ancova
I did ANCOVA with Depended: Posttest/ Fixed: Group/ Covariate: pretest
Levene's test is statistically significant: .002
Is the assumption of homogeneity of variance violated? If yes , what can i do about it?
I am asking since I had done it with Univariate analysis: Postest, Group. pretest and custom model with interaction between group and pretest and interaction Grupa*test1 was not significant.
Your model,
response ~ fixed.A*fixed.B + (1 + cov | fixed.A)
translates to
$$ \begin{split} y_{ijk} & \sim \textrm{N}(\mu_{ijk},\sigma^2_r) \\ \mu_{ijk} & = \beta_{ij} + b_{i1} + b_{i2} x_{ijk} \\ (b_{i1}, b_{i2}) & \sim \textrm{MVN}(\boldsymbol 0,\Sigma(\boldsymbol \theta)) \end{split} $$
The problem is that some of the $\beta_{ij}$ parameters and the $b_{i1}$ parameters are jointly unidentifiable.
If you instead use
response~fixed.A*fixed.B + cov + (0 + cov|fixed.A)
you eliminate the random intercept and get
$$ \begin{split} y_{ijk} & \sim \textrm{N}(\mu_{ijk},\sigma^2_r) \\ \mu_{ijk} & = \beta_{ij} + \beta_x x_{ijk} + b_{i2} x_{ijk} \\ b_{i2} & \sim \textrm{N}(0,\theta^2) \end{split} $$
(For a sensible model you need to include an overall fixed effect of $x$ (cov), because the random effects are zero-centered.)
It's not completely clear to me from that link whether SPSS is applying the Levene test to residuals or to the original observations within each group (I assume it's using residuals, but its quite possible I missed something). In any case Ben is correct, even if you do it with residuals the test doesn't really do what is needed.
In my examples, y is the DV, x is the group variable and z is the covariate.
There's more than one variable in the ANCOVA; you can compare the residual variance in the two groups but equal population variances in the two groups aggregated across the covariate is not sufficient to satisfy the assumptions on which the tests need to be based.
Keeping in mind that the variance assumption on which the usual calculations are based is that all observations have the same population variance, consider an example where the variance is increasing with increasing values of the covariate:
As we see, the regression homoskedasticity assumption on which the test of coefficients are based is not satisfied but the population variance within each group is constant -- even if you look at residuals rather than original observations.
Hopefully it's sufficient to see that in a situation like this the assumptions are not satisfied even though Levene would suggest that there's no problem (that is, a non-rejection is not really a consolation).
If Levene's test is being carried out on the original observations instead of the residuals, we would see another problem:
Here the within-group spreads on the original data are plainly different even though the assumptions required for the test are actually satisfied.
(Again, I assume SPSS isn't doing this, but I couldn't figure out for sure whether that was the case from your link.)
Best Answer
1) The assumption of equality of variance is almost never exactly satisfied
2) ANOVA isn't very sensitive to mild differences in variance, so your ability to reject the null in the Levene test may be almost irrelevant in large samples; correspondingly an inability to reject the null in small samples should be no comfort whatever. Hypothesis tests are simply answering the wrong question [The question: "do the variances differ?" ... to which the correct answer is almost always 'yes, they do' ... is almost useless, since we know the answer already, and a hypothesis test only tells us what we already know in sufficiently large samples. A much better question is How much does that affect my inference? -- which is not answered by a hypothesis test; it's more related to the relative sizes of the variances, and to the relative sizes of the samples they occur in].
3) If sample sizes are equal (or very nearly so), then the ANOVA isn't sensitive to the assumption.
A better choice than testing variance and then choosing what analysis to apply based on the outcome of that test is to not assume equal variances at all (unless you have a reason to do so). A Welch-Satterthwaite type approach works pretty well - just use it whenever you don't have a good reason to do otherwise.