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WEIGHTED LIKELIHOOD ESTIMATION UNDER TWO-PHASE SAMPLING.

ABSTRACT: We develop asymptotic theory for weighted likelihood estimators (WLE) under two-phase stratified sampling without replacement. We also consider several variants of WLEs involving estimated weights and calibration. A set of empirical process tools are developed including a Glivenko-Cantelli theorem, a theorem for rates of convergence of M-estimators, and a Donsker theorem for the inverse probability weighted empirical processes under two-phase sampling and sampling without replacement at the second phase. Using these general results, we derive asymptotic distributions of the WLE of a finite-dimensional parameter in a general semiparametric model where an estimator of a nuisance parameter is estimable either at regular or nonregular rates. We illustrate these results and methods in the Cox model with right censoring and interval censoring. We compare the methods via their asymptotic variances under both sampling without replacement and the more usual (and easier to analyze) assumption of Bernoulli sampling at the second phase.

SUBMITTER: Saegusa T

PROVIDER: S-EPMC3929280 | biostudies-literature | 2013 Feb

REPOSITORIES: biostudies-literature

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WEIGHTED LIKELIHOOD ESTIMATION UNDER TWO-PHASE SAMPLING.

Saegusa Takumi T Wellner Jon A JA

Annals of statistics 20130201 1

We develop asymptotic theory for weighted likelihood estimators (WLE) under two-phase stratified sampling without replacement. We also consider several variants of WLEs involving estimated weights and calibration. A set of empirical process tools are developed including a Glivenko-Cantelli theorem, a theorem for rates of convergence of <i>M</i>-estimators, and a Donsker theorem for the inverse probability weighted empirical processes under two-phase sampling and sampling without replacement at t ...[more]

PMID: 24563559

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Project description:In many public health and medical research settings, information on key covariates may not be readily available or too expensive to gather for all individuals in the study. In such settings, the two-phase design provides a way forward by first stratifying an initial (large) phase I sample on the basis of covariates readily available (including, possibly, the outcome), and sub-sampling participants at phase II to collect the expensive measure(s). When the outcome of interest is binary, several methods have been proposed for estimation and inference for the parameters of a logistic regression model, including weighted likelihood, pseudo-likelihood and maximum likelihood. Although these methods yield consistent estimation and valid inference, they do so solely on the basis of the phase I stratification and the detailed covariate information obtained at phase II. Moreover, they ignore any additional information that is readily available at phase I but was not used as part of the stratified sampling design. Motivated by the potential for efficiency gains, especially concerning parameters corresponding to the additional phase I covariates, we propose a novel augmented pseudo-likelihood estimator for two-phase studies that makes use of all available information. In contrast to recently-proposed weighted likelihood-based methods that calibrate to the influence function of the model of interest, the methods we propose do not require the development of additional models and, therefore, enjoy a degree of robustness. In addition, we expand the broader framework for pseudo-likelihood based estimation and inference to permit link functions for binary regression other than the logit link. Comprehensive simulations, based on a one-time cross sectional survey of 82,887 patients undergoing anti-retroviral therapy in Malawi between 2005 and 2007, illustrate finite sample properties of the proposed methods and compare their performance competing approaches. The proposed method yields the lowest standard errors when the model is correctly specified. Finally, the methods are applied to a large implementation science project examining the effect of an enhanced community health worker program to improve adherence to WHO guidelines for at least four antenatal visits, in Dar es Salaam, Tanzania.

Project description:The Canadian Study of Health and Aging (CSHA) employed a prevalent cohort design to study survival after onset of dementia, where patients with dementia were sampled and the onset time of dementia was determined retrospectively. The prevalent cohort sampling scheme favors individuals who survive longer. Thus, the observed survival times are subject to length bias. In recent years, there has been a rising interest in developing estimation procedures for prevalent cohort survival data that not only account for length bias but also actually exploit the incidence distribution of the disease to improve efficiency. This article considers semiparametric estimation of the Cox model for the time from dementia onset to death under a stationarity assumption with respect to the disease incidence. Under the stationarity condition, the semiparametric maximum likelihood estimation is expected to be fully efficient yet difficult to perform for statistical practitioners, as the likelihood depends on the baseline hazard function in a complicated way. Moreover, the asymptotic properties of the semiparametric maximum likelihood estimator are not well-studied. Motivated by the composite likelihood method (Besag 1974), we develop a composite partial likelihood method that retains the simplicity of the popular partial likelihood estimator and can be easily performed using standard statistical software. When applied to the CSHA data, the proposed method estimates a significant difference in survival between the vascular dementia group and the possible Alzheimer's disease group, while the partial likelihood method for left-truncated and right-censored data yields a greater standard error and a 95% confidence interval covering 0, thus highlighting the practical value of employing a more efficient methodology. To check the assumption of stable disease for the CSHA data, we also present new graphical and numerical tests in the article. The R code used to obtain the maximum composite partial likelihood estimator for the CSHA data is available in the online Supplementary Material, posted on the journal web site.

Dataset Information

WEIGHTED LIKELIHOOD ESTIMATION UNDER TWO-PHASE SAMPLING.

Publications

WEIGHTED LIKELIHOOD ESTIMATION UNDER TWO-PHASE SAMPLING.

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