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.     

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.     

Bias corrected estimation for generalized probit regression with covariate measurement error and censored responses

In this paper, we propose a bias corrected estimate of the regression coefficient for the generalized probit regression model when the covariates are subject to measurement error and the responses are subject to interval censoring. The main improvement of our method is that it reduces most of the bias that the naive estimates have. The great advantage of our method is that it is baseline and censoring distribution free, in a sense that the investigator does not need to calculate the baseline or the censoring distribution to obtain the estimator of the regression coefficient, an important property of Cox regression model. A sandwich estimator for the variance is also proposed. Our procedure can be generalized to general measurement error distribution as long as the first four moments of the measurement error are known. The results of extensive simulations show that our approach is very effective in eliminating the bias when the measurement error is not too large relative to the error term of the regression model.     

A risk set calibration method for failure time regression by using a covariate reliability sample

Regression parameter estimation in the Cox failure time model is considered when regression variables are subject to measurement error. Assuming that repeat regression vector measurements adhere to a classical measurement model, we can consider an ordinary regression calibration approach in which the unobserved covariates are replaced by an estimate of their conditional expectation given available covariate measurements. However, since the rate of withdrawal from the risk set across the time axis, due to failure or censoring, will typically depend on covariates, we may improve the regression parameter estimator by recalibrating within each risk set. The asymptotic and small sample properties of such a risk set regression calibration estimator are studied. A simple estimator based on a least squares calibration in each risk set appears able to eliminate much of the bias that attends the ordinary regression calibration estimator under extreme measurement error circumstances. Corresponding asymptotic distribution theory is developed, small sample properties are studied using computer simulations and an illustration is provided.     

A New Estimator for a Population Proportion Using Group Testing

The maximum likelihood estimator is widely used in estimating the population proportion using group testing. However, it is positive biased and some alternatives have been raised in literatures. In this study, we propose a new estimator by weighted combination of order statistics. Two rules are supplied to determine the unknown weight. Using the rule of minimizing the absolute bias, our estimator is almost unbiased in most cases shown by simulations. Using the rule of minimizing the mean square error, a simple estimator with weight 1 is recommended for its good performance.     

A new biased estimator based on ridge estimation

In this paper we introduce a new biased estimator for the vector of parameters in a linear regression model and discuss its properties. We show that our new biased estimator is superior, in the mean square error(mse) sense, to the ordinary least squares (OLS) estimator, the ordinary ridge regression (ORR) estimator and the Liu estimator. We also compare the performance of our new biased estimator with two other special Liu-type estimators proposed in Liu (2003). We illustrate our findings with a numerical example based on the widely analysed dataset on Portland cement.     

A corrected likelihood method for the proportional hazards model with covariates subject to measurement error

There has been extensive interest in discussing inference methods for survival data when some covariates are subject to measurement error. It is known that standard inferential procedures produce biased estimation if measurement error is not taken into account. With the Cox proportional hazards model a number of methods have been proposed to correct bias induced by measurement error, where the attention centers on utilizing the partial likelihood function. It is also of interest to understand the impact on estimation of the baseline hazard function in settings with mismeasured covariates. In this paper we employ a weakly parametric form for the baseline hazard function and propose simple unbiased estimating functions for estimation of parameters. The proposed method is easy to implement and it reveals the connection between the naive method ignoring measurement error and the corrected method with measurement error accounted for. Simulation studies are carried out to evaluate the performance of the estimators as well as the impact of ignoring measurement error in covariates. As an illustration we apply the proposed methods to analyze a data set arising from the Busselton Health Study [Knuiman, M.W., Cullent, K.J., Bulsara, M.K., Welborn, T.A., Hobbs, M.S.T., 1994. Mortality trends, 1965 to 1989, in Busselton, the site of repeated health surveys and interventions. Austral. J. Public Health 18, 129–135].     

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.     

Analysis of error-prone survival data under additive hazards models: measurement error effects and adjustments

Covariate measurement error occurs commonly in survival analysis. Under the proportional hazards model, measurement error effects have been well studied, and various inference methods have been developed to correct for error effects under such a model. In contrast, error-contaminated survival data under the additive hazards model have received relatively less attention. In this paper, we investigate this problem by exploring measurement error effects on parameter estimation and the change of the hazard function. New insights of measurement error effects are revealed, as opposed to well-documented results for the Cox proportional hazards model. We propose a class of bias correction estimators that embraces certain existing estimators as special cases. In addition, we exploit the regression calibration method to reduce measurement error effects. Theoretical results for the developed methods are established, and numerical assessments are conducted to illustrate the finite sample performance of our methods.     

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.     

A New Estimator for Cox Proportional Hazard Regression Model in Presence of Collinearity

We propose a new approach to estimate the parameters of the Cox proportional hazards model in the presence of collinearity. Generally, a maximum partial likelihood estimator is used to estimate parameters for the Cox proportional hazards model. However, the maximum partial likelihood estimators can be seriously affected by the presence of collinearity since the parameter estimates result in large variances.

In this study, we develop a Liu-type estimator for Cox proportional hazards model parameters and compare it with a ridge regression estimator based on the scalar mean squared error (MSE). Finally, we evaluate its performance through a simulation study.     

SIMEX method for censored quantile regression with measurement error

Censored quantile regression serves as an important supplement to the Cox proportional hazards model in survival analysis. In addition to being exposed to censoring, some covariates may subject to measurement error. This leads to substantially biased estimate without taking this error into account. The SIMulation-EXtrapolation (SIMEX) method is an effective tool to handle the measurement error issue. We extend the SIMEX approach to the censored quantile regression with covariate measurement error. The algorithm is assessed via extensive simulations. A lung cancer study is analyzed to verify the validation of the proposed method.     

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.     

Estimation of Main Effect When Covariates Have Non-Proportional Hazards

The Cox proportional hazards (PH) regression model has been widely used to analyze survival data in clinical trials and observational studies. In addition to estimating the main treatment or exposure group effect, it is common to adjust for additional covariates using the Cox model. It is well known that violation of the PH assumption can lead to estimates that are biased and difficult to interpret, and model checking has become a routine procedure. However, such checking might focus on the primary group comparisons, and the assumption can still be violated when adjusting for many of the potential covariates. We study the effect of violation of the PH assumption of the covariates on the estimation of the main group effect in the Cox model. The results are summarized in terms of the bias and the coverage properties of the confidence intervals. Overall in randomized clinical trials, the bias caused by misspecifying the PH assumption on the covariates is no more than 15% in absolute value regardless of sample size. In observational studies where the covariates are likely correlated with the group variable, however, the bias can be very severe. The coverage properties largely depend on sample size, as expected, as bias becomes dominating with increasing sample size. These findings should serve as cautionary notes when adjusting for potential confounders in observational studies, as the violation of PH assumption on the confounders can lead to erroneous results.     

On bias reduction estimators of skew-normal and skew-t distributions

A particular concerns of researchers in statistical inference is bias in parameters estimation. Maximum likelihood estimators are often biased and for small sample size, the first order bias of them can be large and so it may influence the efficiency of the estimator. There are different methods for reduction of this bias. In this paper, we proposed a modified maximum likelihood estimator for the shape parameter of two popular skew distributions, namely skew-normal and skew-t, by offering a new method. We show that this estimator has lower asymptotic bias than the maximum likelihood estimator and is more efficient than those based on the existing methods.     

Jackknife Estimation for Smooth Functions of the Parametric Component in Partially Linear Regression Models

Abstract

It is known that due to the existence of the nonparametric component, the usual estimators for the parametric component or its function in partially linear regression models are biased. Sometimes this bias is severe. To reduce the bias, we propose two jackknife estimators and compare them with the naive estimator. All three estimators are shown to be asymptotically equivalent and asymptotically normally distributed under some regularity conditions. However, through simulation we demonstrate that the jackknife estimators perform better than the naive estimator in terms of bias when the sample size is small to moderate. To make our results more useful, we also construct consistent estimators of the asymptotic variance, which are robust against heterogeneity of the error variances.     

Markov regression models for count time series with excess zeros: A partial likelihood approach

Count data with excess zeros are common in many biomedical and public health applications. The zero-inflated Poisson (ZIP) regression model has been widely used in practice to analyze such data. In this paper, we extend the classical ZIP regression framework to model count time series with excess zeros. A Markov regression model is presented and developed, and the partial likelihood is employed for statistical inference. Partial likelihood inference has been successfully applied in modeling time series where the conditional distribution of the response lies within the exponential family. Extending this approach to ZIP time series poses methodological and theoretical challenges, since the ZIP distribution is a mixture and therefore lies outside the exponential family. In the partial likelihood framework, we develop an EM algorithm to compute the maximum partial likelihood estimator (MPLE). We establish the asymptotic theory of the MPLE under mild regularity conditions and investigate its finite sample behavior in a simulation study. The performances of different partial-likelihood based model selection criteria are compared in the presence of model misspecification. Finally, we present an epidemiological application to illustrate the proposed methodology.     

Conditional estimation using prior information in 2‐stage group sequential designs assuming asymptotic normality when the trial terminated early

Two‐stage designs are widely used to determine whether a clinical trial should be terminated early. In such trials, a maximum likelihood estimate is often adopted to describe the difference in efficacy between the experimental and reference treatments; however, this method is known to display conditional bias. To reduce such bias, a conditional mean‐adjusted estimator (CMAE) has been proposed, although the remaining bias may be nonnegligible when a trial is stopped for efficacy at the interim analysis. We propose a new estimator for adjusting the conditional bias of the treatment effect by extending the idea of the CMAE. This estimator is calculated by weighting the maximum likelihood estimate obtained at the interim analysis and the effect size prespecified when calculating the sample size. We evaluate the performance of the proposed estimator through analytical and simulation studies in various settings in which a trial is stopped for efficacy or futility at the interim analysis. We find that the conditional bias of the proposed estimator is smaller than that of the CMAE when the information time at the interim analysis is small. In addition, the mean‐squared error of the proposed estimator is also smaller than that of the CMAE. In conclusion, we recommend the use of the proposed estimator for trials that are terminated early for efficacy or futility.     

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.     

Nonparametric Wavelet Regression Based on Biased Data

The estimation of the regression function in the biased nonparametric regression model is investigated. We propose and develop a new wavelet-based methodology for this problem. In particular, an adaptive hard thresholding wavelet estimator is constructed. Under mild assumptions on the model, we prove that it enjoys powerful mean integrated squared error properties over Besov balls.