Rizopoulos 2016 - PBC Basic Joint Model and Extended Longitudinal Outcomes

Summary

Data: Mayo Clinic primary biliary cirrhosis (1974-84); the text says 312 patients randomized to D-penicillamine and 154 to placebo (as printed), while the model output lists 312 subjects, 1,945 serum-bilirubin measurements (6.2 per patient on average) and 169 composite events (54.2%). Basic joint model with natural-spline mixed model for log bilirubin and current-value association: $\alpha =1.41$ (SD 0.094), hazard ratio 2.7 for doubling bilirubin. Student-$t$, binary and left-censored extensions give similar substantive conclusions.

Main Content

The basic joint model jointFit.pbc1 (3.8 min)

Mixed model lme(log(serBilir) ~ ns(year,2), random = ~ ns(year,2) | id); Cox model Surv(years, status2) ~ drug * age; jointModelBayes(..., timeVar="year", n.iter=30000). Fit: LPML -3165.22, DIC 6082.5, pD 929.6.

Component	Estimate	SD	95% CrI
Random effects SD: intercept / ns1 / ns2 / residual	1.006 / 2.225 / 2.078 / 0.303
Longitudinal intercept	0.5116	0.0579	(0.3959, 0.6234)
Longitudinal ns(year,2)1	2.2845	0.1476	(2.0071, 2.5828)
Longitudinal ns(year,2)2	2.0899	0.1516	(1.7947, 2.3801)
drug D-penicil	-0.7776	0.7762	(-2.2608, 0.9006)
age	0.0404	0.0126	(0.0158, 0.0668)
drug:age	0.0125	0.0153	(-0.0196, 0.0412)
Assoct ($\alpha$)	1.4099	0.0935	(1.2317, 1.6008)
tauBs (smoothing)	523.8	272.2	(143.7, 1153.4)

Interpretation: $\exp (0.693\times 1.4099)=2.66$ per doubling of serum bilirubin (95% CI 2.3-3.0); the treatment effect is not distinguished from zero.

Extended longitudinal outcomes via densLong

densLong(y, eta.y, scale, log, data) supplies the log-density of the longitudinal response given its linear predictor, so any family or censoring can be used without changing the sampler.

• Student-$t$ errors (df = 4): dgt(y, mu=eta.y, sigma=scale, df=4). Residual SD falls to 0.209; $\alpha =1.348$; doubling bilirubin gives 2.5-fold risk (CI 2.3-2.9).
• Binary biomarker (bilirubin $>1.8$ mg/dL; mixed logistic model by glmmPQL): $\alpha =0.228$ (SD 0.032) = log hazard ratio per unit increase in the log-odds of high bilirubin.
• Left-censored biomarker (detection limit 0.8 mg/dL): density for uncensored, CDF for censored (CensInd): $\alpha =1.388$ (SD 0.098), close to the uncensored fit.

Comparison across the longitudinal specifications

Model	$\alpha$	SD	95% CrI	Fit time
Normal errors (pbc1)	1.4099	0.0935	(1.2317, 1.6008)	3.8 min
Student-$t$, df=4 (pbc2)	1.3479	0.0896	(1.1786, 1.5178)	3.7 min
Binary (pbc3)	0.2282	0.0316	(0.1774, 0.2947)	2.5 min
Left-censored (pbc4)	1.3883	0.0977	(1.1999, 1.5915)	4.0 min

Only the binary model changes the scale of $\alpha$ (log-odds, not log-bilirubin). The substantive finding (strong positive association) is stable.

Left truncation and exogenous time-varying covariates (Appendix C): fit a counting-process time-varying Cox model first and pass it to jointModelBayes().

Connections

E25 biases (baseline-only, last-observation-carried-forward and two-stage approaches) are the standard foil for the joint model; see Missing Data Models and Rizopoulos 2016 - Joint Model Specification and Posterior. Survival Analysis for the Cox background.

See Also

Rizopoulos 2016 - JMbayes Joint Models Overview