 Solution

For observation $j$ of bird $i$ in year $t(i,j)$, a Gaussian linear mixed model with crossed random intercepts iswhere the $u_i$, $v_t$, and ${\epsilon }_{ij}$ are mutually independent random variables. Here $a_{ij}$ is age, ${\beta }_0$ is the population intercept at age zero, and ${\beta }_1$ is the change in conditional expected value per year of age. The $u_i$ capture persistent between-bird variation, the $v_t$ capture shared annual conditions, and the regression residuals capture remaining observation-level variation. Two observations from the same bird share covariance ${\tau }_b^2$; two from the same year share ${\tau }_y^2$, with both contributions present if both group labels agree. The model also assumes the random effects are independent of the age covariate.