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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