Table 2.

Linear models used in our analyses

Model	Regression equations
Ordinary least squares	$\begin{array}{cc}Heigh{t}_{ij}\hfill & ={\mathit{\beta }}_0+{\mathit{\beta }}_1{t}_{ij}+{\mathit{\beta }}_2{t}_{ij}^2+{\mathit{\beta }}_3{t}_{ij}^3+{\mathit{\beta }}_4{\left(t_{ij}-3\right)}_{+}^3+{\mathit{\beta }}_5{\left(t_{ij}-6\right)}_{+}^3+{\mathit{\beta }}_6{\left(t_{ij}-12\right)}_{+}^3+{\mathit{\beta }}_7{\left(t_{ij}-18\right)}_{+}^3\hfill \\ \hfill & +{\mathit{\beta }}_8{\left(t_{ij}-24\right)}_{+}^3+{\mathit{\beta }}_9{\left(t_{ij}-40\right)}_{+}^3+{\mathit{\beta }}_{10}I\left(t_{ij}>24\right)+{\mathit{\beta }}_{11}I\left(male\right)+{\mathit{\epsilon }}_{ij}\hfill \end{array}$ ${\mathit{\epsilon }}_{ij}\sim N\left(0,{\mathit{\sigma }}^2\right)$
Linear mixed-effect model with random intercept and random slope.	$\begin{array}{cc}Heigh{t}_{ij}\hfill & ={\mathit{\beta }}_0+b_{0i}+{\mathit{\beta }}_1{t}_{ij}+b_{1i}{t}_{ij}+{\mathit{\beta }}_2{t}_{ij}^2+{\mathit{\beta }}_3{t}_{ij}^3+{\mathit{\beta }}_4{\left(t_{ij}-3\right)}_{+}^3+{\mathit{\beta }}_5{\left(t_{ij}-6\right)}_{+}^3+{\mathit{\beta }}_6{\left(t_{ij}-12\right)}_{+}^3\hfill \\ \hfill & +{\mathit{\beta }}_7{\left(t_{ij}-18\right)}_{+}^3+{\mathit{\beta }}_8{\left(t_{ij}-24\right)}_{+}^3+{\mathit{\beta }}_9{\left(t_{ij}-40\right)}_{+}^3+{\mathit{\beta }}_{10}I\left(t_{ij}>24\right)+{\mathit{\beta }}_{11}I\left(male\right)+{\mathit{\epsilon }}_{ij}\hfill \end{array}$ $\left(\begin{array}{c}{b}_0\\ b_1\end{array}\right)\sim \text{MVN}\left(\left[\begin{array}{c}0\\ 0\end{array}\right],\left[\begin{array}{cc}{g}_{11}& g_{21}\\ g_{12}& g_{22}\end{array}\right]\right)$ ${\mathit{\epsilon }}_{ij}\sim N\left(0,{\mathit{\sigma }}^2\right)$
Linear mixed-effect model with random intercept and random slope and first order continuous autoregression (CAR(1))	$\begin{array}{cc}Heigh{t}_{ij}\hfill & ={\mathit{\beta }}_0+b_{0i}+{\mathit{\beta }}_1{t}_{ij}+b_{1i}{t}_{ij}+{\mathit{\beta }}_2{t}_{ij}^2+{\mathit{\beta }}_3{t}_{ij}^3+{\mathit{\beta }}_4{\left(t_{ij}-3\right)}_{+}^3+{\mathit{\beta }}_5{\left(t_{ij}-6\right)}_{+}^3+{\mathit{\beta }}_6{\left(t_{ij}-12\right)}_{+}^3\hfill \\ \hfill & +{\mathit{\beta }}_7{\left(t_{ij}-18\right)}_{+}^3+{\mathit{\beta }}_8{\left(t_{ij}-24\right)}_{+}^3+{\mathit{\beta }}_9{\left(t_{ij}-40\right)}_{+}^3+{\mathit{\beta }}_{10}I\left(t_{ij}>24\right)+{\mathit{\beta }}_{11}I\left(male\right)+{\mathit{\epsilon }}_{ij}\hfill \end{array}$ $\left(\begin{array}{c}{b}_0\\ b_1\end{array}\right)\sim \text{MVN}\left(\left[\begin{array}{c}0\\ 0\end{array}\right],\left[\begin{array}{cc}{g}_{11}& g_{21}\\ g_{12}& g_{22}\end{array}\right]\right)$ ${\mathit{\epsilon }}_{ij}\sim N\left(\left[\begin{array}{c}0\\ ⋮\\ 0\end{array}\right],{\mathit{\sigma }}^2\left[\begin{array}{ccc}1& \dots & {\mathit{\rho }}^{\left|t_{i1}-t_{i{m}_i}\right|}\\ ⋮& \ddots & ⋮\\ {\mathit{\rho }}^{\left|t_{i{m}_i}-t_{i1}\right|}& \dots & 1\end{array}\right]\right)$