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488 CHAPTER 10 MULTIPLE REGRESSION AND CORRELATION
Y
Deviations from the
plane
Regression plane
X 2 X 1
FIGURE 10.2.1
Multiple regression plane and scatter of points.
unit change in X 1 when X 2 remains unchanged, and b 2 measures the average change in
Y for a unit change in X 2 when X 1 remains unchanged. For this reason b 1 and b 2 are
referred to as partial regression coefficients.
10.3 OBTAINING THE MULTIPLE
REGRESSION EQUATION
Unbiased estimates of the parameters b 0 , b 1 , . . . , b k of the model specified in Equation
10.2.1 are obtained by the method of least squares. This means that the sum of the
squared deviations of the observed values of Y from the resulting regression surface is
minimized. In the three-variable case, as illustrated in Figure 10.2.1, the sum of the
squared deviations of the observations from the plane are a minimum when b 0 , b 1 , and
b 2 are estimated by the method of least squares. In other words, by the method of least
squares, sample estimates of b 0 , b 1 , . . . , b k are selected in such a way that the quantity
gP 2
j = g1y j - b 0 - b 1 x 1j - b 2 x 2j - ... - b k x kj 2 2
is minimized. This quantity, referred to as the sum of squares of the residuals, may also
be written as
gP j 2 = g1y j - yN j 2 2
(10.3.1)
indicating the fact that the sum of squares of deviations of the observed values of Y from
the values of Y calculated from the estimated equation is minimized.