Hostname: page-component-7b7567c8c-wxvwp Total loading time: 0 Render date: 2026-11-15T03:04:18.911Z Has data issue: false hasContentIssue false

Interval Estimation of Gamma for an R×S table

Published online by Cambridge University Press:  01 January 2025

Kung-Jong Lui*
Affiliation:
San Diego State University
William G. Cumberland
Affiliation:
University of California, Los Angeles
*
Requests for reprints should be sent to K.-J. Lui, Department of Mathematics and Statistics, San Diego State University, San Diego, CA 92182-7720, U.S.A. E-Mail: kj@rohan.sdsu.edu

Abstract

When the underlying responses are on an ordinal scale, gamma is one of the most frequently used indices to measure the strength of association between two ordered variables. However, except for a brief mention on the use of the traditional interval estimator based on Wald's statistic, discussion of interval estimation of the gamma is limited. Because it is well known that an interval estimator using Wald's statistic is generally not likely to perform well especially when the sample size is small, the goal of this paper is to find ways to improve the finite-sample performance of this estimator. This paper develops five asymptotic interval estimators of the gamma by employing various methods that are commonly used to improve the normal approximation of the maximum likelihood estimator (MLE). Using Monte Carlo simulation, this paper notes that the coverage probability of the interval estimator using Wald's statistic can be much less than the desired confidence level, especially when the underlying gamma is large. Further, except for the extreme case, in which the underlying gamma is large and the sample size is small, the interval estimator using a logarithmic transformation together with a monotonic function proposed here not only performs well with respect to the coverage probability, but is also more efficient than all the other estimators considered here. Finally, this paper notes that applying an ad hoc adjustment procedure—whenever any observed frequency equals 0, we add 0.5 to all cells in calculation of the cell proportions—can substantially improve the traditional interval estimator. This paper includes two examples to illustrate the practical use of interval estimators considered here.

Information

Type
Theory And Methods
Copyright
Copyright © 2004 The Psychometric Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Article purchase

Temporarily unavailable