Diagnostic Measures for Generalized Linear Models with Missing Covariates

Abstract.  In this paper, we carry out an in-depth investigation of diagnostic measures for assessing the influence of observations and model misspecification in the presence of missing covariate data for generalized linear models. Our diagnostic measures include case-deletion measures and conditional residuals. We use the conditional residuals to construct goodness-of-fit statistics for testing possible misspecifications in model assumptions, including the sampling distribution. We develop specific strategies for incorporating missing data into goodness-of-fit statistics in order to increase the power of detecting model misspecification. A resampling method is proposed to approximate the p -value of the goodness-of-fit statistics. Simulation studies are conducted to evaluate our methods and a real data set is analysed to illustrate the use of our various diagnostic measures.     

Local influence for functional comparative calibration models with replicated data

We investigate local influence analysis in functional comparative calibration models with replicated data. A method for selecting appropriate perturbation schemes based on the expected Fisher information matrix with respect to the perturbation vector is proposed. It is shown that arbitrarily perturbing these models may result in misleading inference about the influential subjects. First-order influence measures for identifying the correct influential subjects and replicates on corrected score estimators are defined. We introduce different perturbation schemes including perturbation of subjects and replicates on the corrected likelihood function and obtain the density of the perturbed model from which the methodology is based. Particularly, three perturbation of variances schemes could be a better way to handle badly modeled subjects or replicates. Two real data sets are analyzed to illustrate the use of our local influence measures.     

Estimating infant mortality in Colombia: some overdispersion modelling approaches

It is common to fit generalized linear models with binomial and Poisson responses, where the data show a variability that is greater than the theoretical variability assumed by the model. This phenomenon, known as overdispersion, may spoil inferences about the model by considering significant parameters associated with variables that have no significant effect on the dependent variable. This paper explains some methods to detect overdispersion and presents and evaluates three well-known methodologies that have shown their usefulness in correcting this problem, using random mean models, quasi-likelihood methods and a double exponential family. In addition, it proposes some new Bayesian model extensions that have proved their usefulness in correcting the overdispersion problem. Finally, using the information provided by the National Demographic and Health Survey 2005, the departmental factors that have an influence on the mortality of children under 5 years and female postnatal period screening are determined. Based on the results, extensions that generalize some of the aforementioned models are also proposed, and their use is motivated by the data set under study. The results conclude that the proposed overdispersion models provide a better statistical fit of the 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.     

Robustness of Approaches to ROC Curve Modeling under Misspecification of the Underlying Probability Model

A variety of statistical regression models have been proposed for the comparison of ROC curves for different markers across covariate groups. Pepe developed parametric models for the ROC curve that induce a semiparametric model for the market distributions to relax the strong assumptions in fully parametric models. We investigate the analysis of the power ROC curve using these ROC-GLM models compared to the parametric exponential model and the estimating equations derived from the usual partial likelihood methods in time-to-event analyses. In exploring the robustness to violations of distributional assumptions, we find that the ROC-GLM provides an extra measure of robustness.     

A bivariate regression model for matched paired survival data: local influence and residual analysis

The use of bivariate distributions plays a fundamental role in survival and reliability studies. In this paper, we consider a location scale model for bivariate survival times based on the proposal of a copula to model the dependence of bivariate survival data. For the proposed model, we consider inferential procedures based on maximum likelihood. Gains in efficiency from bivariate models are also examined in the censored data setting. For different parameter settings, sample sizes and censoring percentages, various simulation studies are performed and compared to the performance of the bivariate regression model for matched paired survival data. Sensitivity analysis methods such as local and total influence are presented and derived under three perturbation schemes. The martingale marginal and the deviance marginal residual measures are used to check the adequacy of the model. Furthermore, we propose a new measure which we call modified deviance component residual. The methodology in the paper is illustrated on a lifetime data set for kidney patients.     

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.     

Mixed-effects Models with Skewed Distributions for Time-varying Decay Rate in HIV Dynamics

After initiation of treatment, HIV viral load has multiphasic changes, which indicates that the viral decay rate is a time-varying process. Mixed-effects models with different time-varying decay rate functions have been proposed in literature. However, there are two unresolved critical issues: (i) it is not clear which model is more appropriate for practical use, and (ii) the model random errors are commonly assumed to follow a normal distribution, which may be unrealistic and can obscure important features of within- and among-subject variations. Because asymmetry of HIV viral load data is still noticeable even after transformation, it is important to use a more general distribution family that enables the unrealistic normal assumption to be relaxed. We developed skew-elliptical (SE) Bayesian mixed-effects models by considering the model random errors to have an SE distribution. We compared the performance among five SE models that have different time-varying decay rate functions. For each model, we also contrasted the performance under different model random error assumptions such as normal, Student-t, skew-normal, or skew-t distribution. Two AIDS clinical trial datasets were used to illustrate the proposed models and methods. The results indicate that the model with a time-varying viral decay rate that has two exponential components is preferred. Among the four distribution assumptions, the skew-t and skew-normal models provided better fitting to the data than normal or Student-t model, suggesting that it is important to assume a model with a skewed distribution in order to achieve reasonable results when the data exhibit skewness.     

A new absolute continuous bivariate generalized exponential distribution

The generalized exponential is the most commonly used distribution for analyzing lifetime data. This distribution has several desirable properties and it can be used quite effectively to analyse several skewed life time data. The main aim of this paper is to introduce absolutely continuous bivariate generalized exponential distribution using the method of Block and Basu (1974). In fact, the Block and Basu exponential distribution will be extended to the generalized exponential distribution. We call the new proposed model as the Block and Basu bivariate generalized exponential distribution, then, discuss its different properties. In this case the joint probability distribution function and the joint cumulative distribution function can be expressed in compact forms. The model has four unknown parameters and the maximum likelihood estimators cannot be obtained in explicit form. To compute the maximum likelihood estimators directly, one needs to solve a four dimensional optimization problem. The EM algorithm has been proposed to compute the maximum likelihood estimations of the unknown parameters. One data analysis is provided for illustrative purposes. Finally, we propose some generalizations of the proposed model and compare their models with each other.     

Bayesian estimation and case influence diagnostics for the zero-inflated negative binomial regression model

In recent years, there has been considerable interest in regression models based on zero-inflated distributions. These models are commonly encountered in many disciplines, such as medicine, public health, and environmental sciences, among others. The zero-inflated Poisson (ZIP) model has been typically considered for these types of problems. However, the ZIP model can fail if the non-zero counts are overdispersed in relation to the Poisson distribution, hence the zero-inflated negative binomial (ZINB) model may be more appropriate. In this paper, we present a Bayesian approach for fitting the ZINB regression model. This model considers that an observed zero may come from a point mass distribution at zero or from the negative binomial model. The likelihood function is utilized to compute not only some Bayesian model selection measures, but also to develop Bayesian case-deletion influence diagnostics based on q-divergence measures. The approach can be easily implemented using standard Bayesian software, such as WinBUGS. The performance of the proposed method is evaluated with a simulation study. Further, a real data set is analyzed, where we show that ZINB regression models seems to fit the data better than the Poisson counterpart.     

A general long-term aging model with different underlying activation mechanisms: Modeling,Bayesian estimation,and case influence diagnostics

In this paper we propose a general cure rate aging model. Our approach enables different underlying activation mechanisms which lead to the event of interest. The number of competing causes of the event of interest is assumed to follow a logarithmic distribution. The model is parameterized in terms of the cured fraction which is then linked to covariates. We explore the use of Markov chain Monte Carlo methods to develop a Bayesian analysis for the proposed model. Moreover, some discussions on the model selection to compare the fitted models are given, as well as case deletion influence diagnostics are developed for the joint posterior distribution based on the ψ-divergence, which has several divergence measures as particular cases, such as the Kullback–Leibler (K-L), J-distance, L₁ norm, and χ²-square divergence measures. Simulation studies are performed and experimental results are illustrated based on a real malignant melanoma data.     

Robust Tests Against Smooth Transition Autoregressive Models

In this paper, we consider testing for linearity against a well-known class of regime switching models known as the smooth transition autoregressive (STAR) models. Apart from the model selection issues, one reason for interest in testing for linearity in time-series models is that non-linear models such as the STAR are considerably more difficult to use. This testing problem is non-standard because a nuisance parameter becomes unidentified under the null hypothesis. In this paper, we further explore the class of tests proposed by Luukkonen, Saikonnen and Terasvirta (1988). Luukkonen et al . (1988) proposed LM tests for linearity against STAR models. A potential difficulty here is that the linear approximation introduces high leverage points, and hence outliers are likely to be quite influential. To overcome this difficulty, we use the same approximating linear model of Luukkonen et al . (1988), but we apply Wald and F -tests based on l 1 - and bounded influence estimates. The efficiency gains of this procedure cannot be easily deduced from the existing theoretical results because the test is based on a misspecified model under H 1 . Therefore, we carried out a simulation study, in which we observed that the robust tests have desirable properties compared to the test of Luukkonen et al . (1988) for a range of error distributions in the STAR model, in particular the robust tests have power advantages over the LM test.     

Stratified Gaussian graphical models

Gaussian graphical models represent the backbone of the statistical toolbox for analyzing continuous multivariate systems. However, due to the intrinsic properties of the multivariate normal distribution, use of this model family may hide certain forms of context-specific independence that are natural to consider from an applied perspective. Such independencies have been earlier introduced to generalize discrete graphical models and Bayesian networks into more flexible model families. Here, we adapt the idea of context-specific independence to Gaussian graphical models by introducing a stratification of the Euclidean space such that a conditional independence may hold in certain segments but be absent elsewhere. It is shown that the stratified models define a curved exponential family, which retains considerable tractability for parameter estimation and model selection.     

Bayesian variable selection for the Cox regression model with missing covariates

In this paper, we develop Bayesian methodology and computational algorithms for variable subset selection in Cox proportional hazards models with missing covariate data. A new joint semi-conjugate prior for the piecewise exponential model is proposed in the presence of missing covariates and its properties are examined. The covariates are assumed to be missing at random (MAR). Under this new prior, a version of the Deviance Information Criterion (DIC) is proposed for Bayesian variable subset selection in the presence of missing covariates. Monte Carlo methods are developed for computing the DICs for all possible subset models in the model space. A Bone Marrow Transplant (BMT) dataset is used to illustrate the proposed methodology.     

Influence analysis in multivariate linear general models

In this paper we obtain several influence measures for the multivariate linear general model through the approach proposed by Muñoz-Pichardo et al. (1995), which is based on the concept of conditional bias. An interesting charasteristic of this approach is that it does not require any distributional hypothesis. Appling the obtained results to the multivariate regression model, we obtain some measures proposed by other authors. Nevertheless, on the results obtained in this paper, we emphasize two aspects. First, they provide a theoretical foundation for measures proposed by other authors for the mul¬tivariate regression model. Second, they can be applied to any linear model that can be formulated as a particular case of the multivariate linear general model. In particular, we carry out an application to the multivariate analysis of covariance.     

Censored Exponential Data: Large Deviations for MLEs and Posterior Distributions

In this article, we consider several statistical models for censored exponential data. We prove a large deviation result for the maximum likelihood estimators (MLEs) of each model, and a unique result for the posterior distributions which works well for all the cases. Finally, comparing the large deviation rate functions for MLEs and posterior distributions, we show that a typical feature fails for one model; moreover, we illustrate the relation between this fact and a well-known result for curved exponential models.     

Estimating the grid of time-points for the piecewise exponential model

One of the greatest challenges related to the use of piecewise exponential models (PEMs) is to find an adequate grid of time-points needed in its construction. In general, the number of intervals in such a grid and the position of their endpoints are ad-hoc choices. We extend previous works by introducing a full Bayesian approach for the piecewise exponential model in which the grid of time-points (and, consequently, the endpoints and the number of intervals) is random. We estimate the failure rates using the proposed procedure and compare the results with the non-parametric piecewise exponential estimates. Estimates for the survival function using the most probable partition are compared with the Kaplan-Meier estimators (KMEs). A sensitivity analysis for the proposed model is provided considering different prior specifications for the failure rates and for the grid. We also evaluate the effect of different percentage of censoring observations in the estimates. An application to a real data set is also provided. We notice that the posteriors are strongly influenced by prior specifications, mainly for the failure rates parameters. Thus, the priors must be fairly built, say, really disclosing the expert prior opinion.     

The Power Series Cure Rate Model: An Application to a Cutaneous Melanoma Data

In this article we propose a new cure rate survival model. In our approach the number of competing causes of the event of interest is assumed to follow an exponential discrete power series distribution. An advantage of our model is that it is very flexible, including several particular cases, such as, Bernoulli, geometric, Poisson, etc. Moreover, we derive the appropriate matrices for assessing local influence on the parameter estimates under different perturbation schemes and present some ways to perform global influence analysis. Distribution fitting can be tested for the best fitting in a straightforward way. Maximum likelihood estimation is discussed. Our proposed model is illustrated through cutaneous melanoma data.     

Non-mixture cure correlated frailty models in Bayesian approach

In this article, we develop a Bayesian approach for the estimation of two cure correlated frailty models that have been extended to the cure frailty models introduced by Yin [34]. We used the two different type of frailty with bivariate log-normal distribution instead of gamma distribution. A likelihood function was constructed based on a piecewise exponential distribution function. The model parameters were estimated by the Markov chain Monte Carlo method. The comparison of models is based on the Cox correlated frailty model with log-normal distribution. A real data set of bilateral corneal graft rejection was used to compare these models. The results of this data, based on deviance information criteria, showed the advantage of the proposed models.     

On detecting influential data and selecting regression variables

The analysis of residuals may reveal various functional forms suitable for the regression model. In this paper, we investigate some selection criteria for selecting important regression variables. In doing so, we use statistical selection and ranking procedures. Thus, we derive an appropriate criterion to measure the influence and bias for the reduced models. We show that the reduced models are based on some noncentrality parameters which provide a measure of goodness of fit for the fitted models. In this paper, we also discuss the relationships of influence diagnostics and the statistic proposed earlier by Gupta and Huang (J. Statist. Plann. Inference 20 (1988) 155–167). We introduce a new measure for detecting influential data as an alternative to Cook's measure.