# Conditional response of a linear mixed effects model

Consider the linear mixed effects model:

$$$$X_i(t_{ij}) = \eta + Z_i(t_{ij})w_i + \epsilon_{ij},$$$$

where $$\eta$$ is the mean, $$Z_i(t_{ij}) = [1, \log(t_{ij})]$$, $$w_i = (w_{0i}, w_{1i})' \sim N(0,\Sigma_{w})$$, $$\epsilon_{ij} \sim N(0, \sigma^2)$$, and

$$$$\Sigma_{w} = \begin{pmatrix} \sigma^2_1& \rho\sigma_1\sigma_2 \\ \rho\sigma_1\sigma_2 & \sigma^2_2 \end{pmatrix}.$$$$

The conditional response is given by

$$$$X_i(t_{ij}) \mid w_i, \eta, \theta, \sigma \sim N(\eta + Z_i(t_{ij})w_i, \sigma^2),$$$$

where $$\theta = c(\sigma_1, \sigma_2, \rho)$$.

But, the variance of $$X_i$$ is

\begin{aligned} Cov(X_i(t_i), X_i(t_i)) & = Cov(\eta + Z_i(t_i)w_i + \epsilon_i, \eta + Z_i(t_i)w_i + \epsilon_i) \\ & = Var(Z_i(t_i)w_i) + Var(\epsilon_i) = Z_i(t_i)\Sigma_{w}Z'_i(t_i) + \sigma^2. \end{aligned}

So, why is the conditional response not given by

$$$$X_i(t_{ij}) \mid w_i, \eta, \theta, \sigma \sim N(\eta, Z_i(t_i)\Sigma_{w}Z'_i(t_i) + \sigma^2).$$$$

$$$$X_i(t_{ij}) \mid w_i \sim N(\eta + Z_i(t_{ij})w_i, \sigma^2).$$$$
$$$$X_i(t_{ij}) \sim N(\eta, Z_i(t_i)\Sigma_{w}Z'_i(t_i) + \sigma^2).$$$$