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Table 3 Results from M1 and M4. M1 is the joint model that shares both the random intercept and slope from the longitudinal sub-model with the random intercept from the survival model. M4 is the interval censored survival model with the longitudinal measurements as a time-varying covariate

From: Comparison of a time-varying covariate model and a joint model of time-to-event outcomes in the presence of measurement error and interval censoring: application to kidney transplantation

  M1: Shared Random Effects M4: Time Varying Covariate
    Lower Upper     Lower Upper
Parameter Estimate SD 95% CrI 95% CrI   Estimate SD 95% CrI 95% CrI
Linear Sub-Model          
b 0 7.24 0.06 7.12 7.36 - - - - -
b 1 -0.05 0.005 -0.06 -0.04 - - - - -
\(\sigma ^{2}_{e}\) 7.44 0.09 7.26 7.63 - - - - -
\(\sigma ^{2}_{0}\) 1.66 0.14 1.40 1.94 - - - - -
\(\sigma ^{2}_{1}\) 0.004 0.0006 0.003 0.005 - - - - -
ρ -0.38 0.07 -0.51 -0.24 - - - - -
Survival Sub-Model          
α 0.57 0.06 0.47 0.69 α tvc 0.46 0.03 0.40 0.53
  Hazard      Hazard    
  Ratio      Ratio    
β 0 0.03 0.01 0.01 0.06 γ 0 0.17 0.06 0.08 0.30
β1 (HLA Mismatch Number) 1.26 0.07 1.13 1.40 γ 1 1.29 0.06 1.17 1.43
β2 (African American) 2.09 0.56 1.23 3.39 γ 2 1.79 0.40 1.11 2.70
β3 (Hispanic) 1.60 0.33 1.05 2.31 γ 3 1.67 0.31 1.14 2.33
β4 (Other Race) 1.14 0.56 0.34 2.41 γ 4 1.12 0.47 0.40 2.20
β5 (30-49 years) 0.57 0.15 0.32 0.93 γ 5 0.67 0.17 0.42 1.05
β6 (≥50 years) 0.30 0.08 0.17 0.49 γ 6 0.41 0.10 0.25 0.65
λ 0 0.64 0.06 0.52 0.75 η 0.80 0.03 0.75 0.85
λ1 (per 0.05 change) 0.43 0.07 0.32 0.58 - - - - -
  1. For M1, the posterior mean and 95% credible intervals are presented for the linear portion of the model, the survival portion of the model, and the association parameters. The longitudinal portion is comprised of a fixed intercept (b0), a fixed slope (b1), and a random error term (\(\sim N(0,\sigma ^{2}_{e})\)). There is also a random intercept (\(a_{0i}\sim N(0,\sigma ^{2}_{0})\)) and a random slope for each individual (\(a_{1i}\sim N(0,\sigma ^{2}_{1})\)), which have a correlation parameter ρ. The survival model is comprised of fixed covariates (β), the Weibull association parameter α, and a random intercept for each individual that is related to the random intercept and slope of the longitudinal sub-model through two association parameters. The first association parameter (λ0) links the two sub-models through their shared random intercepts. The second association parameter (λ1) links the two sub-models through the longitudinal random slope and the survival random intercept. M4 contains the Weibull scale parameter, αtvc, baseline covariates with parameter γ, and the coefficient for the time-varying covariate, η
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