Bayesian Analysis of Survival Models with Bathtub Hazard Rates Purushottam W. Laud, Ph.D. Paul Damien, Ph.D. Stephen G. Walker, Ph.D. Technical Report

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1 Bayesian Analysis of Survival Models with Bathtub Hazard Rates Purushottam W. Laud, Ph.D. Paul Damien, Ph.D. Stephen G. Walker, Ph.D. Technical Report 31 June 1998 Division of Biostatistics Medical College of Wisconsin 871 Watertown Plank Road Milwaukee WI Phone: (414)

2 Bayesian Analysis of Survival Models with Bathtub Hazard Rates Purushottam W. Laud 1 Paul Damien 2 Stephen G. Walker 3 1 Division of Biostatistics, Medical College of Wisconsin, Milwaukee, USA. 2 University of Michigan Business School, Ann Arbor, USA. 3 Department of Mathematics, Imperial College, London, UK. SUMMARY Nonparametric Bayesian inference using bathtub hazard rates in survival analysis is developed. A new simulation method, Auxiliary Random Functions, is introduced. When used within a Gibbs sampler, this method enables a unied treatment of exact, right-censored, left-censored, left-trucated and interval censored data, with and without covariates. The models and methods are exemplied via illustrative analyses. Key Words: Extended gamma process, Censored data, Gibbs sampling, Auxiliary functions, Covariates, Bioassay. 1 Introduction In survival analysis, often times, hazard specic prior information and/or hazard specic inference may be available and/or desirable. Modelling increasing or decreasing hazard rates is relatively straightforward; see, for example, Dykstra & Laud (1981). In reality, however, hazard rates are typically both decreasing and increasing, possibly with substantial uncertainty regarding the point where the monotonicity changes. Amman (1984) suggests a model for such hazard rates, appropriately termed bathtubs, using a combined extended gamma process. In this context, a unied nonparametric Bayesian inference solution for bathtub hazard rates given dierent types of censored and/or truncated data with or without covariates has not been available. This is partly due to the complexity of the modelling structure and partly due to the one-o, often dicult, computational approaches needed to sample from the appropriate posterior distributions. To the best of our knowledge, there are only three approaches available 1

3 for hazard rates. The other two, which do not model the shape, are : (i) the conditional gamma martingale of Arjas and Gasbarra (1994) and (ii) the discretized correlated constant hazards of Leonard (1978) and Gamerman (1991). For an excellent review of the general literature on Bayesian survival analysis, see Sinha and Dey (1997). This paper develops a non/semi-parametric Bayesian analysis of bathtub shaped hazard rates given dierent types of censored data. Computations are rendered straightforward based on a new stochastic substitution method. This method, which we term Auxiliary Random Functions (ARF), is particularly suited to simulate from the posterior conditional distributions in a Gibbs sampler (Smith and Roberts, 1993) in the present context. 2 Bathub Processes Here we collect, briey, the key properties of the extended gamma process that will be of use in the rest of the discussion; see, Dykstra & Laud (1981) and Amman (1984) for details. To begin, we consider increasing hazard rates. Let G(; ) denote the gamma distribution with shape parameter and scale parameter 1= >. For =, we dene this distribution to be degenerate at zero. For >, its density is ( x?1 exp(?x) g(xj; ) = ; x > ;?() ; x. Let (t), t, be a non-decreasing left continuous real valued function such that () = and (t); t, be a positive right continuous real valued function with left hand limits existing and bounded away from and 1. Take Z(t), t, to be a gamma process with parameter function (). That is, Z() =, Z(t) has independent increments and for t > s, Z(t)? Z(s) is G((t)? (s); 1). Considering a version of this process such that all its sample paths are non-decreasing and left continuous, let Z r(t) = [(s)]?1 dz(s): [;t) This process fr(t); t g is called the Extended Gamma (EG) (or the Increasing Extended, IEG,)process and we denote it by r(t) I?((); ()): In a manner similar to the description above one can dene an EG process for decreasing hazard rates as in Laud (1977). Take (), () and Z() as above. In addition, let Z(1) G((1); 1) be such that Z(1)?lim t!1 Z(t) is independent of the rest of the process, (1) > lim t!1 (t) and (1) >. Now dene Z Z r(t) = [(s)]?1 dz(s) = [t;1] [t;1) [(s)]?1 dz(s) + [(1)]?1 [Z(1)? lim 2 t!1 Z(t)] :

4 The main purpose of the values at innity is to preclude lim t!1 r(t) = so that R [;1) r(t)dt is innite and the survival time described by r() does not have positive probability at innity. It is clear that such a process generates nonincreasing hazard rates. We call it the Decreasing Extended Gamma (DEG) and denote it by r(t) D?((); ()): Amman (1984) uses the above development and denes a combined process by r(t) = r I (t) + r D (t); where the superscript, inherited also by the parameter functions, indicates the direction of monotonicity, and the two component processes are independent. Amman termed this a U- shaped process whereas elsewhere in the reliability and survival analysis literature one also encounters the usage \bathtub-shaped hazard rates". It should be noted that such a combined process does not necessarily generate hazard rates with this shape. It does, however, add a great deal of exibility in modelling the shape. For example, the expected hazard rate can be arranged to decrease initially and to increase beyond a certain time point by a suitable choice of the parameter functions. A parametric choice for the expectation can be a Weibull distribution with shape parameter less than one for the decreasing component and shape parameter exceeding two for the increasing component. In general, one can require with Since we get the conditions and d 2 d2 E(r(t)) = dt2 dt 2 E(rI (t)) + d2 dt 2 E(rD (t)) Z E(r(t)) = d E(r(t)) = for some t > : dt [;t) Z [ I (s)]?1 d I (s) + [t;1] [ D (s)]?1 d D (s); I (t) I (2) (t)? I (1) (t)i (1) (t) [ I (t)] 2? D (t) D (2) (t)? D (1) (t)d (1) (t) [ D (t)] 2 I (t) (1) I (t)? D (t) (1) = for some t > D (t) where the parenthesized subscript i on a function denotes its i th derivative. Such a prior, while maintaining a bathtub expectation, allows the hazard rates to have more general shapes. The data can then either reinforce the bathtub shape or indicate otherwise via the posterior expectation. 3

5 Several aspects of the posterior distribution are given by Dykstra and Laud (1981), Laud (1977) and Amman (1984). These can be summarized as follows : Fact 2.1 : The EG (or DEG) process is conjugate with respect to right censored data. In particular, given an observation X > x, writing R x r(t)dt as R 1 (x? R t)+ dr(t) (or 1 min(x; t)dr(t)) it follows that the posterior process is again EG (or DEG) with I ()(or D ()) unchanged and the parameter modied as ~ I (t) = I (t) + (x? t) + ~ D (t) = D (t) + min(x; t) : Here the superscript + on a quantity denotes its positive part. For a combined process, the posterior is simply the combined process made with these functions. Fact 2.2 : The posterior with respect to exact data is a mixture of EG(DEG) processes. The mixing measure is n-dimensional, where n is the number of exact observations, and has both continuous and jump components. For a combined process, the posterior is again a mixture of the corresponding combined processes. Fact 2.3 : Although analytic expressions for the posterior mean and variance functions are given by the above mentioned authors, their numerical evaluation is virtually impossible, especially as the sample size increases to more than around 2. They do not even consider the possibility of including covariates as a consequence. Our goal in the rest of this paper is to provide a full Bayesian analysis of the combined hazard rate model for various types of censored data as well as exact observations, both with and without covariates. We note here that the rest of the discussion applies also to data arising in a bioassay because a bioassay can be treated as a censored data problem. In addition, these methods also work for estimating the intensity rate function of a nonhomogeneous Poisson process because the form of the likelihood there is the same as that for survival data (see, for example, Lawless (1982)). Notation: In the following, [Aj:] and [A] denote the conditional and marginal distributions of A, respectively; note that A could be a process as well. Also, where the context is clear, [A] will imply the prior distribution of A. I() refers to the indicator function. 4

6 3 Posterior Computations Via ARFs Damien et al. (1998) develop a general method to sample from posterior distributions using auxiliary variables as follows. Suppose the required conditional distribution for a random variable X is denoted f. The basic idea is to introduce an auxiliary variable U, construct the joint density of U and X, with marginal density for X given by f, and then extend the Gibbs sampler to include the extra full conditional for U. In a Bayesian context, consider the posterior density given by f(x) / l(x)(x) and suppose it is not possible to sample directly from f. The general idea proposed by Damien et al. is to introduce an auxiliary variable Y, dened on the interval (; 1) or more strictly the interval (; l(^x)), where ^x maximises l(:), and dene the joint density with X by f(x; y) / I(y < l(x))(x): The full conditional for Y is U(; l(x)), a uniform variable on the interval (; l(x)), and the full conditional for X is, restricted to the set A y = fx : l(x) > yg. In this paper, we develop a simulation method that is particularly suited to the context of sampling monotone hazard rate processes in a manner similar in spirit to that of Damien et al. (1998). Whereas Damien et al. use latent variables to put a function in place which depends on a nite number of parameters, the latent variables used in this paper put an entire random function into place. Hence we refer to it as the auxiliary random function (ARF) method. To introduce the method, we rst consider only the increasing hazard rate case without covariates, dropping the superscript I. We then implement it for combined processes and various survival data scenarios. Exact observations The contribution to the likelihood resulting from one exact observation at x is given by: Z x f(x) = r(x) F (x) = r(x) exp? r(s)ds ; where F is the survival function. Based on Fact 2.1, it is clear that F (x), being the contribution of a right-censored observation, updates the prior parameters of the EG process. In the above equation, from an ARF perspective, the following question arises: how do we substitute the random function r() so that sampling from the posterior is accomplished readily? Take the cumulative distribution of a random variable Y given the hazard rate process r() as: P (Y yjr()) = r(y) I( y x) + I(y x): r(x) The distribution of Y is dened in terms of a random function, hence the phrase \auxiliary random function". It is in this sense that the ARF algorithm is dierent from Damien et al.'s 5

7 algorithm. To sample Y jr(), let Y = r?1 (Ur(x)) where U U(; 1) and r?1 (w) = infftjr(t) wg. To address sampling r()jy, write the density of Y jr() as [yjr()] = dr(y) I( y x): r(x) Combining this with the prior and the likelihood we arrive at Z x [r()jy] / [r()]dr(y) exp? r(s)ds ; where dr(y) denotes the increment of r() at y. It is clear that this term combines naturally with the EG process, adding a unit jump at y to the shape parameter (). Thus, as a result of the ARF substitution and Fact 2.1, the posterior full conditional for r() is once again an EG process with the parameters updated by: ~(s) = (s) + I(s y) and ~ (s) = (s) + (x? s) + : The case of the decreasing hazard rates is similar with the following change : the auxiliary variable Y has the survival function P (Y > yjr()) = r(y) I(y > x) r(x) and possibly takes the value \1". This results in a DEG process for the full conditional for r() with parameters ~(s) = (s) + I(s y) and ~ (s) = (s) + min(x; s): Turning to the case of the combined process, we write the posterior distribution as [r D (); r I ()jx = x] / [r D ()][r I ()]fr D (x) + r I (x)g exp(? Now the stochastic substitution takes the form Z x fr D (s) + r I (s)gds): [yjx = x; r D (); r I ()] = dri (y)i( y x) + f?dr D (y)gi(y > x) : r D (x) + r I (x) Straightforward algebra results in where [r D (); r I ()jy; X = x] / EG(~ I ; ~ I )EG(~ D ; ~ D ); ~ D (s) = D (s) + I(y > x)i(s y); ~ I (s) = I (s) + I( y x)i(s y) and the two scale parameter functions as given in Fact

8 In each of the cases considered above, the full conditional of the hazard rate after the ARF substitution reduces to EG, DEG or a combination. The increments of the EG(DEG) process can be simulated by using the Bondesson (1981) method as described in Laud, Smith and Damien (1996) or, as is often adequate, simply approximated as gamma variates. Left censored data The contribution to the likelihood from a left-censored observaton, X x, is Z x F (x) = 1? exp? r(s)ds : First, using the auxiliary variate W jr() with hazard rate r() restricted to [; x], we have and [wjr(); X x] = r(w) exp(? R w r(s)ds) 1? exp(? R x I( w x); r(s)ds) Z w [r()jw; X x] / [r()]r(w) exp? r(s)ds : Now, in a manner similar to the exact data scenario, we introduce an ARF Y and proceed as described above. Interval censored data The contribution of an observation, a < X b to the likelihood is Z a Z! b F (a)? F (b) = exp? r(s)ds? exp? r(s)ds : Employing W jr() with hazard rate r() restricted to (a; b] yields the same full conditional for r() as in the left censored data scenario above. Left truncated data Here one observes X (or a censored version of it) only if it exceeds an observed left truncation time L. Thus the contribution of X = x; L = l to the likelihood is Z f(x) x? F (l) I(l x) = r(x) exp r(t)dt I(l x): 7 l

9 Once again, the ARF Y as dened above suces to substitute for r(x). It can be shown easily that the eect of the term e?r x r(t)dt l is to modify the scale functions given in Fact 2.1 as ~ I (t) = I (t) + (x? max(t; l)) + and ~ D (t) = D (t) + (min(x; t)? l) + : The unique conjugacy property of the extended gamma process for the latent model is highlighted in the Appendix. 4 Cox Model Computations Via ARFs Consider now the extension of the nonparametric analysis to include covariates. Employing the Cox model, Cox (1972), and initially tackling a single right-censored observation, let and z denote, respectively, the vectors of regression parameters and covariates. The likelihood function for a right censored observation is given by L(r(); ) = P (X > xjr(); ) = exp?e z Z x r(s)ds : Conditioning on and following the development leading to Fact 2.1, we easily arrive at [r()j; X > x] EG (); () ~ where ~() = () + e z (x? s) + : A similar result obtains for decreasing hazards, yielding ~ () = () + e z min(x; s). In the case of exact data, [r()j; X = x] = [r()]e z r(x) exp?e z Z x r(s)ds : Clearly, the problem now yields to the same ARF substitution employed in the case without covariates. In addition, the decreasing hazards follow suit, leading to the same solution for the combined process as in the previous section. To examine the full conditional distribution of, let denote the usual \noncensoring" indicator and write the likelihood as Z x L(r(); ; x; ) / e z exp?e z r(s)ds : Simulating from [j] is easily accomplished using the auxiliary variable method discussed in Damien et al. Omitting straightforward algebraic details, the simulation of j is given by the following full conditional in the Gibbs sampler : [vjall else] / U(; e z j j ); 8

10 [wjall else] / E(1)I(H(x)e z < w); [ j jall else] / [ j ]fi(a < j < b; z j > ) + I(b < j < a; z j < ) + I(z j = )g; where U and E(1) denote the uniform and standard exponential distributions, respectively and a = ln(v) z j, b = ln(w?ln(h(x))?p z k6=j k k z j ; H(x) = R x r(s)ds. 5 Illustrative Analyses We describe a simulated example without covariates and a case study involving covariates. Example 1 We simulated samples of size 5, 2 and 5 from a mixture of two gamma densities with common scale.25, and shape parameters.5 and 5, with mixing weights.7 and.3 respectively. The corresponding bathtub hazard rates are depicted in Figure 1 (see, Glaser(198) for conditions under which mixtures have bathtub hazards). We consider two dierent prior means: Prior1: r(t) = 4exp(?2t + :8t; Prior2:r(t) = exp(?2t) + :4t; these are depicted in Figure 1. Based on the model described in Section 3, the posterior mean and the sample-based two-standard-deviation limits were calculated; these are also shown in Figure 1. The shape of the posterior mean is seen to respond to the prior modeling via the combined process. Example 2 We consider here data for 548 chronic myelogeneous leukemia (CML) patients between the ages of 15 and 55 years who received an HLA-identical sibling bone marrow transplant. These transplants were reported to the International Bone Marrow Transplant Registry, a working group of over 3 transplant centers worldwide that contribute transplant and follow-up data to a statistical center at the Medical College of Wisconsin. These 548 patients were diagnosed between 1983 and 1991 as suering from CML. Approximately half of the observations are right censored corresponding to patients that are alive or, in a few cases, lost to follow-up. More detailed description of the data are available in Zhang and Klein (1998) and the references therein. For the purposes of this illustration, we consider the survival time from the transplant date as possibly related to the following covariates: (i) Time from diagnosis to transplant. (ii) Gender. (iii) Whether or not the CML diagnosis was made after (It is well recognized in the related medical literature that substantial technical improvements became available in 1988.) (iv) Whether or not the patient's spleen size exceeded 1cm. (v) Whether or not the patient's age exceeded 34. Employing the proportional hazards regression model of Cox (1972), it is thought desirable to model the baseline hazard rate as initially high, reecting complications related to the 9

11 transplant, decreasing thereafter and possibly increasing as patients relapse into the disease. Implementing the analysis outlined in the previous sections, the posterior distributions of the regression coecients indicated that only gender and diagnosis date aected survival. The results presented in Figures 2 through 4 were obtained using these two covariates and their interaction. With the intention of leaving the likelihood nearly unaected by the prior, uniform prior distributions for the regression coecients were chosen, each on the interval (?5; 5). In 5 samples from the Gibbs chain after discarding the initial 1, the values for the interaction were all positive, and those for the rst two coecients all negative, providing substantial evidence that all three terms inuence the survival time. It is customary in survival analysis of this type to address one's inferences to the more readily interpretable relative risk (or risk ratio) as opposed to the original regression coecient. For a binary covariate, it is the ratio of the hazards for the two groups and is constant over time in Cox's model. Figure 2 presents posterior densities of four such ratios to illuminate the nature of the interaction. From the two solid line curves one notes that the gender gap in risk prior to 1988 diminished considerably in the later period. Looked at another way, via the dashed curves, risk for males lowered substantially as shown by the curve to the left lying entirely below.85. For females, however, the procedural improvements of 1988 and later do not appear to have resulted in decreased risk. Albeit by a small amount, the posterior probability of an increased risk indeed exceeds.5. Figure 3 compares post-transplant survival in the four groups. At 25 months, there is much posterior uncertainty in dierentiating among three of the four groups; only males with pre diagnoses can be condently declared to have a lower survival probability than the rest. The posterior mean survival probabilities over eight years are presented in part (b). It should be noted that the posterior uncertainties for these curves are readily available via the posterior samples. However, their clear depiction is somewhat challenging. Finally, Figure 4 addresses the baseline hazard rate and survival. It presents the posterior mean curves and two-sigma limits. The prior means are also shown. The prior two-sigma limits are omitted, however, since the proper but vague choice of prior parameter functions rendered these limits o-scale. 6 Discussion In this paper we have provided a complete Bayesian solution to modelling bathtub hazard rates nonparametrically. A new stochastic simulation method was introduced to sample the posterior processes. We also provided a solution for the semi-parametric model with covariates. Natural extensions are to consider time-dependent covariates and to detect when data contradict the proportional hazards assumption. From the perspective of the physicians and scientists wishing 1

12 to use these methods, based on the stored posterior samples, graphical implementations that can respond to queries regarding substantively interesting functionals would be attractive indeed. Acknowledgements This research was funded in part by an NSF grant, an EPSRC Realising Our Potential Award, an EPSRC Fellowship, and a Faculty Research Award by the University of Michigan Business School. We thank the IBMTR for providing the data in Example 2. Appendix We are working with the joint data model with likelihood component Z f(s; tjr())dtds = dr(s) exp (t? u) + dr(u) I( s t)dt; where the function r() is the parameter and s is latent; i.e. unobserved, whereas t is observed. We show that if r() has independent increments then we have a desirable conjugacy property if these increments are gamma variables. This can be seen if we write the likelihood for one observation as Y dr(u) I(u=s) exp??(t? u) + dr(u) : u Therefore, if dr() G((); ()), the updates given in Section 3 are immediate. The expression above demonstrates the unique conjugacy property of the extended gamma process. 11

13 References Amman, L. (1984). Bayesian nonparametric inference for quantal response data. Annals of Statistics, 12, Arjas, E. and Gasbarra, D. (1994). Nonparametric Bayesian inference from right censored survival data using the Gibbs sampler. Statistica Sinica, 4, Bondesson, L. (1982). On simulation from innitely divisible distributions. Adv. App. Prob. 14, Cox, D.R. (1972). Regression models and life tables (with discussion). J. Roy. Statist. Soc. B, 34, Damien, P., Wakeeld, J.C. and Walker, S.G. (1998). Gibbs sampling for Bayesian nonconjugate models using auxiliary variables. To appear in the J. Roy. Statist. Soc. B. Dykstra, R.L. and Laud, P.W. (1981). A Bayesian nonparametric approach to reliability. Annals of Statistics, 9, Gamerman, D. (1991). Dynamic Bayesian models for survival data. Applied Statistics, 4, Glaser, R.E. (198). Bathtub and related failure rate characterizations. J. Am. Statist. Assoc., 75, Laud, P.W. (1977). Bayesian nonparametric inference in reliability. Ph.D. Dissertation, University of Missouri, Columbia, MO. Laud, P.W., Smith, A.F.M. and Damien, P. (1996). Monte Carlo methods for approximating a posterior hazard process. Statistics and Computing, 6, Lawless, J.F. (1982). New York. Statistical Models & Methods For Lifetime Data, Wiley Publications, Leonard, T. (1978). Density estimation, stochastic processes and prior information. J. Roy. 12

14 Statist. Soc. B, 4, Sinha, D. and Dey, D.K. (1997). Semiparametric Bayesian analysis of survival data. J. Am. Statist. Assoc., 92, Smith, A.F.M. and Roberts, G.O. (1993). Bayesian computations via the Gibbs sampler and related Markov chain Monte Carlo methods. J. Roy. Statist. Soc. B 55, Zhang, M.J. and Klein, J.P. (1998). Condence bands for the dierence of two survival curves under proportional hazards model. Technical Report 29, Division of Biostatistics, Medical College of Wisconsin, Milwaukee, Wisconsin. 13

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