Biostatistics

Biostatistics Advance Access published online on November 19, 2007
Biostatistics, doi:10.1093/biostatistics/kxm038

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The separation of timescales in Bayesian survival modeling of the time-varying effect of a time-dependent exposure

Sebastien J.-P. A. Haneuse^*

Center for Health Studies, 1730 Minor Avenue, Suite 1600, Seattle, WA 98101-1448, USA haneuse.s{at}ghc.org

Kyle D. Rudser

Department of Biostatistics, University of Washington, Seattle, WA, USA

Daniel L. Gillen

Department of Statistics, University of California, Irvine, Irvine, CA, USA

^* To whom correspondence should be addressed.

	SUMMARY

TOP SUMMARY 1. INTRODUCTION 2. PROPOSED FRAMEWORK 3. ANALYSIS OF USRDS... 4. DISCUSSION REFERENCES

 
In this paper, we apply flexible Bayesian survival analysis methods to investigate the risk of lymphoma associated with kidney transplantation among patients with end-stage renal disease. Of key interest is the potentially time-varying effect of a time-dependent exposure: transplant status. Bayesian modeling of the baseline hazard and the effect of transplant requires consideration of 2 timescales: time since study start and time since transplantation, respectively. Previous related work has not dealt with the separation of multiple timescales. Using a hierarchical model for the hazard function, both timescales are incorporated via conditionally independent stochastic processes; smoothing of each process is specified via intrinsic conditional Gaussian autoregressions. Features of the corresponding posterior distribution are evaluated from draws obtained via a Metropolis–Hastings–Green algorithm.

Keywords: Bayesian survival analysis; Conditional autoregression; Nonproportional hazards; Reversible jump Markov chain Monte Carlo

	1. INTRODUCTION

TOP SUMMARY 1. INTRODUCTION 2. PROPOSED FRAMEWORK 3. ANALYSIS OF USRDS... 4. DISCUSSION REFERENCES

 
End-stage renal disease (ESRD) is a condition where the filtration performed by the kidneys has been reduced to a point at which life can no longer adequately be sustained. It is estimated that more than 300 000 persons in the United States have ESRD, and this number has been steadily rising over the past few decades (Jones and others, 1998). The standard of care for patients suffering from ESRD is renal replacement therapy in the form of dialysis or kidney transplantation. Despite numerous studies indicating decreased mortality associated with renal transplantation when compared to dialysis (e.g. Wolfe and others, 1999), the risk of certain comorbidities does rise following surgery.

One such condition for which rates have been observed to increase following transplantation is the occurrence of posttransplant lymphoma (PTL) due to immunosuppression regimes prior to and following surgery. Although prospective viral screening posttransplant may identify patients at high risk for PTL, little is known about the timing of lymphoma onset, making it difficult to create standard surveillance protocols. Recently, Smith and others (2006) reported an investigation of the onset of PTL among 89 260 ESRD patients from the United States Renal Data System (USRDS). To quantify a possible time-varying relationship between renal transplant and risk of lymphoma in a Cox proportional hazards model, time since transplantation was categorized into 6 equally spaced intervals and included in the model as a series of time-dependent dummy variables.

While this approach seems reasonable, there is room for improvement. From a scientific perspective, the choice to split posttransplant time into 6 equally spaced intervals was arbitrary. From a statistical perspective, the rare nature of the outcome poses challenges and with few observed events the extent to which one may rely on asymptotic theory for valid inference is unclear. Further, fixing the partition of the posttransplant timescale may impose unreasonable structure on the model and hence standard error estimates may not reflect additional uncertainty associated with the number and position of interval cutpoints. Allowing the number and position of interval cutpoints to be determined by the data would help to overcome this latter issue. However, within the frequentist statistical framework, such a model would be difficult to maximize and, again, valid standard error estimation would likely be problematic. Faced with these challenges, we consider Bayesian methods to analyze more flexibly the USRDS PTL survival data. In particular, we build on the previous literature and develop a flexible Bayesian hazards regression model to incorporate the potentially time-varying effect of a time-dependent covariate.

In the analysis of failure time data, a common approach is to consider a multiplicative hazards model such as the proportional hazards model (Cox, 1972). While incorporation of time-dependent covariates in these models is relatively straightforward, non-parametric estimation of potentially time-varying regression parameters remains a difficult problem. Fan and others (1997) considered local estimation of a time-varying regression parameter for a multiplicative hazards model. More recently, Tutz and Binder (2004) proposed the use of penalized B-splines for estimating time-varying coefficients in discrete failure time models, while Brown and others (2007) considered the use of P-splines in the context of continuous failure times.

In the frequentist setting, semi-parametric methods such as those referenced above avoid modeling the baseline hazard function by maximizing the partial likelihood. However, the Bayesian paradigm requires explicit parameterization of the baseline hazard. Various approaches to this task have been considered, early approaches employing discrete processes with independent increments (Kalbfleisch, 1978, Hjort, 1990). More recent work has sought to relax the independence framework (Gamerman, 1991; Arjas and Gasbarra, 1994; Gelfand and Mallick, 1995; Mallick and others, 1999; Nieto-Barajas and Walker, 2002, 2005; Pennell and Dunson, 2006); Ibrahim and others (2005) provide a broad overview.

Although these methods permit great flexibility in modeling the baseline hazard function, none incorporate a flexible strategy for modeling a potentially time-varying covariate effect. Recently, McKeague and Tighiouart (2000) considered modeling the log-baseline hazard function via a mixture of step functions, mixing over random jump times. To incorporate time-varying effects, they proposed modeling the log-hazard ratio (HR), ß(·), using the same approach. While this presented a substantial improvement in the ability to model a time-varying treatment effect, the model assumed the same timescale and, more importantly, the same jump times for both the baseline hazard function and the covariate effect. Other recent work has considered the use of splines to parameterize time-varying quantities in the Bayesian framework (Lambert and Eilers, 2005; Hennerfeind and others 2006).

In this paper, we build on the framework proposed by McKeague and Tighiouart (2000). Specifically, we adapt their approach by separating the specification of the 2 underlying timescales as well as relaxing a variety of additional modeling assumptions key to the current application. The remainder of the paper is structured as follows: In Section 2, we define the proposed model. Computation and implementation are briefly outlined although technical details are left to a separate supplementary material document available at Biostatistics online (http://www.biostatistics.oxfordjournals.org). Section 3 provides a description of the USRDS data as well as primary results characterizing the association between the risk of lymphoma and the time since transplantation. Extensive sensitivity analyses examining the impact of various modeling choices are provided in the supplementary material available at Biostatistics online. Finally, Section 4 concludes with a summary and brief discussion.

	2. PROPOSED FRAMEWORK

TOP SUMMARY 1. INTRODUCTION 2. PROPOSED FRAMEWORK 3. ANALYSIS OF USRDS... 4. DISCUSSION REFERENCES

 

2.1 Separation of timescales

In the analysis of survival outcomes, one is required to define the underlying timescale as well as the time of origin. In the context of the USRDS transplant data, the origin is the time at which a subject is placed onto the waiting list. We refer to a subsequent follow-up as being on the "study timescale." Here, the primary scientific interest lies in characterizing the impact of a time-dependent exposure (transplant) on survival outcomes, while allowing for the possibility that the impact of transplant may change over time. To characterize the time-varying effect, we introduce a second timescale, the "transplant timescale." This scale measures observed follow-up posttransplantation and runs in parallel to the study timescale. The origin for the transplant timescale is taken to be the time of transplantation.

2.2 Observed likelihood

For each of n subjects, let T_i and C_i denote the true survival and censoring times for subject i, i = 1, ..., n. The observed survival time for subject i, denoted X_i, is taken to be the minimum of T_i and C_i and is defined on the study timescale. Let _i be a failure indicator taking on a value of 1 if the true survival time is observed and 0 otherwise. Let Z_i denote a vector of time-invariant adjustment covariates, measured at the time of entry, for subject i.

For the ESRD analysis, the primary scientific hypothesis concerns the risk of lymphoma associated with transplantation. Toward this, let R_i(·) denote a time-dependent indicator of transplant status for subject i and for subjects that receive a new kidney, let T denote the time of transplantation. For subjects that do not receive a transplant during the observation period, T is arbitrarily set to . We assume transplant status to be monotone; without loss of generality, R_i(·) takes on a value of 0 prior to transplantation and 1 at transplant and thereafter. Throughout this paper, we distinguish between random variables and their realizations with capital and lower-case letters, respectively.

To evaluate the association between kidney transplantation and risk of lymphoma, we adopt a multiplicative hazards model of the form

(2.1)

where h₀(·) is the baseline hazard function, is a vector of log-HR parameters which are assumed to satisfy proportional hazards, and ß(·) is the (possibly) time-varying effect of transplant. Note that h₀(·) is defined on the study timescale, while ß(·) is defined on the transplant timescale (see Section 2.1). Also, note that for the ith subject, when t < T, ß(·) is defined to equal zero. In this model, we interpret the transplant effect as follows: adjusting for covariates Z, exp{ß(s)} is the HR, for the risk of lymphoma, comparing transplant recipients s time units posttransplant to subjects on the waiting list and yet to receive a transplant.

Assuming independent censoring, the form of the likelihood for the observed data is given by

(2.2)

2.3 Parameterization of time-dependent functions

A fully Bayesian specification requires the explicit parameterization of the time-dependent quantities h₀(·) and ß(·). We build on McKeague and Tighiouart (2000) and specify each of these via mixtures of piecewise constant functions. This approach requires the partitioning of time although, as mentioned above, in the present situation there are 2 timescales. Hence, at the second stage, parallel developments are required for h₀(·) and ß(·) separately.

Let = max{X_i;i = 1,...,n} denote the maximum observed survival time. Consider a partition of the observed study timescale [0, ] into K^L mutually exclusive intervals, for which the end points are denoted by ^L = {,...,,}, where 0 and . For a fixed K^L and ^L, we assume the log-baseline hazard function, (·) = log h₀(·), to be piecewise constant as follows:

(2.3)

where h_0k = exp{_k} is the height of the baseline hazard function on the kth study time interval (, ]. For notational convenience, let = {₁,...,_K^L}.

Similarly, let denote the maximum observation time posttransplant and ^B = {,...,,} the end points which induce a partition of the transplant timescale [0, ] into K^B mutually exclusive intervals. For fixed K^B and ^B, we assume the function ß(·) to be piecewise constant:

(2.4)

where ß_b denotes the height of the function on the bth transplant time interval (, ]. At this point, we note that, due to administrative censoring in the USRDS data (see Section 3.1), = 3 years.

For fixed ^L and ^B, an analysis could proceed by placing priors on the component heights and evaluating the corresponding posterior. In terms of the transplant effect, this would be the Bayesian analogue to the approach taken by Smith and others (2006). Here, we seek to avoid making specific assumptions regarding the 2 partitions and explore a framework within which the data are used to inform their structure. This involves allowing both the number of split times as well as their positions to be random and hence updated via the Bayesian framework. To simplify exposition, we outline the approach for ß(·) on the transplant scale and note that a parallel approach is taken for the log-baseline hazard (·) on the study timescale.

For a fixed , assume the time points ,,... form a time-homogenous Poisson process on [0, ] with rate ^B. That is, the number of splits on [0, ] is distributed according to a Poisson distribution with mean ^B, and the resulting ,..., are the order statistics of K^B – 1 uniformly distributed random variables. Conditional on the partition end points ^B, the K^B heights ß = (ß₁,...,ß_K^B) are assumed to be distributed according to a K^B-dimensional multivariate normal distribution:

(2.5)

where µ_ß is the overall trend (assumed constant across time) in the ß_b heights and > 0 is the overall variability. The K^BxK^B correlation matrix _ß is specified so that there is structure in the ß_b across time. The resulting smoothing may be viewed in the context of (one-dimensional) spatial models, for which the specification of _ß has received much attention (see, e.g. Wakefield and others, 2000,). Modeling of _ß could proceed in a variety of ways; we follow the conditional approach of McKeague and Tighiouart (2000), but adopt a somewhat different weighting scheme (for details, see supplementary material available at Biostatistics online).

2.4 Prior distributional assumptions

For each of the log-HRs in , we assume noninformative flat priors on the real line. In the absence of further information, we set ^B = ^L = 5, which reflects an a priori expectation of 6 intervals on both the timescales. For the transplant effect ß(·), this corresponds to the choice of Smith and others (2006).

In the implementation of their approach, McKeague and Tighiouart (2000) treat the second-stage hyperparamters µ, µ_ß, and as fixed at prespecified levels. Although the discussion of McKeague and Tighiouart (2000) comments on extending their approach, here we provide details on relaxing these assumptions. For the marginal means, µ_ß and µ, we adopt flat priors on the real line. For the variance components, and , a standard approach is to parameterize the model in terms of the precision and adopt a conjugate Gamma prior. Here, we adopt a Gamma(0.5, 0.01) prior for both the baseline hazard and the treatment effect precision parameters. The corresponding induced prior on the standard deviation scale, and _ß, has a median of 0.21 and 95% of its central mass between values of 0.06 and 4.51. In the supplementary material available at Biostatistics online, we examine the impact of this choice via a sensitivity analysis.

Finally, we set the values of the spatial hyperparameters, c and c_ß, equal to 1.0. This value is consistent, in the sense that it represents a limiting form of the model specification, with common choices used in disease mapping and environmental epidemiology applications when expressing prior beliefs about the spatial component of a disease risk model (Besag and others, 1991; Wakefield and others 2000). The impact of alternative choices are considered in the supplementary material available at Biostatistics online.

In practice, it may be difficult to choose values of the hyperparameters introduced above. One option is to perform sensitivity analyses, as we have done; for the Poisson rates, {, _ß}, McKeague and Tighiouart (2002) give recommendations based on a series of simulations studies.

2.5 Posterior

For fixed K^L and K^B, let (K^L,K^B) = (^L,,µ,,,^B,ß,µ_ß,_ß) denote the complete set of unknown parameters in the 3-stage hierarchical model. The posterior distribution, which forms the basis for estimation and inference, is taken to be the combination of the 3 stages as follows:

(2.6)

where N = K^L – 1 and N_ß = K^B – 1 are the number of split times on the study and transplant timescales, respectively.

2.6 Implementation

Suppose that (K^L,K^B) takes values in the (8 + P + 2(K^L + K^B))-dimensional parameter space (K^L,K^B). The full parameter space for the entire specification is therefore where is the set of positive integers. To evaluate features of the posterior distribution, we have implemented a reversible jump Markov chain Monte Carlo (MCMC) algorithm (Green, 1995) in the statistical programming environment R (R Development Core Team, 2006).

For each of the analyses outlined in Section 3, as well as those of the sensitivity analysis, the posterior estimates are based on 2 MCMC chains run simultaneously for at least 1 million scans with a burn-in of 25%, taking just over 6 h to run on an Apple 2.7GHz Power Mac G5. Although the model is complex, the implemented algorithm achieves an overall acceptance rate of around 40%. A detailed description of the algorithm together with diagnostic plots/summary statistics which indicate good mixing of the chains and convergence to the stationary distribution are provided in the supplementary material available at Biostatistics online.

	3. ANALYSIS OF USRDS DATA

TOP SUMMARY 1. INTRODUCTION 2. PROPOSED FRAMEWORK 3. ANALYSIS OF USRDS... 4. DISCUSSION REFERENCES

 

3.1 USRDS transplant data

The USRDS contains demographic and clinical information on all patients with ESRD in the United States who survived more than 90 days from the start of renal replacement therapy (http://www.usrds.org). For the current analysis, we included all caucasian ESRD patients placed on the transplant waiting list between January 1, 1990, and December 31, 2004, who had not previously received a renal transplant and who had Medicare as their primary insurance at the time of placement on the list. Patients were considered at risk from the time of placement onto the transplant waiting list to the first of the following: date of lymphoma diagnosis, date of death, lost-to-follow-up, end of study, or 3 years after the date of transplant. Follow-up was censored 3 years after the date of transplant because transplant recipients younger than 65 who maintain a successful renal transplant for at least 3 years may no longer be eligible for Medicare at that time and are therefore not uniformly included in the USRDS database from that date forward.

Lymphoma is characterized by uncontrolled B-cell proliferation that is often attributed to primary Epstein–Barr virus (EBV) infection. As primary infection is most likely to occur during childhood and adolescence, it is likely that the risk of lymphoma associated with transplantation will also differ by age. As such, we stratified the sample into 4 groups on the basis of age at the initiation of dialysis: less than 25, 25–44, 45–59, and at least 60. Analyses were performed separately for each age stratum, controlling for gender and duration of dialysis prior to being placed on the waiting list as potential confounders. Additional adjustment for age, within strata, was incorporated via a linear term. This differs from the Smith and others (2006) analysis in that they assumed common effects for the confounder variables across the 4 age groups and did not adjust for age beyond the 4-level grouping. Finally, to help with interpretation of the baseline hazard, the referent group for gender was set to male, duration of dialysis was centered at 6 months (the median observed time in the study sample), and age was centered at 20, 35, 50, and 65 in the 4 age strata, respectively.

The final study sample consisted of N = 85056 ESRD patients; 54 740 received a transplant over the course of follow-up. A total of 692 lymphoma diagnoses were observed during the study period, with 431 of these occurring after transplantation. Further details of the USRDS sample together with patient characteristics are available in the supplementary material available at Biostatistics online.

3.2 Results

Table 1 provides the maximum partial likelihood estimates and the corresponding 95% confidence intervals for HRs comparing the risk of lymphoma among transplant recipients during various intervals posttransplant to those patients on the waiting list. For each of these analyses, the partitioning of the transplant timescale was fixed at six 6-month intervals. Among the youngest patients (less than 25 years of age), there is strong evidence to suggest that during the first-year posttransplant, the risk of lymphoma increases but that beyond one year, despite point estimates greater than 1.0, one cannot rule out no effect. As expected, similar but less dramatic results are obtained for the next 2 age groups. For the oldest patients, there is a suggestion of increased risk during the first 6 months (HR = 1.71; 95% CI 1.05, 2.73) with no evidence of differential risk after 6 months. The observed decrease in the HR with increasing time since transplant is consistent with the hypothesis that an increased risk of lymphoma is partially attributable to immunosuppression; prior to and immediately following transplantation, patients are given higher levels of immunosuppression than their long-term stable regimes.

View this table: [in this window] [in a new window]   Table 1. Analysis of the effect of transplantation on the risk of lymphoma, by age at initiation of dialysis, among caucasian ESRD patients placed on the transplant waiting list between January 1, 1990, and December 31, 2004

 
Figure 1 presents posterior results for the time-varying transplant effect for each of the 4 age strata. Within each panel, the lower portion on the plot provides (relative) posterior probabilities associated with the positions of split times for the timescale partitions. The upper portion provides (pointwise) posterior medians and 95% credible intervals (CIs) for the (potentially) time-varying quantities.

View larger version (18K): [in this window] [in a new window] [Download PowerPoint slide]   Fig. 1. Posterior medians (solid) and 95% credible intervals (dashed) for the time-varying transplant effect, ß(·), by (grouped) age at initiation of dialysis. Estimates are adjusted for gender, duration on dialysis prior to placement on waiting list, and age at initiation of dialysis. Lower portions provide (relative) posterior probabilities associated with the positions of the split times.

 
We find that the shape across time posttransplant varies substantially by age group. Among the youngest patients, there is evidence of elevated risk during the first 6 months, which then decreases approximately linearly over the next 6 months. The 95% CIs indicate that during this 12-month period the elevated risk is statistically significant. After the first 12 months, the shape plateaus and, although the risk remains elevated, one can no longer rule out no transplant effect. The shape of the distribution of posterior probabilities of split times on the transplant timescale is consistent with the smooth nature of the transition between the initial 6-month period and the plateau that occurs after 12 months. For the next 2 age groups, there is a general indication of elevated risk early on although the decrease is more linear in time and the estimates are attenuated. In the final age group, although the posterior median is consistently above zero, the pointwise 95% CIs indicate that there is insufficient evidence to rule out no effect of transplant.

In addition to these graphical summaries of Figure 1, the second column of Table 1 provides the corresponding numerical values, on the HR scale, at 3, 9, 15, 21, 27, and 33 months. Overall, we find that the main trends are consistent between the 2 analysis approaches. As expected, the Bayesian analysis produces estimates that are smoother than the frequentist analysis, with the posterior medians exhibiting less variation over time. Comparing the 2 sets of interval estimates in Table 1, we note that the Bayesian analysis suggests greater uncertainty in the estimation of ß(·). The supplementary material available at Biostatistics online provide a more detailed graphical comparison of estimation for the time-varying transplant effect.

Finally, we note that by employing the Bayesian framework, we are able to address a variety of scientific questions not feasible in the original analysis of Smith and others (2006). For example, in posttransplant monitoring, it would be of interest to understand when a patient's risk returns to pretransplant levels. In terms of the model (2.1), this refers to the time at which ß(·) = 0. This can be estimated by considering each scan in the MCMC scheme and noting the first interval for which the height, ß_b, is below zero (see the supplementary material available at Biostatistics online for examples). In scans for which the first interval is below zero, we took the time to be the first instance of the height dropping to zero following an initial increase. For scans in which the height never exceeded zero, as was common for the older age groups, we defined the time to equal zero. For scans in which the height never drops below zero, we censor the time at 3 years. The latter is necessary since we only have 3 years of posttransplant follow-up. This provides a challenge for characterizing the posterior distribution of the time at which ß(·) = 0. However, we are able to report the 2.5 percentiles for the 4 age groups as 0.79, 0.91, 0.82, and 0.33 years, respectively. Based upon our model estimates, transplanted patients in all age groups, with the exception of the 60 above group, experienced a significant increase in the risk PTL for at least the first 9 months following surgery, making this a critical period for clinical monitoring.

	4. DISCUSSION

TOP SUMMARY 1. INTRODUCTION 2. PROPOSED FRAMEWORK 3. ANALYSIS OF USRDS... 4. DISCUSSION REFERENCES

 
In this paper, we model the (potentially) time-varying effect of a time-dependent covariate in the setting of an analysis of lymphoma risk associated with kidney transplant. A key aspect of the problem is that 2 timescales must be considered simultaneously. We have adapted and extended the work of McKeague and Tighiouart (2000) to ensure that (i) the 2 timescales, study time and transplant time, are specified separately and (ii) the estimates of variability fully reflect uncertainty in the 2 timescale partitions. Based on this work, the substantive results from the proposed analysis provide a more thorough understanding of the risk of lymphoma and its association with kidney transplantation. We gain a clearer picture of the shape of the progression of risk over time and find that after an initial period of elevated risk there is an approximately linear decline from 6 to 12 months. Such knowledge can be used, in combination with consideration of established risk factors (e.g. EBV serostatus) and cost, to provide clinical guidance for the schedule of cell proliferation surveillance posttransplant. A variety of avenues for further work are possible. While we adopt a multiplicative hazards framework, the random split-times approach presented here could similarly be extended to the accelerated failure time model setting (Christensen and Johnson, 1988; Walker and Mallick, 1999). More generally, the methods of this paper are applicable to a wide range of problems where interest lies in estimating and characterizing a risk process over time. Recently, Berry and others (2004) considered modeling a time-dependent treatment effect as piecewise constant, but on the same fixed partitioning of the timescale as the baseline hazard. An additional degree of flexibility in this analysis could be achieved by employing a random split-times model and allowing for separate timescales for the baseline hazard and treatment effect functions. Gillen (2006) considers the monitoring of survival in a group sequential framework where the assumption of proportional hazards may not hold. A similar approach to the one presented here is used to characterize the hazard function of each comparison group over time and using this information to predict future survival.

	ACKNOWLEDGMENTS

 
The authors would like to thank the editor, an associate editor, and the 2 reviewers for constructive comments which helped improve this paper. Conflict of Interest: None declared.

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Received November 13, 2006; revised April 27, 2007; revised July 3, 2007; revised August 6, 2007; accepted for publication October 12, 2007.


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