# The Three Assumptions of the Repeated Measures ANOVA

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.

However, before we perform a repeated measures ANOVA we must make sure the following assumptions are met:

1. Independence: Each of the observations should be independent.

2. Normality: The distribution of the response variable is normally distributed.

3. Sphericity: The variances of the differences between all combinations of related groups must be equal.

If one or more of these assumptions are violated, then the results of the repeated measures ANOVA may be unreliable.

In this article we provide an explanation for each assumption, how to determine if the assumption is met, and what to do if the assumption is violated.

## Assumption 1: Independence

A repeated measures ANOVA assumes that each observation in your dataset is independent of every other observation.

### How to Determine if this Assumption is Met

The easiest way to check this assumption is to verify that each individual in the dataset was randomly sampled from the population using a random sampling method.

If a random sampling method was used, it’s safe to assume that each observation is independent.

### What to Do if this Assumption is Violated

If this assumption is violated, then this is a serious issue because the values of each individual may be related to each other in some way.

Often the only remedy in this scenario is to recruit individuals for a new study using a random sampling method.

## Assumption 2: Normality

A repeated measures ANOVA assumes that the distribution of the response variable is normally distributed.

### How to Determine if this Assumption is Met

There are two ways to check if this assumption is met:

1. Create a Histogram or Q-Q Plot

You can visually check if the distribution of the response variable is roughly normally distributed by creating a histogram or Q-Q plot.

If you create a histogram, simply check that the distribution of the response variable roughly follows a “bell” shape. If it does, you can often assume that the normality assumption is met:

If you create a Q-Q plot, check if the data points fall along a straight diagonal line. If they do, then you can typically assume that the normality assumption is met:

2. Perform a Formal Statistical Test

You can also conduct a Shapiro-Wilk Test for normality. If the p-value of the test is less than .05, this suggests that the data is not normally distributed.

However, be aware that when working with extremely large sample sizes then statistical tests like the Shapiro-Wilk test will almost always tell you that your data is non-normal.

For this reason, it’s often best to inspect your data visually using graphs like histograms and Q-Q plots. By simply looking at the graphs, you can get a pretty good idea of whether or not the data is normally distributed.

### What to Do if this Assumption is Violated

In general, a repeated measures ANOVA is considered to be fairly robust against violations of the normality assumption as long as the sample sizes are sufficiently large.

If the normality assumption is severely violated, you have two choices:

1. Transform the response values of your data so that the distributions are more normally distributed.

2. Perform an equivalent non-parametric test such as the Friedman Test that doesn’t require the assumption of normality.

## Assumption 3: Sphericity

A repeated measures ANOVA assumes sphericity – that variances of the differences between all combinations of related groups must be equal.

If this assumption is violated, then the F-ratio becomes inflated and the results of the repeated measures ANOVA become unreliable.

### How to Determine if this Assumption is Met

To test if this assumption is met, we can perform Mauchly’s Test of Sphericity.

This test uses the following null and alternative hypothesis:

• H0: The variances of the differences are equal
• HA: The variances of the differences are not equal

If the p-value of the test is less than some significance level (e.g. α = .05) then we reject the null hypothesis and conclude that the variances of the differences are not equal.

Otherwise, if the p-value is not less than some significance level (e.g. α = .05) then we fail to reject the null hypothesis and conclude that the assumption of sphericity is met.

Depending on which statistical software you use, the results of this test will look something like this:

Since the p-value is not less than .05, we would fail to reject the null hypothesis and conclude that the assumption of sphericity is met in this particular example.

### What to Do if this Assumption is Violated

If we reject the null hypothesis of Mauchly’s test of sphericity, then we typically apply a correction to the degrees of freedom used to calculate the F-value in the repeated measures ANOVA table.

There are three corrections we can apply:

• Huynh-Feldt (least conservative)
• Greenhouse–Geisser
• Lower-bound (most conservative)

Each of these corrections tend to increase the p-values in the output table of the repeated measures ANOVA to account for the fact that the assumption of sphericity is violated.

We can then use these p-values to determine if we should reject or fail to reject the null hypothesis of the repeated measures ANOVA.