# repeated measures ANOVA with MATLAB

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Commented: Adam Danz on 29 Jun 2020
I have a group of n=30 individuals with no repetition. Each individual has three independent measurements (x,y,z) which are measured for each individual at the same time. I wanted to know if x,y, and z are statistically different, then if x and y, y and z, and x and z are statistically different (so, this is a within group comparison).
I thought I could use repated measures ANOVA to do this and tried the code below:
load mydata; %%mydata contains a column of IDs and three other columns associated with x, y, and z.
t = table(mydata.IDs,mydata.x,mydata.y,mydata.z,'VariableNames',{'IDs','d1','d2','d3'});
Meas = dataset([1 2 3]','VarNames',{'Measurements'})
rm = fitrm(t,'d1-d3 ~ IDs-1','WithinDesign',Meas)
ranovatbl = ranova(rm)
ranovatbl =
SumSq DF MeanSq F pValue pValueGG pValueHF pValueLB
______ __ ______ ________ ______ ________ ________ ________
IDs:Measurements 8.6735 48 0.1807 0.037392 1 0.99997 0.99997 0.99997
Error(Measurements) 9.665 2 4.8325
Can someone please first tell me if I did this right and second how to interpret those p-values?
Thank you!
##### 2 CommentsShowHide 1 older comment
Charlotte Ashton on 17 Dec 2019
Hi,
I'm just wondering if you or anyone could tell me why you have specified 'IDs-1'in variable rm, instead of just 'IDs' please?

Adam Danz on 4 Dec 2018
Edited: Adam Danz on 4 Dec 2018
There are 3 parts to the question.
Is repeated measures ANOVA appropriate for these questions?
Yes. The null hypothesis for a repeated measures ANOVA is that the 3+ variables measured on the same subjects have the same means in the population. Is the repeated measures ANOVA appropriate given then data? That I can't say. Repeated measures ANOVA follows these assumptions:
• independent, identically distributed observations
• The test variables approximately follow a normal distribution
Did I use ranova() correctly?
This example (<-link) demonstrates how to prepare tabular data for ranova() and you have followed those steps.
How to interpret the p values.
Your table is provided below, formatted.
pValue: the p-value for the corresponding F-statistic ("F"). The p-value is the probability of obtaining an effect at least as strong as the one in your sample data if the null hypothesis is true. Remember that the null hypothesis is that there is no differences in the means of the actual population. So, the lower the p values, the less probably that your data support the null hypothesis. Usually if p < 0.05 (or 0.01 or 0.001 in some brances of science) we reject the null hypothesis.
Going back to those assumptions I mentioned earlier, the regular p-value is accurate only if all response variables have the same variance and each pair of response variables shares a common correlation. This is dicussed in detail here (<-link). You can use the mauchy method to test of this. When this "compound symmetry" is not satisfied, you can use one of the adjusted p-values.
pValueGG pValueHF pValueLB : Respectively, these are the Greenhouse-Geisser adjustment, Huynh-Feldt adjustment, and the Lower bound adjustment.
ranovatbl =
SumSq DF MeanSq F pValue pValueGG pValueHF pValueLB
______ __ ______ ________ ______ ________ ________ ________
IDs:Measurements 8.6735 48 0.1807 0.037392 1 0.99997 0.99997 0.99997
Error(Measurements) 9.665 2 4.8325
In the example above, p is way above 0.05 so we would not reject the null hypothesis.
Always plot out your data, look at it, and confirm that your statistical test results make sense. If the tests suggest that the means aren't different and your data clearly show two very different distributions, you know you did something wrong. This sanity-check can be used to catch mistakes. However, this sanity check does not necessarily confirm that you performed the statistics correctly or that your data are appropriate for the statistics you chose.
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Adam Danz on 29 Jun 2020
Look into the Friedman test.