Computing the Distribution of the Squared Sample Multiple Correlation Coefficient with S-Systems

Abstract

The multiple correlation coefficient is to measure the maximum correlation between one variable and the linear combination of other correlated variables. It is useful for studying the relationships among multiple covariate observations. The ad hoc approximation methods (Chen, Z.-Y. (2001). On the statistical computation of the sample multiple correlation coefficient. Journal of Statistical Computation and Simulation 70:299–324; Ding, C. G. (1996). On the computation of the distribution of the square of the sample multiple correlation coefficient. Computational Statistics and Data Analysis 22:345–350; Gurland, J., Asiribo, O. (1991). On the distribution of the multiple correlation coefficient and their kinked chi-squared random variable. Journal of Statistical Sinica 1:493–502; Lee, Y. S. (1972). Tables of upper percentage points of the multiple correlation coefficient. Biometrika 59:175–189.) for evaluating various distributional values of interest are represented by different forms and calculated in separate numerical procedures. Here, a unified approach of numerical methods using the S-system form for the squared sample multiple correlation coefficient is derived on the basis of recasting techniques of Rust and Voit (Rust, P. F., Voit, E. O. (1990). Statistical densities, cumulatives, quantiles, and power obtained by S-systems differential equations. Journal of the American Statistical Association. 85:572–578) and Sarageau and Voit (Savageau, M. A., Voit, E. O. (1987). Recasting nonlinear differential equations as S-systems: a canonical nonlinear form. Mathematical Biosciences 87:83–115). Its distributional function form in terms of an infinite weighted sum of the incomplete beta integral is closely related to that of noncentral beta distribution whose S-system form has been investigated by Chen and Chou (Chen, Z. -Y., Chou, Y. -C. (2000). Computing the noncentral beta distribution with S-system. Computational Statistics and Data Analysis 33:343–360). Following their derivations, the S-system formulations for this statistical distribution are studied and demonstrated. Statistical densities, cumulatives, quantiles, and related distributional values can be evaluated in only one S-system form by using its numerical solver PLAS. Compared with the ad hoc evaluations, the S-system demonstration produces more precise results under various situations for computing the statistical values of densities, cumulatives, and quantiles. In addition, this new formulation provides significant numerical advantages over its original form. Related properties are also discussed.