Auxiliary Covariate Information Research Articles

Benchmarking techniques for reconciling Bayesian small area models at distinct geographic levels

In sample surveys, there is often insufficient sample size to obtain reliable direct estimates for parameters of interest for certain domains. Precision can be increased by introducing small area models which ‘borrow strength’ by connecting different areas through use of explicit linking models, area-specific random effects, and auxiliary covariate information. One consequence of the use of small area models is that small area estimates at a lower (for example, county) geographic level typically will not aggregate to the estimate at the corresponding higher (for example, state) geographic level. Benchmarking is the statistical procedure for reconciling these differences. This paper provides new perspectives for the benchmarking problem, especially for complex Bayesian small area models which require Markov Chain Monte Carlo estimation. Two new approaches to Bayesian benchmarking are introduced: one procedure based on minimum discrimination information, and another procedure for fully Bayesian self-consistent conditional benchmarking. Notably the proposed procedures construct adjusted posterior distributions whose first and higher order moments are consistent with the benchmarking constraints. It is shown that certain existing benchmarked estimators are special cases of the proposed methodology under normality, giving a distributional justification for the use of benchmarked estimates. Additionally, a ‘flexible’ benchmarking constraint is introduced, where the higher geographic level estimate is not considered fixed, and is simultaneously adjusted, along with lower level estimates.

Small area estimation under transformed nested-error regression models

The empirical best linear unbiased prediction (EBLUP) based on the nested error regression model (Battese et al. in J Am Stat Assoc 83:28–36, 1988, NER) has been widely used for small area mean estimation. Its so-called optimality largely depends on the normality of the corresponding area level and unit level error terms. To allow departure from normality, we propose a transformed NER model with an invertible transformation, and employ the maximum likelihood method to estimate the underlying parameters of the transformed NER model. Motivated by Duan’s (J Am Stat Assoc 78:605–610, 1983) smearing estimator, we propose two small area mean estimators depending on whether all the population covariates or only the population covariate means are available in addition to sample covariates. We conduct two design-based simulation studies to investigate their finite-sample performance. The simulation results indicate that compared with existing methods such as the empirical best linear unbiased prediction, the proposed estimators are nearly the same reliable when the NER model is valid and become more reliable in general when the NER model is violated. In particular, our method does benefit from incorporating auxiliary covariate information.

A Quantitative Concordance Measure for Comparing and Combining Treatment Selection Markers.

Motivated by an HIV example, we consider how to compare and combine treatment selection markers, which are essential to the notion of precision medicine. The current literature on precision medicine is focused on evaluating and optimizing treatment regimes, which can be obtained by dichotomizing treatment selection markers. In practice, treatment decisions are based not only on efficacy but also on safety, cost and individual preference, making it difficult to choose a single cutoff value for all patients in all settings. It is therefore desirable to have a statistical framework for comparing and combining treatment selection markers without dichotomization. We provide such a framework based on a quantitative concordance measure, which quantifies the extent to which higher marker values are predictive of larger treatment effects. For a given marker, the proposed concordance measure can be estimated from clinical trial data using a U-statistic, which can incorporate auxiliary covariate information through an augmentation term. For combining multiple markers, we propose to maximize the estimated concordance measure among a specified family of combination markers. A cross-validation procedure can be used to remove any re-substitution bias in assessing the quality of an optimized combination marker. The proposed methodology is applied to the HIV example and evaluated in simulation studies.

Continuous auxiliary covariate in additive hazards regression for survival data

This paper considers the additive hazards regression analysis by utilizing continuous auxiliary covariate information to improve the efficiency of the statistical inference when the primary covariate is ascertained only for a randomly selected subsample. The authors construct a martingale-based estimating equation for the regression parameter and establish the asymptotic consistency and normality of the resultant estimators. Simulation study shows that the proposed method can greatly improve the efficiency compared with the estimator which discards the auxiliary covariate information in a variety of settings. A real example is also provided as an illustration.

Analysis of case-cohort designs with binary outcomes: Improving efficiency using whole-cohort auxiliary information.

The case-cohort design has been widely adopted for reducing the cost of covariate measurements in large prospective cohort studies. Under the case-cohort design, complete covariate data are collected only on randomly sampled cases and a subcohort randomly selected from the whole cohort. For the analysis of case-cohort studies with binary outcomes, logistic regression analysis has been routinely used. However, in many applications, certain covariates are readily measured on all samples from the whole cohort, and the case-cohort design may be regarded as a two-phase sampling design. Using this auxiliary covariate information, estimators for the regression parameters can be substantially improved. In this article, we discuss the theoretical basis of the case-cohort design derived from the formulation of the two-phase design and the improved estimators using whole-cohort auxiliary variable information. In particular, we show that the sampling scheme of the case-cohort design is substantially equivalent to that of conventional two-phase case-control studies (also known as two-stage case-control studies for epidemiologists), i.e., the methodologies of two-phase case-control studies can be directly applied to case-cohort data. Under this framework, we review and apply the following improved estimators to the case-cohort design with binary outcomes: (i) weighted estimators, (ii) a semiparametric maximum likelihood estimator, and (iii) a multiple imputation estimator. In addition, based on the framework of the two-phase design, we can obtain risk ratio and risk difference estimators without the rare-disease assumption. We illustrate these methodologies via simulations and the National Wilms Tumor Study data.

Semiparametric estimation in generalized linear mixed models with auxiliary covariates: a pairwise likelihood approach.

Auxiliary covariates are often encountered in biomedical research settings where the primary exposure variable is measured only for a subgroup of study subjects. This article is concerned with generalized linear mixed models in the presence of auxiliary covariate information for clustered data. We propose a novel semiparametric estimation method based on a pairwise likelihood function and develop an estimating equation-based inference procedure by treating both the error structure and random effects as nuisance parameters. This method is robust against misspecification of either error structure or random-effects distribution and allows for dependence between random effects and covariates. We show that the resulting estimators are consistent and asymptotically normal. Extensive simulation studies evaluate the finite sample performance of the proposed estimators and demonstrate their advantage over the validation set based method and the existing method. We illustrate the method with two real data examples.

Auxiliary covariate in additive hazards regression for survival data

We consider the additive hazards regression analysis by utilising auxiliary covariate information to improve the efficiency of the statistical inference when the primary covariate is ascertained only for a randomly selected subsample. We construct a martingale-based estimating equation for the regression parameter and establish the asymptotic consistency and normality of the resultant estimator. Simulation study shows that our proposed method can improve the efficiency compared with the estimator which discards the auxiliary covariate information. A real example is also analysed as an illustration.

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.

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.

Random Effects Logistic Regression Analysis with Auxiliary Covariates

We study a semiparametric estimation method for the random effects logistic regression when there is auxiliary covariate information about the main exposure variable. We extend the semiparametric estimator of Pepe and Fleming (1991, Journal of the American Statistical Association 86, 108-113) to the random effects model using the best linear unbiased prediction approach of Henderson (1975, Biometrics 31, 423-448). The method can be used to handle the missing covariate or mismeasured covariate data problems in a variety of real applications. Simulation study results show that the proposed method outperforms the existing methods. We analyzed a data set from the Collaborative Perinatal Project using the proposed method and found that the use of DDT increases the risk of preterm births among U.S. children.