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I have 18 different groups that I need to compare. I used the raw values from a neuroimaging file to represent synchronization between two brain regions (which ranged from 0 to 1) and Fisher Z transformed them in the hope that I would be able to use parametric statistics (namely an ANOVA to compare the means of these groups.

However, it appears that my fisher z transformed data does not pass the Shapiro-Wilk test. I'm kind of at a loss here, as I know that other non-parametric statistical tests such as the Kruskal-Wallis test exist, I don't think they were made in order to compare Z-scores. I've also read the you shouldn't perform an ANOVA if there is heteroscedasticity, and some of the data forms a slightly skewed distribution (box number 2, 6, the last one). I'm not quite sure what to do now

I've plotted the values using a boxplot (N=50): Boxplot from values gathered from 18 different groups [N=50]

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  • $\begingroup$ Welcome to CV. It's not necessary to conduct a formal hypothesis test of Normality. The Z transform is an approximation to Normality. Why not take a look, with a suitable graphic, at the Z values themselves and decide whether there is evidence that ANOVA might be incorrect? $\endgroup$
    – whuber ♦
    Commented Apr 5, 2023 at 17:39
  • $\begingroup$ Thanks for the response, I've included a box plot up above in the main post to hopefully clarify things, I'm told that heteroscedasticity is an indication to not use the ANOVA. But I've been told (by my advisor of all people) that you're able to go straight to an ANOVA as long as you've fisher transformed the data. Not wanting to go against their wishes, I've been going along with it. But I'm a bit nervous about how a 3rd party may interpret some of these graphs and wonder why I'm using parametric statistics. $\endgroup$
    – ishythefishy
    Commented Apr 6, 2023 at 12:19
  • $\begingroup$ how many subjects do you have per group? Boxplots looks fine, it's obvious there are some effects, I would worry about other possible biases (different regions? scanners? scanner time? preprocessing? data quality? movement? etc.). If you are new to stats, I would try to perform the most standard analysis in the field or follow some authoritative paper, instead of trying to invent my own analysis. $\endgroup$
    – rep_ho
    Commented Apr 6, 2023 at 13:21
  • $\begingroup$ Thanks for the response. Data has been taken from the developing human connectome project, and the scanning time, scanners, and preprocessing have all been accounted for during pre-processing (which the connectome project does itself). They've applied motion correction too. There are 50 subjects per group. The type of analysis I'm using is based on a dissertation from another student. My advisor informed me to use it, and it uses an ANCOVA, with gender as a covariate (gender showed no significant effects). With that in mind, I was told to use an ANOVA after the Fisher Z transformation. $\endgroup$
    – ishythefishy
    Commented Apr 6, 2023 at 15:59

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I suggest you just do it; boxplots look fine, Shapiro-Wilk does not have to be non-significant, the data should just look approximately normal. You can use ANOVA for unequal variances to deal with heteroskedasticity. The difference in variances between groups might not be just a nuisance but also something worth writing a paper about.

Although I think that ANOVA is enough, you can find a version of ANCOVA for unequal variances or just a heteroskedastic/robust linear model, to deal with gender. If you want to go non-parametric, you can use the Kruskal-Wallis test, or a permutation test, neuroimagers love permutation tests. See Winkler et al. 2014 Neuroimage https://www.sciencedirect.com/science/article/pii/S1053811914000913, where they also discuss heteroscedasticity and nuisance variable, so either you can put gender in your model or shuffle within gender. There is also an accompanying (Matlab?) package for it.

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  • $\begingroup$ Thanks so much! $\endgroup$
    – ishythefishy
    Commented Apr 6, 2023 at 16:31

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