Joint modeling of longitudinal health-related quality of life data and survival

Abstract

Purpose

In cancer research, outcome measures may co-vary. Treatment and treatment related impairment of health-related quality of life (HRQoL) may affect survival. When these effects are analyzed separately, bias may arise. Therefore, we investigated the combined effect of treatment and longitudinally measured HRQoL on survival.

Methods

Patients with anaplastic oligodendrogliomas (n = 288) who were randomized (EORTC 26951) to radiotherapy (RT) alone or RT plus procarbazine, lomustine, and vincristine (PCV) chemotherapy were analyzed. HRQoL [appetite loss (AP)] was assessed with the EORTC QLQ-C30. We compared survival results from different analysis strategies: Cox model with treatment only [model 1 (M1)] or with treatment and time-dependent AP score [model 2 (M2)] and the joint model combining longitudinal AP score and survival [model 3 (M3)].

Results

The estimated hazard ratio (HR) for RT plus PCV was 0.76 (95 % CI 0.58–1.00) for M1, 0.72 (0.55–0.96) for M2, and 0.69 (0.52–0.92) for M3. This corresponds to a lower risk of death of 24 % in M1, 28 % in M2, and 31 % in M3, for patients treated with RT plus PCV chemotherapy. AP resulted in an increased risk of death, with estimated HR of 1.06 (1.01–1.12) for M2 and 1.13 (1.03–1.23) for M3: Every 10-point increase of AP resulted in a 13 % increased risk of death in M3 as compared to 6 % in M2.

Conclusion

Part of the survival benefit of treatment with RT plus PCV chemotherapy can be masked by the negative effect that this treatment has on patients’ HRQoL. In our study, up to 7 % of the theoretical treatment efficacy was lost when AP was not adjusted through joint modeling.

Appendix

Formula for model 1

$$h_{i} (t) = h_{0} (t)\exp (\gamma_{1} \omega_{i1} + \gamma_{2} \omega_{i2} + \cdots + \gamma_{p} \omega_{ip} )$$

(1)

where $h_{i} (t)$ denotes the hazard for an event for a patient i at time t, $h_{0} (t)$ denotes the baseline hazard, and $\omega_{i1} , \ldots ,\omega_{ip}$ is a set of covariates.

Formula for model 2

$$h_{i} \left( {t\left| { Y_{i} (t), \omega_{i} } \right.} \right) = h_{0} (t)R_{i} \exp \left\{ {\gamma^{T} \omega_{i} + \alpha y_{i} (t)} \right\},$$

(2)

where $N_{i} (t)$ is the counting process which counts the number of events for patient i by time t, $h_{i} (t)$ denotes the intensity process for $N_{i} (t)$, $R_{i} (t)$ denotes the risk process (‘1’ if patient i still at risk at t), and $y_{i} (t)$ denotes the value of the time-dependent covariate at t.

Formula for model 3

Survival submodel

$$h_{i} \left( {t\left| {M_{i} (t)} \right.} \right) = h_{0} (t)\exp \left\{ {\gamma^{T} \omega_{i} + \alpha m_{i} (t)} \right\},$$

(3a)

where $M_{i} (t) = \left\{ {m_{i} (s),\;0 \le s < t} \right\}$ denotes the history of the true unobserved longitudinal HRQoL score up to time t, $h_{0} ( \cdot )$ denotes the baseline risk function, $\alpha$ quantifies the effect of HRQoL score on the hazard for death, $\omega_{i}$ baseline covariates.

Linear mixed-effect submodel

$$\begin{aligned} y_{i} (t) & = m_{i} (t) + \varepsilon_{i} (t) \\ & = \left( {\beta_{0} + b_{i0} } \right) + \left( {\beta_{1} + b_{i1} } \right)B_{n} \left( {t,\lambda_{1} } \right) + \left( {\beta_{2} + b_{i2} } \right)B_{n} \left( {t,\lambda_{2} } \right) + \left( {\beta_{3} + b_{i3} } \right)B_{n} \left( {t,\lambda_{3} } \right) \\ & \quad + \beta_{4} \left\{ {B_{n} (t,\lambda_{1} ) \times {\text{TTRT}}_{i} } \right\} + \beta_{5} \left\{ {B_{n} (t,\lambda_{2} ) \times {\text{TTRT}}_{i} } \right\} + \beta_{6} \left\{ {B_{n} (t,\lambda_{3} ) \times {\text{TTRT}}_{i} } \right\} \\ & \quad + \beta_{7} {\text{Age}}_{i} + \beta_{8} {\text{TTRT}}_{i} + \beta_{9} {\text{WHO}}_{i} + \beta_{10} {\text{PRES}}_{i} + \beta_{10} {\text{CUSUR}}_{i} + \varepsilon_{i} (t),\quad \varepsilon_{i} (t)\sim N(0,\sigma^{2} ) \\ \end{aligned}$$

(3b)

where $\left\{ {B_{n} \left( {t,\lambda_{k} } \right);k = 1,2,3} \right\}$ denotes a B-splines basis matrix for a natural cubic splines of time 3 knots placed at equally spaced percentiles of the follow-up times, TTRT is the dummy variable for treatment. Age, WHO, PRES, and COSUR are the baseline covariates. $b_{i}$ is the random-effects part, $b_{i} \sim N(0, D)$, $\varepsilon_{i} (t)$ is the measurement error term which is assumed independent of $b_{i}$ with variance $\sigma^{2}$.

Joint distribution

$$p\left( {y_{i} ,T_{i} \delta_{i} } \right) = \mathop \int \nolimits p\left( {y_{i} \left| {b_{i} } \right.} \right)\left\{ {h(T_{i} \left| {b_{i} } \right.)^{{\delta_{i} }} S(T_{i} \left| {b_{i} } \right.)} \right\} p\left( {b_{i} } \right)db_{i} ,$$

(3c)

where b _i is a vector of random effects that explains the interdependencies, $p( \cdot )$ is the density function, and $S( \cdot )$ is the survival function.

Dynamic predictions

$$Y_{i} (t) = \left\{ {y_{i} \left( s \right), 0 \le s \le t} \right\}$$

(4a)

and we are interested in

$$\pi_{i} \left( {u\left| t \right.} \right) = \Pr \left\{ {T_{i}^{*} \ge u|T_{i}^{*} > t, Y_{i} (t), \omega_{i} D_{n} ;\theta^{*} } \right\},\quad t > 0,$$

(4b)

where $u > t, \theta^{*}$ denotes the true parameter values and $D_{n}$ denotes the sample the joint model was fitted.

SAS and R code to fit Cox models and joint models

Description: The SAS codes illustrate basic extended Cox models. The R script illustrates the basic use of the R package JM for fitting joint models for longitudinal and survival data.

• The EORTC 26951 dataset

• # AP score = longitudinal marker

• # SS = Survival times

• # TSS = Survival status

• # TRTT = Randomized treatment

• # QOLTIMES = AP score measurement times

• #Baseline covariates: Age (>40 or ≤40), WHO (WHO performance status 0 or 1 vs. 2), PRES (prior surgery for a low-grade oligodendroglioma; yes or no), and COSUR (surgery; biopsy only vs. debulking surgery/resection).

SAS codes

Basic Cox model (model 1)

Time-dependent Cox model (model 2)

R script

The joint model: jointModel() takes the above fitted models as arguments and fits the joint model; below, we fit a joint model with a relative risk submodel for the event time outcome, in which the baseline risk function is assumed piecewise constant