Regression methods are considered for the analysis of correlated binary data when each binary observation may have its own covariates. It is argued that binary response models that condition on some or all binary responses in a given "block" are useful for studying certain types of dependencies, but not for the estimation of marginal response probabilities or pairwise correlations. Fully parametric approaches to these latter problems appear to be unduly complicated except in such special cases as the analysis of paired binary data. Hence, a generalized estimating equation approach is advocated for inference on response probabilities and correlations. Illustrations involving both small and large block sizes are provided.

Correlated binary regression with covariates specific to each binary observation.

This article review methodologies used for analyzing ordered categorical (ordinal) response variables. We begin by surveying models for data with a single ordinal response variable. We also survey recently proposed strategies for modeling ordinal response variables when the data have some type of clustering or when repeated measurement occurs at various occasions for each subject, such as in longitudinal studies. Primary models in that case includemarginal models andcluster-specific (conditional) models for which effects apply conditionally at the cluster level. Related discussion refers to multi-level and transitional models. The main emphasis is on maximum likelihood inference, although we indicate certain models (e.g., marginal models, multi-level models) for which this can be computationally difficult. The Bayesian approach has also received considerable attention for categorical data in the past decade, and we survey recent Bayesian approaches to modeling ordinal response variables. Alternative, non-model-based, approaches are also available for certain types of inference.

The analysis of ordered categorical data: An overview and a survey of recent developments

1. Introduction 2. Binary regression: the logit model 3. Generalized linear models 4. Modeling of binary data 5. Alternative binary regression models 6. Regularization and variable selection for parametric models 7. Regression analysis of count data 8. Multinomial response models 9. Ordinal response models 10. Semi- and nonparametric generalized regression 11. Tree-based methods 12. The analysis of contingency tables: log-linear and graphical models 13. Multivariate response models 14. Random effects models 15. Prediction and classification Appendix A. Distributions Appendix B. Some basic tools Appendix C. Constrained estimation Appendix D. Kullback-Leibler distance and information-based criteria of model fit Appendix E. Numerical integration and tools for random effects modeling.

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Regression for Categorical Data

Within-cluster resampling is proposed as a new method for analysing clustered data. Although the focus of this paper is clustered binary data, the within-cluster resampling asymptotic theory is general for many types of clustered data. Within-cluster resampling is a simple but computationally intensive estimation method. Its main advantage over other marginal analysis methods, such as generalised estimating equations (Liang & Zeger, 1986; Zeger & Liang, 1986) is that it remains valid when the risk for the outcome of interest is related to the cluster size, which we term nonignorable cluster size. We present theory for the asymptotic normality and provide a consistent variance estimator for the within-cluster resampling estimator. Simulations and an example are developed that assess the finite-sample behaviour of the new method and show that when both methods are valid its performance is similar to that of generalised estimating equations.

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Within‐cluster resampling

Dependent binary response data arise frequently in practice due to repeated measurements in longitudinal studies or to subsampling primary sampling units as in fields such as teratology and ophthalmology. Several classes of approaches have recently been proposed to analyse such repeated binary outcome data. The different classes of approaches measure different effects of covariates on binary responses and address different statistical questions. This article compares the different classes of approaches in terms of parameter interpretation and magnitude, standard errors of model parameters and Wald tests for covariate effects. The results help to clarify the substantive questions which data analysts can address with each approach, as well as why the covariate effects measured by different approaches may be different. Finally, I will provide guidelines to the advantages and disadvantages of alternative approaches for analysing dependent binary responses. Simulations and example data illustrate these findings.

Statistical methods for longitudinal and clustered designs with binary responses

This paper proposes a generalized logistic regression model that can account for the correlation among responses on subunits. The subunits may arise as data on multiple observations within an individual. This method generalizes earlier work by Rosner (1984 a,b) and others. Methodological generalizations include: (1) the use of the more general Polya-Eggenberger distribution instead of the beta-binomial distribution to model the correlation structure, so that cases with negative, positive, or zero intraclass correlation can be handled; (2) a stepwise approach; (3) linear and non-linear regression; and, (4) the inclusion of the case of a truncated distribution. The model can accommodate missing data and covariates on the unit and subunit level. The derivative-free simplex algorithm is used to estimate the parameters. The model is applied to data describing the progression of obstruction in coronary disease where multiple arterial segments are studied for each patient. The correlation in response that may ex...

A generalized model of logistic regression for clustered data

We propose a Bayesian hierarchical model to analyze clustered ordinal data. We describe the method in the context of a clinical trial for examining the efficacy of tretinoin emollient cream in the treatment of photo-aged skin. For each patient there were five ordinal responses (each with 4 ordinal categories) from two locations, four from arms and one from face. Thus the data were nested with two levels: patient level and location level. The Bayesian hierarchical model gives estimates for location-specific treatment effect and credible intervals.

Yinsheng Qu

Papers

A generalized model of logistic regression for clustered data

Analysis of clustered ordinal data with subclusters via a Bayesian hierarchical model