# Methods of Correlation and Regression Analysis, Linear and Curvilinear

By Mordecai Ezekiel; Karl A. Fox | Go to book overview

APPENDIX 3
Technical notes

Note 1. (Chapter 7). To prove for a very simple case that

measures the proportion of variance in Y explained by X. Let a, b, c, etc., be series of variables with σa = σb = σc, and with all intercorrelations such as rab, rac, etc., = 0.

Let Y = a + b + c, and X = a + b. Then

(Here the symbol pvx is used to represent

.) each (y)(x) = (a + b + c)(a + b) = a2 + 2ab + ac + b2 + bc

Since

Σ(ab), Σ(ac), Σ(bc) = 0 Σ(y)(x) = Σa2 + Σb2

Similarly,

(y2) = (a + b + c)2 = a2 + 2ab + 2ac + b2 + 2bc + c2 Σ(y2) = Σa2 + Σb2 + Σc2

By similar proof,

Σ(x)2 = Σa2 + Σb2

Hence

Similar results will be obtained for other combinations of elements.

Note 2. (Chapter 12). It can be proved that the coefficient of multiple determination (R2) measures the percentage of variance ascribable to the several independent factors for certain simple cases. Thus assume four variables, A, B, C, D, with all intercorrelations equal to 0, and all σ equal. Let Y = A + B + C. Then correlate Y with A, B, and D. The regression equation will work out

Y = a + A + B[+0(D)]

-531-

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