Marginal models for correlated binary responses with multiple classes and multiple levels of nesting.

A model for correlated binary data is presented. Marginal probabilities and odds ratios are allowed to have general regression structures that include multiple classes and multiple levels of nesting. Estimation is done through the generalized estimating equations approach of Liang and Zeger (1986, Biometrika 73, 13-22). They are contrasted with conditional models and recommendations for choosing between the two are given. Examples from genetic epidemiology are presented.     

Mixed effects logistic regression models for multiple longitudinal binary functional limitation responses with informative drop-out and confounding by baseline outcomes

In the context of analyzing multiple functional limitation responses collected longitudinally from the Longitudinal Study of Aging (LSOA), we investigate the heterogeneity of these outcomes with respect to their associations with previous functional status and other risk factors in the presence of informative drop-out and confounding by baseline outcomes. We accommodate the longitudinal nature of the multiple outcomes with a unique extension of the nested random effects logistic model with an autoregressive structure to include drop-out and baseline outcome components with shared random effects. Estimation of fixed effects and variance components is by maximum likelihood with numerical integration. This shared parameter selection model assumes that drop-out is conditionally independent of the multiple functional limitation outcomes given the underlying random effect representing an individual's trajectory of functional status across time. Whereas it is not possible to fully assess the adequacy of this assumption, we assess the robustness of this approach by varying the assumptions underlying the proposed model such as the random effects structure, the drop-out component, and omission of baseline functional outcomes as dependent variables in the model. Heterogeneity among the associations between each functional limitation outcome and a set of risk factors for functional limitation, such as previous functional limitation and physical activity, exists for the LSOA data of interest. Less heterogeneity is observed among the estimates of time-level random effects variance components that are allowed to vary across functional outcomes and time. We also note that. under an autoregressive structure, bias results from omitting the baseline outcome component linked to the follow-up outcome component by subject-level random effects.     

A pattern-mixture model for longitudinal binary responses with nonignorable nonresponse

Longitudinal studies frequently incur outcome-related nonresponse. In this article, we discuss a likelihood-based method for analyzing repeated binary responses when the mechanism leading to missing response data depends on unobserved responses. We describe a pattern-mixture model for the joint distribution of the vector of binary responses and the indicators of nonresponse patterns. Specifically, we propose an extension of the multivariate logistic model to handle nonignorable nonresponse. This method yields estimates of the mean parameters under a variety of assumptions regarding the distribution of the unobserved responses. Because these models make unverifiable identifying assumptions, we recommended conducting sensitivity analyses that provide a range of inferences, each of which is valid under different assumptions for nonresponse. The methodology is illustrated using data from a longitudinal study of obesity in children.     

Robust regression for clustered data with application to binary responses

Generalized estimating equations (GEE) can be highly influenced by the presence of unusual data points. A generalization of the GEE procedure, which yields parameter estimates and fitted values that are resistant to influential data, is introduced. Resistant generalized estimating equations (REGEE) include weights in the estimating equations to downweight influential observations or clusters. Influential observations are downweighted according to their leverage or residual in an example of correlated binary regression applied to 137 urinary incontinent elderly patients from 38 medical practices.     

A hybrid model for nonignorable dropout in longitudinal binary responses

This article presents a likelihood-based method for handling nonignorable dropout in longitudinal studies with binary responses. The methodology developed is appropriate when the target of inference is the marginal distribution of the response at each occasion and its dependence on covariates. A "hybrid" model is formulated, which is designed to retain advantageous features of the selection and pattern-mixture model approaches. This formulation accommodates a variety of assumed forms of nonignorable dropout, while maintaining transparency of the constraints required for identifying the overall model. Once appropriate identifying constraints have been imposed, likelihood-based estimation is conducted via the EM algorithm. The article concludes by applying the approach to data from a randomized clinical trial comparing two doses of a contraceptive.     

Joint mapping of quantitative trait Loci for multiple binary characters

Joint mapping for multiple quantitative traits has shed new light on genetic mapping by pinpointing pleiotropic effects and close linkage. Joint mapping also can improve statistical power of QTL detection. However, such a joint mapping procedure has not been available for discrete traits. Most disease resistance traits are measured as one or more discrete characters. These discrete characters are often correlated. Joint mapping for multiple binary disease traits may provide an opportunity to explore pleiotropic effects and increase the statistical power of detecting disease loci. We develop a maximum-likelihood method for mapping multiple binary traits. We postulate a set of multivariate normal disease liabilities, each contributing to the phenotypic variance of one disease trait. The underlying liabilities are linked to the binary phenotypes through some underlying thresholds. The new method actually maps loci for the variation of multivariate normal liabilities. As a result, we are able to take advantage of existing methods of joint mapping for quantitative traits. We treat the multivariate liabilities as missing values so that an expectation-maximization (EM) algorithm can be applied here. We also extend the method to joint mapping for both discrete and continuous traits. Efficiency of the method is demonstrated using simulated data. We also apply the new method to a set of real data and detect several loci responsible for blast resistance in rice.     

Power priors based on multiple historical studies for binary outcomes

Incorporating historical information into the design and analysis of a new clinical trial has been the subject of much recent discussion. For example, in the context of clinical trials of antibiotics for drug resistant infections, where patients with specific infections can be difficult to recruit, there is often only limited and heterogeneous information available from the historical trials. To make the best use of the combined information at hand, we consider an approach based on the multiple power prior that allows the prior weight of each historical study to be chosen adaptively by empirical Bayes. This choice of weight has advantages in that it varies commensurably with differences in the historical and current data and can choose weights near 1 if the data from the corresponding historical study are similar enough to the data from the current study. Fully Bayesian approaches are also considered. The methods are applied to data from antibiotics trials. An analysis of the operating characteristics in a binomial setting shows that the proposed empirical Bayes adaptive method works well, compared to several alternative approaches, including the meta‐analytic prior.     

Exact analysis of dose response for multiple correlated binary outcomes

The neurotoxicity of a substance is often tested using animal bioassays. In the functional observational battery, animals are exposed to a test agent and multiple outcomes are recorded to assess toxicity, using approximately 40 animals measured on up to 30 different items. This design gives rise to a challenging statistical problem: a large number of outcomes for a small sample of subjects. We propose an exact test for multiple binary outcomes, under the assumption that the correlation among these items is equal. This test is based upon an exponential model described by Molenberghs and Ryan (1999, Environmetrics 10, 279-300) and extends the methods developed by Corcoran et al. (2001, Biometrics 57, 941-948) who developed an exact test for exchangeably correlated binary data for groups (clusters) of correlated observations. We present a method that computes an exact p-value testing for a joint dose-response relationship. An estimate of the parameter for dose response is also determined along with its 95% confidence bound. The method is illustrated using data from a neurotoxicity bioassay for the chemical perchlorethylene.     

Bias in estimating association parameters for longitudinal binary responses with drop-outs

This paper considers the impact of bias in the estimation of the association parameters for longitudinal binary responses when there are drop-outs. A number of different estimating equation approaches are considered for the case where drop-out cannot be assumed to be a completely random process. In particular, standard generalized estimating equations (GEE), GEE based on conditional residuals, GEE based on multivariate normal estimating equations for the covariance matrix, and second-order estimating equations (GEE2) are examined. These different GEE estimators are compared in terms of finite sample and asymptotic bias under a variety of drop-out processes. Finally, the relationship between bias in the estimation of the association parameters and bias in the estimation of the mean parameters is explored.     

Likelihood methods for incomplete longitudinal binary responses with incomplete categorical covariates

We consider longitudinal studies in which the outcome observed over time is binary and the covariates of interest are categorical. With no missing responses or covariates, one specifies a multinomial model for the responses given the covariates and uses maximum likelihood to estimate the parameters. Unfortunately, incomplete data in the responses and covariates are a common occurrence in longitudinal studies. Here we assume the missing data are missing at random (Rubin, 1976, Biometrika 63, 581-592). Since all of the missing data (responses and covariates) are categorical, a useful technique for obtaining maximum likelihood parameter estimates is the EM algorithm by the method of weights proposed in Ibrahim (1990, Journal of the American Statistical Association 85, 765-769). In using the EM algorithm with missing responses and covariates, one specifies the joint distribution of the responses and covariates. Here we consider the parameters of the covariate distribution as a nuisance. In data sets where the percentage of missing data is high, the estimates of the nuisance parameters can lead to highly unstable estimates of the parameters of interest. We propose a conditional model for the covariate distribution that has several modeling advantages for the EM algorithm and provides a reduction in the number of nuisance parameters, thus providing more stable estimates in finite samples.     

A character separation index suitable for binary data containing equivocal responses

Character separation indices are useful for determining whichbinary characters best discriminate between two populations.To date these indices are unable to cope sufficiently with binarydata that contain equivocal responses. A character separationindex is described which can be applied to data containing equivocalresponses. Furthermore, for larger numbers of group membersthis index can be used to derive the significance that a givencharacter would have given a particular separation value bychance. The BASIC implementation of a program to calculate characterseparation indices is presented. Received on September 30, 1989; accepted on October 3, 1989     

Multivariate methods for clustered binary data with multiple subclasses, with application to binary longitudinal data.

Clustered binary data occur frequently in biostatistical work. Several approaches have been proposed for the analysis of clustered binary data. In Rosner (1984, Biometrics 40, 1025-1035), a polychotomous logistic regression model was proposed that is a generalization of the beta-binomial distribution and allows for unit- and subunit-specific covariates, while controlling for clustering effects. One assumption of this model is that all pairs of subunits within a cluster are equally correlated. This is appropriate for ophthalmologic work where clusters are generally of size 2, but may be inappropriate for larger cluster sizes. A beta-binomial mixture model is introduced to allow for multiple subclasses within a cluster and to estimate odds ratios relating outcomes for pairs of subunits within a subclass as well as in different subclasses. To include covariates, an extension of the polychotomous logistic regression model is proposed, which allows one to estimate effects of unit-, class-, and subunit-specific covariates, while controlling for clustering using the beta-binomial mixture model. This model is applied to the analysis of respiratory symptom data in children collected over a 14-year period in East Boston, Massachusetts, in relation to maternal and child smoking, where the unit is the child and symptom history is divided into early-adolescent and late-adolescent symptom experience.     

Designs of superiority and noninferiority trials for binary responses are noninterchangeable

To better understand the design of noninferiority trials for binary data, we identify analogies and contrasts between this and the more familiar superiority trial design. We restrict attention to the problem of detecting a difference between experimental and control response rates in the setting where there is no difference (piE - piC = 0) under the noninferiority alternative hypothesis and under the superiority null, and a matching difference between groups under the complementary hypotheses (/piE - piC/ = delta). Our derivation of the constrained maximum likelihood estimates (MLEs) reveals that superiority and noninferiority trials have different nuisance parameters--the marginal response rate and the control-group response rate, respectively. Our empirical results show that when individuals are allocated to treatment groups in the ratio that minimizes the overall sample size, balanced allocation is optimal only for superiority trials when the error rates are equal; otherwise imbalanced allocation is optimal. Different allocation ratios between trial types lead to different variances, and thus to different sample sizes. Finally, since the value of the marginal response rate--a design parameter in noninferiority trials--typically cannot be obtained from preliminary or published studies, we suggest a means of identifying a value that can be used. We conclude that full documentation of the design of a trial requires specification not only of the design parameters but also of the allocation ratio and the nuisance parameter, the value of which is not obvious under unequal allocation.     

Power analysis for cluster randomized trials with multiple binary co-primary endpoints

Cluster randomized trials (CRTs) are widely used in different areas of medicine and public health. Recently, with increasing complexity of medical therapies and technological advances in monitoring multiple outcomes, many clinical trials attempt to evaluate multiple co-primary endpoints. In this study, we present a power analysis method for CRTs with $K\ge 2$ binary co-primary endpoints. It is developed based on the GEE (generalized estimating equation) approach, and three types of correlations are considered: inter-subject correlation within each endpoint, intra-subject correlation across endpoints, and inter-subject correlation across endpoints. A closed-form joint distribution of the K test statistics is derived, which facilitates the evaluation of power and type I error for arbitrarily constructed hypotheses. We further present a theorem that characterizes the relationship between various correlations and testing power. We assess the performance of the proposed power analysis method based on extensive simulation studies. An application example to a real clinical trial is presented.     

On assessing interrater agreement for multiple attribute responses

New methods are developed for assessing the extent of interrater agreement when each unit to be rated is characterized by a (possibly empty) subset of a specified set of distinct nominal attributes. For such multiple attribute response data, a two-rater concordance statistic is derived, and associated statistical inference-making procedures are provided. This concordance statistic is corrected for chance agreement by using an underlying hypergeometric model. Numerical examples are given to illustrate the proposed methodology, and comparisons to other agreement statistics (e.g., kappa) are made.     

Inference about misclassification probabilities from repeated binary responses

Repeated binary responses provide efficient information for two purposes: (1) estimating two misclassification (false-positive and false-negative error) probabilities and (2) testing the hypothesis that either is zero in a reliability study. We focus on the assessment of reliability of a diagnostic test when there is no gold standard. This paper uses a latent class model and illustrates some of its properties. In addition, application to data containing variation among individuals is considered. We apply this model to the serological data on the MNSs blood group of atomic bomb survivors and their children. The results provide valuable information for examining measurement reliability.