Bootstrapping latent variable models for binary response

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.

Binary responses are often present in medical studies. When the dependent variable Y represents a rare event, the logistic regression model shows relevant drawbacks. To overcome these drawbacks, we propose the quantile function of the generalized extreme value regression distribution as a link function and focus our attention on values close to one. One problem arising in the presence of cure fraction is that, it is usually unknown who are the cured and the susceptible subjects, unless the outcome of interest has been observed. In these settings, a logistic regression analysis is no more straightforward. We develop a maximum likelihood estimation procedure, based on the joint modeling of the binary response of interest and the cure status. We investigate the identifiability of the resulting model and establish the asymptotic properties. We conduct a simulation study to investigate its finite-sample behaviour, and application to real data.

Conventional approaches for handling missingness in substance use disorder trials commonly rely upon a single deterministic "worst value" imputation that posits a perfect relationship between missingness and drug use ("missing value = presumed drug use"); this yields biased estimates of treatment effects and their standard errors. Instead, deterministic imputations should be replaced by probabilistic versions that encode researchers prior beliefs that those with missing data are more likely to be using drugs at those occasions. Motivated by this problem, we present a method for handling non-monotone missing binary data in longitudinal studies. Specifically, we consider a joint model that combines a not missing at random (NMAR) selection model with a generalized linear mixed model for longitudinal binary data. The selection model links the distribution of a missing outcome to the corresponding distribution of the outcome for those observed at that occasion via a fixed and known sensitivity parameter. The mixed model for longitudinal binary data assumes the random effects have bridge distributions; the latter yields regression parameters that have both subject-specific and marginal interpretations. This approach is completely transparent about what is being assumed about missing data and can be used as the basis for sensitivity analysis.

The paper proposes a latent variable model for binary data coming from an unobserved heterogeneous population. The heterogeneity is taken into account by replacing the traditional assumption of Gaussian distributed factors by a finite mixture of multivariate Gaussians. The aim of the proposed model is twofold: it allows to achieve dimension reduction when the data are dichotomous and, simultaneously, it performs model based clustering in the latent space. Model estimation is obtained by means of a maximum likelihood method via a generalized version of the EM algorithm. In order to evaluate the performance of the model a simulation study and two real applications are illustrated.

Maximum likelihood estimation of an extended latent Markov model for clustered binary panel data

A pedigree model for binary data, motivated by log-linear modelling, has been developed to examine evidence for familial aggregation in disease status. From an epidemiological point of view a convenient way to express disease concordance between a pair of relatives is in terms of the odds ratio. For a rare disease this is almost equivalent to the relative risk of one family member being affected given that the other is affected, and in extending this to pedigrees it is assumed that these relative risks are multiplicative. In applying the model to the breast cancer data, pedigrees on a rare disease ascertained through an affected proband, it has been shown that estimation of concordance is dependent critically on knowing the probability that a sampled individual is affected. Therefore known population estimates of prevalence or cumulative risk, and an appropriate ascertainment correction, need to be invoked for the model to give proper estimates of disease concordance. The model is flexible in that measured ancillary risk factors, including genetic marker information, can be incorporated into the analysis. Therefore in future studies this information should be collected on all individuals, not just those affected. Suggested statistics for examining a fitted model are presented.

In this study, we have considered the modeling and analyses of binary data. We modeled binary data with categorical predictors, using logistic regression to develop a statistical method. We found that ANOVA-type analyses often performed unsatisfactory, even when using arcsine-square-root trans-formations. We concluded that such methods are not appropriate, especially in cases where the fractions were close to 0 or 1. The logistic transformation of fraction data could be a promising alternative, but it is not desirable in the statistical sense. The major purpose of this paper is to demonstrate that logistic regression with an ANOVA-model like parameterization aids our understanding and provides a somewhat different, but sound, statistical background. We examined a simple real world example to show that we can efficiently test the significance of regression parameters, look for interactions, estimate related confidence intervals, and calculate the difference between the mean values of the referent and experimental subgroups. This paper demonstrates that precise confidence interval estimates can be obtained using the proposed ANOVA-model like approach. The method discussed here can be extended to any type of experimental fraction data analysis, particularly for experimental design.

This study proposes a new model in the family zero-inflated (ZI) binary models, it is called as a ZI Probit Bernoulli (ZIPBer) model. This model can be performed to model or simulate binary data that has an excessive amount of zero counts. At the framework of this research, we first introduce the classical formula, estimating equation (EE) and related functions for the ZIPBer model. Next, the results of asymptotic inferences for the ZIPBer model is studied based on the widespread regularity conditions. For the numerical illustrations, many simulation studies and a practical data set are executed in this paper. Specifically, we perform the fishing data set (FDS) to check the working effectiveness and robustness of the maximum likelihood estimation (MLE) approach in estimating parameters for the ZIPBer model. We also illustrate that this actual data set is more suitable for the proposed model: ZIPBer than other binary models such as logit, probit and ZIBer. The achieved results in this study are very unanimous with reality. It also illustrates that this is a dependable verification for us to know the solution as well as how to proceed to how to catch or take the most fish. This is the contribution of this work in terms of application. Finally, some crucial discussions and conclusions are given in this paper.

Missing data are frequently encountered in longitudinal clinical trials. To better monitor and understand the progress over time, one must handle the missing data appropriately and examine whether the missing data mechanism is ignorable or nonignorable. In this article, we develop a new probit model for longitudinal binary response data. It resolves a challenging issue for estimating the variance of the random effects, and substantially improves the convergence and mixing of the Gibbs sampling algorithm. We show that when improper uniform priors are specified for the regression coefficients of the joint multinomial model via a sequence of one-dimensional conditional distributions for the missing data indicators under nonignorable missingness, the joint posterior distribution is improper. A variation of Jeffreys prior is thus established as a remedy for the improper posterior distribution. In addition, an efficient Gibbs sampling algorithm is developed using a collapsing technique. Two model assessment criteria, the deviance information criterion (DIC) and the logarithm of the pseudomarginal likelihood (LPML), are used to guide the choices of prior specifications and to compare the models under different missing data mechanisms. We report on extensive simulations conducted to investigate the empirical performance of the proposed methods. The proposed methodology is further illustrated using data from an HIV prevention clinical trial.

In this chapter we consider dependent data but move from the linear models of Chap.8 to general regression models. As in Chap.6, we consider generalized linear models (GLMs) and, more briefly, nonlinear models. We first give an outline of this chapter. In Sect.9.2 we describe three motivating datasets to which we return throughout the chapter. The GLMs discussed in Sect.6.3 can be extended to incorporate dependences in observations on the same unit; as with the linear model, an obvious way to carry out modeling in this case is to introduce unit-specific random effects. Within a GLM a natural approach is for these random effects to be included on the linear predictor scale. The resultant conditional models are known as generalized linear mixed models (GLMMs), and these are introduced in Sect.9.3. In Sects.9.4 and9.5 we describe likelihood and conditional likelihood methods of estimation, with Sect.9.6 devoted to a Bayesian treatment. Section9.7 illustrates some of the flexibility of GLMMs by describing and applying a particular model for spatial dependence. An alternative random effects specification, based on conjugacy, is described in Sect.9.8. An important approach to the modeling and analysis of dependent data that is philosophically different from the random effects formulation is via marginal models and generalized estimating equations (GEE), and these are the subject of Sect.9.9. In Sect.9.10, a second GEE approach is described in which the estimating equations for the mean are supplemented with a second set for the variances/covariances. For GLMMs, extra care must be taken with parameter interpretation, and Sect.9.11 discusses this issue, emphasizing how interpretation differs between conditional and marginal models. In PartII of the book, which focused on independent data, Chap.7 was devoted to models for binary data. For dependent data, models binary data are less well developed, and so we do not devote a complete chapter to their description. However, Sect.9.12 introduces the modeling of dependent binary data, and, subsequently, Sects.9.13 and 9.14 describe conditional (mixed) and marginal models for binary data. Section 9.15 considers how nonlinear models, as defined in Sect.6.10, can be extended to the dependent data case. For such models, many applications concentrate on inference for units, and so the introduction of random effects is again suggested. We refer to the resultant class of models as nonlinear mixed models (NLMMs). Section9.16 considers issues related to the parameterization of the nonlinear model. Inference for nonlinear mixed models via likelihood and Bayes approaches is covered in Sects.9.17 and9.18, while GEE is briefly considered in Sect.9.19. The assessment of assumptions for general regression models is described in Sect.9.20, with concluding comments contained in Sect.9.21. Additional references appear in Sect.9.22.

Recently Gupta et al. (Involve J Math, 2013) introduced optional unrelated question randomized response models for both the binary response and quantitative response to sensitive survey questions. Asymptotic normality was established for the mean estimator of the sensitive variable and for the prevalence estimator of the sensitive characteristic. Asymptotic normality was also established for the sensitivity level estimator in each case using first order approximation. These mathematical results were validated using computer simulations. In this paper, the binary and quantitative response models are utilized in surveys of sensitive behaviors to verify that these results hold true in fieldwork applications. The two sensitive questions of interest in the survey are “Have you ever been told by a healthcare professional that you have a sexually transmitted disease?” and “How many sexual partners have you had in the last 12 months?” The target population was undergraduate students enrolled at UNC Greensboro during the 2012–2013 academic year. Subjects were asked these questions by optional unrelated question RRT, check-box survey method, and by direct face-to-face interview. The results of these three methods are compared to each other, as well as to existing published information on these two sensitive behaviors. Estimates provided by the optional unrelated question randomized response models are in line with the mathematical results in Gupta et al. (Involve J Math, 2013). This study also provides the first estimate of sensitivity level through a fieldwork survey.

This study focuses on relational data obtained through object relations. Traditional analysis of relational data often ignores attribute information. Therefore, Mikawa et al. proposed a method to estimate the latent structure of continuous relational data using a generative model and parameter estimation. However, real-world relational data can be discrete, and therefore, we propose a new model for binary relational data using a generative model based on the Bernoulli distribution and the Monte Carlo Expectation-Maximization (EM) algorithm for parameter estimation. We also clarify the effectiveness of the proposed model through simulation experiments using artificial data and real data.

We develop a new class of spatial voting models for binary preference data that can accommodate both monotonic and non-monotonic response functions, and are more flexible than alternative “unfolding” models previously introduced in the literature. We then use these models to estimate revealed preferences for legislators in the U.S. House of Representatives and justices on the U.S. Supreme Court. The results from these applications indicate that the new models provide superior complexity-adjusted performance to various alternatives and that the additional flexibility leads to preferences’ estimates that more closely match the perceived ideological positions of legislators and justices.

This book on multivariate models, statistical inference, and data analysis contains deep coverage of multivariate non-normal distributions for modeling of binary, count, ordinal, and extreme value response data. It is virtually self-contained, and includes many exercises and unsolved problems.

In this work, we aim to introduce a new model belonging to the family of zero-inflated (ZI) models and it is briefly called as a ZI double probit (ZIDP) model. Note that, this model can be implemented to regression analysis and simulation for binary data sets in which there are too numerous zeros in the output variable. First, we introduce how to build the ZIDP model, and then we offer general formula, related and necessary equations of this model. The large-sample theory, numerous simulations and an actual data set are conducted. Specifically, we employ the count of infected blood cells (IBCs) data set to examine the working performance and stability of the maximum likelihood estimation (MLE) approach in evaluating parameters for the ZIDP model. We also show that the ZIDP model fits the IBC data better than the other models considered in this work. The obtained results indicate that smoking status is unlikely to affect the count of IBC, nevertheless gender less or more affects the count of IBC. The final part presents some conclusions.