How do I compare two squared (semi-partial) correlations from the same sample

Cohen and Cohen (1983) show (Appendix 2) that the comparison of squared semi-partial correlations is equivalent to the comparison of the difference in unstandardised regression coefficients, B.

Namely for a sample size of N and k predictors of which two have multiple regression estimates, B(1) and B(2):

[B(1) - B(2)]/Sqrt[V(B(1)) + V(B(2)) - 2COV(B(1),B(2)) ] will follow a t distribution with N-k-1 degrees of freedom under the null hypothesis that the two regression coefficients are equal.

where V() and COV() represent the variance and covariances of the regression estimates respectively.

The covariance matrix of the regression coefficients is routinely outputted in most statistical software. In SPSS it may be requested by clicking on the statistics button and the Regression Coefficients:covariance matrix box when running the regression. The diagonal terms pf the covariance matrix represent variances and off-diagonal terms covariances of the regression coefficients.

Reference

Cohen J and Cohen P (1983) Applied multiple regression/correlation analysis for the behavioral sciences. Second edition. Lawrence Erlbaum:Hillsdale, NJ.

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