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Table 2 Linear models used in our analyses

From: Modelling subject-specific childhood growth using linear mixed-effect models with cubic regression splines

Model Regression equations
Ordinary least squares \(\begin{array}{*{20}ll} Height_{ij} &=\, \beta_{0} + \beta_{1} t_{ij} + \beta_{2} t_{ij}^{2} + \beta_{3} t_{ij}^{3} + \beta_{4} \left( {t_{ij} - 3} \right)_{ + }^{3} + \beta_{5} \left( {t_{ij} - 6} \right)_{ + }^{3} + \beta_{6} \left( {t_{ij} - 12} \right)_{ + }^{3} + \beta_{7} \left( {t_{ij} - 18} \right)_{ + }^{3}\\&\quad + \beta_{8} \left( {t_{ij} - 24} \right)_{ + }^{3} + \beta_{9} \left( {t_{ij} - 40} \right)_{ + }^{3} + \beta_{10} I\left( {t_{ij} > 24} \right) + \beta_{11} I\left( {male} \right) + \varepsilon_{ij} \end{array}\) \(\varepsilon_{ij} \sim N\left( {0, \sigma^{2} } \right)\)
Linear mixed-effect model with random intercept and random slope. \(\begin{array}{*{20}ll} Height_{ij}&=\, \beta_{0} + b_{0i} + \beta_{1} t_{ij} + b_{1i} t_{ij} + \beta_{2} t_{ij}^{2} + \beta_{3} t_{ij}^{3} + \beta_{4} \left( {t_{ij} - 3} \right)_{ + }^{3} + \beta_{5} \left( {t_{ij} - 6} \right)_{ + }^{3} + \beta_{6} \left( {t_{ij} - 12} \right)_{ + }^{3} \\ &\quad + \beta_{7} \left( {t_{ij} - 18} \right)_{ + }^{3} + \beta_{8} \left( {t_{ij} - 24} \right)_{ + }^{3} + \beta_{9} \left( {t_{ij} - 40} \right)_{ + }^{3} + \beta_{10} I\left( {t_{ij} > 24} \right) + \beta_{11} I\left( {male} \right) + \varepsilon_{ij} \end{array}\) \(\left( {\begin{array}{*{20}c} {b_{0} } \\ {b_{1} } \\ \end{array} } \right)\sim {\text{MVN}}\left( {\left[ {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \right],\left[ {\begin{array}{*{20}c} {g_{11} } & {g_{21} } \\ {g_{12} } & {g_{22} } \\ \end{array} } \right]} \right)\) \(\varepsilon_{ij} \sim N\left( {0, \sigma^{2} } \right)\)
Linear mixed-effect model with random intercept and random slope and first order continuous autoregression (CAR(1)) \(\begin{array}{*{20}ll} Height_{ij} &=\, \beta_{0} + b_{0i} + \beta_{1} t_{ij} + b_{1i} t_{ij} + \beta_{2} t_{ij}^{2} + \beta_{3} t_{ij}^{3} + \beta_{4} \left( {t_{ij} - 3} \right)_{ + }^{3} + \beta_{5} \left( {t_{ij} - 6} \right)_{ + }^{3} + \beta_{6} \left( {t_{ij} - 12} \right)_{ + }^{3}\\ &\quad + \beta_{7} \left( {t_{ij} - 18} \right)_{ + }^{3} + \beta_{8} \left( {t_{ij} - 24} \right)_{ + }^{3} + \beta_{9} \left( {t_{ij} - 40} \right)_{ + }^{3} + \beta_{10} I\left( {t_{ij} > 24} \right) + \beta_{11} I\left( {male} \right) + \varepsilon_{ij} \end{array}\) \(\left( {\begin{array}{*{20}c} {b_{0} } \\ {b_{1} } \\ \end{array} } \right)\sim {\text{MVN}}\left( {\left[ {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \right],\left[ {\begin{array}{*{20}c} {g_{11} } & {g_{21} } \\ {g_{12} } & {g_{22} } \\ \end{array} } \right]} \right)\) \(\varepsilon_{ij} \sim N\left( {\left[ {\begin{array}{*{20}c} 0 \\ \vdots \\ 0 \\ \end{array} } \right], \sigma^{2} \left[ {\begin{array}{*{20}c} 1 & \ldots & {\rho^{{\left| {t_{i1} - t_{{im_{i} }} } \right|}} } \\ \vdots & \ddots & \vdots \\ {\rho^{{\left| {t_{{im_{i} }} - t_{i1} } \right|}} } & \ldots & 1 \\ \end{array} } \right]} \right)\)