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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 $k$.)

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