A new extended Birnbaum–Saunders regression model for lifetime modeling

Abstract

A new class of extended Birnbaum–Saunders regression models is introduced. It can be applied to censored data and be used more effectively in survival analysis and fatigue life studies. Maximum likelihood estimation of the model parameters with censored data as well as influence diagnostics for the new regression model are investigated. The normal curvatures for studying local influence are derived under various perturbation schemes and a martingale-type residual is considered to assess departures from the extended Birnbaum–Saunders error assumption as well as to detect outlying observations. Further, a test of homogeneity of the shape parameters of the new regression model is proposed. Two real data sets are analyzed for illustrative purposes.

Introduction

The two-parameter Birnbaum–Saunders (BS) distribution, also known as the fatigue life distribution, was introduced by Birnbaum and Saunders (1969) and has received considerable attention in recent years. It was originally derived from a model for a physical fatigue process where dominant crack growth causes failure. It was later derived by Desmond (1985) using a biological model which followed from relaxing some of the assumptions originally made by Birnbaum and Saunders (1969). The relationship between the BS distribution and the inverse Gaussian distribution was investigated by Desmond (1986) who demonstrated that the BS distribution is an equal-weight mixture of an inverse Gaussian distribution and its complementary reciprocal. For book treatments of inverse Gaussian and BS distributions and their relationships, see Marshall and Olkin (2007, Chapter 13) and especially Saunders (2007, Chapter 10). More recently, Jones (2012) also discussed the relationship between the BS and the inverse Gaussian distributions.

The cumulative distribution function of a random variable $T$ with BS distribution, say $T\sim \mathrm{BS}(\alpha ,\eta )$, is $G\left(t\right)=\Phi \left(v\right)$, with $t>0$, where $\Phi (\cdot )$ is the standard normal cumulative function, $v=v\left(t\right)=\rho (t/\eta )/\alpha$, $\rho \left(z\right)=z^{1/2}-z^{-1/2}$, and $\alpha >0$ and $\eta >0$ are the shape and scale parameters, respectively. The shape of the hazard function of the BS distribution is discussed in Kundu et al. (2008). The authors showed that the hazard function is not monotone and is unimodal for all ranges of the parameter values. Some interesting results on improved statistical inference as well as interval estimation for the BS distribution may be revised in Wu and Wong (2004), Lemonte et al., 2007, Lemonte et al., 2008 and Wang (2012). The BS distribution has been applied in a wide variety of fields. For the applications of the BS distributions, read, for example, Balakrishnan et al. (2007) in reliability and Leiva et al., 2008, Leiva et al., 2009 in other fields. It is worthwhile to mention that there has been a great deal of progress recently in developing statistical methodology for the BS model and its generalizations. Notable contributions include Professor Narayanaswamy Balakrishnan (http://www.math.mcmaster.ca/bala/bala.html) and co-workers, and Professor Victor Leiva (http://staff.deuv.cl/leiva/) and co-workers.

On the basis of the scheme proposed by Marshall and Olkin (1997), Lemonte (2013) introduced a quite flexible distribution which can be used to model failure times for materials subject to fatigue and lifetime data. The new distribution was called by the author as the Marshall–Olkin extended Birnbaum–Saunders (MOEBS) distribution. Hereafter, the random variable $T$ is said to have a MOEBS distribution with shape parameters $\alpha >0$ and $\lambda >0$, and scale parameter $\eta >0$, say $T\sim \mathrm{MOEBS}(\lambda ,\alpha ,\eta )$, if its cumulative function is given by

$$G\left(t\right)=\frac{\Phi \left(v\right)}{1-\bar{\lambda }\Phi (-v)}\text{,}t>0\text{,}$$

where $\bar{\lambda }=1-\lambda$. The survival function is $S\left(t\right)=\lambda \Phi (-v)/[1-\bar{\lambda }\Phi (-v)]$, whereas the probability density function corresponding to (1) takes the form $g\left(t\right)=\lambda \kappa (\alpha ,\eta )t^{-3/2}(t+\eta ){[1-\bar{\lambda }\Phi (-v)]}^{-2}exp\{-\tau (t/\eta )/\left(2{\alpha }^2\right)\}$, where $\kappa (\alpha ,\eta )=exp\left({\alpha }^{-2}\right)/\left(2\alpha \sqrt{2\pi \eta }\right)$ and $\tau \left(z\right)=z+z^{-1}$. It can be shown that if $T\sim \mathrm{MOEBS}(\lambda ,\alpha ,\eta )$, then $kT\sim \mathrm{MOEBS}(\lambda ,\alpha ,k\eta )$, for $k>0$, i.e. the class of MOEBS distributions is closed under scale transformations. The two-parameter BS distribution arises from (1) when $\lambda =1$, that is, $T\sim \mathrm{BS}(\alpha ,\eta )=\mathrm{MOEBS}(1,\alpha ,\eta )$.

Rieck and Nedelman (1991) proposed a log-linear regression model based on the BS distribution. They showed that if $T\sim \mathrm{BS}(\alpha ,\eta )$, then $Y=log\left(T\right)$ is sinh-normal (SN) distributed with shape, location and scale parameters given by $\alpha$, $\mu =log\left(\eta \right)$ and $\sigma =2$, respectively; that is, the log-BS (LBS) distribution is a special case of the SN distribution introduced by them and, in this case, the notation $Y\sim \mathrm{LBS}(\alpha ,\mu )$ is considered. The SN distribution is symmetrical, presents greater and smaller degrees of kurtosis than the normal model and also has bi-modality. Their regression model has received significant attention over the last few years by many researchers. For some recent references about the BS regression model, the reader is refereed to Desmond et al. (2008), Xiao et al. (2010), Lemonte et al. (2010), Lemonte (2011), Lemonte and Ferrari, 2011a, Lemonte and Ferrari, 2011b, Lemonte and Ferrari, 2011c, Qu and Xie (2011) and Li et al. (2012), among others.

Some generalizations of the log-linear BS regression model have been proposed in the statistical literature. For example, some efforts can be found in the works by Barros et al. (2008), Lemonte and Cordeiro (2009), Santana et al. (2011), Lemonte (2012), Desmond et al. (2012) and Villegas et al. (2011). Barros et al. (2008) introduced the generalized BS regression model based on the BS-$t_{\nu }$ distribution (that is, based on the BS Student-$t$ model with $\nu$ degrees of freedom), Lemonte and Cordeiro (2009) proposed a non-linear BS regression model, Santana et al. (2011) and Lemonte (2012) introduced the skewed BS regression model, whereas Villegas et al. (2011) and Desmond et al. (2012) studied a mixed log-linear model based on the BS distribution.

In this paper, in addition to the existing generalizations of the BS regression model, we shall propose the extended BS regression model based on the MOEBS distribution; that is, we will introduce a new class of lifetime regression models in which the errors follow the log-MOEBS distribution. The main motivation for introducing this new class of regression models relies on the fact that the practitioners will have a new BS regression model to use in practical applications. Moreover, the formulas related with the new regression model are manageable and with the use of modern computer resources and its numerical capabilities, the proposed model may prove to be an useful addition to the arsenal of applied statisticians. Additionally, the new model is quite flexible and can be widely applied in analyzing lifetime data. Further, we provide two applications to real data sets which show that the new regression model yields a better fit than the usual BS regression model. Furthermore, the new extended BS regression model can be used for modeling censored data as well as data without censoring. It should be mentioned that censored data is very common in lifetime data because of time limits and other restrictions on data collection. In a engineering life test experiment, for example, it is usually not feasible to continue experimentation until all items under study have failed. In a survival study, patients follow-up may be lost and also data analysis is usually done before all patients have reached the event of interest. The partial information contained in the censored observations is just a lower bound on the lifetime distribution. Reliability studies usually finish before all units have failed, even making use of accelerated tests. This is a special source of difficulty in the analysis of reliability data. Such data are said to be censored at right and they arise when some units are still running at the time of the data analysis, removed from test before they fail or because they failed from an extraneous cause. We refer the reader to Gijbels (2010) for a recent overview on censored data.

It is nowadays a well spread practice, after modeling, to check the model assumptions and conduct diagnostic studies in order to detect possible influential observations that may distort the results of the analysis. Diagnostic analysis is an efficient way to detect influential observations. The first technique developed to assess the individual impact of cases on the estimation process is, perhaps, the case deletion which became a very popular tool. Cook (1977) presents a great development of case deletion diagnostics for a general statistical model. Case deletion is an example of a global influence analysis, that is, the effect of an observation is assessed by completely removing it. However, case deletion excludes all information from an observation and we can hardly say whether this observation has some influence on a specific aspect of the model. To overcome this problem, one can resort to local influence approach where one investigates the model sensitivity under small perturbations. In this context, Cook (1986) proposed a general framework to detect influential observations which gives a measure of this sensitivity under small perturbations on the data or in the model. Many applications of the local influence method may be found in the statistical literature for various models and under different perturbation schemes. For instance, Espinheira et al. (2008), Vasconcellos and Fernandez (2009), Patriota et al. (2010), Lemonte and Patriota (2011), Zevallos et al. (2012) and Matos et al. (2013), among others. In this paper, we also propose a similar methodology to detect influential subjects in the new extended BS regression model. In particular, we obtain explicit formulas for Cook’s (1986) normal curvature measure under three perturbation schemes.

The paper unfolds as follows. The log-MOEBS distribution is proposed in Section 2. In Section 3, we introduce the extended BS regression model and discuss estimation of the model parameters. Specifically, we compute the maximum likelihood estimating equations by assuming random censoring. In Section 4, the normal curvatures of local influence are derived under various perturbation schemes and a kind of deviance residual is proposed to assess departures from the underlying log-MOEBS distribution as well as to detect outlying observations. In Section 5, we propose a likelihood ratio statistic for testing the homogeneity of the shape parameters. Two real data illustrations are considered in Section 6. The paper ends up with some concluding remarks in Section 7.