Table 1: Maximum likelihood estimates of the parameters 𝛼 of the all-cause mortality model 𝜆 𝑖 ( 𝑡 ) = ( 𝛼 4 / 𝛼 0 ) ( 𝑡 / 𝛼 0 ) 𝛼 4 1 e x p { 𝛼 1 𝑏 𝑖 + 𝛼 2 𝑌 𝑖 + 𝛼 3 𝑍 𝑖 } , 𝛽 of the BMI model 𝑌 𝑖 𝑋 𝑖 , 𝑍 𝑖 , 𝛽 𝑁 ( 𝛽 0 + 𝛽 1 𝑏 𝑖 + 𝛽 2 𝑍 𝑖 , 𝛽 2 3 ) , and 𝛾 , the population frequency of the simulated covariate (true value 0.25). Parameters of main interest are 𝛼 2 and 𝛽 2 , which characterize the association between the simulated covariate and primary (all-cause mortality) and secondary (BMI) outcomes, respectively. The values are means and standard deviations over 1000 replications. SE stands for the standard errors of the maximum likelihood estimates.

𝛼 3 𝛽 2 Lik. 𝛼 0 𝛼 1 𝛼 2 𝛼 3 S D ( 𝛼 3 ) S E ( 𝛼 3 ) 𝛼 4 ̂ 𝛽 0 ̂ 𝛽 1 ̂ 𝛽 2 ̂ 𝛽 S D ( 2 ) ̂ 𝛽 S E ( 2 ) ̂ 𝛽 3 ̂ 𝛾

0.00.0 Full23.720.09 0 . 0 1 0 . 0 1 0.10 0.101.33 0.00 0 . 0 8 0.01 0.22 0.224.100.25
Cond.23.740.09 0 . 0 1 0.00 0.10 0.101.33 0.00 0 . 0 8 0.00 0.23 0.234.17
Weighted 0 . 0 1 0 . 0 8 0.03 0.30 0.304.10
0.50.0 Full26.300.09 0 . 0 1 0.49 0.09 0.091.33 0.00 0 . 0 8 0.01 0.22 0.214.100.25
Cond.26.330.09 0 . 0 1 0.50 0.09 0.091.33 0.00 0 . 0 8 0.00 0.22 0.224.17
Weighted 0 . 0 1 0 . 0 8 0.03 0.30 0.304.10
0.0 0.5 Full 23.73 0.09 0 . 0 1 0 . 0 1 0.10 0.10 1.33 0 . 1 3 0 . 0 8 0.53 0.25 0.234.090.25
Cond. 23.75 0.09 0 . 0 1 0.00 0.10 0.10 1.33 0 . 1 3 0 . 0 8 0.52 0.24 0.234.17
Weighted 0 . 1 3 0 . 0 8 0.53 0.31 0.314.09
0.5 0.5 Full 26.32 0.09 0 . 0 2 0.49 0.09 0.09 1.33 0 . 1 4 0 . 0 8 0.55 0.23 0.224.090.25
Cond. 26.34 0.09 0 . 0 1 0.50 0.09 0.09 1.33 0 . 1 4 0 . 0 8 0.54 0.22 0.224.17
Weighted 0 . 1 4 0 . 0 8 0.54 0.31 0.314.09
1.0 0.0 Full 29.62 0.09 0 . 0 1 0.99 0.09 0.09 1.35 0.00 0 . 0 8 0.02 0.21 0.214.100.25
Cond. 29.64 0.09 0 . 0 1 1.00 0.09 0.09 1.35 0.00 0 . 0 8 0.01 0.22 0.214.17
Weighted 0 . 0 1 0 . 0 8 0.03 0.29 0.294.10
0.0 1.0 Full 23.77 0.09 0 . 0 1 0.00 0.10 0.10 1.33 0 . 2 8 0 . 0 8 1.11 0.26 0.234.070.25
Cond. 23.77 0.09 0 . 0 1 0.00 0.10 0.10 1.33 0 . 2 7 0 . 0 8 1.06 0.24 0.234.15
Weighted 0 . 2 6 0 . 0 8 1.05 0.32 0.324.07
1.0 1.0 Full 29.67 0.09 0 . 0 3 1.00 0.09 0.09 1.35 0 . 2 9 0 . 0 8 1.14 0.24 0.224.070.25
Cond. 29.65 0.09 0 . 0 2 1.00 0.09 0.09 1.35 0 . 2 8 0 . 0 8 1.08 0.22 0.214.14
Weighted 0 . 2 7 0 . 0 8 1.07 0.31 0.314.07