Joint modeling of multivariate longitudinal data and the dropout process in a competing risk setting: application to ICU data
- Emmanuelle Deslandes^{1, 2, 3}Email author and
- Sylvie Chevret^{1, 2, 3}
DOI: 10.1186/1471-2288-10-69
© Deslandes and Chevret; licensee BioMed Central Ltd. 2010
Received: 19 April 2010
Accepted: 29 July 2010
Published: 29 July 2010
Abstract
Background
Joint modeling of longitudinal and survival data has been increasingly considered in clinical trials, notably in cancer and AIDS. In critically ill patients admitted to an intensive care unit (ICU), such models also appear to be of interest in the investigation of the effect of treatment on severity scores due to the likely association between the longitudinal score and the dropout process, either caused by deaths or live discharges from the ICU. However, in this competing risk setting, only cause-specific hazard sub-models for the multiple failure types data have been used.
Methods
We propose a joint model that consists of a linear mixed effects submodel for the longitudinal outcome, and a proportional subdistribution hazards submodel for the competing risks survival data, linked together by latent random effects. We use Markov chain Monte Carlo technique of Gibbs sampling to estimate the joint posterior distribution of the unknown parameters of the model. The proposed method is studied and compared to joint model with cause-specific hazards submodel in simulations and applied to a data set that consisted of repeated measurements of severity score and time of discharge and death for 1,401 ICU patients.
Results
Time by treatment interaction was observed on the evolution of the mean SOFA score when ignoring potentially informative dropouts due to ICU deaths and live discharges from the ICU. In contrast, this was no longer significant when modeling the cause-specific hazards of informative dropouts. Such a time by treatment interaction persisted together with an evidence of treatment effect on the hazard of death when modeling dropout processes through the use of the Fine and Gray model for sub-distribution hazards.
Conclusions
In the joint modeling of competing risks with longitudinal response, differences in the handling of competing risk outcomes appear to translate into the estimated difference in treatment effect on the longitudinal outcome. Such a modeling strategy should be carefully defined prior to analysis.
Background
When evaluating the efficacy of a new drug through randomized clinical trials (RCT) in critically ill patients, the primary endpoint of interest is usually death from any cause within some fixed period, generally 28 or 90 days after randomization. However, to better investigate the effect of treatment, one is often interested in evaluating how a biomarker of interest changes over time and how this change may be correlated with the treatment under study; this defines secondary endpoints of interest.
In critically ill patients, the measure of treatment effectiveness is based on the severity of the illness and degree of organ failure, determined using severity scores such as the APACHE (acute physiology and chronic health evaluation) II score [1] or the Glasgow coma score [2] and the SOFA (sequential organ failure assessment) score [3], respectively. However, while the two former scores are mostly used at entry to risk-stratify patients by severity of illness, the latter also applies to quantify evolution of the patient's severity of illness and even benchmark intensive care unit performance [4]. Furthermore, beyond reporting a better record of the course of the disease, it allows for an evaluation of the impact of new treatments on patient outcome [5].
However, to evaluate whether treatment administration influences the course of organ failure, statistical analysis is often based on naive comparisons across randomized groups over time [6–8]. Mixed-effects models, which incorporate repeated measurements of SOFA over time in the same patients, appear to be a well established method for studying the relationship between treatment and the SOFA course. However, given the strong association between organ dysfunction and mortality for critically ill patients, the occurrence of death could result in non-trivial missing data for the longitudinal process. This is likely to provide biased results [9–12].
In a setting where the longitudinal observations may be correlated with survival, joint models of longitudinal and survival processes have been increasingly proposed in the past decade to recover information from these potentially informative censorings [10–26]. Mostly, a Gaussian mixed-effects linear sub-model is assumed for the longitudinal response, although a t-distribution which has a longer tail and thus is more robust to outliers, has been recently proposed [27], and a semi- or fully-parametric survival sub-model fits the survival times. Association between both longitudinal response and survival time is modeled through a zero-mean latent random process, and given all of the random effects, longitudinal measurements and survival times can then be assumed to be conditionally independent.
However, most joint models developed thus far in the literature have focused on univariate time-to-event data, where right censoring of the data acts independently of the survival process under study. In contrast, in the ICU setting, patients discharged alive are likely to be informatively censored. Thus, the analysis of survival data in the ICU in the setting of competing risks has been recently proposed to offer significant advantages over standard survival analyses [28, 29]. Notably, they allow taking the time dependency of risk factors and competing events into account [30].
To study the effects of a covariate in competing risk settings, Cox analysis of cause-specific hazards has long been the technique of choice. Thus, the joint modeling of longitudinal and competing risk data that has been increasingly studied for the past four years first employed the cause-specific hazard sub-model, with a separate latent association between longitudinal measurements and each cause of failure [21, 27, 31–34]. However, although proportional cause-specific hazards modelling is the standard regression model of choice to handle competing risks, results may be difficult to interpret in terms of the cumulative event probabilities. Many authors have noted that the effect of a covariate on the cause-specific hazard function of a particular failure type may be very different from its effect on the cumulative incidence function [28, 35–37]. For supporting clinical decision making, such cause-specific crude cumulative incidence, also known as the cause-specific subdistribution function, which is the probability of the occurrence of a specific event of interest, is widely recognized as clinically useful. This has led to the development of the proportional subdistribution hazards model [36], which offers a synthesis of single cause-specific hazards analyses.
In this paper, we propose a joint random effects model for a longitudinal marker and competing risks data that comprises a proportional subdistribution hazards submodel for the competing risks failure time data. We use the Markov chain Monte Carlo technique of Gibbs sampling to estimate the joint posterior distribution of the unknown parameters of the model, as previously proposed [14, 31, 38]. The paper is organized as follows. First, the ICU data is briefly presented. The next section describes the statistical joint model for the longitudinal and dropout processes.
The performance of our method is evaluated and compared with the cause-specific hazards submodel using both simulated data and the ICU clinical trial. Finally, a discussion is provided in the last section.
Methods
Motivating example
Main characteristics of patients according to randomized arm
N (%) | Arm A (n = 703) | Arm B (n = 698) | p value | |
---|---|---|---|---|
Age (≥63 years) | 360 (51.2) | 347 (49.7) | 0.59 | |
Male Gender | 423 (60.2) | 431 (61.7) | 0.54 | |
Inclusion strata | Trauma patients | 62 (8.8) | 62 (8.9) | |
Sepsis patients | 329 (46.8) | 335 (48.0) | 0.89 | |
Other patients | 312 (44.4) | 301 (43.1) | ||
SOFA score, median[Q1-Q3] | 7.0 [5.0-7.75] | 7.0 [5.0-7.74] | 0.94 |
Separate modeling of SOFA course
Longitudinal | (Posterior mean (95% CI) | |
---|---|---|
Intercept | 6.14 | (5.66, 6.63) * |
Time | -0.22 | (-0.23, -0.20) * |
Treatment group | 0.24 | (-0.16, 0.67) |
Time × Treatment group | -0.02 | (-0.04, -0.003)* |
Age ≥ 63 years | 0.53 | (0.11, 0.94) * |
Male Gender | 0.46 | (0.03, 0.88) * |
Septic patients | 1.45 | (1.03, 1.86) * |
Posterior mean hazard ratio estimates from separate survival models for competing risk data
Cause specific hazards | Subdistribution hazards | ||||
---|---|---|---|---|---|
ICU DEATHS | No deaths/No pts | HR | 95%CI | SHR | 95%CI |
Treatment group | |||||
A | 191/703 | 1 | 1 | ||
B | 182/698 | 1.04 | (0.85,1.27) | 1.03 | (0.85,1.26) |
Age | |||||
<63 years | 151/717 | 1 | 1 | ||
≥63 years | 222/684 | 1.40 | (1.14,1.73)* | 1.58 | (1.28,1.94)* |
Gender | |||||
Female | 150/547 | 1 | 1 | ||
Male | 223/854 | 0.90 | (0.73,1.10) | 0.95 | (0.78,1.16) |
Entry mode | |||||
Other | 185/737 | 1 | 1 | ||
Sepsis | 188/664 | 0.95 | (0.77,1.16) | 1.09 | (0.89,1.33) |
DISCHARGE ALIVE | No events/No pts | ||||
Treatment group | |||||
A | 429/698 | 1 | 1 | ||
B | 431/703 | 0.99 | (0.87,1.13) | 1.00 | 0.87,1.14) |
Age | |||||
<63 years | 493/717 | 1 | 1 | (0.59,0.77)* | |
≥63 years Gender | 367/684 | 0.71 | (0.62,0.81)* | 0.67 | |
Female | 330/547 | 1 | 1 | ||
Male | 530/854 | 0.92 | (0.80,1.06) | 0.95 | (0.83,1.09) |
Entry mode | |||||
Other | 486/737 | 1 | 1 | ||
Sepsis | 374/664 | 0.72 | (0.63,0.83)* | 0.76 | (0.67,0.87)* |
We questioned whether incorporating such prognostic information on dropouts would modify estimates of covariate effects on the SOFA course as exposed above.
The joint model formulation and estimation
Our joint model consists of the two linked submodels, a linear mixed model and a competing risks model.
Longitudinal submodel
where β = (β _{0}, β _{1}) is a parameter vector of regression coefficients commonly referred to as fixed effects in the model; ε _{ ij } ~ N(0, σ^{2}) denotes the zero-mean Gaussian measurement error: we assumed that ε _{ im } was independent of ε _{ is } for any m ≠ s; W _{1i } (t _{ ij } ) refers to the subject-specific random effects, that is, the value at time t _{ ij } of an unobserved zero{mean Gaussian random process. Following previous reports [33, 42], random slope and random-intercept and -slope models were considered, namely W _{1} (t) = U _{1} t or W _{1} (t) = U _{0} + U _{1} t, where (U _{0}, U _{1}) are zero-mean bivariate Gaussian variables.
Competing risks submodel
Let T _{ i } denote the failure time of patient i, and k _{ i } be the cause of failure from two possible causes, where k _{ i } = 1 denotes an ICU death and k _{ i } = 2 denotes a live discharge from the ICU. Let C _{ i } denote the non informative censoring time. Let δ _{ i } = {I[T _{ i } ≤ C _{ i } ] × k _{ i } } be the event indicator, where δ _{ i } = k _{ i } in case of failure and δ _{ i } = 0 for non-informative censoring.
in which λ _{ 0,k } (t) is a non specified baseline subdistribution hazard for failure type k. It appears as a model analogous to the Cox model but based on subdistribution hazards, which is also known as the hazard associated with the crude cumulative incidence function, widely recognized as clinically useful for supporting clinical decision-making [43, 44].
where represent the fixed effects of Z on the two competing risks, k = 1, 2, respectively.
Submodel links
Failure times were associated with the longitudinal response through the latent Gaussian processes W _{1} (t), and , that were assumed to be proportional, i.e.: , where the parameters γ^{(k) }indicate the level of association between the two components of the joint model. Of note, positive values of γ^{(k) }suggests that positive values for associated random effects increase the hazard of "failure", while negative values of γ^{(k) }suggest that the positive values for the random effects decrease the chance of experiencing the event of interest. We further assumed that W _{1} and were independent of the measurement errors ε _{ ij } . At last, the longitudinal measurements and competing risks survival times were assumed to be conditionally independent, given the covariates and random effects.
MCMC sampling procedure
The standard likelihood approach to this problem involves integration of the two sub-models over the distribution of random effects, which requires numerical integration since the two models are not conjugate. As an alternative, to estimate the parameters of interest, we used the Markov-chain Monte-Carlo method of Gibbs sampling to generate the posterior distribution of all unknown parameters of the joint model, given only the observed data.
Results and discussion
Application revisited
Posterior estimates for the ICU data based on joint models
Modeling of dropouts | Cause specific hazards | Subdistribution hazards | ||
---|---|---|---|---|
Sub-model | ||||
LONGITUDINAL | Posterior mean | (95% CrI )) | Posterior mean | (95% CrI )) |
Intercept | 6.29 | (6.05, 6.53)* | 6.24 | (5.74, 6.75)* |
Time | -0.05 | (-0.07, -0.04)* | -0.06 | (-0.07, -0.05)* |
Treatment group | 0.34 | (0.10, 0.58)* | 0.26 | (-0.17, 0.68) |
Time × Treatment group | -0.012 | (-0.04, 0.01) | -0.02 | (-0.03, -0.003)* |
Age (years) | 0.45 | (0.27, 0.63)* | 0.53 | (0.12, 0.96)* |
Male Gender | 0.44 | (0.25, 0.63)* | 0.47 | (0.02, 0.96)* |
Septic patients | 1.33 | (1.15, 1.51)* | 1.44 | (1.01, 1.86)* |
SURVIVAL | HR | (95% CrI)) | SHR | (95% CrI)) |
DEATHS | ||||
Treatment group | 1.22 | (0.98, 1.53) | 1.23 | (1.00, 1.53)* |
Age (years) | 1.93 | (1.55, 2.41)* | 1.38 | (1.10, 1.74)* |
Male Gender | 1.19 | (0.96, 1.50) | 1.35 | (1.07, 1.69)* |
Septic patients | 1.38 | (1.11, 1.72)* | 1.18 | (0.95, 1.46) |
γ^{(1)} | 3.32 | (3.09, 3.57) | 3.42 | (3.24, 3.61) |
DISCHARGES | ||||
Treatment group | 1.17 | (1.02, 1.36)* | 1.25 | (1.08, 1.43)* |
Age (years) | 0.91 | (0.79, 1.05) | 1.08 | (0.93, 1.24) |
Male Gender | 1.25 | (1.07, 1.45)* | 1.30 | (1.11, 1.51)* |
Septic patients | 1.02 | (0.88, 1.18) | 1.12 | (0.96, 1.29) |
γ^{(2)} | -2.96 | (-3.11, -2.80) | -3.35 | (3.48, -3.22) |
A Simulation study
Sampling details
In this section, we conducted a simulation study to illustrate the method, to examine the feasibility as well as properties of the proposed joint model. We simulated the complete data from the following intercept- and slope- random model.
where t _{ ij } = 0, 1,....,7, 14, 28 represent the visit scheduled times, X _{ i } Bernoulli(0:5) acts as the treatment indicator in a randomized clinical trial. The random intercept U _{0i }and slope U _{1i }were assumed normally distributed as N(0, σU _{0} ) and N(0, σU _{1} ), respectively, and independent of the measurement error ε _{ ij } ~N(0, 0.1). Estimates from separate analysis of the longitudinal and the time to event components were reasonable starting values for the model.
which is a unit exponential mixture with mass 1 - p at ∞ when X = 0, and uses the proportional subdistribution hazards model to obtain the subdistribution for nonzero covariate values. The subdistribution for the competing risks failure cause was then obtained using an exponential distribution with rate exp . As detailed above, proportional association between the longitudinal data and the competing risks was generated by setting .
Censoring times were generated from an exponential distribution with rate 0.25. We used the true parameter value of p = 0.25, for n = 100, 500 subjects. This gave 25 per cent cause 1 failures, 65 per cent cause 2 failures, and 10 per cent of censoring.
Based on the collection of previous analyses illustrated in Tables 2 and 3, the following proper priors were used for β _{10}, β _{11}, β _{12}, β _{13}, and . The method used informative priors for some parameters with the prior means (β's) set as the true parameter values. Setting γ = 1 induces a positive association between the competing risks. We also set γ = -1 to induce negatively associated competing risks, that may apply when discharge is a competing cause of failure, as observed in the motivating exemple. Longitudinal responses were missing after the observed or censored event times, with an averaged number of total longitudinal observations of 7:0 per subject. This is consistent with the example findings (6:6 observations per subject). After a burn-in phase of 1,000 iterations, eliminated from the sample to avoid influence of initial parameters, we used means and standard deviations of a single series of 10,000 Gibbs samples as point estimates of the parameters and their standard errors. A total of 100 simulations were performed. Simulations were carried out in R language (R Development Core Team) [48].
Simulation results
Simulation study
Positive association | Negative association | |||
---|---|---|---|---|
Parameter | True value | Bias (SD) | True value | Bias (SD) |
n = 100 | ||||
Longitudinal | ||||
Intercept β _{10} | 6.15 | 0.24 (0.16) | 6.15 | 0.20 (0.16) |
Time β _{11} | -0.25 | 0.22 (0.03) | -0.25 | 0.19 (0.04) |
Binary covariate β _{12} | 0.25 | 0.01 (0.19) | 0.25 | 0.01 (0.20) |
Interaction Time × Binary covariate β _{13} | 0.0 | -0.03 (0.03) | 0.0 | -0.05 (0.03) |
Survival | ||||
Binary covariate β _{2} | 0.0 | 0.25 (0.34) | 0.0 | 0.39 (0.35) |
γ | 1.0 | -0.09 (0.12) | -1.0 | 0.13 (0.12) |
Variances | ||||
σ_{ U0} | 1.0 | 0.04 (0.14) | 1.0 | 0.02 (0.13) |
σ_{ U1} | 1.0 | 0.02 (0.13) | 1.0 | 0.02 (0.13) |
n = 500 | ||||
Longitudinal | ||||
Intercept β _{10} | 6.15 | 0.20 (0.16) | 6.15 | 0.23 (0.04) |
Time β _{11} | -0.25 | 0.02 (0.02) | -0.25 | 0.05 (0.05) |
Binary covariate β _{12} | 0.25 | 0.04 (0.22) | 0.25 | 0.03 (0.06) |
Interaction Time × Binary covariate β _{13} | 0.0 | -0.007 (0.04) | 0.0 | -0.01 (0.01) |
Survival | ||||
Binary covariate β _{2} | 0.0 | 0.12 (0.29) | 0.0 | -0.16 (0.12) |
γ | 1.0 | -0.03 (0.09) | -1.0 | -0.04 (0.10) |
Variances | ||||
σ_{ U0} | 1.0 | 0.02 (0.14) | 1.0 | 0.04 (0.12) |
σ_{ U1 } | 1.0 | 0.03 (0.14) | 1.0 | 0.03 (0.13) |
Discussion
In this paper, we aimed at comparing two treatment groups with respect to the course of the SOFA score in critically ill patients. Analysis was complicated by informative dropouts, since once a patient has been discharged, either alive or dead, from the ICU, no longitudinal measure of the severity score of interest can be collected thereafter. Thus, when analyzing such data, separate modeling of the SOFA score, that is, ignoring the dropout process, is likely to be inappropriate and one should obtain less biased and more efficient inferences using joint models Actually, joint models allow incorporating informative censoring and time by treatment interaction, and provide complementary information when assessing how the treatment manifests itself through the marker [49].
Such joint models in this particular setting required modelling assumptions. First, to assess time by treatment interaction on the SOFA score, a linear time effect was assumed. Indeed, as depicted in Figure 2a, the average SOFA monotonically decreases over time, so that quadratic time pattern was unlikely. Of note, the observed profiles may have suggested a change point at 7 days in the slope whatever the treatment arm, so that a piecewise mixed-effects model could have been fitted. However, we only introduced one linear time trend because of the data, due to the absence of any time points after day 7 except at day 14 and day 28, and due to the increased amount of informative dropouts over time, notably after day 7 (Figure 2b). Secondly, in the particular setting of ICU data, where dropouts result from either deaths or live discharges, models for competing risk failure time data should be used to fit the survival responses [28–30]. Since the SOFA score was actually measured only in patients during their ICU stay, the possibly informative dropout process of interest was clearly that of ICU deaths/discharges. This avoided open issues with regards to the primary outcome to be used in the ICU, as well as on the model to be fitted to such data -either based on binary [50] or survival data analysis techniques [30]. Joint models could also apply to other competing risks settings such as those published by the UCLA team in scleroderma-related interstitial lung disease with intermittent measures of forced vital capacity which were informatively censored by study withdrawal due to disease or treatment related reasons [27, 32, 51]. Based on our simulation study, we observed that an increase in the sample size decreased the estimation bias for the parameters in both submodels. However, we observed, as also noted by Hu [38] in their own model, that the implemented method is sensitive to outliers. A further development will be to implement a more robust joint model for longitudinal and survival data.
A few authors have already proposed a joint modeling of longitudinal and multivariate survival data [31, 33, 38]. Our proposed approach differs from those in two main points. First, a cause-specific hazard sub-model [33] or a frailty model [31] has been conventionally used to handle several types of failures; we decided instead to fit to a sub-distribution hazard sub-model [36] to provide estimates of treatment effect directly related to the cumulative incidence of dropouts [43, 44]. Secondly, an EM algorithm was used for inference purposes in [33]. Actually, the Bayesian framework of Chi in 2006 [31] motivated our investigation of a Bayesian alternative that allows full and exact posterior inference for any parameter or predictive quantity of interest. Thus, we developed a fully Bayesian approach, implemented via MCMC methods using WinBUGS software, as previously reported [14, 31, 52, 53]. Such a Bayesian method for joint modeling of longitudinal and competing risks survival data was very recently reported in the setting of cause-specific hazards sub-model [38]. We illustrated in this paper how the joint model strategy may affect the results. Our results suggest that the treatment effect on the SOFA course in separate modeling of the SOFA course could be evidenced when considering informative censoring modeled by sub-distribution hazards. The significant treatment by time interaction was erased by the modeling of informative dropouts throughout cause-specific hazards.
In the setting of joint modeling of competing risks together with that of longitudinal response, such a difference in the handling of competing risk outcomes based on the Fine and Gray model appears to translate into the observed difference in treatment effect on the longitudinal outcome. This makes clear the requirement for statistical analysis of such data to be clearly planned in the protocol of such studies. Other approaches, such as marginal structural models for non-dynamic treatment regimes, appear to be of prime interest in this setting [54, 55].
Valid inference requires a framework in which potential underlying relationships between the event and longitudinal process are explicitly acknowledged. Latent variable models used in this context do not directly model the association between longitudinal and survival response, but rather focus on correlated, latent random effects. The random intercept-and-slope model was found to give a significantly improved fit (by DIC) above other models examined such as the random intercept-only model. This is similar to that reported by Williamson [33]. Additionally, Henderson et al. [15] described a latent variable association model, and Lin et al. [56] concentrated on latent class mixed models. In a recent paper, Liu [57] showed that the hazard of death may be dependent on random effects from various levels. In this way, Tsiatis and Davidian provided a nice overview of joint models [12], describing further details on underlying assumptions statements and on the likelihood of model parameters in such models.
Conclusions
The consideration of joint models permits useful analysis of very complex data. It could help to improve estimation of the impact of proposed prognostic features on the main endpoints in the trial. We proposed a method that gives accurate estimates, and a Bayesian alternative that permits full and exact posterior inference for any parameter or predictive quantity of interest.
Declarations
Acknowledgements
We gratefully acknowledge the Cristal Study group for providing the data, and the Reviewers for very helpful comments to this manuscript.
Authors’ Affiliations
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