A nonparametric smoothing method for assessing GEE models with longitudinal binary data

Studies involving longitudinal binary responses are widely applied in the health and biomedical sciences research and frequently analyzed by generalized estimating equations (GEE) method. This article proposes an alternative goodness-of-fit test based on the nonparametric smoothing approach for assessing the adequacy of GEE fitted models, which can be regarded as an extension of the goodness-of-fit test of le Cessie and van Houwelingen (Biometrics 1991; 47:1267-1282). The expectation and approximate variance of the proposed test statistic are derived. The asymptotic distribution of the proposed test statistic in terms of a scaled chi-squared distribution and the power performance of the proposed test are discussed by simulation studies. The testing procedure is demonstrated by two real data. Copyright (c) 2008 John Wiley & Sons, Ltd.     

Sensitivity analysis of longitudinal binary quality of life data with drop-out: an example using the EORTC QLQ-C30.

Analysing quality of life data (QOL) may be complicated for several reasons. Quality of life data not only involves repeated measures but is also usually collected on ordered categorical responses. In addition, it is evident that not all patients provide the same number of assessments, due to attrition caused by death or other medical reasons. In the recent statistical literature, increasing attention is given to methods which can handle non-continuous outcomes in the presence of missing data. The aim of this paper is to investigate the effect on statistical conclusions of applying different modelling techniques to QOL data generated from an EORTC phase III trial. Treatment effects and treatment differences are of major concern. First, a random-effects model is fitted, relating a binary longitudinal response (derived from the physical functioning scale of the QLQ-C30) to several covariates. In a second approach, marginal models are fitted, retaining the response variable and the mean structure used before. The fitted marginal models only differ with respect to the considered estimation procedure: generalized estimating equations (GEE); weighted generalized estimating equations (WGEE), and maximum likelihood (ML).     

Partial questionnaire designs, questionnaire non-response, and attributable fraction: applications to adult onset asthma

The attributable fraction (AF) is often used to explore the policy implications of an association between a disease and an exposure. To date, there have been no proposed estimators of AF in the context of partial questionnaire designs (PQD). The PQD, first proposed in a public health context by Wacholder is often used to enhance response rates in questionnaires. It involves eliciting responses from each subject on preassigned subsets of questions, thereby reducing the burden of response. We propose a computationally efficient method of estimating logistic (or more generally, binary) regression parameters from a PQD model where there is non-response to the questionnaire and the rates of non-response differ between sub-populations. Assuming a log-linear model for the distribution of missing covariates, we employ the methods of Wacholder to motivate consistent estimating equations, and weight each subject's contribution to the estimating function by the inverse probability of responding to the questionnaire. We also propose techniques for goodness-of-fit to assist in model selection. We then use the PQD regression estimates to derive an estimate of AF similar to that proposed by Bruzzi. Finally, we demonstrate our methods using data obtained from a study on adult occupational asthma, conducted within a Massachusetts HMO. Although we concentrate on a particular type of missing data mechanism, other missing data techniques can be incorporated into AF estimation in a similar manner.     

Analysis of a cluster randomized trial with binary outcome data using a multi-level model

The use of multi-level logistic regression models was explored for the analysis of data from a cluster randomized trial investigating whether a training programme for general practitioners' reception staff could improve women's attendance at breast screening. Twenty-six general practices were randomized with women nested within them, requiring a two-level model which allowed for between-practice variability. Comparisons were made with fixed effect (FE) and random effects (RE) cluster summary statistic methods, ordinary logistic regression and a marginal model based on generalized estimating equations with robust variance estimates. An FE summary statistic method and ordinary logistic regression considerably understated the variance of the intervention effect, thus overstating its statistical significance. The marginal model produced a higher statistical significance for the intervention effect compared to that obtained from the RE summary statistic method and the multi-level model. Because there was only a moderate number of practices and these had unbalanced cluster sizes, reliable asymptotic properties for the robust standard errors used in the marginal model may not have been achieved. While the RE summary statistic method cannot handle multiple covariates easily, marginal and multi-level models can do so. In contrast to multi-level models however, marginal models do not provide direct estimates of variance components, but treat these as nuisance parameters. Estimates of the variance components were of particular interest in this example. Additionally, parametric bootstrap methods within the multi-level model framework provide confidence intervals for these variance components, as well as a confidence interval for the effect of intervention which allows for the imprecision in the estimated variance components. The assumption of normality of the random effects can be checked, and the models extended to investigate multiple sources of variability.     

Performance of weighted estimating equations for longitudinal binary data with drop-outs missing at random

The generalized estimating equations (GEE) approach is commonly used to model incomplete longitudinal binary data. When drop-outs are missing at random through dependence on observed responses (MAR), GEE may give biased parameter estimates in the model for the marginal means. A weighted estimating equations approach gives consistent estimation under MAR when the drop-out mechanism is correctly specified. In this approach, observations or person-visits are weighted inversely proportional to their probability of being observed. Using a simulation study, we compare the performance of unweighted and weighted GEE in models for time-specific means of a repeated binary response with MAR drop-outs. Weighted GEE resulted in smaller finite sample bias than GEE. However, when the drop-out model was misspecified, weighted GEE sometimes performed worse than GEE. Weighted GEE with observation-level weights gave more efficient estimates than a weighted GEE procedure with cluster-level weights.     

A semiparametric marginal mixture cure model for clustered survival data

We consider a marginal model for the regression analysis of clustered failure time data with a cure fraction. We propose to use novel generalized estimating equations in an expectation–maximization algorithm to estimate regression parameters in a semiparametric proportional hazards mixture cure model. The dependence among the cure statuses and among the survival times of uncured patients within clusters are modeled by working correlation matrices in the estimating equations. We use a bootstrap method to obtain the variances of the estimates. We report a simulation study to demonstrate a substantial efficiency gain of the proposed method over an existing marginal method. Finally, we apply the model and the proposed method to a set of data from a multi‐institutional study of tonsil cancer patients treated with a radiation therapy. Copyright © 2012 John Wiley & Sons, Ltd.     

条件logistic回归模型配合适度研究

为了检验Pregibon提出的条件logistic配合适度理论的可靠性，本文用Monte Carlo方法对有关统计量进行了统计实验研究，并探讨了有关统计量的应用效果和应用条件。分析了配合适度的影响因素，本文将AIC(赤池信息量准则)统计量引入条件logistic回归模型的配合适度检验中，取得了良好效果，可作为选择“最优”模型的常规统计量。对一个1：2匹配的实例进行了分析。本研究用Fortran77语言自编程序实现了条件logistic模型的参数估计及其检验和评价配合适度的整个计算过程，给实际应用提供了一个有效工具。     

Testing proportionality in the proportional odds model fitted with GEE.

Generalized estimating equations (GEE) methodology as proposed by Liang and Zeger has received widespread use in the analysis of correlated binary data. Miller et al. and Lipsitz et al. extended GEE to correlated nominal and ordinal categorical data; in particular, they used GEE for fitting McCullagh's proportional odds model. In this paper, we consider robust (that is, empirically corrected) and model-based versions of both a score test and a Wald test for assessing the assumption of proportional odds in the proportional odds model fitted with GEE. The Wald test is based on fitting separate multiple logistic regression models for each dichotomization of the response variable, whereas the score test requires fitting just the proportional odds model. We evaluate the proposed tests in small to moderate samples by simulating data from a series of simple models. We illustrate the use of the tests on three data sets from medical studies.     

Comparison of alternative regression models for paired binary data

We used simulated data, derived from real ophthalmologic examples, to evaluate the performance of alternative logistic regression approaches for paired binary data. Approaches considered were: standard logistic regression (ignoring the correlation between fellow eyes, treating individuals classified on the basis of their more impaired eye as the unit of analysis, or considering only right eyes); marginal logistic regression models fitted by the maximum likelihood approach of Lipsitz, Laird and Harrington or the estimating equation approach of Liang and Zeger; and conditional logistic regression models fitted by the maximum likelihood approach of Rosner or the estimating equation approach of Connolly and Liang. Taylor series approximations were used to compare conditional and marginal parameter estimates. Consideration of type I and II error rates found application of standard logistic regression to be inferior to methods that treated the eye as the unit of analysis and accounted for the correlation between fellow eyes. Among these latter approaches, none was uniformly superior to the others across the range of conditions considered.     

Evaluating the discriminatory power of a multiple logistic regression model

Various measures for estimating the goodness-of-fit of the multiple logistic regression (MLR) model have been suggested, although there is no clear consensus as to which measure is most suitable. In this paper, a simple measure of the discriminatory power of the fitted MLR model, based on maximization of Youden's J index (J*), is proposed and compared with several goodness-of-fit statistics described previously. The relative effectiveness of the measure is illustrated using data from the Lipid Research Clinics Prevalence Study. It is suggested that J* may be a useful alternative index of goodness-of-fit of an MLR model, with the added advantage of having a simple practical interpretation.     

A comparison of group sequential methods for binary longitudinal data

Interim analyses are conducted to allow for early termination of the trial, for ethical as well as economical reasons. Here we consider interim analyses in repeated measurements studies where the measurements are binary. Two methods for analysing this kind of data are compared according to their operating characteristics. A subject-specific approach based on the logistic random-effects model is compared with the population-averaged approach based on the generalized estimating equations. The comparison is illustrated with simulations using a randomized clinical trial for toenail fungal infection.     

Tree-based model checking for logistic regression

A tree procedure is proposed to check the adequacy of a fitted logistic regression model. The proposed method not only makes natural assessment for the logistic model, but also provides clues to amend its lack-of-fit. The resulting tree-augmented logistic model facilitates a refined model with meaningful interpretation. We demonstrate its use via simulation studies and an application to the Pima Indians diabetes data.     

Graphical model checking with correlated response data

Correlated response data arise often in biomedical studies. The generalized estimation equation (GEE) approach is widely used in regression analysis for such data. However, there are few methods available to check the adequacy of regression models in GEE. In this paper, a graphical method is proposed based on Cook and Weisberg's marginal model plot. A bootstrap method is applied to obtain the reference band to assess statistical uncertainties in comparing two marginal mean functions. We also propose using the generalized additive model (GAM) in a similar fashion. The proposed two methods are easy to implement by taking advantage of existing smoothing and GAM softwares for independent data. The usefulness of the methodology is demonstrated through application to a correlated binary data set drawn from a clinical trial, the Lung Health Study.     

Graphical diagnostics to check model misspecification for the proportional odds regression model

The cumulative logit or the proportional odds regression model is commonly used to study covariate effects on ordinal responses. This paper provides some graphical and numerical methods for checking the adequacy of the proportional odds regression model. The methods focus on evaluating functional misspecification for specific covariate effects, but misspecification of the link function can also be dealt with under the same framework. For the logistic regression model with binary responses, Arbogast and Lin (Statist. Med. 2005; 24:229-247) developed similar graphical and numerical methods for assessing the adequacy of the model using the cumulative sums of residuals. The paper generalizes their methods to ordinal responses and illustrates them using an example from the VA Normative Aging Study. Simulation studies comparing the performance of the different diagnostic methods indicate that some of the graphical methods are more powerful in detecting model misspecification than the Hosmer-Lemeshow-type goodness-of-fit statistics for the class of models studied.     

What can go wrong when ignoring correlation bounds in the use of generalized estimating equations

The analysis of repeated measure or clustered data is often complicated by the presence of correlation. Further complications arise for discrete responses, where the marginal probability‐dependent Fr'echet bounds impose feasibility limits on the correlation that are often more restrictive than the positive definite range. Some popular statistical methods, such as generalized estimating equations (GEE), ignore these bounds, and as such can generate erroneous estimates and lead to incorrect inferential results. In this paper, we discuss two alternative strategies: (i) using QIC to select a data‐driven correlation value within the Fréchet bounds, and (ii) the use of likelihood‐based latent variable modeling, such as multivariate probit, to get around the problem all together. We provide two examples of the repercussions of incorrectly using existing GEE software in the presence of correlated binary responses. Copyright © 2010 John Wiley & Sons, Ltd.     

A Pearson-type goodness-of-fit test for stationary and time-continuous Markov regression models

Markov regression models describe the way in which a categorical response variable changes over time for subjects with different explanatory variables. Frequently it is difficult to measure the response variable on equally spaced discrete time intervals. Here we propose a Pearson-type goodness-of-fit test for stationary Markov regression models fitted to panel data. A parametric bootstrap algorithm is used to study the distribution of the test statistic. The proposed technique is applied to examine the fit of a Markov regression model used to identify markers for disease progression in psoriatic arthritis.     

Estimating Procedure Times for Surgeries by Determining Location Parameters for the Lognormal Model

We present an empirical study of methods for estimating the location parameter of the lognormal distribution. Our results identify the best order statistic to use, and indicate that using the best order statistic instead of the median may lead to less frequent incorrect rejection of the lognormal model, more accurate critical value estimates, and higher goodness-of-fit. Using simulation data, we constructed and compared two models for identifying the best order statistic, one based on conventional nonlinear regression and the other using a data mining/machine learning technique. Better surgical procedure time estimates may lead to improved surgical operations.     

Bayesian semiparametric regression for longitudinal binary processes with missing data

Longitudinal studies with binary repeated measures are widespread in biomedical research. Marginal regression approaches for balanced binary data are well developed, whereas for binary process data, where measurement times are irregular and may differ by individuals, likelihood-based methods for marginal regression analysis are less well developed. In this article, we develop a Bayesian regression model for analyzing longitudinal binary process data, with emphasis on dealing with missingness. We focus on the settings where data are missing at random (MAR), which require a correctly specified joint distribution for the repeated measures in order to draw valid likelihood-based inference about the marginal mean. To provide maximum flexibility, the proposed model specifies both the marginal mean and serial dependence structures using nonparametric smooth functions. Serial dependence is allowed to depend on the time lag between adjacent outcomes as well as other relevant covariates. Inference is fully Bayesian. Using simulations, we show that adequate modeling of the serial dependence structure is necessary for valid inference of the marginal mean when the binary process data are MAR. Longitudinal viral load data from the HIV Epidemiology Research Study are analyzed for illustration.     

POPULATION-AVERAGED AND CLUSTER-SPECIFIC MODELS FOR CLUSTERED ORDINAL RESPONSE DATA

We compare population-averaged and cluster-specific models for clustered ordinal data. We consider generalized estimating equations and constrained equations maximum likelihood estimation of population-averaged cumulative logit regression models, and mixed effects estimation of cluster-specific cumulative logit regression models. A previously reported relationship between population-averaged and cluster-specific parameters for the binary logistic link appears to hold for analogous parameters under the cumulative logit link. We address these issues in the context of data from two cross-over clinical trials.     

A bivariate Bernoulli model for analyzing malnutrition data

Multivariate binary responses from the same subject are usually correlated. For example, malnutrition of children are usually measured using ‘stunting’ (low height-for-age) and ‘wasting’ (low weight-for-age) calculated from their height, weight and age, and hence the status of being stunted may depend on the status of being wasted and vice-versa. For analyzing such malnutrition data, one needs special statistical models allowing for dependence between the responses to avoid misleading inference. The problem of dependence in multivariate binary responses is generally addressed by using marginal models with generalized estimating equation. However, using the marginal models alone, it is difficult to specify the measures of dependence between the responses precisely. Islam et al. (J Appl Stat 40(5):1064–1075, 2013) proposed a joint modeling approach for bivariate binary responses using both the conditional and marginal models where the dependence between the responses can be measured and tested using a link function of the models. However, the author didn’t examine the properties of the regression coefficient except for the dependence parameter. This paper has given further insight into the joint model and investigated the properties of regression coefficients using an extensive simulation study. The simulation results showed that the maximum likelihood estimators (MLEs) of the regression coefficients of the joint model showed well performance in terms of bias, mean squared error and coverage probability particularly when sample size large. Generally speaking, the MLEs of the parameters associated with joint models possessed the same asymptotic properties as the MLEs of those associated with standard generalized linear models, except for the interpretations. Further the paper provided an application of joint model for analyzing malnutrition data from Bangladesh demographic and health survey 2011. The results revealed that the estimates of the both marginal and condition regression coefficients of the joint model have meaningful interpretation and explanation, which will in turn help the policy makers for designing appropriate policies for improving nutrition status.