Measuring accuracy of estimate and degree of correlation for curvilinear multiple regressions
In presenting linear multiple regression methods it was observed that coefficients could be computed to show (1) how closely estimated values of the dependent variable, based on the linear regression equation, could be expected to agree with the actual values; and (2) what proportion of the total observed variance in the dependent factor could be explained or accounted for by its relation to the independent factors considered. These coefficients were, respectively, the standard error of estimate and the coefficient of multiple correlation. Exactly parallel coefficients can be computed to show the importance of the relationship for curvilinear multiple regression, employing curvilinear net regressions such as those discussed in Chapter 14. The term standard error of estimate is again used to indicate the measure of the probable accuracy of estimated values of the dependent factor. In measuring the proportion of variance explained we will follow the usage in simple curvilinear regression, and use the term index to denote the fact that curvilinear regressions have been employed. The proportion of variance accounted for is therefore shown by the index of multiple determination, which is the square of the index of multiple correlation.
Standard Error of Estimate. Values of X1 may be estimated from X2, X3, . . ., Xk by a multiple curvilinear regression equation of the type
X1.f(2,3,. . . k) = a + f2(X2) + f3(X3) + . . . + fk(Xk) (15.1)
where the net regression functions f specify curves fitted simultaneously either as mathematical equations or as graphic curves determined by a successive approximation process. In either case, when estimated values,