An approximation to the distribution of the product of two dependent correlation coefficients

Abstract

Many methodological studies depend on the product of two dependent correlation coefficients. However, the behavior of the distribution of the product of two dependent correlation coefficients is not well known. The distribution of sets of correlation coefficients has been well studied, but not the distribution of the product of two dependent correlation coefficients. The present study derives an approximation to the distribution of the product of two dependent correlation coefficients with a closed form, resulting in a Pearson Type I distribution. A simulation study is also conducted to assess the accuracy of the approximation.

Keywords:

• Approximate distribution
• Correlation coefficient
• Moments
• Pearson distributions

Acknowledgement

The authors are grateful to the Editor, the Associate Editor, and the anonymous reviewers for their valuable comments. This research was partially supported by a grant from the University Research Council of the University of Cincinnati to Wei Pan and by grants from the National Institute of Child Health and Human Development (R01 HD40428-02) and from the National Science Foundation (REC-0126167) to Kenneth A. Frank through the Population Research Center, University of Texas at Austin; Chandra Muller (PI).

Notes

¹Since we only want to obtain the approximate first four moments, the terms of order higher than the fourth will have little effect on the approximation.

²Anderson's equations were tactically used here only for obtaining the higher order product-moments, although Anderson's equations are based on the normal distribution.

³For N < 300, ρ _xc or ρ _yc must be greater than 0.10 for κ to be negative. For smaller correlations, e.g., ρ _xc and ρ _yc  ≤ 0.10, 0 < κ < 1; and the distribution of the product of two dependent correlation coefficients, r _xc r _yc , can be approximated by a Pearson Type IV distribution (cf. Fig. 1).

⁴Here, and later, we use “∼” to distinguish the simulated values from the corresponding observed values.

⁵The approximated moments were calculated by SPSS (see Appendix B for the SPSS code). To make the mode clearer, we computed Eqs. (3.3), (3.6), and (3.7) and substitued them into (3.1), instead of directly using the more complex Eq. (3.9).