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 POISSON-EXPONENTIAL SURVIVAL DISTRIBUTION FOR LIFETIME DATA Vicente G. Cancho Francisco Louzada-Neto Gladys D.C. Barriga RELATÓRIO TÉCNICO DEPARTAMENTO DE ESTATÍSTICA TEORIA E MÉTODO SÉRIE A Maio/2010 nº 217

2 The Poisson-Exponential Survival Distribution For Lifetime Data Vicente G. Cancho a, Francisco Louzada-Neto b and Gladys D.C. Barriga b a Universidade de São Paulo b Universidade de Federal de São Carlos Abstract In this paper we proposed a new two-parameters lifetime distribution with increasing failure rate. The new distribution arises on a latent complementary risk problem base. The properties of the proposed distribution are discussed, including a formal prove of its probability density function and explicit algebraic formulaes for its reliability and failure rate functions, quantiles and moments, including the mean and variance. A simple EM-type algorithm for iteratively computing maximum likelihood estimates is presented. The Fisher information matrix is derived analytically in order to obtaining the asymptotic covariance matrix. The methodology is illustrated on a real data set. Keywords: Complementary Risks, EM Algorithm, Exponential Distribution, Poisson Distribution, Survival Analysis. 1 Introduction The exponential distribution (ED) provides a simple, elegant and close form solution to many problems in lifetime testing and reliability studies. However, the ED does not provide a reasonable parametric fit for some practical applications where the underlying hazard rates are nonconstant, presenting monotone shapes. In recent years, in order to overcame such problem, new classes of models were introduced based on modifications of the ED. Gupta & Kundu (1999) proposed a generalized ED. This extended family can accommodate data with increasing and decreasing failure rate function. Kus (2007) proposed another modification of the ED with decreasing failure rate function. While Barreto-Souza & Cribari-Neto (2009) generalizes the distribution proposed by Kus (2007) by including a power parameter in his distribution. In this paper, we propose a new distribution family based on the ED with increasing failure rate function. Its genesis is stated on a complementary risk problem base (Basu & Klein, 1982) in presence of latent risks, in the sense that there is no information about which factor was 1

3 responsible for the component failure and only the maximum lifetime value among all risks is observed. The paper is organized as follows. In Section 2, we introduce the new distribution and present its properties. Section 3 outlines an EM-type algorithm for maximum likelihood estimation. In Section 4 the methodology is illustrated in a real data set. Some final comments are presented in Section 5. 2 The Poisson-Exponential Distribution Let Y be a nonnegative random variable denoting the lifetime of an component in some population. The random variable Y is said to have a Poisson-Exponential distribution (PED) with parameters λ > 0 and θ > 0 if its probability density function is given by f(y) = θλe λy θe λy 1 e θ, y > 0. (1) The parameter λ controls the scale of the distribution while the parameter θ controls its shape. As θ approaches zero, the PED converges to an ED with parameter λ. Figure 1 (top panel) shows the probability density function (1) for θ = 0.1, 1, 2, 4, 8, which is decreasing if 0 < θ < 1 and unimodal for θ 1. The modal value λe 1 is obtained at y = log θ/λ. Density θ=0.1 θ=1.0 θ=2.0 θ=4.0 θ=8.0 Hazard function θ=0.1 θ=1.0 θ=2.0 θ=4.0 θ= y y Figure 1: Top panel: probability density function of the PE distribution. Bottom panel: failure rate function of the PED. We fixed λ = 1. The reliability function of a PED random varible is given by S(y) = 1 e θe λy 1 e θ, y > 0. (2) 2

4 From (1) and (2) it is easy verify that the failure rate function is given by h(y) = θλ exp( λy θe λy) 1 exp( θe λy, y > 0. (3) ) The failure rate function (3) is increasing. Figure 1 (bottom panel) shows some failure rate function shapes for same fixed values of θ. The initial and long-term failure rate function values are both finite and are given by h(0) = λθ(e θ 1) and h( ) = λ. We simulate the PED considering Q(u) = F 1 (u) = log ( ) θ log(u(e θ 1)) λ 1, where u has the uniform U(0, 1) distribution and F(y) = 1 S(y) is distribution function of Y. 2.1 Genesis Complementary risks problems (Basu & Klein, 1982) arise in several areas, such as public health, actuarial science, biomedical studies, demography and industrial reliability. In the classical Complementary risks scenarios the lifetime associated with a particular risk is not observable, rather we observe only the maximum lifetime value among all risks. Simplistically, in reliability, we observe only the maximum component lifetime of a parallel system. That is, the observable quantities for each component are the maximum lifetime value to failure among all risks, and the cause of failure. Full statistical procedures and extensive literature are available to deal with Complementary risks problems and interested readers can refer to Lawless (2003), Crowder et al. (1991) and Cox & Oakes (1984). A difficulty arises if the risks are latent in the sense that there is no information about which factor was responsible for the component failure, which can be often observed in field data. We call these latent Complementary risks data. On many occasions this information is not available or it is impossible that the true cause of failure is specified by an expert. In reliability, the components can be totally destroyed in the experiment. Further, the true cause of failure can be masked from our view. In modular systems, the need to keep a system running means that a module that contains many components can be replaced without the identification of the exact failing component. Goetghebeur & Ryan (1995) addressed the problem of assessing covariate effects based on a semi-parametric proportional hazards structure for each failure type when the failure type is unknown for some individuals. Reiser et al. (1995) considered statistical procedures for analysing masked data, but their procedure can not be applied when all observations have an unknown cause of failure. Lu & Tsiatis (2001) presents a multiple imputation method for estimating regression coefficients for risks modeling with missing cause of failure. A comparison of two partial likelihood approaches for risks modeling with missing cause of failure is presented in Lu & Tsiatis (2005). 3

5 Our model can be derived as follows. Let M be a random variable denoting the number of Complementary risks related to the occurrence of an event of interest. Further, assume that M has a zero truncated Poisson distribution with probability mass function given by P(M = m) = e θ θ m m!(1 e θ, m = 1, 2..., θ > 0. (4) ) Let T j (j = 1, 2,...) denote the time-to-event due to the j-th Complementary risks, hereafter lifetime. Given M = m, the random variables T j, j = 1,, m are assumed to be independent and identically distributed according to a ED with a common distribution function with probability density function given by f(t; λ) = λ exp( λt), t > 0, λ > 0. (5) In the latent Complementary risks scenario, the number of causes M and the lifetime T j associated with a particular cause are not observable (latent variables), but only the maximum lifetime Y among all causes is usually observed. So, the component lifetime is defined as Y = max (T 1, T 2...,T M ). (6) The following result shows that the random variable Y have probability density function given by (1). Proposition 2.1 If the random variable Y is defined as in (6), then, considering (4) and (5), Y is distributed according to a PED with probability density function given by (1). Proof 2.1 The conditional density function of (6) given M = m is given by f(y M = m, λ) = mλ[1 e λt ] m 1 e λt, t > 0, m = 1,... Them, the marginal probability density function of Y is given by f(y) = mλ[1 e λt ] m 1 e λt θm e θ m!(1 e θ ) m=1 = θλe θ λt [θ(1 e λt )] m 1 1 e θ (m 1)! This completes the proof. m=1 = θλe λy θe λy 1 e θ. The PED parameters have a direct interpretation in terms of Complementary risks. The θ represents the mean of the number of Complementary risks, while λ denotes the failure rate. In comparison with the Kus (2007) formulation we follows an opposite way, since he defines the component lifetime as Y = min(t 1, T 2...,T M ), while we are considering Y = max (T 1, T 2..., T M as stated in (6). 4

6 2.2 Moments Some of the most important features and characteristics of a distribution can be studied through its moments, such mean, variance, tending, dispersion skewness and kurtosis. A general expression for rth ordinary moment µ r = E(Y r ) of the PED can be obtained analytically considering the generalized hypergeometric function denoted by F p,q (a, b, θ) and defined as F p,q (a, b, θ) = j=0 θ j p Γ(a i + j)γ(a i ) 1 Γ(j + 1) q Γ(b i + j)γ(b i ) 1, (7) where a = [a 1,...,a p ], p is the number of operands of a, b = [b 1,...,b q ], and q is the number of operands of b. We then formulate the following result. Proposition 2.2 Suppose Y denotes a random variable from a PED with parameters λ > 0 and θ > 0 and probability density function given by (1), then µ r = θγ(r + 1) λ r (1 e θ ) F r+1,r+1([1,...,1], [2,...,2], θ), (8) where F p,q (a, b, θ) is the generalized hypergeometric function. The proof of 2.2 is obtained by direct integration and it is then omitted. Considering (8), the mean and variance of the of the PED are given, respectively, by 3 Inference θ E(Y ) = λ(1 e θ ) F 2,2([1, 1], [2, 2], θ), [ θ V ar(y ) = λ 2 (1 e θ F 3,3 ([1, 1, 1], [2, 2, 2], θ) ) ] θ (1 e θ ) F 2,2([1, 2 1], [2, 2], θ). 3.1 Estimation by maximum likelihood The log-likelihood function based on an observed sample of size n, y = (y 1,...,y n ), from a PED is given by l(θ) = n log(θλ) λ y i θ e λy i n log(1 e θ ). (9) The MLEs of θ = (θ, λ) can be directly obtained by maximizing the log-likelihood function 5

7 (9) or, alternatively, by finding the solution for the following two nonlinear equations, l(θ) θ l(θ) λ = n θ n e θ 1 e λy i = 0, (10) = n λ y i + θ y i e λy i = 0. (11) The equation (11) can be solved exactly for θ, i.e., ˆθ = n y i n λ 1 n y ie ˆλy i, (12) conditional upon the value of ˆλ, where ˆθ and ˆλ are the MLEs for θ and λ, respectively. Therefore, it is sufficient to solve the equation (10) iteratively in order to find the MLE for λ. 3.2 An EM-algorithm In this subsection we develop an EM-algorithm (Dempster et al., 1977) for obtain the MLEs of the PED parameters. Let y = (y 1,...,y n ) and m = (m 1,...,m n ). It follows that the complete log-likelihood function associated with (y,m) is given by l c (θ y,m) = log(f(y i, m i θ)), (13) where f(y i, m i θ) = m i θ m i λ[1 e λy i ] m i 1 e θ λy i /Γ(m i + 1)(1 e θ ), m i = 1, 2,..., y i > 0. Considering m i = E[M i θ = θ, y i ] we can direct to the implementation of the M-step, which consists in maximizing the expected complete data function or the Q function over θ, given by Q(θ θ (k) ) = E[l c (θ) y, θ (k) ] = c + n log λ + ( m i 1)log(1 e λy i ) λ y i + m i log θ n log(e θ 1), where c is a constant that is independent of θ, m i = θ(1 e λy i ) and θ (k) is an updated value of θ. Then, the E-step and the M-step of the algorithm are given by: E-step: Given a current estimate θ (k), compute m (k) i 6

8 M-step: Update θ (k) by maximizing Q(θ θ (k) ) over θ, which leads to the following nice expressions θ (k+1) = λ (k+1) = 3.3 Interval Estimation m (k) i (1 e θ (k+1) ), n n ( y i n [ ( m (k) i 1)y i e λ (k+1) y i 1 e λ (k+1) y i Large sample inference for the parameters can be based, in principle, on the MLEs and their estimated standard errors. Following (Cox & Hinkley, 1974), under suitable regularity conditions, it can be shown that ) ( n ( θ θ N 2 0,I 1 (θ) ), ]). where I(θ) is Fisher information matrix, i.e, ( ) I(θ) = E 2 l(θ) θ 2 ) E ( 2 l(θ) θ λ E E ( 2 l(θ) θ λ ) ( ) 2 l(θ) λ 2, where the second derivatives are given by 2 l(θ)/ θ 2 = n/θ 2 + ne θ /(e θ 1) 2, 2 l(θ)/ θ λ = n y ie λy i and 2 l(θ)/ λ 2 = nλ 2 θ n y2 i e λy i. Them, the elements of the Fisher Information matrix are derived as, ( 2 ) l(θ) nθ neθ E θ 2 = 2 (e θ 1) 2, ( 2 ) l(θ) nθ E = θ λ 4λ(1 e θ ) F 2,2([2, 2], [3, 3], θ), ( 2 ) l(θ) nθ 2 E λ 2 = 4λ 2 (1 e θ ) F 3,3([2, 2, 2], [3, 3, 3], θ) + n λ 2. 4 Application In this section we reanalyze the data set extracted from (Lawless, 2003). The lifetimes are the number of million revolutions before failure for each one of the 23 ball bearings on an endurance test of deep groove ball bearings. Firstly, in order to identify the shape of a lifetime data failure rate function we shall consider a graphical method based on the TTT plot (Aarset, 1987). In its empirical version the TTT plot is given by G(r/n) = [( r Y i:n) (n r)y r:n ]/( r Y i:n), 7

9 where r = 1,...,n and Y i:n represent the order statistics of the sample. It has been shown that the failure rate function is increasing (decreasing) if the TTT plot is concave (convex). Although, the TTT plot is only a sufficient condition, not a necessary one for indicating the failure rate function shape, it is used here as a crude indicative of its shape. Figure 2 (top panel) shows the TTT plot for the considered data, which is concave indicating an increasing failure rate function, which can be properly accommodated by a PED. G(r n) Survival Function K M Poisson Exponential Exponential r/n Time Figure 2: Top panel: Empirical scaled TTT-Transform for the data. Bottom panel: Kaplan- Meier curve with estimated RFs of the PED and ED. Then, the PED was fitted to the data. In order to obtain the MLEs of the parameters we used the EM-algorithm, considering as initial guess θ = 1 and λ = 1/ȳ (a moment estimator for λ from an independent identically distributed exponential observation). Of course, this choice is not foolproof and it is advisable to run the method several times from different starting values. Table 1 presents the MLEs and the corresponding 95% confidence intervals of the PED parameters, which were based on the Fisher information matrix at the MLEs, given by, [ ] I( θ) = For comparison of nested models, which is the case when comparing the PED with the ED, we can compute the maximum values of the unrestricted and restricted log-likelihoods to obtain the LRS. For testing H 0 : θ = 0 versus H 1 : θ > 0 we consider the LRS, w n = 2(l PE l E ), where l ED and l PED are the log-likelihoods for the model under the restricted hipothesis H 0 and under the unrestricted hipothesis H 1 under a sample of size n. Taking into account that the test is 8

10 Table 1: MLEs and their corresponding 95% confidence intervals. Parameter Dsitribution θ λ l( ) PED ( ; ) (0.0241; ) ED ( ; ) performed in the boundary of the parameter space, following Maller & Zhou (1995), the LRS, w n, is assumed to be asymptotically distributed as a symmetric mixture of a chi-squared distribution with one degree of freedom and a point-mass at zero. Then, lim n P(w n c) = 1/2 + 1/2 P(χ 2 1 c), where P(χ2 1 c) denotes a random variable with a chi-square distribution with one degree of freedom. Large positive values of w n give favourable evidence to the full model. The last column of Table 1 presents the log-likelihood values for both models. Thus, w n is equals to , with a p-value < , which is a strong evidence in favour of the PED. Also, we compare the PED and ED by inspection of the Akaike s information criterion (AIC), 2l( θ) + 2q, and Schawarz s Bayesian information criterion (BIC), 2l( θ) + q log(n), where q is the number of parameters in the model and n is size sample. The preferred model is the one with the smaller value on each criterion. The PED overcome the corresponding ED in the two considered criterions. The estimated statistics AIC and BIC for the PED are equal to and , respectively. While, the estimated statistics AIC and BIC for the ED are equal to and , respectively. These results are corroborated by the plot in the bottom panel of Figure 2, which compares the estimated rate function obtained via ED and PED fitting on the empirical Kaplan-Meier rate function base. 5 Concluding remarks In this paper we proposed the PED, as a possible extension to the well known ED. We provide a mathematical treatment of this distribution including a formal prove of its probability density function and explicit algebraic formulae for its rate function and failure rate function, quantiles and moments (particularly, for the mean and variance). The parameters of the PED has a direct interpretation in terms of Complementary risks. We discussed MLE, providing an EM-algorithm. The PED allows a straightforwardly nested hypothesis testing procedure for comparison with its ED particular case. The practical relevance and applicability of the PED were demonstrated in a real data set. 9

11 Acknowledgments: The research of Francisco Louzada-Neto is funded by the Brazilian organization CNPq. References Aarset, M. V. (1987). How to identify a bathtub hazard rate. IEEE Transactions on Reliability, 2(1), Barreto-Souza, W. & Cribari-Neto, F. (2009). A generalization of the exponential-poisson distribution. Statistics and Probability Letters, 79, Basu, A. & Klein, J. (1982). Some recent development in competing risks theory. In Survival Analysis, Edited by Crowley, J. and Johnson, R.A., Hayward: IMS, 1(1), Cox, D. & Oakes, D. (1984). Analysis of Survival Data. Chapman and Hall, London. Cox, D. R. & Hinkley, D. V. (1974). Theoretical statistics. Chapman and Hall, London. Crowder, M., Kimber, A., Smith, R. & Sweeting, T. (1991). Statistical Analysis of Reliability Data. Chapman and Hall, London. Dempster, A., Laird, N. & Rubin, D. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the Royal Statistical Society, Series B,, 39, Goetghebeur, E. & Ryan, L. (1995). A modified log rank test for competing risks with missing failure type. Biometrika, 77(2), Gupta, R. & Kundu, D. (1999). Generalized exponential distributions. Australian and New Zealand Journal of Statistics, 41(2), Kus, C. (2007). A new lifetime distribution distributions. Computational Statistics & Data analysis, 11, Lawless, J. F. (2003). Statistical Models and Methods for Lifetime Data. Wiley, New York, NY, second edition. Lu, K. & Tsiatis, A. A. (2001). Multiple imputation methods for estimating regression coefficients in the competing risks model with missing cause of failure. Biometrics, 57(4), Lu, K. & Tsiatis, A. A. (2005). Comparison between two partial likelihood approaches for the competing risks model with missing cause of failure. Lifetime Data Analysis, 11(1),

12 Maller, R. A. & Zhou, S. (1995). Testing for the presence of immune or cured individuals in censored survival data. Biometrics, 51(4), Reiser, B., Guttman, I., Lin, D., Guess, M. & Usher, J. (1995). Bayesian inference for masked system lifetime data. Applied Statistics, 44(1),

13 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 Mais informações sobre publicações anteriores ao ano de 2009 podem ser obtidas via dfln@ufscar.br