# Table 1 Specifications of the four models (simple linear regression and RE)

Model

Conditional Distribution

Specification of the mean

M1- Linear regression

Yi~N(Î¼i,â€‰Ïƒ2)

Îµi~N(0,â€‰Ïƒ2)

Î¼iâ€‰=â€‰Î²0â€‰+â€‰Parai

M2- Random effect

Yij~N(Î¼ij,â€‰Ïƒ2)

$${u}_i\sim N\left(0,{\upsigma}_u^2\right)$$

$${e}_{ij}\sim N\left(0,{\upsigma}_e^2\right)$$

Î¼ijâ€‰=â€‰Î²0â€‰+â€‰Paraijâ€‰+â€‰ui

M3- Random effect: intercept forced to unity

Yij~N(Î¼ij,â€‰Ïƒ2)

$${u}_i\sim N\left(0,{\upsigma}_u^2\right)$$

$${e}_{ij}\sim N\left(0,{\upsigma}_e^2\right)$$

Î¼ijâ€‰=â€‰Paraijâ€‰+â€‰ui

M4- Random effect: intercept forced to unity and inclusion of most

Yij~N(Î¼ij,â€‰Ïƒ2)

$${u}_i\sim N\left(0,{\upsigma}_u^2\right)$$

$${e}_{ij}\sim N\left(0,{\upsigma}_e^2\right)$$

Î¼ijâ€‰=â€‰Paraijâ€‰+â€‰Î²mostmostijâ€‰+â€‰ui

1. Paraijâ€‰=â€‰Î²PF2PF2ijâ€‰+â€‰Î²PF3PF3ijâ€‰+â€‰Î²PF4PF4ijâ€‰+â€‰Î²PF5PF5ijâ€‰+â€‰Î²PF6PF6ijâ€‰+â€‰Î²RL2RL2ijâ€‰+â€‰Î²RL3RL3ijâ€‰+â€‰Î²RL4RL4ijâ€‰+â€‰Î²SF2SF2ijâ€‰+â€‰Î²SF3SF3ijâ€‰+â€‰Î²SF4SF4ijâ€‰+â€‰Î²SF5SF5ijâ€‰+â€‰Î²PAIN2PAIN2ijâ€‰+â€‰Î²PAIN3PAIN3ijâ€‰+â€‰Î²PAIN4PAIN4ijâ€‰+â€‰Î²PAIN5PAIN5ijâ€‰+â€‰Î²PAIN6PAIN6ijâ€‰+â€‰Î²MH2MH2ijâ€‰+â€‰Î²MH3MH3ijâ€‰+â€‰Î²MH4MH4ijâ€‰+â€‰Î²MH5MH5ijâ€‰+â€‰Î²VIT2VIT2ijâ€‰+â€‰Î²VIT3VIT3ijâ€‰+â€‰Î²VIT4VIT4ijâ€‰+â€‰Î²VIT5VIT5ij