Influence diagnostics in nonlinear reproductive dispersion mixed models

The class of nonlinear reproductive dispersion mixed models (NRDMMs) is an extension of nonlinear reproductive dispersion models and generalized linear mixed models. This paper discusses the influence analysis of the model based on Laplace approximation. The equivalence of case-deletion models and mean-shift outlier models in NRDMMs is investigated, and some diagnostic measures are proposed via the case-deletion method. We also investigate the assessment of local influence of various perturbation schemes. The proposed method is illustrated with an example.     

Scale mixtures log-Birnbaum–Saunders regression models with censored data: a Bayesian approach

The main objective of this paper is to develop a full Bayesian analysis for the Birnbaum–Saunders (BS) regression model based on scale mixtures of the normal (SMN) distribution with right-censored survival data. The BS distributions based on SMN models are a very general approach for analysing lifetime data, which has as special cases the Student-t-BS, slash-BS and the contaminated normal-BS distributions, being a flexible alternative to the use of the corresponding BS distribution or any other well-known compatible model, such as the log-normal distribution. A Gibbs sample algorithm with Metropolis–Hastings algorithm is used to obtain the Bayesian estimates of the parameters. Moreover, some discussions on the model selection to compare the fitted models are given and case-deletion influence diagnostics are developed for the joint posterior distribution based on the Kullback–Leibler divergence. The newly developed procedures are illustrated on a real data set previously analysed under BS regression models.     

Bayesian analysis of censored linear regression models with scale mixtures of normal distributions

As is the case of many studies, the data collected are limited and an exact value is recorded only if it falls within an interval range. Hence, the responses can be either left, interval or right censored. Linear (and nonlinear) regression models are routinely used to analyze these types of data and are based on normality assumptions for the errors terms. However, those analyzes might not provide robust inference when the normality assumptions are questionable. In this article, we develop a Bayesian framework for censored linear regression models by replacing the Gaussian assumptions for the random errors with scale mixtures of normal (SMN) distributions. The SMN is an attractive class of symmetric heavy-tailed densities that includes the normal, Student-t, Pearson type VII, slash and the contaminated normal distributions, as special cases. Using a Bayesian paradigm, an efficient Markov chain Monte Carlo algorithm is introduced to carry out posterior inference. A new hierarchical prior distribution is suggested for the degrees of freedom parameter in the Student-t distribution. 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 the q-divergence measure. The proposed Bayesian methods are implemented in the R package BayesCR. The newly developed procedures are illustrated with applications using real and simulated data.     

Influence Diagnostics of Semiparametric Nonlinear Reproductive Dispersion Models

This article proposes a semiparametric nonlinear reproductive dispersion model (SNRDM) which is an extension of nonlinear reproductive dispersion model and semiparametric regression model. Maximum penalized likelihood estimators (MPLEs) of unknown parameters and nonparametric functions in SNRDMs are presented. Some novel diagnostic statistics such as Cook distance and difference deviance for parametric and nonparametric parts are developed to identify influence observations in SNRDMs on the basis of case-deletion method, and some formulae readily computed with the MPLEs algorithm for diagnostic measures are given. The equivalency of case-deletion models and mean-shift outlier models in SNRDM is investigated. A simulation study and a real example are used to illustrate the proposed diagnostic measures.     

Influence analysis of additive mixed-effects nonlinear regression models via EM algorithm

This paper presents a unified method for influence analysis to deal with random effects appeared in additive nonlinear regression models for repeated measurement data. The basic idea is to apply the Q-function, the conditional expectation of the complete-data log-likelihood function obtained from EM algorithm, instead of the observed-data log-likelihood function as used in standard influence analysis. Diagnostic measures are derived based on the case-deletion approach and the local influence approach. Two real examples and a simulation study are examined to illustrate our methodology.     

Statistical diagnostics for nonlinear regression models based on scale mixtures of skew-normal distributions

The purpose of this paper is to develop diagnostics analysis for nonlinear regression models (NLMs) under scale mixtures of skew-normal (SMSN) distributions introduced by Garay et al. [Nonlinear regression models based on SMSN distributions. J. Korean Statist. Soc. 2011;40:115–124]. This novel class of models provides a useful generalization of the symmetrical NLM [Vanegas LH, Cysneiros FJA. Assessment of diagnostic procedures in symmetrical nonlinear regression models. Comput. Statist. Data Anal. 2010;54:1002–1016] since the random terms distributions cover both symmetric as well as asymmetric and heavy-tailed distributions such as the skew-t, skew-slash, skew-contaminated normal distributions, among others. Motivated by the results given in Garay et al. [Nonlinear regression models based on SMSN distributions. J. Korean Statist. Soc. 2011;40:115–124], we presented a score test for testing the homogeneity of the scale parameter and its properties are investigated through Monte Carlo simulations studies. Furthermore, local influence measures and the one-step approximations of the estimates in the case-deletion model are obtained. The newly developed procedures are illustrated considering a real data set.     

CORRECTED SCORE FUNCTIONS IN CLASSICAL ERROR-IN-VARIABLES AND INCIDENTAL PARAMETER MODELS

Nakamura (1990) introduced an approach to estimation in measurement error models based on a corrected score function, and claimed that the estimators obtained are consistent for functional models. Proof of the claim essentially assumed the existence of a corrected log-likelihood for which differentiation with respect to model parameters can be interchanged with conditional expectation taken with respect to the measurement error distributions, given the response variables and true covariates. This paper deals with simple yet practical models for which the above assumption is false, i.e. a corrected score function for the model may not be obtained through differentiating a corrected log-likelihood although it exists. Alternative regularity conditions with no reference to log-likelihood are given, under which the corrected score functions yield consistent and asymptotically normal estimators. Application to functional comparative calibration yields interesting results.     

Estimation and diagnostics for heteroscedastic nonlinear regression models based on scale mixtures of skew-normal distributions

An extension of some standard likelihood based procedures to heteroscedastic nonlinear regression models under scale mixtures of skew-normal (SMSN) distributions is developed. This novel class of models provides a useful generalization of the heteroscedastic symmetrical nonlinear regression models (Cysneiros et al., 2010), since the random term distributions cover both symmetric as well as asymmetric and heavy-tailed distributions such as skew-t, skew-slash, skew-contaminated normal, among others. A simple EM-type algorithm for iteratively computing maximum likelihood estimates of the parameters is presented and the observed information matrix is derived analytically. In order to examine the performance of the proposed methods, some simulation studies are presented to show the robust aspect of this flexible class against outlying and influential observations and that the maximum likelihood estimates based on the EM-type algorithm do provide good asymptotic properties. Furthermore, local influence measures and the one-step approximations of the estimates in the case-deletion model are obtained. Finally, an illustration of the methodology is given considering a data set previously analyzed under the homoscedastic skew-t nonlinear regression model.     

Assessing case influence on confidence intervals in nonlinear regression

Diagnostic techniques are proposed for assessing the influence of individual cases on confidence intervals in nonlinear regression. The technique proposed uses the method of profile t-plots applied to the case-deletion model. The effect of the geometry of the statistical model on the influence measures is assessed, and an algorithm for computing case-deleted confidence intervals is described. This algorithm provides a direct method for constructing a simple diagnostic measure based on the ratio of the lengths of confidence intervals. The generalization of these methods to multiresponse models is discussed.     

Case-deletion Influence Measures for the Data from Multivariate t Distributions

For the data from multivariate t distributions, it is very hard to make an influence analysis based on the probability density function since its expression is intractable. In this paper, we present a technique for influence analysis based on the mixture distribution and EM algorithm. In fact, the multivariate t distribution can be considered as a particular Gaussian mixture by introducing the weights from the Gamma distribution. We treat the weights as the missing data and develop the influence analysis for the data from multivariate t distributions based on the conditional expectation of the complete-data log-likelihood function in the EM algorithm. Several case-deletion measures are proposed for detecting influential observations from multivariate t distributions. Two numerical examples are given to illustrate our methodology.     

Mean shift and influence measures in linear measurement error models with stochastic linear restrictions

We present influence diagnostics for linear measurement error models with stochastic linear restrictions using the corrected likelihood of Nakamura in 1990. The case deletion and mean shift outlier models are developed to identify outlying and influential observations. We derive a corrected score test statistic for outlier detection based on mean shift outlier models. The analogs of Cook's distance and likelihood distance are proposed to determine influential observations based on case deletion models. A parametric bootstrap procedure is used to obtain empirical distributions of the test statistics and a simulation study has been used to evaluate the performance of the proposed estimators based on the mean squares error criterion and the score test statistic. Finally, a numerical example is given to illustrate the theoretical results.     

Bayesian inference and diagnostics in zero-inflated generalized power series regression model

ABSTRACT

The paper provides a Bayesian analysis for the zero-inflated regression models based on the generalized power series distribution. The approach is based on Markov chain Monte Carlo methods. The residual analysis is discussed and case-deletion influence diagnostics are developed for the joint posterior distribution, based on the ψ-divergence, which includes several divergence measures such as the Kullback–Leibler, J-distance, L₁ norm, and χ²-square in zero-inflated general power series models. The methodology is reflected in a data set collected by wildlife biologists in a state park in California.     

CASE-DELETION DIAGNOSTICS FOR TEST STATISTICS IN MULTIVARIATE REGRESSION

Influence measures in multivariate regression analysis have been widely developed, especially through use of the case-deletion approach. However, there seem to be few accounts of the influence of observations on test statistics in hypothesis testing. This paper examines four common multivariate tests, namely the Wilks' ratio, Lawley-Hotelling trace, Pillai's trace and Roy's greatest root for testing a general linear hypothesis of the regression coefficients in multivariate regression. The influence of observations is measured using the case-deletion approach. The proposed diagnostic measures, except that of Roy's greatest root, can be expressed in terms of statistics without involving the actual deletion of observations. An illustrative example is given with satisfactory results.     

Influence diagnostics in constrained general linear models

ABSTRACT

Constrained general linear models (CGLMs) have wide applications in practice. Similar to other data analysis, the identification of influential observations that may be potential outliers is an important step beyond in the CGLMs. We develop multiple case-deletion diagnostics for detecting influential observations in the CGLMs. The diagnostics are functions of basic building blocks: studentized residuals, error contrast matrix, and the inverse of the response variable covariance matrix. The basic building blocks are computed only once from the complete data analysis and provide information on the influence of the data on different aspects of the model fit. Computational formulas are given which make the procedures feasible. An illustrative example with a real data set is also reported.     

Maximum-likelihood estimation and influence analysis in multivariate skew-normal reproductive dispersion mixed models for longitudinal data

Various mixed models were developed to capture the features of between- and within-individual variation for longitudinal data under the normality assumption of the random effect and the within-individual random error. However, the normality assumption may be violated in some applications. To this end, this article assumes that the random effect follows a skew-normal distribution and the within-individual error is distributed as a reproductive dispersion model. An expectation conditional maximization (ECME) algorithm together with the Metropolis-Hastings (MH) algorithm within the Gibbs sampler is presented to simultaneously obtain estimates of parameters and random effects. Several diagnostic measures are developed to identify the potentially influential cases and assess the effect of minor perturbation to model assumptions via the case-deletion method and local influence analysis. To reduce the computational burden, we derive the first-order approximations to case-deletion diagnostics. Several simulation studies and a real data example are presented to illustrate the newly developed methodologies.     

Influence diagnostics for Student-t censored linear regression models

In this paper, we extend the censored linear regression model with normal errors to Student-t errors. A simple EM-type algorithm for iteratively computing maximum-likelihood estimates of the parameters is presented. To examine the performance of the proposed model, case-deletion and local influence techniques are developed to show its robust aspect against outlying and influential observations. This is done by the analysis of the sensitivity of the EM estimates under some usual perturbation schemes in the model or data and by inspecting some proposed diagnostic graphics. The efficacy of the method is verified through the analysis of simulated data sets and modelling a real data set first analysed under normal errors. The proposed algorithm and methods are implemented in the R package CensRegMod.     

Diagnostics for quasi-likelihood nonlinear models

Quasi-likelihood nonlinear models (QLNMs) are an extension of generalized linear model and include a widen class of models as special cases. This article investigates some diagnostic methods in QLNMs. An equivalency between a case-deletion model and a mean-shift outlier model in QLNM is established. Two simulation study and a real dataset are used to illustrate the proposed diagnostic methods.     

Detection of influential observations in nonlinear reproductive dispersion models

A robust case-deletion diagnostic which could avoid masking is presented for nonlinear reproductive dispersion models (NRDM) (Jorgensen, 1997) that include generalized linear models and exponential family nonlinear models as special cases. Based on the second-order approximation of log-likelihood displacement and Poon & Poon's (1999) conformal normal curvature, a measure of local influence ranging from 0 to 1 is constructed. An example illustrates application of the techniques.     

Influence on tests with focus on linear models

To assess the influence of single observations on the parameter estimates, case-deletion diagnostics are commonly used in linear regression models; one example is Cook's distance. For nested parametric models we consider a deletion diagnostic for evaluating the influence of a single observation on the likelihood ratio (LR) test. In order to have a common scale as reference, the asymptotic distribution of the diagnostic is derived and the values of the diagnostic are converted to percentiles. We focus on linear models and general linear models, and in these cases explicit results are derived. The performance of the diagnostic is explored in two small bench mark examples from linear regression and in a larger linear mixed model example.     

Estimation of variance components in linear mixed measurement error models

In this paper, we consider a linear mixed model with measurement errors in fixed effects. We find the corrected score function estimators for the variance components. An iterative algorithm is proposed for estimating the parameters. The computations on each iteration of this algorithm are those associated with computing estimates of fixed and random effects for given values of the variance components. We also derive the consistency of the estimators under regularity conditions. The simulation study shows that for relatively small sample size the corrected estimators perform very well. Finally, an example of real data is given for illustration.