# Residuals of a 2x2 between-subjects factorial ANOVA are not normally distributed

Dependent variable: Emotional impact – measured by two ordinal variables (questions) on a Likert scale [1 (Least) to 10 (Most)] A composite mean score of the two variables was created named meanEMOTION which became the DV.

IV1: Gender (Male, Female)

IV2: Relationship type (Stranger, In a relationship)

I would like to conduct a 2x2 between-subjects ANOVA but my model is not normally distributed. The Shapiro-Wilk test for the residuals of my DV was significant for all of my factors. I worry this may affect my ANOVA results.

There are some posts related to this on this site but the answers are inconsistent and there does not seem to be an agreement. In general, some people on the internet suggest using different non-parametric tests while others suggest not doing anything. I have read in the Andy Field (Discovering statistics using IBM SPSS statistics) book that there isn't a non-parametric equivalent to a 2x2 factorial ANOVA. And some say that factorial ANOVA may be robust enough to deal with non-normally distributed residuals. Perhaps I do not have to do anything with it?

I'm attaching the outputs from the test of normality and residuals:

[Redacted by moderator upon request]

• The most useful plots seem to be missing: a normal q-q plot of all residuals in the model, and a plot of residuals versus predicted for the entire model. You show separate plots for combinations of predictors, but those don't really matter so much as the overall plots.
– EdM
Commented Sep 15, 2022 at 21:07

The nonparametric (actually semiparametric) alternative is the proportional odds ordinal logistic model, which handles Wilcoxon and Kruskal-Wallis tests as special cases but allows for interaction terms. An overview and resources may be found here.

• I've been trying to understand ordinal regression for another question for the past few weeks. I even posted a question on it here but did not help. There is no clear explanation anywhere and especially not how to deal with the interaction. There is no way I will ever be able to interpret an ordinal regression let alone do further calculations with it so I will do anything possible to avoid it. Thank you for the link but I'd need a simple explanation in terms of SPSS (the way Andy Field does it; unfortunately he does not give an example of ordinal regression in his book).
– user361794
Commented Jul 21, 2022 at 6:08
• I have amended my question and I would again need some help, please. I worry that because there are already some answers, people may think the query has already been answered and thus they might not read it and answer it? Thank you.
– user361794
Commented Jul 22, 2022 at 3:44

Tests of normality are not very useful, because they are heavily dependent on sample size and because linear models are quite robust to moderate deviations from normality. So with a large sample, small deviations from normality will be detected even though it doesn't really affect the model results.

A better solution is to plot your model output. QQplots, histograms of residuals and plots of residuals vs. fitted points are all common and useful ways to evaluate whether the assumptions of the model are sufficiently met for you to rely on the output.

If you show some of these plots, we may be able to offer more advice about whether you need be concerned.

Also, non-parametric tests are less powerful than parametric ones and provide less informative output; I would not turn to them unless there are strong reasons to. If the deviation from normality is large enough to be a concern, I'd consider a transformation of the data (followed by a parametric model) before considering a nonparametric test.

• Nonparametric tests, depending on the test, tend to be slightly less powerful when parametric assumptions are met but can be considerably more power when assumptions are violated.
– Dave
Commented Jul 16, 2022 at 15:47
• @mkt Thank you so much for your response. I'm amending my question above and adding tables and plots of normality for my variables. Let me know what you think.
– user361794
Commented Jul 21, 2022 at 4:13