series, online surveys and other observational studies, clinical trials, and narrative reviews help to generate hypotheses. Observational and interventional studies help to test hypotheses. A good hypothesis is usually based on previous evidence-based reports. Hypotheses without evidence-based justification and a priori ideas are not received favourably by the scientific community. Original research to test a hypothesis should be carefully planned to ensure appropriate methodology and adequate statistical power. While hypotheses can challenge conventional thinking and may be controversial, they should not be destructive. A hypothesis should be tested by ethically sound experiments with meaningful ethical and clinical implications. The coronavirus disease 2019 pandemic has brought into sharp focus numerous hypotheses, some of which were proven (e.g. effectiveness of corticosteroids in those with hypoxia) while others were disproven (e.g. ineffectiveness of hydroxychloroquine and ivermectin).

Graphical Abstract

An external file that holds a picture, illustration, etc.
Object name is jkms-36-e338-abf001.jpg

DEFINING WORKING AND STANDALONE SCIENTIFIC HYPOTHESES

Science is the systematized description of natural truths and facts. Routine observations of existing life phenomena lead to the creative thinking and generation of ideas about mechanisms of such phenomena and related human interventions. Such ideas presented in a structured format can be viewed as hypotheses. After generating a hypothesis, it is necessary to test it to prove its validity. Thus, hypothesis can be defined as a proposed mechanism of a naturally occurring event or a proposed outcome of an intervention. 1 , 2

Hypothesis testing requires choosing the most appropriate methodology and adequately powering statistically the study to be able to “prove” or “disprove” it within predetermined and widely accepted levels of certainty. This entails sample size calculation that often takes into account previously published observations and pilot studies. 2 , 3 In the era of digitization, hypothesis generation and testing may benefit from the availability of numerous platforms for data dissemination, social networking, and expert validation. Related expert evaluations may reveal strengths and limitations of proposed ideas at early stages of post-publication promotion, preventing the implementation of unsupported controversial points. 4

Thus, hypothesis generation is an important initial step in the research workflow, reflecting accumulating evidence and experts' stance. In this article, we overview the genesis and importance of scientific hypotheses and their relevance in the era of the coronavirus disease 2019 (COVID-19) pandemic.

DO WE NEED HYPOTHESES FOR ALL STUDY DESIGNS?

Broadly, research can be categorized as primary or secondary. In the context of medicine, primary research may include real-life observations of disease presentations and outcomes. Single case descriptions, which often lead to new ideas and hypotheses, serve as important starting points or justifications for case series and cohort studies. The importance of case descriptions is particularly evident in the context of the COVID-19 pandemic when unique, educational case reports have heralded a new era in clinical medicine. 5

Case series serve similar purpose to single case reports, but are based on a slightly larger quantum of information. Observational studies, including online surveys, describe the existing phenomena at a larger scale, often involving various control groups. Observational studies include variable-scale epidemiological investigations at different time points. Interventional studies detail the results of therapeutic interventions.

Secondary research is based on already published literature and does not directly involve human or animal subjects. Review articles are generated by secondary research. These could be systematic reviews which follow methods akin to primary research but with the unit of study being published papers rather than humans or animals. Systematic reviews have a rigid structure with a mandatory search strategy encompassing multiple databases, systematic screening of search results against pre-defined inclusion and exclusion criteria, critical appraisal of study quality and an optional component of collating results across studies quantitatively to derive summary estimates (meta-analysis). 6 Narrative reviews, on the other hand, have a more flexible structure. Systematic literature searches to minimise bias in selection of articles are highly recommended but not mandatory. 7 Narrative reviews are influenced by the authors' viewpoint who may preferentially analyse selected sets of articles. 8

In relation to primary research, case studies and case series are generally not driven by a working hypothesis. Rather, they serve as a basis to generate a hypothesis. Observational or interventional studies should have a hypothesis for choosing research design and sample size. The results of observational and interventional studies further lead to the generation of new hypotheses, testing of which forms the basis of future studies. Review articles, on the other hand, may not be hypothesis-driven, but form fertile ground to generate future hypotheses for evaluation. Fig. 1 summarizes which type of studies are hypothesis-driven and which lead on to hypothesis generation.

An external file that holds a picture, illustration, etc.
Object name is jkms-36-e338-g001.jpg

STANDARDS OF WORKING AND SCIENTIFIC HYPOTHESES

A review of the published literature did not enable the identification of clearly defined standards for working and scientific hypotheses. It is essential to distinguish influential versus not influential hypotheses, evidence-based hypotheses versus a priori statements and ideas, ethical versus unethical, or potentially harmful ideas. The following points are proposed for consideration while generating working and scientific hypotheses. 1 , 2 Table 1 summarizes these points.

Evidence-based data

A scientific hypothesis should have a sound basis on previously published literature as well as the scientist's observations. Randomly generated (a priori) hypotheses are unlikely to be proven. A thorough literature search should form the basis of a hypothesis based on published evidence. 7

Unless a scientific hypothesis can be tested, it can neither be proven nor be disproven. Therefore, a scientific hypothesis should be amenable to testing with the available technologies and the present understanding of science.

Supported by pilot studies

If a hypothesis is based purely on a novel observation by the scientist in question, it should be grounded on some preliminary studies to support it. For example, if a drug that targets a specific cell population is hypothesized to be useful in a particular disease setting, then there must be some preliminary evidence that the specific cell population plays a role in driving that disease process.

Testable by ethical studies

The hypothesis should be testable by experiments that are ethically acceptable. 9 For example, a hypothesis that parachutes reduce mortality from falls from an airplane cannot be tested using a randomized controlled trial. 10 This is because it is obvious that all those jumping from a flying plane without a parachute would likely die. Similarly, the hypothesis that smoking tobacco causes lung cancer cannot be tested by a clinical trial that makes people take up smoking (since there is considerable evidence for the health hazards associated with smoking). Instead, long-term observational studies comparing outcomes in those who smoke and those who do not, as was performed in the landmark epidemiological case control study by Doll and Hill, 11 are more ethical and practical.

Balance between scientific temper and controversy

Novel findings, including novel hypotheses, particularly those that challenge established norms, are bound to face resistance for their wider acceptance. Such resistance is inevitable until the time such findings are proven with appropriate scientific rigor. However, hypotheses that generate controversy are generally unwelcome. For example, at the time the pandemic of human immunodeficiency virus (HIV) and AIDS was taking foot, there were numerous deniers that refused to believe that HIV caused AIDS. 12 , 13 Similarly, at a time when climate change is causing catastrophic changes to weather patterns worldwide, denial that climate change is occurring and consequent attempts to block climate change are certainly unwelcome. 14 The denialism and misinformation during the COVID-19 pandemic, including unfortunate examples of vaccine hesitancy, are more recent examples of controversial hypotheses not backed by science. 15 , 16 An example of a controversial hypothesis that was a revolutionary scientific breakthrough was the hypothesis put forth by Warren and Marshall that Helicobacter pylori causes peptic ulcers. Initially, the hypothesis that a microorganism could cause gastritis and gastric ulcers faced immense resistance. When the scientists that proposed the hypothesis themselves ingested H. pylori to induce gastritis in themselves, only then could they convince the wider world about their hypothesis. Such was the impact of the hypothesis was that Barry Marshall and Robin Warren were awarded the Nobel Prize in Physiology or Medicine in 2005 for this discovery. 17 , 18

DISTINGUISHING THE MOST INFLUENTIAL HYPOTHESES

Influential hypotheses are those that have stood the test of time. An archetype of an influential hypothesis is that proposed by Edward Jenner in the eighteenth century that cowpox infection protects against smallpox. While this observation had been reported for nearly a century before this time, it had not been suitably tested and publicised until Jenner conducted his experiments on a young boy by demonstrating protection against smallpox after inoculation with cowpox. 19 These experiments were the basis for widespread smallpox immunization strategies worldwide in the 20th century which resulted in the elimination of smallpox as a human disease today. 20

Other influential hypotheses are those which have been read and cited widely. An example of this is the hygiene hypothesis proposing an inverse relationship between infections in early life and allergies or autoimmunity in adulthood. An analysis reported that this hypothesis had been cited more than 3,000 times on Scopus. 1

LESSONS LEARNED FROM HYPOTHESES AMIDST THE COVID-19 PANDEMIC

The COVID-19 pandemic devastated the world like no other in recent memory. During this period, various hypotheses emerged, understandably so considering the public health emergency situation with innumerable deaths and suffering for humanity. Within weeks of the first reports of COVID-19, aberrant immune system activation was identified as a key driver of organ dysfunction and mortality in this disease. 21 Consequently, numerous drugs that suppress the immune system or abrogate the activation of the immune system were hypothesized to have a role in COVID-19. 22 One of the earliest drugs hypothesized to have a benefit was hydroxychloroquine. Hydroxychloroquine was proposed to interfere with Toll-like receptor activation and consequently ameliorate the aberrant immune system activation leading to pathology in COVID-19. 22 The drug was also hypothesized to have a prophylactic role in preventing infection or disease severity in COVID-19. It was also touted as a wonder drug for the disease by many prominent international figures. However, later studies which were well-designed randomized controlled trials failed to demonstrate any benefit of hydroxychloroquine in COVID-19. 23 , 24 , 25 , 26 Subsequently, azithromycin 27 , 28 and ivermectin 29 were hypothesized as potential therapies for COVID-19, but were not supported by evidence from randomized controlled trials. The role of vitamin D in preventing disease severity was also proposed, but has not been proven definitively until now. 30 , 31 On the other hand, randomized controlled trials identified the evidence supporting dexamethasone 32 and interleukin-6 pathway blockade with tocilizumab as effective therapies for COVID-19 in specific situations such as at the onset of hypoxia. 33 , 34 Clues towards the apparent effectiveness of various drugs against severe acute respiratory syndrome coronavirus 2 in vitro but their ineffectiveness in vivo have recently been identified. Many of these drugs are weak, lipophilic bases and some others induce phospholipidosis which results in apparent in vitro effectiveness due to non-specific off-target effects that are not replicated inside living systems. 35 , 36

Another hypothesis proposed was the association of the routine policy of vaccination with Bacillus Calmette-Guerin (BCG) with lower deaths due to COVID-19. This hypothesis emerged in the middle of 2020 when COVID-19 was still taking foot in many parts of the world. 37 , 38 Subsequently, many countries which had lower deaths at that time point went on to have higher numbers of mortality, comparable to other areas of the world. Furthermore, the hypothesis that BCG vaccination reduced COVID-19 mortality was a classic example of ecological fallacy. Associations between population level events (ecological studies; in this case, BCG vaccination and COVID-19 mortality) cannot be directly extrapolated to the individual level. Furthermore, such associations cannot per se be attributed as causal in nature, and can only serve to generate hypotheses that need to be tested at the individual level. 39

IS TRADITIONAL PEER REVIEW EFFICIENT FOR EVALUATION OF WORKING AND SCIENTIFIC HYPOTHESES?

Traditionally, publication after peer review has been considered the gold standard before any new idea finds acceptability amongst the scientific community. Getting a work (including a working or scientific hypothesis) reviewed by experts in the field before experiments are conducted to prove or disprove it helps to refine the idea further as well as improve the experiments planned to test the hypothesis. 40 A route towards this has been the emergence of journals dedicated to publishing hypotheses such as the Central Asian Journal of Medical Hypotheses and Ethics. 41 Another means of publishing hypotheses is through registered research protocols detailing the background, hypothesis, and methodology of a particular study. If such protocols are published after peer review, then the journal commits to publishing the completed study irrespective of whether the study hypothesis is proven or disproven. 42 In the post-pandemic world, online research methods such as online surveys powered via social media channels such as Twitter and Instagram might serve as critical tools to generate as well as to preliminarily test the appropriateness of hypotheses for further evaluation. 43 , 44

Some radical hypotheses might be difficult to publish after traditional peer review. These hypotheses might only be acceptable by the scientific community after they are tested in research studies. Preprints might be a way to disseminate such controversial and ground-breaking hypotheses. 45 However, scientists might prefer to keep their hypotheses confidential for the fear of plagiarism of ideas, avoiding online posting and publishing until they have tested the hypotheses.

SUGGESTIONS ON GENERATING AND PUBLISHING HYPOTHESES

Publication of hypotheses is important, however, a balance is required between scientific temper and controversy. Journal editors and reviewers might keep in mind these specific points, summarized in Table 2 and detailed hereafter, while judging the merit of hypotheses for publication. Keeping in mind the ethical principle of primum non nocere, a hypothesis should be published only if it is testable in a manner that is ethically appropriate. 46 Such hypotheses should be grounded in reality and lend themselves to further testing to either prove or disprove them. It must be considered that subsequent experiments to prove or disprove a hypothesis have an equal chance of failing or succeeding, akin to tossing a coin. A pre-conceived belief that a hypothesis is unlikely to be proven correct should not form the basis of rejection of such a hypothesis for publication. In this context, hypotheses generated after a thorough literature search to identify knowledge gaps or based on concrete clinical observations on a considerable number of patients (as opposed to random observations on a few patients) are more likely to be acceptable for publication by peer-reviewed journals. Also, hypotheses should be considered for publication or rejection based on their implications for science at large rather than whether the subsequent experiments to test them end up with results in favour of or against the original hypothesis.

Hypotheses form an important part of the scientific literature. The COVID-19 pandemic has reiterated the importance and relevance of hypotheses for dealing with public health emergencies and highlighted the need for evidence-based and ethical hypotheses. A good hypothesis is testable in a relevant study design, backed by preliminary evidence, and has positive ethical and clinical implications. General medical journals might consider publishing hypotheses as a specific article type to enable more rapid advancement of science.

Disclosure: The authors have no potential conflicts of interest to disclose.

Author Contributions:

Data curation: Gasparyan AY, Misra DP, Zimba O, Yessirkepov M, Agarwal V, Kitas GD.

IMAGES

PPT
How to Write a Hypothesis in 12 Steps 2023
PPT
PPT
How to Write a Hypothesis
Formulating hypotheses

VIDEO

Formulating Hypothesis || CAPSTONE
W2D2 Tutorial4 Formulating a Hypothesis
Formulating Research Question and Hypothesis
DAILY HYPO
ECON 3406: Formulating a Hypothesis for a One-Sample Mean, Examining Critical Values
Who is known for formulating the theory of relativity? #youtubeshorts

COMMENTS

How will you write the moderating hypothesis and direction of the
You need to explore your model further and decide a moderated mediation or mediated moderation. Once you are clear, then you need to write a complex hypotheses for moderating effect and remember ...
The Three Most Common Types of Hypotheses
With moderation, it is important to note that the moderating variable can be a category (e.g., sex) or it can be a continuous variable (e.g., scores on a personality questionnaire). ... It's been really helpful but I still don't know how to formulate the hypothesis with my mediating variable. I have one dependent variable DV which is formed ...
Section 7.3: Moderation Models, Assumptions, Interpretation, and Write
Moderation Assumptions. The dependent and independent variables should be measured on a continuous scale. There should be a moderator variable that is a nominal variable with at least two groups. The variables of interest (the dependent variable and the independent and moderator variables) should have a linear relationship, which you can check ...
Section 7.3: Moderation Models, Assumptions, Interpretation, and Write
From these results, it can be concluded that the effect of perceived stress on mental distress is partially moderated by cyberbullying. Section 7.3: Moderation Models, Assumptions, Interpretation, and Write Up is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by LibreTexts.
How to Write a Strong Hypothesis
Step 3. Formulate your hypothesis. Now you should have some idea of what you expect to find. Write your initial answer to the question in a clear, concise sentence. Example: Formulating your hypothesis Attending more lectures leads to better exam results. Tip AI tools like ChatGPT can be effectively used to brainstorm potential hypotheses.
How to Write a Strong Hypothesis
Step 5: Phrase your hypothesis in three ways. To identify the variables, you can write a simple prediction in if … then form. The first part of the sentence states the independent variable and the second part states the dependent variable. If a first-year student starts attending more lectures, then their exam scores will improve.
Addressing Moderated Mediation Hypotheses: Theory, Methods, and
and moderation may be modeled. Various sources refer to some of these effects as mediated moderation or moderated mediation (e.g., Baron & Kenny, 1986), but there is a fair amount of confusion over precisely what pattern of causal relationships constitutes each kind of effect and how to assess the presence, strength, and signiﬁcance of these ...
5.2
5.2 - Writing Hypotheses. The first step in conducting a hypothesis test is to write the hypothesis statements that are going to be tested. For each test you will have a null hypothesis ( H 0) and an alternative hypothesis ( H a ). When writing hypotheses there are three things that we need to know: (1) the parameter that we are testing (2) the ...
Chapter 17 Moderation
17. Moderation. Spotlight Analysis: Compare the mean of the dependent of the two groups (treatment and control) at every value ( Simple Slopes Analysis) Floodlight Analysis: is spotlight analysis on the whole range of the moderator ( Johnson-Neyman intervals) Simple slope: when a continuous independent variable interact with a moderating ...
7 Fundamentals of Moderation Analysis
7 Fundamentals of Moderation Analysis. The effect of $X$ on some variable $Y$ is moderated by $W$ if its size, sign, or strength depends on or can be predicted by $W$.In that case, $W$ is said to be a moderator of $X$ 's effect on $Y$, or that $W$ and $X$ interact in their influence on $Y$.Identifying a moderator of an effect helps to establish the boundary conditions of ...
Chapter 14: Mediation and Moderation
1 What are Mediation and Moderation?. Mediation analysis tests a hypothetical causal chain where one variable X affects a second variable M and, in turn, that variable affects a third variable Y. Mediators describe the how or why of a (typically well-established) relationship between two other variables and are sometimes called intermediary variables since they often describe the process ...
Section 7.1: Mediation and Moderation Models
Section 7.1: Mediation and Moderation Models. Learning Objectives. At the end of this section you should be able to answer the following questions: Define the concept of a moderator variable. Define the concept of a mediator variable. As we discussed in the lesson on correlations and regressions, understanding associations between psychological ...
Chapter 6 Moderation and mediation
The hypothesis test of this effect is now also significant, indicating that we have reliable evidence for this moderation. This shows that by including more predictors in a model, it is possible to increase the reliability of the estimates for other predictors. There is also a significant interaction between $\texttt{fun}$ and $\texttt{intel}.$
Understanding and Using Mediators and Moderators
Mediation and moderation are two theories for refining and understanding a causal relationship. Empirical investigation of mediators and moderators requires an integrated research design rather than the data analyses driven approach often seen in the literature. This paper described the conceptual foundation, research design, data analysis, as well as inferences involved in a mediation and/or ...
Moderation in Management Research: What, Why, When, and How
Moderation in Statistical Terms. The simplest form of moderation is where a relationship between an independent variable, X, and a dependent variable, Y, changes according to the value of a moderator variable, Z. A straightforward test of a linear relationship between X and Y would be given by the regression equation of Y on X: $$ Y \, = \, b ...
How Do You Formulate (Important) Hypotheses?
Building on the ideas in Chap. 1, we describe formulating, testing, and revising hypotheses as a continuing cycle of clarifying what you want to study, making predictions about what you might find together with developing your reasons for these predictions, imagining tests of these predictions, revising your predictions and rationales, and so ...
How do we write the moderated moderation hypothesis?
3.54 MB. Cite. 1 Recommendation. Amalia Raquel Pérez Nebra. University of Zaragoza. Dear Sanjeev Kumar, The usual answer will be as you said: H1: x will explain y, moderated by w. However, in the ...
8 Extending the Fundamental Principles of Moderation Analysis
Although this approach works, it is a widely believed myth that it is necessary to use this approach in order to test a moderation hypothesis. (p. 289, emphasis in the original) Although this method is not necessary, it can be handy to slowly build your model. This method can also serve nice rhetorical purposes in a paper.
Scientific Hypotheses: Writing, Promoting, and Predicting Implications
A snapshot analysis of citation activity of hypothesis articles may reveal interest of the global scientific community towards their implications across various disciplines and countries. As a prime example, Strachan's hygiene hypothesis, published in 1989,10 is still attracting numerous citations on Scopus, the largest bibliographic database ...
Addressing Moderated Mediation Hypotheses: Theory, Methods, and
1 There is a distinction between simultaneous and nonsimultaneous confidence bands and regions of significance (Citation Pothoff, 1964).For nonsimultaneous bands, rejection rates are accurate for any given conditional value of the moderator. Simultaneous bands, on the other hand, describe regions of the moderator for which the simple slope will be significant for all values of the moderator ...
Formulating Hypotheses for Different Study Designs
Formulating Hypotheses for Different Study Designs. Generating a testable working hypothesis is the first step towards conducting original research. Such research may prove or disprove the proposed hypothesis. Case reports, case series, online surveys and other observational studies, clinical trials, and narrative reviews help to generate ...
PDF 4 Hypotheses Complex Relationships and
Simple Hypotheses. Figure 4.1 presents a diagram illustrating a simple hypothesis. In this figure, the hypothesis predicts a relationship between one IV and one DV (outcome). In Figure 4.1, the IV is the presumed cause, influ-ence, or antecedent of DV, which is the predicted outcome, effect, or consequence.
How to Write Direct, Mediating & Moderating Hypothesis in the ...
Website link - https://researchphd.in/Workshop on Data Analysis using Excel - https://www.kudoselearning.com/s/store/courses/description/A-Live-Workshop-on-D...

The Three Most Common Types of Hypotheses

37 replies on “The Three Most Common Types of Hypotheses”

Leave a Reply Cancel reply

Section 7.3: Moderation Models, Assumptions, Interpretation, and Write Up

Moderation Models

Moderation Assumptions

Moderation Interpretation

Interpretation

Moderation Write Up

Have a language expert improve your writing

How to Write a Strong Hypothesis | Steps & Examples

Example: Hypothesis

Table of contents

Variables in hypotheses

Here's why students love Scribbr's proofreading services

Step 1. Ask a question

Step 2. Do some preliminary research

Step 3. Formulate your hypothesis

4. Refine your hypothesis

5. Phrase your hypothesis in three ways

6. Write a null hypothesis

Receive feedback on language, structure, and formatting

Cite this Scribbr article

Is this article helpful?

Shona McCombes

Have a language expert improve your writing

How to Write a Strong Hypothesis | Guide & Examples

Table of contents

Variables in hypotheses

Prevent plagiarism, run a free check.

Step 2: Do some preliminary research

Step 3: Formulate your hypothesis

Step 4: Refine your hypothesis

Step 5: Phrase your hypothesis in three ways

Step 6. Write a null hypothesis

Cite this Scribbr article

Is this article helpful?

Shona McCombes

User Preferences

Keyboard Shortcuts

Chapter 14: Mediation and Moderation

1.1 Getting Started

2 Mediation Analyses

2.1 Example Mediation Data

2.2 Method 1: Baron & Kenny

2.3 Interpreting Barron & Kenny Results

2.4 Method 2: The Mediation Pacakge Method

2.5 Interpreting Mediation Results

3 Moderation Analyses

3.1 Example Moderation Data

3.2 Moderation Analysis

3.3 Interpreting Moderation Results

4 References and Further Reading

Section 7.1: Mediation and Moderation Models

Examples of Research Questions

Share This Book

Statistics: Data analysis and modelling

6.1 Moderation

6.1.2 Conditional slopes

6.1.3 Modeling slopes with linear models

6.1.4 Simple slopes and centering

6.1.5 Don’t forget about fun! A model with multiple interactions

6.2 Mediation

6.2.2 Causal steps

6.2.2.1 Testing mediation of legacy motive by intention with the causal steps approach

6.2.3 Estimating the mediated effect

How Do You Formulate (Important) Hypotheses?

Cite this chapter

Part I. Getting Started

Exercise 2.1

Part II. Paths from a General Interest to an Informed Hypothesis

Beginning with a Prediction If You Have One

Developing a Rationale for Your Predictions

Imagining Testing Your Hypotheses

Cycles of Building Rationales and Planning to Test Your Predictions

Beginning by Asking Questions to Clarify Your Interests

Imagining Answers to Your Questions

Shifting to the Hypothesis Formulation and Testing Path

Cycles of Refining Questions and Predicting Answers

Beginning with a Research Problem