Assessing large sample bias in misspecified model scenarios with reference to exposure model misspecification in errors-in-variable regression: A new computational approach

In this paper, we develop a numerical method for evaluating the large sample bias in estimated regression coefficients arising due to exposure model misspecification while adjusting for measurement errors in errors-in-variable regression. The application of the proposed method has been demonstrated in the case of a logistic errors-in-variable regression model. The method is based on the combination of Monte-Carlo, numerical and, in some special cases, analytic integration techniques. The proposed method facilitates the investigation of the limiting bias in the estimated regression parameters based on a single data set rather than on repeated data sets as required by the conventional repeated sample method. Simulation studies demonstrate that the proposed method provides very similar estimates of bias in the estimated regression parameters under exposure model misspecification in logistic errors-in-variable regression with a higher degree of precision as compared to the conventional repeated sample method.     

A Simple Corrected Score for Logistic Regression with Errors-in-Covariates

We develop a simple corrected score for logistic regression with errors-in-covariates. The new method is an extension of the consistent functional methods proposed by Huang and Wang (2001) and is closely related to the corrected score method by Nakamura (1990 Nakamura, T. (1990). Corrected score function for errors-in-variables models: Methodology and application to generalized linear models. Biometrika. 77:127–137.[Crossref], [Web of Science ®] , [Google Scholar]) and Stefanski (1989 Stefanski, L.A. (1989). Unbiased estimation of a nonlinear function a normal mean with application to measurement error models. Commun. Stat. Ser. A - Theory Methods. 18:4335–4358.[Taylor & Francis Online], [Web of Science ®] , [Google Scholar]). The new method requires that the measurement error distribution is known, but does not require normality. The new method yields a consistent and asymptotically normal estimator under regularity conditions. We examine the finite-sample performance of the new estimator through simulation studies. Finally, we illustrate the new method by applying it to an AIDS study.     

Estimating Survival Curves Under Proportional Hazards Model with Covariate Measurement Errors

For the Cox proportional hazards model with additive covariate measurement errors, we propose a corrected cumulative baseline hazard estimator that reduces the bias of the na]ve Breslow estimator. We also derive corresponding modified estimators for the hazard functions and the survival functions of individuals with particular covariate values. Using a Monte Carlo technique developed by Lin et al . (1994), we construct confidence bands for such hazard and survival functions.     

Objective Priors for Parameters in a Normal Linear Regression with Measurement Error

We investigate certain objective priors for the parameters in a normal linear regression models with one of the explanatory variables subject to measurement error. We first show that the use of the standard non informative prior for normal linear regression without measurement error leads to an improper posterior in the measurement error model. We then derive the Jeffreys prior and reference priors, and show that they lead to proper posteriors. We use simulation study to compare the frequentist performance of the estimates derived using these priors, and the MLE.     

Accounting for Response Misclassification and Covariate Measurement Error Using a Random Effects Logit Model

Often in longitudinal data arising out of epidemiologic studies, measurement error in covariates and/or classification errors in binary responses may be present. The goal of the present work is to develop a random effects logistic regression model that corrects for the classification errors in binary responses and/or measurement error in covariates. The analysis is carried out under a Bayesian set up. Simulation study reveals the effect of ignoring measurement error and/or classification errors on the estimates of the regression coefficients.     

A Class of Estimators Using Auxiliary Information for Estimating Finite Population Variance in Presence of Measurement Errors

This article addresses the problem of estimating the population variance using auxiliary information in the presence of measurement errors. When the measurement error variance associated with study variable is known, a class of estimators of the population variance using auxiliary information has been proposed. We obtain the bias and mean squared errors of the suggested class of estimators upto the terms of order n ^?1, and also optimum estimators in asymptotic sense of the class with approximate mean squared error formula.     

Regression calibration for logistic regression with multiple surrogates for one exposure

Methods have been developed by several authors to address the problem of bias in regression coefficients due to errors in exposure measurement. These approaches typically assume that there is one surrogate for each exposure. Occupational exposures are quite complex and are often described by characteristics of the workplace and the amount of time that one has worked in a particular area. In this setting, there are several surrogates which are used to define an individual's exposure. To analyze this type of data, regression calibration methodology is extended to adjust the estimates of exposure-response associations for the bias and additional uncertainty due to exposure measurement error from multiple surrogates. The health outcome is assumed to be binary and related to the quantitative measure of exposure by a logistic link function. The model for the conditional mean of the quantitative exposure measurement in relation to job characteristics is assumed to be linear. This approach is applied to a cross-sectional epidemiologic study of lung function in relation to metal working fluid exposure and the corresponding exposure assessment study with quantitative measurements from personal monitors. A simulation study investigates the performance of the proposed estimator for various values of the baseline prevalence of disease, exposure effect and measurement error variance. The efficiency of the proposed estimator relative to the one proposed by Carroll et al. [1995. Measurement Error in Nonlinear Models. Chapman & Hall, New York] is evaluated numerically for the motivating example. User-friendly and fully documented Splus and SAS routines implementing these methods are available (http://www.hsph.harvard.edu/faculty/spiegelman/multsurr.html).     

Two Pitfalls of Using Standard Regression Diagnostics When Both X and Y Have Measurement Error

Modern exploratory data analysis produces models that are not based on physical theory but that are consistent with pictures of the data. When both X and Y have error this can be risky, because important features are hidden. Two examples are given that show that systematic model departures and heteroscedasticity may not be detectable with standard regression diagnostics.     

Transformations for Smooth Regression Models with Multiplicative Errors

We consider whether one should transform to estimate nonparametrically a regression curve sampled from data with a constant coefficient of variation, i.e. with multiplicative errors. Kernel-based smoothing methods are used to provide curve estimates from the data both in the original units and after transformation. Comparisons are based on the mean-squared error (MSE) or mean integrated squared error (MISE), calculated in the original units. Even when the data are generated by the simplest multiplicative error model, the asymptotically optimal MSE (or MISE) is surprisingly not always obtained by smoothing transformed data, but in many cases by directly smoothing the original data. Which method is optimal depends on both the regression curve and the distribution of the errors. Data-based procedures which could be useful in choosing between transforming and not transforming a particular data set are discussed. The results are illustrated on simulated and real data.     

Optimal Designs for Random Coefficient Regression Models with Heteroscedastic Errors

The purpose of this article is to present the optimal designs based on D-, G-, A-, I-, and D _β-optimality criteria for random coefficient regression (RCR) models with heteroscedastic errors. A sufficient condition for the heteroscedastic structure is given to make sure that the search of optimal designs can be confined at extreme settings of the design region when the criteria satisfy the assumption of the real valued monotone design criteria. Analytical solutions of D-, G-, A-, I-, and D _β-optimal designs for the RCR models are derived. Two examples are presented for random slope models with specific heteroscedastic errors.     

Selecting the Optimal Transformation of a Continuous Covariate in Cox's Regression: Implications for Hypothesis Testing

ABSTRACT

Background: Many exposures in epidemiological studies have nonlinear effects and the problem is to choose an appropriate functional relationship between such exposures and the outcome. One common approach is to investigate several parametric transformations of the covariate of interest, and to select a posteriori the function that fits the data the best. However, such approach may result in an inflated Type I error. Methods: Through a simulation study, we generated data from Cox's models with different transformations of a single continuous covariate. We investigated the Type I error rate and the power of the likelihood ratio test (LRT) corresponding to three different procedures that considered the same set of parametric dose-response functions. The first unconditional approach did not involve any model selection, while the second conditional approach was based on a posteriori selection of the parametric function. The proposed third approach was similar to the second except that it used a corrected critical value for the LRT to ensure a correct Type I error. Results: The Type I error rate of the second approach was two times higher than the nominal size. For simple monotone dose-response, the corrected test had similar power as the unconditional approach, while for non monotone, dose-response, it had a higher power. A real-life application that focused on the effect of body mass index on the risk of coronary heart disease death, illustrated the advantage of the proposed approach. Conclusion: Our results confirm that a posteriori selecting the functional form of the dose-response induces a Type I error inflation. The corrected procedure, which can be applied in a wide range of situations, may provide a good trade-off between Type I error and power.     

Empirical Bayes Estimation of Small Area Means under a Nested Error Linear Regression Model with Measurement Errors in the Covariates

Abstract.  Previously, small area estimation under a nested error linear regression model was studied with area level covariates subject to measurement error. However, the information on observed covariates was not used in finding the Bayes predictor of a small area mean. In this paper, we first derive the fully efficient Bayes predictor by utilizing all the available data. We then estimate the regression and variance component parameters in the model to get an empirical Bayes (EB) predictor and show that the EB predictor is asymptotically optimal. In addition, we employ the jackknife method to obtain an estimator of mean squared prediction error (MSPE) of the EB predictor. Finally, we report the results of a simulation study on the performance of our EB predictor and associated jackknife MSPE estimators. Our results show that the proposed EB predictor can lead to significant gain in efficiency over the previously proposed EB predictor.     

Quasi-Minimax Estimation in the Linear Model with Measurement Errors

In this article, we consider quasi-minimax estimation in the linear regression model where some covariates are measured with additive errors. When measurement errors are directly ignored the minimax risk of the resulting estimator can be large. By correcting the attenuation we propose a penalized quadratic risk function. A simulation study is conducted to illustrate the performance of the proposed estimators.     

Likelihood-Based Inference for Multivariate Skew-Normal Regression Models

In this article, we present EM algorithms for performing maximum likelihood estimation for three multivariate skew-normal regression models of considerable practical interest. We also consider the restricted estimation of the parameters of certain important special cases of two models. The methodology developed is applied in the analysis of longitudinal data on dental plaque and cholesterol levels.     

Polynomial Regression and Estimating Functions in the Presence of Multiplicative Measurement Error

We consider the polynomial regression model in the presence of multiplicative measurement error in the predictor. Two general methods are considered, with the methods differing in their assumptions about the distributions of the predictor and the measurement errors. Consistent parameter estimates and asymptotic standard errors are derived by using estimating equation theory. Diagnostics are presented for distinguishing additive and multiplicative measurement error. Data from a nutrition study are analysed by using the methods. The results from a simulation study are presented and the performances of the methods are compared.     

An Estimate of a Change Point in Variance of Measurement Errors and Its Convergence Rate

In this article, an estimate of a change point in variance of measurement errors (ME) is given in terms of characteristic functions when the variances are known. Its modification is also given for the case that the variances are unknown. In addition, the consistency and convergence rates of the estimator and its modification are investigated. The simulation study shows that the proposed estimators perform well.     

A Semiparametric Approach for Accelerated Failure Time Models with Covariates Subject to Measurement Error

There are relatively few discussions about measurement error in the accelerated failure time (AFT) model, particularly for the semiparametric AFT model. In this article, we propose an adjusted estimation procedure for the semiparametric AFT model with covariates subject to measurement error, based on the profile likelihood approach and simulation and exploration (SIMEX) method. The simulation studies show that the proposed semiparametric SIMEX approach performs well. The proposed approach is applied to a coronary heart disease dataset from the Busselton Health study for illustration.     

Phi-Divergence Statistics for Testing Linear Hypotheses in Logistic Regression Models

In this paper we introduce and study two new families of statistics for the problem of testing linear combinations of the parameters in logistic regression models. These families are based on the phi-divergence measures. One of them includes the classical likelihood ratio statistic and the other the classical Pearson's statistic for this problem. It is interesting to note that the vector of unknown parameters, in the two new families of phi-divergence statistics considered in this paper, is estimated using the minimum phi-divergence estimator instead of the maximum likelihood estimator. Minimum phi-divergence estimators are a natural extension of the maximum likelihood estimator.     

A Goodness-of-Fit Test for Logistic Regression Models in Stratified Case-Control Studies via Empirical Likelihood

In the literature, there were only a few reports on goodness-of-fit tests on logistic regression models specifically derived for case-control studies. In this article, we propose a goodness-of-fit test for logistic regression models in stratified case-control studies using an empirical likelihood approach. The proposed statistic is an alternative to the statistic G _o , recently proposed by Arbigast and Lin (2005 Arbigast , P. G. , Lin , D. Y. ( 2005 ). Model-checking techniques for stratified case-control studies . Statist. Med. 24 : 229 – 247 . [Google Scholar]). Simulation results show that the proposed statistic is often slightly more powerful than G _o , although their performances are always close to each other. Moreover, implementation of our method is easy since the usual stratified logistic regression procedures in many statistical softwares can be employed. Some asymptotic results and application of the proposed statistic to two real datasets are also presented.