# Between grouping-factor variance-covariance matrix

I am working with lmer in R and am unsure on the assumptions on the variance-covariance matrix for the random effects in a mixed effects model.

If I have a 2 factor model, say of the form (in mixed effects R language):

Y ~ x + (1|factor1) + (variable1|factor1) + (1|factor2) + (variable1|factor2)

I know that in the underlying math, this is represented as a mixed effects model:

$$Y=X\beta+Zb + \epsilon$$

Where:

• $$X$$ is an ($$n\times p$$) design matrix.
• $$\beta$$ is a $$(p \times 1)$$ parameter vector.
• $$Z$$ is a ($$n \times (s_1l_1+s_2l_2)$$) matrix containing the information about grouping factor variables (where $$s_i$$ is the number of variables listed for factor $$i$$ and $$l_i$$ is the number of levels of factor $$i$$).
• $$b$$ is an ($$(s_1l_1+s_2l_2) \times 1$$) matrix of random effects.
• $$\epsilon$$ is an $$(n\times 1)$$ error term.

I know that it is assumed that the covariance between levels of a factor are zero. For example, if you had a grouping factor of subject and variables of say weight and height over time, then $$cov(w_{s_i},h_{s_j})=0$$ for all $$i\neq j$$, where $$w_{s_k}$$ and $$h_{s_k}$$ represents the weight and height over time measurement random effects for subject $$k$$.

If there is one factor in the analysis, the zero covariance between levels of a factor, means that $$\Sigma=cov(b)$$ is block-diagonal. What I am unsure on however (and this is my question), is;

Is it also assumed that if you have multiple grouping factors that the covariance between the variables belonging to those factors is still 0? i.e. for a multi-factor random effects model is $$\Sigma=cov(b)$$ still block-diagonal?

(e.g. for example if you have a grouping factor of say subject and also say, location the subjects reading was taken then, is it assumed that: $$cov(w_{s_i},h_{l_j})=0$$ for all $$i \neq j$$? Where $$w_{s_k}$$ represents the weight over time measurement random effect for subject $$k$$ and $$h_{l_k}$$ represents the height over time measurement random effect for location $$k$$.)