The impact of dichotomization in longitudinal data analysis: a simulation study

In this paper, a simulation study is conducted to systematically investigate the impact of dichotomizing longitudinal continuous outcome variables under various types of missing data mechanisms. Generalized linear models (GLM) with standard generalized estimating equations (GEE) are widely used for longitudinal outcome analysis, but these semi‐parametric approaches are only valid under missing data completely at random (MCAR). Alternatively, weighted GEE (WGEE) and multiple imputation GEE (MI‐GEE) were developed to ensure validity under missing at random (MAR). Using a simulation study, the performance of standard GEE, WGEE and MI‐GEE on incomplete longitudinal dichotomized outcome analysis is evaluated. For comparisons, likelihood‐based linear mixed effects models (LMM) are used for incomplete longitudinal original continuous outcome analysis. Focusing on dichotomized outcome analysis, MI‐GEE with original continuous missing data imputation procedure provides well controlled test sizes and more stable power estimates compared with any other GEE‐based approaches. It is also shown that dichotomizing longitudinal continuous outcome will result in substantial loss of power compared with LMM. Copyright © 2009 John Wiley & Sons, Ltd.     

Diagnostic tests for bias of estimating equations in weighted regression with missing covariates

The authors propose two tests, one parametric and the other semiparametric, for testing bias of estimating equations in weighted regression with partially missing covariates when the primary regression model is correctly specified. More generally, the proposed tests may be thought of as a diagnostic tool for the combined package of the primary regression model and the missingness assumptions. The asymptotic null distributions of the two test statistics are derived under the assumption of missingness at random for the partially missing covariates. A small scale simulation study completes the work.     

ROBUST NON-PARAMETRIC ESTIMATION OF GENOTYPE–PHENOTYPE RELATIONSHIPS IN HUMAN PEDIGREES

Non‐parametric estimation of functional relationships is an important part of data analysis, particularly in the exploratory stages. This paper considers non‐parametric estimation of the mean functions in family studies using weighted robust estimating equations while retaining a fully parametric model for the covariance structure. The proposed procedure allows an exploratory examination of complex pedigree data that is an invaluable aid in determining appropriate models. This is illustrated by an examination of the relationship between IQ and the level of a particular protein in individuals collected as part of a large family study.     

Estimation in Partially Linear Single-Index Models with Missing Covariates

In this article, we consider a partially linear single-index model Y = g(Z ^τθ₀) + X ^τβ₀ + ? when the covariate X may be missing at random. We propose weighted estimators for the unknown parametric and nonparametric part by applying weighted estimating equations. We establish normality of the estimators of the parameters and asymptotic expansion for the estimator of the nonparametric part when the selection probabilities are unknown. Simulation studies are also conducted to illustrate the finite sample properties of these estimators.     

Additive hazards regression with censoring indicators missing at random

In this article, the authors consider a semiparametric additive hazards regression model for right‐censored data that allows some censoring indicators to be missing at random. They develop a class of estimating equations and use an inverse probability weighted approach to estimate the regression parameters. Nonparametric smoothing techniques are employed to estimate the probability of non‐missingness and the conditional probability of an uncensored observation. The asymptotic properties of the resulting estimators are derived. Simulation studies show that the proposed estimators perform well. They motivate and illustrate their methods with data from a brain cancer clinical trial. The Canadian Journal of Statistics 38: 333–351; 2010 © 2010 Statistical Society of Canada     

A general quantile residual life model for length‐biased right‐censored data

The quantile residual lifetime function provides comprehensive quantitative measures for residual life, especially when the distribution of the latter is skewed or heavy‐tailed and/or when the data contain outliers. In this paper, we propose a general class of semiparametric quantile residual life models for length‐biased right‐censored data. We use the inverse probability weighted method to correct the bias due to length‐biased sampling and informative censoring. Two estimating equations corresponding to the quantile regressions are constructed in two separate steps to obtain an efficient estimator. Consistency and asymptotic normality of the estimator are established. The main difficulty in implementing our proposed method is that the estimating equations associated with the quantiles are nondifferentiable, and we apply the majorize–minimize algorithm and estimate the asymptotic covariance using an efficient resampling method. We use simulation studies to evaluate the proposed method and illustrate its application by a real‐data example.     

Relative risk regression for current status data in case‐cohort studies

We propose using the weighted likelihood method to fit a general relative risk regression model for the current status data with missing data as arise, for example, in case‐cohort studies. The missingness probability is either known or can be reasonably estimated. Asymptotic properties of the weighted likelihood estimators are established. For the case of using estimated weights, we construct a general theorem that guarantees the asymptotic normality of the M‐estimator of a finite dimensional parameter in a class of semiparametric models, where the infinite dimensional parameter is allowed to converge at a slower than parametric rate, and some other parameters in the objective function are estimated a priori. The weighted bootstrap method is employed to estimate the variances. Simulations show that the proposed method works well for finite sample sizes. A motivating example of the case‐cohort study from an HIV vaccine trial is used to demonstrate the proposed method. The Canadian Journal of Statistics 39: 557–577; 2011. © 2011 Statistical Society of Canada     

A Model Validation Procedure when Covariate Data are Missing at Random

Abstract. In the presence of missing covariates, standard model validation procedures may result in misleading conclusions. By building generalized score statistics on augmented inverse probability weighted complete‐case estimating equations, we develop a new model validation procedure to assess the adequacy of a prescribed analysis model when covariate data are missing at random. The asymptotic distribution and local alternative efficiency for the test are investigated. Under certain conditions, our approach provides not only valid but also asymptotically optimal results. A simulation study for both linear and logistic regression illustrates the applicability and finite sample performance of the methodology. Our method is also employed to analyse a coronary artery disease diagnostic dataset.     

Partial Linear Models for Longitudinal Data Based on Quadratic Inference Functions

In this paper, we consider improved estimating equations for semiparametric partial linear models (PLM) for longitudinal data, or clustered data in general. We approximate the non‐parametric function in the PLM by a regression spline, and utilize quadratic inference functions (QIF) in the estimating equations to achieve a more efficient estimation of the parametric part in the model, even when the correlation structure is misspecified. Moreover, we construct a test which is an analogue to the likelihood ratio inference function for inferring the parametric component in the model. The proposed methods perform well in simulation studies and real data analysis conducted in this paper.     

ASYMPTOTIC DISTRIBUTIONS OF SEMIPARAMETRIC MAXIMUM LIKELIHOOD ESTIMATORS WITH ESTIMATING EQUATIONS FOR GROUP-CENSORED DATA

Semiparametric maximum likelihood estimation with estimating equations (SMLE) is more flexible than traditional methods; it has fewer restrictions on distributions and regression models. The required information about distribution and regression structures is incorporated in estimating equations of the SMLE to improve the estimation quality of non‐parametric methods. The likelihood of SMLE for censored data involves complicated implicit functions without closed‐form expressions, and the first derivatives of the log‐profile‐likelihood cannot be expressed as summations of independent and identically distributed random variables; it is challenging to derive asymptotic properties of the SMLE for censored data. For group‐censored data, the paper shows that all the implicit functions are well defined and obtains the asymptotic distributions of the SMLE for model parameters and lifetime distributions. With several examples the paper compares the SMLE, the regular non‐parametric likelihood estimation method and the parametric MLEs in terms of their asymptotic efficiencies, and illustrates application of SMLE. Various asymptotic distributions of the likelihood ratio statistics are derived for testing the adequacy of estimating equations and a partial set of parameters equal to some known values.     

Longitudinal data analysis for generalized linear models with follow‐up dependent on outcome‐related variables

In longitudinal studies, observation times are often irregular and subject‐specific. Frequently they are related to the outcome measure or other variables that are associated with the outcome measure but undesirable to condition upon in the model for outcome. Regression analyses that are unadjusted for outcome‐dependent follow‐up then yield biased estimates. The authors propose a class of inverse‐intensity rate‐ratio weighted estimators in generalized linear models that adjust for outcome‐dependent follow‐up. The estimators, based on estimating equations, are very simple and easily computed; they can be used under mixtures of continuous and discrete observation times. The predictors of observation times can be past observed outcomes, cumulative values of outcome‐model covariates and other factors associated with the outcome. The authors validate their approach through simulations and they illustrate it using data from a supported housing program from the US federal government.     

Extending the Mann–Whitney–Wilcoxon rank sum test to longitudinal regression analysis

Outliers are commonly observed in psychosocial research, generally resulting in biased estimates when comparing group differences using popular mean-based models such as the analysis of variance model. Rank-based methods such as the popular Mann–Whitney–Wilcoxon (MWW) rank sum test are more effective to address such outliers. However, available methods for inference are limited to cross-sectional data and cannot be applied to longitudinal studies under missing data. In this paper, we propose a generalized MWW test for comparing multiple groups with covariates within a longitudinal data setting, by utilizing the functional response models. Inference is based on a class of U-statistics-based weighted generalized estimating equations, providing consistent and asymptotically normal estimates not only under complete but missing data as well. The proposed approach is illustrated with both real and simulated study data.     

Rate/Mean Regression for Multiple-Sequence Recurrent Event Data with Missing Event Category

Abstract.  Censored recurrent event data frequently arise in biomedical studies. Often, the events are not homogenous, and may be categorized. We propose semiparametric regression methods for analysing multiple-category recurrent event data and consider the setting where event times are always known, but the information used to categorize events may be missing. Application of existing methods after censoring events of unknown category (i.e. 'complete-case' methods) produces consistent estimators only when event types are missing completely at random, an assumption which will frequently fail in practice. We propose methods, based on weighted estimating equations, which are applicable when event category missingness is missing at random. Parameter estimators are shown to be consistent and asymptotically normal. Finite sample properties are examined through simulations and the proposed methods are applied to an end-stage renal disease data set obtained from a national organ failure registry.     

Empirical likelihood weighted composite quantile regression with partially missing covariates

This paper develops a novel weighted composite quantile regression (CQR) method for estimation of a linear model when some covariates are missing at random and the probability for missingness mechanism can be modelled parametrically. By incorporating the unbiased estimating equations of incomplete data into empirical likelihood (EL), we obtain the EL-based weights, and then re-adjust the inverse probability weighted CQR for estimating the vector of regression coefficients. Theoretical results show that the proposed method can achieve semiparametric efficiency if the selection probability function is correctly specified, therefore the EL weighted CQR is more efficient than the inverse probability weighted CQR. Besides, our algorithm is computationally simple and easy to implement. Simulation studies are conducted to examine the finite sample performance of the proposed procedures. Finally, we apply the new method to analyse the US news College data.     

Semi-empirical likelihood inference for the ROC curve with missing data

The receiver operating characteristic (ROC) curve is one of the most commonly used methods to compare the diagnostic performance of two or more laboratory or diagnostic tests. In this paper, we propose semi-empirical likelihood based confidence intervals for ROC curves of two populations, where one population is parametric and the other one is non-parametric and both have missing data. After imputing missing values, we derive the semi-empirical likelihood ratio statistic and the corresponding likelihood equations. It is shown that the log-semi-empirical likelihood ratio statistic is asymptotically scaled chi-squared. The estimating equations are solved simultaneously to obtain the estimated lower and upper bounds of semi-empirical likelihood confidence intervals. We conduct extensive simulation studies to evaluate the finite sample performance of the proposed empirical likelihood confidence intervals with various sample sizes and different missing probabilities.     

Estimation and inference of combining quantile and least-square regressions with missing data

In this paper, we consider how to incorporate quantile information to improve estimator efficiency for regression model with missing covariates. We combine the quantile information with least-squares normal equations and construct an unbiased estimating equations (EEs). The lack of smoothness of the objective EEs is overcome by replacing them with smooth approximations. The maximum smoothed empirical likelihood (MSEL) estimators are established based on inverse probability weighted (IPW) smoothed EEs and their asymptotic properties are studied under some regular conditions. Moreover, we develop two novel testing procedures for the underlying model. The finite-sample performance of the proposed methodology is examined by simulation studies. A real example is used to illustrate our methods.     

Semiparametric inference for estimating equations with nonignorably missing covariates

We consider statistical inference of unknown parameters in estimating equations (EEs) when some covariates have nonignorably missing values, which is quite common in practice but has rarely been discussed in the literature. When an instrument, a fully observed covariate vector that helps identifying parameters under nonignorable missingness, is available, the conditional distribution of the missing covariates given other covariates can be estimated by the pseudolikelihood method of Zhao and Shao [(2015), ‘Semiparametric pseudo likelihoods in generalised linear models with nonignorable missing data’, Journal of the American Statistical Association, 110, 1577–1590)] and be used to construct unbiased EEs. These modified EEs then constitute a basis for valid inference by empirical likelihood. Our method is applicable to a wide range of EEs used in practice. It is semiparametric since no parametric model for the propensity of missing covariate data is assumed. Asymptotic properties of the proposed estimator and the empirical likelihood ratio test statistic are derived. Some simulation results and a real data analysis are presented for illustration.     

A robust pairwise likelihood method for incomplete longitudinal binary data arising in clusters

Clustered longitudinal data feature cross‐sectional associations within clusters, serial dependence within subjects, and associations between responses at different time points from different subjects within the same cluster. Generalized estimating equations are often used for inference with data of this sort since they do not require full specification of the response model. When data are incomplete, however, they require data to be missing completely at random unless inverse probability weights are introduced based on a model for the missing data process. The authors propose a robust approach for incomplete clustered longitudinal data using composite likelihood. Specifically, pairwise likelihood methods are described for conducting robust estimation with minimal model assumptions made. The authors also show that the resulting estimates remain valid for a wide variety of missing data problems including missing at random mechanisms and so in such cases there is no need to model the missing data process. In addition to describing the asymptotic properties of the resulting estimators, it is shown that the method performs well empirically through simulation studies for complete and incomplete data. Pairwise likelihood estimators are also compared with estimators obtained from inverse probability weighted alternating logistic regression. An application to data from the Waterloo Smoking Prevention Project is provided for illustration. The Canadian Journal of Statistics 39: 34–51; 2011 © 2010 Statistical Society of Canada     

Parametric regression models for continuous time removal and recapture studies

We use a class of parametric counting process regression models that are commonly employed in the analysis of failure time data to formulate the subject-specific capture probabilities for removal and recapture studies conducted in continuous time. We estimate the regression parameters by modifying the conventional likelihood score function for left-truncated and right-censored data to accommodate an unknown population size and missing covariates on uncaptured subjects, and we subsequently estimate the population size by a martingale-based estimating function. The resultant estimators for the regression parameters and population size are consistent and asymptotically normal under appropriate regularity conditions. We assess the small sample properties of the proposed estimators through Monte Carlo simulation and we present an application to a bird banding exercise.     

Estimation of conditional cumulative incidence functions under generalized semiparametric regression models with missing covariates,with application to analysis of biomarker correlates in vaccine trials

This article presents generalized semiparametric regression models for conditional cumulative incidence functions with competing risks data when covariates are missing by sampling design or happenstance. A doubly robust augmented inverse probability weighted (AIPW) complete-case approach to estimation and inference is investigated. This approach modifies IPW complete-case estimating equations by exploiting the key features in the relationship between the missing covariates and the phase-one data to improve efficiency. An iterative numerical procedure is derived to solve the nonlinear estimating equations. The asymptotic properties of the proposed estimators are established. A simulation study examining the finite-sample performances of the proposed estimators shows that the AIPW estimators are more efficient than the IPW estimators. The developed method is applied to the RV144 HIV-1 vaccine efficacy trial to investigate vaccine-induced IgG binding antibodies to HIV-1 as correlates of acquisition of HIV-1 infection while taking account of whether the HIV-1 sequences are near or far from the HIV-1 sequences represented in the vaccine construct.