Why are there no covariance terms between random effects?

Why are there no covariance terms between random effects?

Because the two random effects are separated into their own terms, no covariance term is included between them. In other words, only the diagonal of the variance-covariance matrix is estimated.

How are covariance terms specified in a formula?

Covariance terms are specified in the same formula as the fixed and random effects. Covariance terms are specified by the way the formula is written. Here there are two fixed effects that are allowed to vary randomly, and one grouping factor g.

Are there any fixed effects that are allowed to vary randomly?

Here there are two fixed effects that are allowed to vary randomly, and one grouping factor g. Because the two random effects are separated into their own terms, no covariance term is included between them. In other words, only the diagonal of the variance-covariance matrix is estimated.

How to specify models with multiple random effects?

I was working in R packages nlme and lme4, trying to specify the models with multiple random effects. I found, that only nlme allows to specify the heterogeneous structure of the variance. Therefore, I got a model, where temperature (Y) depends on time (in hours), intercept varies by date and year, and variance also varies by year:

How to fit a model with separate random effects?

Separate random effects terms are considered independent, however, so if you want to fit (e.g.) a model with random intercept and slope where the intercept and slope are uncorrelated (not necessarily a good idea), you can use the formula (1|g) + (0+x|g), where g is the grouping factor; the 0 in the second term suppresses the intercept.

How are the intercept and X1 random effects related?

Here the intercept and x1 random effects are allowed to vary together while a zero correlation is imposed between the x2 random effect and each of the other two. Again a 0 is included in the x2 random effect term only to explicitly avoid including a random intercept that covaries with the x2 random effect.

Which is the default structure for variance covariance?

The default variance-covariance structure is unstructured — that is, the only constraint on the variance-covariance matrix for an vector random effect with n levels is that is positive definite.