UNIVERSIDADE FEDERAL DE SÃO CARLOS CENTRO DE CIÊNCIAS EXATAS E TECNOLÓGICAS

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1 ISSN UNIVERSIDADE FEDERAL DE SÃO CARLOS CENTRO DE CIÊNCIAS EXATAS E TECNOLÓGICAS THE FGM BIVARIATE LIFETIME COPULA MODEL: A BAYESIAN APPROACH Francisco Louzada-Neto Vicente G. Cancho Adriano K. Suzuki Gladys D.C. Barriga RELATÓRIO TÉCNICO DEPARTAMENTO DE ESTATÍSTICA TEORIA E MÉTODO SÉRIE A Novembro/2010 nº 234

2 The FGM Bivariate Lifetime Copula Model: A Bayesian Approach Francisco Louzada a, Vicente G. Cancho b Adriano K. Suzuki a and Gladys D.C. Barriga c a Universidade Federal de São Carlos b Universidade de São Paulo c Universidade Estadual Paulista Abstract. In this paper we propose a bivariate distribution for the bivariate survival times based on Farlie-Gumbel-Morgenstern copula to model the dependence on a bivariate survival data. The proposed model allows for the presence of censored data and covariates. For inferential purpose a Bayesian approach via Markov Chain Monte Carlo (MCMC) were considered. Further, some discussions on the model selection criteria are given. In order to examine outlying and influential observations, we present a Bayesian case deletion influence diagnostics based on the Kullback-Leibler divergence. The newly developed procedures are illustrated via a simulation study and a real data set. 1 Introduction In survival and reliability analysis we can observe two lifetimes for a same patient or equipment. For instance, in the medical area we can have interest in studying the lifetimes of matched human organs, as kidneys and eyes, and double recurrence of a certain disease. In industrial applications this type of data can occur for systems whose duration times depend on the durability of two components. For instance, the damage of dual generators in a power plant or the lifetime of motors in a twin-engine airplane. The examples set up above are illustrations of bivariate survival data. Bivariate survival data in general are correlated and the study of that dependence has been focus of many researches. The most popular approach is the frailty models in that one or more random effects are included in the model in order to model the dependence between the observations (see for instance, Clayton, 1978; Vaupel, et al. 1979; and Hougaard, 1986; Oakes, 1989). In this case, the marginal times are conditionally independent given the fragility variable. Details on early developments of the such models may be seen in Clayton (1978). Sahu and Dey (2000) considered a comparative study between the fragility models and the bivariate models based on the exponential and Weibull distributions for bivariate survival data according to Bayesian criteria. Recently, the copula models (see for instance, Shih and Louis, 1995; Nelsen, 2006; and Embrechts, et al., 2003) have become a popular tools to model the dependence of multi- AMS 2000 subject classifications. Primary 60K35, 60K35; secondary 60K35 Keywords and phrases. sample, L A TEX 2ε 1

3 2 Louzada et al. variate data, especially in biological areas, actuarial sciences and finances. A copula is a function that connects the marginal distributions to restore the joint distribution. Different copula functions represent different dependence structures between variables (Nelsen, 2006). By comparing to the joint distribution approach, a copula model is a more convenient tool in studying the dependence structure. Indeed it is more flexible in applications, since, when the scatter of the data does not fit any known family of joint distributions, it may be difficult to specify the joint distribution. Using copulas, however, we can first estimate the marginal distributions and then estimate the copula. Another advantage of the copula modeling is its relatively mathematical simplicity. Also, it is possible to build a variety of dependence structures based on parametric or non-parametric models for the marginal distributions. In survival analysis, models based on copulas are considered in Hougaard (1989), Oakes (1989), Shih and Louis (1995) and Gustafson et al. (2003). Particularly, from a Bayesian perspective, Romeo et al. (2006) considered an application of the Archimedian copula family for modeling the dependence of bivariate survival. In their analysis, the author considered a Weibull distribution for the marginal distributions of the bivariate lifetime components. In this paper, we consider a copula model based on the Farlie-Gumbel-Morgenstern (FGM) distribution (see, Conway, 1979). The motivation for considering the FGM copula is modeling bivariate data with a weak dependence between marginal times. Also, the FGM copula can measure both positive and negative dependence, while the usual copulas, such as the Archimedian copula family, can only measure positive one. The main objective of the paper is to present the FGM bivariate survival copula model and to develop diagnostic measures from a Bayesian perspective based on the Kullback-Leibler (K-L) divergence as proposed by Peng and Dey (1995). For each individual lifetime variable we consider Weibull marginal distributions. For inferential purpose joint estimation was performed and a sampling based approach via Markov Chain Monte Carlo (MCMC) was considered. A simulation study and an application on real data illustrates our approach. We compare the methodology propose with the Positive stable frailty (PSF) model (Hougaard, 1986). The paper is organized as follows. Section 2 presents a survival bivariate model by considering a FGM copula distribution. Section 3 presents the Bayesian inferential procedure as well as some discussions on the model selection criteria are given. Section 4 a Bayesian case deletion influence diagnostics based on the K-L divergence is presented. Section 5 presents the results of a simulation study and re-analyses the dataset on the efficacy of photocoagulation treatment for proliferative retinopathy on diabetic patients reported by The Diabetic Retinopathy Study (1976). Final comments in Section 6 conclude the paper. 2 The bivariate survival model based on FGM copula In order to define a copula we first suppose that C φ is a distribution function with density function c φ on [0, 1] 2 for φ R. Then, let (T 1, T 2 ) denote the pared failure times and S j e

4 The FGM Bivariate Lifetime Copula Model 3 f j denoting respectively the marginal survival functions an the marginal density function of T j, j = 1, 2. If (T 1, T 2 ) comes from the C φ copula for some φ then the joint survival and density function of (T 1, T 2 ) are given by (Shih and Louis, 1995) and S(t 1, S 2 ) = C φ (S 1 (t 1 ), S 2 (t 2 )), t 1, t 2 > 0, (2.1) f(t 1, t 2 ) = c φ (S 1 (t 1 ), S 2 (t 2 ))f 1 (t 1 )f 2 (t 2 ), t 1, t 2 > 0, (2.2) respectively. Notice that the marginal distributions and the dependence structure can be visualized separately and this dependence structure is represented by a copula. Following Conway (1979), considering the Farlie-Gumbel-Morgenstern distribution, hereafter FGM copula, we have the copula distribution function given as C φ (u, v) = uv[1 + φ(1 u)(1 v)], (2.3) where 0 u, v 1 e 1 φ 1. The survival bivariate function induced by FGM copula (2.3) is given then by S(t 1, t 2 ) = S 1 (t 1 )S 2 (t 2 )(1 + φ(1 S 1 (t 1 ))(1 S 2 (t 2 ))), (2.4) where φ parameter measures the intensity of the dependence between the lifetimes. Observe that when φ = 0, S(t 1, t 2 ) = S 1 (t 1 )S 2 (t 2 ), leading to the indicative that the random variables T 1 and T 2 are independentes. For the FGM copula, C φ (2.3), the Kendall s tau (Nelsen, 2006) is given by τ φ = 2φ 9, (2.5) for φ = 0, τ φ is equals 0. Other dependence measures between the variables T 1 e T 2 can be found in Nelsen (2006). For instance, if the random variables T 1 e T 2 have Weibull distribution with density function f(t j ) = α j λ j t α j 1 j exp { λ j t α j j }, j = 1, 2, (2.6) it can be proved that Pearson s correlation coefficient between T 1 e T 2, is given by ) ( ) φ (1 2 1 α α 2 ρ = ( ) ( ), (2.7) Γ( 2 +1) α 1 1 Γ( 2 +1) α 2 1 Γ( 1 +1) α 2 Γ( 1 +1) 1 α 2 2 observe when φ = 0 the correlation coefficient, ρ is equals 0. When T j has exponential distribution, with parameter α j = 1, j = 1, 2, the correlation coefficient is given by, ρ = φ/4.

5 4 Louzada et al. 3 Inference For inference, we adopt a full Bayesian approach. The likelihood function, prior distributions for the parameters in the model, details of the MCMC algorithm and the model comparison are described below. Let (T i1, T i2 ) the ith bivariate lifetime and (C i1, C i2 ) the censored bivariate lifetimes, for i = 1,..., n. Suppose that (T i1 ; T i2 ) and (C i1 ; C i2 ) are independent. For each individual i observed quantities are represented by the random variables t ij = min(t ij, C ij ) and δ ij = I(y ij = Y ij ), which denotes a censorship indicator, j = 1, 2. Let S(t 1 γ 1 ) and S(t 2 γ 1 ) be the survival functions of T i1 and T i2, respectively, where γ 1 and γ 2 are parameter vectors of q 1 and q 2 elements associated to each one of the marginal distributions. Considering the bivariate survival function S(t 1, t 2 φ, γ 1, γ 2 ) given in (2.4), following Lawless (2003), the contribution of the ith individual for the log-likelihood of θ = (φ, γ 1, γ 2 ) is given by ( 2 ) ( ) S(t 1, t 2 θ) S(t1, t 2 θ) l i (θ) = δ i1 δ i2 log + δ i1 (1 δ i2 ) log t i1 t i2 t i1 ( ) S(t1, t 2 θ) +δ i2 (1 δ i1 ) log + (1 δ i1 )(1 δ i2 ) log S(t 1, t 2 θ). (3.1) 3.1 Prior and posterior densities t i2 The use of the Bayesian method besides being an alternative statistical approach, it allows the incorporation of previous knowledge of the parameters through an informative priori distribution. When there is not such previous knowledge one may consider a noninformative priori structure. For instance, in order to carry out a Bayesian inference procedure for the bivariate survival model of giving in (2.4), consider as in Section 2, that the parametric distribution family of the marginal lifetimes T 1 and T 2 are known and indexed by the parameter vectors, γ 1 and γ 2, respectively. In order to guarantee proper posteriors, we adopt proper priors with known hyper-parameters. Thus, we consider a prior distribution of θ = (φ, γ 1, γ 2 ) given by 2 π(θ) (1 φ) r1 1 (1 + φ) r 2 1 π(γ j ). (3.2) Equation (3.2) implies that (1 φ)/2 follows a Beta(r 1, r 2 ) distribution, π(γ j ) is the priori distribution of γ j and that γ 1, γ 2 e φ are mutually independent. Besides, φ ( 1, 1). Combining (3.2) with the likelihood function, L(θ)) = exp( n l i(θ)), where l i (θ)) is given in (3.1), we straightforwardly obtain the joint posterior distribution of θ), π(θ) D), where D is the observed dataset. For parametric specification of the marginals it is possible to introduce covariates in each one of the components, i.e., γ j = ϕ(x j ). For simplicity and to avoid restrictions in the parametric space, we consider ϕ(x j ) = exp(x j β j), where β j is corresponding to the vector of unknown coefficients of order q j 1, associated to the covariates x j, j = 1, 2. j=1

6 3.2 Computation The FGM Bivariate Lifetime Copula Model 5 In the Bayesian approach, the target distribution for inference is the posterior of the parameters of interest. For this, we need to obtain the marginal posterior densities of each parameter, which are obtained by integrating the joint posterior density with respect to each parameter. We point out that for any marginal distribution of T 1 and T 2 the joint posterior distribution is not tractable analytically but Markov chain Monte Carlo (MCMC) methods such as the Gibbs sampler, can be used to draw samples, from which features of the marginal posterior distributions of interest can be inferred (Gilks et al., 1996). For estimation procedure we consider joint estimation where all the model parameters simultaneously in the MCMC algorithm. The Gibbs sampler is an iterative procedure of abroad class of methods generically named Markov Chain Monte Carlo (MCMC). Many practical aspects of the MCMC methodology are described in Gelfand and Smith (1990) and Gamerman and Lopes (2006). This method is applicable in situations where one is not able to generate samples directly from the joint posterior density. It however requires the full conditional densities for generating samples. Assuming that T j has the Weibull distribution given in (2.6) with parameters α j and λ ij = exp(β 0j + β 1j x i ), j = 1, 2. We choose the following independent prior distributions: β kj N(µ kj, σkj 2 ), α j Gamma(a j, b j ) where k = 0, 1 and j = 1, 2. Combining the likelihood function (3.1) and the prior distribution (3.2), we can obtained the joint posterior density of all unobservable which is not tractable analytically but MCMC methods such as the Gibbs samples, can be used to draw samples, from which features of marginal posterior distributions of interest can be inferred. The full conditional posterior densities for each parameter is given( by n ) π(φ D, θ ( φ) ) (1 φ) r1 1 (1 + φ) r 2 1 i and ( n ) ( n π(β 0j D, θ ( β0j )) i exp { δ ji + µ0j }β σ0j 2 0j β2 0j 2σ0j 2 ( n ) ( n π(β 1j D, θ ( β1j )) i exp { δ jix i + µ1j }β σ1j 2 1j β2 1j 2σ1j 2 π(α j D, θ ( αj )) ( n it (α j 1)δ ji ji ) ( α (a j 1)( n δ ji) j exp n n e (β 0j +β 1j x i ) t α j ji e (β 0j +β 1j x i ) t α j ji ) ) ) n (β 0j + β 1jx i)t α j ji bjαj for j = 1, 2 with i = ν (1 δ 1)(1 δ2) ν 1i = 1 + φ 2 (1 s j (t ji )), j=1 ν δ 1δ 2 1i 2i ν δ 1(1 δ2) 3i ν δ 2(1 δ 1 ) 4i,

7 6 Louzada et al. ( ) ν 2i = 1 + φ (1 2s j (t ji )), ( j=1 ) ν 3i = 1 + φ 1 2s 1 (t ji ) s 2 (t ji ) s j (t ji ) j=1 and ( ) ν 4i = 1 + φ 1 s 1 (t ji ) 2s 2 (t ji ) s j (t ji ) j=1. The conditional densities above do not belong to any known parametric density family. In order to generate our samples we then implement an Metropolis-Hasting algorithm within Gibbs iterations (Chib and Greenberg, 1995). The simulations were performed using the Winbugs software (Spiegelhalter et al. 2003). Winbugs codes are available mailing to one of the authors. 3.3 Model comparison criteria In the literature, there are various methodologies which intend to analyze the suitability of a distribution, as well as selecting the best fit among a collection of distributions. In this paper we shall inspect some of the Bayesian model selection criteria; namely, the deviance information criterion (DIC) proposed by Spiegelhalter et al. (2002), the expected Akaike information criterion (EAIC) - Brooks (2002), and the expected Bayesian (or Schwarz) information criterion (EBIC) - Carlin and Louis (2001) was used. These criteria are based on the posterior mean of the deviance, E{D(θ)}, which is also a measure of fit and can be approximated from the MCMC output by Dbar = 1 V D(θ v ) V where the index v indicates the v-th realization of a total of V realizations and n D(θ) = 2 log(g((t 1i, t 2i θ)), where g(.) is pdf and to our model, for an observed data we have that g(t 1i, t 2i ) = f 1 (t 1i )f 2 (t 2i ) i, for i = 1,..., n, with i = 1+α(1 2S 1 (t 1i ) 2S 2 (t 2i )+4S 1 (t 1i )S 2 (t 2i )), f j (.) and S j (.), j = 1, 2, are Weibull density function and Weibull survival function, respectively. The EAIC, EBIC and DIC criteria can be calculated using the MCMC output by means of ÊAIC = Dbar + 2q, ÊBIC = Dbar + q log(n) and DIC = Dbar + ρ D = 2Dbar Dhat, respectively, where q is the number of parameters in the model and ρ D is the effective number of parameters, defined as E{D(θ} D(E(θ)), where D(E(θ)) is the deviance evaluated at the expected values of the posterior distributions, which can be estimated as Dhat = D ( 1 V V φ (v), 1 V v=1 V α (v) 1, 1 V v=1 v=1 V α (v) 2, 1 V v=1 V β (v) 1, 1 V v=1 V v=1 β 2 (v) ).

8 The FGM Bivariate Lifetime Copula Model 7 Comparing alternative models, the preferred model is the one with the smallest criteria values. Another criteria which is one of the most used in applied works is derived from the conditional predictive ordinate (CP O) statistics. For a detailed discussion on the CP O statistics and its applications to model selection, see Gelfand et al. (1992). Let D the full data and D ( i) denote the data with the ith observation deleted. We denote the posterior density of θ given D ( i) by π(θ D ( i) ), for i = 1,..., n. For the ith observation, the CP O i can be written as { } 1 CP O i = g((t 1i, t 2i ) θ)π(θ D ( i) π(θ D ( i) ) ) = g((t 1i, t 2i ) θ) dθ, i = 1,..., n. Θ For the proposed model a closed form of the CP O i is not available. However, a Monte Carlos estimate of CP O i can be obtained by using a single MCMC sample from the posterior distribution π(θ D). Let θ 1, θ 2,..., θ Q be a sample of size Q of π(θ D) after the burn-in. A Monte Carlo approximation of CP O i (Chen et al., 2000) is given by 1 ĈP O i = q q=1 A summary statistics of the CP O i s is B = n the better is the fit of the model. 4 Bayesian case influence diagnostics Θ Q 1 g((t 1i, t 2i ) θ q ) 1. log(ĉp O). The larger is the value of B, The best known pertubation schemes are based on case deletion (Cook and Weisberg, 1982) in which the effects are studied of completely removing cases from the analysis. This reasoning will form the basis for our Bayesian global influence methodology and in doing so it will be possible to determine which subjects might be influential for the analysis. In order to investigate if some of the observations are influential for the analysis, we considered a Bayesian case influence diagnostic procedure based on the KullbackLeibler (K-L) divergence between P and P ( i), denoted by K(P, P ( i) ), where P denotes the posterior distribution of Θ for the full data, and P ( i) denotes the posterior distribution of Θ dropping the i-th observation. Specifically, [ ] π(θ D) K(P, P ( i) ) = π(θ D) log π(θ D ( i) dθ. (4.1) ) The K(P, P ( i) ) measures the effect of deleting of i-th observation from the full data on the joint posterior distribution of Θ. As pointed by Peng and Dey (1995), calibration of K(P, P ( i) ) can be done by solving for p i the equation K(P, P ( i) ) = K[B(0.5), B(p i )] =

9 8 Louzada et al. log[4p i (1 p i )]/2 where B(p) denotes the Bernoulli distribution with success probability p. This implies that describing outcomes using π(θ D) instead of π(θ D ( i) ) is compatible with describing an unobserved event as having probability p i when correct probability is 0.5. After some algebraic manipulation it can be shown that p i = 1 2 {1+ 1 exp[ 2K(P, P ( i) )]}. This equation implies that 0.5 p i 1. That is, if p i 0.5 then the ith case is considered influential. For our case it can be shown that (4.1) can be expressed as a posterior expectation K(P, P ( i) ) = log E θ D {[g((t 1i, t 2i ) θ)] 1 } + E θ D {log[g((t 1i, t 2i ) θ)]} (4.2) = log(cp O i ) + E θ D {log[g((t 1i, t 2i ) θ)]}, where E θ D (.) denotes the expectation with respect to the joint posterior π(θ D). Thus (4.2) can be computed by sampling from the posterior distribution of θ via MCMC methods. Let θ 1, θ 2,..., θ Q be a sample of size Q of π(θ D). Then, a Monte Carlo estimate of K(P, P ( i) ) is given by K(P, P ( i) ) = log(ĉp O i ) + 1 Q Q log[g((t 1i, t 2i ) θ q )]. (4.3) q=1 5 Applications In this section, results from simulation studies and a real example are presented in order to illustrate the performance of the proposed methodology. 5.1 Frequentist properties Initially we employ simulated data to study the frequentist properties of the parameter estimates when the structure for the data is known. The goal of this simulation study is to show the good behavior of the Bayesian estimates, based on the frequentist mean squared error (MSE) and the frequentist mean (Mean), and the measurement used for model comparison (EAIC, EBIC, DIC, B). The censoring times C ij were generated from an uniform distribution U(0, τ j ), with τ j, j = 1, 2 controlling the percentage of censoring observation. In the study we considered the survival model given in (2.4), assuming that T j has the Weibull distribution given in (2.6) with parameters α j and λ ij = exp(β 0j + β 1j x i ), j = 1, 2, where the covariates x i were generated from a Bernoulli distribution with parameter 0.5. The bivariate data (T i1,t i2 ), i = 1,..., n, were generated n = 100 data according to the following steps: First we generated T i1 = ( log(1 u i1 )/λ i1 ) 1/α 1 where u i1 U(0, 1). Then T i1 was compared with the censoring value C i1 in order to determine the censoring indicator δ i1 and the observed value given by min(t i1, C i1 ). Next T i2 was generated using a random variable u i2 U(0, 1) and the solution of the nonlinear equation, w i + φ(2u 1i 1)(w 2 i w i) u i2 = 0, considering

10 The FGM Bivariate Lifetime Copula Model 9 T i2 = ( log(1 w i )/λ i2 ) 1/α 2. Then T i2 was compared with the censoring value C i2 in order to obtain the censoring indicator and the observed value given by min(y i2, C i2 ). In our study we considered the following values for parameters of the model β 01 = 1, β 11 = 1, α 1 = 2, β 02 = 1.5, β 12 = 0.5, α 2 = 1.5 and φ = 0.5. We simulated datasets assuming (0%, 0%) and (15%, 15%) of censuring. Therefore two different simulation settings were performed according to the censuring percentage, each on with 100 Monte Carlo generated data sets. For sake of illustration we also compare our FGM bivariate copula model with the well known positive stable frailty (PSF) model (Hougaard, 1986). The joint survival function of the PSF model is given by (Hougaard, 1986), S(t 1, t 2 ) = (( ln(s 1 (t 1 ))) 1 φ + ( ln(s 2 (t 2 ))) 1 φ ) φ. (5.1) When φ 1, we obtain S(t 1, t 2 ) = S(t 1, t 2 ). The Kendall s tau (Nelsen, 2006) is given by τ φ = 1 φ, for φ = 1, τ φ is equals 0. For more details on frailty models where the random effects have a positive stable distribution interested readers can refer to Ibrahim et al. (2001) and Manatunga and Oakes (1999). The following independent priors were considered to perform the Gibbs sampler: β ij N(0, 10 3 ) and α j Gamma(0.01, 0.01) i = 0, 1, j = 1, 2. For parameter of PSF model we assume φ Beta(1, 1) and for FGM copula that 1 2π(φ) Beta(1, 1), such choices guarantee that φ ( 1, 1) and ensure non-informativeness. For each generated data set we simulate two chain of size 50, 000 for each parameter, disregarding the first 10, 000 iterations to eliminate the effect of the initial values and to avoid correlation problems, we consider a spacing of size 20, obtaining a effective sample of size 4, 000 upon which the posterior inference is based on. For each sample the posterior mean of the parameter and the EAIC, EBIC, DIC and B are recorded. The simulations were performed using the Winbugs software (Spiegelhalter et al. 2003). The Bayes estimates were based on posterior samples recorded every 20th iteration from 50, 000 Gibbs samples after a burn-in of 10, 000 samples. The MCMC convergence was monitored according to the methods recommended by Cowles and Carlin (1996) (CODA package). The number of iterations is considered sufficient for the approximate convergence since in all cases the Gelman-Rubin diagnostic is very close to 1 ( 1.02). Table 1 shows the simulation summary statistics for the parameters fitting model based on FGM copula, crossed with the two settings of the censuring. The true values are giving in parenthesis, the MC Mean denotes the arithmetic average of the 100 estimates given ˆθ kj /200, and the MC MSE denotes the empirical mean squared error given by by 100 j=1 100 (ˆθ kj θ k ) 2 /200, where θ k is the respective true value of the estimated parameter ˆθ k. We j=1 can observe that in both cases (with and without censuring) the obtained estimates are close to the true value on average and have a low MSE value.

11 10 Louzada et al. In Table 2 we present the arithmetic average (MC Mean) of the measurements used for model comparison (EAIC, EBIC, DIC and B) for both bivariate survival models induced by the FGM copula and the PSF model. As expected, by considering all comparison criteria the model based on the FGM copula present the better fitting. Also, the FGM copula model was pointed out as the best fitting in more than 95% of all simulations. Table 1 Monte Carlo results based on 100 generated samples crossed with two settings of censuring. The MC mean and the MC MSE denote the mean average and the MSE average of the posterior mean estimates, respectively. % of Censored (0;0) (15;15) Parameter MC Mean MC MSE MC Mean MC MSE β 01(1) β 11(1) r 1(2) β 02(1.5) β 12(0.5) r 2(1.5) φ(0.5) Table 2 Comparison between the two bivariate model based on the FGM and PSF copula models by considering different Bayesian criteria. The MC denotes the arithmetic average of the respective criterion. % of Censored Parameter Criterion MC EAIC MC EBIC MC DIC MC B (0;0) FGM PSF (15;15) FGM PSF Influence of outlying observations One of our main goals in this study is to show the need for robust models to deal with the presence of outliers in the data. In order to do so, we generated a sample of length 100 with fixed parameters β 01 = 1, β 11 = 1, α 1 = 2, β 02 = 1.5, β 12 = 0.5, α 2 = 1.5 and φ = 0.5. In this sample, we also considered that 15% of each time was censured. We selected cases 12, 28, 50 (all observed data in both times) and 77 (censured data in both times) for perturbation. The perturbation scheme were structured as following. To create influential observation artificially in the dataset, we choose one, two or three of these selected cases. For each case we perturbed one or both lifetimes as follows t i = t i + 4S t, i = 1, 2, where S t is the standard deviations of the t i s. For case 12 we perturbed only the lifetime t 1 and for case 50, the lifetime t 2, and for both cases 28 and 77, both lifetimes were perturbed. The MCMC computations were made in a similar fashion to those in the last section and further to monitor the convergence of the Gibbs samples we also used the methods recommended by Cowles and Carlin (1996). Table 3 shows that the posterior inferences are

12 The FGM Bivariate Lifetime Copula Model 11 sensitive to the perturbation of the selected case(s). In Table 3, Dataset (a) denotes the original simulated data set with no perturbation, and Datasets (b) to (i) denote datasets with perturbed cases. Table 3 Simulated data. Posterior means and standard deviations (SD) for the FGM bivariate survival model parameters according to different perturbation schemes. Dataset Perturbed case β 01 β 11 α 1 β 02 β 12 α 2 φ Mean Mean Mean Mean Mean Mean Mean [SD] [SD] [SD] [SD] [SD] [SD] [SD] a Original Sample [0.156] [0.230] [0.184] [0.174] [0.225] [0.120] [0.28] b [0.154] [0.234] [0.167] [0.173] [0.224] [0.122] [0.273] c [0.160] [0.230] [0.185] [0.166] [0.228] [0.112] [0.277] d [0.154] [0.219] [0.147] [0.166] [0.223] [0.112] [0.272] e {12,50} [0.151] [0.228] [0.164] [0.166] [0.226] [0.115] [0.287] f {12,77} [0.152] [0.222] [0.139] [0.167] [0.222] [0.111] [0.272] g {12,50,77} [0.149] [0.225] [0.139] [0.165] [0.223] [0.106] [0.279] h [0.156] [0.220] [0.134] [0.171] [0.223] [0.110] [0.283] i {28,77} [0.151] [0.218] [0.132] [0.169] [0.228] [0.105] [0.263] Table 4 displays the fit of different cases of the perturbed data set. We can observe that the original simulated data (dataset a) had the best fit. Table 4 Simulated data. Bayesian criteria for each perturbed version fitting bivariate model based on the FGM bivariate survival model parameters according to different perturbation schemes. Dataset Bayesian criteria EAIC EBIC DIC B a b c d e f g h i Now we consider the sample from the posterior distributions of the parameters of the model based on the FGM bivariate survival model to compute the K-L divergence and calibration of this divergence, as describe in Section 4. The results in Table 5 show, be-

13 12 Louzada et al. fore perturbation (Dataset a), that all the selected cases are not influential with small K(P, P ( i) ) and related calibration close to However, after perturbations (Dataset (b) to (i) ), the K(P, P ( i) ) increases and the corresponding calibrations become larger than 0.5, indicating those cases are influential. Table 5 Case influence diagnostics for the simulated data. Data names Case number K(P, P ( i) ) Calibration a b c d e f g h i In the Figure 1 shows the K(P, P ( i) ) for the FGM bivariate survival model. Clearly we can see that K(P, P ( i) ) performed well to identifying influential case(s), providing larger K(P, P ( i) ) when compared to the other cases. 5.3 Real data In this section we consider a Bayesian analysis of the dataset presented in The diabetic retinopathy study data (1976) that consist of follow up times for 197 diabetic patients under 60 years of age. The main purpose of the study is to assess the efficacy of photocoagulation treatment for proliferative retinopathy. The treatment was randomly assigned to one eye of each patient and the other, was considered as a control. The first component is a vector of lifetimes which represents the time up to visual loss for the treatment eye (T 1 ), while the second component (T 2 ) is the time up to visual loss for the control eye. We consider as covariate the dichotomized patient age (less than 20 years and older than 20 years). The data set was then reanalyzed though our approach by adopting the model in (2.4) with Weibull marginal as in (2.6). The Bayes estimates were based on posterior samples recorded every 20th iteration from 50, 000 Gibbs samples after a burn-in of 10, 000 samples. We choose the following independent prior distributions: β ji N(0, 100), α j Gamma(0.1, 0.01) i = 0, 1 j = 1, 2 and 1 2π(φ) Beta(1, 1), such choices guarantee that φ ( 1, 1) and ensure non-informativeness. To monitor the convergence of the Gibbs samples we uses the methods recommended by Cowles and Carlin (1996).

14 The FGM Bivariate Lifetime Copula Model 13 a b c K L divergence K L divergence K L divergence Index d Index e Index f K L divergence K L divergence K L divergence Index g Index h Index i K L divergence K L divergence K L divergence Index Index Figure 1 Simulation data. Index plot of K(P, P ( i) ) from fitting a bivariate survival model based on FGM copula. Index

15 14 Louzada et al. In Table 6 we report posterior summaries for the parameters under model based FGM bivariate survival copula model. Following Gelman et al. (2006), we have checked the sensitivity of the routine use for gamma prior on the variance components and found that results are fairly robust under different prior. We also checked the sensitivity analysis for the variance components parameters for various choices of prior parameters by changing only on parameter at a time and keeping all other parameters constant to their default values. The posterior summaries of the parameters do not present remarkable difference and not impair the results in Table 6. We compute the K-L divergences and related calibrations for the diabetic retinopathy study data, but we do not find highly influential cases. The K(P, P ( i) ) are smaller that and the corresponding calibrations are smaller than 1, as it is shown in Figure 2. Table 6 Diabetic retinopathy study data. Summary results from the posterior distribution, mean and standard deviation (SD) for parameter under model based on FGM copula. Parameter Mean SD HPD (90%) Time 1 r (0.641; 0.971) β (-4.788; ) β (-0.954; ) Time 2 r (0.723; 0.962) β (-4.215; ) β (0.038; 0.691) Copula φ (0.460; 0.984) Based on Kendall s tau we can observe which exists a weak dependence between the marginal quantities. The Kendall s tau equals to Table 7 presents the comparison model criteria as discussed in Section 3.3 for comparing the FGM bivariate survival copula model with Weibull marginal distributions with its particular independence case. The FGM model fitting is better then the model fitting which considers independence. We also fitted the FGM bivariate survival copula model with exponential marginal distributions. The EAIC, EBIC, DIC and B criteria are equal to , , 1669 and , respectively, which is evidence in favor of the FGM bivariate survival copula model with Weibull marginal distributions. Besides, we fitted the PSF model for the diabetic retinopathy study data. The EAIC, EBIC, DIC and B criteria are equal to , , 1665 and , respectively, indicating that the FGM bivariate survival copula model can be seen as a competitor to the PSF model, which is a well known model commonly used in literature for fitting bivariate lifetime data. Table 7 Diabetic retinopathy study data. Bayesian criteria. Copula Structure EAIC EBIC DIC B FGM Independence Figure 3 shows the Kaplan-Meier curves for the variables T 1 (left panel) and T 2 (center panel), dichotomized by the patient age, up to 60 months of follow-up, along with the fit

16 The FGM Bivariate Lifetime Copula Model 15 K L divergence Index Figure 2 Diabetic Retinopathy study data. Index plot of K(P, P ( i) ) from fitting the bivariate survival model based on FGM copula. of the FGM bivariate survival copula model to diabetic retinopathy study data. Also, the scatter plot between the variables was displayed (right panel). Survival >= 20 years < 20 years Survival >= 20 years < 20 years Time 2 Control observed both censured only t1 censured only t2 censured Time 1 Treatment Time 2 Control Time 1 Treatment Figure 3 Kaplan-Meier curves for the variables T 1 (left panel) and T 2 (center panel), up to 60 months of follow-up, along with the fit of the FGM Copula model with Weibull marginals and Scatter plot of T 1 vs T 2 (right panel).

17 16 Louzada et al. 6 Some final remarks In this paper, we presented the FGM bivariate survival copula model. Parameter estimation is based on a Bayesian approach via MCMC. We have used Bayesian case influence diagnostic based on the Kullback-Leibler divergence in order to study the sensitivity of the Bayesian estimates under perturbations in the model/data. Finally, we demonstrate our approach with a simulation study and a real data set. As pointed out one of the referees, in the two-step approach, the marginal parameters are estimated firstly and then, the copula parameters are estimated in a second step. This approach provides consistent, but not efficient estimators using a frequentist approach. However, from the Bayesian perspective, one can estimate all the model parameters simultaneously in the MCMC algorithm such that the assumption of independence in the first step is avoided. We also considered a two-step estimation method, where the marginal parameters are estimated firstly and then, the copula parameter is estimated in a second step. However, in this paper we also considered a two-step estimation method but we omit the results since they are similar to those obtained by considering the joint estimation. Several extensions are possible. Firstly, although we consider Weibull marginal distributions, the model is, at least in principle, very flexible with respect to the choice of the marginal distributions and other marginal distributions, such as the log-logistic and exponentiated-weibull distributions among other lifetime distributions, could be easily considered using a similar framework. Secondly, the implications of such models in a competing risk analyses structure need to be study. Thirdly, a bivariate model with cure rate needs to be proposed. Fourthly, at least in principle, at expense of moderate complexity of implementation, this methodology can be extended to multivariate settings. These extensions shall be reported elsewhere. Acknowledgements The research was partially supported by the Brazilian Organizations CNPQ and CAPES. References Brooks, S.P. (2002) Discussion on the paper by Spiegelhalter, Best, Carlin, and van der Linde. 64, Carlin, B.P. and Louis, T.A. (2001) Bayes and Empirical Bayes Methods for Data Analysis, 2nd ed. Boca Raton: Chapman & Hall/CRC. Chen, M.H., Shao, Q.M. and Ibrahim, J. (2000) Monte Carlo Methods in Bayesian Computation. New York: Springer-Verlag. Chib, S. and Greenberg, E. (1995) Understanding the metropolis-hastings algorithm. The American Statistician, 49, Clayton, D.G. (1978) A model for association in bivariate life-tables and its application in epidemiological studies of familial tendency in chronic disease incidence. Biometrika, 65, Conway, D.A. (1979) Farlie-Gumbel-Morgenstern distributions. In: Encyclopedia of Statistical Sciences 3. S.Kotz and N.L. Johnson, editors (John Wiley & Sons, New York),

18 The FGM Bivariate Lifetime Copula Model 17 Cook, R.D. and Weisberg, S. (1982) Residuals and Influence in Regression. Boca Raton: Chapman & Hall/CRC. Cowless, M.K. and Carlim, B.P. (1996) Markov chain Monte Carlo convergence diagnostics: a comparative review. Journal of the American Statistical Association, 91, Embrechts, P., Linskog, F. e McNiel, A. (2003) Modelling dependence with copulas and applications to risks management. Gamerman, D. and Lopes, H. F. (2006) Markov Chain Monte Carlo: Stochastic Simulation for Bayesian Inference, 2nd. Boca Raton: Chapman and Hall. Gelfand, A.E., Dey, D.K. and Chang, H. (1992) Model determination using predictive distributions with implementation via sampling-based methods. Bayesian Statistics, 4. Pescola, Oxford Univ. Press, New York, Gelfand, A.E. and Smith, A.F.M. (1990) J. Amer. Statist. Assoc., 85(410), Gelman, A., Carlin, J. and Rubin, D. (2006) Bayesian Data Analysis. New York: Chapman & Hall/CRC. Gilks, W.R., Richardson, S. and Spiegelhater, D.J. (1996) Markov Chain Monte Carlo in Practice. London: Chapman & Hall. Gustafson, P., Aeschliman, D. and Levy, A.R. (2003) Lifetime Data Analysis, 9, Hougaard, P. (1986) A class of multivariate failure time distributions. Biometrika, 73, Hougaard, P. (1989) Fitting a multivariate failure time distribution. IEEE Transactions on Reliability, 38, Ibrahim J. G., Chen M-H., Sinha D. (2001) A Bayesian survival analysis. New York: Springer Verlag. Lawless, J. F.(2003) Statistical Models and Methods for lifetime data. New York: Wiley. Manatunga, A.K. and Oakes, D. (1999) Parametric Analysis of Matched Pair Survival Data. Lifetime Data Analysis, 5, Nelsen, R. (2006) An Introduction to Copulas, 2nd edn. New York: Springer. Romeo, J.S., Tanaka, N. I. and Pedroso de Lima, A.C. (2006) Bivariate survival modeling: a Bayesian approach based on Copulas. Lifetime Data Analysis, 12, Oakes, D. (1989) Bivariate survival models induced by frailties. Journal of the American Statistical Association, 84, Peng, F. and Dey, D. (1995) Bayesian analysis of outlier problems using divergence measures. The Canadian Journal of Statistics. La Revue Canadienne de Statistique 23 (2), Sahu, S.K. and Dey, D.K. (2000) A comparison of frailty and other models for bivariate survival data. Lifetime Data Analysis, 6, Shih, J.H. and Louis, T.A. (1995) Inferences on the association parameter in copula models for bivariate survival data. Biometrics, 51, Spiegelhalter, D. J., Best, N. G., Carlin, B. P., and van der Linde, A. (2002) Bayesian measures of model complexity and fit. Journal of the Royal Statistical Society B, 644, Spiegelhalter, D.J., Thomas, A. and Best, N.G. (2003) WinBugs Computer Program. Imperial College & MRC Biostatistics Unit, IPH. Cambridge, UK. in The Diabetic Retinopathy Study Research Group (1976) Preliminary report on the efeect of photocoagulation therapy. American Journal of Ophthalmology, 81, Vaupel, J.W., Manton, K.G. and Stallard, E. (1979) The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography, 16, Francisco Louzada Adriano K. Suzuki DEs, Universidade Federal de São Carlos, SP, Brazil dfln@ufscar.br; adrianokamimura@gmail.com Vicente G. Cancho DMAE - ICMC - Universidade de São Paulo, São Carlos - SP, Brazil garibay@icmc.usp.br

19 18 Louzada et al. Gladys D.C. Barriga FEB - Universidade Estadual Paulista, Bauru - SP, Brazil gladyscacsire@yahoo.com.br

20 PUBLICAÇÕES RODRIGUES, J.; CASTRO, M.; BLAKRISHNAN, N.; GARIBAY, V.; Destructive weighted Poisson cure rate models Novembro/2009 Nº 210. COBRE, J.; LOUZADA-NETO, F., PERDONÁ, G.; A Bayesian Analysis for the Generalized Negative Binomial Weibull Cure Fraction Survival Model: Estimating the Lymph Nodes Metastasis Rates Janeiro/2010 Nº 211. DINIZ, C. A. R. ; LOUZADA-NETO, F.; MORITA, L. H. M.; The Multiplicative Heteroscedastic Von Bertalan_y Model Fevereiro/2010 Nº 212. DINIZ, C. A. R. ; MORITA, L. H. M, LOUZADA-NETO, F.; Heteroscedastic Von Bertalanffy Growth Model and an Application to a Kubbard female chicken corporeal weight growth data Fevereiro/2010 Nº 213. FURLAN, C. P. R.; DINIZ, C. A. R.; FRANCO, M. A. P.; Estimation of Lag Length in Distributed Lag Models: A Comparative Study Março/2010 Nº 214. RODRIGUES, J., CANCHO, V. G., CASTRO, M., BALAKRISHNAN, N., A Bayesian destructive weighted Poisson cure rate model and an application to a cutaneous melanoma data, Março/2010 Nº CORDEIRO, G. M.; RODRIGUES, J.; CASTRO, M. The exponential COM-Poisson distribution, Abril/2010 Nº 216. CANCHO, V. G.; LOUZADA-NETO, F.; BARRIGA, G. D. C. The Poisson-Exponential Survival Distribution For Lifetime Data, Maio/2010 Nº 217. DIINIZ, C. A. R.; FURLAN, C. P. R.; LEITE, J. G. A Bayesian Estimation of Lag Length in Distributed Lag Models, Julho/2010 Nº 218. CONCEIÇÃO, K. S.; PIRES, R. M.; LOUZADA-NETO, F.; ANDRADE, M. G.; DINIZ, C. A. R. A Generalized Species-Area Relationship for Estimating Species Diversity: The Poisson Distribution Case Julho/2010 Nº 219. LOUZADA-NETO, F.; CANCHO, V. G.; BARRIGA, G. D. C.; A Bayesian Analysis For The Poisson-Exponential Distribution Agosto/2010 Nº 220. SCACABAROZI, F.N., DINIZ C. A. R., FRANCO M. A. P., A Comparative Study of Credibility an Confidence Intervals for the Parameter of a Poisson Distribuition Setembro/2010 Nº 221. TOMAZELLA, V. L., BERNARDO, J. M.; Testing for Hardy-Weinberg Equilibrium in a Biological Population: An Objective Bayesian Analysis Setembro/2010 Nº 222. LOUZADA-NETO F., DINIZ C. A. R., COSTA, C. C., SILVA P. H. F., DESTEFANI, C. R., TEMPONI A. P. O. Procedimentos Estatísticos para Segmentação de Base de Dados Setembro/2010 Nº 223. ARA-SOUZA, A. L.; LOUZADA-NETO, F.; Caracterização dos Docentes e Necessidade de Doutores dentro das Graduações de Estatística do Brasil Setembro/2010 Nº 224. LOUZADA-NEDTO, F.; ROMAN, M.; CANCHO, V. G.; The Complementary Exponential Geometric Distribution: Model, Properties and a Comparison With Its Counterpart Setembro/2010 Nº 225. LOUZADA-NETO,F.; BORGES, P.; The Exponential Negative Binomial Distribuition Setembro/2010 Nº 226. PEREIRA, G.A; LOUZADA-NETO, F.; MORAES-SOUZA, H. ; FERREIRA-SILVA, M. M.; BARBOSA, V. F.; General Bayesian latent class model for the evaluation of the performance of L diagnostic tests for Chagas disease in the absence of a gold standard considering M covariates and V different disease prevalences Setembro/2010 Nº 227. SARAIVA, E. F.; MILAN, L. A.; LOUZADA-NETO, F.; A Posterior Split-Merge MCMC Algorithm for Mixture Models with an unknown number of components Setembro/2010 Nº 228. BORGES, P.; ROMAN, M.; TOJEIRO, C. A. V.; LOUZADA-NETO, F.; The complementary exponential logarithmic distribution: A two-parameter lifetime distribution with increasing failure rate Outubro/2010 Nº 229. TOJEIRO, C.; A. V. LOUZADA-NETO, F.; A General Threshold Stress Hybrid Hazard Model for Lifetime Data Outubro/2010 Nº 230. TOMAZELLA, V. L.D.; CANCHO, V. G.; LOUZADA-NETO, F.; Objective Bayesian Reference Analysis for the Poissonexponential lifetime distribution Outubro/2010 Nº 231. DELGADO, J. F.; BORGES, P.; CANCHO, V.G.; LOUZADA-NETO, F.; A compound class of exponential and power series distributions with increasing failure rate Novembro/2010 Nº 232. DELGADO. J. J. F., CANCHO V. G., LOUZADA-NETO F.; The power series cure rate model: an application to a cutaneous melanoma data Novembro/2010 Nº 233 Os recentes relatórios poderão ser obtidos pelo endereço Mais informações sobre publicações anteriores ao ano de 2010 podem ser obtidas via dfln@ufscar.br