# Homogeneity of variance in linear mixed model

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!

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 Orthodont datasset in R's nlme.)

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.

• Actually my experiment got one predictor at 4 levels and I did a repeated measurement. Therefore, I use random intercept and slope for each subject. Levene's test on the predictor variable is not significant, however, for the day is significant. Do I need to have homogeneity of variance for the day as well? I thought I had modeled the variance by AR(1)/Unstructure covariance matrix already. Thank you so much for your generous help. – Kam Aug 25 '14 at 4:57

Assumption of Homogeneity of variance means that the variance of residual should be constant at each value of the predictor variables. Students residual is used to check for outliers. While residual vs predicted values is used to check for assumption of linear regression.

• May I know how to test the homogeneity of variance? Is it by Levene's test on the residual by each predicted variable? I am using spss, but the linear mixed model does not contain this test. Many thanks! – Kam Aug 20 '14 at 5:10