Nonparametric Comparison for Multivariate Panel Count Data

Multivariate panel count data often occur when there exist several related recurrent events or response variables defined by occurrences of related events. For univariate panel count data, several nonparametric treatment comparison procedures have been developed. However, it does not seem to exist a nonparametric procedure for multivariate cases. Based on differences between estimated mean functions, this article proposes a class of nonparametric test procedures for multivariate panel count data. The asymptotic distribution of the new test statistics is established and a simulation study is conducted. Moreover, the new procedures are applied to a skin cancer problem that motivated this study.     

Bayesian estimation and prediction based on lognormal record values

In this paper we consider the problems of estimation and prediction when observed data from a lognormal distribution are based on lower record values and lower record values with inter-record times. We compute maximum likelihood estimates and asymptotic confidence intervals for model parameters. We also obtain Bayes estimates and the highest posterior density (HPD) intervals using noninformative and informative priors under square error and LINEX loss functions. Furthermore, for the problem of Bayesian prediction under one-sample and two-sample framework, we obtain predictive estimates and the associated predictive equal-tail and HPD intervals. Finally for illustration purpose a real data set is analyzed and simulation study is conducted to compare the methods of estimation and prediction.     

Pseudo-empirical Bayes estimation of small area means under a nested error linear regression model with functional measurement errors

Small area estimation is studied under a nested error linear regression model with area level covariate subject to measurement error. Ghosh and Sinha (2007) obtained a pseudo-Bayes (PB) predictor of a small area mean and a corresponding pseudo-empirical Bayes (PEB) predictor, using the sample means of the observed covariate values to estimate the true covariate values. In this paper, we first derive an efficient PB predictor by using all the available data to estimate true covariate values. We then obtain a corresponding PEB predictor and show that it is asymptotically “optimal”. In addition, we employ a jackknife method to estimate the mean squared prediction error (MSPE) of the PEB predictor. Finally, we report the results of a simulation study on the performance of our PEB predictor and associated jackknife MSPE estimator. Our results show that the proposed PEB predictor can lead to significant gain in efficiency over the previously proposed PEB predictor. Area level models are also studied.     

Partially functional linear varying coefficient model

In this paper, we introduce a new partially functional linear varying coefficient model, where the response is a scalar and some of the covariates are functional. By means of functional principal components analysis and local linear smoothing techniques, we obtain the estimators of coefficient functions of both function-valued variable and real-valued variables. Then the rates of convergence of the proposed estimators and the mean squared prediction error are established under some regularity conditions. Moreover, we develop a hypothesis test for the model and employ the bootstrap procedure to evaluate the null distribution of test statistic and the p-value of the test. At last, we illustrate the finite sample performance of our methods with some simulation studies and a real data application.     

Linear models with an inverse gaussian poisson error distribution

The inverse Gaussian-Poisson (two-parameter Sichel) distribution is useful in fitting overdispersed count data. We consider linear models on the mean of a response variable, where the response is in the form of counts exhibiting extra-Poisson variation, and assume an IGP error distribution. We show how maximum likelihood estimation may be carried out using iterative Newton-Raphson IRLS fitting, where GLIM is used for the IRLS part of the maximization. Approximate likelihood ratio tests are given.     

A modified generalized lasso algorithm to detect local spatial clusters for count data

Detecting local spatial clusters for count data is an important task in spatial epidemiology. Two broad approaches—moving window and disease mapping methods—have been suggested in some of the literature to find clusters. However, the existing methods employ somewhat arbitrarily chosen tuning parameters, and the local clustering results are sensitive to the choices. In this paper, we propose a penalized likelihood method to overcome the limitations of existing local spatial clustering approaches for count data. We start with a Poisson regression model to accommodate any type of covariates, and formulate the clustering problem as a penalized likelihood estimation problem to find change points of intercepts in two-dimensional space. The cost of developing a new algorithm is minimized by modifying an existing least absolute shrinkage and selection operator algorithm. The computational details on the modifications are shown, and the proposed method is illustrated with Seoul tuberculosis data.     

Empirical likelihood test for equality of two distributions using distance of characteristic functions

In this paper, we investigate the problem of testing for the equality of two distributions. We employ a two-sample Jackknife Empirical Likelihood (JEL) approach to construct a test statistic whose limiting distribution is Chi-square distribution with degree of freedom 1, no matter what the data dimension (fixed) is. A variety of synthetic data experiments demonstrate that our JEL test statistic performs very well, with a very neat asymptotic distribution under the null hypothesis. Furthermore, we apply the test procedure to a real dataset to obtain competitive results.     

Construction of Fuzzy Control Charts Based on Weighted Possibilistic Mean

The problem of constructing control charts for fuzzy data has been considered in literature. The proposed transformation approaches and direct fuzzy approaches have their advantages and disadvantages. The representative values charts based on transformation methods are often recommended in practical application. For representing a fuzzy set by a crisp value, the weight of importance of the members assigned with some membership levels in a fuzzy set should be considered, and the possibility theory can be employed to deal with such problem. In this article, we propose to employ the weighted possibilistic mean (WPM), weighted interval valued possibilistic mean (WIVPM) of fuzzy number as a sort of representative values for the fuzzy attribute data, and establish new fuzzy control charts with WPM and WIVPM. The performance of the charts is compared to the existing fuzzy charts with a fuzzy c-chart example via newly defined average number of inspection for variation of control state.     

Bayesian spatial quantile regression for areal count data,with application on substitute care placements in Texas

Quantile regression (QR) allows one to model the effect of covariates across the entire response distribution, rather than only at the mean, but QR methods have been almost exclusively applied to continuous response variables and without considering spatial effects. Of the few studies that have performed QR on count data, none have included random spatial effects, which is an integral facet of the Bayesian spatial QR model for areal counts that we propose. Additionally, we introduce a simplifying alternative to the response variable transformation currently employed in the QR for counts literature. The efficacy of the proposed model is demonstrated via simulation study and on a real data application from the Texas Department of Family and Protective Services (TDFPS). Our model outperforms a comparable non-spatial model in both instances, as evidenced by the deviance information criterion (DIC) and coverage probabilities. With the TDFPS data, we identify one of four covariates, along with the intercept, as having a nonconstant effect across the response distribution.     

Efficient Bayesian inference for COM-Poisson regression models

COM-Poisson regression is an increasingly popular model for count data. Its main advantage is that it permits to model separately the mean and the variance of the counts, thus allowing the same covariate to affect in different ways the average level and the variability of the response variable. A key limiting factor to the use of the COM-Poisson distribution is the calculation of the normalisation constant: its accurate evaluation can be time-consuming and is not always feasible. We circumvent this problem, in the context of estimating a Bayesian COM-Poisson regression, by resorting to the exchange algorithm, an MCMC method applicable to situations where the sampling model (likelihood) can only be computed up to a normalisation constant. The algorithm requires to draw from the sampling model, which in the case of the COM-Poisson distribution can be done efficiently using rejection sampling. We illustrate the method and the benefits of using a Bayesian COM-Poisson regression model, through a simulation and two real-world data sets with different levels of dispersion.     

Nonlinear Logistic Discrimination Via Regularized Gaussian Basis Expansions

We consider the problem of constructing multi-class classification methods for analyzing data with complex structure. A nonlinear logistic discriminant model is introduced based on Gaussian basis functions constructed by the self-organizing map. In order to select adjusted parameters, we employ model selection criteria derived from information-theoretic and Bayesian approaches. Numerical examples are conducted to investigate the performance of the proposed multi-class discriminant procedure. Our modeling procedure is also applied to protein structure recognition in life science. The results indicate the effectiveness of our strategy in terms of prediction accuracy.     

Isotone additive latent variable models

For manifest variables with additive noise and for a given number of latent variables with an assumed distribution, we propose to nonparametrically estimate the association between latent and manifest variables. Our estimation is a two step procedure: first it employs standard factor analysis to estimate the latent variables as theoretical quantiles of the assumed distribution; second, it employs the additive models’ backfitting procedure to estimate the monotone nonlinear associations between latent and manifest variables. The estimated fit may suggest a different latent distribution or point to nonlinear associations. We show on simulated data how, based on mean squared errors, the nonparametric estimation improves on factor analysis. We then employ the new estimator on real data to illustrate its use for exploratory data analysis.     

On selection with restriction

In this paper, we .consider the problem of prediction of an unobserved variable and then selecting a group of individuals which are superior with respect to this variable. It is desired at the same time that for this group, mean values of other unobserved variables or some other function of some variable are above certain prespecified levels. We provide the methods of computing the decision rules for these problems. These problems may be termed as the problems of restricted prediction.     

Estimation of group means in generalized linear mixed models

In this study, we investigate the concept of the mean response for a treatment group mean as well as its estimation and prediction for generalized linear models with a subject‐wise random effect. Generalized linear models are commonly used to analyze categorical data. The model‐based mean for a treatment group usually estimates the response at the mean covariate. However, the mean response for the treatment group for studied population is at least equally important in the context of clinical trials. New methods were proposed to estimate such a mean response in generalized linear models; however, this has only been done when there are no random effects in the model. We suggest that, in a generalized linear mixed model (GLMM), there are at least two possible definitions of a treatment group mean response that can serve as estimation/prediction targets. The estimation of these treatment group means is important for healthcare professionals to be able to understand the absolute benefit vs risk. For both of these treatment group means, we propose a new set of methods that suggests how to estimate/predict both of them in a GLMMs with a univariate subject‐wise random effect. Our methods also suggest an easy way of constructing corresponding confidence and prediction intervals for both possible treatment group means. Simulations show that proposed confidence and prediction intervals provide correct empirical coverage probability under most circumstances. Proposed methods have also been applied to analyze hypoglycemia data from diabetes clinical trials.     

Bayesian density regression for discrete outcomes

We develop Bayesian models for density regression with emphasis on discrete outcomes. The problem of density regression is approached by considering methods for multivariate density estimation of mixed scale variables, and obtaining conditional densities from the multivariate ones. The approach to multivariate mixed scale outcome density estimation that we describe represents discrete variables, either responses or covariates, as discretised versions of continuous latent variables. We present and compare several models for obtaining these thresholds in the challenging context of count data analysis where the response may be over‐ and/or under‐dispersed in some of the regions of the covariate space. We utilise a nonparametric mixture of multivariate Gaussians to model the directly observed and the latent continuous variables. The paper presents a Markov chain Monte Carlo algorithm for posterior sampling, sufficient conditions for weak consistency, and illustrations on density, mean and quantile regression utilising simulated and real datasets.     

A latent variable model for analyzing mixed longitudinal (k,l)-inflated count and ordinal responses

A random effects model for analyzing mixed longitudinal count and ordinal data is presented where the count response is inflated in two points (k and l) and an (k,l)-Inflated Power series distribution is used as its distribution. A full likelihood-based approach is used to obtain maximum likelihood estimates of parameters of the model. For data with non-ignorable missing values models with probit model for missing mechanism are used.The dependence between longitudinal sequences of responses and inflation parameters are investigated using a random effects approach. Also, to investigate the correlation between mixed ordinal and count responses of each individuals at each time, a shared random effect is used. In order to assess the performance of the model, a simulation study is performed for a case that the count response has (k,l)-Inflated Binomial distribution. Performance comparisons of count-ordinal random effect model, Zero-Inflated ordinal random effects model and (k,l)-Inflated ordinal random effects model are also given. The model is applied to a real social data set from the first two waves of the national longitudinal study of adolescent to adult health (Add Health study). In this data set, the joint responses are the number of days in a month that each individual smoked as the count response and the general health condition of each individual as the ordinal response. For the count response there is incidence of excess values of 0 and 30.     

A Semi-Nonparametric Approach to Model Panel Count Data

In count data models, overdispersion of the dependent variable can be incorporated into the model if a heterogeneity term is added into the mean parameter of the Poisson distribution. We use a nonparametric estimation for the heterogeneity density based on a squared Kth-order polynomial expansion, that we generalize for panel data. A numerical illustration using an insurance dataset is discussed. Even if some statistical analyses showed no clear differences between these new models and the standard Poisson with gamma random effects, we show that the choice of the random effects distribution has a significant influence for interpreting our results.     

Bayesian approach to errors-in-variables in count data regression models with departures from normality and overdispersion

In most practical applications, the quality of count data is often compromised due to errors-in-variables (EIVs). In this paper, we apply Bayesian approach to reduce bias in estimating the parameters of count data regression models that have mismeasured independent variables. Furthermore, the exposure model is misspecified with a flexible distribution, hence our approach remains robust against any departures from normality in its true underlying exposure distribution. The proposed method is also useful in realistic situations as the variance of EIVs is estimated instead of assumed as known, in contrast with other methods of correcting bias especially in count data EIVs regression models. We conduct simulation studies on synthetic data sets using Markov chain Monte Carlo simulation techniques to investigate the performance of our approach. Our findings show that the flexible Bayesian approach is able to estimate the values of the true regression parameters consistently and accurately.     

Response prediction in mixed effects models

Although prediction in mixed effects models usually concerns the random effects, in this paper we deal with the problem of prediction of a future, or yet unobserved, response random variable, belonging to a given cluster. In particular, the aim is to define computationally tractable prediction intervals, with conditional and unconditional coverage probability close to the target nominal value. This solution involves the conditional density of the future response random variable given the observed data, or a suitable high-order approximation based on the Laplace method. We prove that, unless the amount of data is very limited, the estimative or naive predictive procedure gives a relatively simple, feasible solution for response prediction. An application to generalized linear mixed models is presented.     

Adaptively transformed mixed‐model prediction of general finite‐population parameters

For estimating area‐specific parameters (quantities) in a finite population, a mixed‐model prediction approach is attractive. However, this approach strongly depends on the normality assumption of the response values, although we often encounter a non‐normal case in practice. In such a case, transforming observations to make them suitable for normality assumption is a useful tool, but the problem of selecting a suitable transformation still remains open. To overcome the difficulty, we here propose a new empirical best predicting method by using a parametric family of transformations to estimate a suitable transformation based on the data. We suggest a simple estimating method for transformation parameters based on the profile likelihood function, which achieves consistency under some conditions on transformation functions. For measuring the variability of point prediction, we construct an empirical Bayes confidence interval of the population parameter of interest. Through simulation studies, we investigate the numerical performance of the proposed methods. Finally, we apply the proposed method to synthetic income data in Spanish provinces in which the resulting estimates indicate that the commonly used log transformation would not be appropriate.