Update of loevh2 (on SSC)

[Dirk Enzmann]

Thanks to Kit Baum an update of loevh2 (now version 3.2) is available on SSC.

loevh2 calculates the H-coefficient of homogeneity according to Loevinger (Loevinger, 1947, pp. 29–31) for two dichotomous (0/1) variables; this coefficient is not influenced by the base rates (marginal proportions). The program returns standard errors, z-statistics, p-values, and various types of confidence intervals for the H-coefficient. H can be used as a base-rate-independent measure of the overlap in the 1/1 cell in a 2×2 table of two positively correlated dichotomous variables, corrected for chance and ceiling effect; that is, it is independent of how rare or common either of the two variables is. H is particularly useful when comparing H values between groups (see also Yule, 1912).

Loevinger's H is identical to RIOC (relative improvement over chance, see Copas & Loeber, 1990), see also rioc (SSC) by daniel klein.

The update

• adds the option to test for equality of H across subsamples using the by: prefix command: The compare option tests the homogeneity of effect sizes, as in meta-analyses, under the null hypothesis that all H values in the population are equal;
• the calculation of the standard error of H for large samples (Copas & Loeber, Eq. 11) becomes the default;
• adds the option to calculate the standard error of H for small samples (Copas & Loeber, Eqs. 20–23),
• the calculation of the Pearson standard error of H (optional, previously default) is corrected by using the denominator n (instead of the denominator (n-1), as in loevh (SSC) by Jean-Benoit Hardouin).
• corrects the display of H in the loevh2_boot program (see below) so that Loevinger’s H is now displayed instead of the mean of the bootstrap replicates.

The program loevh2_boot (within the package) provides bootstrap confidence intervals for Loevinger's H, taking into account systematic biases in the sample distribution of H as well as non-constant variance and skewness. Bootstrap confidence intervals can be more robust when sample sizes are small or asymptotic assumptions are not met.

The programs loevh2i and loevh2_booti (within the package) are immediate commands of loevh2 and loevh2_boot (analogous to rioci in the package rioc (SSC)).

The program descriptions in the help files have been improved and substantially expanded.

References:

• Copas, J. B., & Loeber, R. (1990). Relative improvement over chance (RIOC) for 2×2 tables. British Journal of Mathematical and Statistical Psychology, 43(2), 293–307.
• Loevinger, J. A. (1947). A systematic approach to the construction and evaluation of tests of ability. Psychological Monographs, 61(4), i–49.
• Yule, G. U. (1912). On the methods of measuring association between two attributes. Journal of the Royal Statistical Society, 75(6), 579–642.

[Dirk Enzmann]

daniel klein correctly pointed to an issue (thanks!):

Both loevh2 and loevh2_boot allow “aweight.” However, when “aweight” is used, the standard errors and confidence intervals are no longer valid—this is true even if the use of “aweight” was intended solely to allow non-integer frequency weights. Allowing “aweight,” however, leads the user to expect that the weights are inversely proportional to the variance of the observation, which is not the case when using “aweight” with non-integer weights (instead, “iweight” should have been used).

Therefore: Do not use “aweight” with this version of loevh2 and loevh2_boot. I am currently working on a revised version that will allow only “fweight” for loevh and both “fweight” and “pweight” for loevh2_boot. As soon as the revised version is available, I will announce it in a new thread.