I am confused by the assumption of homogeneity of variance of the Linear mixed model.
Does homogeneity of variance equal to homogeneity of error? May I know is the homogeneity of variance referring to the variance of the dependent variable? Or the residual of the model developed?
I have read some online source using the levene's test for equality of variances? There is a spread vs level option in SPSS and several options (power estimation, transformed, untransformed), would anyone please tell me what are the difference and how to use?
While some suggests to conduct a scatterplot with predicted value vs standardized residual of the model, I have also read papers using the unstandardized residual and studentized residual to plot against the predicted value. May I know are there any differences?
If there is slight violation of the assumption, could I still use linear mixed model?
Thank you very much for all the help!
Best Answer
I just want to add to @mrcet007's answer. I assume that you have a categorical predictor with i levels, and that you are fitting random intercepts and slopes and allowing them to covary. Then you should also check the variabilities of the intercepts and slopes as well as their relationship across the levels of the categorical predictor. Why? Because such a model assumes that the random intercepts $b_{0i}$ and slopes $b_{1i}$ for group i are drawn from a bivariate distribution
$\left[ \begin{matrix} b_{0i} \\ b_{1i} \end{matrix} \right] \sim \mathcal{N} \left(\left[ \begin{matrix} 0 \\ 0 \end{matrix} \right], \left[ \begin{matrix} \sigma^2_0 ~ \sigma_{01} \\ \sigma_{01} ~ \sigma^2_1 \end{matrix} \right] \right) $
where $\sigma^2_0$ and $\sigma^2_1$ are the variances of $b_{0i}$ and $b_{1i}$, respectively, and $\sigma_{01}$ is their covariance.
This simply means that for all groups the random intercepts and slopes are assumed to have similar distributions (and a similar covariance/correlation). Otherwise, this assumption is violated. In the figure below I drew boxplots for intercept and age slope coefficients. (I used the
Orthodontdatasset inR'snlme.)So how can you do this with SPSS? If it allows you to save intercepts/slopes coefficients, you can draw parallel boxplots and scatter plots for them.