Consistent Small-Sample Variances for Six Gamma-Family Measures of Ordinal Association

Abstract

Gamma-family measures are bivariate ordinal correlation measures that form a family because they all reduce to Goodman and Kruskal's gamma in the absence of ties (1954). For several gamma-family indices, more than one variance estimator has been introduced. In previous research, the “consistent” variance estimator described by Cliff and colleagues produced more accurate confidence intervals (CIs) than 3 other types of variances. However, the Cliff consistent (CC) variance is available for only 4 of 10 gamma-family measures. In the present research, a CC variance is derived for the remaining 6 gamma-family measures, and CIs constructed with the CC variance are compared with CIs constructed using other available variance estimators. The CIs are evaluated and compared in simulations and illustrated with scores on the Disinhibition and Avoidance scales of the Schedule for Nonadaptive and Adaptive Personality. Computer code for calculating all 10 gamma-family measures and their CC variances (using the R program) is available on a Web site (http://www.artsci.wustl.edu/~cwoods/CCvarR.html).

Notes

¹The appropriateness of a statistical method for a particular data set is affected by the researchers' philosophies, characteristics of the data, and the nature of the research questions. Although most social scientists are familiar with the discrete measurement types—nominal, ordinal, interval, and ratio—these are not universally adopted. For example, some scholars believe that a measurement level in between ordinal and interval exists (Takane, Young, & de Leeuw, 1977). These researchers also feel that the measurement level of the data is dependent upon the interaction between the data and the model chosen to describe the data rather than a characteristic of the data alone (Takane et al., 1977).

²Although some measurement theorists would question whether addition should be used with ordinal items and others suggest that a continuum underlies measurement level itself, so that Likert scales could be considered intermediate between ordinal and interval levels, these theoretical issues are not critical to the topics treated in this article.

³ Somers's (1962a) d _yx and d _xy are (respectively) equivalent to d ^K _xy and d ^K _yx (Kim, 1971).

⁴I am grateful to Thomas F. Oltmanns and Eric Turkheimer for providing data for the empirical example.

⁵The data were originally collected by Carol M. Woods as part of a collaborative project with Jonathan Abramowitz and David Tolin.