Biased Sampling Model Research Articles

Large sample theory of maximum likelihood estimates in semiparametric biased sampling models

Vardi [Ann.Statist.13 178-203 (1985)] introduced an $s$-sample biased sampling model with known selection weight functions, gave a condition under which the common underlying probability distribution $G$ is uniquely estimable and developed simple procedure for computing the nonparametric maximum likelihood estimator (NPMLE) $\mathbb{G}_n$ of $G$. Gill, Vardi and Wellner thoroughly described the large sample properties of Vardi’s NPMLE, giving results on uniform consistency, convergence of $\sqrt{n}(\mathbb{G}-G)$ to a Gaussian process and asymptotic efficiency of $\mathbb{G}_n$. Gilbert, Lele and Vardi considered the class of semiparametric $s$-sample biased sampling models formed by allowing the weight functions to depend on an unknown finite-dimensional parameter $\theta$ .They extended Vardi’s estimation approach by developing a simple two-step estimation procedure in which $\hat{\theta}_n$ is obtained by maximizing a profile partial likelihood and $\mathbb{G}_n \equiv \mathbb{G}_n(\hat{\theta}_n)$ is obtained by evaluating Vardi’s NPMLE at $\hat{\theta}_n$. Here we examine the large sample behavior of the resulting joint MLE $(\hat{\theta}_n,\mathbb{G}_n)$, characterizing conditions on the selection weight functions and data in order that $(\hat{\theta}_n, \mathbb{G}_n)$ is uniformly consistent, asymptotically Gaussian and efficient. Examples illustrated here include clinical trials (especially HIV vaccine efficacy trials), choice-based sampling in econometrics and case-control studies in biostatistics.

A goodness-of-fit test for logistic regression models based on case-control data

SUMMARY We test the logistic regression assumption under a case-control sampling plan. After reparameterisation, the assumed logistic regression model is equivalent to a two-sample semiparametric model in which the log ratio of two density functions is linear in data. By identifying this model with a biased sampling model, we propose a Kolmogorov-Smirnovtype statistic to test the validity of the logistic link function. Moreover, we point out that this test statistic can also be used in mixture sampling. We present a bootstrap procedure along with some results on simulation and on analysis of two real datasets.

An identity for the nonparametric maximum likelihood estimator in missing data and biased sampling models

We derive an identity for the maximum likelihood estimator in nonparametric missing data models and biased sampling models, which almost says that this estimator is efficient. Application of empirical process theory to the identity provides us with a straightforward consistency and efficiency proof. The identity is illustrated with the random truncation model.

Nonparametric Inference Under Biased Sampling from a Finite Population

We consider a biased sampling model that has been found useful in incorporating size biases inherent in many types of discovery data. The model postulates that the data are obtained from a finite population by selecting successively without replacement and with probability proportional to some measure of size. Unlike the ppswor scheme in survey sampling, it is assumed here that the size measure is a function of the unknown population values. In this article, we consider maximum likelihood estimation of the finite population parameters under this biased sampling model. We study the large sample behavior of the MLE's and derive a simple, asymptotically efficient approximation to the MLE. The approximate MLE is structurally similar to the Horvitz-Thompson estimator. We show that information about the order in the sample can be used to make inference even when the population size is unknown, which in fact can be estimated. Small sample behavior of the estimators is investigated through a limited simulation study, and the results are used to analyze oil and gas discovery data from the North Sea basin.

Large Sample Theory of Empirical Distributions in Biased Sampling Models

Vardi (1985a) introduced an $s$-sample model for biased sampling, gave conditions which guarantee the existence and uniqueness of the nonparametric maximum likelihood estimator $\mathbb{G}_n$ of the common underlying distribution $G$ and discussed numerical methods for calculating the estimator. Here we examine the large sample behavior of the NPMLE $\mathbb{G}_n$, including results on uniform consistency of $\mathbb{G}_n$, convergence of $\sqrt n (\mathbb{G}_n - G)$ to a Gaussian process and asymptotic efficiency of $\mathbb{G}_n$ as an estimator of $G$. The proofs are based upon recent results for empirical processes indexed by sets and functions and convexity arguments. We also give a careful proof of identifiability of the underlying distribution $G$ under connectedness of a certain graph $\mathbf{G}$. Examples and applications include length-biased sampling, stratified sampling, enriched stratified sampling, choice-based sampling in econometrics and case-control studies in biostatistics. A final section discusses design issues and further problems.

Pooling of unshared information in group decision making: Biased information sampling during discussion.

Decision-making groups can potentially benefit from pooling members' information, particularly when members individually have partial and biased information but collectively can compose an unbiased characterization of the decision alternatives. The proposed biased sampling model of group discussion, however, suggests that group members often fail to effectively pool their information because discussion tends to be dominated by (a) information that members hold in common before discussion and (b) information that supports members' existent preferences. In a political caucus simulation, group members individually read candidate descriptions that contained partial information biased against the most favorable candidate and then discussed the candidates as a group. Even though groups could have produced unbiased composites of the candidates through discussion, they decided in favor of the candidate initially preferred by a plurality rather than the most favorable candidate. Group members' pre- and postdiscussion recall of candidate attributes indicated that discussion tended to perpetuate, not to correct, members' distorted pictures of the candidates.