An empirical assessment of ranking accuracy in ranked set sampling

Abstract

Ranked set sampling (RSS) involves ranking of potential sampling units on the variable of interest using judgment or an auxiliary variable to aid in sample selection. Its effectiveness depends on the success in this ranking. We provide an empirical assessment of RSS ranking accuracy in estimation of a population proportion.

Introduction

Ranked set sampling (RSS) employs ranking of the characteristic of interest prior to sampling via auxiliary information to improve estimation of a population parameter. It was first proposed by McIntyre (1952) for estimating mean pasture yield. Generally, the most promising setting for RSS is where sample units can be readily ranked (either through subjective judgment or via the use of a concomitant variable) but where the exact measurement of sample units is costly in time or effort. Patil (1995) provided more details about appropriate settings for the use of RSS.

A general RSS procedure can be carried out as follows. For $r=1,\dots ,m$, sample at random $n_r>0$ sets of size m (m is referred to as the set size) units each from the population of interest and rank the units within each set on the variable of interest, say X, without making actual measurements on X. The rankings can be the result of judgment ranking or obtained via a concomitant variable or several concomitant variables (regression approach). Then the $r$th smallest ranked item in each of the $n_r$ sets is quantified for $r=1,\dots ,m$. The remaining $\left(n_{r}m-n_r\right)$ units are used solely for the purpose of the ranking and are not quantified. We note that in the RSS protocol the unquantified units provide additional information about the characteristics of interest through the ranking process and therefore lead to improved estimation of population parameters.

When rankings are not perfect, we obtain judgment order statistics as opposed to the true order statistics obtained under perfect rankings. Thus, the notations $[]$ and $()$ are used in the subscripts to represent possibly erroneous and perfect rankings, respectively. Let $X_{[r]}$ represent the collection of $r$th judgment order statistics and let $X_{[r]j}$ denote the $j$th observation from that collection. This sampling scheme yields the following RSS of size n with ${\sum }_{r=1}^m{n}_r=n$.

$$$$

We note that $n_r$ is actually the sample size for rank r. In the situation where $n_1=n_2=\cdots =n_m$, the procedure leads to a balanced RSS; otherwise, it leads to an unbalanced RSS.

In the case where the population parameter of interest is the mean of X, the RSS estimator of the population mean $\mu$ is defined as

$${\widehat{\mu }}_{\mathrm{RSS}}=\frac{1}{m}\sum _{r=1}^m\frac{1}{n_r}\sum _{j=1}^{n_r}{X}_{[r]j}\text{.}$$

Theoretical results have shown that the RSS estimator of a population mean is unbiased regardless of ranking errors (Dell and Clutter, 1972). That is,

$$E\left({\widehat{\mu }}_{\mathrm{RSS}}\right)=\mu \text{.}$$

The ranking process is an important aspect of the RSS procedure. The precision of RSS relative to simple random sampling (SRS) with the same number of quantifications, defined as the ratio of the two variances for the corresponding estimators, depends to a great extent on the success in ranking. In general, for a fixed sample allocation of $n_r$, the more accurate the ranking within each set, the more precise the RSS procedure will be. In the case of using a single concomitant variable for ranking, an extensive literature has shown that the amount of increase in the precision of the RSS estimator is dependent on the association between the auxiliary variable and the continuous variable of interest (see, for example, Stokes, 1977, Bohn, 1996; Husby et al., 2005). Furthermore, it does not matter whether the association is positive or negative.

We note that an assessment of the accuracy in ranking has not been addressed previously in the RSS literature. In this paper, we present an empirical comparison of judgment rankings versus the true ranks of the observations in the context of RSS for binary variables since the application of RSS for estimating population proportions has not been studied thoroughly (Lacayo et al., 2002, Chen, 2006).

When the variable of interest is binary, the rankings can be accomplished through a concomitant variable (Terpstra and Liudahl, 2004) or, more generally, a logistic regression model that could be previously developed or fit from an auxiliary data set (Chen et al., 2005). Specifically, let ${\pi }_i$ denote the probability of success for an individual i in a set of size m from a Bernoulli(p) distribution. The estimated probability of success for this individual, ${\widehat{\pi }}_i$, can be obtained from a logistic regression model fit from a training sample. Then, the m sample units can be ranked according to their estimated probabilities of success, ${\widehat{\pi }}_1,\dots ,{\widehat{\pi }}_m$. In this study, we use a substantial data set, the third National Health and Nutrition Examination Survey (NHANES III), to evaluate empirically the accuracy in rankings obtained in this manner.

In the following section, we describe our criteria for evaluating the accuracy in rankings. Section 3 describes the data set and the variables used in the simulation study. Section 4 presents the results of our study, and in Section 5 we discuss conclusions.