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    $\begingroup$ That ANOVA is sensitive to extreme observations is why it is important to keep them. $\endgroup$ Commented Dec 20, 2021 at 4:18
  • $\begingroup$ @Dave but you're supposed to check assumptions of ANOVA before you do them, right? See here: statistics.laerd.com/spss-tutorials/… The fact mine has violated one of these key assumptions... isn't that a problem? My analysis will no longer be valid, right? So I'm not sure what to do. I have to report the main effects, simple effects, etc. But then presumably I'd be all 'but yeah all this is nonsense, because of the outliers, so...' which doesn't sound ideal. $\endgroup$ Commented Dec 20, 2021 at 4:35
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    $\begingroup$ Then you change your modeling approach to reflect the reality of the data, rather than changing your data to reflect the assumptions of a mathematical procedure. $\endgroup$ Commented Dec 20, 2021 at 4:38
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    $\begingroup$ With more information about your proposed experimental design, the type of data, and your objectives, it might be easier to give more relevant advice . [(i) "$2\times 2$ repeated measures" could mean several different things. (ii) How do the outliers arise? What do they mean? What do you most want to know from your data?] An ANOVA on overall ranks would not ensure exact normality of residuals, but it would lessen the effect of outliers without censoring them. A regression approach as suggested by @Dave might work. It might be possible to find an appropriate metric for a permutation test, Etc. $\endgroup$ Commented Dec 20, 2021 at 6:08
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    $\begingroup$ @BruceET Hmm, well, it's a 2 x 2 Stroop task measuring reaction time as the dependent variable, with posture and congruence as dependent variables, each with two levels. I'm running it in SPSS, a standard within subjects repeated measures ANOVA. When I generate boxplots to assess outliers of reaction time, I get like 4 or 5 out of around 200. I've read that ANOVAs are robust against normality violations, but not outliers. I want to know whether there are effects of posture and congruence on reaction time, mainly. But will I be able to determine any main effects with these outliers? $\endgroup$ Commented Dec 21, 2021 at 2:42