Multivariate Frailty Models for Exchangeable ... - Semantic Scholar

Multivariate Frailty Models for Exchangeable Survival Data With Covariates Catalina S TEFANESCU

Bruce W. T URNBULL

London Business School London NW1 4SA United Kingdom ([email protected] )

School of Operations Research Cornell University Ithaca, NY 14853 ([email protected] )

We consider a multivariate lognormal frailty model for correlated exchangeable failure time data, where the marginal lifetimes have conditional Weibull distributions. We discuss Bayesian statistical methods to fit this model to experimental data with varying cluster sizes. The Bayesian inferential approach arises naturally from the hierarchical structure of the frailty model. In contrast, implementation of the maximum likelihood approach encounters practical difficulties. The methodology is illustrated with the analyses of three datasets. KEY WORDS: Correlated survival data; Gibbs sampling; Load-sharing models; Markov chain Monte Carlo; Multivariate lognormal frailty; Reliability.

1. INTRODUCTION In this article, we consider parametric models for correlated failure time data and discuss Bayesian methods for fitting these models to experimental data. An important example of such data is provided by component lifetimes in reliability studies (Crowder, Kimber, Smith, and Sweeting 1991, sec. 7.2; Roy 2001; Lawless 2002, sec. 11.2). Association may arise between component lifetimes because they share a common operating environment. Also, the failure of one part of a system may place a higher stress on the surviving components, potentially leading to earlier failure. Such ideas of load sharing may occur in many situations (e.g., in software reliability studies, power plant safety assessment, or materials testing with fiber composites); see Hollander and Pena (1995) and Kvam and Pena (2005). Correlated failure time data may be regarded as grouped in clusters, where the clusters are lifetimes of components belonging to the same system. The observations from different clusters are independent, but those from the same cluster are associated. In many such cases, it is reasonable to assume that the responses within a cluster are exchangeable. For example, in the previous application, the load-sharing mechanism may be quite complex to model; if the components are identical, however, considerations of symmetry would imply the reasonable minimal assumption that the failure times be exchangeable. Sometimes it is the association itself between the clustered responses that is of scientific interest; its magnitude and dependence on explanatory variables may then need to be estimated. In other situations, the intracluster association is regarded merely as a nuisance characteristic of the data, whereas interest focuses on the marginal parameters. In these cases, the association still needs to be taken into account in order to make correct inference about the regression parameters. Indeed, the standard analytic tools that ignore the intracluster association may lead to inefficient, inconsistent, and biased estimates, as well as to erroneous standard errors. A number of different models for multivariate survival data have been proposed in the literature—an extensive survey of these models and their applications is presented by Hougaard (2000). One particularly flexible modeling tool for correlated

survival data is the frailty approach. The models in this class assume that, conditional on some unobserved quantity, W say, the lifetimes are independent. When the unknown W is integrated out, the lifetimes become dependent; the dependence is induced by the common value of W. When W is a scalar, we say we have a univariate or “shared” frailty model. In the reliability literature, several articles have focused on shared frailty models (Lindley and Singpurwalla 1986; Nayak 1987; Hougaard 1989; Whitmore and Lee 1991; Jaisingh, Dey, and Griffith 1993). The theory of multivariate frailty models, where W is a vector, has received less attention. Yet these models have several advantages; they can take into account individual-level explanatory variables (discussed later) and allow a more flexible dependence structure within clusters of observations. Indeed, the univariate frailty model can only model positive dependence between lifetimes. In practice, negative dependence could also arise, for example, when components may be competing for resources. Xue and Brookmeyer (1996, sec. 2) discussed several modeling situations where a multivariate rather than a univariate frailty approach is needed. In addition to the modeling of association, a second issue concerns the introduction of covariates for multivariate survival data. It may be reasonable to assume that the clustered survival times are exchangeable only after taking the presence of explanatory variables into account. We shall say that the clustered lifetimes T1 , T2 , . . . , Tr with covariates x1 , . . . , xr are exchangeable after adjustment for covariates if their joint survival function satisfies S(t1 , . . . , tr ; x1 , . . . , xr )

= S tπ(1) , . . . , tπ(r) ; xπ(1) , . . . , xπ(r)

(1)

for any t1 , . . . , tr and any permutation π of the indices 1, 2, . . . , r. The inclusion of covariates is important because their omission could bias the results of the analysis and because their

411

© 2006 American Statistical Association and the American Society for Quality TECHNOMETRICS, AUGUST 2006, VOL. 48, NO. 3 DOI 10.1198/004017006000000048

412

CATALINA STEFANESCU AND BRUCE W. TURNBULL

effects on the marginal survival times distributions may themselves be of interest. Covariates may act either at a cluster level or at an individual level. For example, the covariates specific to the operating conditions of the system (such as temperature, load, and pressure) act at the cluster level, whereas the covariates specific to each component (e.g., lot number and manufacturer) act at the individual level (Costigan and Klein 1993). There is a need for models rich enough to allow both types of covariates while keeping the exchangeable association structure fairly flexible. The main contributions of this article are to investigate a class of parametric multivariate lognormal frailty models for failure time data that can incorporate covariates and to implement a Bayesian approach to model fitting when clusters of varying sizes are available. Hougaard (2000, sec. 10.6) mentioned the multivariate frailty model as a promising way of describing negative dependence and more general complex dependence structures. Xue and Brookmeyer (1996) discussed the case of a bivariate lognormal frailty and proposed a modified expectation–maximization (EM) algorithm for estimation. In the multivariate case, direct maximization of the likelihood is generally difficult to achieve because of the computational intractability of the high-dimensional Laplace transforms involved in the likelihood function. As an alternative to maximum likelihood estimation, we propose a Bayesian framework for inference. This arises naturally from the inherent hierarchical structure of the models and may be implemented by means of Gibbs sampling. Similar Bayesian approaches to inference for univariate frailty models with Weibull marginals have been discussed by Sahu, Dey, Aslanidou, and Sinha (1997), Sahu and Dey (2000), and Ibrahim, Chen, and Sinha (1991, chap. 4). The article is structured as follows. Section 2 examines parameterizations of models for clustered exchangeable survival variables. The Bayesian framework for estimation is developed in Section 3, and Section 4 describes three applications.

Here σ 2 ≥ 0 is the frailty variance and ρ is the frailty correlation. The restriction ρ > −1/(r − 1) is needed to ensure that r is positive definite. We consider the case where the conditional lifetime distributions are Weibull: Ti |(Wi , xi )

∼ Weibull α, exp(Wi + xi β)/µ ,

EXCHANGEABLE MODELS FOR SURVIVAL DATA

Let (T1 , T2 , . . . , Tr ) be a cluster of lifetimes and denote by S(t1 , . . . , tr ) their joint survival function. Let xi ∈ p be a vector of covariates corresponding to the ith observation of the cluster and let β be the vector of regression parameters. We assume that the lifetimes are exchangeable after adjustment for covariates. Let W = (W1 , . . . , Wr ) be an r vector of frailties representing the unmeasured effects of the environment on component life lengths. Given W, the lifetimes are assumed to be independent. The exchangeability of Ti implies that the frailties Wi are also exchangeable. Possible choices for the frailty distribution are the multivariate normal, the multivariate logistic (Kotz, Balakrishnan, and Johnson 2000, p. 551), and the multivariate t (Spiegelhalter, Thomas, Best, and Lunn 2003). Following Hougaard (2000, p. 379), we take the frailties to have a multivariate normal distribution W ∼ Nr (0, r ), with the covariance matrix r parameterized by (r )ii = σ 2 , (r )ij = ρσ 2 ,

i = 1, . . . , r, i, j = 1, . . . , r, i = j.

TECHNOMETRICS, AUGUST 2006, VOL. 48, NO. 3

(2)

(3)

where α, µ > 0. The focus on the Weibull distribution is motivated by mathematical convenience and by the fact that Weibull is perhaps the most widely used lifetime distribution in reliability (Crowder et al. 1991; Lawless 2002). It provides a useful approximating model for a monotone failure rate, which is often a reasonable assumption. Its mathematical properties make it a particularly convenient and flexible generalization of the exponential distribution. It is sometimes also used in biological and medical applications as an alternative to nonparametric and semiparametric specifications. Note that the conditional hazard function, given by h(ti |Wi ; xi ) = αµ−α exp[α(Wi + xi β)]tiα−1 , has a Cox (1972) proportional hazards structure. The marginal moments and correlations of (T1 , . . . , Tr ) may be expressed as functions of the model parameters by way of successive conditioning. Thus, for example, on the log scale we have E(log(T); x) = E E(log(T)|W; x) γ W+xβ = E − − log(e /µ) α γ (4) = − + log(µ) − xβ, α where γ = .5772 . . . is Euler’s constant. Equation (4) holds because log(T) has an extreme-value type I conditional distribution. Similarly, var(log(T); x) = var E(log(T)|W; x) + E var(log(T)|W; x) =

2.

i = 1, . . . , r,

(1) + σ 2, α2

where (1) = π 2 /6 is the variance of the extreme-value type I distribution. It can also be shown that, for i = j, corr log(Ti ), log(Tj ); xi , xj =

ρσ 2 (1)/α 2

+ σ2

.

(5)

On the original time scale, the expressions for the marginal moments are more complicated. They are given by E(T; x) = E[E(T|W; x)] = (1 + 1/α)µ exp(σ 2 /2 − xβ), ∞ where (z) = 0 tz−1 e−t dt, z > 0, is the gamma function, and 2 2 var(T; x) = eσ −2xβ µ2 eσ (1 + 2/α) − (1 + 1/α)2 . For i = j, we have E(Ti Tj ; xi , xj ) = E[E(Ti Tj |W; xi , xj )] = E E(Ti |W; xi )E(Tj |W; xj ) =

((1 + 1/α))2 2 −(Wi +Wj ) µ E e e(xi +xj )β

=

((1 + 1/α))2 2 (ρ+1)σ 2 µ e , e(xi +xj )β

MULTIVARIATE FRAILTY MODELS

and, hence, corr(Ti , Tj ; xi , xj ) =

E(Ti Tj ; xi , xj ) − E(Ti ; xi )E(Tj ; xj ) var(Ti ; xi ) var(Tj ; xj ) 2

=

((1 + 1/α))2 (eρσ − 1) eσ (1 + 2/α) − (1 + 1/α)2 2

.

(6)

Note that the correlation between lifetimes does not depend on the values of the covariates. In particular, when the conditional distributions are exponential (α = 1), we have 2 2 2 E(T; x) = µeσ /2−xβ , var(T; x) = µ2 eσ −2xβ 2eσ − 1 , and corr(Ti , Tj ) =

2 eρσ

−1

2 2eσ

−1

.

The bivariate correlation when σ 2 = 1 in the exponential case is graphed in figure 2 of Lindeboom and Van Den Berg (1994). In general, corr(Ti , Tj ) is monotone in ρ. When ρ = 1 and we have a shared frailty model, the correlation tends to .5 as σ 2 → ∞. When ρ = −1/(r − 1), the lower bound for the correlation depends on the cluster size r, and it is obtained for σ 2 = σ02 , where σ02 is the root of the equation

r 1 2 r − 2 exp(σ 2 ) 2 exp σ + = 0. r−1 r−1 r−1 For example, when r = 2, we have −.1716 ≤ corr(Ti , Tj ) ≤ .5. In general, in all survival models with frailties, the intracluster correlation between lifetimes does not vary over the whole range [−1, 1]. For example, the shared univariate frailty model can only describe positive association of lifetimes. Note, however, that the correlation on the logarithmic scale is not similarly restricted. Indeed, from (5) it follows that corr(log(Ti ), log(Tj )) can take any value in the (−1, 1) range. Alternative measures of intracluster dependence include Spearman’s correlation coefficient ρS , the median concordance, and Kendall’s τ coefficient (Hougaard 2000, secs. 4.2–4.4). Numerical integration is necessary to compute Spearman’s ρS . Also, the median concordance and Kendall’s τ both depend on the Laplace transform of the lognormal distribution. Because no simple formulas are available for it, approximations or numerical integration must also be used to compute τ and the median concordance (Hougaard 2000, pp. 229–230, 244). Independence between clustered observations can be obtained in the multivariate frailty model when either ρ = 0 or σ 2 = 0. The two conditions lead, however, to different interpretations. If σ 2 = 0, then the frailties are constant, the frailty correlation parameter ρ is not identifiable, and the clustered lifetimes are independent. If ρ = 0 but σ 2 > 0, then the clustered lifetimes T1 , . . . , Tr are again independent, but their hazard rates are still affected by the univariate (nonshared) frailties W1 , . . . , Wr . These frailties represent now the unmeasurable covariates that lead to unobserved heterogeneity among clustered observations (Hougaard 1995). The multivariate frailty model defined by (3) is an extension of the shared univariate frailty model with Weibull hazards, which is obtained when ρ = 1. The main advantage of the multivariate model is given by its more flexible dependence

413

structure, in particular, its ability to describe negative dependence. However, the multivariate model is sometimes to be preferred to the univariate model even when the dependence is positive—see the example in Section 4.1. The multivariate model is particularly useful for the analysis of data with several sources of variation giving rise to different degrees of dependence (Hougaard 2000, p. 345). Indeed, the model can be extended to accommodate a more complex random effects structure—see the application in Section 4.3. 3. ESTIMATION In this section we discuss a Bayesian approach for fitting model (3) to experimental data. Suppose that K independent clusters of varying sizes are available for inference. Denote by {((Tki , Cki , δki ), xki )|1 ≤ k ≤ K, 1 ≤ i ≤ rk } the data from the kth cluster. The lifetimes Tk1 , . . . , Tkrk are possibly right censored by Ck1 , . . . , Ckrk , with censoring indicators δ k = (δk1 , . . . , δkrk ) given by 1, if Tki ≤ Cki δki = 0, otherwise. The observed data consist of {((Yki , δki ), xki )|1 ≤ k ≤ K, 1 ≤ i ≤ rk }, where Yki = min(Tki , Cki ). Let the maximum cluster size be R = max{rk ; 1 ≤ k ≤ K}. The model parameters must satisfy the constraints α, µ > 0, σ 2 ≥ 0, and also ρ ≥ −1/(R − 1) in order to ensure that r is positive definite for all 1 ≤ r ≤ R. Let wk = (wk1 , . . . , wkrk ) be the frailty vector for the kth cluster. We assume that conditionally on w the censoring is independent and noninformative of w. The observed but incomplete data for the kth cluster are (yk , δ k ); the complete data are (yk , δ k , wk ). For j = 1, . . . , rk , we have δkj f (ykj , δkj |wk ) = αµ−α yα−1 kj exp{α(wkj + xkj β)}

× exp −µ−α yαkj exp[α(wkj + xkj β)] , and the complete data density in the kth cluster is f (yk1 , δk1 ), . . . , ykrk , δkrk , wk = φrk (wk ; ρ, σ ) 2

rk

f (ykj , δkj |wk ),

j=1

where φr (·; ρ, σ 2 ) is the density function of the multivariate normal distribution with mean 0 and covariance matrix r given by (2). Hence, the full likelihood of the sample in terms of parameters θ is given by L(θ, w) =

K

φrk (wk )

k=1

×

rk K k=1 j=1

αµ−α eα(wkj +xkj β) yα−1 kj

δkj

× exp −µ−α eα(wkj +xkj β) yαkj . (7)

The likelihood of the sample is obtained from (7) by integrating out the unobserved frailties with respect to their lognormal TECHNOMETRICS, AUGUST 2006, VOL. 48, NO. 3

414


densities. Maximum likelihood estimators are difficult to obtain directly because no closed-form analytical expressions are available for this likelihood and the multivariate integrals become increasingly more difficult to compute with higher cluster sizes. We propose the use of a Bayesian approach for estimation. This is natural because of the inherent hierarchical structure of the multivariate lognormal frailty model (3). The first level of the hierarchy is given by the parameters of the lognormal distribution ρ and σ 2 , on the second level lie the unobserved random effects w and the Weibull scale and shape parameters µ and α, and the third level consists of the observed data given by the clustered observations. The parameters and hyperparameters of the model are θ = (α, µ, β, ρ, σ 2 , w). A Bayesian specification requires prior distributions to be chosen for all parameters and hyperparameters in the hierarchy. We assume that the parameters are a priori independent, and with little external information available, we generally would like to specify noninformative priors p(·) for the components of θ . The prior distribution of θ then becomes p(θ ) = p(α)p(µ)p(β)p(ρ)p(σ 2 ), and the joint posterior density p(θ, {wk }|{(y, δ)}) is proportional to the product of the prior and the augmented likelihood given by (7). The marginal posterior density of each parameter is obtained by integrating out the other parameters from the joint posterior density. This is difficult to achieve analytically; therefore, we propose the use of Gibbs sampling (Geman and Geman 1984; Gelfand and Smith 1990) for generation of the marginal posterior distributions. Details of its implementation are given in the Appendix. For all parameters, 95% credible intervals can be computed from the samples of observations generated from the posterior densities, and these can then be used in testing specific hypotheses about the parameters. In particular, the hypotheses that β = 0 (no covariate effect), ρ = 1 (univariate frailty), ρ = 0 (independence with unobserved heterogeneity), or σ 2 = 0 (independence) may be of interest. Ibrahim et al. (1991, chap. 6) described several methods for model comparison in a Bayesian framework. Following Spiegelhalter et al. (2003), in the applications described in Sections 4 and 5 we shall use the deviance information criterion (DIC) to choose among different models fitted to the same dataset. 4. APPLICATIONS 4.1 Load-Sharing Data Kim and Kvam (2004) in their table 1 described two simulated failure time datasets, which were then used to illustrate order-restricted estimation of load-sharing models. Each of the datasets consists of 20 clusters of equal size R = 3. None of the failure times is censored and there are no covariates. We analyzed the data from their sample 1 using several frailty models, in particular, those discussed in Section 2. Kim and Kvam (2004) discussed many load-sharing models including local and nonmonotone load sharing. However, in the absence of an indication of any particular load-sharing mechanism, it is useful to TECHNOMETRICS, AUGUST 2006, VOL. 48, NO. 3

Table 1. Load-Sharing Data: Parameter Estimates Variable

Mean

Standard error

Median

95% credible intervals

Model 1: multivariate lognormal frailty (DIC = 285.271) α 3.56 3.92 2.24 (1.39, 15.33) µ 6.384 1.417 6.276 (3.929, 9.430) ρ .784 .151 .800 (.463, .994) σ2 .966 .429 .925 (.280, 1.920) Model 2: multivariate lognormal frailty, ρ = 0 (DIC = 366.906) α 1.21 .15 1.20 (.953, 1.519) µ 8.079 .980 8.038 (6.264, 10.130) σ2 .065 .083 .034 (.004, .314) Model 3: univariate lognormal frailty (DIC = 332.312) α 1.75 .24 1.75 (1.29, 2.25) µ 7.004 1.520 6.867 (4.458, 10.310) σ2 .735 .370 .674 (.197, 1.626) Model 4: multivariate t -distributed frailty (DIC = 314.242) α 2.42 2.48 1.98 (1.41, 6.38) µ 7.839 1.376 7.784 (5.228, 10.710) ρ .84 .14 .88 (.49, .99) σ2 .368 .249 .305 (.080, 1.036)

consider a “minimalist” model for which the correlation structure is based only on the assumption of exchangeability of component lifetimes. In particular, this can serve as a benchmark against which to judge the fit of more specific load-sharing models. The frailty correlation is restricted by −1/2 ≤ ρ ≤ 1, and we specified a uniform prior for ρ on (−1/2, 1) (Ibrahim et al. 1991, p. 138). The posterior estimates of the parameters given in Table 1 are based on an inverse (.1, .1) prior for α and σ 2 , and a N(0, 103 ) prior for log(µ). As in the example of Section 4.3 we have investigated different choices of priors, but they had little influence on the posterior estimates. The Gibbs sampler was started with initial values α0 = 1, µ0 = 1, σ02 = 1, and ρ0 = 0. The chains were run for 30,000 iterations, with the first 10,000 iterations discarded as the burn-in period. We fit the multivariate lognormal frailty model with general correlation ρ (model 1), then the same model with the constraint ρ = 0 (model 2), then a univariate lognormal frailty model (model 3), and finally, a multivariate-t frailty model with k = 3 degrees of freedom (model 4). The models can be compared using the deviance information criterion (DIC). Not surprisingly, model 2, which implies independence of failure times, has the worst fit. This is consistent with the results of testing that ρ = 0 in the other models; because none of the credible intervals for ρ contain 0, the hypothesis of independence can be rejected. Model 1 has the best fit, followed by model 4. Therefore, the multivariate lognormal frailty seems preferable to the multivariate-t frailty in this case. Note that the difference in the goodness of fit between models 1 and 3 (the multivariate versus univariate frailty) is quite substantial and that the estimated ρ is rather high. This provides an example of a case when a multivariate frailty model gives a much better fit to the data than a univariate frailty model, even though the association is strong and positive. 4.2 Tumorigenesis Data Mantel, Bohidar, and Ciminera (1977) reported data from a litter-matched tumorigenesis experiment. The experiment involved 50 male and 50 female litters, each of three rats. Two rats


in each litter served as controls, and the remaining rat received a drug. The data recorded are the time to tumor appearance. Censoring was induced by death from other causes, as well as by the end of study after 104 weeks. Common genetic factors and shared carcinogenic exposure induce association in the times to tumor appearance between litter mates. To assess the treatment effect, it is, therefore, important to account for intralitter dependence. Note, in particular, that it seems reasonable to assume exchangeability of the responses within a litter. The sample of male rats was heavily censored, because there were only two male rats that developed tumors. Therefore, we restrict our analysis to the subset of the data concerning the female rats. Here treatment is an individuallevel covariate. We now fit the multivariate lognormal frailty model (3) to the female rats’ tumor time data, with treatment as a binary individual-level covariate (control x = 0 or drug x = 1). The results reported here are based on an inverse (.1, .1) prior for σ 2 , a (.1, .1) prior for α, a N(0, 105 ) prior for log(µ), and a N(0, 106 ) prior for β. Because the cluster size is R = 3, the frailty correlation is restricted by −1/2 ≤ ρ ≤ 1, and an appropriate noninformative prior for ρ is given by the uniform distribution on (−1/2, 1) (Ibrahim et al. 1991, p. 138). To study the sensitivity of the results to other choices of prior for ρ, however, we also fit the model with diffuse beta-type priors on (−1/2, 1). The results are displayed in Table 2. It can be seen that the estimates were robust to the prior specification. Also, not shown, we found that different choices of diffuse priors for α, σ 2 , β, and log(µ) had little influence on the estimates of µ and β. The Gibbs sampler was started with initial values α0 = 1, µ0 = 1, σ02 = 1, ρ0 = 0, and β0 = 0. The chain was run for 50,000 iterations, with the first 10,000 iterations discarded as the burn-in period. Table 2 presents the mean, standard deviation, and median of the marginal posterior distributions of the parameters, as well as the 90% credible intervals. It appears that the chain of samples from the marginal posterior distributions of σ 2 and ρ has very slow mixing properties, and this is corroborated by the relatively large standard errors. These litter tumorigenesis data were also analyzed by Klein and Moeschberger (2003, pp. 429–435) who fit a Cox proporTable 2. Female Rat Tumor Time Data: Parameter Estimates With Different Priors for Frailty Correlation ρ Prior

ρ ∼ U(−.5, 1)

ρ = −.5 + 1.5u u ∼ beta(ζ , ζ ) ζ ∼ (.1, .1) ρ = −.5 + 1.5u u ∼ beta(ζ , ζ ) ζ ∼ (1, .01)

Variable Mean

α µ β ρ σ2 α µ β ρ σ2 α µ β ρ σ2

Standard error Median


6.620 5.110 4.830 (3.130, 16.600) 163.000 19.140 160.700 (136.200, 198.500) .260 .103 .255 (.097, .437) .443 .257 .434 (.043, .889) .171 .092 .158 (.052, .340) 5.904 4.491 4.205 (2.954, 19.950) 164.600 17.890 161.800 (140.000, 208.300) .243 .107 .236 (.045, .480) .591 .340 .607 (−.040, .999) .152 .080 .138 (.039, .340) 5.588 3.481 4.431 (2.948, 16.460) 163.600 15.920 162.800 (136.300, 197.700) .264 .095 .263 (.086, .452) .506 .249 .503 (.022, .925) .147 .082 .133 (.030, .321)

415

tional hazards model first assuming independence and then with a univariate gamma frailty. They found no evidence of a litter effect in this experiment, and their estimate of the treatment effect β is close to the estimate in Table 2. Ripatti, Larsen, and Palmgren (2002) fit a univariate frailty model with a lognormal frailty to the litter tumorigenesis data. Hougaard (2000, p. 284) reported analyses of this dataset using Weibull models with univariate stable and gamma frailty. All of these analyses found only a slight intracluster dependence. Unlike the univariate frailty models, our approach using a multivariate frailty model allows for negative correlation within a litter. The independence case no longer lies on the boundary of the parameter space, and, thus, testing whether the correlation is statistically significant becomes straightforward. (When the null value is on the boundary of the parameter space, testing is more complicated—see, e.g., Andersen, Borgan, Gill, and Keiding 1993, p. 663.) For the rat example, the 90% credible interval for the frailty correlation ρ does not include 0, and, hence, there is some suggestive evidence of a positive dependence among litter mates. 4.3 Breaking Strengths of Rigging Lines Crowder et al. (1991, table 7.1) reported data from an experiment on the breaking strengths of parachute rigging lines. The test was carried out on eight lines on each of six parachutes, and the strength was measured on each line at six equally spaced positions at increasing distances from the hem. There is a clear difference in strength between the parachutes due to different exposure times to weathering. In an attempt to determine how breaking strength varies with line and position, we fit the frailty model (3) to the parachute dataset. In fact, we need to use a slight generalization to allow one of the covariates to be modeled as a random effect rather than a fixed effect. The correlation between measurements on the same line is modeled by choosing each line to be a cluster. We model the breaking strengths as exchangeable after accounting for the variable “distance from the hem” (coded as “positions” with values 1, . . . , 6)—an individual-level covariate. As suggested in the analysis of Crowder et al. (1991, p. 141), we might start by considering a linear effect of this covariate. A parachute random effect is introduced in order to model the heterogeneity in breaking strengths among parachutes. Model (3) thus becomes Tkji |(Wkj , xkji ) ∼ Weibull α, exp(Wkji + uk + βxkji )/µ (8) for i = 1, . . . , 6, j = 1, . . . , 8, and k = 1, . . . , 6, where uk ∼ N(0, σu2 ). Here k is the index for parachutes, j for lines, and i for position on the line. The parachute random effects u1 , . . . , u6 will also be sampled at each iteration of the Gibbs sampler, and the posterior conditional distributions (A.2)–(A.7) will include the variance of the random effect σu2 as well. Because R = 6, the frailty correlation is restricted by −1/5 ≤ ρ ≤ 1, and we specified a uniform prior for ρ on (−1/5, 1) (Ibrahim et al. 1991, p. 138). To check the sensitivity of our estimates, we specified a range of different priors—gamma and inverse gamma for α and σ 2 , normal with variances between 103 and 106 for β and log(µ). These choices of priors had little influence on the estimates of µ and β. The results reported TECHNOMETRICS, AUGUST 2006, VOL. 48, NO. 3

416


Table 3. Rigging-Line Breaking-Strength Data: Parameter Estimates Variable

α µ β ρ σ2 σu2

Mean

Standard error

42.13 1,412.00 −.0092 .896 .0051 .178

5.54 199.50 .0013 .050 .0012 .220

Median


41.49 (34.52, 52.09) 1,382.00 (1,132.00, 1,709.00) −.0092 (−.0113, −.0071) .903 (.803, .962) .0050 (.0035, .0073) .118 (.032, .512)

• The random variables w1 , . . . , wK are independent, and wk |yk , δ k , θ has density proportional to r k φrk (wk ) exp α wkj δkj j=1

− µ−α

rk

exp{α(wkj + xkj β)}yαkj .

(A.1)

j=1

here are based on an inverse (.1, .1) prior for α, σ 2 , and σu2 , a N(0, 105 ) prior for log(µ), and a N(0, 106 ) prior for β. The Gibbs sampler was started with initial values α0 = 1, µ0 = 1, 2 = 1, and β = 0. σ02 = 1, ρ0 = 0, σu0 0 The chain was run for 25,000 iterations, with the first 10,000 iterations discarded as the burn-in period. The posterior estimates are given in Table 3. The chain of samples from the marginal posterior distributions of σ 2 and ρ has very slow mixing properties, mainly due to the small number of clusters in the dataset. The estimated β is negative, which agrees with Crowder’s conclusion that the strength of the line decreases as one moves from position 6 down to position 1. The estimate for the frailty correlation is high: ρˆ = .90 with 90% credible interval (.80, .96).

• The posterior distribution of α conditional on (µ, β, ρ, σ 2 ), {yk }, {δ k }, {xk }, and {wk } has density proportional to rk K −α α(w +x β) α−1 δkj p(α) αµ e kj kj ykj k=1 j=1

× exp −µ−α eα(wkj +xkj β) yαkj .

(A.2)

• The posterior distribution of µ conditional on (α, β, ρ, σ 2 ), {yk }, {δ k }, {xk }, and {wk } has density proportional to p(µ)µ−α

k,j δkj

× exp −µ

−α

rk K

yαkj exp[α(wkj

+ xkj β)] .

(A.3)

k=1 j=1

5. CONCLUSION The model discussed in this article extends both the shared univariate frailty model, which only allows for positive dependence, and the univariate nonshared frailty model, which captures unobserved heterogeneity due to neglected individual covariates. More complex dependence structures could easily be modeled by relaxing the exchangeability assumption and allowing different values for the frailty correlations in r . Our computational experience shows that Gibbs sampling is an efficient approach to estimation in the Bayesian framework. As expected, the computational complexity and the time to convergence of the Markov chains increased with cluster size, but they remained reasonable in all the applications that we investigated.

• The posterior distribution of β conditional on (α, µ, ρ, σ 2 ), {yk }, {δ k }, {xk }, and {wk } has density proportional to K r k p(β) exp xkj δkj αβ k=1 j=1

−µ

−α

rk K

yαkj exp[α(wkj

+ xkj β)] .

(A.4)

k=1 j=1

• The posterior distribution of ρ conditional on (α, µ, β, σ 2 ), {yk }, {δ k }, {xk }, and {wk } has density proportional to p(ρ)σ −

K

k=1 rk

×

(1 − ρ)−

K

K

k=1 (rk −1)/2

{1 + (rk − 1)ρ}−1/2 f (ρ, σ 2 , {wk }),

(A.5)

k=1

ACKNOWLEDGMENTS This research was supported in part by grant R01 CA66218 from the U.S. National Institutes of Health and by an RAMD grant from London Business School. APPENDIX: IMPLEMENTATION OF THE GIBBS SAMPLER The Gibbs sampler proceeds by successively updating each variable by sampling from its conditional distribution given the current values of all other variables. Under mild conditions, it can be proven that convergence is achieved after a sufficiently large number of iterations, and the values of the updated variables so obtained form a sample from the joint distribution— see, for example, Robert and Casella (1999). The Gibbs sampler requires all the conditional posterior distributions, and these can be derived based on the joint posterior: TECHNOMETRICS, AUGUST 2006, VOL. 48, NO. 3

where f (ρ, σ 2 , {wk })

= exp − rk

K {1 + (rk − 2)ρ}Sk1 − 2ρSk2 k=1

2σ 2 (1 − ρ){1 + (rk − 1)ρ}

,

and Sk2 = i=j wki wkj . To see this, with Sk1 = note that the posterior conditional distribution of ρ is proportional to K 2 p(ρ) φrk (wk ; ρ, σ ) 2 j=1 wkj

k=1

∝ p(ρ)

K k=1

1 1 −1 w exp − w k . 2 k rk (2π)rk /2 |rk |1/2 (A.6)


But |r | = σ 2r (1 − ρ)r−1 {1 + (r − 1)ρ} and r−1 =

1 σ 2 (1 − ρ){1 + (r − 1)ρ}

Ir {1 + (r − 1)ρ} − ρ1r ,

where Ir is the identity matrix of order r and 1r is the r × r matrix of 1’s. Hence, (A.5) may now be derived from (A.6). • The posterior distribution of σ 2 conditional on (α, µ, β, ρ), {yk }, {δ k }, {xk }, and {wk } has density proportional to p(σ 2 )(σ 2 )−

K

k=1 rk /2

f (ρ, σ 2 , {wk }).

(A.7)

This follows from arguments similar to those used to derive (A.5). Sampling from the conditional posterior distributions can be realized using a griddy Gibbs approach (Ritter and Tanner 1992). In practice, the Gibbs sampler may be implemented in WinBugs (Spiegelhalter et al. 2003), and convergence diagnostics are usually computed with CODA (Cowles and Carlin 1996). [Received ????. Revised ????.]

REFERENCES Andersen, P. K., Borgan, O., Gill, R., and Keiding, N. (1993), Statistical Models Based on Counting Processes, New York: Springer-Verlag. Costigan, T. M., and Klein, J. P. (1993), “Multivariate Survival Analysis Based on Frailty Models,” in Advances in Reliability, ed. A. P. Basu, Amsterdam: Elsevier, pp. 43–58. Cowles, M. K., and Carlin, B. P. (1996), “Markov Chain Monte Carlo Convergence Diagnostics: A Comparative Review,” Journal of the American Statistical Association, 91, 883–904. Cox, D. R. (1972), “Regression Models and Life Tables” (with discussion), Journal of the Royal Statistical Society, Ser. B, 34, 187–220. Crowder, M. J., Kimber, A. C., Smith, R. L., and Sweeting, T. J. (1991), Statistical Analysis of Reliability Data, London: Chapman & Hall. Gelfand, A. E., and Smith, A. F. M. (1990), “Sampling-Based Approaches to Calculating Marginal Densities,” Journal of the American Statistical Association, 85, 398–409. Geman, S., and Geman, D. (1984), “Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images,” IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721–741. Hollander, M., and Pena, E. A. (1995), “Dynamic Reliability Models With Conditional Proportional Hazards,” Lifetime Data Analysis, 1, 377–401.

417

Hougaard, P. (1989), “Fitting a Multivariate Failure Time Distribution,” IEEE Transactions on Reliability, 38, 444–448. (1995), “Frailty Models for Survival Data,” Lifetime Data Analysis, 1, 255–273. (2000), Analysis of Multivariate Survival Data, New York: SpringerVerlag. Ibrahim, J. G., Chen, M. H., and Sinha, D. (1991), Bayesian Survival Analysis, New York: Springer-Verlag. Jaisingh, L. R., Dey, D. K., and Griffith, W. S. (1993), “Properties of a Multivariate Survival Distribution Generated by a Weibull and Inverse Gaussian Mixture,” IEEE Transactions on Reliability, 42, 618–622. Kim, H., and Kvam, P. H. (2004), “Reliability Estimation Based on System Data With an Unknown Load Share Rule,” Lifetime Data Analysis, 10, 83–94. Klein, J. P., and Moeschberger, M. (2003), Survival Analysis, New York: Springer-Verlag. Kotz, S., Balakrishnan, N., and Johnson, N. L. (2000), Continuous Multivariate Distributions, Vol. 1 (2nd ed.), New York: Wiley. Kvam, P. H., and Pena, E. A. (2005), “Estimating Load-Sharing Properties in a Dynamic Reliability System,” Journal of the American Statistical Association, 100, 262–272. Lawless, J. F. (2002), Statistical Models and Methods for Lifetime Data (2nd ed.), New York: Wiley. Lindeboom, M., and Van Den Berg, G. J. (1994), “Heterogeneity in Models for Bivariate Survival: The Importance of the Mixing Distribution,” Journal of the Royal Statistical Society, Ser. B, 56, 1016–1022. Lindley, D. V., and Singpurwalla, N. D. (1986), “Multivariate Distributions for the Life Lengths of Components of a System Sharing a Common Environment,” Journal of Applied Probability, 23, 418–431. Mantel, N., Bohidar, N. R., and Ciminera, J. L. (1977), “Mantel–Haenszel Analyses of Litter-Matched Time-to-Response Data, With Modifications for Recovery of Interlitter Information,” Cancer Research, 37, 3863–3868. Nayak, T. K. (1987), “Multivariate Lomax Distribution: Properties and Usefulness in Reliability Theory,” Journal of Applied Probability, 24, 170–177. Ripatti, S., Larsen, K., and Palmgren, J. (2002), “Maximum Likelihood Inference for Multivariate Frailty Models Using an Automated Monte Carlo EM Algorithm,” Lifetime Data Analysis, 8, 349–360. Ritter, C., and Tanner, M. A. (1992), “The Gibbs Stopper and the Griddy Gibbs Sampler,” Journal of the American Statistical Association, 87, 861–868. Robert, C. P., and Casella, G. (1999), Monte Carlo Statistical Methods, New York: Springer-Verlag. Roy, D. (2001), “Some Properties of a Classification System for Multivariate Life Distributions,” IEEE Transactions on Reliability, 50, 214–220. Sahu, S. K., and Dey, D. K. (2000), “A Comparison of Frailty and Other Models for Bivariate Survival Data,” Lifetime Data Analysis, 6, 207–227. Sahu, S. K., Dey, D. K., Aslanidou, H., and Sinha, D. (1997), “A Weibull Regression Model With Gamma Frailties for Multivariate Survival Data,” Lifetime Data Analysis, 3, 123–137. Spiegelhalter, D. G., Thomas, A., Best, N. G., and Lunn, D. (2003), WinBUGS Version 1.4 User Manual, Cambridge, U.K.: MRC Biostatistics Unit. Whitmore, G. A., and Lee, M. T. (1991), “A Multivariate Survival Distribution Generated by an Inverse Gaussian Mixture of Exponentials,” Technometrics, 33, 39–50. Xue, X., and Brookmeyer, R. (1996), “Bivariate Frailty Model for the Analysis of Multivariate Survival Time,” Lifetime Data Analysis, 2, 277–289.

TECHNOMETRICS, AUGUST 2006, VOL. 48, NO. 3