An omnibus two-sample test for ranked-set sampling data

We develop an omnibus two-sample test for ranked-set sampling (RSS) data. The test statistic is the conditional probability of seeing the observed sequence of ranks in the combined sample, given the observed sequences within the separate samples. We compare the test to existing tests under perfect rankings, finding that it can outperform existing tests in terms of power, particularly when the set size is large. The test does not maintain its level under imperfect rankings. However, one can create a permutation version of the test that is comparable in power to the basic test under perfect rankings and also maintains its level under imperfect rankings. Both tests extend naturally to judgment post-stratification, unbalanced RSS, and even RSS with multiple set sizes. Interestingly, the tests have no simple random sampling analog.     

A note on ranked-set sampling using a covariate

Ranked-set sampling (RSS) and judgment post-stratification (JPS) use ranking information to obtain more efficient inference than is possible using simple random sampling. Both methods were developed with subjective, judgment-based rankings in mind, but the idea of ranking using a covariate has received a lot of attention. We provide evidence here that when rankings are done using a covariate, the standard RSS and JPS mean estimators no longer make efficient use of the available information. We first show that when rankings are done using a covariate, the standard nonparametric mean estimators in JPS and unbalanced RSS are inadmissible under squared error loss. We then show that when rankings are done using a covariate, nonparametric regression techniques yield mean estimators that tend to be significantly more efficient than the standard RSS and JPS mean estimators. We conclude that the standard estimators are best reserved for settings where only subjective, judgment-based rankings are available.     

Two-sample post-stratified or subgroup analysis with restricted randomization rules

In a clinical trial to compare two treatments, subjects may be allocated sequentially to treatment groups by a restricted randomization rule. Suppose that at the end of the trial, the investigator is interested in a post-stratified or subgroup analysis with respect to a particular demographic or clinical factor which was not selected prior to the trial for stratified randomization. Under a randomization model, large sample theory of two-sample post-stratified permutational tests is developed with a broad class of restricted randomization treatment allocation rules. The test procedures proposed here are illustrated with a real-life example. The results of this example indicate that it is not always possible to ignore the treatment rule used in the trial in the design-based analysis.     

Stratified Case-Cohort Analysis of General Cohort Sampling Designs

Abstract.  It is shown that variance estimates for regression coefficients in exposure-stratified case-cohort studies (Borgan et al ., Lifetime Data Anal., 6, 2000, 39–58) can easily be obtained from influence terms routinely calculated in the standard software for Cox regression. By allowing for post-stratification on outcome we also place the estimators proposed by Chen ( J. R. Statist. Soc. Ser. B , 63, 2001, 791–809) for a general class of cohort sampling designs within the Borgan et al. 's framework, facilitating simple variance estimation for these designs. Finally, the Chen approach is extended to accommodate stratified designs with surrogate variables available for all cohort members, such as stratified case-cohort and counter-matching designs.     

Estimating variances of strata in ranked set sampling

In RSS, the variance of observations in each ranked set plays an important role in finding an optimal design for unbalanced RSS and in inferring the population mean. The empirical estimator (i.e., the sample variance in a given ranked set) is most commonly used for estimating the variance in the literature. However, the empirical estimator does not use the information in the entire data over different ranked sets. Further, it is highly variable when the sample size is not large enough, as is typical in RSS applications. In this paper, we propose a plug-in estimator for the variance of each set, which is more efficient than the empirical one. The estimator uses a result in order statistics which characterizes the cumulative distribution function (CDF) of the rth order statistics as a function of the population CDF. We analytically prove the asymptotic normality of the proposed estimator. We further apply it to estimate the standard error of the RSS mean estimator. Both our simulation and empirical study show that our estimators consistently outperform existing methods.     

AN INVESTIGATION OF GENERALIZED RAKINGIN THE SYNTHETIC ESTIMATION OFPOPULATION SIZE

The problem of counting a population that is cross-classified with respect to demographic and geographic attributes is considered. A census is conducted in which individuals are “captured” with probabilities that are believed to be relatively constant within demographic categories. The census is followed by a random sample in which individuals are “recaptured” independently of the census. Using the two counts, capture-recapture estimates of the demographic category populations are obtained. A synthetic estimate of population size for a geographic entity is obtained by summing the corresponding adjustment factors (capture-recapture estimates divided by census counts) across all individuals captured by the census in the entity. The use of generalized raking is considered as a method for smoothing adjustment factors. It is found that generalized raking differs little from a class of weighted least squares regression models. This suggests that generalized raking does not offer an improvement over regression for smoothing adjustment factors. The efficiency loss of generalized raking relative to the best regression-based procedures can be substantial.