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I am currently analysing my quantitative data. My sample size is 157 students who sat for a survey in 2 phases after an intervention. The same students sat for the same survey on positive outcomes on peer feedback (this is just one of the 7 dependent variables). I did RM ANOVA and test on the normality. It shows that there is a non normal distribution and a few with no value at all not even p<.01. Someone suggested that I should do transformation and the result is still non normal distribution. Then I did the non parametric Friedman test but because it is only 2 phases, it automatically used the Wilcoxon test. I understand that the Wilcoxon test will not allow me to test my variables based on my independent variables which are gender and class. I am interested to find out whether under the dependent variable postive outcomes on peer feedback, whether female or male students change in terms of whether they find peer feedback helpful or which class is more favourable towards peer feedback. My question is should I go back to 2 Way RM ANOVA for this even though my test for normality is non normal? since Wilcoxon does not give me the option to use my independent variables. Are there any reference that can support my decision to go back to RM ANOVA?

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  • $\begingroup$ In what sense aren't your data normal? Note also that only the residuals (and the random effect for rmANOVA) need to be normal, not the actual data themselves. $\endgroup$ – gung - Reinstate Monica Jan 9 '13 at 3:32
  • $\begingroup$ When I look at the shapiro wilk result all were less than .00001. i checked on the kurtosis and skewness - all do not fall under the normal range. The histograms looked normal though. $\endgroup$ – Dan Jan 10 '13 at 1:49
  • $\begingroup$ So did you run the Shapiro-Wilk test on the raw Y data, or on the residuals? What were the skew & kurtosis values? $\endgroup$ – gung - Reinstate Monica Jan 10 '13 at 2:05
  • $\begingroup$ I ran the ShapiroWilk test on the raw data. the Kurtosis are between the range of 11.12 (the highest) to -0.15 (the lowest). Skew 0.2 (the highest) and -1.55. When I looked at my histogram, i was surprise that they looked pretty normal except that the kurtosis is really too sharp. I did a transformation, log and squareroot but still the same non normal distribution. Is there any transformation for kurtosis. the skew is pretty ok just the kurtosis. $\endgroup$ – Dan Jan 11 '13 at 21:40
  • $\begingroup$ I don't understand what you mean by saying that the kurtosis or the skew "ranged". You should only be concerned about the normality of the residuals (you can see my answer here: What if residuals are normally distributed, but y is not?), & there should only be 1 skew value & 1 kurtosis value. In reality, even then your residuals can be a little non-normal; w/ N=157 the central limit theorem will give you a little leeway. $\endgroup$ – gung - Reinstate Monica Jan 12 '13 at 5:32
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(Just to summarize the comments so that this question isn't counted as officially 'unanswered'... )

The raw data, $y$, needn't be normally distributed. Primarily, the residuals need to be normally distributed (this is discussed in my answer here). However, even when discussing the residuals, with a big enough sample, they can deviate somewhat from true normality and the Central Limit Theorem will provide you with a little slack. In addition, repeated measures ANOVA typically assumes that the random effects are normally distributed, although I believe it is possible to use other distributions.

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