Dynamic Mixed Models for Familial Longitudinal Data

This book provides a theoretical foundation for the analysis of discrete data such as count and binary data in the longitudinal setup. Unlike the existing books, this book uses a class of auto-correlation structures to model the longitudinal correlations for the repeated discrete data that accommodates all possible Gaussian type auto-correlation models as special cases including the equi-correlation models. This new dynamic modelling approach is utilized to develop theoretically sound inference techniques such as the generalized quasi-likelihood (GQL) technique for consistent and efficient es

Longitudinal categorical data analysis

This is the first book in longitudinal categorical data analysis with parametric correlation models developed based on dynamic relationships among repeated categorical responses. This book is a natural generalization of the longitudinal binary data analysis to the multinomial data setup with more than two categories. Thus, unlike the existing books on cross-sectional categorical data analysis using log linear models, this book uses multinomial probability models both in cross-sectional and longitudinal setups. A theoretical foundation is provided for the analysis of univariate multinomial responses, by developing models systematically for the cases with no covariates as well as categorical covariates, both in cross-sectional and longitudinal setups. In the longitudinal setup, both stationary and non-stationary covariates are considered. These models have also been extended to the bivariate multinomial setup along with suitable covariates. For the inferences, the book uses the generalized quasi-likelihood as well as the exact likelihood approaches. The book is technically rigorous, and, it also presents illustrations of the statistical analysis of various real life data involving univariate multinomial responses both in cross-sectional and longitudinal setups. This book is written mainly for the graduate students and researchers in statistics and social sciences, among other applied statistics research areas. However, the rest of the book, specifically the chapters from 1 to 3, may also be used for a senior undergraduate course in statistics. Brajendra Sutradhar is a University Research Professor at Memorial University in St. John's, Canada. He is author of the book Dynamic Mixed Models for Familial Longitudinal Data, published in 2011 by Springer, New York. Also, he edited the special issue of the Canadian Journal of Statistics (2010, Vol. 38, June Issue, John Wiley) and the Lecture Notes in Statistics (2013, Vol. 211, Springer), with selected papers from two symposiums: ISS-2009 and ISS-2012, respectively

A multivariate approach for estimating the random effects variance component in one-way random effects model

It is well known that the ANOVA estimator of the random effects variance component in one-way random effects model can assume negative values. It is also well known that nonnegative quadratic unbiased estimators do not exist for estimating the random effects variance component (LaMotte, 1973). LaMotte (1985) indicated the possibility that nonnegative invariant quadratic estimator of the random effects variance component uniformly better than the ANOVA estimator may exist for the balanced one-way random effects model. Mathew et al. (1992a) have shown that such estimator exists only when the number of treatments is 9 or less. As noted in Herbach (1959) (see also Thompson, 1962), a simple truncation of the ANOVA estimator at zero yields uniform improvement over the ANOVA estimator. The estimators suggested by Herbach and Thompson are, in fact, restricted maximum likelihood estimators, and they are nonquadratic by nature. In this paper, we discuss a multivariate technique which always yields positive estimate of the random effects variance component in one-way random effects model. The multivariate approach exploits the estimates of the eigenvalues of the covariance matrix of the model in estimating the variance components including the error variance. The resulting estimates are nonquadratic. The success of this multivariate approach depends on the precise estimation of the eigenvalues. Since there does not exist any unbiased estimation procedure in the small sample case for the estimation of the eigenvalues, we use a delete-d jackknife procedure to estimate them. This delete-d jackknife based multivariate approach yields better estimates (in terms of mean squared error) for the random effects variance component than the restricted maximum likelihood estimation as well as Chow and Shao's (1988) nonquadratic estimation approaches, which is shown through a simulation study for the cases with number of treatments up to 20.Covariance matrix of the model Eigenvalues Positive estimates of variance components Delete jackknife Restricted maximum likelihood estimates Monte-Carlo experiment

Semi-parametric Dynamic Models for Longitudinal Ordinal Categorical Data

The over all regression function in a semi-parametric model involves a partly specified regression function in some primary covariates and a non-parametric function in some other secondary covariates. This type of semi-parametric models in a longitudinal setup has recently been discussed extensively both for repeated Poisson and negative binomial count data. However, when it is appropriate to interpret the longitudinal binary responses through a binary dynamic logits model, the inferences for semi-parametric Poisson and negative binomial models cannot be applied to such binary models as these models unlike the count data models produce recursive means and variances containing the dynamic dependence or correlation parameters. In this paper, we consider a general multinomial dynamic logits model in a semi-parametric setup first to analyze nominal categorical data in a semi-parametric longitudinal setup, and then modify this model to analyze ordinal categorical data. The ordinal responses are fitted by using a cumulative semi-parametric multinomial dynamic logits model. For the benefits of practitioners, a step by step estimation approach is developed for the non-parametric function, and for both regression and dynamic dependence parameters. In summary, a kernel-based semi-parametric weighted likelihood approach is used for the estimation of the non-parametric function. This weighted likelihood estimate for the non-parametric function is shown to be consistent. The regression and dynamic dependence parameters of the model are estimated by maximizing an approximate semi-parametric likelihood function for the parameters, which is constructed by replacing the non-parametric function with its consistent estimate. Asymptotic properties including the proofs for the consistency of the likelihood estimators of the regression and dynamic dependence parameters are discussed

A parameter dimension-split based asymptotic regression estimation theory for a multinomial panel data model

In this paper we revisit the so-called non-stationary regression models for repeated categorical/multinomial data collected from a large number of independent individuals. The main objective of the study is to obtain consistent and efficient regression estimates after taking the correlations of the repeated multinomial data into account. The existing (1) ‘working’ odds ratios based GEE (generalized estimating equations) approach has both consistency and efficiency drawbacks. Specifically, the GEE-based regression estimates can be inconsistent which is a serious limitation. Some other existing studies use a MDL (multinomial dynamic logits) model among the repeated responses. As far as the estimation of the regression effects and dynamic dependence (i.e., correlation) parameters is concerned, they use either (2) a marginal or (3) a joint likelihood approach. In the marginal approach, the regression parameters are estimated for known correlation parameters by solving their respective marginal likelihood estimating equations, and similarly the correlation parameters are estimated by solving their likelihood equations for known regression estimates. Thus, this marginal approach is an iterative approach which may not provide quick convergence. In the joint likelihood approach, the regression and correlation parameters are estimated simultaneously by searching the maximum value of the likelihood function with regard to these parameters together. This approach may encounter computational drawback, specially when the number of correlation parameters gets large. In this paper, we propose a new estimation approach where the likelihood function for the regression parameters is developed from the joint likelihood function by replacing the correlation parameter with a consistent estimator involving unknown regression parameters. Thus the new approach relaxes the dimension issue, that is, the dimension of the correlation parameters does not affect the estimation of the main regression parameters. The asymptotic properties of the estimates of the main regression parameters (obtained based on consistent estimating functions for correlation parameters) are studied in detail

An Overview on Econometric Models for Linear Spatial Panel Data

When spatial data are repeatedly collected from the same spatial locations over a short period of time, a spatial panel/longitudinal data set is generated. Thus, this type of spatial longitudinal data must exhibit both spatial and longitudinal correlations, which are not easy to model. This work is motivated by existing studies in statistics and econometrics literature but the proposed model and inference procedures should be applicable to the spatial panel data encountered in other fields as well such as environmental and/or ecological setups. Specifically, unlike the existing studies, we propose a new dynamic mixed model to accommodate both spatial and panel correlations. A complete theoretical analysis is given for the estimation of regression effects, and spatial and panel correlations by exploiting second and higher order moments based quasi-likelihood methods. Asymptotic properties are also studied in details. The step by step estimation results developed in the paper should be useful to the practitioners dealing with spatial panel data

On familial longitudinal Poisson mixed models with gamma random effects

Poisson mixed models are used to analyze a wide variety of cluster count data. These models are commonly developed based on the assumption that the random effects have either the log-normal or the gamma distribution. Obtaining consistent as well as efficient estimates for the parameters involved in such Poisson mixed models has, however, proven to be difficult. Further problem gets mounted when the data are collected repeatedly from the individuals of the same cluster or family. In this paper, we introduce a generalized quasilikelihood approach to analyze the repeated familial data based on the familial structure caused by gamma random effects. This approach provides estimates of the regression parameters and the variance component of the random effects after taking the longitudinal correlations of the data into account. The estimators are consistent as well as highly efficient.Multi-dimensional count data Mixed effects Quasilikelihood Efficiency Repeated count responses Regression effects Variance component of the random effects Longitudinal correlations

A higher-order approximation to likelihood inference in the Poisson mixed model

Sutradhar and Qu (Canad. J. Statist. 26 (1998) 169) have introduced a small variance component (for random effects) based likelihood approximation (LA) approach to estimate the parameters of the Poisson mixed models, and have shown that their LA approach performs better compared to other leading approaches. This paper further improves the LA of Sutradhar and Qu (1998) to accommodate larger values of the variance component, and provides the improved LA (ILA) based estimators for the regression parameters as well as the variance component of the random effects of the model. The results of a simulation study show that the ILA approach leads to significant improvement over the LA approach in estimating the parameters of the model, the variance component of the random effects in particular.Count data Fixed effects Overdispersion Likelihood approximations Consistent estimates Asymptotic distribution