A weighted estimating equation for linear regression with missing covariate data

Linear regression is one of the most popular statistical techniques. In linear regression analysis, missing covariate data occur often. A recent approach to analyse such data is a weighted estimating equation. With weighted estimating equations, the contribution to the estimating equation from a complete observation is weighted by the inverse 'probability of being observed'. In this paper, we propose a weighted estimating equation in which we wrongly assume that the missing covariates are multivariate normal, but still produces consistent estimates as long as the probability of being observed is correctly modelled. In simulations, these weighted estimating equations appear to be highly efficient when compared to the most efficient weighted estimating equation as proposed by Robins et al. and Lipsitz et al. However, these weighted estimating equations, in which we wrongly assume that the missing covariates are multivariate normal, are much less computationally intensive than the weighted estimating equations given by Lipsitz et al. We compare the weighted estimating equations proposed in this paper to the efficient weighted estimating equations via an example and a simulation study. We only consider missing data which are missing at random; non-ignorably missing data are not addressed in this paper.     

Semiparametric regression models for repeated measures of mortal cohorts with non‐monotone missing outcomes and time‐dependent covariates

We propose a semiparametric marginal modeling approach for longitudinal analysis of cohorts with data missing due to death and non‐response to estimate regression parameters interpreted as conditioned on being alive. Our proposed method accommodates outcomes and time‐dependent covariates that are missing not at random with non‐monotone missingness patterns via inverse‐probability weighting. Missing covariates are replaced by consistent estimates derived from a simultaneously solved inverse‐probability‐weighted estimating equation. Thus, we utilize data points with the observed outcomes and missing covariates beyond the estimated weights while avoiding numerical methods to integrate over missing covariates. The approach is applied to a cohort of elderly female hip fracture patients to estimate the prevalence of walking disability over time as a function of body composition, inflammation, and age. Copyright © 2010 John Wiley & Sons, Ltd.     

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.     

A new estimation with minimum trace of asymptotic covariance matrix for incomplete longitudinal data with a surrogate process

Missing data is a very common problem in medical and social studies, especially when data are collected longitudinally. It is a challenging problem to utilize observed data effectively. Many papers on missing data problems can be found in statistical literature. It is well known that the inverse weighted estimation is neither efficient nor robust. On the other hand, the doubly robust (DR) method can improve the efficiency and robustness. As is known, the DR estimation requires a missing data model (i.e., a model for the probability that data are observed) and a working regression model (i.e., a model for the outcome variable given covariates and surrogate variables). Because the DR estimating function has mean zero for any parameters in the working regression model when the missing data model is correctly specified, in this paper, we derive a formula for the estimator of the parameters of the working regression model that yields the optimally efficient estimator of the marginal mean model (the parameters of interest) when the missing data model is correctly specified. Furthermore, the proposed method also inherits the DR property. Simulation studies demonstrate the greater efficiency of the proposed method compared with the standard DR method. A longitudinal dementia data set is used for illustration. Copyright © 2013 John Wiley & Sons, Ltd.     

Variable selection in competing risks models based on quantile regression

The proportional subdistribution hazard regression model has been widely used by clinical researchers for analyzing competing risks data. It is well known that quantile regression provides a more comprehensive alternative to model how covariates influence not only the location but also the entire conditional distribution. In this paper, we develop variable selection procedures based on penalized estimating equations for competing risks quantile regression. Asymptotic properties of the proposed estimators including consistency and oracle properties are established. Monte Carlo simulation studies are conducted, confirming that the proposed methods are efficient. A bone marrow transplant data set is analyzed to demonstrate our methodologies.     

Multinomial logistic regression with missing outcome data: An application to cancer subtypes

Many diseases such as cancer and heart diseases are heterogeneous and it is of great interest to study the disease risk specific to the subtypes in relation to genetic and environmental risk factors. However, due to logistic and cost reasons, the subtype information for the disease is missing for some subjects. In this article, we investigate methods for multinomial logistic regression with missing outcome data, including a bootstrap hot deck multiple imputation (BHMI), simple inverse probability weighted (SIPW), augmented inverse probability weighted (AIPW), and expected estimating equation (EEE) estimators. These methods are important approaches for missing data regression. The BHMI modifies the standard hot deck multiple imputation method such that it can provide valid confidence interval estimation. Under the situation when the covariates are discrete, the SIPW, AIPW, and EEE estimators are numerically identical. When the covariates are continuous, nonparametric smoothers can be applied to estimate the selection probabilities and the estimating scores. These methods perform similarly. Extensive simulations show that all of these methods yield unbiased estimators while the complete-case (CC) analysis can be biased if the missingness depends on the observed data. Our simulations also demonstrate that these methods can gain substantial efficiency compared with the CC analysis. The methods are applied to a colorectal cancer study in which cancer subtype data are missing among some study individuals.     

Weighted quantile regression for analyzing health care cost data with missing covariates

Analysis of health care cost data is often complicated by a high level of skewness, heteroscedastic variances and the presence of missing data. Most of the existing literature on cost data analysis have been focused on modeling the conditional mean. In this paper, we study a weighted quantile regression approach for estimating the conditional quantiles health care cost data with missing covariates. The weighted quantile regression estimator is consistent, unlike the naive estimator, and asymptotically normal. Furthermore, we propose a modified BIC for variable selection in quantile regression when the covariates are missing at random. The quantile regression framework allows us to obtain a more complete picture of the effects of the covariates on the health care cost and is naturally adapted to the skewness and heterogeneity of the cost data. The method is semiparametric in the sense that it does not require to specify the likelihood function for the random error or the covariates. We investigate the weighted quantile regression procedure and the modified BIC via extensive simulations. We illustrate the application by analyzing a real data set from a health care cost study. Copyright © 2013 John Wiley & Sons, Ltd.     

Assessment of stratum-covariate interactions in Cox's proportional hazards regression model

We assess stratum (e.g. treatment) interactions with covariates and with the baseline hazard function in the proportional hazards (PH) regression model for lifetime data. We consider models incorporating stratum interactions both with and without stratification of the risk sets in the likelihood function, and describe likelihood ratio statistics for tests of the presence of these interactions. We also present step-down methods for building reduced models which include stratum-specific parameters corresponding to covariates which interact with treatment. We apply PH models with such interactions to a clinical trial of DES in the treatment of prostate cancer to determine optimal treatment conditional on each patient's covariates.     

Best linear inverse probability weighted estimation for two-phase designs and missing covariate regression

The inverse probability weighted estimator is often applied to two-phase designs and regression with missing covariates. Inverse probability weighted estimators typically are less efficient than likelihood-based estimators but, in general, are more robust against model misspecification. In this paper, we propose a best linear inverse probability weighted estimator for two-phase designs and missing covariate regression. Our proposed estimator is the projection of the SIPW onto the orthogonal complement of the score space based on a working regression model of the observed covariate data. The efficiency gain is from the use of the association between the outcome variable and the available covariates, which is the working regression model. One advantage of the proposed estimator is that there is no need to calculate the augmented term of the augmented weighted estimator. The estimator can be applied to general missing data problems or two-phase design studies in which the second phase data are obtained in a subcohort. The method can also be applied to secondary trait case-control genetic association studies. The asymptotic distribution is derived, and the finite sample performance of the proposed estimator is examined via extensive simulation studies. The methods are applied to a bladder cancer case-control study.     

Meta‐regression with partial information on summary trial or patient characteristics

We present a model for meta‐regression in the presence of missing information on some of the study level covariates, obtaining inferences using Bayesian methods. In practice, when confronted with missing covariate data in a meta‐regression, it is common to carry out a complete case or available case analysis. We propose to use the full observed data, modelling the joint density as a factorization of a meta‐regression model and a conditional factorization of the density for the covariates. With the inclusion of several covariates, inter‐relations between these covariates are modelled. Under this joint likelihood‐based approach, it is shown that the lesser assumption of the covariates being Missing At Random is imposed, instead of the more usual Missing Completely At Random (MCAR) assumption. The model is easily programmable in WinBUGS, and we examine, through the analysis of two real data sets, sensitivity and robustness of results to the MCAR assumption. Copyright © 2010 John Wiley & Sons, Ltd.     

A comparison of multiple imputation and fully augmented weighted estimators for Cox regression with missing covariates

Several approaches exist for handling missing covariates in the Cox proportional hazards model. The multiple imputation (MI) is relatively easy to implement with various software available and results in consistent estimates if the imputation model is correct. On the other hand, the fully augmented weighted estimators (FAWEs) recover a substantial proportion of the efficiency and have the doubly robust property. In this paper, we compare the FAWEs and the MI through a comprehensive simulation study. For the MI, we consider the multiple imputation by chained equation and focus on two imputation methods: Bayesian linear regression imputation and predictive mean matching. Simulation results show that the imputation methods can be rather sensitive to model misspecification and may have large bias when the censoring time depends on the missing covariates. In contrast, the FAWEs allow the censoring time to depend on the missing covariates and are remarkably robust as long as getting either the conditional expectations or the selection probability correct due to the doubly robust property. The comparison suggests that the FAWEs show the potential for being a competitive and attractive tool for tackling the analysis of survival data with missing covariates. Copyright © 2010 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.     

Model validation and influence diagnostics for regression models with missing covariates

Missing covariate values are prevalent in regression applications. While an array of methods have been developed for estimating parameters in regression models with missing covariate data for a variety of response types, minimal focus has been given to validation of the response model and influence diagnostics. Previous research has mainly focused on estimating residuals for observations with missing covariates using expected values, after which specialized techniques are needed to conduct proper inference. We suggest a multiple imputation strategy that allows for the use of standard methods for residual analyses on the imputed data sets or a stacked data set. We demonstrate the suggested multiple imputation method by analyzing the Sleep in Mammals data in the context of a linear regression model and the New York Social Indicators Status data with a logistic regression model.     

Fitting stratified proportional odds models by amalgamating conditional likelihoods

Classical methods for fitting a varying intercept logistic regression model to stratified data are based on the conditional likelihood principle to eliminate the stratum-specific nuisance parameters. When the outcome variable has multiple ordered categories, a natural choice for the outcome model is a stratified proportional odds or cumulative logit model. However, classical conditioning techniques do not apply to the general K-category cumulative logit model (K>2) with varying stratum-specific intercepts as there is no reduction due to sufficiency; the nuisance parameters remain in the conditional likelihood. We propose a methodology to fit stratified proportional odds model by amalgamating conditional likelihoods obtained from all possible binary collapsings of the ordinal scale. The method allows for categorical and continuous covariates in a general regression framework. We provide a robust sandwich estimate of the variance of the proposed estimator. For binary exposures, we show equivalence of our approach to the estimators already proposed in the literature. The proposed recipe can be implemented very easily in standard software. We illustrate the methods via three real data examples related to biomedical research. Simulation results comparing the proposed method with a random effects model on the stratification parameters are also furnished.     

Doubly robust conditional logistic regression

Epidemiologic research often aims to estimate the association between a binary exposure and a binary outcome, while adjusting for a set of covariates (eg, confounders). When data are clustered, as in, for instance, matched case-control studies and co-twin-control studies, it is common to use conditional logistic regression. In this model, all cluster-constant covariates are absorbed into a cluster-specific intercept, whereas cluster-varying covariates are adjusted for by explicitly adding these as explanatory variables to the model. In this paper, we propose a doubly robust estimator of the exposure-outcome odds ratio in conditional logistic regression models. This estimator protects against bias in the odds ratio estimator due to misspecification of the part of the model that contains the cluster-varying covariates. The doubly robust estimator uses two conditional logistic regression models for the odds ratio, one prospective and one retrospective, and is consistent for the exposure-outcome odds ratio if at least one of these models is correctly specified, not necessarily both. We demonstrate the properties of the proposed method by simulations and by re-analyzing a publicly available dataset from a matched case-control study on induced abortion and infertility.     

Regression models for unconstrained, partially or fully constrained continuation odds ratios

Epidemiologists frequently encounter studies with ordered responses. Standard ordered response logit models, such as the continuation ratio model, constrain exposure to have a homogeneous effect across thresholds of the ordered response. We demonstrate a method for fitting regression models for unconstrained, partially or fully constrained continuation odds ratios using a 'person-threshold' data set. For each subject, we create a separate record for each response threshold the subject is 'at risk' of passing and then apply standard binary logistic regression to estimate the continuation-ratio model. An example demonstrates the unconstrained, partially and fully constrained continuation-ratio model, while a small simulation study examines some properties of the proposed 'person-threshold' approach. Finally, we present a brief discussion of statistical software to implement the method.     

A multistate Markov chain model for longitudinal, categorical quality-of-life data subject to non-ignorable missingness

Quality-of-life (QOL) is an important outcome in clinical research, particularly in cancer clinical trials. Typically, data are collected longitudinally from patients during treatment and subsequent follow-up. Missing data are a common problem, and missingness may arise in a non-ignorable fashion. In particular, the probability that a patient misses an assessment may depend on the patient's QOL at the time of the scheduled assessment. We propose a Markov chain model for the analysis of categorical outcomes derived from QOL measures. Our model assumes that transitions between QOL states depend on covariates through generalized logit models or proportional odds models. To account for non-ignorable missingness, we incorporate logistic regression models for the conditional probabilities of observing measurements, given their actual values. The model can accommodate time-dependent covariates. Estimation is by maximum likelihood, summing over all possible values of the missing measurements. We describe options for selecting parsimonious models, and we study the finite-sample properties of the estimators by simulation. We apply the techniques to data from a breast cancer clinical trial in which QOL assessments were made longitudinally, and in which missing data frequently arose.     

Combining fractional polynomial model building with multiple imputation

Multivariable fractional polynomial (MFP) models are commonly used in medical research. The datasets in which MFP models are applied often contain covariates with missing values. To handle the missing values, we describe methods for combining multiple imputation with MFP modelling, considering in turn three issues: first, how to impute so that the imputation model does not favour certain fractional polynomial (FP) models over others; second, how to estimate the FP exponents in multiply imputed data; and third, how to choose between models of differing complexity. Two imputation methods are outlined for different settings. For model selection, methods based on Wald‐type statistics and weighted likelihood‐ratio tests are proposed and evaluated in simulation studies. The Wald‐based method is very slightly better at estimating FP exponents. Type I error rates are very similar for both methods, although slightly less well controlled than analysis of complete records; however, there is potential for substantial gains in power over the analysis of complete records. We illustrate the two methods in a dataset from five trauma registries for which a prognostic model has previously been published, contrasting the selected models with that obtained by analysing the complete records only. © 2015 The Authors. Statistics in Medicine Published by John Wiley & Sons Ltd.     

Semiparametric transformation models for panel count data with correlated observation and follow‐up times

The statistical analysis of panel count data has recently attracted a great deal of attention, and a number of approaches have been developed. However, most of these approaches are for situations where the observation and follow‐up processes are independent of the underlying recurrent event process unconditional or conditional on covariates. In this paper, we discuss a more general situation where both the observation and the follow‐up processes may be related with the recurrent event process of interest. For regression analysis, we present a class of semiparametric transformation models and develop some estimating equations for estimation of regression parameters. Numerical studies under different settings conducted for assessing the proposed methodology suggest that it works well for practical situations, and the approach is applied to a skin cancer study that motivated the study. Copyright © 2013 John Wiley & Sons, Ltd.     

Using modified approaches on marginal regression analysis of longitudinal data with time‐dependent covariates

Quadratic inference functions (QIFs) and estimating equations using the conjugate gradient method (CGM) for fitting marginal models to longitudinal data show appealing features in improving the efficiency without making assumptions on the correlation structure. However, our simulation study shows that both methods produce biased and inefficient estimates of regression parameters when time‐dependent covariates are present. In this paper, we extend both the QIF and CGM methods for fitting marginal models to longitudinal data with time‐dependent covariates. The idea is to restrict the moment conditions to the ones that are only valid to certain types of time‐dependent covariates. Our simulations show that efficiency on estimating regression parameters is achieved using modified approaches. Furthermore, we apply the modified approach to anthropometric screening data to evaluate the association between body mass index and morbidity in children in the Philippines. Copyright © 2014 John Wiley & Sons, Ltd.