For my current project I need to compare means of four groups by one-way ANOVA. In order to test whether my data come from normal distribution or not, I have checked each group for normality by Shapiro-Wilk test and now I have four p-values, i.e. one p-value per group. Should I apply Bonferroni correction to these p-values?
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$\begingroup$ A pre-test for normality does invalidate (in terms of type 1 error control) subsequent tests, so I am not sure to what extent this would help. $\endgroup$– BjörnCommented Feb 8, 2017 at 7:49
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$\begingroup$ @Björn But I read in a textbook that testing for normality of the dataset should be done before doing ANOVA. How such a pre-test would invalidate subsequent tests? $\endgroup$– user46212Commented Feb 13, 2017 at 5:16
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2$\begingroup$ When you compare means a bit of non-normality does not really matter and changing your analysis based on the data does weird things to your type 1 and 2 errors. See e.g. Rasch, D., Kubinger, K. D., & Moder, K. (2011). The two-sample t test: pre-testing its assumptions does not pay off. Statistical papers, 52(1), 219-231. There's a bunch of publications like that (plus of course all the people trying to sell their latest test for normality ). $\endgroup$– BjörnCommented Feb 14, 2017 at 6:15
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$\begingroup$ Tests of normality are typicallt based on residuals, that way you avoid the problem by having just one test. For a fourwau anova that is just: for each group, subtract its mean. $\endgroup$– kjetil b halvorsen ♦Commented Dec 23, 2017 at 0:26
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2 Answers
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Bonferroni is used to control false discoveries (Type I errors). Your 4 p-values, if I'm interpreting your question correctly, are from assumption tests, not from tests to demonstrate the significance of your discoveries, and therefore don't call for Bonferroni correction. I doubt that you are trying to demonstrate non-normality, or that you would claim a "discovery" for detecting non-normality. In fact, the goal of assumption tests is typically NONSIGNIFICANCE, not significance. Therefore, there is no reason to apply Bonferroni correction to assumption tests under typical circumstances.
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In Shapiro-Wilk test, the null hypothesis is that the sample come from a normal distribution. You can use your p-value to reject the null hypothesis. If your tests show large departure from normality assumptions, ANOVA might not be an appropriate model.
There's no reason to apply Bonferroni correction here. You either go ahead with ANOVA if you believe the assumptions are met or choose another model if the assumption are not met. The normality test only guide you if you want to perform the the model.
You should check your QQ plot.
answered Feb 27, 2017 at 5:07