# Normality test for repeated measures data

I've measured the motor function of the same subjects (n=6) over 7 different time points. I would like to know whether the mean motor function varies significantly with time (day 1 versus day 2 etc) and I know that a 1 way repeated measures anova will allow me to make this assessment. However, before I perform the repeated measures 1 way anova I'd like to assess whether my data are normally distributed. Which test would be most applicable for assessing whether repeated measures data are normally distributed?

Thank you

• You can't tell that your data are normally distributed. But in any case, (a) what exactly assumed to be normal for this analysis, (b) where is that assumption used, and (c) why would a test (per your title) be the best way of deciding whether your data were suitable rather than some diagnostic check say? – Glen_b -Reinstate Monica Sep 17 '14 at 1:43
• Thank you for your replies. What do you mean by diagnostic check? Do you just mean simply looking at the frequency distribution histogram and assessing skewness? – Emily Oct 14 '14 at 14:30
• I added the goodness-of-fit tag which provides you more information. – Horst Grünbusch Feb 6 '15 at 13:19

Assuming that your measure of the motor function is continuous, you need to check not only multivariate normality but also sphericity and independence.

1. You can test multivariate normality with the various tests in the mvnormtest package if you use R. Otherwise you can check whether residuals based on your repeated-measures ANOVA model is normally distributed.
2. Sphericity assumes, roughly speaking, that the difference between two adjacent measures are constant (e.g., $Y_{Day_2} - Y_{Day_1} \approx Y_{Day_3} - Y_{Day_2} \approx ... \approx Y_{Day_7} - Y_{Day_6}$, in your example). You can use Mauchly's test to test sphericity.
3. Independence assumes that values obtained from one subjects are not in any way related to the values obtained from the other subjects.

• Ad 2: Sphericity is not necessary, except for textbook level homoskedastic ANOVA. Ad 3: This is typically a matter of design, nothing to test. We may think about dependence, if the subjects originate e.g. from the same litter. – Horst Grünbusch Feb 6 '15 at 14:10

In order to test whether the distribution of a variable is normal you would generally test whether the following characteristics are present:

1. Skewness - does the distribution have a long tail to the left or right (often called outliers)
2. Kurtosis - is the peak of the distribution too high (too peaked) or too low (too flat)

Both of these can be tested for in R using the moments package. However, these tests are affected by the sample size (the larger the sample the more likely the test is to highlight deviations from the ideal), therefore they are only really reliable when comparing different distributions with similar numbers of participants. However, as suggested in this post here, if you have a large enough sample size just have a look at a histogram. This post also mentioned that ANOVA is quite robust to non-normal distributions, so if the histogram looks OK then that's all you'll need to know.

You are right that it is generally good habit to check for the underlying assumptions. But besides the reasoning about goodness-of-fit tests that you can explore by browsing the questions marked with the goodness-of-fit tag, in your particular case the normality assumption is not so critical, because --as ChrisP pointed out-- you would use one-way repeated measures ANOVA, favourably with heteroskedasticity assumption, which is quite robust against departures from the normality assumption.

Research on this topic is still pending, but expecially skewness is a criterion you should focus on, as low skewness will prove the distribution of the real data's ANOVA statistic to be close to the normal ANOVA statistic. Kurtosis doesn't have that much impact any more.

However, instead of testing the point hypothesis "skewness = 0" I would suggest to do compute confidence intervals for the skewness. If this interval reaches to quite extreme skewnesses, your test statistic may exceed it's $\alpha$ error rate. If not, you can be keen.

This adresses the fact (at which ChrisP pointed) that small sample sizes would spuriously suggest your data's skewness might be 0, as the point hypothesis has not been rejected. This would be quite nasty since if you already have a small sample size, then even a small skewness is disturbing.

To find the level of skewness you can tolerate, you may simulate your dataset with various skewed distributions under your $H_0$-hypothesis that the motor function is constant and see how good it meets the $\alpha$-level.