Deviance residuals in generalised log-gamma regression models with censored observations

Abstract

In this article, we compare three residuals based on the deviance component in generalised loggamma regression models with censored observations. For different parameter settings, sample sizes and censoring percentages, various simulation studies are performed and the empirical distribution of each residual is displayed and compared with the standard normal distribution. For all cases studied, the empirical distributions of the proposed residuals are in general symmetric around zero, but only a martingale-type residual presented negligible kurtosis for the majority of the cases studied. These studies suggest that the residual analysis usually performed in normal linear regression models can be straightforwardly extended for the martingale-type residual in generalised log-gamma regression models with censored data. A lifetime data set is analysed under log-gamma regression models and a model checking based on the martingale-type residual is performed.

Semi-parametric generalized log-gamma regression models

“…We will consider censoring under the assumption of independent random censoring. The expression for the observed information matrix is consistent with the one given in Ortega et al (2008).…”

“…For instance, Lawless (1978) built confidence intervals for the parameter of the subfamily of extreme value distributions, DiCiccio (1987) derived approximate inferences for the quantiles and scale parameter, Young and Bakir (1987) obtained the bias of order n −1 , where n is the sample size, for the parameter estimates in generalized log-gamma regression models without censoring. More recently, Ortega et al (2008) studied three residuals based on the deviance component in generalized log-gamma regression models under the presence of censored observations.…”

“…In Ortega et al (2003) was presented some estimation procedures and derived di-agnostic measures based on the local influence approach under the presence of censored observations, Cox et al (2007) presented a taxonomy of the hazard function of generalized gamma distribution with application to study of survival after diagnostic of clinical AIDS during different phases of HIV therapy, whereas Zhang et al (2008) assumed generalized log-gamma distributions for the random effect in linear mixed models. More recently, Ortega et al (2008) compared three types of residuals in generalized log-gamma regression models under the presence of censored observations. In Fabio et al (2012) is assumed a generalized log-gamma distribution for the random effect in random intercept Poisson models, whereas Ortega et al (2009) proposed the log-gamma regression model with cure fraction, giving emphasis to the local influence approach.…”

The McBurr XII and Log-McBurr XII Models with Applications to Lifetime Data

“…The martingale residuals are skew, have maximum value +1 and minimum value −∞. In parametric lifetime models, the martingale residual can be expressed as r M i = δ i + log[S (y i ;θ)], where δ i = 0 if the ith observation is censored and δ i = 1 if the ith observation is uncensored (see, for example, Klein and Moeschberger, 1997;Ortega et al, 2008;Silva et al, 2011). Hence, the martingale residuals for the LMcBXII model can be expressed as…”

“…Ortega et al (2008) and Hashimoto et al (2010) investigated the empirical distributions of r M i and r D i for the generalized log-gamma and LEW regression models for interval-censored data varying the sample sizes and censoring proportions, respectively.…”