= Independent random-intercept and random-slope model
{title2=$Y_{ij}=\beta_0+\beta_1t_{ij}+u_j+v_jt_{ij}+\varepsilon_{ij}$}
Independent Gaussian <random intercepts> $u_j$ and <random slopes> $v_j$ give marginal within-group <covariance> $\tau_0^2+\tau_1^2t_{ij}t_{kj}+\sigma^2\mathbf1_{\{i=k\}}$. In `lme4`, separate terms `(1 | group)` and `(0 + x | group)` impose zero intercept-slope covariance; `(1 + x | group)` estimates it. This independence restriction depends on the predictor origin: replacing $t$ by $t-c$ transforms the intercept effect to $u_j+cv_j$, generally correlated with $v_j$.
Back to article page
