Independent random-intercept and random-slope model ($Y_{ij}={\beta }_0+{\beta }_1{t}_{ij}+u_j+v_j{t}_{ij}+{\epsilon }_{ij}$)

Independent Gaussian random intercepts $u_j$ and random slopes $v_j$ give marginal within-group covariance ${\tau }_0^2+{\tau }_1^2{t}_{ij}{t}_{kj}+{\sigma }^2{1}_{\{i=k\}}$. 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 $u_j+c{v}_j$, generally correlated with $v_j$.