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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 estimation of the underlying regression effects involved in the model, whereas the existing ‘working’ correlations based GEE (generalized
estimating equations) approach has serious theoretical limitations both for consistent and efficient estimation, and the existing random effects based correlations approach is not suitable to model the longitudinal correlations. The book has exploited the random effects carefully only to model the correlations of the familial data. Subsequently, this book has modelled the correlations of the longitudinal data collected from the members of a large number of independent families by using the class of auto-correlation structures conditional on the random effects. The book also provides models and inferences for discrete longitudinal data in the adaptive clinical trial set up.
The book is mathematically rigorous and provides details for the development of estimation approaches under selected familial and longitudinal models. Further, while the book provides special cares for mathematics behind the correlation models, it also presents the
illustrations of the statistical analysis of various real life data.
This book will be of interest to the researchers including graduate students in biostatistics and econometrics, among other applied statistics research areas.
Brajendra Sutradhar is a University Research Professor at Memorial University in St. John’s, Canada. He is an elected member of the International Statistical Institute and a fellow of the American Statistical Association. He has published about 110 papers in statistics journals in the area of multivariate analysis, time series analysis including forecasting, sampling, survival analysis for correlated failure times, robust inferences in generalized linear mixed models with outliers, and generalized linear longitudinal mixed models with bio-statistical and econometric applications. He has served as an associate editor for six years for Canadian Journal of Statistics and for four years for the Journal of Environmental and Ecological Statistics. He has served for 3 years as a member of the advisory committee on statistical methods in Statistics Canada. Professor Sutradhar was awarded 2007 distinguished service award of Statistics Society of Canada for his many years of services to the
society including his special services for society’s annual meetings.

Inhaltsverzeichnis

Frontmatter

Chapter 1. Introduction

Discrete data analysis such as count or binary clustered data analysis has been an important research topic over the last three decades. In general, two types of clusters are frequently encountered. First, a cluster may be formed with the responses along with associated covariates from the members of a group/family. These clustered responses are supposed to be correlated as the members of the cluster share a common random group/family effect. In this book, we refer to this type of correlation among the responses of members of same family as the familial correlation. Second, a cluster may be formed with the repeated responses along with associated covariates collected from an individual. These repeated responses from the same individual are also supposed to be correlated as there may be a dynamic relationship between the present and past responses. In this book, we refer to these correlations among the repeated responses collected from the same individual as the longitudinal correlations. It is of interest to fit a suitable parametric or semi-parametric familial and/or longitudinal correlation model primarily to analyze the means and variances of the data. Note that the familial and longitudinal correlations, however, play an important role in a respective setup to analyze the means and variances of the data efficiently.
Brajendra C. Sutradhar

Chapter 2. Overview of Linear Fixed Models for Longitudinal Data

In a longitudinal setup, a small number of repeated responses along with certain multidimensional covariates are collected from a large number of independent individuals. Let y yi1, …,y it , …,y iT i be T i ≥ 2 repeated responses collected from the ith individual, for i = 1, …, K, where K → ∞. Furthermore, let x it = (x it 1 , …,x it p ) be the p-dimensional covariate vector corresponding to y it , and β denote the effects of the components of xit it on y it . For example, in a biomedical study, to examine the effects of two treatments and other possible covariates on blood pressure, the physician may collect blood pressure for T i = T = 10 times from K = 200 independent subjects.
Brajendra C. Sutradhar

Chapter 3. Overview of Linear Mixed Models for Longitudinal Data

Recall from the last chapter [eqn. (2.48)] that there exists [Verbeke and Molenberghs (2000, Chapter 3, eqn. (3.11)); Diggle, Liang, and Zeger (1994)] a random effects based longitudinal mixed model given by
$$ y_{it} = x^{\prime}_{it}\beta + z_{i}\gamma_{i} + \varepsilon_{it}, $$
(3.1)
where the ε it are independent errors for all t =1, …, T i for the ith (i=1, …, K) individual.
Brajendra C. Sutradhar

Chapter 4. Familial Models for Count Data

Familial models for count data are also known as Poisson mixed models for count data. In this setup, count responses along with a set of multidimensional covariates are collected from the members of a large number of independent families. Let y ij denote the count response for the jth (j =1, …,n i ) member on the ith (i=1, …, K) family/cluster. Also, let x ij = (x ij1, …,x ijp )′ denote the p covariates associated with the count response y ij . For example, in a health economics study, a state government may be interested to know the effects of certain socioeconomic and epidemiological covariates such as gender, education level, and age on the number of visits by a family member to the house physician in a particular year. Note that in this problem it is also likely that the count responses of the members of a family are influenced by a common random family effect, say γ i. This makes the count responses of any two members of the same family correlated, and this correlation is usually referred to as the familial correlation. It is of scientific interest to find the effects of the covariates on the count responses of an individual member after taking the familial correlations into account.
Brajendra C. Sutradhar

Chapter 5. Familial Models for Binary Data

As opposed to Chapter 4, we now consider y ij as the binary response for the jth (j = 1, …,ni) member of the ith (i = 1, …,K) family/cluster. Suppose that x ij = (x ij1 , …,x ijp )′ is the p-dimensional covariate vector associated with the binary response y ij . For example, in a chronic obstructive pulmonary disease (COPD) study, y ij denotes the impaired pulmonary function (IPF) status (yes or no), and x ij is the vector of covariates such as gender, race, age, and smoking status, for the jth sibling of the ith COPD patient. Note that in this problem it is likely that the IPF status for n i siblings of the ith patient may be influenced by an unobservable random effect (γi) due to the ith COPD patient.
Brajendra C. Sutradhar

Chapter 6. Longitudinal Models for Count Data

In longitudinal studies for count data, a small number of repeated count responses along with a set of multidimensional covariates are collected from a large number of independent individuals. For example, in a health care utilization study, the number of visits to a physician by a large number of independent individuals may be recorded annually over a period of several years. Also, the information on the covariates such as gender, number of chronic conditions, education level, and age, may be recorded for each individual.
Brajendra C. Sutradhar

Chapter 7. Longitudinal Models for Binary Data

In Chapter 6, we have discussed the stationary and nonstationary correlation models for count data, and estimated the effects of the covariates on the count responses, by taking the correlation structure into account. In this chapter, we deal with repeated binary responses. For example, there exists a longitudinal study on the health effects of air pollution, where wheezing status (1 = yes, 0 = no) of a large number of independent children are repeatedly recorded, along with maternal smoking status, family cleanliness status, level of chemicals used, and pet-owning status of the family.
Brajendra C. Sutradhar

Chapter 8. Longitudinal Mixed Models for Count Data

Recall that in Chapter 6, a class of correlation models was discussed for the analysis of longitudinal count data collected from a large number of independent individuals, whereas in Chapter 4, we discussed the analysis of count data collected from the members of a large number of independent families. Thus, in Chapter 4, familial correlations among the responses of the members of a given family were assumed to be caused by the influence of the same family effect on the members of the family, whereas in Chapter 6, longitudinal correlations were assumed to be generated through a dynamic relationship among the repeated counts collected from the same individual. A comparison between the models in these two chapters (4 and 6) clearly indicates that modelling the longitudinal correlations for count data through a common individual random effect would be inappropriate. If it is, however, thought that the longitudinal count responses may also be influenced by an invisible random effect due to the individual, this will naturally create a complex correlation structure where repeated responses will satisfy a longitudinal correlation structure but conditional on the individual random effect.
Brajendra C. Sutradhar

Chapter 9. Longitudinal Mixed Models for Binary Data

Recall that various stationary and nonstationary correlated binary fixed models were discussed in Chapter 7. In this chapter, we consider a generalization of some of these fixed models to the mixed model setup by assuming that the repeated binary responses of an individual may also be influenced by the individual’s random effect. Thus, this generalization will be similar to that for the repeated count data subject to the influence of the individual’s random effect that we have discussed in Chapter 8. Note that in this chapter, we concentrate mainly on the nonstationary models, stationary models being the special cases.
Brajendra C. Sutradhar

Chapter 10. Familial Longitudinal Models for Count Data

In Chapter 4, we discussed familial models for count data, where count responses along with a set of covariates are collected from the members of a large number of independent families. In Chapter 6, we discussed longitudinal models for count data, where count responses along with a set of covariates are collected from a large number of independent individuals over a small period of time. In practice there are situations where the count responses and their corresponding covariates are collected in a familial longitudinal setup. In this setup, count responses and the associated covariates are collected from the members of a large number of independent families over a small period of time. For example, in health care utilization data, the number of visits to the physician by the members of a large number of independent families may be recorded over a period of several years. Also the information on the covariates: gender, number of chronic conditions, education level, and age may be recorded for the members of each family.
Brajendra C. Sutradhar

Chapter 11. Familial Longitudinal Models for Binary Data

In the familial longitudinal setup, binary responses along with a set of multidimensional time-dependent covariates are collected from the members of a large number of independent families. For example, in a clinical study, the asthma status of each of the family members of a large number of independent families may be collected every year over a period of four years. Also, the covariates such as gender, age, education level, and life style of the individual member may be collected. In this setup, it is likely that the responses from the members of the same family at a given year will be correlated. This is due to the fact that every member of the family shares certain common family effects which are latent or invisible. Also, the repeated asthma status collected over several years will be longitudinally correlated. It is of interest to take these two types of familial and longitudinal correlations into account and then find the effects of the covariates on the responses.
Brajendra C. Sutradhar

Backmatter

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