# Can covariance parameter estimates and fixed effect estimates of a Generalized Linear Mixed Model be correlated?

Specifically, I am fitting a Poisson Mixed Model with random coefficients and intercepts on longitudinal count data. The rate ratio between time points is log normal with expected value $e^{\mu}e^{\sigma^2 / 2}$. I can estimate $\mu$ from fixed effect estimates, and $\sigma$ from the random effect covariance estimate. The best option I can think of to get a confidence interval of the rate ratio is to use the variance estimates of $\hat{\mu}$ and $\hat{\sigma^2}$ which I can get from the output of model estimates (in this case from SAS proc glimmix) and by summing terms get the combined variance estimate necessary to calculate the endpoints. However I do not know whether Cov($\hat{\mu}$,$\hat{\sigma^2}$) = 0, so this will not work if this covariance is not zero. My question is, am I justified in assuming such covariance = 0?

The theoretical covariance between $$\hat\mu$$ and $$\hat\sigma^2$$ is not, in general, zero in the case of GLMMs only in the case of linear mixed models. Note also that in the case of GLMMs, typically the inverse of the observed information matrix (negative hessian matrix) is used to calculate standard errors than the inverse of the Fisher information matrix. Working with the observed information matrix is even better when you have missing data.