Performance of the marginal structural models under various scenarios of incomplete marker's values: A simulation study

Marginal structural models (MSMs) have been proposed for estimating a treatment's effect, in the presence of time‐dependent confounding. We aimed to evaluate the performance of the Cox MSM in the presence of missing data and to explore methods to adjust for missingness. We simulated data with a continuous time‐dependent confounder and a binary treatment. We explored two classes of missing data: (i) missed visits, which resemble clinical cohort studies; (ii) missing confounder's values, which correspond to interval cohort studies. Missing data were generated under various mechanisms. In the first class, the source of the bias was the extreme treatment weights. Truncation or normalization improved estimation. Therefore, particular attention must be paid to the distribution of weights, and truncation or normalization should be applied if extreme weights are noticed. In the second case, bias was due to the misspecification of the treatment model. Last observation carried forward (LOCF), multiple imputation (MI), and inverse probability of missingness weighting (IPMW) were used to correct for the missingness. We found that alternatives, especially the IPMW method, perform better than the classic LOCF method. Nevertheless, in situations with high marker's variance and rarely recorded measurements none of the examined method adequately corrected the bias.     

The Generalised Estimating Equations: A Comparison of Procedures Available in Commercial Statistical Software Packages

The Generalised Estimating Equations (GEE) proposed by Liang and Zeger (1986) and Zeger and Liang (1986) have found considerable attention in the last decade (for an overview see e.g. Ziegler, and Blettner , 1998). Several self-made programs for solving the GEE are available. This paper presents a comparison of three GEE procedures that are already available in SAS PROC GENMOD, STATA procedure XTGEE and SUDAAN PROC MULTILOG. We show that the estimation results may be quite distinct due to different implementations. Summing up, it is pleasant that GEE is becoming established in commercial software packages. However, some aspects of the implementations should be improved.     

A bivariate pseudolikelihood for incomplete longitudinal binary data with nonignorable nonmonotone missingness

For analyzing longitudinal binary data with nonignorable and nonmonotone missing responses, a full likelihood method is complicated algebraically, and often requires intensive computation, especially when there are many follow-up times. As an alternative, a pseudolikelihood approach has been proposed in the literature under minimal parametric assumptions. This formulation only requires specification of the marginal distributions of the responses and missing data mechanism, and uses an independence working assumption. However, this estimator can be inefficient for estimating both time-varying and time-stationary effects under moderate to strong within-subject associations among repeated responses. In this article, we propose an alternative estimator, based on a bivariate pseudolikelihood, and demonstrate in simulations that the proposed method can be much more efficient than the previous pseudolikelihood obtained under the assumption of independence. We illustrate the method using longitudinal data on CD4 counts from two clinical trials of HIV-infected patients.     

A marginalized pattern-mixture model for longitudinal binary data when nonresponse depends on unobserved responses

This paper proposes a method for modeling longitudinal binary data when nonresponse depends on unobserved responses. The proposed method presumes that the target of inference is the marginal distribution of the response at each occasion and its dependence on covariates, and can accommodate both monotone and non-monotone missingness. The approach involves a marginally specified pattern-mixture model that directly parameterizes both the marginal means at each occasion and the dependence of each response on indicators of nonresponse pattern. This formulation readily incorporates a variety of nonresponse processes assumed within a sensitivity analysis. Once identifying restrictions have been made, estimation of model parameters proceeds via solution to a set of modified generalized estimating equations. The proposed method provides an alternative to standard selection and pattern-mixture modeling frameworks, while featuring certain advantages of each. The paper concludes with application of the method to data from a contraceptive clinical trial with substantial dropout.     

Marginalized models for moderate to long series of longitudinal binary response data

Marginalized models (Heagerty, 1999, Biometrics 55, 688-698) permit likelihood-based inference when interest lies in marginal regression models for longitudinal binary response data. Two such models are the marginalized transition and marginalized latent variable models. The former captures within-subject serial dependence among repeated measurements with transition model terms while the latter assumes exchangeable or nondiminishing response dependence using random intercepts. In this article, we extend the class of marginalized models by proposing a single unifying model that describes both serial and long-range dependence. This model will be particularly useful in longitudinal analyses with a moderate to large number of repeated measurements per subject, where both serial and exchangeable forms of response correlation can be identified. We describe maximum likelihood and Bayesian approaches toward parameter estimation and inference, and we study the large sample operating characteristics under two types of dependence model misspecification. Data from the Madras Longitudinal Schizophrenia Study (Thara et al., 1994, Acta Psychiatrica Scandinavica 90, 329-336) are analyzed.     

Prediction using partly conditional time-varying coefficients regression models

Data collected longitudinally in time provide the opportunity to develop predictive models of future observations given current data for an individual. Such models may be of particular value in defining individuals at high risk and thereby in suggesting subgroups for targeting of prevention intervention research efforts. In this paper, we propose a method for estimating predictive functions. The method uses an extension of the marginal regression analysis methods of Liang and Zeger (1986, Biometrika 73, 13-22) and is implemented using simple estimating equations. A key feature of the models is that regression coefficients are modelled as smooth functions of the times both at and for prediction. Data from a study of obesity in childhood and early adulthood is used to demonstrate the methodology. Criteria for defining individuals to be at high risk can be defined on the basis of estimated predictive functions. We suggest methods for evaluating the diagnostic accuracy (sensitivity and specificity) of such rules using cross-validation. The method holds promise as a robust and technically easy way of evaluating information about future prognosis that may be gleaned from a patient's current and past clinical status.     

Measurement error in covariates in the marginal hazards model for multivariate failure time data

We consider measurement error in covariates in the marginal hazards model for multivariate failure time data. We explore the bias implications of normal additive measurement error without assuming a distribution for the underlying true covariate. To correct measurement-error-induced bias in the regression coefficient of the marginal model, we propose to apply the SIMEX procedure and demonstrate its large and small sample properties for both known and estimated measurement error variance. We illustrate this method using the Lipid Research Clinics Coronary Primary Prevention Trial data with total cholesterol as the covariate measured with error and time until angina and time until nonfatal myocardial infarction as the correlated outcomes of interest.     

Modified Gaussian estimation for correlated binary data

In this paper, we develop a Gaussian estimation (GE) procedure to estimate the parameters of a regression model for correlated (longitudinal) binary response data using a working correlation matrix. A two‐step iterative procedure is proposed for estimating the regression parameters by the GE method and the correlation parameters by the method of moments. Consistency properties of the estimators are discussed. A simulation study was conducted to compare 11 estimators of the regression parameters, namely, four versions of the GE, five versions of the generalized estimating equations (GEEs), and two versions of the weighted GEE. Simulations show that (i) the Gaussian estimates have the smallest mean square error and best coverage probability if the working correlation structure is correctly specified and (ii) when the working correlation structure is correctly specified, the GE and the GEE with exchangeable correlation structure perform best as opposed to when the correlation structure is misspecified.     

Modeling microarray data using a threshold mixture model

An important goal of microarray studies is the detection of genes that show significant changes in expression when two classes of biological samples are being compared. We present an ANOVA-style mixed model with parameters for array normalization, overall level of gene expression, and change of expression between the classes. For the latter we assume a mixing distribution with a probability mass concentrated at zero, representing genes with no changes, and a normal distribution representing the level of change for the other genes. We estimate the parameters by optimizing the marginal likelihood. To make this practical, Laplace approximations and a backfitting algorithm are used. The performance of the model is studied by simulation and by application to publicly available data sets.     

Doubly robust estimation in missing data and causal inference models

The goal of this article is to construct doubly robust (DR) estimators in ignorable missing data and causal inference models. In a missing data model, an estimator is DR if it remains consistent when either (but not necessarily both) a model for the missingness mechanism or a model for the distribution of the complete data is correctly specified. Because with observational data one can never be sure that either a missingness model or a complete data model is correct, perhaps the best that can be hoped for is to find a DR estimator. DR estimators, in contrast to standard likelihood-based or (nonaugmented) inverse probability-weighted estimators, give the analyst two chances, instead of only one, to make a valid inference. In a causal inference model, an estimator is DR if it remains consistent when either a model for the treatment assignment mechanism or a model for the distribution of the counterfactual data is correctly specified. Because with observational data one can never be sure that a model for the treatment assignment mechanism or a model for the counterfactual data is correct, inference based on DR estimators should improve upon previous approaches. Indeed, we present the results of simulation studies which demonstrate that the finite sample performance of DR estimators is as impressive as theory would predict. The proposed method is applied to a cardiovascular clinical trial.     

The Generalised Estimating Equations: An Annotated Bibliography

The Generalised Estimating Equations (GEE) proposed by Liang and Zeger (1986) and Zeger and Liang (1986) have found considerable attention in the last ten years and several extensions have been proposed. In this annotated bibliography we describe the development of the GEE and its extensions during the last decade. Additionally, we discuss advantages and disadvantages of the different parametrisations that have been proposed in the literature. Furthermore, we review regression diagnostic techniques and approaches for dealing with missing data. We give an insight to the different fields of application in biometry. We also describe the software available for the GEE.     

The marginal sinus in the perifollicular region of the rat spleen

Summary The marginal sinus in the spleen of the Wistar rat surrounds the follicle and has more numerous PAS positive fibers on the inner wall than on the outer wall. India ink- and lead oxide-gelatin were injected into the abdominal aorta. It was found that much of the india ink-gelatin accumulated in the marginal sinus, the marginal zone, and part of the red pulp, while most of the lead oxide-gelatin collected in the marginal sinus.Ultrastructurally, the capillaries of the follicle were found to open into the marginal sinus. Regions not perforated by the marginal sinus lie between the follicle and the marginal zone. The wall of the marginal sinus is discontinuous and the discontinuities are wider on the marginal zone side than on the follicle side. The relationship of these findings is discussed.A part of this study was presented at the 64th Annual Meeting of the Japanese Pathological Society, Takatsuki, April, 1975     

Association models for a multivariate binary response

Models for a multivariate binary response are parameterized by univariate marginal probabilities and dependence ratios of all orders. The w-order dependence ratio is the joint success probability of w binary responses divided by the joint success probability assuming independence. This parameterization supports likelihood-based inference for both regression parameters, relating marginal probabilities to explanatory variables, and association model parameters, relating dependence ratios to simple and meaningful mechanisms. Five types of association models are proposed, where responses are (1) independent given a necessary factor for the possibility of a success, (2) independent given a latent binary factor, (3) independent given a latent beta distributed variable, (4) follow a Markov chain, and (5) follow one of two first-order Markov chains depending on the realization of a binary latent factor. These models are illustrated by reanalyzing three data sets, foremost a set of binary time series on auranofin therapy against arthritis. Likelihood-based approaches are contrasted with approaches based on generalized estimating equations. Association models specified by dependence ratios are contrasted with other models for a multivariate binary response that are specified by odds ratios or correlation coefficients.     

Marginal modeling of multilevel binary data with time-varying covariates

We propose and compare two approaches for regression analysis of multilevel binary data when clusters are not necessarily nested: a GEE method that relies on a working independence assumption coupled with a three-step method for obtaining empirical standard errors, and a likelihood-based method implemented using Bayesian computational techniques. Implications of time-varying endogenous covariates are addressed. The methods are illustrated using data from the Breast Cancer Surveillance Consortium to estimate mammography accuracy from a repeatedly screened population.