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variation [a], [b], and [c] defined in Fig. 10.10. For the example data set,

R2 = 0.5835, as already shown.

• Compute the multiple regression of y against X. The corresponding R2 measures [a + b], which is the sum of the fractions of variation [a] and [b]. For the example, this R2 = 0.4793. The vector of fitted values corresponding to fraction [a + b], which is required to plot Fig. 10.12 (below), is also computed.

• Compute the multiple regression of y against W. The corresponding R2 measures [b + c], which is the sum of the fractions of variation [b] and [c]. For the example, this R2 = 0.3878. The vector of fitted values corresponding to fraction [b + c], which is required to plot Fig. 10.12 (below), is also computed.

• If needed, fraction [d] may be computed by subtraction. It is equal to 1 - [a + b + c], or 1 - 0.5835 = 0.4165 for the example data set.

• Fraction [b] of the variation may be obtained by subtraction, in the same way as the quantity B used for comparing two qualitative descriptors in Section 6.2:

For the example data set,

Note that no fitted vector can be estimated for fraction [b], which is obtained by subtraction and not by estimation of an explicit parameter in the regression model. For this reason, fraction [b] may be negative and, as such, it is not a rightful measure of variance; this is why it is referred to by the looser term variation.

Correlations

Path coefficients symmetric model

Path coefficients asymmetric model

Coefficients of determination (R2)