# Repeated measures ANOVA or Friedman test?

I have a study with 35 subjects who received hypercoagulability testing at 4 different times (before surgery, after, at 1 week, and at 1 month). Each subject has 9 different test of hyper coagulability at the 4 time points. I want to compare each of these 9 tests over time (i.e.: clot time_preop vs. clot time_postop vs. clot time_1wk vs clot time_1mo) to see if they change.

One issue I am having is how to decide if my data is normal. How do I assess for normality in the data? I have explored the data in SPSS and see that although most are normal, some are not. If one of the measurements in a variable (i.e.: clot time_1wk) is not normal (kurtosis=10 and Shapiro-Wilk <0.01) but the others are, should I use Friedman non-parametric instead? Or should I be assessing normality another way?

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Another relevant question: stats.stackexchange.com/questions/2492/… –  Gaël Laurans Sep 19 at 8:34

Those tests didn't tell you the data were normal, just that they were generally not significantly deviating from normal. Those are two different things. There is no test for normal.

One way I help my students come to a conclusion about satisfying normality is getting them to calculate the summary statistics of the data in question and then look at quantile-quantile plots of the sample data and simulations of the sample data. So, you could get the standard deviation you found and the n of the sample and generate several sets of simulated data and look at what typical normal data should look like and compare it what was found. That helps with the typical advice to really just look at the data and see if it is normal.

Directly addressing what you actually did... I'm guessing you examined each condition across subject? That's OK but consider the multiple testing issue. Some of those data were bound to come out looking non-normal, it's inevitable. Furthermore, the data that you should be testing the normality of is the residuals of the ANOVA. You should be plotting and examining that in order to see if you met the assumptions... not each condition of the experiment. Besides getting around the multiple testing issue it's the actual assumption of the ANOVA.

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I was under the assumption that I needed to assess normality BEFORE I went on to do the repeated measures ANOVA or Friedman. If I examine the normality of the residuals of the ANOVA and one of them is non-normal (i.e.: clot time at 1 week) should I examine the clot time (pre vs. post vs. 1 wk vs. 1 mo) using nonparametric statistics? –  Chad Thorson May 12 '12 at 15:01
The residuals don't work that way. It's one set of residuals from the model. –  John May 12 '12 at 15:26
@John In my answer when I mentioned contrasts i intended to point to the multiplicity issue +1 for you. I also like the simple way you made the point about testing for normality and the statement "my data are normal" or my "data are not normal". I do think that you can test normality on the original sample separate from any linear model assumptions and not just test the model residuals. This could indicate whether or not there might be problems with the normal assumption. Perhaps not in a repeated measures situation due to the within patient correlation structure to the data. –  Michael Chernick May 12 '12 at 16:35
@Chad Thorson There are no rules about when to check assumptions. The important thing is that if you have reason to think assumptions might not be valis you should do the diagnostic checks that you have available. Rules about specifying things in advance of looking at the data are generally made to prevent bias in the results. In a clinical trial if you specify how you are going to check your model assumptions and proceed based on these check in your protocol you should be okay. –  Michael Chernick May 12 '12 at 16:48
@Chad Thorson. Out of curiosity are your tests looking to identify hemophilia in the patients? –  Michael Chernick May 12 '12 at 16:49

I would never ask "Are my data normal?" Taken literally a safe answwer without looking at the data is to say "no!" This goes back to the wise and often quoted statement by George Box "All models are wrong, but some are useful. Real data are complex. Sometimes a normal iid model is a convenient approximation to describe the data but unless your data is simulated by selection from a normal it will not be a sample from a normal population. Also random number generation on the computer is not purely random and so are called "pseudo-random". To get to the specifics of your problem it is clear that you want to assess the data to see if the behavior of the sample is close enough to that from a normal distribution to make that model useful for your inference problem. Goodness of fit tests have a problem in large sample sizes because you can reject slight departures from normality that are so slight that the normal model might still be very useful. Since your sample size is small, Shapiro-Wilk is detecting a large difference which is also indicated by th kurtosis estimate = 10 and probably many other univariate tests or exploratory measures would indicate problems with the normal model for that one variable. There are tests for multivariate normality but that is not what I would suggest here. My feeling is that if all the other variables appear to be close to normal doing the parametric ANOVA would be okay. What I think is the best approach is to leave this one problem variable out and assess the eqaulity of means just on the others. After doing that along with contrasts to find which means are different if the F tests rejects equality you could look at comparisons between the "non-normal" variable and comparing it with other variables that you think might be interesting to look at and potentially might have different means. For this do a Wilcoxon. My general advice is "when in doubt do both tests to see if the normality assumptions affect your p-values very much."

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Using the repeated-measures anova and non-parametric Friedman does not affect the p value much. For 7 of the 9 variables assessed over time, there is a significant (all p<0.01 and most p<0.001) change, both statistically and clinically. I think the issue may lie in the nature of the hyper coagulability tests. Most values are "normal" when comparing them to the reference ranges. Some tests have very narrow distribution (i.e: "angle of deflection" can range from 0-100 and ref range is 70-79) so of course the results will be kurtotic –  Chad Thorson May 12 '12 at 15:08
@Chad Thorson It may be in your case that the p-values don't change much but that will not always be the case. I suspected it would be in your case since you said that practically all the variable fit well to a normal. There is a difference between rejecting normality and thus suspecting that the assumptions for the parametric F tests do not hold versus rejecting normality and thus expect that the results will be sensitive to it. Comparing the p-values for Friedman's test to the p-value for the parametric repeated measures F test is a way to do sensitivity analysis. –  Michael Chernick May 12 '12 at 16:30
For a publication it is wise to spell out what you tried and why you did it and to report both results. Also since the assumptions appear to be violated I would prefer the nonparametric result. –  Michael Chernick May 12 '12 at 16:30
Thanks for all of your comments. I feel more comfortable with the strength of my conclusions based on the data. –  Chad Thorson May 12 '12 at 16:49