A Joint Model for
Multiple Longitudinal Outcomes,
Recurrent and Terminal Events
using CF Patient Registry Data

Dimitris Rizopoulos (PM Afonso & ER Andrinopoulou)
Department of Biostatistics, Erasmus University Medical Center Rotterdam


ENAR Spring Meeting | March 29, 2022

Cystic Fibrosis

  Chronic respiratory problems, and lung infections.

  Poor growth, and low weight.


Cystic Fibrosis



Research Goals

  Outcomes:

    · Lung function decline (FEV1)

    · Nutritional status evolution (BMI)

    · Pulmonary exacerbations

    · Death or transplantation


  • Research questions
    • How FEV1 and BMI relate to the risk of pulmonary exacerbations?
    • How FEV1 and BMI relate to the risk of death/transplantation?
    • Are pulmonary exacerbations related to the risk of death/transplantation?

Data

  27K CF patients

  1,400K observations

  317K years of follow-up


Data

  50% of patients reported ≥50 measurements, and ≥12.5 years of follow-up.

Data

  50% of patients experienced ≥4 acute respiratory events.

Data

  26% of patients died or received a transplant.

Joint Model

  Why?

    · Survival analysis: endogenous time-varying covariates

    · Longitudinal analysis: nonrandom dropout

Joint Model

  Why?

    · Survival analysis: endogenous time-varying covariates

    · Longitudinal analysis: nonrandom dropout


{yi(t)=xi(t)⊤β+zi(t)⊤bi+εi(t)=ηi(t)+εi(t)Longitudinal outcomehi(t)=h0(t)exp{wi(t)⊤γ+f{ηi(t)}α}Time-to-event outcome


bi∼N(0,D),εi(t)∼N(0,σ2)

Extended Joint Model

  Recurrent failure times + Multiple longitudinal markers + Multiple functional forms


{yi1(t)=xi1(t)⊤β1+zi1(t)⊤bi1+εi1(t)=ηi1(t)+εi1(t)FEV1yi2(t)=xi2(t)⊤β2+zi2(t)⊤bi2+εi2(t)=ηi2(t)+εi2(t)BMIhi(t)=h0(t)exp{wit(t)⊤γt+f1{ηi1(t)}αt1+f2{ηi2(t)}αt2+viαF}Death or TXri(t)=r0(t)exp{wir(t)⊤γr+g1{ηi1(t)}αr1+g2{ηi2(t)}αr2+vi}PEx


(bi1bi2vi)∼N(0,(D00σ2F)),εij(t)∼N(0,σ2j)

Extended Joint Model

  Functional Forms


f{ηi(t)}={ηi(t)valueddtηi(t)slopeηi(t)−ηi(t−1)last year increase1t∫t0ηi(s)ds(normalized) cumulative effect


  • Combinations of the above
    • interactions with covariates

Extended Joint Model

  Hazard timescale

Extended Joint Model

  Discontinuous risk intervals

Estimation & Software

Estimation & Software



##   id tstart tstop status strata sex
## 1  1      0     2      1      1   m
## 2  1      3     4      1      1   m
## 3  1      5     7      0      1   m
## 4  1      0     7      0      2   m
## 5  2      0     1      1      1   f
## 6  2      2     4      0      1   f
## 7  2      0     4      1      2   f

Estimation & Software


{yi(t)=β0+tβ+b0i+tb1i+εi(t)=ηi(t)+εi(t)hi(t)=h0(t)exp{groupiγt+ηi(t)αt+viαF}ri(t)=r0(t)exp{groupiγr+ηi(t)αr+vi}


            library("JMbayes2")

            lme_fit <- lme(fixed = y ~ time, random = ~ time | id, data = data_long)
            
            cox_fit <- coxph(Surv(tstart, tstop, status) ~ group * strata(strata),
                               data = data_strt)
            
            jm_fit <- jm(cox_fit, lme_fit, time_var = "time", 
                         functional_forms = ~ value(y) * strata,
                         recurrent = "gap")

Simulation Results

  100 replications

{yi(t)=β0+b0i+∑4j=1(βj+bji)nsj(t)+ε(t)=ηi(t)+ε(t)hi(t)=h0(t)exp{wi1γt1+wi2γt2+ηi(t)αt+viαF}ri(t)=r0(t)exp{wi1γr1+wi2γr2+ηi(t)αr+vi}


(bivi)∼N(0,(D00σ2F)),εi(t)∼N(0,σ2),i=1,…,500

Simulation Results

Application Results

  Running time: 9.3 hours 

FEV1_fit <- lme(fixed = FEV1 ~ age, random = ~ age | ID, data = dataL$long)

BMI_fit  <- lme(fixed = BMI ~ age, random = ~ age | ID, data = dataL$long)

Cox_fit <- coxph(Surv(tstart, tstop, status) ~ Gender * strata(process),
                 data = dataL$strt)
        
jm_fit <- jm(Cox_fit, list(FEV1_fit, BMI_fit), time_var = "age",
             functional_forms = ~ value(FEV1):process + value(BMI):process
             recurrent = "gap")

Application Results



  • Pulmonary Exacerbations
    • a 10-unit decrease of FEV1 increases the hazard of an exacerbation by 21% (16; 26%)
    • a 5-unit increase of BMI increases the hazard of an exacerbation by 4% (-9; 21%)


  • Death/Transplantation
    • a 10-unit decrease of FEV1 increases the hazard of death by 60% (38; 98%)
    • a 5-unit increase of BMI decreases the hazard of death by 36% (20; 57%)


  • Frailty
    • an one-SD increase of the frailty term, increases the hazard of death by 160% (96; 278%)

Discussion


  • Extended Joint Model
    • multiple longitudinal outcomes
    • recurrent events
    • terminating event
    • gap and calendar time scales
    • discontinuous risk intervals
    • different functional forms


  • Challenging Estimation
    • high-dimensional random effects and frailties
    • efficient MCMC implementation

JMbayes2 - Additional Options

  • Various distributions for longitudinal outcomes
    • Normal & censored Normal, Student’s-t
    • Beta
    • Gamma
    • Binomial, Beta-Binomial
    • Poisson & Negative Binomial


  • Comparison between models
    • DIC, WAIC, LPML


  • Dynamic predictions
    • still under implementation











Thanks for your attention!

@drizopoulos
d.rizopoulos @ erasmus mc.nl