Joint models for predicting transplant-related mortality from quality of life data

Abstract

Purpose

To test whether longitudinally measured health-related quality of life (HRQL) predicts transplant-related mortality (TRM) in pediatric hematopoietic stem cell transplant (HSCT).

Methods

The predictors of interest were emotional functioning, physical functioning, role functioning, and global HRQL, as rated by the parent about the child up to 6 times over 12 months of follow-up and measured by the Child Health Ratings Inventories. We used joint models, specifically shared parameter models, with time to TRM as the outcome of interest and other causes of mortality as a competing risk, via the JM software package in R. Choosing shared parameter models instead of standard survival models, such as Cox models with time-dependent covariates, enabled us to address measurement error in the HRQL trajectories and appropriately handle missing data. The nonlinear trajectories for each HRQL domain were modeled by random spline functions. The survival submodels were adjusted for baseline patient, family, and transplant characteristics.

Results

Hazard ratios per one-half standard deviation difference in emotional, physical, and role functioning, and global HRQL were 0.61 (95 % CI 0.46–0.81; p < 0.001), 0.70 (0.51–0.96; p = 0.03), 0.54 (0.34–0.85; p = 0.007), and 0.57 (0.41–0.79; p < 0.001), respectively.

Conclusions

HRQL trajectories were predictive of TRM in pediatric HSCT, even after adjusting the survival outcome for baseline characteristics.

Appendix

Model building for the longitudinal submodels

Appendix Table 3 shows which covariates were considered for each longitudinal submodel. Variables that were associated with a longitudinal outcome at p < 0.10 were tested in multivariable models using backward elimination with p < 0.05.

Table 3 Selection of covariates for the longitudinal submodels

The joint model

The notation is defined in Appendix Table 4. The longitudinal submodel for the ith individual at time t is

$$Y_{i} \left( t \right) = \eta_{i} \left( t \right) + \varepsilon_{i} \left( t \right)$$

where in the case of emotional functioning, for example, $\eta_{i} \left( t \right)$is the sum of a natural cubic spline (ns) with fixed effects coefficients and a ns with random effects coefficients. The random effects have multivariate normal distribution with mean zero. The errors are independent and normally distributed with mean zero and are also independent of the random effects.

Table 4 Model notation

The survival submodel for the ith individual is defined by the cause-specific hazards for TRM

$$\lambda_{i} \left( t \right) = \lambda_{0} \left( t \right)\exp \left( {\gamma_{1} {\text{gender}}_{i} + \gamma_{2} {\text{HSCTtype}}_{i} + \alpha \eta_{i} \left( t \right)} \right)$$

and disease-related mortality

$$\lambda_{i}^{'} \left( t \right) = \lambda_{0}^{'} \left( t \right)\exp \left( {\gamma_{1}^{'} {\text{gender}}_{i} + \alpha_{i}^{'} \eta_{i} \left( t \right)} \right)$$

The log baseline hazard is approximated by b-splines.

The joint distribution of HRQL (one domain at a time), TRM, and disease-related mortality was modeled as

$$\left[ {T,Y} \right] = \int {\left[ {T|\eta } \right]} \left[ {Y|\eta } \right]\left[ \eta \right]d\eta$$

The joint model assumes that observed HRQL and mortality are conditionally independent, given the latent HRQL.

R Code for emotional functioning joint model

• #### Cox PH Competing Risk Model ####

• #Data for Cox PH (before competing risk format)

• data_surv

id	event	eventtime	childfemale	hscttype
141	1	76	1	1
143	0	447	0	0
145	0	360	0	0
148	0	374	1	1
152	0	378	0	0
154	1	12	1	0
155	0	388	0	0
156	0	466	1	1
158	0	465	1	0
160	0	399	0	0
:

• #create competing risk dataset

• data_surv_cr <-crLong(data_surv, “event”,”0”)

• data_surv_cr

	id	event	eventtime	childfemale	hscttype	strata	status2
141	141	1	76	1	1	1	1
141.1	141	1	76	1	1	2	0
143	143	0	447	0	0	1	0
143.1	143	0	447	0	0	2	0
145	145	0	360	0	0	1	0
145.1	145	0	360	0	0	2	0
148	148	0	374	1	1	1	0
148.1	148	0	374	1	1	2	0
152	152	0	378	0	0	1	0
152.1	152	0	378	0	0	2	0
:

• #Fit Cox PH competing risk model

• fitCoxef_cr <-coxph(Surv(eventtime, status2)   ~   (childfemale   +   hscttype)*strata   +   strata(strata), data   =   data_surv_cr, x   =   T)

• #### Code for LME Model ####

• #Data for LME model

• data_long

id	time	childEF
141	−7	0.000
141	42	5.000
143	−25	5.714
143	107	8.571
143	155	8.929
143	360	8.929
143	416	8.571
145	−18	6.429
145	45	4.643
145	88	5.714
:

• #Fit LME model

• fitLMEef <-lme(childEF   ~   ns(time, df   =   4),

• random   =   list(id   =   pdDiag(form   =   ~   ns(time, df   =   4))),

• na.action   =   na.omit, data   =   data_long)

• #### Code for JM Model ####

• fitJMef <-jointModel(fitLMEef, fitCoxef_cr, timeVar   =   ”time”, method   =   ”spline-PH-aGH”, interFact   =   list(value   =   ~strata, data   =   data_surv_cr), CompRisk   =   T)