Covariance structure for random intercepts and slopes Can you please help me figuring out what the covariance structure of a model with random intercept and random slope is?
Here is my model $Y_{ij} = \beta_0 + \beta_1 t_{ij} + b_{0,i} + b_{1,i}t_{ij} + e_{ij}$ 
where $b_{0,i}$ and $b_{1,i}$ are random intercept and random slope and we know that $e_{ij}$ are mutually independent and also:
$b_i$ ~ $N(0,D)$ where $b_i = (b_{0,i},b_{1,i})$ and $e_{ij}$ ~ $N(0,\sigma^2)$
I appreciate your help.
 A: Using the properties of the covariance operator we have
$$$$
$$\begin{align*}
\textrm{Cov}(Y_{ij}, Y_{ik}) 
& = \textrm{Cov}(\beta_0 + \beta_1 t_{ij} + b_{0i} + b_{1i} t_{ij} + e_{ij}, \;
                 \beta_0 + \beta_1 t_{ik} + b_{0i} + b_{1i} t_{ik} + e_{ik}) \\
& = \textrm{Cov}(b_{0i} + b_{1i} t_{ij} + e_{ij}, \; 
                 b_{0i} + b_{1i} t_{ik} + e_{ik}) \\
& = \textrm{Cov}(b_{0i}, \; b_{0i}) 
        + t_{ik} \, \textrm{Cov}(b_{0i}, \; b_{1i}) 
        + \textrm{Cov}(b_{0i}, \; e_{ik}) \\
& \quad +\, t_{ij} \, \textrm{Cov}(b_{1i}, \; b_{0i}) 
        + t_{ij} t_{ik} \, \textrm{Cov}(b_{1i}, \; b_{1i}) 
        + t_{ij} \, \textrm{Cov}(b_{1i}, \; e_{ik}) \\
& \quad +\, \textrm{Cov}(e_{ij}, \; b_{0i}) 
        + t_{ik} \, \textrm{Cov}(e_{ij}, \; b_{1i}) 
        + \textrm{Cov}(e_{ij}, \; e_{ik})  \\
& = \textrm{Var}(b_{0i})
        +\, (t_{ij} + t_{ik}) \textrm{Cov}(b_{0i}, \; b_{1i})
        +\, t_{ij}t_{ik} \textrm{Var}(b_{1i})
        +\, \textrm{Cov}(e_{ij}, \; e_{ik})
\end{align*}$$
$$$$
If $j \neq k$, then
$$\begin{align*}
\textrm{Cov}(Y_{ij}, Y_{ik})
 & = \textrm{Var}(b_{0i}) 
        + (t_{ij} + t_{ik}) \, \textrm{Cov}(b_{0i}, \; b_{1i})
        + t_{ij} t_{ik} \textrm{Var}(b_{1i})
\end{align*}$$
If $j = k$, then
$$\begin{align*}
\textrm{Cov}(Y_{ij}, Y_{ik})
 & = \textrm{Var}(Y_{ij}) \\
 & = \textrm{Var}(b_{0i}) 
        + 2 \, t_{ij} \, \textrm{Cov}(b_{0i}, \; b_{1i})
        + t_{ij}^2 \textrm{Var}(b_{1i})
        + \sigma^2
\end{align*}$$
$$$$
$\textrm{Var}(b_{0i})$, $\textrm{Var}(b_{1i})$, and $\textrm{Cov}(b_{0i}, \; b_{1i})$ are found in your matrix $D$.
