Bivariate categorical data analysis using normal linear conditional multinomial probability model

Bivariate multinomial data such as the left and right eyes retinopathy status data are analyzed either by using a joint bivariate probability model or by exploiting certain odds ratio‐based association models. However, the joint bivariate probability model yields marginal probabilities, which are complicated functions of marginal and association parameters for both variables, and the odds ratio‐based association model treats the odds ratios involved in the joint probabilities as ‘working’ parameters, which are consequently estimated through certain arbitrary ‘working’ regression models. Also, this later odds ratio‐based model does not provide any easy interpretations of the correlations between two categorical variables. On the basis of pre‐specified marginal probabilities, in this paper, we develop a bivariate normal type linear conditional multinomial probability model to understand the correlations between two categorical variables. The parameters involved in the model are consistently estimated using the optimal likelihood and generalized quasi‐likelihood approaches. The proposed model and the inferences are illustrated through an intensive simulation study as well as an analysis of the well‐known Wisconsin Diabetic Retinopathy status data. Copyright © 2014 John Wiley & Sons, Ltd.     

Bivariate logistic regression: modelling the association of small for gestational age births in twin gestations.

Clustered binary responses, such as disease status in twins, frequently arise in perinatal health and other epidemiologic applications. The scientific objective involves modelling both the marginal mean responses, such as the probability of disease, and the within-cluster association of the multivariate responses. In this regard, bivariate logistic regression is a useful procedure with advantages that include (i) a single maximization of the joint probability distribution of the bivariate binary responses, and (ii) modelling the odds ratio describing the pairwise association between the two binary responses in relation to several covariates. In addition, since the form of the joint distribution of the bivariate binary responses is assumed known, parameters for the regression model can be estimated by the method of maximum likelihood. Hence, statistical inferences may be based on likelihood ratio tests and profile likelihood confidence intervals. We apply bivariate logistic regression to a perinatal database comprising 924 twin foetuses resulting from 462 pregnancies to model obstetric and clinical risk factors for the association of small for gestational age births in twin gestations.     

Comparison of different maximum likelihood estimators in a small sample logistic regression with two independent binary variables

In order to examine the bias of the estimate of the log odds ratio in a 2 x 2 contingency table, Walter computed the entire distribution of the estimated log odds ratio using various small sample sizes. This is equivalent to computing the distribution of the estimated parameter b1 in a logistic regression with one independent binary variable. In this paper, the distributions of the estimated parameters b1 and b2 for two independent binary variables are computed for some small sample logistic regressions using six different estimation methods based on maximum likelihood. These estimates are then compared to the true parameter values. The best estimation method depends on the frequency of the outcome of interest and on whether the bias or mean square error is considered more important.     

Regression analysis of discrete time survival data under heterogeneity

This paper concerns the regression analysis of discrete time survival data for heterogeneous populations by means of frailty models. We express the survival time for each individual as a sequence of binary variables that indicate if the individual survived at each time point. The main result is that the likelihood for these indicators can be factored into contributions that involve the conditional survival probabilities integrated over the frailty distribution of the risk set (population-averaged). We then model these population-averaged conditional probabilities as a function of covariates. The result justifies the practice of treating the failure indicators as independent Bernoulli trials and fitting binary regression models for the conditional failure probabilities at each time point. However, we must interpret the regression coefficients as population-averaged rather than subject-specific parameters. We apply the method to the Framingham Heart Study on risk factors for cardiovascular disease. © 1997 by John Wiley & Sons, Ltd.     

Pseudo-likelihood methods for longitudinal binary data with non-ignorable missing responses and covariates

In this paper we consider longitudinal studies in which the outcome to be measured over time is binary, and the covariates of interest are categorical. In longitudinal studies it is common for the outcomes and any time-varying covariates to be missing due to missed study visits, resulting in non-monotone patterns of missingness. Moreover, the reasons for missed visits may be related to the specific values of the response and/or covariates that should have been obtained, i.e. missingness is non-ignorable. With non-monotone non-ignorable missing response and covariate data, a full likelihood approach is quite complicated, and maximum likelihood estimation can be computationally prohibitive when there are many occasions of follow-up. Furthermore, the full likelihood must be correctly specified to obtain consistent parameter estimates. We propose a pseudo-likelihood method for jointly estimating the covariate effects on the marginal probabilities of the outcomes and the parameters of the missing data mechanism. The pseudo-likelihood requires specification of the marginal distributions of the missingness indicator, outcome, and possibly missing covariates at each occasions, but avoids making assumptions about the joint distribution of the data at two or more occasions. Thus, the proposed method can be considered semi-parametric. The proposed method is an extension of the pseudo-likelihood approach in Troxel et al. to handle binary responses and possibly missing time-varying covariates. The method is illustrated using data from the Six Cities study, a longitudinal study of the health effects of air pollution.     

Analysis of clustered failure time data with cure fraction using copula

Clustered survival data in the presence of cure has received increasing attention. In this paper, we consider a semiparametric mixture cure model which incorporates a logistic regression model for the cure fraction and a semiparametric regression model for the failure time. We utilize Archimedean copula (AC) models to assess the strength of association for both susceptibility and failure times between susceptible individuals in the same cluster. Instead of using the full likelihood approach, we consider a composite likelihood function and a two-stage estimation procedure for both marginal and association parameters. A Jackknife procedure that takes out one cluster at a time is proposed for the variance estimation of the estimators. Akaike information criterion is applied to select the best model among ACs. Simulation studies are performed to validate our estimating procedures, and two real data sets are analyzed to demonstrate the practical use of our proposed method.     

Value of multinomial model in epidemiology: application to the comparison of risk factors for severely and moderately preterm births

BACKGROUND: The multinomial logistic regression model is employed to model the relationship between an outcome variable with more than two categories and a set of covariates. This model is not widely used in epidemiology. We discuss the value of the multinomial model by comparing it with the binary logistic model, and we present a statistical comparison of odds ratios (OR) using the multinomial model. We studied the associations between obstetric history and very (< 33 weeks of amenorrhea) and moderate (33-36 weeks) preterm births. METHODS: Parameters (lnOR) of very and moderate preterm births, associated with the severity of obstetric history (none=0, moderate=1, severe=2), were estimated using two logistic binary models (moderate preterm births vs full-term births (>=37 weeks), and very preterm births vs full-term births) and one logistic multinomial model which compared very and moderate preterm births to full-term births. These analyses were performed before and after adjustment for a covariate: the country of survey. Parameters of very preterm birth and moderate preterm birth, estimated from multinomial model, were compared using Wald test. These analyses were performed using data from a large case-control survey in Europe, the EUROPOP survey; 1 675 very preterm births, 3 652 moderate preterm births and 7 965 full-term births were included. RESULTS: Crude parameters of very and moderate preterm births were similar, regardless the logistic regression model, binary or multinomial. The estimated parameters slightly differ after adjustment for the covariate, but lower variance estimates were obtained using multinomial logistic regression model. Parameters of very preterm birth associated with moderate obstetric history, B(gp)=0.5040, and severe obstetric history, B(gp)'=1.545, differ significantly from those of moderate preterm birth, B(pm)=0.4434 and B(pm)'=1.223 respectively (p < 0.001). CONCLUSION: Parameters obtained in separate logistic binary models are close to those obtained in a multinomial model. The multinomial model is useful for testing the heterogeneity of risk factors for distinct health problems.     

Estimation for paired binomial data with application to radiation therapy.

We compare and contrast several different methods for estimating the effect of treatment when responses are paired binomial observations. The ratio of binomial probabilities is the parameter of interest, while the binomial probabilities are nuisance parameters which may vary between pairs. The application is a meta-analysis of the treatment of rectal cancer, with observations in each study indicating the number of recurrences of the cancer in each of two groups, one with radiation therapy and one without. The ratio of the probabilities of recurrence in the radiation to non-radiation groups is of substantive interest, and is modelled as a logistic or complementary log-log function of an unknown linear combination of the covariates. The three methods we consider are maximum likelihood, a Bayesian approach and an approach based on estimating equations. For the MLE and Bayesian approach the potentially large number of nuisance parameters are estimated together with the parameters of interest, whereas for the estimating equation approach only the parameters of interest are estimated. A simulation study is performed to compare the methods and evaluate the impact of overdispersion.     

Meta‐analysis of studies with bivariate binary outcomes: a marginal beta‐binomial model approach

When conducting a meta‐analysis of studies with bivariate binary outcomes, challenges arise when the within‐study correlation and between‐study heterogeneity should be taken into account. In this paper, we propose a marginal beta‐binomial model for the meta‐analysis of studies with binary outcomes. This model is based on the composite likelihood approach and has several attractive features compared with the existing models such as bivariate generalized linear mixed model (Chu and Cole, 2006) and Sarmanov beta‐binomial model (Chen et al., 2012). The advantages of the proposed marginal model include modeling the probabilities in the original scale, not requiring any transformation of probabilities or any link function, having closed‐form expression of likelihood function, and no constraints on the correlation parameter. More importantly, because the marginal beta‐binomial model is only based on the marginal distributions, it does not suffer from potential misspecification of the joint distribution of bivariate study‐specific probabilities. Such misspecification is difficult to detect and can lead to biased inference using currents methods. We compare the performance of the marginal beta‐binomial model with the bivariate generalized linear mixed model and the Sarmanov beta‐binomial model by simulation studies. Interestingly, the results show that the marginal beta‐binomial model performs better than the Sarmanov beta‐binomial model, whether or not the true model is Sarmanov beta‐binomial, and the marginal beta‐binomial model is more robust than the bivariate generalized linear mixed model under model misspecifications. Two meta‐analyses of diagnostic accuracy studies and a meta‐analysis of case–control studies are conducted for illustration. Copyright © 2015 John Wiley & Sons, Ltd.     

Goodness-of-fit for GEE: an example with mental health service utilization

Suppose we use generalized estimating equations to estimate a marginal regression model for repeated binary observations. There are no established summary statistics available for assessing the adequacy of the fitted model. In this paper we propose a goodness-of-fit test statistic which has an approximate chi-squared distribution when we have specified the model correctly. The proposed statistic can be viewed as an extension of the Hosmer and Lemeshow goodness-of-fit statistic for ordinary logistic regression to marginal regression models for repeated binary responses. We illustrate the methods using data from a study of mental health service utilization by children. The repeated responses are a set of binary measures of service use. We fit a marginal logistic regression model to the data using generalized estimating equations, and we apply the proposed goodness-of-fit statistic to assess the adequacy of the fitted model.     

Conditional pseudolikelihood methods for clustered ordinal,multinomial, or count outcomes with complex survey data

In order to adjust individual‐level covariate effects for confounding due to unmeasured neighborhood characteristics, we have recently developed conditional pseudolikelihood methods to estimate the parameters of a proportional odds model for clustered ordinal outcomes with complex survey data. The methods require sampling design joint probabilities for each within‐neighborhood pair. In the present article, we develop a similar methodology for a baseline category logit model for clustered multinomial outcomes and for a loglinear model for clustered count outcomes. All of the estimators and asymptotic sampling distributions we present can be conveniently computed using standard logistic regression software for complex survey data, such as sas proc surveylogistic . We demonstrate validity of the methods theoretically and also empirically by using simulations. We apply the new method for clustered multinomial outcomes to data from the 2008 Florida Behavioral Risk Factor Surveillance System survey in order to investigate disparities in frequency of dental cleaning both unadjusted and adjusted for confounding by neighborhood. Copyright © 2012 John Wiley & Sons, Ltd.     

Efficient semiparametric mean-association estimation for longitudinal binary responses

Semiparametric methods for longitudinal data with association within subjects have recently received considerable attention. However, existing methods for semiparametric longitudinal binary regression modeling (i) mainly concern mean structures with association parameters treated as nuisance; (ii) generally require a correct specification of the covariance structure for misspecified covariance structure may lead to inefficient mean parameter estimates; and (iii) usually run into computation and estimation problems when the time points are irregularly and possibly subject specific. In this article, we propose a semiparametric logistic regression model, which simultaneously takes into account both the mean and response-association structures (via conditional log-odds ratio) for multivariate longitudinal binary outcomes. Our main interest lies in efficient estimation of both the marginal and association parameters. The estimators of the parameters are obtained via the profile kernel approach. We evaluate the proposed methodology through simulation studies and apply it to a real dataset. Both theoretical and empirical results demonstrate that the proposed method yields highly efficient estimators and performs satisfactorily.     

Missing data in the 2 x 2 table: patterns and likelihood-based analysis for cross-sectional studies with supplemental sampling

Standard measures of crude association in the context of a cross-sectional study are the risk difference, relative risk and odds ratio as derived from a 2x 2 table. Most such studies are subject to missing data on disease, exposure, or both, introducing bias into the usual complete-case analysis. We describe several scenarios distinguished by the manner in which missing data arise, and for each we adjust the natural multinomial likelihood to properly account for missing data. The situations presented allow for increasing levels of generality with regard to the missing data mechanism. The final case, quite conceivable in epidemiologic studies, assumes that the probability of missing exposure depends on true exposure and disease status, as well as upon whether disease status is missing (and conversely for the probability of missing disease information). When parameters relating to the missing data process are inestimable without strong assumptions, we propose maximum likelihood analysis subsequent to collecting supplemental data in the spirit of a validation study. Analytical results give insight into the bias inherent in complete-case analysis for each scenario, and numerical results illustrate the performance of likelihood-based point and interval estimates in the most general case. Adjustment for potential confounders via stratified analysis is also discussed.     

The analysis of binary longitudinal data with time-dependent covariates

We consider longitudinal studies with binary outcomes that are measured repeatedly on subjects over time. The goal of our analysis was to fit a logistic model that relates the expected value of the outcomes with explanatory variables that are measured on each subject. However, additional care must be taken to adjust for the association between the repeated measurements on each subject. We propose a new maximum likelihood method for covariates that may be fixed or time varying. We also implement and make comparisons with two other approaches: generalized estimating equations, which may be more robust to misspecification of the true correlation structure, and alternating logistic regression, which models association via odds ratios that are subject to less restrictive constraints than are correlations. The proposed estimation procedure will yield consistent and asymptotically normal estimates of the regression and correlation parameters if the correlation on consecutive measurements on a subject is correctly specified. Simulations demonstrate that our approach can yield improved efficiency in estimation of the regression parameter; for equally spaced and complete data, the gains in efficiency were greatest for the parameter associated with a time-by-group interaction term and for stronger values of the correlation. For unequally spaced data and with dropout according to a missing-at-random mechanism, MARK1ML with correctly specified consecutive correlations yielded substantial improvements in terms of both bias and efficiency. We present an analysis to demonstrate application of the methods we consider. We also offer an R function for easy implementation of our approach.     

A goodness‐of‐fit test for the proportional odds regression model

We examine goodness‐of‐fit tests for the proportional odds logistic regression model—the most commonly used regression model for an ordinal response variable. We derive a test statistic based on the Hosmer–Lemeshow test for binary logistic regression. Using a simulation study, we investigate the distribution and power properties of this test and compare these with those of three other goodness‐of‐fit tests. The new test has lower power than the existing tests; however, it was able to detect a greater number of the different types of lack of fit considered in this study. Moreover, the test allows for the results to be summarized in a contingency table of observed and estimated frequencies, which is a useful supplementary tool to assess model fit. We illustrate the ability of the tests to detect lack of fit using a study of aftercare decisions for psychiatrically hospitalized adolescents. The test proposed in this paper is similar to a recently developed goodness‐of‐fit test for multinomial logistic regression. A unified approach for testing goodness of fit is now available for binary, multinomial, and ordinal logistic regression models. Copyright © 2012 John Wiley & Sons, Ltd.     

USE OF A STOCHASTIC MODEL FOR REPEATED BINARY ASSESSMENT

We discuss a stochastic model appropriate for binary data in clinical studies where assessments are made at various nominal times during a treatment phase. The model is then applied to data on headache relief, nausea and photophobia/phonophobia in a migraine study. The transition rates and probabilities during the initial 240 minutes after treatment administration are derived using the method of maximum likelihood. The results are then compared with analysis at each nominal time point. Stochastic modelling is considered more appropriate for the analysis of repeated binary assessments than analysis at each nominal time since each patient's assessments are modelled simultaneously.     

A mixed effects Markov model for repeated binary outcomes with non-ignorable dropout

In many areas of research, repeated binary measures often represent a two-state stochastic process, where individuals can transition among two states. In a behavioural or physical disability setting, individuals can flow from susceptible or subthreshold state, to an infectious or symptomatic state, and back to a subthreshold state. Quite often the transition among the states happens in continuous time but is observed at discrete, irregularly spaced timepoints which may be unique to each individual. Methods for analyses of such data are typically based on the Markov assumption. Cook (Biometrics 1999; 55:915-920) introduced a conditional Markov model that accommodates the subject-to-subject variation in the model parameters with random effects. We extend this model by adding a non-ignorable dropout component to the model. Specification of the distribution of the random effects is made to guarantee a closed form expression of the marginal likelihood. This methodology is illustrated by applications to a data set from a parasitic field infection survey, a data set from a cocaine treatment study, and a data set from an aging study. Simulations suggest that the shared parameter model is robust with respect to at least one alternative non-ignorable model.     

A new non-randomized model for analysing sensitive questions with binary outcomes

We propose a new non-randomized model for assessing the association of two sensitive questions with binary outcomes. Under the new model, respondents only need to answer a non-sensitive question instead of the original two sensitive questions. As a result, it can protect a respondent's privacy, avoid the usage of any randomizing device, and be applied to both the face-to-face interview and mail questionnaire. We derive the constrained maximum likelihood estimates of the cell probabilities and the odds ratio for two binary variables associated with the sensitive questions via the EM algorithm. The corresponding standard error estimates are then obtained by bootstrap approach. A likelihood ratio test and a chi-squared test are developed for testing association between the two binary variables. We discuss the loss of information due to the introduction of the non-sensitive question, and the design of the co-operative parameters. Simulations are performed to evaluate the empirical type I error rates and powers for the two tests. In addition, a simulation is conducted to study the relationship between the probability of obtaining valid estimates and the sample size for any given cell probability vector. A real data set from an AIDS study is used to illustrate the proposed methodologies.     

A comparison of adaptive allocation rules for group-sequential binary response clinical trials

In clinical trials to compare two or more treatments with dichotomous responses, group-sequential designs may reduce the total number of patients involved in the trial and response-adaptive designs may result in fewer patients being assigned to the inferior treatments. In this paper, we combine group-sequential and response-adaptive designs, extending recent work on sample size re-estimation in trials to compare two treatments with normally distributed responses, to analogous binary response trials. We consider the use of two parameters of interest in the group-sequential design, the log odds ratio and the simple difference between the probabilities of success. In terms of the adaptive sampling rules, we study two urn models, the drop-the-loser rule and the randomized Pólya urn rule, and compare their properties with those of two sequential maximum likelihood estimation rules, which minimize the expected number of treatment failures. We investigate two ways in which adaptive urn designs can be used in conjunction with group-sequential designs. The first method updates the urn at each interim analysis and the second method continually updates the urn after each patient response, assuming immediate patient responses. Our simulation results show that the group-sequential design, which uses the drop-the-loser rule, applied fully sequentially, is the most effective method for reducing the expected number of treatment failures and the average sample number, whilst still maintaining the nominal error rates, over a range of success probabilities.     

Estimating vaccination coverage for the trivalent measles-mumps-rubella vaccine from trivariate serological data

The effectiveness of childhood immunization programs depends on the vaccination coverage actually achieved. Routinely collected coverage data are not always available, and comparability between countries is often compromised because of different data collection methods. In 2000, Gay developed a method to estimate trivalent vaccination coverage from readily available trivariate serological data on the basis of parametric assumptions related to the rate of seroconversion for each vaccine component and probabilities of natural exposure to infection. Gay's work was indirectly published in a paper by Altmann and Altmann, who derived exact solutions for the parameters on the basis of Gay's modeling equations. In this paper, we propose a general likelihood‐based marginal model framework to extend Gay's model by relaxing two of its main assumptions. We use the Bahadur model for trivariate binary data to explicitly account for an association between the disease‐specific exposure probabilities. We fit several correlation structures to measles, mumps, and rubella serology from Belgium and Ireland. For both countries, we estimate a small positive pairwise exposure correlation, which improves the fit to the data. However, the effect on the estimated vaccination coverage and its associated variability is fairly moderate. For both Belgium and Ireland, all models reveal that the vaccination coverage achieved during the first 15 years since the introduction of measles, mumps, and rubella immunization is insufficient to eliminate measles. Copyright © 2012 John Wiley & Sons, Ltd.