SEMIPARAMETRIC REGRESSION AND GRAPHICAL MODELS

Semiparametric regression models that use spline basis functions with penalization have graphical model representations. This link is more powerful than previously established mixed model representations of semiparametric regression, as a larger class of models can be accommodated. Complications such as missingness and measurement error are more naturally handled within the graphical model architecture. Directed acyclic graphs, also known as Bayesian networks, play a prominent role. Graphical model-based Bayesian 'inference engines', such as bugs and vibes , facilitate fitting and inference. Underlying these are Markov chain Monte Carlo schemes and recent developments in variational approximation theory and methodology.     

Invariant co-ordinate selection

Summary.  A general method for exploring multivariate data by comparing different estimates of multivariate scatter is presented. The method is based on the eigenvalue–eigenvector decomposition of one scatter matrix relative to another. In particular, it is shown that the eigenvectors can be used to generate an affine invariant co-ordinate system for the multivariate data. Consequently, we view this method as a method for invariant co-ordinate selection . By plotting the data with respect to this new invariant co-ordinate system, various data structures can be revealed. For example, under certain independent components models, it is shown that the invariant co- ordinates correspond to the independent components. Another example pertains to mixtures of elliptical distributions. In this case, it is shown that a subset of the invariant co-ordinates corresponds to Fisher's linear discriminant subspace, even though the class identifications of the data points are unknown. Some illustrative examples are given.     

The generalized log-gamma mixture model with covariates: local influence and residual analysis

In a sample of censored survival times, the presence of an immune proportion of individuals who are not subject to death, failure or relapse, may be indicated by a relatively high number of individuals with large censored survival times. In this paper the generalized log-gamma model is modified for the possibility that long-term survivors may be present in the data. The model attempts to separately estimate the effects of covariates on the surviving fraction, that is, the proportion of the population for which the event never occurs. The logistic function is used for the regression model of the surviving fraction. Inference for the model parameters is considered via maximum likelihood. Some influence methods, such as the local influence and total local influence of an individual are derived, analyzed and discussed. Finally, a data set from the medical area is analyzed under the log-gamma generalized mixture model. A residual analysis is performed in order to select an appropriate model. The authors would like to thank the editor and referees for their helpful comments. This work was supported by CNPq, Brazil.     

A copula-based Markov chain model for the analysis of binary longitudinal data

A fully parametric first-order autoregressive (AR(1)) model is proposed to analyse binary longitudinal data. By using a discretized version of a copula, the modelling approach allows one to construct separate models for the marginal response and for the dependence between adjacent responses. In particular, the transition model that is focused on discretizes the Gaussian copula in such a way that the marginal is a Bernoulli distribution. A probit link is used to take into account concomitant information in the behaviour of the underlying marginal distribution. Fixed and time-varying covariates can be included in the model. The method is simple and is a natural extension of the AR(1) model for Gaussian series. Since the approach put forward is likelihood-based, it allows interpretations and inferences to be made that are not possible with semi-parametric approaches such as those based on generalized estimating equations. Data from a study designed to reduce the exposure of children to the sun are used to illustrate the methods.     

Forecast mean squared error reductionin the VAR(1) process

When VAR models are used to predict future outcomes, the forecast error can be substantial. Through imposition of restrictions on the off-diagonal elements of the parameter matrix, however, the information in the process may be condensed to the marginal processes. In particular, if the cross-autocorrelations in the system are small and only a small sample is available, then such a restriction may reduce the forecast mean squared error considerably.

In this paper, we propose three different techniques to decide whether to use the restricted or unrestricted model, i.e. the full VAR(1) model or only marginal AR(1) models. In a Monte Carlo simulation study, all three proposed tests have been found to behave quite differently depending on the parameter setting. One of the proposed tests stands out, however, as the preferred one and is shown to outperform other estimators for a wide range of parameter settings.     

Modelling udder infection data using copula models for quadruples

We study copula models for correlated infection times in the four udder quarters of dairy cows. Both a semi-parametric and a nonparametric approach are considered to estimate the marginal survival functions, taking into account the effect of a binary udder quarter level covariate. We use a two-stage estimation approach and we briefly discuss the asymptotic behaviour of the estimators obtained in the first and the second stage of the estimation. A pseudo-likelihood ratio test is used to select an appropriate copula from the power variance copula family that describes the association between the outcomes in a cluster. We propose a new bootstrap algorithm to obtain the p-value for this test. This bootstrap algorithm also provides estimates for the standard errors of the estimated parameters in the copula. The proposed methods are applied to the udder infection data. A small simulation study for a setting similar to the setting of the udder infection data gives evidence that the proposed method provides a valid approach to select an appropriate copula within the power variance copula family.     

A comparison of nonparametric priors in hierarchical mixture modelling for AFT regression

We will pursue a Bayesian nonparametric approach in the hierarchical mixture modelling of lifetime data in two situations: density estimation, when the distribution is a mixture of parametric densities with a nonparametric mixing measure, and accelerated failure time (AFT) regression modelling, when the same type of mixture is used for the distribution of the error term. The Dirichlet process is a popular choice for the mixing measure, yielding a Dirichlet process mixture model for the error; as an alternative, we also allow the mixing measure to be equal to a normalized inverse-Gaussian prior, built from normalized inverse-Gaussian finite dimensional distributions, as recently proposed in the literature. Markov chain Monte Carlo techniques will be used to estimate the predictive distribution of the survival time, along with the posterior distribution of the regression parameters. A comparison between the two models will be carried out on the grounds of their predictive power and their ability to identify the number of components in a given mixture density.     

Copula-based regression models: A survey

In this review paper we collect several results about copula-based models, especially concerning regression models, by focusing on some insurance applications.     

Nonparametric estimation of waiting time distributions in a Markov model based on current status data

We construct nonparametric estimators of state waiting time distribution functions in a Markov multistate model using current status data. This is a particularly difficult problem since neither the entry nor the exit times of a given state are directly observed. These estimators are obtained, using the Markov property, from estimators of counting processes of state entry and exit times, as well as, the size of “at risk” sets of state entry and transitions out of that state. Consistency of our estimators is established. Finite-sample behavior of our estimators is studied by simulation, in which we show that our estimators based on current status data compare well with those based on complete data. We also illustrate our method using a pubertal development data set obtained from the NHANES III [1997. NHANES III Reference Manuals and Reports (CD-ROM). Analytic and Reporting Guidelines: The Third National Health and Nutrition Examination Survey (1988–94). National Center for Health Statistics, Centers for Disease Control and Prevention, Hyattsville, MD] study.     

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.