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I have a set of 9 different factor levels from my independent variable to be compared against each other. Here are the results of the different assumption tests in R. I'm just going through my methodology in the hope that someone can guide me to the right conclusion.

  • The Shapiro-Wilk test is significant and shows that the my dependent variable is not normally distributed. I read somewhere that the assumption of normality doesn't always have to be met if the size of the dataset is substantial as it will almost always deviate from normality when it's the case.
  • Mauchly's test for sphericity is significant, therefore the assumption of sphericity has been violated. My Greenhouse-Geiser correction confirmed it, but it's p-value is significant, which indicates that the levels of my independent variable significantly affects the dependent variable.

Where do I go from here?

The example I'm basing my methodology on in a textbook goes on to use these methods on the same dataset every time (though that dataset is different than mine)

  • Repeated measures ANOVA
  • A multilevel approach
  • Robust test

Based on what I know, I think I have to go for a robust method given the assumptions above, but I have no idea if this is the right call.

bolleke
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  • See https://stats.stackexchange.com/questions/176869/what-is-assuming-sphericity-in-anova-function. I'd suggest you switch to a multilelvel model (MLM), using R's `lmer()`. With regards to normality, what you are worried about in MLM is the normality of the residuals. You can easily look at these after running your `lmer` model. See https://www.ssc.wisc.edu/sscc/pubs/MM/MM_DiagInfer.html – Erik Ruzek Dec 29 '20 at 20:58
  • Thank you very much for the insights, I do have an additional question if you don't mind. Basically, when is it a good idea to use a robust method? And should I use the F-ratio of the GG correction? I stumbled upon [this article](https://cran.r-project.org/web/packages/WRS2/vignettes/WRS2.pdf) by Wilcox in which he writes that when normality and homoscedasticity are not met, classic inferential methods based on means (e.g., the ANOVA, F-test) are not the way to go, and you should switch to a robust method. – bolleke Dec 30 '20 at 08:48
  • In some fields, such as econometrics, analysts almost uniformly employ robust methods when estimating a regression because it protects against some of the potential violations of linear regression (OLS, multilevel models, etc.), including those you mention. Others don't like it because it is a bit of black box. With good sleuthing, you can often figure out why your data violates these assumptions. I am not familiar w/ ANOVA/ANCOVA approaches and would suggest that you instead use multilevel modeling for your data, which is much more flexible. – Erik Ruzek Dec 30 '20 at 19:22

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