I am running a 2x2 mixed method ANOVA (with one between-group factor and one within-group factor - time 1 and time 2). I want to check whether the assumption of normality of residuals is met. When I run the ANOVA in SPSS I get two sets of residuals if I save the unstandardised residuals - one for time 1 and one for time 2. I was wondering why there are two sets of residuals, and which set of residuals should be normally distributed (or is it both?).

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    $\begingroup$ I am not an SPSS expert, but it would almost surely help if you posted your code. It sounds like you are doing something wrong, but I am not sure. $\endgroup$ – Peter Flom Jan 16 '13 at 20:44

I'd suggest that in that case you'd want to look at the standardised residuals for normality rather than the raw residuals. We'd be checking for the normality of the error term, not the raw residuals.

Since you could rewrite the two way ANOVA as a a multi-variable linear regression. In that case the expected variance in our residuals is...

$\newcommand{\Var}{\mathrm{Var}}\boldsymbol e = (\boldsymbol I-\boldsymbol H)y$

Where $\boldsymbol H = \boldsymbol X (\boldsymbol X^T \boldsymbol X) \boldsymbol X^T$

And $(\boldsymbol I-\boldsymbol H)(\boldsymbol I-\boldsymbol H)^T=(\boldsymbol I-\boldsymbol H)$

So that (probably with some abuse of notation):

$\Var(\boldsymbol e) = \Var((\boldsymbol I-\boldsymbol H)\boldsymbol y) = (\boldsymbol I-\boldsymbol H)\Var(\boldsymbol y) = (\boldsymbol I-\boldsymbol H)\sigma^2$

That means we can't expect the $i$ th residual to be standard normally distributed unless we divide through by the square root of the $i$,$i$ th entry of the last expression. That's what the standardized residuals are intended to provide.


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