On a Criterion for Combining Correlational Data

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 N as the sample size N increases.