Cox Regression with Covariate Measurement Error

This article deals with parameter estimation in the Cox proportional hazards model when covariates are measured with error. We consider both the classical additive measurement error model and a more general model which represents the mis-measured version of the covariate as an arbitrary linear function of the true covariate plus random noise. Only moment conditions are imposed on the distributions of the covariates and measurement error. Under the assumption that the covariates are measured precisely for a validation set, we develop a class of estimating equations for the vector-valued regression parameter by correcting the partial likelihood score function. The resultant estimators are proven to be consistent and asymptotically normal with easily estimated variances. Furthermore, a corrected version of the Breslow estimator for the cumulative hazard function is developed, which is shown to be uniformly consistent and, upon proper normalization, converges weakly to a zero-mean Gaussian process. Simulation studies indicate that the asymptotic approximations work well for practical sample sizes. The situation in which replicate measurements (instead of a validation set) are available is also studied.     

Estimation of dependence between paired correlated failure times in the presence of covariate measurement error

Summary. In many biomedical studies, covariates are subject to measurement error. Although it is well known that the regression coefficients estimators can be substantially biased if the measurement error is not accommodated, there has been little study of the effect of covariate measurement error on the estimation of the dependence between bivariate failure times. We show that the dependence parameter estimator in the Clayton–Oakes model can be considerably biased if the measurement error in the covariate is not accommodated. In contrast with the typical bias towards the null for marginal regression coefficients, the dependence parameter can be biased in either direction. We introduce a bias reduction technique for the bivariate survival function in copula models while assuming an additive measurement error model and replicated measurement for the covariates, and we study the large and small sample properties of the dependence parameter estimator proposed.     

Estimation of regression parameters in generalized linear models for cluster correlated data with measurement error

Liang and Zeger (1986) introduced a class of estimating equations that gives consistent estimates of regression parameters and of their asymptotic variances in the class of generalized linear models for cluster correlated data. When the independent variables or covariates in such models are subject to measurement errors, the parameter estimates obtained from these estimating equations are no longer consistent. To correct for the effect of measurement errors, an estimator with smaller asymptotic bias is constructed along the lines of Stefanski (1985), assuming that the measurement error variance is either known or estimable. The asymptotic distribution of the bias-corrected estimator and a consistent estimator of its asymptotic variance are also given. The special case of a binary logistic regression model is studied in detail. For this case, methods based on conditional scores and quasilikelihood are also extended to cluster correlated data. Results of a small simulation study on the performance of the proposed estimators and associated tests of hypotheses are reported.     

GMM nonparametric correction methods for logistic regression with error‐contaminated covariates and partially observed instrumental variables

We consider logistic regression with covariate measurement error. Most existing approaches require certain replicates of the error‐contaminated covariates, which may not be available in the data. We propose generalized method of moments (GMM) nonparametric correction approaches that use instrumental variables observed in a calibration subsample. The instrumental variable is related to the underlying true covariates through a general nonparametric model, and the probability of being in the calibration subsample may depend on the observed variables. We first take a simple approach adopting the inverse selection probability weighting technique using the calibration subsample. We then improve the approach based on the GMM using the whole sample. The asymptotic properties are derived, and the finite sample performance is evaluated through simulation studies and an application to a real data set.     

Approximate estimations in capture–recapture model with heteroscedastic measurement errors

In a capture–recapture experiment, the number of measurements for individual covariates usually equals the number of captures. This creates a heteroscedastic measurement error problem and the usual surrogate condition does not hold in the context of a measurement error model. This study adopts a small measurement error assumption to approximate the conventional estimating functions and the population size estimator. This study also investigates the biases of the resulting estimators. In addition, modifications for two common approximation methods, regression calibration and simulation extrapolation, to accommodate heteroscedastic measurement error are also discussed. These estimation methods are examined through simulations and illustrated by analysing a capture–recapture data set.     

Weighted denoised minimum distance estimation in a regression model with autocorrelated measurement errors

This paper deals with the linear regression model with measurement errors in both response and covariates. The variables are observed with errors together with an auxiliary variable, such as time, and the errors in response are autocorrelated. We propose a weighted denoised minimum distance estimator (WDMDE) for the regression coefficients. The consistency, asymptotic normality, and strong convergence rate of the WDMDE are proved. Compared with the usual denoised least squares estimator (DLSE) in the previous literature, the WDMDE is asymptotically more efficient in the sense of having smaller variances. It also avoids undersmoothing the regressor functions over the auxiliary variable, so that data-driven optimal choice of the bandwidth can be used. Furthermore, we consider the fitting of the error structure, construct the estimators of the autocorrelation coefficients and the error variances, and derive their large-sample properties. Simulations are conducted to examine the finite sample performance of the proposed estimators, and an application of our methodology to analyze a set of real data is illustrated as well.     

Statistical inference for a semiparametric measurement error regression model with heteroscedastic errors

Efficient inference for regression models requires that the heteroscedasticity be taken into account. We consider statistical inference under heteroscedasticity in a semiparametric measurement error regression model, in which some covariates are measured with errors. This paper has multiple components. First, we propose a new method for testing the heteroscedasticity. The advantages of the proposed method over the existing ones are that it does not need any nonparametric estimation and does not involve any mismeasured variables. Second, we propose a new two-step estimator for the error variances if there is heteroscedasticity. Finally, we propose a weighted estimating equation-based estimator (WEEBE) for the regression coefficients and establish its asymptotic properties. Compared with existing estimators, the proposed WEEBE is asymptotically more efficient, avoids undersmoothing the regressor functions and requires less restrictions on the observed regressors. Simulation studies show that the proposed test procedure and estimators have nice finite sample performance. A real data set is used to illustrate the utility of our proposed methods.     

SMALL AREA ESTIMATION USING SURVEY WEIGHTS WITH FUNCTIONAL MEASUREMENT ERROR IN THE COVARIATE

Nested error linear regression models using survey weights have been studied in small area estimation to obtain efficient model‐based and design‐consistent estimators of small area means. The covariates in these nested error linear regression models are not subject to measurement errors. In practical applications, however, there are many situations in which the covariates are subject to measurement errors. In this paper, we develop a nested error linear regression model with an area‐level covariate subject to functional measurement error. In particular, we propose a pseudo‐empirical Bayes (PEB) predictor to estimate small area means. This predictor borrows strength across areas through the model and makes use of the survey weights to preserve the design consistency as the area sample size increases. We also employ a jackknife method to estimate the mean squared prediction error (MSPE) of the PEB predictor. Finally, we report the results of a simulation study on the performance of our PEB predictor and associated jackknife MSPE estimator.     

Nonlinear regression models with general distortion measurement errors

This paper considers nonlinear regression models when neither the response variable nor the covariates can be directly observed, but are measured with both multiplicative and additive distortion measurement errors. We propose conditional variance and conditional mean calibration estimation methods for the unobserved variables, then a nonlinear least squares estimator is proposed. For the hypothesis testing of parameter, a restricted estimator under the null hypothesis and a test statistic are proposed. The asymptotic properties for the estimator and test statistic are established. Lastly, a residual-based empirical process test statistic marked by proper functions of the regressors is proposed for the model checking problem. We further suggest a bootstrap procedure to calculate critical values. Simulation studies demonstrate the performance of the proposed procedure and a real example is analysed to illustrate its practical usage.     

Efficient estimation for time series following generalized linear models

In this paper, we consider James–Stein shrinkage and pretest estimation methods for time series following generalized linear models when it is conjectured that some of the regression parameters may be restricted to a subspace. Efficient estimation strategies are developed when there are many covariates in the model and some of them are not statistically significant. Statistical properties of the pretest and shrinkage estimation methods including asymptotic distributional bias and risk are developed. We investigate the relative performances of shrinkage and pretest estimators with respect to the unrestricted maximum partial likelihood estimator (MPLE). We show that the shrinkage estimators have a lower relative mean squared error as compared to the unrestricted MPLE when the number of significant covariates exceeds two. Monte Carlo simulation experiments were conducted for different combinations of inactive covariates and the performance of each estimator was evaluated in terms of its mean squared error. The practical benefits of the proposed methods are illustrated using two real data sets.     

A simulation study of estimators for generalized linear measurement error models

The purpose of this paper is to examine the properties of several bias-corrected estimators for generalized linear measurement error models, along with the naive estimator, in some special settings. In particular, we consider logistic regression, poisson regression and exponential-gamma models where the covariates are subject to measurement error. Monte Carlo experiments are conducted to compare the relative performance of the estimators in terms of several criteria. The results indicate that the naive estimator of slope is biased towards zero by a factor increasing with the magnitude of slope and measurement error as well as the sample size. It is found that none of the biased-corrected estimators always outperforms the others, and that their small sample properties typically depend on the underlying model assumptions.     

Failure time regression with continuous covariates measured with error

We consider failure time regression analysis with an auxiliary variable in the presence of a validation sample. We extend the nonparametric inference procedure of Zhou and Pepe to handle a continuous auxiliary or proxy covariate. We estimate the induced relative risk function with a kernel smoother and allow the selection probability of the validation set to depend on the observed covariates. We present some asymptotic properties for the kernel estimator and provide some simulation results. The method proposed is illustrated with a data set from an on-going epidemiologic study.     

On partial linear additive isotonic regression

In this paper we discuss semiparametric additive isotonic regression models. We discuss the efficiency bound of the model and the least squares estimator under this model. We show that the ordinary least square estimator studied by Huang (2002) and Cheng (2009) for the semiparametric isotonic regression achieves the efficiency bound for the regular estimator when the true parameter belongs to the interior of the parameter space. We also show that the result by Cheng (2009) can be generalized to the case that the covariates are dependent on each other.     

On the simple linear regression model with correlated measurement errors

The simple linear regression model with measurement error has been subject to much research. In this work we will focus on this model when the error in the explanatory variable is correlated with the error in the regression equation. Specifically, we are interested in the comparison between the ordinary errors-in-variables estimator of the regression coefficient β$\beta$ and the estimator that takes account of the correlation between the errors. Based on large sample approximations, we compare the estimators and find that the estimator that takes account of the correlation should be preferred in most situations. We also compare the estimators in small sample situations. This is done by stochastic simulation. The results show that the estimators behave quite similarly in most of the simulated situations, but that the ordinary errors-in-variables estimator performs considerably worse than the estimator that takes account of the correlation for certain parameter combinations. In addition, we look briefly into the bias introduced by ignoring correlated errors when computing sample correlations, and in predictions.     

Approaches to regression analysis with multiple measurements from individual sampling units

Application of ordinary least-squares regression to data sets which contain multiple measurements from individual sampling units produces an unbiased estimator of the parameters but a biased estimator of the covariance matrix of the parameter estimates. The present work considers a random coefficient, linear model to deal with such data sets: this model permits many senses in which multiple measurements are taken from a sampling unit, not just when it is measured at several times. Three procedures to estimate the covariance matrix of the error term of the model are considered. Given these, three procedures to estimate the parameters of the model and their covariance matrix are considered; these are ordinary least-squares, generalized least-squares, and an adjusted ordinary least-squares procedure which produces an unbiased estimator of the covariance matrix of the parameters with small samples. These various procedures are compared in simulation studies using three examples from the biological literature. The possibility of testing hypotheses about the vector of parameters is also considered. It is found that all three procedures for regression estimation produce estimators of the parameters with bias of no practical consequence, Both generalized least-squares and adjusted ordinary least-squares generally produce estimators of the covariance matrix of the parameter estimates with bias of no practical consequence, while ordinary least-squares produces a negatively biased estimator. Neither ordinary nor generalized least-squares provide satisfactory hypothesis tests of the vector of parameter estimates. It is concluded that adjusted ordinary least-squares, when applied with either of two of the procedures used to estimate the error coveriance matrix, shows promise for practical application with data sets of the nature considered here.     

Estimation in zero-inflated binomial regression with missing covariates

We investigate inverse-probability-weighted (IPW) maximum likelihood estimation in zero-inflated binomial regression with missing-at-random covariates. Large sample properties (consistency, asymptotic normality) of the IPW estimator are established. Finite sample properties are assessed via simulations. The methodology is illustrated on a real data set.     

Generalized M-estimation for the accelerated failure time model

The accelerated failure time (AFT) model is an important regression tool to study the association between failure time and covariates. In this paper, we propose a robust weighted generalized M (GM) estimation for the AFT model with right-censored data by appropriately using the Kaplan–Meier weights in the GM–type objective function to estimate the regression coefficients and scale parameter simultaneously. This estimation method is computationally simple and can be implemented with existing software. Asymptotic properties including the root-n consistency and asymptotic normality are established for the resulting estimator under suitable conditions. We further show that the method can be readily extended to handle a class of nonlinear AFT models. Simulation results demonstrate satisfactory finite sample performance of the proposed estimator. The practical utility of the method is illustrated by a real data example.     

A GEOMETRIC CHARACTERIZATION OF LINEAR REGRESSION

This paper concerns the asymptotic properties of the maximum likelihood estimators of the parameters in a non regular Cox model involving a change-point in the regression on time-dependent covariates. The global consistency derives from the uniform convergence of the partial log-likelihood. We prove that the estimator of the change-point is n -consistent and the estimator of the regression parameter n 1/2 -consistent, and their asymptotic distributions are established.     

Some conditions for optimality of the almost unbiased estimators of regression coefficients

The purpose of this paper is to examine the asymptotic properties of the operational almost unbiased estimator of regression coefficients which includes almost unbiased ordinary ridge estimator a s a special case. The small distrubance approximations for the bias and mean square error matrix of the estimator are derived. As a consequence, it is proved that, under certain conditions, the estimator is more efficient than a general class of estimators given by Vinod and Ullah (1981). Also it is shown that, if the ordinary ridge estimator (ORE) dominates the ordinary least squares estimator then the almost unbiased ordinary ridge estimator does not dominate ORE under the mean square error criterion.     

The LAD estimation of the change-point linear model with randomly censored data

Abstract

In this paper, a change-point linear model with randomly censored data is investigated. We propose the least absolute deviation estimation procedure for regression and change-point parameters simultaneously. The asymptotic properties of the change-point and regression parameter estimators are obtained. We show that the resulting regression parameter estimator is asymptotically normal, and the change-point estimator converges weakly to the minimizer of a given random process. The extensive simulation studies and the analysis of an acute myocardial infarction data set are conducted to illustrate the finite sample performance of the proposed method.