Corrected Score Estimator Research Articles

Ridge estimation in linear mixed measurement error models using generalized maximum entropy

In this article, we concentrate on the generalized maximum entropy (GME) estimators and their asymptotic properties in linear mixed models (LMMs) with measurement error in the fixed effects (MEFE) variables. Moreover, we obtain the Ridge-GME estimator in these models as a remedy to collinearity. Finally, a simulation study and an example of real data are given to characterize the superiority of the Ridge-GME estimator over the corrected score estimator (CSE) and ridge estimator (RE), using the mean squared error matrix (MSEM).

Corrected score estimators in linear multivariate multiple regression models with heteroscedastic measurement errors

Corrected score estimators in linear multivariate multiple regression models with heteroscedastic measurement errors

An improved corrected score estimator for the proportional hazards model with time-dependent covariates measured with error at informative observation times

An improved corrected score estimator for the proportional hazards model with time-dependent covariates measured with error at informative observation times

Asymptotic properties of the corrected score estimator in the autoregressive model with measurement errors

The autoregressive model with errors in variables with normally distributed control se- quence is considered. For the main sequence, two cases are dealt with: (a) main sequence has station- ary distribution, and (b) initial distribution is arbitrary, independent of the control sequence and has finite fourth moment. Here the elements of the main sequence are not observed directly, but surrogate data that include a normally distributed additive error are observed. Errors and main sequence are assumed to be mutually independent. We estimate unknown parameter using the Corrected Score method and in both cases prove strict consistency and asymptotic normality of the estimator. To prove asymptotic normality we apply the theory of strong mixing sequences. Finally, we compare the efficiency of the Least Squares (naive) estimator and the Corrected Score estimator in the forecasting problem and conclude that the naive estimator gives better forecast.

Bias Correction Methods for Misclassified Covariates in the Cox Model: comparison offive correction methods by simulation and data analysis.

Measurement error/misclassification is commonplace in research when variable(s) can notbe measured accurately. A number of statistical methods have been developed to tackle this problemin a variety of settings and contexts. However, relatively few methods are available to handlemisclassified categorical exposure variable(s) in the Cox proportional hazards regression model. Inthis paper, we aim to review and compare different methods to handle this problem - naïvemethods, regression calibration, pooled estimation, multiple imputation, corrected score estimation,and MC-SIMEX - by simulation. These methods are also applied to a life course study with recalleddata and historical records. In practice, the issue of measurement error/misclassification should beaccounted for in design and analysis, whenever possible. Also, in the analysis, it could be moreideal to implement more than one correction method for estimation and inference, with properunderstanding of underlying assumptions.

Local influence for functional comparative calibration models with replicated data

We investigate local influence analysis in functional comparative calibration models with replicated data. A method for selecting appropriate perturbation schemes based on the expected Fisher information matrix with respect to the perturbation vector is proposed. It is shown that arbitrarily perturbing these models may result in misleading inference about the influential subjects. First-order influence measures for identifying the correct influential subjects and replicates on corrected score estimators are defined. We introduce different perturbation schemes including perturbation of subjects and replicates on the corrected likelihood function and obtain the density of the perturbed model from which the methodology is based. Particularly, three perturbation of variances schemes could be a better way to handle badly modeled subjects or replicates. Two real data sets are analyzed to illustrate the use of our local influence measures.

Influence measures on corrected score estimators in functional heteroscedastic measurement error models

This paper deals with the local influence assessment of the effects of minor perturbations of data on corrected score estimators in the functional heteroscedastic measurement error models with known variances. By extending to the context of measurement error models the differential-geometrical framework proposed by Zhu et al. [H.T. Zhu, J.G. Ibrahim, S. Lee, H. Zhang, Perturbation selection and influence measures in local influence analysis, The Annals of Statistics 35 (2007) 2565–2588], an n-dimensional Riemannian manifold is defined. The associated metric tensor is utilized for the selection of appropriate perturbation schemes. The Levi-Civita connection and first and second derivatives of the corrected score estimators are used to construct influence measures. Simple formulas are obtained under different perturbation schemes. A comparison with the slope and curvature based diagnostics defined from the surface of the corrected score estimators formed by perturbation is included. A real data application and a simulated example illustrate the performance of the proposed diagnostics.

Asymptotic properties of simple linear measurement error models

The main ob jective of this paper is to study estimators of regression models on the independent variable X which is not directly observed for some reasons. In such a situation, a substitute variable W is observed instead. This substitution complicates the statistical analysis of the observed data when the purpose of the analysis is inference about a model defined in terms of X. The substitution causes a inconsistent estimator; this is defined as a measurement error problem. To correct this problem, the conditional score and corrected score methods are proposed by Stefanski&Carroll (1985) and Nakamura (1990), respectively. In this study, large sample distribution theory for both the conditional score and corrected score estimators are derived and the performance of the estimators and the adequacy of the large sample distribution theory are obtained via Monte Carlo simulation.

Joint modeling of survival time and longitudinal data with subject‐specific changepoints in the covariates

Joint models are frequently used in survival analysis to assess the relationship between time-to-event data and time-dependent covariates, which are measured longitudinally but often with errors. Routinely, a linear mixed-effects model is used to describe the longitudinal data process, while the survival times are assumed to follow the proportional hazards model. However, in some practical situations, individual covariate profiles may contain changepoints. In this article, we assume a two-phase polynomial random effects with subject-specific changepoint model for the longitudinal data process and the proportional hazards model for the survival times. Our main interest is in the estimation of the parameter in the hazards model. We incorporate a smooth transition function into the changepoint model for the longitudinal data and develop the corrected score and conditional score estimators, which do not require any assumption regarding the underlying distribution of the random effects or that of the changepoints. The estimators are shown to be asymptotically equivalent and their finite-sample performance is examined via simulations. The methods are applied to AIDS clinical trial data.

Optimality of the quasi-score estimator in a mean–variance model with applications to measurement error models

We consider a regression of y on x given by a pair of mean and variance functions with a parameter vector θ to be estimated that also appears in the distribution of the regressor variable x . The estimation of θ is based on an extended quasi-score (QS) function. We show that the QS estimator is optimal within a wide class of estimators based on linear-in- y unbiased estimating functions. Of special interest is the case where the distribution of x depends only on a subvector α of θ , which may be considered a nuisance parameter. In general, α must be estimated simultaneously together with the rest of θ , but there are cases where α can be pre-estimated. A major application of this model is the classical measurement error model, where the corrected score (CS) estimator is an alternative to the QS estimator. We derive conditions under which the QS estimator is strictly more efficient than the CS estimator.

Semiparametric Approaches for Joint Modeling of Longitudinal and Survival Data with Time‐Varying Coefficients

We study joint modeling of survival and longitudinal data. There are two regression models of interest. The primary model is for survival outcomes, which are assumed to follow a time-varying coefficient proportional hazards model. The second model is for longitudinal data, which are assumed to follow a random effects model. Based on the trajectory of a subject's longitudinal data, some covariates in the survival model are functions of the unobserved random effects. Estimated random effects are generally different from the unobserved random effects and hence this leads to covariate measurement error. To deal with covariate measurement error, we propose a local corrected score estimator and a local conditional score estimator. Both approaches are semiparametric methods in the sense that there is no distributional assumption needed for the underlying true covariates. The estimators are shown to be consistent and asymptotically normal. However, simulation studies indicate that the conditional score estimator outperforms the corrected score estimator for finite samples, especially in the case of relatively large measurement error. The approaches are demonstrated by an application to data from an HIV clinical trial.

Corrected score estimation in the proportional hazards model with misclassified discrete covariates.

We consider Cox proportional hazards regression when the covariate vector includes error-prone discrete covariates along with error-free covariates, which may be discrete or continuous. The misclassification in the discrete error-prone covariates is allowed to be of any specified form. Building on the work of Nakamura and his colleagues, we present a corrected score method for this setting. The method can handle all three major study designs (internal validation design, external validation design, and replicate measures design), both functional and structural error models, and time-dependent covariates satisfying a certain 'localized error' condition. We derive the asymptotic properties of the method and indicate how to adjust the covariance matrix of the regression coefficient estimates to account for estimation of the misclassification matrix. We present the results of a finite-sample simulation study under Weibull survival with a single binary covariate having known misclassification rates. The performance of the method described here was similar to that of related methods we have examined in previous works. Specifically, our new estimator performed as well as or, in a few cases, better than the full Weibull maximum likelihood estimator. We also present simulation results for our method for the case where the misclassification probabilities are estimated from an external replicate measures study. Our method generally performed well in these simulations. The new estimator has a broader range of applicability than many other estimators proposed in the literature, including those described in our own earlier work, in that it can handle time-dependent covariates with an arbitrary misclassification structure. We illustrate the method on data from a study of the relationship between dietary calcium intake and distal colon cancer.

Hypothesis Testing in Functional Comparative Calibration Models

The purpose of this article is to investigate hypothesis testing in functional comparative calibration models. Wald type statistics are considered which are asymptotically distributed according to the chi-square distribution. The statistics are based on maximum likelihood, corrected score approach, and method of moment estimators of the model parameters, which are shown to be consistent and asymptotically normally distributed. Results of analytical and simulation studies seem to indicate that the Wald statistics based on the method of moment estimators and the corrected score estimators are, as expected, less efficient than the Wald type statistic based on the maximum likelihood estimators for small n. Wald statistic based on moment estimators are simpler to compute than the other Wald statistics tests and their performance improves significantly as n increases. Comparisons with an alternative F statistics proposed in the literature are also reported.

An Exact Corrected Log‐Likelihood Function for Cox's Proportional Hazards Model under Measurement Error and Some Extensions

Abstract. This paper studies Cox's proportional hazards model under covariate measurement error. Nakamura's [Biometrika 77 (1990) 127] methodology of corrected log‐likelihood will be applied to the so‐called Breslow likelihood, which is, in the absence of measurement error, equivalent to partial likelihood. For a general error model with possibly heteroscedastic and non‐normal additive measurement error, corrected estimators of the regression parameter as well as of the baseline hazard rate are obtained. The estimators proposed by Nakamura [Biometrics 48 (1992) 829], Kong et al. [Scand. J. Statist. 25 (1998) 573] and Kong & Gu [Statistica Sinica 9 (1999) 953] are re‐established in the special cases considered there. This sheds new light on these estimators and justifies them as exact corrected score estimators. Finally, the method will be extended to some variants of the Cox model.

Poisson regression models with errors-in-variables: implication and treatment

Overdispersion has been a common phenomenon in count data and usually treated with the negative binomial model. This paper shows that measurement errors in covariates in general also lead to overdispersion on the observed data if the true data generating process is indeed the Poisson regression. This kind of overdispersion cannot be treated using the negative binomial model, as otherwise, biases will occur. To provide consistent estimates, we propose a new type of corrected score estimator assuming that the distribution of the latent variables is known. The consistency and asymptotic normality of the proposed estimator are established. Simulation results show that this estimator has good finite sample performance. We also illustrate that the Akaike information criterion and Bayesian information criterion work well for selecting the correct model if the true model is the errors-in-variables Poisson regression.