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Using maximum likelihood estimation as described by R. A. Fisher (1912), a new estimator for the population squared multiple correlation was developed. This estimator ( ρcirc;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 of ρcirc; ² _(ML) in terms of providing the basis for the development of small sample significance tests is demonstrated. A Microsoft Excel workbook for computing ρcirc; ² _(ML) and its regions of nonsignificance and for computing a normal transformation for R² is offered.

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