Bivariate Binary Data Research Articles

TG‐compound optimal design for bivariate binary responses modelled via Copulas

AbstractUnivariate data has been the focus of optimal design for generalized linear models. It is common practice to conduct experiments with multiple dependent responses that are modelled using regression techniques. To design the experiment for all these responses, one needs a multivariate distribution supporting a previously chosen data model. The T‐optimal criterion is used to discriminate between two competing models. The designs that are optimal for discrimination between two models may be unsuitable for describe the quality of predictions for test data. G‐optimal criterion overcomes this restriction by minimizing the maximum variance of the predicted values. The design of experiments with dependent bivariate binary data is a topic we discuss in this paper. For dependent bivariate binary data, a technique for deriving TG‐optimal experimental designs using Copula is shown. The new compound TG‐criterion satisfy the dual aim to selecting the true model and minimizing the maximum prediction variance. The equivalence theorem is demonstrated and proved for the new compound criterion. A numerical example is presented to show the properties of the new compound criterion for bivariate binary setting and its value in meeting the dual aims.

Modelling Correlated Bivariate Binary Data: A Comparative View

Modelling Correlated Bivariate Binary Data: A Comparative View

Binary segmentation procedures using the bivariate binomial distribution for detecting streakiness in sports data

Streakiness is an important measure in many sports data for individual players or teams in which the success rate is not a constant over time. That is, there are many successes/failures during some periods and few or no successes/failures during other periods. In this paper we propose a Bayesian binary segmentation procedure using a bivariate binomial distribution to locate the changepoints and estimate the associated success rates. The proposed method consists of a series of nested hypothesis tests based on the Bayes factors or posterior probabilities. At each stage, we compare three different changepoint models to the constant success rate model using the bivariate binary data. The proposed method is applied to analyze real sports datasets on baseball and basketball players as illustration. Extensive simulation studies are performed to demonstrate the usefulness of the proposed methodologies.

Predictive ability of covariate-dependent Markov models and classification tree for analyzing rainfall data in Bangladesh

This study attempts to make comparison between different parametric regressive models for the bivariate binary data with a machine learning technique. The data on sequential occurrence of rainfall in consecutive days is considered. The outcomes are classified as rainfall in both days, rainfall in one of the consecutive days, and no rainfall in both days. The occurrence of rainfall in consecutive days is analyzed by using statistical models with covariate dependence and classification tree for the period from 1980 to 2014. We have used relative humidity, minimum temperature, maximum temperature, sea level pressure, sunshine hour, and cloud cover in the model as covariates. The binary outcome variable is defined as the occurrence or non-occurrence of rainfall. Five regions of Bangladesh are considered in this study and one station from each region is selected on the basis of two criteria: (i) contains fewer missing values and (ii) representative of the regional characteristics geographically. Several measures are used to compare the models based on Markov chain and classification tree. It is found that for yearly data, both the Markov model and classification tree performed satisfactorily. However, the seasonal data show variation of rainfall. In some seasons, both models perform equally good such as monsoon, pre-monsoon, and post-monsoon, but in the winter season, the Markov model works poorly whereas classification tree fails to work. Additionally, we also observe that the Markov model performed consistently for each season and performs better compared with the classification tree. It has been demonstrated that the covariate-dependent Markov models can be used as classifiers alternative to the classification tree. It is revealed that the predictive ability of the covariate-dependent Markov model based on Markovian assumption performs either better or equally good compared with the classification tree. The joint models also consistently showed better predictive performance compared with regressive model for whole year data as well as for several seasonal data.

Measures for concordance and discordance with applications in disease control and prevention

Bivariate binary response data appear in many applications. Interest goes most often to a parameterization of the joint probabilities in terms of the marginal success probabilities in combination with a measure for association, most often being the odds ratio. Using, for example, the bivariate Dale model, these parameters can be modelled as function of covariates. But the odds ratio and other measures for association are not always measuring the (joint) characteristic of interest. Agreement, concordance, and synchrony are in general facets of the joint distribution distinct from association, and the odds ratio as in the bivariate Dale model can be replaced by such an alternative measure. Here, we focus on the so-called conditional synchrony measure. But, as indicated by several authors, such a switch of parameter might lead to a parameterization that does not always lead to a permissible joint bivariate distribution. In this contribution, we propose a new parameterization in which the marginal success probabilities are replaced by other conditional probabilities as well. The new parameters, one homogeneity parameter and two synchrony/discordance parameters, guarantee that the joint distribution is always permissible. Moreover, having a very natural interpretation, they are of interest on their own. The applicability and interpretation of the new parameterization is shown for three interesting settings: quantifying HIV serodiscordance among couples in Mozambique, concordance in the infection status of two related viruses, and the diagnostic performance of an index test in the field of major depression disorders.

A joint regression modeling framework for analyzing bivariate binary data in R

Abstract We discuss some of the features of the R add-on package GJRM which implements a flexible joint modeling framework for fitting a number of multivariate response regression models under various sampling schemes. In particular,we focus on the case inwhich the user wishes to fit bivariate binary regression models in the presence of several forms of selection bias. The framework allows for Gaussian and non-Gaussian dependencies through the use of copulae, and for the association and mean parameters to depend on flexible functions of covariates. We describe some of the methodological details underpinning the bivariate binary models implemented in the package and illustrate them by fitting interpretable models of different complexity on three data-sets.

Properties of Ideal Point Classification Models for Bivariate Binary Data.

The ideal point classification (IPC) model was originally proposed for analysing multinomial data in the presence of predictors. In this paper, we studied properties of the IPC model for analysing bivariate binary data with a specific focus on three quantities: (1) the marginal probabilities; (2) the association structure between the two binary responses; and (3) the joint probabilities. We found that the IPC model with a specific class point configuration represents either the marginal probabilities or the association structure. However, the IPC model is not able to represent both quantities at the same time. We then derived a new parametrization of the model, the bivariate IPC (BIPC) model, which is able to represent both the marginal probabilities and the association structure. Like the standard IPC model, the results of the BIPC model can be displayed in a biplot, from which the effects of predictors on the binary responses and on their association can be read. We will illustrate our findings with a psychological example relating personality traits to depression and anxiety disorders.

A marginalized multilevel model for bivariate longitudinal binary data

This study considers analysis of bivariate longitudinal binary data. We propose a model based on marginalized multilevel model framework. The proposed model consists of two levels such that the first level associates the marginal mean of responses with covariates through a logistic regression model and the second level includes subject/time specific random intercepts within a probit regression model. The covariance matrix of multiple correlated time-specific random intercepts for each subject is assumed to represent the within-subject association. The subject-specific random effects covariance matrix is further decomposed into its dependence and variance components through modified Cholesky decomposition method and then the unconstrained version of resulting parameters are modelled in terms of covariates with low-dimensional regression parameters. This provides better explanations related to dependence and variance parameters and a reduction in the number of parameters to be estimated in random effects covariance matrix to avoid possible identifiability problems. Marginal correlations between responses of subjects and within the responses of a subject are derived through a Taylor series-based approximation. Data cloning computational algorithm is used to compute the maximum likelihood estimates and standard errors of the parameters in the proposed model. The validity of the proposed model is assessed through a Monte Carlo simulation study, and results are observed to be at acceptable level. Lastly, the proposed model is illustrated through Mother’s Stress and Children’s Morbidity study data, where both population-averaged and subject-specific interpretations are drawn through Emprical Bayes estimation of random effects.

Generating Bivariate Uniform Data with a Full Range of Correlations and Connections to Bivariate Binary Data

In this article, random number generation algorithms for generating bivariate uniform data based on a known class of symmetric bivariate uniform distributions that allow the entire correlation range are given, and its previously unrecognized connection with bivariate binary data is established via matching the cumulative distribution functions.

Analysis of bivariate binary data with possible chances of wrong ascertainment

In familial data, ascertainment correction is often necessary to decipher genetic bases of complex human diseases. This is because families usually are not drawn at random or are not selected according to well-defined rules. While there has been much progress in identifying genes associated with a certain phenotype, little attention has been paid so far for familial studies on exploring common genetic influences on different phenotypes of interest. In this study, we develop a powerful bivariate analytical approach that can be used for a complex situation with paired binary traits. In addition, our model has been framed to accommodate the possibility of imperfect diagnosis as traits may be wrongly observed. Thus, the primary focus is to see whether a particular gene jointly influences both phenotypes. We examine the plausibility of this theory in a sample of families ascertained on the basis of at least one affected individual. We propose a bivariate binary mixed model that provides a novel and flexible way to account for wrong ascertainment in families collected with multiple cases. A hierarchical Bayesian analysis using Markov Chain Monte Carlo (MCMC) method has been carried out to investigate the effect of covariates on the disease status. Results based on simulated data indicate that estimates of the parameters are biased when classification errors and/or ascertainment are ignored.

Bayesian approach to analysing longitudinal bivariate binary data with informative dropout

To study the effect of methadone treatment in reducing multiple drug uses while controlling for their joint dependency and non-random dropout, we propose a bivariate binary model with a separate informative dropout (ID) model. In the model, the logit of the probabilities of each type of drug-use and dropout indicator as well as the log of the odds ratio of both drug-uses are linear in some covariates and outcomes. The model allows the evaluation of the joint probabilities of bivariate outcomes. To account for the heterogeneity of drug use across patients, the model is further extended to incorporate mixture and random effects. Parameter estimation is conducted using a Bayesian approach and is demonstrated using a methadone treatment data. A simulation experiment is conducted to evaluate the effect of including an ID modeling to parameters in the outcome models.

Design of experiments for bivariate binary responses modelled by Copula functions

Optimal design for generalized linear models has primarily focused on univariate data. Often experiments are performed that have multiple dependent responses described by regression type models, and it is of interest and of value to design the experiment for all these responses. This requires a multivariate distribution underlying a pre-chosen model for the data. Here, we consider the design of experiments for bivariate binary data which are dependent. We explore Copula functions which provide a rich and flexible class of structures to derive joint distributions for bivariate binary data. We present methods for deriving optimal experimental designs for dependent bivariate binary data using Copulas, and demonstrate that, by including the dependence between responses in the design process, more efficient parameter estimates are obtained than by the usual practice of simply designing for a single variable only. Further, we investigate the robustness of designs with respect to initial parameter estimates and Copula function, and also show the performance of compound criteria within this bivariate binary setting.

Bayesian Dose‐Finding in Phase I/II Clinical Trials Using Toxicity and Efficacy Odds Ratios

A Bayesian adaptive design is proposed for dose-finding in phase I/II clinical trials to incorporate the bivariate outcomes, toxicity and efficacy, of a new treatment. Without specifying any parametric functional form for the drug dose-response curve, we jointly model the bivariate binary data to account for the correlation between toxicity and efficacy. After observing all the responses of each cohort of patients, the dosage for the next cohort is escalated, deescalated, or unchanged according to the proposed odds ratio criteria constructed from the posterior toxicity and efficacy probabilities. A novel class of prior distributions is proposed through logit transformations which implicitly imposes a monotonic constraint on dose toxicity probabilities and correlates the probabilities of the bivariate outcomes. We conduct simulation studies to evaluate the operating characteristics of the proposed method. Under various scenarios, the new Bayesian design based on the toxicity-efficacy odds ratio trade-offs exhibits good properties and treats most patients at the desirable dose levels. The method is illustrated with a real trial design for a breast medical oncology study.

Bayesian neural networks for bivariate binary data: an application to prostate cancer study

Prostate cancer is one of the most common cancers in American men. The cancer could either be locally confined, or it could spread outside the organ. When locally confined, there are several options for treating and curing this disease. Otherwise, surgery is the only option, and in extreme cases of outside spread, it could very easily recur within a short time even after surgery and subsequent radiation therapy. Hence, it is important to know, based on pre-surgery biopsy results how likely the cancer is organ-confined or not. The paper considers a hierarchical Bayesian neural network approach for posterior prediction probabilities of certain features indicative of non-organ confined prostate cancer. In particular, we find such probabilities for margin positivity (MP) and seminal vesicle (SV) positivity jointly. The available training set consists of bivariate binary outcomes indicating the presence or absence of the two. In addition, we have certain covariates such as prostate specific antigen (PSA), gleason score and the indicator for the cancer to be unilateral or bilateral (i.e. spread on one or both sides) in one data set and gene expression microarrays in another data set. We take a hierarchical Bayesian neural network approach to find the posterior prediction probabilities for a test and validation set, and compare these with the actual outcomes for the first data set. In case of the microarray data we use leave one out cross-validation to access the accuracy of our method. We also demonstrate the superiority of our method to the other competing methods through a simulation study. The Bayesian procedure is implemented by an application of the Markov chain Monte Carlo numerical integration technique. For the problem at hand, our Bayesian bivariate neural network procedure is shown to be superior to the classical neural network, Radford Neal's Bayesian neural network as well as bivariate logistic models to predict jointly the MP and SV in a patient in both the data sets as well as in the simulation study.

A Conditional Autoregressive Gaussian Process for Irregularly Spaced Multivariate Data with Application to Modelling Large Sets of Binary Data

A Gaussian conditional autoregressive (CAR) formulation is presented that permits the modelling of the spatial dependence and the dependence between multivariate random variables at irregularly spaced sites so capturing some of the modelling advantages of the geostatistical approach. The model benefits not only from the explicit availability of the full conditionals but also from the computational simplicity of the precision matrix determinant calculation using a closed form expression involving the eigenvalues of a precision matrix submatrix. The introduction of covariates into the model adds little computational complexity to the analysis and thus the method can be straightforwardly extended to regression models. The model, because of its computational simplicity, is well suited to application involving the fully Bayesian analysis of large data sets involving multivariate measurements with a spatial ordering. An extension to spatio-temporal data is also considered. Here, we demonstrate use of the model in the analysis of bivariate binary data where the observed data is modelled as the sign of the hidden CAR process. A case study involving over 450 irregularly spaced sites and the presence or absence of each of two species of rain forest trees at each site is presenteds Markov chain Monte Carlo (MCMC) methods are implemented to obtain posterior distributions of all unknowns. The MCMC method works well with simulated data and the tree biodiversity data set.

Bivariate Binary Data with Missing Values: Analysis of a Field Experiment to Investigate Chemical Attractants of Wild Dogs

This article considers design of a field experiment to investigate the effectiveness of various chemical lures or attractants for dingoes, Canis lupus dingo, Australia's native wild dog, and the subsequent analysis of the resulting data. Chemicals were located 50 meters (m) apart at each of 135 sites equally spaced at 500m apart along an approximate straight path about 70 kilometers (km) long, a so-called transect design. Successful attraction to the chemicals was noted each day for seven days. Analysis of the resulting bivariate binary data (successful or not) are carried out to obtain estimates of the effectiveness of the chemicals. Where there was no response at a site this could either have been due to failure of the chemical to attract a dingo, which was present, or absence of the dingo from the site. In order to analyze the resulting data, a model conditioning on dingo presence/absence and hypothesizing a distribution for dingo presence/absence is introduced and estimates of the attractiveness of chemicals (defined as a probability) are obtained using an EM algorithm. Standard errors of estimates are obtained using both asymptotic approximations and a bootstrap for dependent data. An analysis that conditions on observing at least one chemical at a site being visited by a dingo is investigated and estimates obtained. An investigation is made of the information lost by the conditional analysis. Both empirical and theoretical results infer precision is gained by considering the unconditional analyses.

Accommodating negative intracluster correlation with a mixed effects logistic model for bivariate binary data

We extend the random intercept logistic model to accommodate negative intracluster correlations for bivariate binary response data. This approach assumes a single random effect per cluster but entails separate affine transformations of this random effect for the two responses of the pair. We show this approach works for two data sets and a simulation whereas other mixed effects approaches fail. The two data sets are from a crossover trial and a developmental toxicity study of the effects of chemical exposure on malformation risk among rat pups. Comparisons are made with the conditional likelihood approach and with generalized estimating equations estimation of the population-averaged logit model. Simulations show the conditional likelihood approach does not perform well for moderate to strong negative correlations as a positive intracluster correlation is assumed. The proposed mixed effects approach appears to be slightly more conservative than the population-averaged approach with respect to coverage of confidence intervals. Nonetheless, the statistical literature suggests that mixed effects models provide information in addition to that provided by population-averaged models under scientific contexts such as crossover trials. Extensions to trivariate and higher-dimensional responses also are addressed. However, such extensions require certain constraints on the correlation structure.

Hypothesis testing and confidence interval construction in 2 × 2 tables of correlated proportions

The 2 × 2 table is an invaluable tool for displaying bivariate binary data. It is easy to find examples of correlated binary response in biopharmaceutical experiments and clinical research and analysis of these data is a current research topic. The most common hypothesis tested for 2 × 2 tables of correlated proportions is that of homogeneity of the marginal proportions or, equivalently, the hypothesis of table symmetry. The 2 × 2 table of correlated proportions is rich with information and we present a survey of some of the analyses relevant for these data. Using asymptotic theory, we develop estimators of relevant parameters and associated test statistics that are of interest. We discuss interval estimation using arguments proposed by Quesenberry and Hurst (1) and Goodman (2). These interval estimators do not rely on estimation of the covariance matrix and are not necessarily equivalent to those obtained using modified chi-square statistics.

A Likelihood Approach to Analysing Longitudinal Bivariate Binary Data

AbstractTo study the effect of methadone treatment in reducing multiple drug use, say heroin and benzodiazepines while controlling for their possible interaction, we analyse the results of urine drug screens from patients in treatment at a Sydney clinic in 1986. Weekly tests are either positive or negative for each type of drug and a bivariate binary model was developed to analyse such repeated bivariate binary outcomes. It models simultaneously the logit of each type of drug use and their log odds ratio linearly in some covariates. The serial correlation within subject is accounted for by including the ‘previous outcome’ of both drugs and their interaction as covariates. Our main conclusion is that drug use is reduced over time and the interaction between dose and time effects is not significant. It also suggests that while methadone maintenance is effective in reducing heroin use (CHAN et al., 1995), it does not suppress non‐opioid drug use. Concerning the association between the two drugs, it is found that the present strength of their association depends on the previous outcomes only through a measure of concordance. The proposed model has a tractable likelihood function and so a full likelihood analysis is possible. It can be easily extended to incorporate mixture effects. The EM algorithm is used for the estimation of parameters in the mixture model and model selection can be based on the Akaike Information Criterion.

Interim monitoring of bivariate responses using repeated confidence intervals

A method for interim monitoring is described for bivariate response data arising in clinical trials. This procedure centers on a sequence of rectangular joint confidence regions obtained by inverting the hypothesis testing-confidence interval relationship when the testing procedure is governed by a bivariate group sequential design described by Cook and Farewell [4]. The advantage of this type of design over other possible multivariate sequential designs is that the marginal analyses are maintained throughout the trial. This approach proves to be particularly advantageous for active control trials in which a region of therapeutic equivalence is specified for two responses related to efficacy. Two examples illustrate the design and analysis aspects of this approach. The procedure is presented in the context of normally distributed responses; however, adaptations to handle bivariate binary and bivariate failure time data are indicated. Generalizations of this procedure to handle multivariate response data are straightforward.