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I got a dataset of 60 people from a between-subject approach that I was trying to check for normality. (Since I need information about normality for ANOVA post-hoc tests.) They were split in three different groups. So, let's say I take the variable "visualDistance" (just as an example, since it really matters there), the descriptives of that variable say the following: enter image description here

You can see, that for the group 2 and 3 the distribution seems to be clearly non-normal. However, when I do an ANOVA, it returns the following value: enter image description here

And a One-Way ANOVA (which theoretically should be the same) yields: enter image description here

I'm not completely solid with ANOVAs, but I thought the normality assumption has to apply to all test groups.

  • Why is it then that the assumption checks in R for ANOVAs result in a confirmed assumption for normality, when clearly two out of three groups don't fit the assumption?
  • Which results should I listen to? Can I savely use post-hoc tests based on normality, with such results?
  • And how come that the results slightly differ between ANOVA and One-Way ANOVA? (in this case it seems minor, but in another case one results in a p-value of 0.472, while the other results in a p-value of 0.466)
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    $\begingroup$ Typically normality is tested using the residuals when you test it within a formula like ANOVA or regression. Check out Kim (2013) doi.org/10.5395/rde.2013.38.1.52. The article discusses the alternative thresholds you can use for ANOVA with skew and kurtosis (minding whether you have kurt or excess kurt). It's definitely a good idea to reassess this. Either way, ANOVA is really pretty robust against violations of normality (for which it is equivalently weak against variance violations). $\endgroup$
    – Kat
    Commented May 9, 2022 at 14:28
  • $\begingroup$ Why do you say that groups 2 and 3 are clearly non-normal? $\endgroup$
    – Dave
    Commented May 9, 2022 at 14:28

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There's a marvelous thread here discussing whether normality testing is essentially useless. Don't get hung up on normality testing, particularly within individual groups with small numbers of observations like the 2 that seem to fail the Shapiro-Wilk test based on p-values (evidently due to one with high and one with low within-group skewness).

As @Kat says in a comment, it's best to evaluate the entire set of residuals. Although I'm not sure what software you are using, I suspect that the overall Shapiro-Wilk tests with the p-values of 0.46 are on the entire set of residuals. That suggests you don't have much to worry about. I think that most who frequent this site would prefer visual examination of a normal q-q plot of residuals, comparing their distribution to the ideal normal distribution. You should also examine the other diagnostic plots for regression models (of which ANOVA represents a subset). Those plots can be more helpful for identifying the nature and sources of potential problems.

Remember that it's not the normality of the data within each group that matters for ANOVA: it's the normality of the distribution of estimates of the mean value. The central limit theorem helps with that, and as noted in comments ANOVA can thus be pretty robust to violatons of normality.

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