Mixed Models in SPSS and interpretation of Random Effects

Question

I am trying to run a mixed model using SPSS. The example I am using is taken from the book "Linear Mixed Models, a practical guide using statistical software" of West, Welch and Galecki.

The example I am using is the rat pup example, in which the dependent variable is birth weight of rat pups. The independent variables are sex, treatment that the mother got (control, low dose, high dose), the litter size, and the interaction between the sex and the treatment. The litter is a random effect, since there is correlation between pups born to the same mother.

My model in mathematical terms is:

$weight=\beta _{0}+\beta _{1}Treat1+\beta _{2}Treat2+\beta _{3}Sex1+\beta _{4}LitterSize+\beta _{5}Sex1\times Treat1+\beta _{6}Sex1\times Treat2+u+\varepsilon $

where $u:N(0,\sigma _{litter}^{2})$ and $\varepsilon : N(0,\sigma ^{2})$

I attach the main output and have two questions.

1. How do I make interpretation of the random effect ? I got two numbers there: 0.0965 and 0.0328. I understand that 0.0328 is probably the estimate of the square root of $\sigma _{litter}^{2}$, but what does the second number tells me ? Which part of the model equation does it estimate ?

2. In the book, after running this model, they tried another model in which $\sigma _{\varepsilon }^{2}$ is different for each level of treatment. They wrote that this can be done using R and SAS only and version 13 of SPSS doesn't support it. I am using version 25. Does anyone knows if it can be done now, and how ?

Thank you.

Answer 1

The research question is:

I want to check the relation of the fixed effects to the dependent variable, taking into account the fact that the design is clustered.

This is answered by the fixed effects estimates.

How do I make interpretation of the random effect

Generally there is no requirement to interpret the random effects - you are controlling for clustering / repeated measures by fitting random intercepts for litter. That said, there is obviously nothing wrong with understanding what the output means. 0.0965 is the estimate for the variance of the random intercept for litter (I am not familiar with SPSS, so it's possible this is the standard deviation, not the variance, so please check with the documentation). 0.0328 is the standard error of this estimate (which can be used, given some assumotions/caveats) to test the significance of the estimate (ie whether it is truly greater than zero). Assuming that they are variances, not standard deviations, then you can compute the intra-class correlation coefficient (ICC) as 0.096517 / (0.096517 + 0.163488) and this is a measure of the dependence (correlation) within the clusters, because observations within one cluster are more similar to (corrrelated with) other observations within the same cluster, compared to other clusters. If the estimates are standard deviations, then you will need to square them before making the calculation for the ICC.

Regarding the 2nd question, this is a software specific question, and is off-topic here. Do they show the R syntax for such a model ?