Probability Of Missingness Research Articles

Local linear estimating equations: uniform consistency and rate of convergence

In this paper, uniform consistency of estimators obtained from kernel-based local linear estimating equations is derived. Furthermore, it is shown that the rate of convergence is at least n 1/3 for a suitable choice of bandwidth. These results are used to find asymptotic results for the integral of the estimator. As an application, we consider an ‘inverse probability of missingness reweighted estimating equation’ estimator.

Analysis of longitudinal data with intermittent missing values using the stochastic EM algorithm

Longitudinal data are not uncommon in many disciplines where repeated measurements on a response variable are collected for all subjects. Some intended measurements may not be available for some subjects resulting in a missing data pattern. Dropout pattern occurs when some subjects leave the study prematurely. The missing data pattern is defined as intermittent if a missing value followed by an observed value. When the probability of missingness depends on the missing value, and may be on the observed values, the missing data mechanism is termed as nonrandom. Ignoring the missing values in this case leads to biased inferences. The stochastic EM (SEM) algorithm is proposed and developed to find parameters estimates in the presence of intermittent missing values. Also, in this setting, the Monte Carlo method is developed to find the standard errors of parameters estimates. Finally, the proposed techniques are applied to a real data from the International Breast Cancer Study Group.

An index of local sensitivity to nonignorable drop-out in longitudinal modelling

In longitudinal studies with potentially nonignorable drop-out, one can assess the likely effect of the nonignorability in a sensitivity analysis. Troxel et al. proposed a general index of sensitivity to nonignorability, or ISNI, to measure sensitivity of key inferences in a neighbourhood of the ignorable, missing at random (MAR) model. They derived detailed formulas for ISNI in the special case of the generalized linear model with a potentially missing univariate outcome. In this paper, we extend the method to longitudinal modelling. We use a multivariate normal model for the outcomes and a regression model for the drop-out process, allowing missingness probabilities to depend on an unobserved response. The computation is straightforward, and merely involves estimating a mixed-effects model and a selection model for the drop-out, together with some simple arithmetic calculations. We illustrate the method with three examples.

Estimation in Partially Linear Models With Missing Covariates

The partially linear model Y = XTβ + ν(Z) + ϵ has been studied extensively when data are completely observed. In this article, we consider the case where the covariate X is sometimes missing, with missingness probability π depending on (Y, Z). New methods are developed for estimating β and ν(·). Our methods are shown to outperform asymptotically methods based only on the complete data. Asymptotic efficiency is discussed, and the semiparametric efficient score function is derived. Justification of the use of the nonparametric bootstrap in this context is sketched. The proposed estimators are extended to a working independence analysis of longitudinal/clustered data and applied to analyze an AIDS clinical trial dataset. The results of a simulation experiment are also given to illustrate our approach.

Nonignorable missingness in matched case-control data analyses.

Matched case-control data analysis is often challenged by a missing covariate problem, the mishandling of which could cause bias or inefficiency. Satten and Carroll (2000, Biometrics56, 384-388) and other authors have proposed methods to handle missing covariates when the probability of missingness depends on the observed data, i.e., when data are missing at random. In this article, we propose a conditional likelihood method to handle the case when the probability of missingness depends on the unobserved covariate, i.e., when data are nonignorably missing. When the missing covariate is binary, the proposed method can be implemented using standard software. Using the Northern Manhattan Stroke Study data, we illustrate the method and discuss how sensitivity analysis can be conducted.

Estimation of the mean of a K-sample U-statistic with missing outcomes and auxiliaries

We propose estimators of the mean of a K-sample U-statistic when data on the outcomes of interest are missing in some sampled units and auxiliary variables are available in the entire sample. The proposed estimators exploit the information available in the auxiliaries without requiring assumptions about the joint distribution of the auxiliaries and outcomes. The properties of the proposed estimators are derived as a consequence of general results on efficient semiparametric estimation of the mean of a K-sample U-statistic with missing at random outcomes, observed auxiliary variables and known missingness probabilities. We illustrate the methods with the estimation of the area under a receiver operating characteristic curve when the biomarker data are observed only in subsamples of the diseased and healthy subjects.

Regression Analysis Under Non-standard Situations: A Pairwise Pseudolikelihood Approach

Summary Regression analysis is one of the most used statistical methods for data analysis. There are, however, many situations in which one cannot base inference solely on f(y∣x; β), the conditional probability (density) function for the response variable Y, given x, the covariates. Examples include missing data where the missingness is non-ignorable, sampling surveys in which subjects are selected on the basis of the Y-values and meta-analysis where published studies are subject to ‘selection bias’. The conventional approaches require the correct specification of the missingness mechanism, sampling probability and probability for being published respectively. In this paper, we propose an alternative estimating procedure for β based on an idea originated by Kalbfleisch. The novelty of this method is that no assumption on the missingness probability mechanisms etc. mentioned above is required to be specified. Asymptotic efficiency calculations and simulation studies were conducted to compare the method proposed with the two existing methods: the conditional likelihood and the weighted estimating function approaches.

Estimating equations with nonignorably missing response data.

Troxel, Lipsitz, and Brennan (1997, Biometrics 53, 857-869) considered parameter estimation from survey data with nonignorable nonresponse and proposed weighted estimating equations to remove the biases in the complete-case analysis that ignores missing observations. This paper suggests two alternative modifications for unbiased estimation of regression parameters when a binary outcome is potentially observed at successive time points. The weighting approach of Robins, Rotnitzky, and Zhao (1995, Journal of the American Statistical Association 90, 106-121) is also modified to obtain unbiased estimating functions. The suggested estimating functions are unbiased only when the missingness probability is correctly specified, and misspecification of the missingness model will result in biases in the estimates. Simulation studies are carried out to assess the performance of different methods when the covariate is binary or normal. For the simulation models used, the relative efficiency of the two new methods to the weighting methods is about 3.0 for the slope parameter and about 2.0 for the intercept parameter when the covariate is continuous and the missingness probability is correctly specified. All methods produce substantial biases in the estimates when the missingness model is misspecified or underspecified. Analysis of data from a medical survey illustrates the use and possible differences of these estimating functions.

Analysis of Longitudinal Data with Non-Ignorable Non-Monotone Missing Values

SUMMARY A full likelihood method is proposed to analyse continuous longitudinal data with non-ignorable (informative) missing values and non-monotone patterns. The problem arose in a breast cancer clinical trial where repeated assessments of quality of life were collected: patients rated their coping ability during and after treatment. We allow the missingness probabilities to depend on unobserved responses, and we use a multivariate normal model for the outcomes. A first-order Markov dependence structure for the responses is a natural choice and facilitates the construction of the likelihood; estimates are obtained via the Nelder–Mead simplex algorithm. Computations are difficult and become intractable with more than three or four assessments. Applying the method to the quality-of-life data results in easily interpretable estimates, confirms the suspicion that the data are non-ignorably missing and highlights the likely bias of standard methods. Although treatment comparisons are not affected here, the methods are useful for obtaining unbiased means and estimating trends over time.

Generalized Estimating Equation Model for Binary Outcomes with Missing Covariates

This paper presents an approach to handling missing covariates in the generalized estimating equation (GEE) model for binary outcomes when the probability of missingness depends on the observed outcomes and covariates. The proposed method is to replace the missing quantities in the estimating function with consistent estimates. In special cases, the proposed model reduces to a weighted GEE model for the completely observed units, where the weight is the inverse of the probability of missingness. Our method can be viewed as an extension of the mean score method by Reilly and Pepe (1995, Biometrika 82, 299-314) to the GEE context. Under certain regularity conditions, the estimates of the regression coefficients obtained by the proposed method are consistent and asymptotically normally distributed. The finite sample properties of the estimates are illustrated via computer simulations. An application to the study of dementia among stroke patients is presented.

On using the Cox proportional hazards model with missing covariates

SUMMARY We propose two methods for handling missing covariates in using the Cox proportional hazards model. The maximum partial likelihood estimator based only on study subjects having complete covariates does not utilise all available information. Also it is biased when the probability of missingness depends on the failure or censoring time. Our suggestion is to impute the conditional expectation of the statistic involving missing covariates given the available information. The proposed method provides a consistent regression parameter estimator when the probability of missingness depends on the failure or censoring time as well as on the observed covariates. Also the proposed estimator is more efficient than the estimator suggested previously by Lin & Ying (1993), when data are missing completely at random.

Estimation of Regression Coefficients When Some Regressors Are Not Always Observed

Abstract In applied problems it is common to specify a model for the conditional mean of a response given a set of regressors. A subset of the regressors may be missing for some study subjects either by design or happenstance. In this article we propose a new class of semiparametric estimators, based on inverse probability weighted estimating equations, that are consistent for parameter vector α0 of the conditional mean model when the data are missing at random in the sense of Rubin and the missingness probabilities are either known or can be parametrically modeled. We show that the asymptotic variance of the optimal estimator in our class attains the semiparametric variance bound for the model by first showing that our estimation problem is a special case of the general problem of parameter estimation in an arbitrary semiparametric model in which the data are missing at random and the probability of observing complete data is bounded away from 0, and then deriving a representation for the efficient score, the semiparametric variance bound, and the influence function of any regular, asymptotically linear estimator in this more general estimation problem. Because the optimal estimator depends on the unknown probability law generating the data, we propose locally and globally adaptive semiparametric efficient estimators. We compare estimators in our class with previously proposed estimators. We show that each previous estimator is asymptotically equivalent to some, usually inefficient, estimator in our class. This equivalence is a consequence of a proposition stating that every regular asymptotic linear estimator of α0 is asymptotically equivalent to some estimator in our class. We compare various estimators in a small simulation study and offer some practical recommendations.

Estimation of Regression Coefficients When Some Regressors are not Always Observed

Abstract In applied problems it is common to specify a model for the conditional mean of a response given a set of regressors. A subset of the regressors may be missing for some study subjects either by design or happenstance. In this article we propose a new class of semiparametric estimators, based on inverse probability weighted estimating equations, that are consistent for parameter vector α0 of the conditional mean model when the data are missing at random in the sense of Rubin and the missingness probabilities are either known or can be parametrically modeled. We show that the asymptotic variance of the optimal estimator in our class attains the semiparametric variance bound for the model by first showing that our estimation problem is a special case of the general problem of parameter estimation in an arbitrary semiparametric model in which the data are missing at random and the probability of observing complete data is bounded away from 0, and then deriving a representation for the efficient score...