This article develops equations for determining the asymptotic confidence limits for the difference between 2 squared multiple correlation coefficients. The present procedure uses the delta method described by I. Olkin and J. D. Finn (1995) but does not require the variance-covariance matrix and the partial derivatives for all the zero-order correlations that enter into the expression for the difference, as does their procedure. This simplified approach can lead to an extreme reduction in the calculations required, as well as a reduction in the mathematical complexity of the solution. This approach also demonstrates clearly that in some cases, it may be inappropriate to use the asymptotic confidence limits in tests of significance.
Olkin & Finn (1995) developed expressions for confidence intervals for functions of simple, partial, and multiple correlations. This paper describes procedures and computer programs for solving these problems using the methods described by Olkin and Finn. The programs extendthe methods for any number of predictors or for partialing out any number of variables.
Using maximum likelihood estimation as described by R. A. Fisher (1912), a new estimator for the population squared multiple correlation was developed. This estimator (ρ 2 (ML) ) was derived by examining all possible values of the population squared multiple correlation for a given sample size and number of predictors, and finding the one for which the observed sample value had the highest probability of occurring. The new estimator is shown to have greater accuracy than other estimators and to generate values that always fall within the parameter space. The utility ofρ 2 (ML) in terms of providing the basis for the development of small sample significance tests is demonstrated. A Microsoft Excel workbook for computingρ 2 (ML) and its regions of nonsignificance and for computing a normal transformation for R 2 is offered.
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