Solved – 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.

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 $D$.

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