# Non-normal distribution in anova - a big problem or not

Please don't close my question, it really is not a duplicate, no other answer on this forum is relevant to my case. Also, I have been advised that if I submit the spss output of my non-normally distributed model, someone will have a look at it and advise me. I have been waiting for some help for weeks. It's very important, please.

My data show deviation from normality of the residuals (as evidenced by the Shapiro results) but they say that anova is strong enough to 'survive' that. My dependent variable (emotion mean) is a mean score done from two ordinal variables. I wonder if this is what causes the normality problem. I have two IVs (gender and relationship type).

Could someone have a peek at my results and advise me if I can ignore it (the non-normal distribution) and still do my two-way anova, please?

The following are the results of the normality testing for residuals:

– DevD
Commented Aug 1, 2022 at 18:53
• @DevD I did read that. It is a very different anova. I need help with the residuals of my particular DV and anova. I was told to post the output. Commented Aug 1, 2022 at 19:02
• What is the model you have fit? Is there just one predictor?
– mkt
Commented Aug 1, 2022 at 19:11
• @mkt That's not a predictor. That's normality results for the residuals for my DV. My previous question had the normality distribution of the DV for both of my IVs but it got closed. And it was ignored before that and people asked me for the results for the residuals. So I posted the residuals this time. They said that is what's important. But if you would like to see it, it is here stats.stackexchange.com/questions/582159/… Commented Aug 1, 2022 at 19:35

The QQ plot looks "reasonably" Normal given that there are implicit constraints on the values the residuals can take because:

• The independent variables are gender (two levels) and relationship type (also two levels).
• The dependent variable is meanEmotion defined as the average of two Likert-type ordinal variables.

That's why there are 107 observations (the degrees of freedom of the Shapiro Wilk test) but we see only about 20 distinct residuals in the QQ plot: there is overplotting (points plotted on top of each other).

Since you know — even before doing the analysis — that the Normal distribution is only an approximation in your case, you shouldn't over-interpret the result of the Normality tests. Your data would be better modeled with an ordinal logistic regression instead of the ANOVA, which assumes (among other things) that the response is continuous. @EdM points to the UCLA Stats tutorials which have a section about ordinal logistic regression.

This answer by @ChristianHennig discusses some pitfalls of checking model assumptions: Testing Model Assumptions in R.

• It's more of a misunderstanding rather than an error: given the nature of your data, the Normal distribution can only be an approximation. But then you perform not one but two tests for Normality. Commented Aug 1, 2022 at 20:19
• A Normal random variable can take any value of the real line. Your response can take only a limited set of distinct values. If the original variables have levels 1, 2, 3, 4, 5, the average of two of those can be 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5. And there are 4 possible combinations of the IVs. Hence the limited possibilities for the residual values. This looks suspicious to the Kolmogorov Smirnov test, hence the low p-value. But it's to be expected given the nature of your data! That's why I say not to read too much into that p-value. Commented Aug 1, 2022 at 20:31
• +1 in particular for the suggestion to try ordinal logistic regression. The OP might want to look at this UCLA OARC web page for links to examples of how to do it.
– EdM
Commented Aug 1, 2022 at 20:51
• @lisaarthur ordinal logistic regression can be very useful in general, as it only requires an ordered set of outcome values without any assumptions about spacing. Frank Harrell recommends this as a useful approach even when the outcome values are continuous. See Chapters 13-15 of his Regression Modeling Strategies.
– EdM
Commented Oct 19, 2023 at 14:03
• I didn't consider this 1 year ago but now I would have questions / concerns about the averaging of two ordinal variables. I would guess this step could be reasonable enough if the individual differences between the two emotional scales are small: eg. "average" 5 and 6 to give 5.5 on a derived scale of 10. But to average 0 and 10 and call this emotional state of 5 seems less reasonable. That would be particularly the case if the differences between the two measurements varies by the IV variables (gender and/or relationship type). This could be a complementary analysis to look into. Commented Oct 19, 2023 at 14:45