Auxiliary Covariates Research Articles

Nonparametric regression with the scale depending on auxiliary variable

The paper is devoted to the problem of estimation of a univariate component in a heteroscedastic nonparametric multiple regression under the mean integrated squared error (MISE) criteria. The aim is to understand how the scale function should be used for estimation of the univariate component. It is known that the scale function does not affect the rate of the MISE convergence, and as a result sharp constants are explored. The paper begins with developing a sharp-minimax theory for a pivotal model $Y=f(X)+\sigma(X,\mathbf{Z})\varepsilon$, where $\varepsilon$ is standard normal and independent of the predictor X and the auxiliary vector-covariate $\mathbf{Z}$. It is shown that if the scale $\sigma(x,\mathbf{z})$ depends on the auxiliary variable, then a special estimator, which uses the scale (or its estimate), is asymptotically sharp minimax and adaptive to unknown smoothness of f(x). This is an interesting conclusion because if the scale does not depend on the auxiliary covariate $\mathbf{Z}$, then ignoring the heteroscedasticity can yield a sharp minimax estimation. The pivotal model serves as a natural benchmark for a general additive model $Y=f(X)+g(\mathbf{Z})+\sigma(X,\mathbf{Z})\varepsilon$, where $\varepsilon$ may depend on $(X,\mathbf{Z})$ and have only a finite fourth moment. It is shown that for this model a data-driven estimator can perform as well as for the benchmark. Furthermore, the estimator, suggested for continuous responses, can be also used for the case of discrete responses. Bernoulli and Poisson regressions, that are inherently heteroscedastic, are particular considered examples for which sharp minimax lower bounds are obtained as well. A numerical study shows that the asymptotic theory sheds light on small samples.

Latent Class Analysis of Peer Conformity

This study used latent class analysis to examine typologies of peer conformity in a community sample of middle school students. Students responded to 31 items assessing diverse facets of conformity dispositions. The most parsimonious model produced three qualitatively distinct classes that differed on the basis of conformity to recreational activities, deviant behaviors, style conformity, and social comparison. Gender comparisons suggested relatively stable class proportions for males and females but also significant parameter differences in tests of measurement invariance for latent class indicators. Multinomial logistic regression models predicting class membership from auxiliary covariates and psychosocial risk indicated that compared to mild conformists deviance conforming youth were more likely to be White, have low self-esteem, refrain from using adaptive coping skills, and be more socially anxious. Socially conforming youth were more likely to be male, White, and have low self-esteem. Findings are discussed with regard to classic definitions of conformity and its role as a developmental phenomenon.

Nonparametric Modeling Auxiliary Covariates in Random Coefficient Models

Random coefficient model (RCM) is a powerful statistical tool in analyzing correlated data collected from studies with different clusters or from longitudinal studies. In practice, there is a need for statistical methods that allow biomedical researchers to adjust for the measured and unmeasured covariates that might affect the regression model. This article studies two nonparametric methods dealing with auxiliary covariate data in linear random coefficient models. We demonstrate how to estimate the coefficients of the models and how to predict the random effects when the covariates are missing or mismeasured. We employ empirical estimator and kernel smoother to handle a discrete and continuous auxiliary, respectively. Simulation results show that the proposed methods perform better than an alternative method that only uses data in the validation data set and ignores the random effects in the random coefficient model.

On the covariate-adjusted estimation for an overall treatment difference with data from a randomized comparative clinical trial

To estimate an overall treatment difference with data from a randomized comparative clinical study, baseline covariates are often utilized to increase the estimation precision. Using the standard analysis of covariance technique for making inferences about such an average treatment difference may not be appropriate, especially when the fitted model is nonlinear. On the other hand, the novel augmentation procedure recently studied, for example, by Zhang and others (2008. Improving efficiency of inferences in randomized clinical trials using auxiliary covariates. Biometrics 64, 707-715) is quite flexible. However, in general, it is not clear how to select covariates for augmentation effectively. An overly adjusted estimator may inflate the variance and in some cases be biased. Furthermore, the results from the standard inference procedure by ignoring the sampling variation from the variable selection process may not be valid. In this paper, we first propose an estimation procedure, which augments the simple treatment contrast estimator directly with covariates. The new proposal is asymptotically equivalent to the aforementioned augmentation method. To select covariates, we utilize the standard lasso procedure. Furthermore, to make valid inference from the resulting lasso-type estimator, a cross validation method is used. The validity of the new proposal is justified theoretically and empirically. We illustrate the procedure extensively with a well-known primary biliary cirrhosis clinical trial data set.

Accelerated Failure Time Models with Auxiliary Covariates

In this paper we study semi-parametric inference procedure for accelerated failure time models with auxiliary information about a main exposure variable. We use a kernel smoothing method to introduce the auxiliary covariate to the likelihood function. The regression parameters are then estimated through maximization of the estimated likelihood function. A consistent estimator of the variance of the estimator of the regression coefficients is proposed. Simulation studies show that the efficiency gain is remarkable when compared to just using the validation sample. The method is applied to the PBC data from the Mayo Clinic trial in primary biliary cirrhosis as an illustration.

Marginal hazard regression for correlated failure time data with auxiliary covariates

Marginal hazard regression for correlated failure time data with auxiliary covariates

Marginal hazard regression for correlated failure time data with auxiliary covariates

In many biomedical studies, it is common that due to budget constraints, the primary covariate is only collected in a randomly selected subset from the full study cohort. Often, there is an inexpensive auxiliary covariate for the primary exposure variable that is readily available for all the cohort subjects. Valid statistical methods that make use of the auxiliary information to improve study efficiency need to be developed. To this end, we develop an estimated partial likelihood approach for correlated failure time data with auxiliary information. We assume a marginal hazard model with common baseline hazard function. The asymptotic properties for the proposed estimators are developed. The proof of the asymptotic results for the proposed estimators is nontrivial since the moments used in estimating equation are not martingale-based and the classical martingale theory is not sufficient. Instead, our proofs rely on modern empirical process theory. The proposed estimator is evaluated through simulation studies and is shown to have increased efficiency compared to existing methods. The proposed method is illustrated with a data set from the Framingham study.

Estimated Generalized Estimating Equation for Correlated Failure Time Data with Auxiliary Covariates

In this article, we study a marginal hazard model with common baseline hazard for correlated failure time data. We assume that the true covariate is measured precisely in a subset of the whole study cohort, whereas an auxiliary information for the true covariate is available for the whole cohort. We first estimate the relative risk function empirically. Then we obtain the estimator for the regression parameter by replacing the relative risk function with its estimator in a generalized estimating equation (GEE) proposed by Cai (1992). A key feature of this method is that it is nonparametric with respect to the association between the missing covariate and the observed auxiliary covariate. The proposed estimator is shown to be consistent and asymptotically normal. Furthermore, we present a corrected Breslow-type estimator for the cumulative hazard function. Simulation studies are conducted to evaluate the proposed method.

Semiparametric estimation of treatment effect with time-lagged response in the presence of informative censoring

In many randomized clinical trials, the primary response variable, for example, the survival time, is not observed directly after the patients enroll in the study but rather observed after some period of time (lag time). It is often the case that such a response variable is missing for some patients due to censoring that occurs when the study ends before the patient's response is observed or when the patients drop out of the study. It is often assumed that censoring occurs at random which is referred to as noninformative censoring; however, in many cases such an assumption may not be reasonable. If the missing data are not analyzed properly, the estimator or test for the treatment effect may be biased. In this paper, we use semiparametric theory to derive a class of consistent and asymptotically normal estimators for the treatment effect parameter which are applicable when the response variable is right censored. The baseline auxiliary covariates and post-treatment auxiliary covariates, which may be time-dependent, are also considered in our semiparametric model. These auxiliary covariates are used to derive estimators that both account for informative censoring and are more efficient then the estimators which do not consider the auxiliary covariates.

Efficiency improvement in a class of survival models through model-free covariate incorporation

In randomized clinical trials, we are often concerned with comparing two-sample survival data. Although the log-rank test is usually suitable for this purpose, it may result in substantial power loss when the two groups have nonproportional hazards. In a more general class of survival models of Yang and Prentice (Biometrika 92:1-17, 2005), which includes the log-rank test as a special case, we improve model efficiency by incorporating auxiliary covariates that are correlated with the survival times. In a model-free form, we augment the estimating equation with auxiliary covariates, and establish the efficiency improvement using the semiparametric theories in Zhang et al. (Biometrics 64:707-715, 2008) and Lu and Tsiatis (Biometrics, 95:674-679, 2008). Under minimal assumptions, our approach produces an unbiased, asymptotically normal estimator with additional efficiency gain. Simulation studies and an application to a leukemia study show the satisfactory performance of the proposed method.

Semiparametric inference for a 2-stage outcome-auxiliary-dependent sampling design with continuous outcome

Two-stage design has long been recognized to be a cost-effective way for conducting biomedical studies. In many trials, auxiliary covariate information may also be available, and it is of interest to exploit these auxiliary data to improve the efficiency of inferences. In this paper, we propose a 2-stage design with continuous outcome where the second-stage data is sampled with an "outcome-auxiliary-dependent sampling" (OADS) scheme. We propose an estimator which is the maximizer for an estimated likelihood function. We show that the proposed estimator is consistent and asymptotically normally distributed. The simulation study indicates that greater study efficiency gains can be achieved under the proposed 2-stage OADS design by utilizing the auxiliary covariate information when compared with other alternative sampling schemes. We illustrate the proposed method by analyzing a data set from an environmental epidemiologic study.

Small area estimation under Fay–Herriot models with non-parametric estimation of heteroscedasticity

Fay–Herriot models relate direct estimators of small area means to vectors of area-level auxiliary covariates. Estimation of error variances in these models is a problem because of the lack of data within areas. A non-parametric approach is proposed for estimating these variances. Estimators of the remaining model parameters are derived and their asymptotic properties are studied. Moreover, small area estimators that incorporate the estimated error variances are obtained and several simple estimators of the mean squared error of these estimators are proposed. Simulation experiments study the small sample performance of the new small area estimators and compare the different estimators of the mean squared errors. Finally, the results are applied to the estimation of unemployment proportions in Spanish domains.

Cox Regression in Nested Case–Control Studies with Auxiliary Covariates

Nested case-control (NCC) design is a popular sampling method in large epidemiological studies for its cost effectiveness to investigate the temporal relationship of diseases with environmental exposures or biological precursors. Thomas' maximum partial likelihood estimator is commonly used to estimate the regression parameters in Cox's model for NCC data. In this article, we consider a situation in which failure/censoring information and some crude covariates are available for the entire cohort in addition to NCC data and propose an improved estimator that is asymptotically more efficient than Thomas' estimator. We adopt a projection approach that, heretofore, has only been employed in situations of random validation sampling and show that it can be well adapted to NCC designs where the sampling scheme is a dynamic process and is not independent for controls. Under certain conditions, consistency and asymptotic normality of the proposed estimator are established and a consistent variance estimator is also developed. Furthermore, a simplified approximate estimator is proposed when the disease is rare. Extensive simulations are conducted to evaluate the finite sample performance of our proposed estimators and to compare the efficiency with Thomas' estimator and other competing estimators. Moreover, sensitivity analyses are conducted to demonstrate the behavior of the proposed estimator when model assumptions are violated, and we find that the biases are reasonably small in realistic situations. We further demonstrate the proposed method with data from studies on Wilms' tumor.

Modeling continuous auxiliary covariate data in generalized linear mixed models using the kernel smoother

Auxiliary covariate data are often collected in biomedical studies when the primary exposure variable is only assessed on a subset of the study subjects. In this study, we investigate a semiparametric-estimated likelihood estimation for the generalized linear mixed models (GLMM) in the presence of a continuous auxiliary variable. We use a kernel smoother to handle continuous auxiliary data. The method can be used to deal with missing or mismeasured covariate data problems in a variety of applications when an auxiliary variable is available and cluster sizes are not too small. Simulation study results show that the proposed method performs better than that which ignores the random effects in GLMM and that which only uses data in the validation data set. We illustrate the proposed method with a real data set from a recent environmental epidemiology study on the maternal serum 1,1-dichloro-2,2-bis(p-chlorophenyl) ethylene level in relationship to preterm births.

Adjustment for Missingness Using Auxiliary Information in Semiparametric Regression

In this article, we study the estimation of mean response and regression coefficient in semiparametric regression problems when response variable is subject to nonrandom missingness. When the missingness is independent of the response conditional on high-dimensional auxiliary information, the parametric approach may misspecify the relationship between covariates and response while the nonparametric approach is infeasible because of the curse of dimensionality. To overcome this, we study a model-based approach to condense the auxiliary information and estimate the parameters of interest nonparametrically on the condensed covariate space. Our estimators possess the double robustness property, i.e., they are consistent whenever the model for the response given auxiliary covariates or the model for the missingness given auxiliary covariate is correct. We conduct a number of simulations to compare the numerical performance between our estimators and other existing estimators in the current missing data literature, including the propensity score approach and the inverse probability weighted estimating equation. A set of real data is used to illustrate our approach.

Multivariate failure times regression with a continuous auxiliary covariate

Multivariate failure times regression with a continuous auxiliary covariate

Estimated pseudopartial-likelihood method for correlated failure time data with auxiliary covariates.

As biological studies become more expensive to conduct, statistical methods that take advantage of existing auxiliary information about an expensive exposure variable are desirable in practice. Such methods should improve the study efficiency and increase the statistical power for a given number of assays. In this article, we consider an inference procedure for multivariate failure time with auxiliary covariate information. We propose an estimated pseudopartial likelihood estimator under the marginal hazard model framework and develop the asymptotic properties for the proposed estimator. We conduct simulation studies to evaluate the performance of the proposed method in practical situations and demonstrate the proposed method with a data set from the studies of left ventricular dysfunction (SOLVD Investigators, 1991, New England Journal of Medicine 325, 293-302).

Multivariate failure times regression with a continuous auxiliary covariate

How to take advantage of the available auxiliary covariate information when the primary covariate of interest is not measured is a frequently encountered question in biomedical study. In this paper, we consider the multivariate failure times regression analysis in which the primary covariate is assessed only in a validation set, but a continuous auxiliary covariate for it is available for all subjects in the study cohort. Under the frame of marginal hazard model, we propose to estimate the induced relative risk function in the non-validation set through kernel smoothing method and then obtain an estimated pseudo-partial likelihood function. The proposed estimator which maximizes the estimated pseudo-partial likelihood is shown to be consistent and asymptotically normal. We also give an estimator of the marginal cumulative baseline hazard function. Simulation studies are conducted to evaluate the finite sample performance of our proposed estimator. The proposed method is illustrated by analyzing a heart disease data from the Study of Left Ventricular Dysfunction (SOLVD).

Consistency of the estimator of cumulative hazard function in estimated pseudo-partial-likelihood approach

For multivariate failure time with auxiliary covariate information, an estimated pseudo-partial-likelihood estimator under the marginal hazard model with distinguishable baseline hazard has been proposed. However, the asymptotic properties of the corresponding estimated cumulative hazard function have not been studied. In this paper, based on counting process martingale, we use the continuous mapping theorem and Lenglart inequality and prove the consistency of the estimated cumulative hazard function in estimated pseudo-partial-likelihood approach.

Estimated Pseudopartial-Likelihood Method for Correlated Failure Time Data with Auxiliary Covariates

Estimated Pseudopartial-Likelihood Method for Correlated Failure Time Data with Auxiliary Covariates