The Hotelling/Williams Test for the Difference Between Two Dependent Correlations

The Hotelling/Williams procedure tests the null hypothesis of equality between two dependent product-moment correlations. Specifically, it tests whether the correlation between Z and X differs from the correlation between Z and Y, where X, Y and Z are three variables measured on the same set of observational units.

HWTEST.ZIP contains SAS code, and also Splus/R code, to carry out the Hotelling/Williams test. Full documentation is contained in the comment lines of each of the two text files included in HWTEST.ZIP.

The following article describes the Hotelling/Williams test, along with other statistical tools for assessing dependent correlations between stream and watershed attributes.

Van Sickle, John. 2003. Analyzing correlations between stream and watershed attributes.

Journal of the American Water Resources Association 39(3):717-726.

ABSTRACT:

Bivariate correlation analysis has been widely used to explore relationships between stream and watershed attributes that have all been measured on the same set of watersheds or sampling locations. Researchers routinely test H0: rho=0 for each correlation in a large table and then go on to discuss only those that are declared “significant”. Such test results are inaccurate because no allowance is made for multiple testing, and also because the tests are not mutually independent. This paper reviews the Bonferroni approach to controlling the overall error rate in multiple testing and shows how the approach becomes impractical for large correlation tables. The Hotelling/Williams test is introduced for comparing two dependent correlations that share a variable, and numerical constraints for two such correlations are illustrated. References are also given for testing other hypothesized patterns among dependent correlations, and links to dependent–correlation software are provided.  The methods are illustrated for watershed and stream variables sampled in 23 small agricultural watersheds of the Willamette Valley, Oregon.

NOTE: The above article contains an incorrect statement of the Hotelling/Williams formula. For the correct formula, please see:

Van Sickle, J. (2005) Errata.
Journal of the American Water Resources Association 41: 741.

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