This might be a question of within-group comparison in different situations but I'm not quite sure. The data are time lags measured by hour. I sampled lags from the same group of students. Each student participated in a various formats of discussion including debates, role play, Q&A and Web quest. I want to find out if response lags are different in different types of discussions. The distribution of lags for any given discussion are non-normal. Should I use a non-parametric test? What methods are there to test for group differences in dependent, non-normal response data?
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Why go nonparametric? – Dave Mar 31 '21 at 20:18
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Thanks, Dave. Because it's not normal distribution, and it's not independent identical distribution. – AritaL Mar 31 '21 at 20:29
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Log transform the response because it's far easier to interpret differences in "lags" as ratios. For instance, a regression coefficient of about 0.05 has $\exp(0.05) \approx 1.05$ which is interpreted as 5% longer lag time (compared to reference group). Also, I bet the data are going to be much normaler if you log transform. Fit a repeated measures ANOVA with a random intercept for each person.
Non-normal responses do not require non-parametric tests. Not all parametric tests are for normally distributed responses. And many of the tests that are exact in small samples for normally distributed responses, like the t-test and linear regression, provide powerful tests and unbiased estimates when the sample size is moderately large.

AdamO
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This is so helpful! Thanks so much for explaining it too! I'll go try it out! :-) – AritaL Mar 31 '21 at 21:48
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A follow-up question if you don't mind! I tried Shapiro-Wilk test and the p value is .004 for one of the discussion types. Forgot to mention sample sizes are different for the four groups. Can I still use repeated measures ANOVA? Thanks! – AritaL Mar 31 '21 at 22:25
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Related: https://stats.stackexchange.com/questions/2492/is-normality-testing-essentially-useless. repeated measures ANOVA does not depend on balanced classes. Ironically, if you *had* balanced classes, you wouldn't need methods for dependent data. – AdamO Apr 01 '21 at 15:23