# Check normality assumption for a 3 by 2 within-between (mixed) ANOVA

I'm conducting a 3 (within) x 2 (between) mixed ANOVA in SPSS and would like to assess whether my data satisfy normality assumptions. Do to so, I have saved the unstandardized residuals from the repeated measures module after specifying my model following the syntax below. This gives me 3 sets of residuals - one for each participant for each level of my within-group factor.

GLM @level1 @level2 @level3 BY group
/WSFACTOR=timepoint 3 Polynomial
/METHOD=SSTYPE(3)
/SAVE=RESID
/CRITERIA=ALPHA(.05)
/WSDESIGN=timepoint
/DESIGN=group.

Question

To assess normality, do I generate QQ-plots and/or perform Shapiro-Wilk tests for each within-subject factor (1 within-factor with 3 levels = 3 plots & 3 tests)

OR

Do I do this for every cell mean (1 within-factor with 3 levels and 2 groups = 3 x 2 groups = 6 plots and 6 tests)?

This is going to be a very pedantic reply, but I hope it helps. Basically, you are saying that your data are realised from the following model:

Y = global mean + group_mean_1 + group_mean_2 + error

If you are using this model, the big assumptions are about random errors (not data per se). You need to assume that errors are: - independent - mean of 0 - constant variance (given by say, sigma^2) You don't have to assume errors are Normal, although many people do.

You might make assumptions about the conditional distribution of Y - that it is Normal(global_mean + group_mean_1 + group_mean_2, sigma^2), but it is important to note that this is a conditional distribution. It won't have a Normal distribution.

The errors are some fictional entity, but you hope that the residuals are very similar to the errors, and you will carry out checks on the residuals. Constant variance (at all values of group_mean_1 and group_mean_2) is the most important check.

If you want to do a Normality check, you can look at a qq plot of all the residuals in one go. IMHO, any formal testing of Normality is overkill. The residuals can't actually be Normal, even if the errors are Normal.

Allowing for my pedantry over residuals =! errors, this webpage gives you a few practical ideas about model assumptions that may be helpful:

https://www.theanalysisfactor.com/assumptions-of-linear-models/

• Thank you for clarifying this for me. I appreciate your pedantry. I just want to ensure that my p values and F statistics are valid, which I understand rely on normally distributed residuals. That I should perform a QQ plot of all the residuals in one go answers my question.
– grug
Commented Oct 21, 2019 at 2:21