BAYESIAN ANALYSIS OF BIVARIATE COMPETING RISKS MODELS
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1 Sankhyā : The Indian Journal of Statistics 2000, Volume 62, Series B, Pt. 3, pp BAYESIAN ANALYSIS OF BIVARIATE COMPETING RISKS MODELS By CHEN-PIN WANG University of South Florida, Tampa, U.S.A. and MALAY GHOSH University of Florida, Gainsville, U.S.A. SUMMARY. Absolutely continuous bivariate exponential (ACBVE) models have been widely used in the analysis of competing risks data involving two risk components. For such an analysis, frequentist approach often runs into difficulty due to a likelihood containing some nonidentifiable parameters. With an end to overcome this nonindentifiability, we consider Bayesian procedures. Utilization of informative priors in the Bayesian analysis is attractive in the presence of historical data. However, systematic prior elicitation becomes difficult when no such data is available. The present study focuses instead on Bayesian analysis with noninformative priors. The ACBVE model structure often leads to a likelihood function with a nonregular Fisher information matrix impeding thereby the calculation of standard noninformative priors such as Jeffreys s prior and its variants. As a remedy, a stagewise noninformative prior elicitation strategy is proposed. A variety of noninformative priors are developed, and are used for data analysis. These priors are evaluated according to the frequentist probability matching criterion for identifiable parameters. In addition, this paper examines the asymptotic property of nonidentifiable parameters under certain informative setting. 1. Introduction Often in life testing situations, each individual is subject to several possible causes of failure, and the diagnosed outcomes consist of individual lifetime T and the cause of failure I. This is what is usually referred to in the literature as the competing risks framework. Bivariate exponential (BVE) distributions were developed initially for modeling correlated bivariate lifetimes. They are suitable also for the analysis of competing risks data when the two risk components are correlated. The most commonly used BVE is the Marshall-Olkin (1967) BVE which features positive probability of simultaneous failure. Block and Basu (1975), Sarkar (1986), and Ryu (1993), Paper received February 2000; revised September AMS (1991) subject classification. 62C10, 62F15. Key words and phrases. Bivariate exponential distribution, competing risks, nonidentifiability, Bayesian analysis, two-stage priors, matching.
2 bayesian analysis of bivariate competing risks models 389 among others, modified the singularity (or non-absolute continuity) of this distribution, and constructed a variety of bivariate lifetime processes that allowed no simultaneous failure. Absolutely continuous BVE models of Gumbel (1960) and Freund (1961) have also enjoyed popularity in many reliability applications. In this paper, we refer to these absolutely continuous BVE models as ACBVE. Numerous analyses for BVE competing risks data have been addressed in the literature from various angles. Among others, we may refer to David and Moeschberger (1971,1978), Moeschberger (1974), Lagakos (1978), Nakao and Liu (1990, 1991), Lu and Bhattacharyya (1992), and Wada, Sen, and Shimakura (1996). The inferential problem associated with the analysis of competing risks data based on ACBVE s often arises from nonidentifiable parameters in the likelihood. This was recognized by Tsiatis (1975) and Basu and Ghosh (1980). Then the MLE s for the nonidentifiable parameters are not unique, and the usual asymptotics does not hold. One way to overcome the nonindentifiability is the Bayesian approach. Utilization of informative priors in the Bayesian analysis is attractive in the presence of historical data. Otherwise, Bayesian methods are still applicable by employing noninformative priors. Such priors have clear pragmatic appeal, and are often useful for making reliable inference. In this paper, we focus primarily on Bayesian analysis with noninformative priors. However, we have considered also a class of partially informative priors. In Section 2, we begin with a general likelihood function for the observed outcomes (T, I) resulting from the ACBVE models, which are commonly characterized by (a) conditional independence of T and I, and (b) T is memoryless. These examples are introduced in Section 2. Due to presence of nonidentifiable parameters in the likelihood, the Fisher information matrix either cannot be fully specified, or is singular, impeding thereby the calculation of standard noninformative priors such as Jeffreys s prior and its variants. A stagewise noninformative prior elicitation strategy is proposed, and is utilized for the ACBVE s. In order to develop these priors, we split the parameter vector into two parts: nonidentifiable and identifiable parameters. At the first stage, a conditional version of Laplace s prior is assigned to the nonidentifiable parameters given the identifiable ones. Then, at the second stage, we consider both Laplace s and Jeffreys s priors as marginal priors for the identifiable parameters. A variety of noninformative priors and the corresponding posteriors are developed in this section. We compare the performance of the newly developed noninformative priors in Section 3 based on the criterion of matching the coverage probabilities of the Bayesian credible intervals with the corresponding frequentist confidence intervals. Such priors were orginally developed by Welch and Peers (1963) for random variables with continuous distributions. The needed modification in the discrete case is pointed out in Ghosh (1994, p97). In our framework, the modified version of Jeffreys s prior emerges as the matching prior for identifiable parameters. This is also evidenced in the simulation study performed in this section. Since the posterior distributions have explicit form, it is possible to calculate the posterior means, standard deviations, percentiles as well as the credible intervals explicitly in this paper.
3 390 chen-pin wang and malay ghosh In Section 4, we modify our arguments with informative priors at both stages. A simulation study for the Block-Basu (or Sarkar) ACBVE was conducted, where conditional beta prior and degenerate prior are assumed at the first and the second stages, respectively. As is well-known, under usual regularity conditions, in the calculation of the posterior, the likelihood dominates the prior when the sample size is large. For the ACBVE examples, due to presence of nonidentifiability, these regularity conditions do not hold. Indeed, our simulation study reveals that the posterior of the nonidentifiable parameter is closer to the prior even for large sample sizes. An intuitive explanation of this phenomenon is provided in Section 4. In the rest of this section, we introduce a catalog of notations that are used throughout this paper. 1 [E] : indicator function of event E. n: sample size. : failure due to primary cause indicator, i.e., 1 [I=1]. (t i, δ i ): realization of (T i, i ), for1 i n. d i (t i, δ i ): i th observed pair. d {d i : 1 i n}: the entire data set s: realization of S n T i. i=1 n 1 : realization of N 1 n i. i=1 n 2 : realization of N 2 n (1 i ). i=1 θ: the vector of model based parameters. f(x θ): probability density function (pdf) of x given parameters θ. F (x θ): x f(y θ) dy. F (.): 1 F (.). X G(a, b): f(x) = [Γ(a)] 1 b a x a 1 exp ( bx). X B(a, b):f(x) = Γ(a+b) Γ(a)Γ(b) xa 1 (1 x) b 1. X Bin(m, p): f(x) = m! x!(m x)! px (1 p) m x. L(θ d): likelihood function of θ given data d. P (.): probability measure. π(θ): prior density. π(θ d): posterior density of θ given data d and prior π. 2. Bayesian Analysis for the ACBVE Models Likelihood, noninformative priors and posteriors. Examples considered in this paper are ACBVE s due to Freund, Block-Basu (B-B), Sarkar, and Gumbel. In all these cases, the likelihood function for (T 1, I 1 ),, (T n, I n ) has the general form L(λ, ρ, φ d) λ n e λs ρ n 1 (1 ρ) n 2 1 [φ R(λ,ρ)]. (1)
4 bayesian analysis of bivariate competing risks models 391 Their original joint survival functions or pdf s have different structures in all these cases. The Gumbel ACBVE has joint survival function given by F (t 1, t 2 ) = e [(λ 1t 1 ) 1/r +(λ 2 t 2 ) 1/r ] r. (2) For Freund s ACBVE, the joint pdf f(t 1, t 2 ) is f(t 1, t 2 ) = { αβ e (α+β β )t 1 β t 2 0 < t 1 t 2, βα e (β+α α )t 2 α t 1 0 < t 2 t 1. (3) The joint ACBVE pdf given by Block and Basu is f(t 1, t 2 ) = { λλ1 (λ (λ 1+λ 2) 2 + λ 12 ) e λ1t1 (λ2+λ12)t2 0 < t 1 t 2, λλ 2 (λ 1 +λ 2 ) (λ 1 + λ 12 ) e λ2t2 (λ1+λ12)t1 0 < t 2 t 1. The joint survival function of Sarkar s ACBVE is e (λ 1+λ 12 )t 2 {1 [1 e λ 1t 2 ] w [1 e λ 1t 1 ] 1+w } 0 < t 1 t 2, F (t 1, t 2 ) = e (λ2+λ12)t1 {1 [1 e λ2t1 ] w [1 e λ2t2 ] 1+w } 0 < t 2 t 1. (4) (5) where w = λ 12 /(λ 1 +λ 2 ). Let θ denote the generic vector of model parameters (e.g., θ = (λ 1, λ 2, λ 12 ) for the Sarkar or Block-Basu ACBVE). The relationship between θ and (λ, ρ, φ) and between φ and R(λ, ρ) is given in Table 1. Table 1. The ACBVE Reparameterization Model B-B & Sarkar Gumbel-B Freund λ λ 1 + λ 2 + λ 12 (λ r λ r 1 2 ) r α + β λ ρ 1 λ r 1 α λ 1 +λ 2 λ r 1 +λ 1 r 1 α+β 2 λ 1 + λ 2 r (α α, β β) R(λ, ρ) (0, λ) (0, 1) (0, λ λρ) (0, λ ρ) 1 The likelihood function given in (1) impedes one from calculating the noninformative priors arising out of the Fisher information matrix. As a remedy, a two-stage prior elicitation strategy is proposed to find noninformative priors. At the first stage, we consider the uniform prior π U1 π U1 (φ λ, ρ) 1 [φ R(λ,ρ)]. (6)
5 392 chen-pin wang and malay ghosh At the second stage, we first compute the integrated likelihood function L(λ, ρ, φ d)π U1 dφ = L 1 (λ, ρ d) λ n e λs ρ n1 (1 ρ) n2. (7) Next, the marginal priors for (λ, ρ) are derived. The common choices are Laplace s prior, Jeffreys s prior, and the reference prior to pair with π U1. The calculations are provided below. The Fisher information matrix I 1 (λ, ρ) derived from (7), is given by [ ] λ 2 0 I 1 (λ, ρ) = 0 ρ 1 (1 ρ) 1. Hence, the second-stage Jeffreys s prior π J2 (λ, ρ) is given by π J2 (λ, ρ) λ 1 ρ.5 (1 ρ).5. (8) Since I 1 (λ, ρ) is diagonal, the second-stage reference prior with rectangular compacts for λ and ρ is unique and equals Jeffreys s prior (Datta and Ghosh,M. 1995). Thus, we obtain the two-stage prior π JU π JU (λ, ρ, φ) = π U1 π J2 λ 1 ρ 1 2 (1 ρ) 1 2 1[φ R(λ,ρ)]. (9) A second two-stage prior, when Laplace s rule is applied on both (φ λ, ρ) and (λ, ρ), is given by π UU π UU (λ, ρ, φ) = π U1 π U2 1 [φ R(λ,ρ)]. (10) The resulting joint posteriors under π JU and π UU are π JU (λ, ρ, φ d) λ n 1 e λs ρ n1.5 (1 ρ) n2.5 1 [φ R(λ,ρ)] ; (11) π UU (λ, ρ, φ d) λ n e λs ρ n 1 (1 ρ) n 2 1 [φ R(λ,ρ)]. (12) Observe the fact that R(λ, ρ) is compact and π U1 is bounded. Hence, the propriety of (11) and (12) follow from the finiteness of gamma and beta integrals provided that n 1 1 and n 2 1. In Table 2, we summarize the posterior distributions for λ and ρ. Posterior distributions for φ can often be complicated. These are mixtures of standard distributions. Nevertheless, all posterior moments are finite and can be obtained in closed forms. These results are enlisted in Table 3. Table 2. Posterior Distributions for λ and ρ of ACBVE s θ λ ρ π UU (θ d) G(n + 1, t t ) B(n 1 + 1, n 2 + 1) π JU (θ d) G(n, t t) B(n , n )
6 bayesian analysis of bivariate competing risks models 393 Table 3. Posterior Distributions (Moments) for of ACBVE s ACBVE B-B & Sarkar Gumbel-B Freund λ 1 + λ 2 r d 1 = α α d 2 = β β π UU G (n) (t t ) U 0,1 G w (n + 1, t t, G w (n + 1, t t, π( d) n 2 + 1, n 2 + 1) n 1 + 1, n 2 +.5) π JU G (n 1) (t t) U 0,1 G w(n, t t, G w(n, t t, n 2 +.5, n 1 +.5) n 1 +.5, n 2 + 1) E( d, π) π UU (n + 1)/2t t 1/2 q (E) U (n, n 2, t t) q (E) U (n, n 1, t t) π JU n/2t t 1/2 q (E) J (n, n 2, t t ) q (E) J (n, n 1, t t ) V ( d, π) π UU (n + 1)(n + 5)/12t t 1/12 q (V ) U (n, n 2, t t ) q (V ) U (n, n 1, t t ) π JU n(n + 4)/12t 2 t 1/12 q (V ) J (n, n 2, t t ) q (V ) J (n, n 1, t t ) where U 0,1 Uniform (0, 1) G (m) (x) f Gw(r,u,a,b)(x) m 1 m G(i, x), = x i=1 1 Γ(r) ur x r 1 e rx Γ(a)Γ(b) Γ(a+b) ( x y )a 1 (1 x y )b 1 dy, q (E) m k+1 U (m, k, x) = m+2 2x, q (E) J (m, k, x) = m 1 q (V ) U m+1 k x, ( m+1 k+1 (m, k, x) = (m+2)(k+2) m+2 2x 2 3(m+3) q (V ) J (m, k, x) = m k+ 1 2 m+1 ) (m+1)(k+1) 4(m+2), ( (m+1)(k+ 3 2x 2 2 ) 3(m+2) m(k+ 1 2 ) 4(m+1) ). 3. Prior Comparison As a means to evaluate the asymptotic performance of the proposed noninformative priors, we compare the coverage probabilities of Bayesian credible intervals with the corresponding frequentist confidence intervals for parameters λ and ρ. In other words, we shall examine the closeness between the Bayesian posterior tail probability and the corresponding probability. Write θ = (η, ω), where η is the parameter of interest. Let η π (α) (d n) denote the posterior α-quantile of η under prior π and data d n of size n. Thus, P (η < η (α) π (d n) π, d n) = α. (13)
7 394 chen-pin wang and malay ghosh We say that π is a first order probability matching prior if P (η < η (α) π (d n) θ) = α + o(n.5 ), (14) while π is a second order probability matching prior if P (η < η (α) π (d n) θ) = α + o(n 1 ). (15) These matching criteria were originally introduced by Welch and Peers (1963). Further development is due to Stein (1985), Tibshirani (1989), Ghosh and Mukerjee (1992), Mukerjee and Dey (1993), Datta and Ghosh, J. K. (1995), Datta and Ghosh, M, (1995, 1996), Datta (1996) and Mukerjee and Ghosh (1997). Since P (η < η π (α) (d n) θ) can be explicitly derived in these examples, we suggest the following approach to examine the matching properties. First, let λ be the parameter of interest. Then we map λ to η and (ρ, φ) to ω in the ACBVE examples. First, consider the posterior given in (11). Here, λs S is distributed as G(n, 1). Also, we may recall that S λ G(n, λ). Hence, λs λ has the same G(n, 1) distribution. Thus, in this example, there is perfect frequentist-bayes agreement for interval estimation of λ, and the prior π JU is the ideal matching prior. Next from (12), λs S is distributed as G(n + 1, 1). In this case, matching holds only asymptotically. The probabilities at various posterior quantiles are computed using SPLUS. The theoretical and simulation results are presented respectively in Tables 4 and 5. Columns 3-11 of the two tables provide the frequentist distributions evaluated at the posterior α-quantiles (α =.05,.2,.3,.4,.5,.6,.7,.8,.95) given π (π = π UU, π JU ) and samples of sizes 10, 50, and 100. The simulation results confirm the conclusions made earlier. Table 4. Exact Frequentist Coverage Probabilities for λ of BVE s under π UU and π JU n π α =.05 α =.2 α =.3 α =.4 α =.5 α =.6 α =.7 α =.8 α =.95 5 π UU π JU π UU π JU π UU π JU π UU π JU Next, we consider interval estimation of ρ. Since N 1 Bin(n; ρ), the notion of matching priors based on continuous likelihoods is not valid. A suitable modification of the matching criterion is proposed in Ghosh (1994, p.97, (9.50)). Based on this modification, Jeffreys s prior continues to be a second order matching prior when there are no nuisance parameters. This fact is borne out in our simulation results. First, however, we provide a general formula for computing the probabilities when the underlying random variables are discrete, and marginal
8 bayesian analysis of bivariate competing risks models 395 Table 5. Simulated Frequentist Coverage Probabilities for λ of BVE s under π UU and π JU n π α =.05 α =.2 α =.3 α =.4 α =.5 α =.6 α =.7 α =.8 α =.95 5 π UU π JU π UU π JU π UU π JU π UU π JU posterior densities of model parameters are standard or posterior quantiles can be numerically found. Later on, we illustrate this technique through the ACBVE examples. Let V (V 1,, V l ) be a discrete random vector with joint pdf f(v θ) = P (V = v θ), where v = (v 1,, v l ), θ = (η, ω), and η is the parameter of primary interest. Denote the proper posterior density by π(θ v). Suppose that given v and prior (v), is obtainable for all 0 < α < 1 and v. Then for any given α, the frequentist distribution evaluated at the posterior α-quantile is given by π(θ) the posterior quantile of η, η (α) π P (η < η (α) π θ) = {v:η<η (α) π (v)} f(v θ) (16) For the ACBVE s, we find that N 1 Bin(n; ρ). From Table 2, ρ (π JU, n 1 ) B(n 1 +.5, n 2 +.5) and ρ (π UU, n 1 ) B(n 1 + 1, n 2 + 1). Let ξ ρ (α) (a, b) denote the α quantile for B(a, b) distribution. Applying (16), the probabilities are obtained as = = P (ρ < ξ ρ (α) (i +.5, n i +.5) ρ) ( {i:ρ<ξ ρ (α) (i+.5,n i+.5)} P ρ (ρ < ξ ρ (α) {i:ρ<ξ ρ (α) (i+1,n i+1)} (i + 1, n i + 1) ρ) ( n! i!(n i)! ρi (1 ρ) n i n! i!(n i)! ρi (1 ρ) n i ) ) (17) (18) The numerical computations are carried out by using the SPLUS software package. These results are presented with the scatter plots of the exact frequentist versus posterior probabilities for ρ. Figures 1-3 respectively demonstrate the results for n = 10, 50, 100, and ρ =.2,.4,.6,.8.
9 396 chen-pin wang and malay ghosh Jeffreys Laplace Figure 1. Transformed Q-Q Plots: Posterior Probabilities of ρ vs Frequentist Probabilities of ρ under Jeffreys s and Laplace s priors and samples of size 10. (a) transformed Q-Q plot for ρ =.2; (b) transformed Q-Q plot for ρ =.4; (c) transformed Q-Q plot for ρ =.6; (d) transformed Q-Q plot for ρ =.8. Figures 1-3 suggest that the coverage discrepancy for ρ between π JU and π UU is associated with n, the parameter values, and α. In general, the matching properties between π JU and π UU are similar for ρ within the central part of the range (e.g., ρ =.4,.6 etc.) as compared to those near the boundary (e.g., ρ =.2,.8). This difference decreases as sample size increases. It, however, remains non-negligible for the selected α s within (.05,.95). For α =.05 or α =.95, the associated posterior quantiles are usually close to their frequentist counterparts for arbitrary parameter values given either of the selected priors. Thus, frequentist (1 2α) confidence interval for ρ can be found from the (1 2α) Bayesian credible set under either prior even though π UU is not a second order matching prior.
10 bayesian analysis of bivariate competing risks models 397 Jeffreys Laplace Figure 2. Transformed Q-Q Plots: Posterior Probabilities of φ vs Frequentist Probabilities of φ under Jeffreys s and Laplace s priors and samples of size 50. (a) transformed Q-Q plot for ρ =.2; (b) transformed Q-Q plot for ρ =.4; (c) transformed Q-Q plot for ρ =.6; (d) transformed Q-Q plot for ρ = Informative Priors and Asymptotics of Nonidentifiable Parameters So far, we have considered analyses based on several noninformative priors. In contrast, we will consider some informative alternatives in this section. This modification can also help study the asymptotic properties of nonidentifiable parameters. To this end, we first assume a class of informative priors with π(φ λ, ρ) s that are generalizations of π U1. A simulation study for the Block-Basu (or Sarkar) ACBVE is conducted. Since in this case φ is a scale parameter and the associated posterior is now independent of ρ and δ (cf. Table 3), we proceed by generating independent samples {t 1,, t n } from G(1, λ), and B(a, b) prior for φ/λ for fixed λ. Thus the conditional prior for φ is given by ( ) a 1 ( φ π (φ λ) 1 φ ) b 1 1 λ λ λ. (19)
11 398 chen-pin wang and malay ghosh Jeffreys Laplace Figure 3. Transformed Q-Q Plots: Posterior Probabilities of ρ vs Frequentist Probabilities of ρ under Jeffreys s and Laplace s priors and samples of size 100. (a) transformed Q-Q plot for ρ =.2; (b) transformed Q-Q plot for ρ =.4; (c) transformed Q-Q plot for ρ =.6; (d) transformed Q-Q plot for ρ =.8. The unconditional posterior of φ, though without an explicitly expressed form, can be drawn via the Gibbs sampling scheme. Without loss of generality, we let λ = 1. Consider samples of sizes 10 and 50 generated from G(1, 1) paired with priors B(4, 16), B(20, 80), B(10, 10), B(50, 50), B(14, 6), and B(70, 30) assumed for φ, accordingly. Posterior means and standard deviations of φ are summarized in Table 6. Table 6. Posterior Estimates for φ under three-parameter ACBVE s and Beta Priors π(φ λ) E π (φ) E π (φ d)(vπ 1/2 (φ d)) n = 10 n = 50 B(4, 16) (.1213).2043 (.0945) B(20, 80) (.0862).2042 (.0502) B(10, 10) (.2098).5088 (.1330) B(50, 50) (.1782).5086 (.0882) B(14, 6) (.2480).7142 (.1413) B(70, 30) (.2281).7066 (.1076)
12 bayesian analysis of bivariate competing risks models 399 posterior distribution B(4,16) n=10 n= posterior distribution B(20,80) n=10 n= posterior distribution B(10,10) n=10 n=50 posterior distribution B(50,50) n=10 n= posterior distribution B(14,6) n=10 n= posterior distribution B(70,30) n=10 n= Figure 4. Conditional beta Prior Distributions and the Associated Posterior Distributions of φ under n = 10 and n = 50: (a) prior: B(4, 16); (b) prior: B(20, 80); (c) prior: B(10, 10); (d) prior: B(50, 50); (e) prior: B(14, 6); (f) prior: B(70, 30). Figure 4 provides a graphic representation for the simulation results. In each of the sub-plots, a specified prior distribution of φ is drawn along with the associated posterior distributions given samples of sizes 10 and 50. These graphs indicate the increasingly dominating influence of the priors on the posteriors as the sample sizes increase. These results apparently do not conform with the regular asymptotics, which formalizes the notion that the importance of the prior distribution diminishes as the sample size increases. This contradiction can partly be explained by the nonregularity of the likelihood. To be specific, since in this simulation study, data were generated from Block-Basu ACBVE with λ = 1, the posterior of λ is nearly degenerate at 1 as the sample sizes increase. Hence, for large sample sizes, the marginal posterior π(φ d) is nearly the same as π(φ 1, d). Also, since the likelihood L(λ, ρ, φ d) does not depend on φ except through its range, as pointed out in Remark 1 of Ghosh, Ghosh, Chen and Agresti (2000), the posterior π(φ λ, d) = π(φ λ), and does not depend on d. Thus, the marginal posterior of φ essentially depends only on the prior, and not on the data when the sample size gets very large. As a side remark,
13 400 chen-pin wang and malay ghosh the fact that π(φ λ, d) = π(φ λ) implies that the parameter φ is nonidentifiable in the sense of Dawid (1979). Heuristically, this result can be extented to a more general setting when π is a matching prior for λ, e.g., π (λ, ρ, φ) π (φ λ, ρ)π J (λ, ρ) since lim λ d = ˆλ a.s., where ˆλ is the maximum n likelihood estimate of λ. 5. Conclusion Tsiatis (1975) and Basu and Ghosh (1980) have addressed the nonidentifiability issue for bivariate competing risks models. This undesirable feature has trapped analysts for decades. The Bayesian methodology with two-stage noninformative priors proposed in Section 2 overcomes this nonidentifiability. This strategy can be readily extended to any bivariate competing risks model, where the lifetime random variable and the cause of failure are conditionally independent, and the distribution of the former satisfies the memoryless property. An important topic for future research will be a similar study for multivariate competing risks models. Acknowledgements. This research was partially supported by NSF Grant Number SBR Thanks are due to a referee for very helpful constructive comments. References Basu, A. P. and Ghosh, J. K. (1980). Identifiability of distributions under competing risks and complementary risks model, Communications Statistics, Part A Theory and Method, 14, Berger, J. O. and Bernardo, J. M. (1989). Estimating a product of means: Bayesian analysis with reference priors, Journal of the American Statistical Association, 84, Bernardo, J. M. (1979). Reference posterior distributions for Bayesian inference (with discussion), Journal of the Royal Statistical Society, Series B, 41, Block, H. W. and Basu, A. P. (1974). A continuous bivariate exponential extension, Journal of the American Statistical Association, 69, Datta, G. S. (1996). On priors providing frequentist validity for Bayesian inference for multiple parametric functions, Biometrika, 83, Datta, G. S. and Ghosh, J. K. (1995). On priors providing frequentist validity for Bayesian inference, Biometrika, 82, Datta, G. S. and Ghosh, M. (1995). Some remarks on noninformative priors, Journal of the American Statistical Association, 90, (1996). On the invariance of noninformative priors, Annals of Statistics, 24, David H. A. and Moeschberger, M. L. (1978). The Theory of Competing Risks. New York: Macmillan. (1971). Lifetests under competing causes of failure and the theory of competing risks, Biometrics, 27, Dawid, A. P. (1979). Conditional independence in statistical theory (with discussion), Journal of the Royal Statistical Society, Series B, bf 41, Freund, J. E. (1961). A bivariate extension of the exponential distribution, Journal of the American Statistical Association, 56,
14 bayesian analysis of bivariate competing risks models 401 Ghosh, J. K. (1994). Higher Order asymptomatics. institute of mathematical statistics, Series: NSF-CBMS Regional Conference Series in Probability and Statistics 4. Ghosh, J. K. and Mukerjee, R. (1992). Non-informative priors. In Bayesian Statistics, 4. Editors. J. M. Bernardo, J. O. Berger, A. P. Dawid, and A. F. M. Smith. Oxford Science Publications, Oxford, pp Ghosh, M., Ghosh, A., Chen, M-H., and Agresti, A. (2000). Noninformative priors for one-parameter item response models, Journal of Statistical Planning and Inference, 88, Gumbel, E. J. (1960). Bivariate exponential distributions, Journal of the American Statistical Association, 56, Jeffreys, H. (1961). Theory of Probability. Oxford, Oxford University Press. Lagakos, S. W. (1978). A covariate models for partially censored data subject to competing causes of failures, Applied Statistics, 27, Laplace, P. (1812). Theorie Analytique des Probabilities. paris, courcier. Lu, J. C. and Bhattacharyya, G. K. (1991). Inference procedures for a bivariate exponential model of gumbel based on life test of component and system, Journal of Statistical Planning and Inference, 27, Marshall, A. W. and Olkin, I. (1967). A multivariate exponential distribution, Journal of the American Statistical Association, 62, Moeschberger, M. L. (1971). Lifetests under competing causes of failure and the theory of competing risks, Biometrics, 27, Moeschberger, M. L. (1974). Life tests under competing causes of failure. Technometrics, 16(1), Mukerjee, R. and Dey, D. (1993). Frequentist validity of psterior quantiles in the presence of a nuisance parameter: higher order asymptotics, Biometrika, 80, Mukerjee, R. and Ghosh, M. (1997). Second-order probability matching priors, Biometrika, 84, Nakao, Z. and Liu, Z. Z. (1990). On Bayesian parameters and reliability estimation in a bivariate exponential model, Japonica, 35, (1991). Empirical Bayesian interval estimation for reliability function of a bivariate exponential model, Japonica, 37, Ryu, K. (1993). An extension of marshall and Olkin s bivariate exponential distribution, Journal of the American Statistical Association, 87, Sarkar, S. K. (1987). A continuous bivariate exponential distribution. Journal of the American Statistical Association, 82, Stein, C. (1985). On the ocverage pobability of confidence sets based on a prior distribution, Sequntial Methods in Statistics. Banach Center Publications 16, Warsaw PWN, Polish Scientific Publishers, Tibshirani, R. (1989). Noninformative priors for one parameter of many, Biometrika, 76, Tsiatis, A. (1975). A nonidentifiability aspect of the problem of competing risks, National Academy of Sciences., 72, Wada, C. Y., Sen, P. K., and Shimakura, S. E. (1996). A bivariate exponential model with covariates in competing risks data, Calcutta Statistical Association Bulletin 46, Welch, B. and Peers., H. W. (1963). On formulae for confidence points based on integrals of weighted likelihoods, Journal of the Royal Statistical Society, Series B, 24, Chen-Pin Wang Department of Epidemiology and Biostatistics University of South Florida College of Public Health Bruce B. Downs Blvd- MDC 56 Tampa, FL USA cwang@hsc.usf.edu Malay Ghosh Department of Statistics University of Florida 103, Griffin-Floyd Hall P.O. Box Gainsville, FL USA ghoshm@stat.ufl.edu
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