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Supplemental Digital Content 2
APPENDIX
The trajectory equations for the bivariate growth curve model are given by
yij  0iy  1iyij  2iyij2   yij
(1)
zij  0iz  1izij  2izij2  gzij
(2)
where
y jj
and
z jj denotes the PCS and MCS scores for individual i, and measurement occasion j (j =
1,2,…,6).
0iy and 0iz are the PCS and MCS random intercepts at time t0 for individual i 1iy and
1iz
 2iy
are the PCS and MCS random linear slopes or rates of change per year for individual i
and
2iz are the PCS and MCS random quadratic slopes or acceleration in the rate of change
per year for individual i

ij
is the time score in years for individual i on measurement occasion j
 yij and  zij are the individual PCS and MCS disturbance terms, or random errors, for
individual i on occasion j.
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The intercept equations are given by
 0iy       0 y xi   0 yi
(3)
0iz     0z xi   0zi
(4)
0y
0z
where

0y
and

0z
are the mean PCS and MCS intercepts
xi are the time-invariant covariates or predictors of the random intercepts and slopes (gender,
cohort and cohort2) for individual i
 0 y and  0 z are the regression coefficients for xi on the PCS and MCS random intercepts
 0 yi and  0 zi are the disturbances in the PCS and MCS random intercepts
The linear slope equations are given by
1iy    1y xi  1yz 0iz  1yi
(5)
1iz       1z xi   1zy  0iy   1zi
(6)
1y
1z
where

1y
and

1z
are the mean PCS and MCS linear slopes
 1 y and  1 z are the regression coefficients for xi on the PCS and MCS random linear slopes
 1 yz and  1zy are the regression coefficients for  0iz and 0iy , the PCS and MCS random
intercepts on the PCS and MCS random linear slopes
 1 yi and  1zi are the disturbances in the PCS and MCS random linear slopes.
The quadratic slope equations are given by
2iy     2 y xi   2 yz 1iz   2 yi
(7)
 2iz       2 z xi   2 zy 1iy   2 zi
(8)
2y
2z
where

1y
and

1z
are the mean PCS and MCS quadratic slopes
 1 y and  1 z are the regression coefficients for xi on the PCS and MCS random quadratic
slopes
 2 yz and  2 zy are the regression coefficients for 1iz and 1iy , the PCS and MCS random
linear slopes on the PCS and MCS random quadratic slopes
 2 yi and  2 zi are the disturbances in the PCS and MCS random quadratic slopes.
Then   0 y ,  0 z ,  1 y ,  1 z ,   2 y and  2 z represent individual differences or
conditional variance in the random PCS and MCS intercept and linear and quadratic slopes, with
covariance  0 y 0 z ,  0 y 1 y ,  1 y 2 y ,   0 z 1 z and  1 z  2 z , while the between occasion
variance in individual PCS and MCS scores are represented by
yij and zij .
The equation for loss to follow-up is given by
 u 
logit j   0uj   1ujxi  uyj1 yij1  uzj1 zij1   uij
1 u 
j 

(9)
where
uj
1 u j
are the odds of loss to follow-up on occasion j
0uj is the intercept on occasion j
xi are the time-invariant covariates or predictors of loss to follow-up (gender, cohort and
cohort2) for individual i
1uj
the regression coefficients for
xi
uy
j 1
and
uz
j 1
are the regression coefficients for the PCS and MCS scores,
occasion j-1
 uij is the random error for individual i on occasion j.
yij 1 and zij 1 , on