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ANOVA (Analysis of variance) – Formulas, Types, and Examples

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Analysis of Variance (ANOVA)

Analysis of Variance (ANOVA) is a statistical method used to test differences between two or more means. It is similar to the t-test, but the t-test is generally used for comparing two means, while ANOVA is used when you have more than two means to compare.

ANOVA is based on comparing the variance (or variation) between the data samples to the variation within each particular sample. If the between-group variance is high and the within-group variance is low, this provides evidence that the means of the groups are significantly different.

ANOVA Terminology

When discussing ANOVA, there are several key terms to understand:

• Factor : This is another term for the independent variable in your analysis. In a one-way ANOVA, there is one factor, while in a two-way ANOVA, there are two factors.
• Levels : These are the different groups or categories within a factor. For example, if the factor is ‘diet’ the levels might be ‘low fat’, ‘medium fat’, and ‘high fat’.
• Response Variable : This is the dependent variable or the outcome that you are measuring.
• Within-group Variance : This is the variance or spread of scores within each level of your factor.
• Between-group Variance : This is the variance or spread of scores between the different levels of your factor.
• Grand Mean : This is the overall mean when you consider all the data together, regardless of the factor level.
• Treatment Sums of Squares (SS) : This represents the between-group variability. It is the sum of the squared differences between the group means and the grand mean.
• Error Sums of Squares (SS) : This represents the within-group variability. It’s the sum of the squared differences between each observation and its group mean.
• Total Sums of Squares (SS) : This is the sum of the Treatment SS and the Error SS. It represents the total variability in the data.
• Degrees of Freedom (df) : The degrees of freedom are the number of values that have the freedom to vary when computing a statistic. For example, if you have ‘n’ observations in one group, then the degrees of freedom for that group is ‘n-1’.
• Mean Square (MS) : Mean Square is the average squared deviation and is calculated by dividing the sum of squares by the corresponding degrees of freedom.
• F-Ratio : This is the test statistic for ANOVAs, and it’s the ratio of the between-group variance to the within-group variance. If the between-group variance is significantly larger than the within-group variance, the F-ratio will be large and likely significant.
• Null Hypothesis (H0) : This is the hypothesis that there is no difference between the group means.
• Alternative Hypothesis (H1) : This is the hypothesis that there is a difference between at least two of the group means.
• p-value : This is the probability of obtaining a test statistic as extreme as the one that was actually observed, assuming that the null hypothesis is true. If the p-value is less than the significance level (usually 0.05), then the null hypothesis is rejected in favor of the alternative hypothesis.
• Post-hoc tests : These are follow-up tests conducted after an ANOVA when the null hypothesis is rejected, to determine which specific groups’ means (levels) are different from each other. Examples include Tukey’s HSD, Scheffe, Bonferroni, among others.

Types of ANOVA

Types of ANOVA are as follows:

One-way (or one-factor) ANOVA

This is the simplest type of ANOVA, which involves one independent variable . For example, comparing the effect of different types of diet (vegetarian, pescatarian, omnivore) on cholesterol level.

Two-way (or two-factor) ANOVA

This involves two independent variables. This allows for testing the effect of each independent variable on the dependent variable , as well as testing if there’s an interaction effect between the independent variables on the dependent variable.

Repeated Measures ANOVA

This is used when the same subjects are measured multiple times under different conditions, or at different points in time. This type of ANOVA is often used in longitudinal studies.

Mixed Design ANOVA

This combines features of both between-subjects (independent groups) and within-subjects (repeated measures) designs. In this model, one factor is a between-subjects variable and the other is a within-subjects variable.

Multivariate Analysis of Variance (MANOVA)

This is used when there are two or more dependent variables. It tests whether changes in the independent variable(s) correspond to changes in the dependent variables.

Analysis of Covariance (ANCOVA)

This combines ANOVA and regression. ANCOVA tests whether certain factors have an effect on the outcome variable after removing the variance for which quantitative covariates (interval variables) account. This allows the comparison of one variable outcome between groups, while statistically controlling for the effect of other continuous variables that are not of primary interest.

Nested ANOVA

This model is used when the groups can be clustered into categories. For example, if you were comparing students’ performance from different classrooms and different schools, “classroom” could be nested within “school.”

ANOVA Formulas

ANOVA Formulas are as follows:

Sum of Squares Total (SST)

This represents the total variability in the data. It is the sum of the squared differences between each observation and the overall mean.

• yi represents each individual data point
• y_mean represents the grand mean (mean of all observations)

Sum of Squares Within (SSW)

This represents the variability within each group or factor level. It is the sum of the squared differences between each observation and its group mean.

• yij represents each individual data point within a group
• y_meani represents the mean of the ith group

Sum of Squares Between (SSB)

This represents the variability between the groups. It is the sum of the squared differences between the group means and the grand mean, multiplied by the number of observations in each group.

• ni represents the number of observations in each group
• y_mean represents the grand mean

Degrees of Freedom

The degrees of freedom are the number of values that have the freedom to vary when calculating a statistic.

For within groups (dfW):

For between groups (dfB):

For total (dfT):

• N represents the total number of observations
• k represents the number of groups

Mean Squares

Mean squares are the sum of squares divided by the respective degrees of freedom.

Mean Squares Between (MSB):

Mean Squares Within (MSW):

F-Statistic

The F-statistic is used to test whether the variability between the groups is significantly greater than the variability within the groups.

If the F-statistic is significantly higher than what would be expected by chance, we reject the null hypothesis that all group means are equal.

Examples of ANOVA

Examples 1:

Suppose a psychologist wants to test the effect of three different types of exercise (yoga, aerobic exercise, and weight training) on stress reduction. The dependent variable is the stress level, which can be measured using a stress rating scale.

Here are hypothetical stress ratings for a group of participants after they followed each of the exercise regimes for a period:

• Yoga: [3, 2, 2, 1, 2, 2, 3, 2, 1, 2]
• Aerobic Exercise: [2, 3, 3, 2, 3, 2, 3, 3, 2, 2]
• Weight Training: [4, 4, 5, 5, 4, 5, 4, 5, 4, 5]

The psychologist wants to determine if there is a statistically significant difference in stress levels between these different types of exercise.

To conduct the ANOVA:

1. State the hypotheses:

• Null Hypothesis (H0): There is no difference in mean stress levels between the three types of exercise.
• Alternative Hypothesis (H1): There is a difference in mean stress levels between at least two of the types of exercise.

2. Calculate the ANOVA statistics:

• Compute the Sum of Squares Between (SSB), Sum of Squares Within (SSW), and Sum of Squares Total (SST).
• Calculate the Degrees of Freedom (dfB, dfW, dfT).
• Calculate the Mean Squares Between (MSB) and Mean Squares Within (MSW).
• Compute the F-statistic (F = MSB / MSW).

3. Check the p-value associated with the calculated F-statistic.

• If the p-value is less than the chosen significance level (often 0.05), then we reject the null hypothesis in favor of the alternative hypothesis. This suggests there is a statistically significant difference in mean stress levels between the three exercise types.

4. Post-hoc tests

• If we reject the null hypothesis, we conduct a post-hoc test to determine which specific groups’ means (exercise types) are different from each other.

Examples 2:

Suppose an agricultural scientist wants to compare the yield of three varieties of wheat. The scientist randomly selects four fields for each variety and plants them. After harvest, the yield from each field is measured in bushels. Here are the hypothetical yields:

The scientist wants to know if the differences in yields are due to the different varieties or just random variation.

Here’s how to apply the one-way ANOVA to this situation:

• Null Hypothesis (H0): The means of the three populations are equal.
• Alternative Hypothesis (H1): At least one population mean is different.
• Calculate the Degrees of Freedom (dfB for between groups, dfW for within groups, dfT for total).
• If the p-value is less than the chosen significance level (often 0.05), then we reject the null hypothesis in favor of the alternative hypothesis. This would suggest there is a statistically significant difference in mean yields among the three varieties.
• If we reject the null hypothesis, we conduct a post-hoc test to determine which specific groups’ means (wheat varieties) are different from each other.

How to Conduct ANOVA

Conducting an Analysis of Variance (ANOVA) involves several steps. Here’s a general guideline on how to perform it:

• Null Hypothesis (H0): The means of all groups are equal.
• Alternative Hypothesis (H1): At least one group mean is different from the others.
• The significance level (often denoted as α) is usually set at 0.05. This implies that you are willing to accept a 5% chance that you are wrong in rejecting the null hypothesis.
• Data should be collected for each group under study. Make sure that the data meet the assumptions of an ANOVA: normality, independence, and homogeneity of variances.
• Calculate the Degrees of Freedom (df) for each sum of squares (dfB, dfW, dfT).
• Compute the Mean Squares Between (MSB) and Mean Squares Within (MSW) by dividing the sum of squares by the corresponding degrees of freedom.
• Compute the F-statistic as the ratio of MSB to MSW.
• Determine the critical F-value from the F-distribution table using dfB and dfW.
• If the calculated F-statistic is greater than the critical F-value, reject the null hypothesis.
• If the p-value associated with the calculated F-statistic is smaller than the significance level (0.05 typically), you reject the null hypothesis.
• If you rejected the null hypothesis, you can conduct post-hoc tests (like Tukey’s HSD) to determine which specific groups’ means (if you have more than two groups) are different from each other.
• Regardless of the result, report your findings in a clear, understandable manner. This typically includes reporting the test statistic, p-value, and whether the null hypothesis was rejected.

When to use ANOVA

ANOVA (Analysis of Variance) is used when you have three or more groups and you want to compare their means to see if they are significantly different from each other. It is a statistical method that is used in a variety of research scenarios. Here are some examples of when you might use ANOVA:

• Comparing Groups : If you want to compare the performance of more than two groups, for example, testing the effectiveness of different teaching methods on student performance.
• Evaluating Interactions : In a two-way or factorial ANOVA, you can test for an interaction effect. This means you are not only interested in the effect of each individual factor, but also whether the effect of one factor depends on the level of another factor.
• Repeated Measures : If you have measured the same subjects under different conditions or at different time points, you can use repeated measures ANOVA to compare the means of these repeated measures while accounting for the correlation between measures from the same subject.
• Experimental Designs : ANOVA is often used in experimental research designs when subjects are randomly assigned to different conditions and the goal is to compare the means of the conditions.

Here are the assumptions that must be met to use ANOVA:

• Normality : The data should be approximately normally distributed.
• Homogeneity of Variances : The variances of the groups you are comparing should be roughly equal. This assumption can be tested using Levene’s test or Bartlett’s test.
• Independence : The observations should be independent of each other. This assumption is met if the data is collected appropriately with no related groups (e.g., twins, matched pairs, repeated measures).

Applications of ANOVA

The Analysis of Variance (ANOVA) is a powerful statistical technique that is used widely across various fields and industries. Here are some of its key applications:

Agriculture

ANOVA is commonly used in agricultural research to compare the effectiveness of different types of fertilizers, crop varieties, or farming methods. For example, an agricultural researcher could use ANOVA to determine if there are significant differences in the yields of several varieties of wheat under the same conditions.

Manufacturing and Quality Control

ANOVA is used to determine if different manufacturing processes or machines produce different levels of product quality. For instance, an engineer might use it to test whether there are differences in the strength of a product based on the machine that produced it.

Marketing Research

Marketers often use ANOVA to test the effectiveness of different advertising strategies. For example, a marketer could use ANOVA to determine whether different marketing messages have a significant impact on consumer purchase intentions.

Healthcare and Medicine

In medical research, ANOVA can be used to compare the effectiveness of different treatments or drugs. For example, a medical researcher could use ANOVA to test whether there are significant differences in recovery times for patients who receive different types of therapy.

ANOVA is used in educational research to compare the effectiveness of different teaching methods or educational interventions. For example, an educator could use it to test whether students perform significantly differently when taught with different teaching methods.

Psychology and Social Sciences

Psychologists and social scientists use ANOVA to compare group means on various psychological and social variables. For example, a psychologist could use it to determine if there are significant differences in stress levels among individuals in different occupations.

Biology and Environmental Sciences

Biologists and environmental scientists use ANOVA to compare different biological and environmental conditions. For example, an environmental scientist could use it to determine if there are significant differences in the levels of a pollutant in different bodies of water.

Advantages of ANOVA

Here are some advantages of using ANOVA:

Comparing Multiple Groups: One of the key advantages of ANOVA is the ability to compare the means of three or more groups. This makes it more powerful and flexible than the t-test, which is limited to comparing only two groups.

Control of Type I Error: When comparing multiple groups, the chances of making a Type I error (false positive) increases. One of the strengths of ANOVA is that it controls the Type I error rate across all comparisons. This is in contrast to performing multiple pairwise t-tests which can inflate the Type I error rate.

Testing Interactions: In factorial ANOVA, you can test not only the main effect of each factor, but also the interaction effect between factors. This can provide valuable insights into how different factors or variables interact with each other.

Handling Continuous and Categorical Variables: ANOVA can handle both continuous and categorical variables . The dependent variable is continuous and the independent variables are categorical.

Robustness: ANOVA is considered robust to violations of normality assumption when group sizes are equal. This means that even if your data do not perfectly meet the normality assumption, you might still get valid results.

Provides Detailed Analysis: ANOVA provides a detailed breakdown of variances and interactions between variables which can be useful in understanding the underlying factors affecting the outcome.

Capability to Handle Complex Experimental Designs: Advanced types of ANOVA (like repeated measures ANOVA, MANOVA, etc.) can handle more complex experimental designs, including those where measurements are taken on the same subjects over time, or when you want to analyze multiple dependent variables at once.

Disadvantages of ANOVA

Some limitations or disadvantages that are important to consider:

Assumptions: ANOVA relies on several assumptions including normality (the data follows a normal distribution), independence (the observations are independent of each other), and homogeneity of variances (the variances of the groups are roughly equal). If these assumptions are violated, the results of the ANOVA may not be valid.

Sensitivity to Outliers: ANOVA can be sensitive to outliers. A single extreme value in one group can affect the sum of squares and consequently influence the F-statistic and the overall result of the test.

Dichotomous Variables: ANOVA is not suitable for dichotomous variables (variables that can take only two values, like yes/no or male/female). It is used to compare the means of groups for a continuous dependent variable.

Lack of Specificity: Although ANOVA can tell you that there is a significant difference between groups, it doesn’t tell you which specific groups are significantly different from each other. You need to carry out further post-hoc tests (like Tukey’s HSD or Bonferroni) for these pairwise comparisons.

Complexity with Multiple Factors: When dealing with multiple factors and interactions in factorial ANOVA, interpretation can become complex. The presence of interaction effects can make main effects difficult to interpret.

Requires Larger Sample Sizes: To detect an effect of a certain size, ANOVA generally requires larger sample sizes than a t-test.

Equal Group Sizes: While not always a strict requirement, ANOVA is most powerful and its assumptions are most likely to be met when groups are of equal or similar sizes.

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Muhammad Hassan

Researcher, Academic Writer, Web developer

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The Ultimate Guide to ANOVA

Get all of your ANOVA questions answered here

ANOVA is the go-to analysis tool for classical experimental design, which forms the backbone of scientific research.

In this article, we’ll guide you through what ANOVA is, how to determine which version to use to evaluate your particular experiment, and provide detailed examples for the most common forms of ANOVA.

This includes a (brief) discussion of crossed, nested, fixed and random factors, and covers the majority of ANOVA models that a scientist would encounter before requiring the assistance of a statistician or modeling expert.

What is ANOVA used for?

ANOVA, or (Fisher’s) analysis of variance, is a critical analytical technique for evaluating differences between three or more sample means from an experiment. As the name implies, it partitions out the variance in the response variable based on one or more explanatory factors.

As you will see there are many types of ANOVA such as one-, two-, and three-way ANOVA as well as nested and repeated measures ANOVA. The graphic below shows a simple example of an experiment that requires ANOVA in which researchers measured the levels of neutrophil extracellular traps (NETs) in plasma across patients with different viral respiratory infections.

Many researchers may not realize that, for the majority of experiments, the characteristics of the experiment that you run dictate the ANOVA that you need to use to test the results. While it’s a massive topic (with professional training needed for some of the advanced techniques), this is a practical guide covering what most researchers need to know about ANOVA.

When should I use ANOVA?

If your response variable is numeric, and you’re looking for how that number differs across several categorical groups, then ANOVA is an ideal place to start. After running an experiment, ANOVA is used to analyze whether there are differences between the mean response of one or more of these grouping factors.

ANOVA can handle a large variety of experimental factors such as repeated measures on the same experimental unit (e.g., before/during/after).

If instead of evaluating treatment differences, you want to develop a model using a set of numeric variables to predict that numeric response variable, see linear regression and t tests .

What is the difference between one-way, two-way and three-way ANOVA?

The number of “ways” in ANOVA (e.g., one-way, two-way, …) is simply the number of factors in your experiment.

Although the difference in names sounds trivial, the complexity of ANOVA increases greatly with each added factor. To use an example from agriculture, let’s say we have designed an experiment to research how different factors influence the yield of a crop.

An experiment with a single factor

In the most basic version, we want to evaluate three different fertilizers. Because we have more than two groups, we have to use ANOVA. Since there is only one factor (fertilizer), this is a one-way ANOVA. One-way ANOVA is the easiest to analyze and understand, but probably not that useful in practice, because having only one factor is a pretty simplistic experiment.

What happens when you add a second factor? 

If we have two different fields, we might want to add a second factor to see if the field itself influences growth. Within each field, we apply all three fertilizers (which is still the main interest). This is called a crossed design. In this case we have two factors, field and fertilizer, and would need a two-way ANOVA.

As you might imagine, this makes interpretation more complicated (although still very manageable) simply because more factors are involved. There is now a fertilizer effect, as well as a field effect, and there could be an interaction effect, where the fertilizer behaves differently on each field.

How about adding a third factor?

Finally, it is possible to have more than two factors in an ANOVA. In our example, perhaps you also wanted to test out different irrigation systems. You could have a three-way ANOVA due to the presence of fertilizer, field, and irrigation factors. This greatly increases the complication.

Now in addition to the three main effects (fertilizer, field and irrigation), there are three two-way interaction effects (fertilizer by field, fertilizer by irrigation, and field by irrigation), and one three-way interaction effect.

If any of the interaction effects are statistically significant, then presenting the results gets quite complicated. “Fertilizer A works better on Field B with Irrigation Method C ….”

In practice, two-way ANOVA is often as complex as many researchers want to get before consulting with a statistician. That being said, three-way ANOVAs are cumbersome, but manageable when each factor only has two levels.

What are crossed and nested factors?

In addition to increasing the difficulty with interpretation, experiments (or the resulting ANOVA) with more than one factor add another level of complexity, which is determining whether the factors are crossed or nested.

With crossed factors, every combination of levels among each factor is observed. For example, each fertilizer is applied to each field (so the fields are subdivided into three sections in this case).

With nested factors, different levels of a factor appear within another factor. An example is applying different fertilizers to each field, such as fertilizers A and B to field 1 and fertilizers C and D to field 2. See more about nested ANOVA here .

What are fixed and random factors?

Another challenging concept with two or more factors is determining whether to treat the factors as fixed or random. 

Fixed factors are used when all levels of a factor (e.g., Fertilizer A, Fertilizer B, Fertilizer C) are specified and you want to determine the effect that factor has on the mean response. 

Random factors are used when only some levels of a factor are observed (e.g., Field 1, Field 2, Field 3) out of a large or infinite possible number (e.g., all fields), but rather than specify the effect of the factor, which you can’t do because you didn’t observe all possible levels, you want to quantify the variability that’s within that factor (variability added within each field).

Many introductory courses on ANOVA only discuss fixed factors, and we will largely follow suit other than with two specific scenarios (nested factors and repeated measures). 

What are the (practical) assumptions of ANOVA?

These are one-way ANOVA assumptions, but also carryover for more complicated two-way or repeated measures ANOVA.

• Categorical treatment or factor variables - ANOVA evaluates mean differences between one or more categorical variables (such as treatment groups), which are referred to as factors or “ways.”
• Three or more groups - There must be at least three distinct groups (or levels of a categorical variable) across all factors in an ANOVA. The possibilities are endless: one factor of three different groups, two factors of two groups each (2x2), and so on. If you have fewer than three groups, you can probably get away with a simple t-test.
• Numeric Response - While the groups are categorical, the data measured in each group (i.e., the response variable) still needs to be numeric. ANOVA is fundamentally a quantitative method for measuring the differences in a numeric response between groups. If your response variable isn’t continuous, then you need a more specialized modelling framework such as logistic regression or chi-square contingency table analysis to name a few.
• Random assignment - The makeup of each experimental group should be determined by random selection.
• Normality - The distribution within each factor combination should be approximately normal, although ANOVA is fairly robust to this assumption as the sample size increases due to the central limit theorem.

What is the formula for ANOVA?

The formula to calculate ANOVA varies depending on the number of factors, assumptions about how the factors influence the model (blocking variables, fixed or random effects, nested factors, etc.), and any potential overlap or correlation between observed values (e.g., subsampling, repeated measures). 

The good news about running ANOVA in the 21st century is that statistical software handles the majority of the tedious calculations. The main thing that a researcher needs to do is select the appropriate ANOVA.

An example formula for a two-factor crossed ANOVA is:

How do I know which ANOVA to use?

As statisticians, we like to imagine that you’re reading this before you’ve run your experiment. You can save a lot of headache by simplifying an experiment into a standard format (when possible) to make the analysis straightforward.

Regardless, we’ll walk you through picking the right ANOVA for your experiment and provide examples for the most popular cases. The first question is:

Do you only have a single factor of interest?

If you have only measured a single factor (e.g., fertilizer A, fertilizer B, .etc.), then use one-way ANOVA . If you have more than one, then you need to consider the following:

Are you measuring the same observational unit (e.g., subject) multiple times?

This is where repeated measures come into play and can be a really confusing question for researchers, but if this sounds like it might describe your experiment, see repeated measures ANOVA . Otherwise:

Are any of the factors nested, where the levels are different depending on the levels of another factor?

In this case, you have a nested ANOVA design. If you don’t have nested factors or repeated measures, then it becomes simple:

Do you have two categorical factors?

Then use two-way ANOVA.

Do you have three categorical factors?

Use three-way ANOVA.

Do you have variables that you recorded that aren’t categorical (such as age, weight, etc.)?

Although these are outside the scope of this guide, if you have a single continuous variable, you might be able to use ANCOVA, which allows for a continuous covariate. With multiple continuous covariates, you probably want to use a mixed model or possibly multiple linear regression.

Prism  does  offer multiple linear regression but assumes that all factors are fixed. A full “mixed model” analysis is not yet available in Prism, but is offered as options within the one- and two-way ANOVA parameters.

How do I perform ANOVA?

Once you’ve determined which ANOVA is appropriate for your experiment, use statistical software to run the calculations. Below, we provide detailed examples of one, two and three-way ANOVA models.

How do I read and interpret an ANOVA table?

Interpreting any kind of ANOVA should start with the ANOVA table in the output. These tables are what give ANOVA its name, since they partition out the variance in the response into the various factors and interaction terms. This is done by calculating the sum of squares (SS) and mean squares (MS), which can be used to determine the variance in the response that is explained by each factor.

If you have predetermined your level of significance, interpretation mostly comes down to the p-values that come from the F-tests. The null hypothesis for each factor is that there is no significant difference between groups of that factor. All of the following factors are statistically significant with a very small p-value.

One-way ANOVA Example

An example of one-way ANOVA is an experiment of cell growth in petri dishes. The response variable is a measure of their growth, and the variable of interest is treatment, which has three levels: formula A, formula B, and a control.

Classic one-way ANOVA assumes equal variances within each sample group. If that isn’t a valid assumption for your data, you have a number of alternatives .

Calculating a one-way ANOVA

Using Prism to do the analysis, we will run a one-way ANOVA and will choose 95% as our significance threshold. Since we are interested in the differences between each of the three groups, we will evaluate each and correct for multiple comparisons (more on this later!). 

For the following, we’ll assume equal variances within the treatment groups. Consider 

The first test to look at is the overall (or omnibus) F-test, with the null hypothesis that there is no significant difference between any of the treatment groups. In this case, there is a significant difference between the three groups (p<0.0001), which tells us that at least one of the groups has a statistically significant difference.

Now we can move to the heart of the issue, which is to determine which group means are statistically different. To learn more, we should graph the data and test the differences (using a multiple comparison correction).

Graphing one-way ANOVA

The easiest way to visualize the results from an ANOVA is to use a simple chart that shows all of the individual points. Rather than a bar chart, it’s best to use a plot that shows all of the data points (and means) for each group such as a scatter or violin plot.

As an example, below you can see a graph of the cell growth levels for each data point in each treatment group, along with a line to represent their mean. This can help give credence to any significant differences found, as well as show how closely groups overlap.

Determining statistical significance between groups

In addition to the graphic, what we really want to know is which treatment means are statistically different from each other. Because we are performing multiple tests, we’ll use a multiple comparison correction . For our example, we’ll use Tukey’s correction (although if we were only interested in the difference between each formula to the control, we could use Dunnett’s correction instead). 

In this case, the mean cell growth for Formula A is significantly  higher  than the control (p<.0001) and Formula B ( p=0.002 ), but there’s no significant difference between Formula B and the control.

Two-way ANOVA example

For two-way ANOVA, there are two factors involved. Our example will focus on a case of cell lines. Suppose we have a 2x2 design (four total groupings). There are two different treatments (serum-starved and normal culture) and two different fields. There are 19 total cell line “experimental units” being evaluated, up to 5 in each group (note that with 4 groups and 19 observational units, this study isn’t balanced). Although there are multiple units in each group, they are all completely different replicates and therefore not repeated measures of the same unit.

As with one-way ANOVA, it’s a good idea to graph the data as well as look at the ANOVA table for results.

Graphing two-way ANOVA

There are many options here. Like our one-way example, we recommend a similar graphing approach that shows all the data points themselves along with the means.

Determining statistical significance between groups in two-way ANOVA

Let’s use a two-way ANOVA with a 95% significance threshold to evaluate both factors’ effects on the response, a measure of growth.

Feel free to use our two-way ANOVA checklist as often as you need for your own analysis.

First, notice there are three sources of variation included in the model, which are interaction, treatment, and field. 

The first effect to look at is the interaction term, because if it’s significant, it changes how you interpret the main effects (e.g., treatment and field). The interaction effect calculates if the effect of a factor depends on the other factor. In this case, the significant interaction term (p<.0001) indicates that the treatment effect depends on the field type.

A significant interaction term muddies the interpretation, so that you no longer have the simple conclusion that “Treatment A outperforms Treatment B.” In this case, the graphic is particularly useful. It suggests that while there may be some difference between three of the groups, the precise combination of serum starved in field 2 outperformed the rest.

To confirm whether there is a statistically significant result, we would run pairwise comparisons (comparing each factor level combination with every other one) and account for multiple comparisons.

Do I need to correct for multiple comparisons for two-way ANOVA?

If you’re comparing the means for more than one combination of treatment groups, then absolutely! Here’s more information about multiple comparisons for two-way ANOVA .

Repeated measures ANOVA

So far we have focused almost exclusively on “ordinary” ANOVA and its differences depending on how many factors are involved. In all of these cases, each observation is completely unrelated to the others. Other than the combination of factors that may be the same across replicates, each replicate on its own is independent.

There is a second common branch of ANOVA known as repeated measures . In these cases, the units are related in that they are matched up in some way. Repeated measures are used to model correlation between measurements within an individual or subject. Repeated measures ANOVA is useful (and increases statistical power) when the variability within individuals is large relative to the variability among individuals.

It’s important that all levels of your repeated measures factor (usually time) are consistent. If they aren’t, you’ll need to consider running a mixed model, which is a more advanced statistical technique.

There are two common forms of repeated measures:

• You observe the same individual or subject at different time points. If you’re familiar with paired t-tests, this is an extension to that. (You can also have the same individual receive all of the treatments, which adds another level of repeated measures.)
• You have a randomized block design, where matched elements receive each treatment. For example, you split a large sample of blood taken from one person into 3 (or more) smaller samples, and each of those smaller samples gets exactly one treatment.

Repeated measures ANOVA can have any number of factors. See analysis checklists for one-way repeated measures ANOVA and two-way repeated measures ANOVA .

What does it mean to assume sphericity with repeated measures ANOVA?

Repeated measures are almost always treated as random factors, which means that the correlation structure between levels of the repeated measures needs to be defined. The assumption of sphericity means that you assume that each level of the repeated measures has the same correlation with every other level.

This is almost never the case with repeated measures over time (e.g., baseline, at treatment, 1 hour after treatment), and in those cases, we recommend not assuming sphericity. However, if you used a randomized block design, then sphericity is usually appropriate .

Example two-way ANOVA with repeated measures

Say we have two treatments (control and treatment) to evaluate using test animals. We’ll apply both treatments to each two animals (replicates) with sufficient time in between the treatments so there isn’t a crossover (or carry-over) effect. Also, we’ll measure five different time points for each treatment (baseline, at time of injection, one hour after, …). This is repeated measures because we will need to measure matching samples from the same animal under each treatment as we track how its stimulation level changes over time.

The output shows the test results from the main and interaction effects. Due to the interaction between time and treatment being significant (p<.0001), the fact that the treatment main effect isn’t significant (p=.154) isn’t noteworthy.

Graphing repeated measures ANOVA

As we’ve been saying, graphing the data is useful, and this is particularly true when the interaction term is significant. Here we get an explanation of why the interaction between treatment and time was significant, but treatment on its own was not. As soon as one hour after injection (and all time points after), treated units show a higher response level than the control even as it decreases over those 12 hours. Thus the effect of time depends on treatment. At the earlier time points, there is no difference between treatment and control.

Graphing repeated measures data is an art, but a good graphic helps you understand and communicate the results. For example, it’s a completely different experiment, but here’s a great plot of another repeated measures experiment with before and after values that are measured on three different animal types.

What if I have three or more factors?

Interpreting three or more factors is very challenging and usually requires advanced training and experience . 

Just as two-way ANOVA is more complex than one-way, three-way ANOVA adds much more potential for confusion. Not only are you dealing with three different factors, you will now be testing seven hypotheses at the same time. Two-way interactions still exist here, and you may even run into a significant three-way interaction term.

It takes careful planning and advanced experimental design to be able to untangle the combinations that will be involved ( see more details here ). 

Non-parametric ANOVA alternatives

As with t-tests (or virtually any statistical method), there are alternatives to ANOVA for testing differences between three groups. ANOVA is means-focused and evaluated in comparison to an F-distribution. 

The two main non-parametric cousins to ANOVA are the Kruskal-Wallis and Friedman’s tests. Just as is true with everything else in ANOVA, it is likely that one of the two options is more appropriate for your experiment.

Kruskal-Wallis tests the difference between medians (rather than means) for 3 or more groups. It is only useful as an “ordinary ANOVA” alternative, without matched subjects like you have in repeated measures. Here are some tips for interpreting Kruskal-Wallis test results. 

Friedman’s Test is the opposite, designed as an alternative to repeated measures ANOVA with matched subjects. Here are some tips for interpreting Friedman's Test . 

What are simple, main, and interaction effects in ANOVA?

Consider the two-way ANOVA model setup that contains two different kinds of effects to evaluate:

The 𝛼 and 𝛽 factors are “main” effects, which are the isolated effect of a given factor. “Main effect” is used interchangeably with “simple effect” in some textbooks.

The interaction term is denoted as “𝛼𝛽”, and it allows for the effect of a factor to depend on the level of another factor. It can only be tested when you have replicates in your study. Otherwise, the error term is assumed to be the interaction term.

What are multiple comparisons?

When you’re doing multiple statistical tests on the same set of data, there’s a greater propensity to discover statistically significant differences that aren’t true differences. Multiple comparison corrections attempt to control for this, and in general control what is called the familywise error rate. There are a number of multiple comparison testing methods , which all have pros and cons depending on your particular experimental design and research questions.

What does the word “way” mean in one-way vs two-way ANOVA?

In statistics overall, it can be hard to keep track of factors, groups, and tails. To the untrained eye “two-way ANOVA” could mean any of these things.

The best way to think about ANOVA is in terms of factors or variables in your experiment. Suppose you have one factor in your analysis (perhaps “treatment”). You will likely see that written as a one-way ANOVA. Even if that factor has several different treatment groups, there is only one factor, and that’s what drives the name. 

Also, “way” has absolutely nothing to do with “tails” like a t-test. ANOVA relies on F tests, which can only test for equal vs unequal because they rely on squared terms. So ANOVA does not have the “one-or-two tails” question .

What is the difference between ANOVA and a t-test?

ANOVA is an extension of the t-test. If you only have two group means to compare, use a t-test. Anything more requires ANOVA.

What is the difference between ANOVA and chi-square?

Chi-square is designed for contingency tables, or counts of items within groups (e.g., type of animal). The goal is to see whether the counts in a particular sample match the counts you would expect by random chance.

ANOVA separates subjects into groups for evaluation, but there is some numeric response variable of interest (e.g., glucose level).

Can ANOVA evaluate effects on multiple response variables at the same time?

Multiple response variables makes things much more complicated than multiple factors. ANOVA (as we’ve discussed it here) can obviously handle multiple factors but it isn’t designed for tracking more than one response at a time. 

Technically, there is an expansion approach designed for this called Multivariate (or Multiple) ANOVA, or more commonly written as MANOVA. Things get complicated quickly, and in general requires advanced training.

Can ANOVA evaluate numeric factors in addition to the usual categorical factors?

It sounds like you are looking for ANCOVA (analysis of covariance). You can treat a continuous (numeric) factor as categorical, in which case you could use ANOVA, but this is a common point of confusion .

What is the definition of ANOVA?

ANOVA stands for analysis of variance, and, true to its name, it is a statistical technique that analyzes how experimental factors influence the variance in the response variable from an experiment.

What is blocking in Anova?

Blocking is an incredibly powerful and useful strategy in experimental design when you have a factor that you think will heavily influence the outcome, so you want to control for it in your experiment. Blocking affects how the randomization is done with the experiment. Usually blocking variables are nuisance variables that are important to control for but are not inherently of interest. 

A simple example is an experiment evaluating the efficacy of a medical drug and blocking by age of the subject. To do blocking, you must first gather the ages of all of the participants in the study, appropriately bin them into groups (e.g., 10-30, 30-50, etc.), and then randomly assign an equal number of treatments to the subjects within each group.

There’s an entire field of study around blocking. Some examples include having multiple blocking variables, incomplete block designs where not all treatments appear in all blocks, and balanced (or unbalanced) blocking designs where equal (or unequal) numbers of replicates appear in each block and treatment combination.

What is ANOVA in statistics?

For a one-way ANOVA test, the overall ANOVA null hypothesis is that the mean responses are equal for all treatments. The ANOVA p-value comes from an F-test.

Can I do ANOVA in R?

While Prism makes ANOVA much more straightforward, you can use open-source coding languages like R as well. Here are some examples of R code for repeated measures ANOVA, both one-way ANOVA in R and two-way ANOVA in R .

Perform your own ANOVA

Are you ready for your own Analysis of variance? Prism makes choosing the correct ANOVA model simple and transparent .

Start your 30 day free trial of Prism   and get access to:

• A step by step guide on how to perform ANOVA
• Sample data to save you time
• More tips on how Prism can help your research

With Prism, in a matter of minutes you learn how to go from entering data to performing statistical analyses and generating high-quality graphs.

Teach yourself statistics

One-Way Analysis of Variance: Example

In this lesson, we apply one-way analysis of variance to some fictitious data, and we show how to interpret the results of our analysis.

Note: Computations for analysis of variance are usually handled by a software package. For this example, however, we will do the computations "manually", since the gory details have educational value.

Problem Statement

A pharmaceutical company conducts an experiment to test the effect of a new cholesterol medication. The company selects 15 subjects randomly from a larger population. Each subject is randomly assigned to one of three treatment groups. Within each treament group, subjects receive a different dose of the new medication. In Group 1, subjects receive 0 mg/day; in Group 2, 50 mg/day; and in Group 3, 100 mg/day.

The treatment levels represent all the levels of interest to the experimenter, so this experiment used a fixed-effects model to select treatment levels for study.

After 30 days, doctors measure the cholesterol level of each subject. The results for all 15 subjects appear in the table below:

In conducting this experiment, the experimenter had two research questions:

• Does dosage level have a significant effect on cholesterol level?
• How strong is the effect of dosage level on cholesterol level?

To answer these questions, the experimenter intends to use one-way analysis of variance.

Is One-Way ANOVA the Right Technique?

Before you crunch the first number in one-way analysis of variance, you must be sure that one-way analysis of variance is the correct technique. That means you need to ask two questions:

• Is the experimental design compatible with one-way analysis of variance?
• Does the data set satisfy the critical assumptions required for one-way analysis of variance?

Let's address both of those questions.

Experimental Design

As we discussed in the previous lesson (see One-Way Analysis of Variance: Fixed Effects ), one-way analysis of variance is only appropriate with one experimental design - a completely randomized design. That is exactly the design used in our cholesterol study, so we can check the experimental design box.

Critical Assumptions

We also learned in the previous lesson that one-way analysis of variance makes three critical assumptions:

• Independence . The dependent variable score for each experimental unit is independent of the score for any other unit.
• Normality . In the population, dependent variable scores are normally distributed within treatment groups.
• Equality of variance . In the population, the variance of dependent variable scores in each treatment group is equal. (Equality of variance is also known as homogeneity of variance or homoscedasticity.)

Therefore, for the cholesterol study, we need to make sure our data set is consistent with the critical assumptions.

Independence of Scores

The assumption of independence is the most important assumption. When that assumption is violated, the resulting statistical tests can be misleading.

The independence assumption is satisfied by the design of the study, which features random selection of subjects and random assignment to treatment groups. Randomization tends to distribute effects of extraneous variables evenly across groups.

Normal Distributions in Groups

Violations of normality can be a problem when sample size is small, as it is in this cholesterol study. Therefore, it is important to be on the lookout for any indication of non-normality.

There are many different ways to check for normality. On this website, we describe three at: How to Test for Normality: Three Simple Tests . Given the small sample size, our best option for testing normality is to look at the following descriptive statistics:

• Central tendency. The mean and the median are summary measures used to describe central tendency - the most "typical" value in a set of values. With a normal distribution, the mean is equal to the median.
• Skewness. Skewness is a measure of the asymmetry of a probability distribution. If observations are equally distributed around the mean, the skewness value is zero; otherwise, the skewness value is positive or negative. As a rule of thumb, skewness between -2 and +2 is consistent with a normal distribution.
• Kurtosis. Kurtosis is a measure of whether observations cluster around the mean of the distribution or in the tails of the distribution. The normal distribution has a kurtosis value of zero. As a rule of thumb, kurtosis between -2 and +2 is consistent with a normal distribution.

The table below shows the mean, median, skewness, and kurtosis for each group from our study.

In all three groups, the difference between the mean and median looks small (relative to the range ). And skewness and kurtosis measures are consistent with a normal distribution (i.e., between -2 and +2). These are crude tests, but they provide some confidence for the assumption of normality in each group.

Note: With Excel, you can easily compute the descriptive statistics in Table 1. To see how, go to: How to Test for Normality: Example 1 .

Homogeneity of Variance

When the normality of variance assumption is satisfied, you can use Hartley's Fmax test to test for homogeneity of variance. Here's how to implement the test:

where X i, j is the score for observation i in Group j , X j is the mean of Group j , and n j is the number of observations in Group j .

Here is the variance ( s 2 j ) for each group in the cholesterol study.

F RATIO = s 2 MAX / s 2 MIN

F RATIO = 1170 / 450

F RATIO = 2.6

where s 2 MAX is the largest group variance, and s 2 MIN is the smallest group variance.

where n is the largest sample size in any group.

Note: The critical F values in the table are based on a significance level of 0.05.

Here, the F ratio (2.6) is smaller than the Fmax value (15.5), so we conclude that the variances are homogeneous.

Note: Other tests, such as Bartlett's test , can also test for homogeneity of variance. For the record, Bartlett's test yields the same conclusion for the cholesterol study; namely, the variances are homogeneous.

Analysis of Variance

Having confirmed that the critical assumptions are tenable, we can proceed with a one-way analysis of variance. That means taking the following steps:

• Specify a mathematical model to describe the causal factors that affect the dependent variable.
• Write statistical hypotheses to be tested by experimental data.
• Specify a significance level for a hypothesis test.
• Compute the grand mean and the mean scores for each group.
• Compute sums of squares for each effect in the model.
• Find the degrees of freedom associated with each effect in the model.
• Based on sums of squares and degrees of freedom, compute mean squares for each effect in the model.
• Compute a test statistic , based on observed mean squares and their expected values.
• Find the P value for the test statistic.
• Accept or reject the null hypothesis , based on the P value and the significance level.
• Assess the magnitude of the effect of the independent variable, based on sums of squares.

Now, let's execute each step, one-by-one, with our cholesterol medication experiment.

Mathematical Model

For every experimental design, there is a mathematical model that accounts for all of the independent and extraneous variables that affect the dependent variable. In our experiment, the dependent variable ( X ) is the cholesterol level of a subject, and the independent variable ( β ) is the dosage level administered to a subject.

For example, here is the fixed-effects model for a completely randomized design:

X i j = μ + β j + ε i ( j )

where X i j is the cholesterol level for subject i in treatment group j , μ is the population mean, β j is the effect of the dosage level administered to subjects in group j ; and ε i ( j ) is the effect of all other extraneous variables on subject i in treatment j .

Statistical Hypotheses

For fixed-effects models, it is common practice to write statistical hypotheses in terms of the treatment effect β j . With that in mind, here is the null hypothesis and the alternative hypothesis for a one-way analysis of variance:

H 0 : β j = 0 for all j

H 1 : β j ≠ 0 for some j

If the null hypothesis is true, the mean score (i.e., mean cholesterol level) in each treatment group should equal the population mean. Thus, if the null hypothesis is true, mean scores in the k treatment groups should be equal. If the null hypothesis is false, at least one pair of mean scores should be unequal.

Significance Level

The significance level (also known as alpha or α) is the probability of rejecting the null hypothesis when it is actually true. The significance level for an experiment is specified by the experimenter, before data collection begins.

Experimenters often choose significance levels of 0.05 or 0.01. For this experiment, let's use a significance level of 0.05.

Mean Scores

Analysis of variance begins by computing a grand mean and group means:

X  = ( 1 / 15 ) * ( 210 + 210 + ... + 270 + 240 )

• Group means. The mean of group j ( X j ) is the mean of all observations in group j , computed as follows:

X  1  = 258

X  2  = 246

X  3  = 210

In the equations above, n is the total sample size across all groups; and n  j is the sample size in Group j  .

Sums of Squares

A sum of squares is the sum of squared deviations from a mean score. One-way analysis of variance makes use of three sums of squares:

SSB = 5 * [ ( 238-258 ) 2 + ( 238-246) 2 + ( 238-210 ) 2 ]

SSW = 2304 + ... + 900 = 9000

• Total sum of squares. The total sum of squares (SST) measures variation of all scores around the grand mean. It can be computed from the following formula: SST = k Σ j=1 n j Σ i=1 ( X  i j  -  X  ) 2

SST = 784 + 4 + 1084 + ... + 784 + 784 + 4

SST = 15,240

It turns out that the total sum of squares is equal to the between-groups sum of squares plus the within-groups sum of squares, as shown below:

SST = SSB + SSW

15,240 = 6240 + 9000

Degrees of Freedom

The term degrees of freedom (df) refers to the number of independent sample points used to compute a statistic minus the number of parameters estimated from the sample points.

To illustrate what is going on, let's find the degrees of freedom associated with the various sum of squares computations:

Here, the formula uses k independent sample points, the sample means X   j  . And it uses one parameter estimate, the grand mean X , which was estimated from the sample points. So, the between-groups sum of squares has k - 1 degrees of freedom ( df BG  ).

df BG = k - 1 = 5 - 1 = 4

Here, the formula uses n independent sample points, the individual subject scores X  i j  . And it uses k parameter estimates, the group means X   j  , which were estimated from the sample points. So, the within-groups sum of squares has n - k degrees of freedom ( df WG  ).

n = Σ n i = 5 + 5 + 5 = 15

df WG = n - k = 15 - 3 = 12

Here, the formula uses n independent sample points, the individual subject scores X  i j  . And it uses one parameter estimate, the grand mean X , which was estimated from the sample points. So, the total sum of squares has n  - 1 degrees of freedom ( df TOT  ).

df TOT  = n - 1 = 15 - 1 = 14

The degrees of freedom for each sum of squares are summarized in the table below:

Mean Squares

A mean square is an estimate of population variance. It is computed by dividing a sum of squares (SS) by its corresponding degrees of freedom (df), as shown below:

MS = SS / df

To conduct a one-way analysis of variance, we are interested in two mean squares:

MS WG = SSW / df WG

MS WG = 9000 / 12 = 750

MS BG = SSB / df BG

MS BG = 6240 / 2 = 3120

Expected Value

The expected value of a mean square is the average value of the mean square over a large number of experiments.

Statisticians have derived formulas for the expected value of the within-groups mean square ( MS WG  ) and for the expected value of the between-groups mean square ( MS BG  ). For one-way analysis of variance, the expected value formulas are:

Fixed- and Random-Effects:

E( MS WG  ) = σ ε 2

Fixed-Effects:

Random-effects:.

E( MS BG  ) = σ ε 2 + nσ β 2

In the equations above, E( MS WG  ) is the expected value of the within-groups mean square; E( MS BG  ) is the expected value of the between-groups mean square; n is total sample size; k is the number of treatment groups; β  j is the treatment effect in Group j ; σ ε 2 is the variance attributable to everything except the treatment effect (i.e., all the extraneous variables); and σ β 2 is the variance due to random selection of treatment levels.

Notice that MS BG should equal MS WG when the variation due to treatment effects ( β  j for fixed effects and σ β 2 for random effects) is zero (i.e., when the independent variable does not affect the dependent variable). And MS BG should be bigger than the MS WG when the variation due to treatment effects is not zero (i.e., when the independent variable does affect the dependent variable)

Conclusion: By examining the relative size of the mean squares, we can make a judgment about whether an independent variable affects a dependent variable.

Test Statistic

Suppose we use the mean squares to define a test statistic F as follows:

F(v 1 , v 2 ) = MS BG / MS WG

F(2, 12) = 3120 / 750 = 4.16

where MS BG is the between-groups mean square, MS WG is the within-groups mean square, v 1 is the degrees of freedom for MS BG , and v 2 is the degrees of freedom for MS WG .

Defined in this way, the F ratio measures the size of MS BG relative to MS WG . The F ratio is a convenient measure that we can use to test the null hypothesis. Here's how:

• When the F ratio is close to one, MS BG is approximately equal to MS WG . This indicates that the independent variable did not affect the dependent variable, so we cannot reject the null hypothesis.
• When the F ratio is significantly greater than one, MS BG is bigger than MS WG . This indicates that the independent variable did affect the dependent variable, so we must reject the null hypothesis.

What does it mean for the F ratio to be significantly greater than one? To answer that question, we need to talk about the P-value.

In an experiment, a P-value is the probability of obtaining a result more extreme than the observed experimental outcome, assuming the null hypothesis is true.

With analysis of variance, the F ratio is the observed experimental outcome that we are interested in. So, the P-value would be the probability that an F statistic would be more extreme (i.e., bigger) than the actual F ratio computed from experimental data.

We can use Stat Trek's F Distribution Calculator to find the probability that an F statistic will be bigger than the actual F ratio observed in the experiment. Enter the between-groups degrees of freedom (2), the within-groups degrees of freedom (12), and the observed F ratio (4.16) into the calculator; then, click the Calculate button.

From the calculator, we see that the P ( F > 4.16 ) equals about 0.04. Therefore, the P-Value is 0.04.

Hypothesis Test

Recall that we specified a significance level 0.05 for this experiment. Once you know the significance level and the P-value, the hypothesis test is routine. Here's the decision rule for accepting or rejecting the null hypothesis:

• If the P-value is bigger than the significance level, accept the null hypothesis.
• If the P-value is equal to or smaller than the significance level, reject the null hypothesis.

Since the P-value (0.04) in our experiment is smaller than the significance level (0.05), we reject the null hypothesis that drug dosage had no effect on cholesterol level. And we conclude that the mean cholesterol level in at least one treatment group differed significantly from the mean cholesterol level in another group.

Magnitude of Effect

The hypothesis test tells us whether the independent variable in our experiment has a statistically significant effect on the dependent variable, but it does not address the magnitude of the effect. Here's the issue:

• When the sample size is large, you may find that even small differences in treatment means are statistically significant.
• When the sample size is small, you may find that even big differences in treatment means are not statistically significant.

With this in mind, it is customary to supplement analysis of variance with an appropriate measure of effect size. Eta squared (η 2 ) is one such measure. Eta squared is the proportion of variance in the dependent variable that is explained by a treatment effect. The eta squared formula for one-way analysis of variance is:

η 2 = SSB / SST

where SSB is the between-groups sum of squares and SST is the total sum of squares.

Given this formula, we can compute eta squared for this drug dosage experiment, as shown below:

η 2 = SSB / SST = 6240 / 15240 = 0.41

Thus, 41 percent of the variance in our dependent variable (cholesterol level) can be explained by variation in our independent variable (dosage level). It appears that the relationship between dosage level and cholesterol level is significant not only in a statistical sense; it is significant in a practical sense as well.

ANOVA Summary Table

It is traditional to summarize ANOVA results in an analysis of variance table. The analysis that we just conducted provides all of the information that we need to produce the following ANOVA summary table:

Analysis of Variance Table

This ANOVA table allows any researcher to interpret the results of the experiment, at a glance.

The P-value (shown in the last column of the ANOVA table) is the probability that an F statistic would be more extreme (bigger) than the F ratio shown in the table, assuming the null hypothesis is true. When the P-value is bigger than the significance level, we accept the null hypothesis; when it is smaller, we reject it. Here, the P-value (0.04) is smaller than the significance level (0.05), so we reject the null hypothesis.

To assess the strength of the treatment effect, an experimenter might compute eta squared (η 2 ). The computation is easy, using sum of squares entries from the ANOVA table, as shown below:

η 2 = SSB / SST = 6,240 / 15,240 = 0.41

For this experiment, an eta squared of 0.41 means that 41% of the variance in the dependent variable can be explained by the effect of the independent variable.

An Easier Option

In this lesson, we showed all of the hand calculations for a one-way analysis of variance. In the real world, researchers seldom conduct analysis of variance by hand. They use statistical software. In the next lesson, we'll analyze data from this problem with Excel. Hopefully, we'll get the same result.

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

Section 6.2: One-Way ANOVA Assumptions, Interpretation, and Write Up

Learning Objectives

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

• What are assumptions that need to be met before performing a Between Groups ANOVA?
• How would you interpret a Main Effect in a One-Way ANOVA?

One-Way ANOVA Assumptions

There are a number of assumptions that need to be met before performing a Between Groups ANOVA:

• The dependent variable (the variable of interest) needs to be a continuous scale (i.e., the data needs to be at either an interval or ratio measurement).
• The independent variable needs to have two independent groups with two levels. When testing three or more independent, categorical groups it is best to use a one-way ANOVA, The test could be used to test the difference between just two groups, however, an independent samples t-test would be more appropriate.
• The data should have independence of observations (i.e., there shouldn’t be the same participants who are in both groups.)
• The dependent variable should be normally or near-to-normally distributed for each group. It is worth noting that while the t-test is robust for minor violations in normality, if your data is very non-normal, it would be worth using a non-parametric test or bootstrapping (see later chapters).
• There should be no spurious outliers.
• The data must have homogeneity of variances. This assumption can be tested using Levene’s test for homogeneity of variances in the statistics package. which is shown in the output included in the next chapter.

Sample Size

A consideration for ANOVA is homogeneity. Homogeneity, in this context, just means that all of the groups’ distribution and errors differ in approximately the same way, regardless of the mean for each group. The more incompatible or unequal the group sizes are in a simple one-way between-subjects ANOVA, the more important the assumption of homogeneity is. Unequal group sizes in factorial designs can create ambiguity in results. You can test for homogeneity in PSPP and SPSS. In this class, a significant result indicates that homogeneity has been violated.

Equal cell Sizes

It is preferable to have similar or the same number of observations in each group. This provides a stronger model that tends not to violate any of the assumptions. Having unequal groups can lead to violations in normality or homogeneity of variance.

One-Way ANOVA Interpretation

Below you click to see the output for the ANOVA test of the Research Question, we have included the research example and hypothesis we will be working through is: Is there a difference in reported levels of mental distress for full-time, part-time, and casual employees?

PowerPoint: One Way ANOVA

Please have a look at the following slides:

• Chapter Six – One Way ANOVA

Main Effects

As can be seen in the circled section in red on Slide 3, the main effect was significant. By looking at the purple circle, we can see the means for each group. In the light blue circle is the test statistic, which in this case is the F value. Finally, in the dark blue circle, we can see both values for the degrees of freedom.

Posthoc Tests

In order to run posthoc tests, we need to enter some syntax. This will be covered in the slides for this section, so please do go and have a look at the syntax that has been used. The information has also been included on Slide 4.

Posthoc Test Results

These are the results. There are a number of different tests that can be used in posthoc differences tests, to control for type 1 or type 2 errors, however, for this example none have been used.

As can be seen in the red and green circles on Slide 6, both part-time and casual workers reported higher mental distress than full-time workers. This can be cross-referenced with the means on the results slide. As be seen in blue, there was not a significant difference between casual and part-time workers.

One-Way ANOVA Write Up

The following text represents how you may write up a One Way ANOVA:

A one-way ANOVA was conducted to determine if levels of mental distress were different across employment status. Participants were classified into three groups: Full-time (n = 161), Part-time (n = 83), Casual (n = 123). There was a statistically significant difference between groups as determined by one-way ANOVA ( F (2,364) = 13.17, p < .001). Post-hoc tests revealed that mental distress was significantly higher in participants who were part-time and casually employed, when compare to full-time ( Mdiff   = 4.11, p = .012, and Mdiff   = 7.34, p < .001, respectively). Additionally, no difference was found between participants who were employed part-time and casually ( Mdiff   =3.23, p = .06).

Statistics for Research Students Copyright © 2022 by University of Southern Queensland is licensed under a Creative Commons Attribution 4.0 International License , except where otherwise noted.

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Lesson 10: introduction to anova, overview section  .

In the previous lessons, we learned how to perform inference for a population mean from one sample and also how to compare population means from two samples (independent and paired). In this Lesson, we introduce Analysis of Variance or ANOVA. ANOVA is a statistical method that analyzes variances to determine if the means from more than two populations are the same. In other words, we have a quantitative response variable and a categorical explanatory variable with more than two levels. In ANOVA, the categorical explanatory is typically referred to as the factor.

• Describe the logic behind analysis of variance.
• Set up and perform one-way ANOVA.
• Identify the information in the ANOVA table.
• Interpret the results from ANOVA output.
• Perform multiple comparisons and interpret the results, when appropriate.

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Unit 16: Analysis of variance (ANOVA)

About this unit.

Analysis of variance, or ANOVA, is an approach to comparing data with multiple means across different groups, and allows us to see patterns and trends within complex and varied data. See three examples of ANOVA in action as you learn how it can be applied to more complex statistical analyses.

Analysis of variance (ANOVA)

• ANOVA 1: Calculating SST (total sum of squares) (Opens a modal)
• ANOVA 2: Calculating SSW and SSB (total sum of squares within and between) (Opens a modal)
• ANOVA 3: Hypothesis test with F-statistic (Opens a modal)

What Is An ANOVA Test In Statistics: Analysis Of Variance

Julia Simkus

Editor at Simply Psychology

BA (Hons) Psychology, Princeton University

Julia Simkus is a graduate of Princeton University with a Bachelor of Arts in Psychology. She is currently studying for a Master's Degree in Counseling for Mental Health and Wellness in September 2023. Julia's research has been published in peer reviewed journals.

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Saul Mcleod, PhD

Editor-in-Chief for Simply Psychology

BSc (Hons) Psychology, MRes, PhD, University of Manchester

Saul Mcleod, PhD., is a qualified psychology teacher with over 18 years of experience in further and higher education. He has been published in peer-reviewed journals, including the Journal of Clinical Psychology.

On This Page:

An ANOVA test is a statistical test used to determine if there is a statistically significant difference between two or more categorical groups by testing for differences of means using a variance.

Another key part of ANOVA is that it splits the independent variable into two or more groups.

For example, one or more groups might be expected to influence the dependent variable, while the other group is used as a control group and is not expected to influence the dependent variable.

Assumptions of ANOVA

The assumptions of the ANOVA test are the same as the general assumptions for any parametric test:

• An ANOVA can only be conducted if there is no relationship between the subjects in each sample. This means that subjects in the first group cannot also be in the second group (e.g., independent samples/between groups).
• The different groups/levels must have equal sample sizes .
• An ANOVA can only be conducted if the dependent variable is normally distributed so that the middle scores are the most frequent and the extreme scores are the least frequent.
• Population variances must be equal (i.e., homoscedastic). Homogeneity of variance means that the deviation of scores (measured by the range or standard deviation, for example) is similar between populations.

Types of ANOVA Tests

There are different types of ANOVA tests. The two most common are a “One-Way” and a “Two-Way.”

The difference between these two types depends on the number of independent variables in your test.

One-way ANOVA

A one-way ANOVA (analysis of variance) has one categorical independent variable (also known as a factor) and a normally distributed continuous (i.e., interval or ratio level) dependent variable.

The independent variable divides cases into two or more mutually exclusive levels, categories, or groups.

The one-way ANOVA test for differences in the means of the dependent variable is broken down by the levels of the independent variable.

An example of a one-way ANOVA includes testing a therapeutic intervention (CBT, medication, placebo) on the incidence of depression in a clinical sample.

Note : Both the One-Way ANOVA and the Independent Samples t-Test can compare the means for two groups. However, only the One-Way ANOVA can compare the means across three or more groups.

Two-way (factorial) ANOVA

A two-way ANOVA (analysis of variance) has two or more categorical independent variables (also known as a factor) and a normally distributed continuous (i.e., interval or ratio level) dependent variable.

The independent variables divide cases into two or more mutually exclusive levels, categories, or groups. A two-way ANOVA is also called a factorial ANOVA.

An example of factorial ANOVAs include testing the effects of social contact (high, medium, low), job status (employed, self-employed, unemployed, retired), and family history (no family history, some family history) on the incidence of depression in a population.

What are “Groups” or “Levels”?

In ANOVA, “groups” or “levels” refer to the different categories of the independent variable being compared.

For example, if the independent variable is “eggs,” the levels might be Non-Organic, Organic, and Free Range Organic. The dependent variable could then be the price per dozen eggs.

ANOVA F -value

The test statistic for an ANOVA is denoted as F . The formula for ANOVA is F = variance caused by treatment/variance due to random chance.

The ANOVA F value can tell you if there is a significant difference between the levels of the independent variable, when p < .05. So, a higher F value indicates that the treatment variables are significant.

Note that the ANOVA alone does not tell us specifically which means were different from one another. To determine that, we would need to follow up with multiple comparisons (or post-hoc) tests.

When the initial F test indicates that significant differences exist between group means, post hoc tests are useful for determining which specific means are significantly different when you do not have specific hypotheses that you wish to test.

Post hoc tests compare each pair of means (like t-tests), but unlike t-tests, they correct the significance estimate to account for the multiple comparisons.

What Does “Replication” Mean?

Replication requires a study to be repeated with different subjects and experimenters. This would enable a statistical analyzer to confirm a prior study by testing the same hypothesis with a new sample.

How to run an ANOVA?

For large datasets, it is best to run an ANOVA in statistical software such as R or Stata. Let’s refer to our Egg example above.

Non-Organic, Organic, and Free-Range Organic Eggs would be assigned quantitative values (1,2,3). They would serve as our independent treatment variable, while the price per dozen eggs would serve as the dependent variable. Other erroneous variables may include “Brand Name” or “Laid Egg Date.”

Using data and the aov() command in R, we could then determine the impact Egg Type has on the price per dozen eggs.

ANOVA vs. t-test?

T-tests and ANOVA tests are both statistical techniques used to compare differences in means and spreads of the distributions across populations.

The t-test determines whether two populations are statistically different from each other, whereas ANOVA tests are used when an individual wants to test more than two levels within an independent variable.

Referring back to our egg example, testing Non-Organic vs. Organic would require a t-test while adding in Free Range as a third option demands ANOVA.

Rather than generate a t-statistic, ANOVA results in an f-statistic to determine statistical significance.

What does anova stand for?

ANOVA stands for Analysis of Variance. It’s a statistical method to analyze differences among group means in a sample. ANOVA tests the hypothesis that the means of two or more populations are equal, generalizing the t-test to more than two groups.

It’s commonly used in experiments where various factors’ effects are compared. It can also handle complex experiments with factors that have different numbers of levels.

When to use anova?

ANOVA should be used when one independent variable has three or more levels (categories or groups). It’s designed to compare the means of these multiple groups.

What does an anova test tell you?

An ANOVA test tells you if there are significant differences between the means of three or more groups. If the test result is significant, it suggests that at least one group’s mean differs from the others. It does not, however, specify which groups are different from each other.

Why do you use chi-square instead of ANOVA?

You use the chi-square test instead of ANOVA when dealing with categorical data to test associations or independence between two categorical variables. In contrast, ANOVA is used for continuous data to compare the means of three or more groups.

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Hypothesis Testing - Analysis of Variance (ANOVA)

Lisa Sullivan, PhD

Professor of Biostatistics

Boston University School of Public Health

Introduction

This module will continue the discussion of hypothesis testing, where a specific statement or hypothesis is generated about a population parameter, and sample statistics are used to assess the likelihood that the hypothesis is true. The hypothesis is based on available information and the investigator's belief about the population parameters. The specific test considered here is called analysis of variance (ANOVA) and is a test of hypothesis that is appropriate to compare means of a continuous variable in two or more independent comparison groups. For example, in some clinical trials there are more than two comparison groups. In a clinical trial to evaluate a new medication for asthma, investigators might compare an experimental medication to a placebo and to a standard treatment (i.e., a medication currently being used). In an observational study such as the Framingham Heart Study, it might be of interest to compare mean blood pressure or mean cholesterol levels in persons who are underweight, normal weight, overweight and obese.  

The technique to test for a difference in more than two independent means is an extension of the two independent samples procedure discussed previously which applies when there are exactly two independent comparison groups. The ANOVA technique applies when there are two or more than two independent groups. The ANOVA procedure is used to compare the means of the comparison groups and is conducted using the same five step approach used in the scenarios discussed in previous sections. Because there are more than two groups, however, the computation of the test statistic is more involved. The test statistic must take into account the sample sizes, sample means and sample standard deviations in each of the comparison groups.

If one is examining the means observed among, say three groups, it might be tempting to perform three separate group to group comparisons, but this approach is incorrect because each of these comparisons fails to take into account the total data, and it increases the likelihood of incorrectly concluding that there are statistically significate differences, since each comparison adds to the probability of a type I error. Analysis of variance avoids these problemss by asking a more global question, i.e., whether there are significant differences among the groups, without addressing differences between any two groups in particular (although there are additional tests that can do this if the analysis of variance indicates that there are differences among the groups).

The fundamental strategy of ANOVA is to systematically examine variability within groups being compared and also examine variability among the groups being compared.

Learning Objectives

After completing this module, the student will be able to:

• Perform analysis of variance by hand
• Appropriately interpret results of analysis of variance tests
• Distinguish between one and two factor analysis of variance tests
• Identify the appropriate hypothesis testing procedure based on type of outcome variable and number of samples

The ANOVA Approach

Consider an example with four independent groups and a continuous outcome measure. The independent groups might be defined by a particular characteristic of the participants such as BMI (e.g., underweight, normal weight, overweight, obese) or by the investigator (e.g., randomizing participants to one of four competing treatments, call them A, B, C and D). Suppose that the outcome is systolic blood pressure, and we wish to test whether there is a statistically significant difference in mean systolic blood pressures among the four groups. The sample data are organized as follows:

The hypotheses of interest in an ANOVA are as follows:

• H 0 : μ 1 = μ 2 = μ 3 ... = μ k
• H 1 : Means are not all equal.

where k = the number of independent comparison groups.

In this example, the hypotheses are:

• H 0 : μ 1 = μ 2 = μ 3 = μ 4
• H 1 : The means are not all equal.

The null hypothesis in ANOVA is always that there is no difference in means. The research or alternative hypothesis is always that the means are not all equal and is usually written in words rather than in mathematical symbols. The research hypothesis captures any difference in means and includes, for example, the situation where all four means are unequal, where one is different from the other three, where two are different, and so on. The alternative hypothesis, as shown above, capture all possible situations other than equality of all means specified in the null hypothesis.

Test Statistic for ANOVA

The test statistic for testing H 0 : μ 1 = μ 2 = ... =   μ k is:

and the critical value is found in a table of probability values for the F distribution with (degrees of freedom) df 1 = k-1, df 2 =N-k. The table can be found in "Other Resources" on the left side of the pages.

NOTE: The test statistic F assumes equal variability in the k populations (i.e., the population variances are equal, or s 1 2 = s 2 2 = ... = s k 2 ). This means that the outcome is equally variable in each of the comparison populations. This assumption is the same as that assumed for appropriate use of the test statistic to test equality of two independent means. It is possible to assess the likelihood that the assumption of equal variances is true and the test can be conducted in most statistical computing packages. If the variability in the k comparison groups is not similar, then alternative techniques must be used.

The F statistic is computed by taking the ratio of what is called the "between treatment" variability to the "residual or error" variability. This is where the name of the procedure originates. In analysis of variance we are testing for a difference in means (H 0 : means are all equal versus H 1 : means are not all equal) by evaluating variability in the data. The numerator captures between treatment variability (i.e., differences among the sample means) and the denominator contains an estimate of the variability in the outcome. The test statistic is a measure that allows us to assess whether the differences among the sample means (numerator) are more than would be expected by chance if the null hypothesis is true. Recall in the two independent sample test, the test statistic was computed by taking the ratio of the difference in sample means (numerator) to the variability in the outcome (estimated by Sp).  

The decision rule for the F test in ANOVA is set up in a similar way to decision rules we established for t tests. The decision rule again depends on the level of significance and the degrees of freedom. The F statistic has two degrees of freedom. These are denoted df 1 and df 2 , and called the numerator and denominator degrees of freedom, respectively. The degrees of freedom are defined as follows:

df 1 = k-1 and df 2 =N-k,

where k is the number of comparison groups and N is the total number of observations in the analysis.   If the null hypothesis is true, the between treatment variation (numerator) will not exceed the residual or error variation (denominator) and the F statistic will small. If the null hypothesis is false, then the F statistic will be large. The rejection region for the F test is always in the upper (right-hand) tail of the distribution as shown below.

Rejection Region for F   Test with a =0.05, df 1 =3 and df 2 =36 (k=4, N=40)

For the scenario depicted here, the decision rule is: Reject H 0 if F > 2.87.

The ANOVA Procedure

We will next illustrate the ANOVA procedure using the five step approach. Because the computation of the test statistic is involved, the computations are often organized in an ANOVA table. The ANOVA table breaks down the components of variation in the data into variation between treatments and error or residual variation. Statistical computing packages also produce ANOVA tables as part of their standard output for ANOVA, and the ANOVA table is set up as follows: 

where  

• X = individual observation,
• k = the number of treatments or independent comparison groups, and
• N = total number of observations or total sample size.

The ANOVA table above is organized as follows.

• The first column is entitled "Source of Variation" and delineates the between treatment and error or residual variation. The total variation is the sum of the between treatment and error variation.
• The second column is entitled "Sums of Squares (SS)" . The between treatment sums of squares is

and is computed by summing the squared differences between each treatment (or group) mean and the overall mean. The squared differences are weighted by the sample sizes per group (n j ). The error sums of squares is:

and is computed by summing the squared differences between each observation and its group mean (i.e., the squared differences between each observation in group 1 and the group 1 mean, the squared differences between each observation in group 2 and the group 2 mean, and so on). The double summation ( SS ) indicates summation of the squared differences within each treatment and then summation of these totals across treatments to produce a single value. (This will be illustrated in the following examples). The total sums of squares is:

and is computed by summing the squared differences between each observation and the overall sample mean. In an ANOVA, data are organized by comparison or treatment groups. If all of the data were pooled into a single sample, SST would reflect the numerator of the sample variance computed on the pooled or total sample. SST does not figure into the F statistic directly. However, SST = SSB + SSE, thus if two sums of squares are known, the third can be computed from the other two.

• The third column contains degrees of freedom . The between treatment degrees of freedom is df 1 = k-1. The error degrees of freedom is df 2 = N - k. The total degrees of freedom is N-1 (and it is also true that (k-1) + (N-k) = N-1).
• The fourth column contains "Mean Squares (MS)" which are computed by dividing sums of squares (SS) by degrees of freedom (df), row by row. Specifically, MSB=SSB/(k-1) and MSE=SSE/(N-k). Dividing SST/(N-1) produces the variance of the total sample. The F statistic is in the rightmost column of the ANOVA table and is computed by taking the ratio of MSB/MSE.  

A clinical trial is run to compare weight loss programs and participants are randomly assigned to one of the comparison programs and are counseled on the details of the assigned program. Participants follow the assigned program for 8 weeks. The outcome of interest is weight loss, defined as the difference in weight measured at the start of the study (baseline) and weight measured at the end of the study (8 weeks), measured in pounds.  

Three popular weight loss programs are considered. The first is a low calorie diet. The second is a low fat diet and the third is a low carbohydrate diet. For comparison purposes, a fourth group is considered as a control group. Participants in the fourth group are told that they are participating in a study of healthy behaviors with weight loss only one component of interest. The control group is included here to assess the placebo effect (i.e., weight loss due to simply participating in the study). A total of twenty patients agree to participate in the study and are randomly assigned to one of the four diet groups. Weights are measured at baseline and patients are counseled on the proper implementation of the assigned diet (with the exception of the control group). After 8 weeks, each patient's weight is again measured and the difference in weights is computed by subtracting the 8 week weight from the baseline weight. Positive differences indicate weight losses and negative differences indicate weight gains. For interpretation purposes, we refer to the differences in weights as weight losses and the observed weight losses are shown below.

Is there a statistically significant difference in the mean weight loss among the four diets?  We will run the ANOVA using the five-step approach.

• Step 1. Set up hypotheses and determine level of significance

H 0 : μ 1 = μ 2 = μ 3 = μ 4 H 1 : Means are not all equal              α=0.05

• Step 2. Select the appropriate test statistic.  

The test statistic is the F statistic for ANOVA, F=MSB/MSE.

• Step 3. Set up decision rule.  

The appropriate critical value can be found in a table of probabilities for the F distribution(see "Other Resources"). In order to determine the critical value of F we need degrees of freedom, df 1 =k-1 and df 2 =N-k. In this example, df 1 =k-1=4-1=3 and df 2 =N-k=20-4=16. The critical value is 3.24 and the decision rule is as follows: Reject H 0 if F > 3.24.

• Step 4. Compute the test statistic.  

To organize our computations we complete the ANOVA table. In order to compute the sums of squares we must first compute the sample means for each group and the overall mean based on the total sample.  

We can now compute

So, in this case:

Next we compute,

SSE requires computing the squared differences between each observation and its group mean. We will compute SSE in parts. For the participants in the low calorie diet:  

For the participants in the low fat diet:  

For the participants in the low carbohydrate diet:  

For the participants in the control group:

We can now construct the ANOVA table .

• Step 5. Conclusion.  

We reject H 0 because 8.43 > 3.24. We have statistically significant evidence at α=0.05 to show that there is a difference in mean weight loss among the four diets.    

ANOVA is a test that provides a global assessment of a statistical difference in more than two independent means. In this example, we find that there is a statistically significant difference in mean weight loss among the four diets considered. In addition to reporting the results of the statistical test of hypothesis (i.e., that there is a statistically significant difference in mean weight losses at α=0.05), investigators should also report the observed sample means to facilitate interpretation of the results. In this example, participants in the low calorie diet lost an average of 6.6 pounds over 8 weeks, as compared to 3.0 and 3.4 pounds in the low fat and low carbohydrate groups, respectively. Participants in the control group lost an average of 1.2 pounds which could be called the placebo effect because these participants were not participating in an active arm of the trial specifically targeted for weight loss. Are the observed weight losses clinically meaningful?

Another ANOVA Example

Calcium is an essential mineral that regulates the heart, is important for blood clotting and for building healthy bones. The National Osteoporosis Foundation recommends a daily calcium intake of 1000-1200 mg/day for adult men and women. While calcium is contained in some foods, most adults do not get enough calcium in their diets and take supplements. Unfortunately some of the supplements have side effects such as gastric distress, making them difficult for some patients to take on a regular basis.  

 A study is designed to test whether there is a difference in mean daily calcium intake in adults with normal bone density, adults with osteopenia (a low bone density which may lead to osteoporosis) and adults with osteoporosis. Adults 60 years of age with normal bone density, osteopenia and osteoporosis are selected at random from hospital records and invited to participate in the study. Each participant's daily calcium intake is measured based on reported food intake and supplements. The data are shown below.   

Is there a statistically significant difference in mean calcium intake in patients with normal bone density as compared to patients with osteopenia and osteoporosis? We will run the ANOVA using the five-step approach.

H 0 : μ 1 = μ 2 = μ 3 H 1 : Means are not all equal                            α=0.05

In order to determine the critical value of F we need degrees of freedom, df 1 =k-1 and df 2 =N-k.   In this example, df 1 =k-1=3-1=2 and df 2 =N-k=18-3=15. The critical value is 3.68 and the decision rule is as follows: Reject H 0 if F > 3.68.

To organize our computations we will complete the ANOVA table. In order to compute the sums of squares we must first compute the sample means for each group and the overall mean.  

 If we pool all N=18 observations, the overall mean is 817.8.

We can now compute:

Substituting:

SSE requires computing the squared differences between each observation and its group mean. We will compute SSE in parts. For the participants with normal bone density:

For participants with osteopenia:

For participants with osteoporosis:

We do not reject H 0 because 1.395 < 3.68. We do not have statistically significant evidence at a =0.05 to show that there is a difference in mean calcium intake in patients with normal bone density as compared to osteopenia and osterporosis. Are the differences in mean calcium intake clinically meaningful? If so, what might account for the lack of statistical significance?

One-Way ANOVA in R

The video below by Mike Marin demonstrates how to perform analysis of variance in R. It also covers some other statistical issues, but the initial part of the video will be useful to you.

Two-Factor ANOVA

The ANOVA tests described above are called one-factor ANOVAs. There is one treatment or grouping factor with k > 2 levels and we wish to compare the means across the different categories of this factor. The factor might represent different diets, different classifications of risk for disease (e.g., osteoporosis), different medical treatments, different age groups, or different racial/ethnic groups. There are situations where it may be of interest to compare means of a continuous outcome across two or more factors. For example, suppose a clinical trial is designed to compare five different treatments for joint pain in patients with osteoarthritis. Investigators might also hypothesize that there are differences in the outcome by sex. This is an example of a two-factor ANOVA where the factors are treatment (with 5 levels) and sex (with 2 levels). In the two-factor ANOVA, investigators can assess whether there are differences in means due to the treatment, by sex or whether there is a difference in outcomes by the combination or interaction of treatment and sex. Higher order ANOVAs are conducted in the same way as one-factor ANOVAs presented here and the computations are again organized in ANOVA tables with more rows to distinguish the different sources of variation (e.g., between treatments, between men and women). The following example illustrates the approach.

Consider the clinical trial outlined above in which three competing treatments for joint pain are compared in terms of their mean time to pain relief in patients with osteoarthritis. Because investigators hypothesize that there may be a difference in time to pain relief in men versus women, they randomly assign 15 participating men to one of the three competing treatments and randomly assign 15 participating women to one of the three competing treatments (i.e., stratified randomization). Participating men and women do not know to which treatment they are assigned. They are instructed to take the assigned medication when they experience joint pain and to record the time, in minutes, until the pain subsides. The data (times to pain relief) are shown below and are organized by the assigned treatment and sex of the participant.

Table of Time to Pain Relief by Treatment and Sex

The analysis in two-factor ANOVA is similar to that illustrated above for one-factor ANOVA. The computations are again organized in an ANOVA table, but the total variation is partitioned into that due to the main effect of treatment, the main effect of sex and the interaction effect. The results of the analysis are shown below (and were generated with a statistical computing package - here we focus on interpretation). 

 ANOVA Table for Two-Factor ANOVA

There are 4 statistical tests in the ANOVA table above. The first test is an overall test to assess whether there is a difference among the 6 cell means (cells are defined by treatment and sex). The F statistic is 20.7 and is highly statistically significant with p=0.0001. When the overall test is significant, focus then turns to the factors that may be driving the significance (in this example, treatment, sex or the interaction between the two). The next three statistical tests assess the significance of the main effect of treatment, the main effect of sex and the interaction effect. In this example, there is a highly significant main effect of treatment (p=0.0001) and a highly significant main effect of sex (p=0.0001). The interaction between the two does not reach statistical significance (p=0.91). The table below contains the mean times to pain relief in each of the treatments for men and women (Note that each sample mean is computed on the 5 observations measured under that experimental condition).  

Mean Time to Pain Relief by Treatment and Gender

Treatment A appears to be the most efficacious treatment for both men and women. The mean times to relief are lower in Treatment A for both men and women and highest in Treatment C for both men and women. Across all treatments, women report longer times to pain relief (See below).  

Notice that there is the same pattern of time to pain relief across treatments in both men and women (treatment effect). There is also a sex effect - specifically, time to pain relief is longer in women in every treatment.  

Suppose that the same clinical trial is replicated in a second clinical site and the following data are observed.

Table - Time to Pain Relief by Treatment and Sex - Clinical Site 2

The ANOVA table for the data measured in clinical site 2 is shown below.

Table - Summary of Two-Factor ANOVA - Clinical Site 2

Notice that the overall test is significant (F=19.4, p=0.0001), there is a significant treatment effect, sex effect and a highly significant interaction effect. The table below contains the mean times to relief in each of the treatments for men and women.  

Table - Mean Time to Pain Relief by Treatment and Gender - Clinical Site 2

Notice that now the differences in mean time to pain relief among the treatments depend on sex. Among men, the mean time to pain relief is highest in Treatment A and lowest in Treatment C. Among women, the reverse is true. This is an interaction effect (see below).  

Notice above that the treatment effect varies depending on sex. Thus, we cannot summarize an overall treatment effect (in men, treatment C is best, in women, treatment A is best).    

When interaction effects are present, some investigators do not examine main effects (i.e., do not test for treatment effect because the effect of treatment depends on sex). This issue is complex and is discussed in more detail in a later module. 

Repeated Measures ANOVA

Introduction.

Repeated measures ANOVA is the equivalent of the one-way ANOVA, but for related, not independent groups, and is the extension of the dependent t-test . A repeated measures ANOVA is also referred to as a within-subjects ANOVA or ANOVA for correlated samples. All these names imply the nature of the repeated measures ANOVA, that of a test to detect any overall differences between related means. There are many complex designs that can make use of repeated measures, but throughout this guide, we will be referring to the most simple case, that of a one-way repeated measures ANOVA. This particular test requires one independent variable and one dependent variable. The dependent variable needs to be continuous (interval or ratio) and the independent variable categorical (either nominal or ordinal ).

When to use a Repeated Measures ANOVA

We can analyse data using a repeated measures ANOVA for two types of study design. Studies that investigate either (1) changes in mean scores over three or more time points, or (2) differences in mean scores under three or more different conditions. For example, for (1), you might be investigating the effect of a 6-month exercise training programme on blood pressure and want to measure blood pressure at 3 separate time points (pre-, midway and post-exercise intervention), which would allow you to develop a time-course for any exercise effect. For (2), you might get the same subjects to eat different types of cake (chocolate, caramel and lemon) and rate each one for taste, rather than having different people taste each different cake. The important point with these two study designs is that the same people are being measured more than once on the same dependent variable (i.e., why it is called repeated measures).

In repeated measures ANOVA, the independent variable has categories called levels or related groups . Where measurements are repeated over time, such as when measuring changes in blood pressure due to an exercise-training programme, the independent variable is time . Each level (or related group ) is a specific time point. Hence, for the exercise-training study, there would be three time points and each time-point is a level of the independent variable (a schematic of a time-course repeated measures design is shown below):

Where measurements are made under different conditions, the conditions are the levels (or related groups) of the independent variable (e.g., type of cake is the independent variable with chocolate, caramel, and lemon cake as the levels of the independent variable). A schematic of a different-conditions repeated measures design is shown below. It should be noted that often the levels of the independent variable are not referred to as conditions, but treatments . Which one you want to use is up to you. There is no right or wrong naming convention. You will also see the independent variable more commonly referred to as the within-subjects factor .

The above two schematics have shown an example of each type of repeated measures ANOVA design, but you will also often see these designs expressed in tabular form, such as shown below:

This particular table describes a study with six subjects (S 1 to S 6 ) performing under three conditions or at three time points (T 1 to T 3 ). As highlighted earlier, the within-subjects factor could also have been labelled "treatment" instead of "time/condition". They all relate to the same thing: subjects undergoing repeated measurements at either different time points or under different conditions/treatments.

Hypothesis for Repeated Measures ANOVA

The repeated measures ANOVA tests for whether there are any differences between related population means. The null hypothesis (H 0 ) states that the means are equal:

H 0 : µ 1 = µ 2 = µ 3 = … = µ k

where µ = population mean and k = number of related groups. The alternative hypothesis (H A ) states that the related population means are not equal (at least one mean is different to another mean):

H A : at least two means are significantly different

For our exercise-training example, the null hypothesis (H 0 ) is that mean blood pressure is the same at all time points (pre-, 3 months, and 6 months). The alternative hypothesis is that mean blood pressure is significantly different at one or more time points. A repeated measures ANOVA will not inform you where the differences between groups lie as it is an omnibus statistical test. The same would be true if you were investigating different conditions or treatments rather than time points, as used in this example. If your repeated measures ANOVA is statistically significant, you can run post hoc tests that can highlight exactly where these differences occur. You can learn how to run appropriate post-hoc tests for a repeated measures ANOVA in SPSS Statistics on page 2 of our guide: One-Way Repeated Measures ANOVA in SPSS Statistics .

Logic of the Repeated Measures ANOVA

The logic behind a repeated measures ANOVA is very similar to that of a between-subjects ANOVA. Recall that a between-subjects ANOVA partitions total variability into between-groups variability (SS b ) and within-groups variability (SS w ), as shown below:

In this design, within-group variability (SS w ) is defined as the error variability (SS error ). Following division by the appropriate degrees of freedom, a mean sum of squares for between-groups (MS b ) and within-groups (MS w ) is determined and an F -statistic is calculated as the ratio of MS b to MS w (or MS error ), as shown below:

A repeated measures ANOVA calculates an F -statistic in a similar way:

The advantage of a repeated measures ANOVA is that whereas within-group variability (SS w ) expresses the error variability (SS error ) in an independent (between-subjects) ANOVA, a repeated measures ANOVA can further partition this error term, reducing its size, as is illustrated below:

This has the effect of increasing the value of the F -statistic due to the reduction of the denominator and leading to an increase in the power of the test to detect significant differences between means (this is discussed in more detail later). Mathematically, and as illustrated above, we partition the variability attributable to the differences between groups (SS conditions ) and variability within groups (SS w ) exactly as we do in a between-subjects (independent) ANOVA. However, with a repeated measures ANOVA, as we are using the same subjects in each group, we can remove the variability due to the individual differences between subjects, referred to as SS subjects , from the within-groups variability (SS w ). How is this achieved? Quite simply, we treat each subject as a block. That is, each subject becomes a level of a factor called subjects. We then calculate this variability as we do with any between-subjects factor. The ability to subtract SS subjects will leave us with a smaller SS error term, as highlighted below:

Now that we have removed the between-subjects variability, our new SS error only reflects individual variability to each condition. You might recognise this as the interaction effect of subject by conditions; that is, how subjects react to the different conditions. Whether this leads to a more powerful test will depend on whether the reduction in SS error more than compensates for the reduction in degrees of freedom for the error term (as degrees of freedom go from ( n - k ) to ( n - 1 )( k - 1 ) (remembering that there are more subjects in the independent ANOVA design).

The next page of our guide deals with how to calculate a repeated measures ANOVA.

Statistics Made Easy

Repeated Measures ANOVA: Definition, Formula, and Example

A  repeated measures ANOVA  is used to determine whether or not there is a statistically significant difference between the means of three or more groups in which the same subjects show up in each group.

A repeated measures ANOVA is typically used in two specific situations:

1. Measuring the mean scores of subjects during three or more time points. For example, you might want to measure the resting heart rate of subjects one month before they start a training program, during the middle of the training program, and one month after the training program to see if there is a significant difference in mean resting heart rate across these three time points.

2. Measuring the mean scores of subjects under three different conditions. For example, you might have subjects watch three different movies and rate each one based on how much they enjoyed it. 

One-Way ANOVA vs. Repeated Measures ANOVA

In a typical one-way ANOVA , different subjects are used in each group. For example, we might ask subjects to rate three movies, just like in the example above, but we use different subjects to rate each movie:

In this case, we would conduct a typical one-way ANOVA to test for the difference between the mean ratings of the three movies. 

In real life there are two benefits of using the same subjects across multiple treatment conditions:

1. It’s cheaper and faster for researchers to recruit and pay a smaller number of people to carry out an experiment since they can just obtain data from the same people multiple times. 

2.  We are able to attribute some of the variance in the data to the subjects themselves, which makes it easier to obtain a smaller p-value.

One potential drawback of this type of design is that subjects might get bored or tired if an experiment lasts too long, which could skew the results. For example, subjects might give lower movie ratings to the third movie they watch because they’re tired and ready to go home.

Repeated Measures ANOVA: Example

Suppose we recruit five subjects to participate in a training program. We measure their resting heart rate before participating in a training program, after participating for 4 months, and after participating for 8 months. 

The following table shows the results:

We want to know whether there is a difference in mean resting heart rate at these three time points so we conduct a repeated measures ANOVA at the .05 significance level using the following steps:

Step 1. State the hypotheses. 

The null hypothesis (H 0 ):  µ 1 = µ 2 = µ 3 (the population means are all equal)

The alternative hypothesis: (Ha): at least one population mean is different from the rest

Step 2. Perform the repeated measures ANOVA.

We will use the Repeated Measures ANOVA Calculator using the following input:

Once we click “Calculate” then the following output will automatically appear:

Step 3. Interpret the results. 

From the output table we see that the F test statistic is  9.598  and the corresponding p-value is  0.00749 .

Since this p-value is less than 0.05, we reject the null hypothesis. This means we have sufficient evidence to say that there is a statistically significant difference between the mean resting heart rate at the three different points in time.

Additional Resources

The following articles explain how to perform a repeated measures ANOVA using different statistical softwares:

Repeated Measures ANOVA in Excel Repeated Measures ANOVA in R Repeated Measures ANOVA in Stata Repeated Measures ANOVA in Python Repeated Measures ANOVA in SPSS Repeated Measures ANOVA in Google Sheets Repeated Measures ANOVA By Hand Repeated Measures ANOVA Calculator

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Hey there. My name is Zach Bobbitt. I have a Masters of Science degree in Applied Statistics and I’ve worked on machine learning algorithms for professional businesses in both healthcare and retail. I’m passionate about statistics, machine learning, and data visualization and I created Statology to be a resource for both students and teachers alike.  My goal with this site is to help you learn statistics through using simple terms, plenty of real-world examples, and helpful illustrations.

2 Replies to “Repeated Measures ANOVA: Definition, Formula, and Example”

thank you for the easy to understand explanation. are there post-hoc tests like the bonferroni for one way anova?

This was so helpful, thankyou. However, wanted to ask if the alternative non parametrics tests determines mean differences or MEDIAN.?

Somewhere, I read that non parametric are used to determine median difference since it is non outlier.

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ANOVA Test (Analysis of Variance) is used to compare the means of different groups using various estimate methodologies. ANOVA is an abbreviation for the analysis of variance. The ANOVA analysis is a statistical relevance tool designed to evaluate whether or not the null hypothesis can be rejected while testing hypotheses. It is used to determine whether or not the means of three or more groups are equal.

Whenever there are more than two or more independent groups, the ANOVA test is used. The ANOVA test is used to look for heterogeneity within groups as well as variability across groupings. The f test returns the ANOVA test statistic.

Table of Content

ANOVA Formula

Types of anova  formula, anova formula example, anova  formula – faqs.

ANOVA formula is made up of numerous parts. The best way to tackle an ANOVA test problem is to organize the formulae inside an ANOVA table. Below are the ANOVA formulae.

• F = ANOVA Coefficient
• MSB = Mean of the total of squares between groupings
• MSW = Mean total of squares within groupings
• MSE = Mean sum of squares due to error
• SST = total Sum of squares
• p = Total number of populations
• n = The total number of samples in a population
• SSW = Sum of squares within the groups
• SSB = Sum of squares between the groups
• SSE = Sum of squares due to error
• s = Standard deviation of the samples
• N = Total number of observations

Examples of the use of ANOVA Formula

• Assume it is necessary to assess whether consuming a specific type of tea will result in a mean weight decrease. Allow three groups to use three different varieties of tea: green tea, Earl Grey tea, and Jasmine tea. Thus, the ANOVA test (one way) will be utilized to examine if there was any mean weight decrease displayed by a certain group.
• Assume a poll was undertaken to see if there is a relationship between salary and gender and stress levels during job interviews. A two-way ANOVA will be utilized to carry out such a test.
• One-Way ANOVA: This test is used to see if there is a variation in the mean values of three or more groups. Such a test is used where the data set has only one independent variable. If the test statistic exceeds the critical value, the null hypothesis is rejected, and the averages of at least two different groups are significant statistically.
• Two-Way ANOVA: Two independent variables are used in the two-way ANOVA. As a result, it can be viewed as an extension of a one-way ANOVA in which only one variable influences the dependent variable. A two-way ANOVA test is used to determine the main effect of each independent variable and whether there is an interaction effect. Each factor is examined independently to determine the main effect, as in a one-way ANOVA. Furthermore, all components are analyzed at the same time to test the interaction impact.  

Example 1: Three different kinds of food are tested on three groups of rats for 5 weeks. The objective is to check the difference in mean weight(in grams) of the rats per week. Apply one-way ANOVA using a 0.05 significance level to the following data:

H 0 : μ 1 = μ 2 =μ 3 H 1 : The means are not equal Since, X̄ 1 = 5, X̄ 2 = 9, X̄ 3 = 10 Total mean = X̄ = 8 SSB = 6(5 – 8) 2 + 6(9 – 8) 2 + 6(10 – 8) 2 = 84 SSE = 68 MSB = SSB/df 1 = 42 MSE = SSE/df 2 = 4.53 f = MSB/MSE = 42/4.53 = 9.33 Since f > F, the null hypothesis stands rejected.

Example 2: Calculate the ANOVA coefficient for the following data:

Plant n x s s 2 Hibiscus 5 12 2 4 Marigold 5 16 1 1 Rose 5 20 4 16 p = 3 n = 5 N = 15 x̄ = 16 SST = Σn(x−x̄) 2 SST= 5(12 − 16) 2 + 5(16 − 16) 2 + 11(20 − 16) 2 = 160 MST = SST/p-1 = 160/3-1 = 80 SSE = ∑ (n−1) = 4 (4 + 1) + 4(16) = 84 MSE = 7 F = MST/MSE = 80/7   F = 11.429

Example 3: The following data show the number of worms quarantined from the GI areas of four groups of muskrats in a carbon tetrachloride anthelmintic study. Conduct a two-way ANOVA test.

Source of Variation Sum of Squares Degrees of Freedom Mean Square Between the groups 62111.6 8 9078.067 Within the groups 98787.8 16 4567.89 Total 167771.4 24   Since F = MST / MSE            = 9.4062 / 3.66         F = 2.57

Example 4: Enlist the results in APA format after performing ANOVA on the following data set:

[Tex]\begin{bmatrix}  \textbf{n} & \textbf{mean} & \textbf{sd} \\  30 & 50.26 & 10.45 \\  30 & 45.32 & 12.76 \\  30 & 53.67 & 11.47 \\ \end{bmatrix} [/Tex]

Variance of first set = (10.45) 2 = 109.2 Variance of second set = (12.76) 2 = 162.82 Variance of third set = (11.47) 2 = 131.56 MS error = {109.2 + 162.82 + 131.56} / {3}            = 134.53 MS between = (17.62)(30) = 528.75 F = MS between   /  MS error    = 528.75 / 134.53 F = 4.86 APA writeup: F(2, 87)=3.93, p >=0.01, η 2 =0.08.

Related Resources:

Variance and Standard Deviation How to Calculate Variance? Frequency Distribution

How does one set the hypothesis for ANOVA?

The equality of the means of distinct groups must be tested in an ANOVA test. As a result, the hypothesis is as follows: H 0 = 1 = 2 = 3 =… = k = Null Hypothesis H 1 Alternative Hypothesis: The means are not equal.

How do you calculate the ANOVA?

ANOVA compares group means’ differences. Calculate Grand Mean, Between-Group Variability (SSB), and Within-Group Variability (SSW). Determine significance through variance comparison.

What is meant by ANOVA statistic?

The sample mean of the jth treatment of a grouping or a mass data sample is called the ANOVA statistic. It is denoted by the alphabet f. 

What is the p value in ANOVA?

In ANOVA, a shared P-value is initially obtained. A significant P-value in the ANOVA test suggests statistical significance in at least one pair’s mean difference. Multiple comparisons are then employed to identify these significant pair(s).

What do you mean by one-way ANOVA?

One-way ANOVA is a form of ANOVA test that is used when just one independent variable is present. It compares the means of the various test groups. A test of this type can only provide information on the statistical significance of the means; it cannot establish which groups have different means.

Comment on the accuracy of the ANOVA test.

Since it is more versatile and requires fewer observations, ANOVA analysis is sometimes thought to be more accurate than t-testing. It is also more suited to employ in more sophisticated studies than those that can be evaluated by testing.

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Employers ask about your ideal work environment to assess fit.

Predicting what you'll be asked in a job interview is challenging. One common question that may leave you stumped if caught off guard is, “What is your ideal work environment?” Another version of this question is, "What type of work environment do you prefer?" Obviously, there is no perfect workplace. But preparing an answer to this question in advance will accomplish two things. First, it will force you to dig deep to identify what’s important to you. Best of all, articulating your vision in a clear and succinct way that aligns with the company’s values will also leave a lasting impression on your future employer.

Your ideal work environment refers to the type of workplace where you will be the most productive and satisfied. Employers ask this question for several reasons. One is to assess cultural fit. They want to know that your desires match what they have to offer. Why? Employees who fit well within an organization are more likely to feel motivated and engaged, resulting in higher productivity. It also gives the hiring manager insight into your personality—something more difficult to glean from a résumé.

Finding an ideal culture match matters just as much to you, the job seeker, as to the employer. So much so that a Glassdoor survey polling over 5,000 respondents from the U.S., U.K., France and Germany found that 73% said they wouldn’t even apply to a company unless its values align with their own. The next time you prepare for an interview, follow these steps to respond to the question, “What is your ideal work environment?”

Reflect on past experiences

The first step is to define your preferences. Look back on past work experiences to identify the environments in which you thrived. Remember, it’s about more than just describing the physical location. Think about factors such as:

• Flexibility
• Work-life balance
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• Collaboration vs. working independently
• Structured vs. ambiguous environments

Then, make a list and prioritize these attributes. Are there any elements on which you could be flexible? Also, note any characteristics you consider deal breakers.

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Some employers ask about your ideal work environment to ensure you researched the company. Check the job description for keywords like creative, fast-paced or team-oriented. To learn more about the company culture , review the corporate website. Pay special attention to the mission statement and careers section. Also, look at social media channels to get a glimpse into the organization’s priorities. Another idea is to create a Google alert to stay on top of breaking news or announcements. Finally, talk to current employees. By scheduling informational interviews, you can get an insider perspective on what it’s like to work there.

Prepare your response

In a job interview, you always want to appear energetic and enthusiastic about the role. So, when you respond, frame your answer in a positive light. For example, instead of describing how you hated working for your micromanaging boss who tracked your every move, focus on the fact that you’re a self-starter who thrives on flexibility. Highlight what is most important to you and connect it to the organization you’re interviewing with. To make your response more compelling, use real-life examples. By using a storytelling approach, your interview will be engaging and memorable.

Example answers

Here are a few sample responses to this increasingly common interview question:

You enjoy a team-based environment

My ideal work environment is one where I can express my creativity while using my problem-solving skills to overcome obstacles. I enjoy collaborating with team members on challenging assignments. Working in a rewarding environment is also important to me. That’s why I was impressed that you recently created a program to recognize employees who go above and beyond. I find that I’m most productive and motivated when I’m part of a team that celebrates each other’s wins.

You prefer a balance between group and independent projects

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You thrive in a remote setting

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Job interviews are a two-way conversation. If you determine that the company culture and your expectations don’t align, that’s okay. The role may not be a good fit. However, if there is overlap, you can decide whether some preferences are worth compromising. Most importantly, be authentic. It will make you a more attractive candidate and increase the likelihood of finding a job opportunity that is the best fit for you.

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