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m clean up spacing around commas and other punctuation fixes, replaced: ,K → , K (4), ,M → , M (3), ,P → , P (4), ,Y → , Y, ,b → , b (4), ,j → , j (3), ,l → , l (4), ; → ; (2) |
→Bayesian nonlinear mixed-effects model: spaced to improve page alignment on some monitors due to the large box next to this section |
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'''''Stage 1: Individual-Level Model'''''
<math>\begin{align}
\phantom{spacer} \\
&\epsilon_{ij} \sim N(0, \sigma^2), \\
\phantom{spacer} \\
&i =1,\ldots, N, \, j = 1,\ldots, M_i.
\end{align}</math>
'''''Stage 2: Population Model'''''
<math>\begin{align}
\phantom{spacer} \\
&\eta_{li} \sim N(0, \omega_l^2), \\
\phantom{spacer} \\
&i =1,\ldots, N, \, l=1,\ldots, K.
\end{align}</math>
'''''Stage 3: Prior'''''
<math>\begin{align}
<math> \sigma^2 \sim \pi(\sigma^2),\quad \alpha_l \sim \pi(\alpha_l), \quad (\beta_{l1},\ldots,\beta_{lb},\ldots,\beta_{lP}) \sim \pi(\beta_{l1},\ldots,\beta_{lb},\ldots,\beta_{lP}), \quad \omega_l^2 \sim \pi(\omega_l^2), \quad l=1,\ldots, K.</math>▼
&\sigma^2 \sim \pi(\sigma^2),\\
\phantom{spacer} \\
&\alpha_l \sim \pi(\alpha_l), \\
\phantom{spacer} \\
▲
\phantom{spacer} \\
&\omega_l^2 \sim \pi(\omega_l^2), \\
\phantom{spacer} \\
&l=1,\ldots, K.
\end{align}</math>
Here, <math>y_{ij}</math> denotes the continuous response of the <math>i</math>-th subject at the time point <math>t_{ij}</math>, and <math>x_{ib}</math> is the <math>b</math>-th covariate of the <math>i</math>-th subject. Parameters involved in the model are written in Greek letters. <math>f(t ; \theta_{1},\ldots,\theta_{K})</math> is a known function parameterized by the <math>K</math>-dimensional vector <math>(\theta_{1},\ldots,\theta_{K})</math>. Typically, <math>f</math> is a `nonlinear' function and describes the temporal trajectory of individuals. In the model, <math>\epsilon_{ij}</math> and <math>\eta_{li}</math> describe within-individual variability and between-individual variability, respectively. If '''''Stage 3: Prior''''' is not considered, then the model reduces to a frequentist nonlinear mixed-effect model.
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