On a Criterion for Combining Correlational Data

Abstract

To test the hypothesis of a positive correlation, consider the following plans: Plan A consists of combining k >1 independent sample correlations, each one based on N bivariate observations and a common correlation parameter. Plan B consists of observing a single correlation based on the same total kN bivariate observations. Plan C also requires the total of kN observations and differs from A only in that the sample sizes are not homogeneous. It is shown that B is the relatively optimal plan, whether the corresponding testing procedures are compared in terms of relative large sample power of the test, based on a weighted sum of the corresponding Fisher’s z transformations, or in terms of their Bayes relative risks; the interpretation is that correlations “avoid” averaging. In contrast, it is also shown that certain statistics “prefer” averaging, while others are “indifferent.” A characterization of these classes was obtained in terms of the order of magnitude of the rate of change in the signal-to-noise ratio of the corresponding statistic, with respect to $$\sqrt{\mathrm{N}}$$ as the sample size N increases.