Exponentiated Weibull regression for time-to-event data

The Weibull, log-logistic and log-normal distributions are extensively used to model time-to-event data. The Weibull family accommodates only monotone hazard rates, whereas the log-logistic and log-normal are widely used to model unimodal hazard functions. The increasing availability of lifetime data with a wide range of characteristics motivate us to develop more flexible models that accommodate both monotone and nonmonotone hazard functions. One such model is the exponentiated Weibull distribution which not only accommodates monotone hazard functions but also allows for unimodal and bathtub shape hazard rates. This distribution has demonstrated considerable potential in univariate analysis of time-to-event data. However, the primary focus of many studies is rather on understanding the relationship between the time to the occurrence of an event and one or more covariates. This leads to a consideration of regression models that can be formulated in different ways in survival analysis. One such strategy involves formulating models for the accelerated failure time family of distributions. The most commonly used distributions serving this purpose are the Weibull, log-logistic and log-normal distributions. In this study, we show that the exponentiated Weibull distribution is closed under the accelerated failure time family. We then formulate a regression model based on the exponentiated Weibull distribution, and develop large sample theory for statistical inference. We also describe a Bayesian approach for inference. Two comparative studies based on real and simulated data sets reveal that the exponentiated Weibull regression can be valuable in adequately describing different types of time-to-event data.     

Model diagnostics for marginal regression analysis of correlated binary data

We propose several diagnostic methods for checking the adequacy of marginal regression models for analyzing correlated binary data. We use a parametric marginal model based on latent variables and derive the projection (hat) matrix, Cook's distance, various residuals and Mahalanobis distance between the observed binary responses and the estimated probabilities for a cluster. Emphasized are several graphical methods including the simulated Q-Q plot, the half-normal probability plot with a simulated envelope, and the partial residual plot. The methods are illustrated with a real life example.     

A Parametric Model for the Interval Censored Survival Times of Acacia Mangium Plantation in a Spacing Trial

Survival times for the Acacia mangium plantation in the Segaliud Lokan Project, Sabah, East Malaysia were analysed based on 20 permanent sample plots (PSPs) established in 1988 as a spacing experiment. The PSPs were established following a complete randomized block design with five levels of spacing randomly assigned to units within four blocks at different sites. The survival times of trees in years are of interest. Since the inventories were only conducted annually, the actual survival time for each tree was not observed. Hence, the data set comprises censored survival times. Initial analysis of the survival of the Acacia mangium plantation suggested there is block by spacing interaction; a Weibull model gives a reasonable fit to the replicate survival times within each PSP; but a standard Weibull regression model is inappropriate because the shape parameter differs between PSPs. In this paper we investigate the form of the non-constant Weibull shape parameter. Parsimonious models for the Weibull survival times have been derived using maximum likelihood methods. The factor selection for the parameters is based on a backward elimination procedure. The models are compared using likelihood ratio statistics. The results suggest that both Weibull parameters depend on spacing and block.     

Nonparametric Bayesian survival analysis using mixtures of Weibull distributions

Bayesian nonparametric methods have been applied to survival analysis problems since the emergence of the area of Bayesian nonparametrics. However, the use of the flexible class of Dirichlet process mixture models has been rather limited in this context. This is, arguably, to a large extent, due to the standard way of fitting such models that precludes full posterior inference for many functionals of interest in survival analysis applications. To overcome this difficulty, we provide a computational approach to obtain the posterior distribution of general functionals of a Dirichlet process mixture. We model the survival distribution employing a flexible Dirichlet process mixture, with a Weibull kernel, that yields rich inference for several important functionals. In the process, a method for hazard function estimation emerges. Methods for simulation-based model fitting, in the presence of censoring, and for prior specification are provided. We illustrate the modeling approach with simulated and real data.     

Diagnostic tools in generalized Weibull linear regression models

We propose some statistical tools for diagnosing the class of generalized Weibull linear regression models [A.A. Prudente and G.M. Cordeiro, Generalized Weibull linear models, Comm. Statist. Theory Methods 39 (2010), pp. 3739–3755]. This class of models is an alternative means of analysing positive, continuous and skewed data and, due to its statistical properties, is very competitive with gamma regression models. First, we show that the Weibull model induces ma-ximum likelihood estimators asymptotically more efficient than the gamma model. Standardized residuals are defined, and their statistical properties are examined empirically. Some measures are derived based on the case-deletion model, including the generalized Cook's distance and measures for identifying influential observations on partial F-tests. The results of a simulation study conducted to assess behaviour of the global influence approach are also presented. Further, we perform a local influence analysis under the case-weights, response and explanatory variables perturbation schemes. The Weibull, gamma and other Weibull-type regression models are fitted into three data sets to illustrate the proposed diagnostic tools. Statistical analyses indicate that the Weibull model fitted into these data yields better fits than other common alternative models.     

A multicollinearity diagnostic for models fit to censored data

A multicollinearity diagnostic is discussed for parametric models fit to censored data. The models considered include the Weibull, exponential and lognormal models as well as the Cox proportional hazards model. This diagnostic is an extension of the diagnostic proposed by Belsley, Kuh, and Welsch (1980). The diagnostic is based on the condition indicies and variance proportions of the variance covariance matrix. Its use and properties are studied through a series of examples. The effect of centering variables included in model is also discussed.     

Fisher Information for Two Gamma Frailty Bivariate Weibull Models

The asymptotic properties of frailty models for multivariate survival data are not well understood. To study this aspect, the Fisher information is derived in the standard bivariate gamma frailty model, where the survival distribution is of Weibull form conditional on the frailty. For comparison, the Fisher information is also derived in the bivariate gamma frailty model, where the marginal distribution is of Weibull form.     

Additive Transformation Models for Recurrent Events

In this article, we propose a class of additive transformation models for recurrent event data, which includes the additive rates model as a special case. The new models offer great flexibility in formulating the effects of covariates on the mean function of recurrent events. Estimating equation approaches are developed for the model parameters, and asymptotic properties of the resulting estimators are established. In addition, a model checking procedure is presented to assess the adequacy of the model. The finite sample performance of the proposed estimators is examined through simulation studies, and an application to a bladder cancer study is presented.     

Predictive comparison of joint longitudinal-survival modeling: a case study illustrating competing approaches

The joint modeling of longitudinal and survival data has received extraordinary attention in the statistics literature recently, with models and methods becoming increasingly more complex. Most of these approaches pair a proportional hazards survival with longitudinal trajectory modeling through parametric or nonparametric specifications. In this paper we closely examine one data set previously analyzed using a two parameter parametric model for Mediterranean fruit fly (medfly) egg-laying trajectories paired with accelerated failure time and proportional hazards survival models. We consider parametric and nonparametric versions of these two models, as well as a proportional odds rate model paired with a wide variety of longitudinal trajectory assumptions reflecting the types of analyses seen in the literature. In addition to developing novel nonparametric Bayesian methods for joint models, we emphasize the importance of model selection from among joint and non joint models. The default in the literature is to omit at the outset non joint models from consideration. For the medfly data, a predictive diagnostic criterion suggests that both the choice of survival model and longitudinal assumptions can grossly affect model adequacy and prediction. Specifically for these data, the simple joint model used in by Tseng et al. (Biometrika 92:587–603, 2005) and models with much more flexibility in their longitudinal components are predictively outperformed by simpler analyses. This case study underscores the need for data analysts to compare on the basis of predictive performance different joint models and to include non joint models in the pool of candidates under consideration.     

Location-scale mixed models and goodness-of-fit assessment applied to insect ecology

Survival models have been extensively used to analyse time-until-event data. There is a range of extended models that incorporate different aspects, such as overdispersion/frailty, mixtures, and flexible response functions through semi-parametric models. In this work, we show how a useful tool to assess goodness-of-fit, the half-normal plot of residuals with a simulated envelope, implemented in the hnp package in R, can be used on a location-scale modelling context. We fitted a range of survival models to time-until-event data, where the event was an insect predator attacking a larva in a biological control experiment. We started with the Weibull model and then fitted the exponentiated-Weibull location-scale model with regressors both for the location and scale parameters. We performed variable selection for each model and, by producing half-normal plots with simulated envelopes for the deviance residuals of the model fits, we found that the exponentiated-Weibull fitted the data better. We then included a random effect in the exponentiated-Weibull model to accommodate correlated observations. Finally, we discuss possible implications of the results found in the case study.     

A comparison of parametric and semi-parametric survival models with artificial neural networks

Survival models are used to examine data in the event of an occurrence. These are discussed in various types including parametric, non-parametric and semi-parametric models. Parametric models require a clear distribution of survival time, and semi-parametric models assume proportional hazards. Among these models, the non-parametric model of artificial neural network has the fewest assumptions and can be often replaced by other models. Given the importance of distribution Weibull survival models in this study of simulation shape parameter of the Weibull distribution have been assumed as 1, 2 and 3, and also the average rate at levels of 0%–75% have been censored. The values predicted by the neural network forecasting model with parametric survival and Cox regression models were compared. This comparison considering levels of complexity due to the hazard model using the ROC curve and the corresponding tests have been carried out.     

Frailty models of manufacturing effects

The median service lifetime of respirator safety devices produced by different manufacturers is determined using frailty models to account for unobserved differences in manufacturing processes and raw materials. The gamma and positive stable frailty distributions are used to obtain survival distribution estimates when the baseline hazard is assumed to be Weibull. Frailty distributions are compared using laboratory test data of the failure times for 104 respirator cartridges produced by 10 different manufacturers tested with three different challenge agents. Likelihood ratio tests indicate that both frailty models provide a significant improvement over a Weibull model assuming independence. Results are compared to fixed effects approaches for analysis of this data.     

Hierarchical likelihood approach for the Weibull frailty model

In this article, the proportional hazard model with Weibull frailty, which is outside the range of the exponential family, is used for analysing the right-censored longitudinal survival data. Complex multidimensional integrals are avoided by using hierarchical likelihood to estimate the regression parameters and to predict the realizations of random effects. The adjusted profile hierarchical likelihood is adopted to estimate the parameters in frailty distribution, during which the first- and second-order methods are used. The simulation studies indicate that the regression-parameter estimates in the Weibull frailty model are accurate, which is similar to the gamma frailty and lognormal frailty models. Two published data sets are used for illustration.     

Bayesian Weibull tree models for survival analysis of clinico-genomic data

An important goal of research involving gene expression data for outcome prediction is to establish the ability of genomic data to define clinically relevant risk factors. Recent studies have demonstrated that microarray data can successfully cluster patients into low- and high-risk categories. However, the need exists for models which examine how genomic predictors interact with existing clinical factors and provide personalized outcome predictions. We have developed clinico-genomic tree models for survival outcomes which use recursive partitioning to subdivide the current data set into homogeneous subgroups of patients, each with a specific Weibull survival distribution. These trees can provide personalized predictive distributions of the probability of survival for individuals of interest. Our strategy is to fit multiple models; within each model we adopt a prior on the Weibull scale parameter and update this prior via Empirical Bayes whenever the sample is split at a given node. The decision to split is based on a Bayes factor criterion. The resulting trees are weighted according to their relative likelihood values and predictions are made by averaging over models. In a pilot study of survival in advanced stage ovarian cancer we demonstrate that clinical and genomic data are complementary sources of information relevant to survival, and we use the exploratory nature of the trees to identify potential genomic biomarkers worthy of further study.     

A log-linear regression model for the odd Weibull distribution with censored data

We introduce the log-odd Weibull regression model based on the odd Weibull distribution (Cooray, 2006). We derive some mathematical properties of the log-transformed distribution. The new regression model represents a parametric family of models that includes as sub-models some widely known regression models that can be applied to censored survival data. We employ a frequentist analysis and a parametric bootstrap for the parameters of the proposed model. We derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some ways to assess global influence. Further, for different parameter settings, sample sizes and censoring percentages, some simulations are performed. In addition, the empirical distribution of some modified residuals are given and compared with the standard normal distribution. These studies suggest that the residual analysis usually performed in normal linear regression models can be extended to a modified deviance residual in the proposed regression model applied to censored data. We define martingale and deviance residuals to check the model assumptions. The extended regression model is very useful for the analysis of real data.     

Heteroscedastic log-exponentiated Weibull regression model

We introduce a new class of heteroscedastic log-exponentiated Weibull (LEW) regression models. The class of regression models can be applied to censored data and be used more effectively in survival analysis. Maximum likelihood estimation of the model parameters with censored data as well as influence diagnostics for the new regression model is investigated. For different parameter settings, sample sizes and censoring percentages, various simulation studies are performed and compared to the performance of the heteroscedastic LEW regression model. The normal curvatures for studying local influence are derived under various perturbation schemes. An empirical application to a real data set is provided to illustrate the usefulness of the new class of heteroscedastic regression models.     

Truncated location-scale non linear regression models

We present a class of truncated non linear regression models for location and scale where the truncated nature of the data is incorporated into the statistical model by assuming that the response variable follows a truncated distribution. The location parameter of the response variable is assumed to be modeled by a continuous non linear function of covariates and unknown parameters. In addition, the proposed model also allows for the scale parameter of the responses to be characterized by a continuous function of the covariates and unknown parameters. Three particular cases of the proposed models are presented by considering the response variable to follow a truncated normal, truncated skew normal, and truncated beta distribution. These truncated non linear regression models are constructed assuming fixed known truncation limits and model parameters are estimated by direct maximization of the log-likelihood using a non linear optimization algorithm. Standardized residuals and diagnostic metrics based on the cases deletion are considered to verify the adequacy of the model and to detect outliers and influential observations. Results based on simulated data are presented to assess the frequentist properties of estimates, and a real data set on soil-water retention from the Buriti Vermelho River Basin database is analyzed using the proposed methodology.     

Bayesian analysis of multivariate survival data using Monte Carlo methods

This paper deals with the analysis of multivariate survival data from a Bayesian perspective using Markov-chain Monte Carlo methods. The Metropolis along with the Gibbs algorithm is used to calculate some of the marginal posterior distributions. A multivariate survival model is proposed, since survival times within the same group are correlated as a consequence of a frailty random block effect. The conditional proportional-hazards model of Clayton and Cuzick is used with a martingale structured prior process (Arjas and Gasbarra) for the discretized baseline hazard. Besides the calculation of the marginal posterior distributions of the parameters of interest, this paper presents some Bayesian EDA diagnostic techniques to detect model adequacy. The methodology is exemplified with kidney infection data where the times to infections within the same patients are expected to be correlated.     

A family of multiplicative survival models incorporating a long-term survivorship parameter c as a function of covariates

A family of multiplicative survival models is obtained for the analysis of clinical trial data in which a substantial proportion c of patients respond favorably to treatment with longterm survivorship. The model (MWSM) representing the hazard function for all patients as a function of c and a Weibull density is developed in which the distributional parameters and c are regressed on covariates and estimated by the method of maximum likelihood. The MWSM hazard function is monotonically increasing, peaking, then decreasing thereafter if the shape parameter exceeds unity, and is monotonically decreasing otherwise. Mortality rates with similar behavior have been empirically observed in cancer clinical trial data. A nonproportional or proportional hazards model results depending upon whether or not any Weibull parameter is regressed on covariates. Tests of hypotheses and confidence intervals for c and p-quantiles are obtained.     

Correlated gamma frailty models for bivariate survival data

Frailty models are used in the survival analysis to account for the unobserved heterogeneity in individual risks to disease and death. To analyze the bivariate data on related survival times (e.g., matched pairs experiments, twin or family data) the shared frailty models were suggested. Shared frailty models are used despite their limitations. To overcome their disadvantages correlated frailty models may be used. In this article, we introduce the gamma correlated frailty models with two different baseline distributions namely, the generalized log logistic, and the generalized Weibull. We introduce the Bayesian estimation procedure using Markov chain Monte Carlo (MCMC) technique to estimate the parameters involved in these models. We present a simulation study to compare the true values of the parameters with the estimated values. Also we apply these models to a real life bivariate survival dataset related to the kidney infection data and a better model is suggested for the data.