Diagnostic checks for discrete data regression models using posterior predictive simulations

Model checking with discrete data regressions can be difficult because the usual methods such as residual plots have complicated reference distributions that depend on the parameters in the model. Posterior predictive checks have been proposed as a Bayesian way to average the results of goodness-of-fit tests in the presence of uncertainty in estimation of the parameters. We try this approach using a variety of discrepancy variables for generalized linear models fitted to a historical data set on behavioural learning. We then discuss the general applicability of our findings in the context of a recent applied example on which we have worked. We find that the following discrepancy variables work well, in the sense of being easy to interpret and sensitive to important model failures: structured displays of the entire data set, general discrepancy variables based on plots of binned or smoothed residuals versus predictors and specific discrepancy variables created on the basis of the particular concerns arising in an application. Plots of binned residuals are especially easy to use because their predictive distributions under the model are sufficiently simple that model checks can often be made implicitly. The following discrepancy variables did not work well: scatterplots of latent residuals defined from an underlying continuous model and quantile–quantile plots of these residuals.     

Two ways of modelling overdispersion in non-normal data

For non-normal data assumed to have distributions, such as the Poisson distribution, which have an a priori dispersion parameter, there are two ways of modelling overdispersion: by a quasi-likelihood approach or with a random-effect model. The two approaches yield different variance functions for the response, which may be distinguishable if adequate data are available. The epilepsy data of Thall and Vail and the fabric data of Bissell are used to exemplify the ideas.     

Practical Maximum Pseudolikelihood for Spatial Point Patterns(with Discussion)

This paper describes a technique for computing approximate maximum pseudolikelihood estimates of the parameters of a spatial point process. The method is an extension of Berman & Turner's (1992) device for maximizing the likelihoods of inhomogeneous spatial Poisson processes. For a very wide class of spatial point process models the likelihood is intractable, while the pseudolikelihood is known explicitly, except for the computation of an integral over the sampling region. Approximation of this integral by a finite sum in a special way yields an approximate pseudolikelihood which is formally equivalent to the (weighted) likelihood of a loglinear model with Poisson responses. This can be maximized using standard statistical software for generalized linear or additive models, provided the conditional intensity of the process takes an 'exponential family' form. Using this approach a wide variety of spatial point process models of Gibbs type can be fitted rapidly, incorporating spatial trends, interaction between points, dependence on spatial covariates, and mark information.     

ASSOCIATIONS BETWEEN AIR POLLUTION AND HOSPITAL VISITS FOR CARDIOVASCULAR DISEASES IN THE ELDERLY IN SYDNEY USING BAYESIAN STATISTICAL METHODS

Using generalized linear models (GLMs), Jalaludin  et al. (2006;  J. Exposure Analysis and Epidemiology   16 , 225–237) studied the association between the daily number of visits to emergency departments for cardiovascular disease by the elderly (65+) and five measures of ambient air pollution. Bayesian methods provide an alternative approach to classical time series modelling and are starting to be more widely used. This paper considers Bayesian methods using the dataset used by Jalaludin  et al.  (2006) , and compares the results from Bayesian methods with those obtained by Jalaludin  et al.  (2006) using GLM methods.     

Conditional variance estimation in heteroscedastic regression models

First, we propose a new method for estimating the conditional variance in heteroscedasticity regression models. For heavy tailed innovations, this method is in general more efficient than either of the local linear and local likelihood estimators. Secondly, we apply a variance reduction technique to improve the inference for the conditional variance. The proposed methods are investigated through their asymptotic distributions and numerical performances.     

Transformed generalized linear models

The estimation of data transformation is very useful to yield response variables satisfying closely a normal linear model. Generalized linear models enable the fitting of models to a wide range of data types. These models are based on exponential dispersion models. We propose a new class of transformed generalized linear models to extend the Box and Cox models and the generalized linear models. We use the generalized linear model framework to fit these models and discuss maximum likelihood estimation and inference. We give a simple formula to estimate the parameter that index the transformation of the response variable for a subclass of models. We also give a simple formula to estimate the r$r$th moment of the original dependent variable. We explore the possibility of using these models to time series data to extend the generalized autoregressive moving average models discussed by Benjamin et al. [Generalized autoregressive moving average models. J. Amer. Statist. Assoc. 98, 214–223]. The usefulness of these models is illustrated in a simulation study and in applications to three real data sets.     

Statistical inference for partially time-varying coefficient errors-in-variables models

This paper studies the partially time-varying coefficient models where some covariates are measured with additive errors. In order to overcome the bias of the usual profile least squares estimation when measurement errors are ignored, we propose a modified profile least squares estimator of the regression parameter and construct estimators of the nonlinear coefficient function and error variance. The proposed three estimators are proved to be asymptotically normal under mild conditions. In addition, we introduce the profile likelihood ratio test and then demonstrate that it follows an asymptotically χ²${\chi }^2$ distribution under the null hypothesis. Finite sample behavior of the estimators is investigated via simulations too.     

Weighted local linear composite quantile estimation for the case of general error distributions

It is known that for nonparametric regression, local linear composite quantile regression (local linear CQR) is a more competitive technique than classical local linear regression since it can significantly improve estimation efficiency under a class of non-normal and symmetric error distributions. However, this method only applies to symmetric errors because, without symmetric condition, the estimation bias is non-negligible and therefore the resulting estimator is inconsistent. In this paper, we propose a weighted local linear CQR method for general error conditions. This method applies to both symmetric and asymmetric random errors. Because of the use of weights, the estimation bias is eliminated asymptotically and the asymptotic normality is established. Furthermore, by minimizing asymptotic variance, the optimal weights are computed and consequently the optimal estimate (the most efficient estimate) is obtained. By comparing relative efficiency theoretically or numerically, we can ensure that the new estimation outperforms the local linear CQR estimation. Finite sample behaviors conducted by simulation studies further illustrate the theoretical findings.