Page 292 - Solid Waste Analysis and Minimization a Systems Approach

270 SOLID WASTE ESTIMATION AND PREDICTION

The above procedure was applied to all business groups. The mathematics, statistical
tests, and example calculations for the transportation equipment manufacturer’s waste
group are discussed in the following paragraphs.
For the multiple regression equation (Walpole and Myers, Theorem 12.1, 1993):

y = X +βε

an unbiased estimator of σ 2 is given by the error or residual mean square

s = SSE
2
nk 1−−

where

n
i ∑
SSE = ∑ e = n ( y − ˆ ) 2
2
y
i
i
i=1 i=1
Sum of square calculations for multiple linear regression is similar to the previous
simple linear regression equation discussed earlier. One difference is the degrees of
freedom discussed previously in this chapter.
One criterion that is commonly used to illustrate the adequacy of a fitted regres-
2
sion line is the coefficient of multiple determination (R ) (Walpole and Myers, 1993):

n
∑ ( ˆ i y) 2
y −
=
R = SSR = i 1
2
SST n
∑ ( y − y) 2
i
=
i 1
The coefficient of determination indicates what proportion of the total variation in
the response is explained by the fitted model. The regression sum of squares can be
used to give some indication concerning whether or not the model is an adequate expla-
2
nation of the true situation (Walpole and Myers, 1993). The R value is the percent of
2
variation explained by each independent variable. The higher an R for a dependent and
independent variable is, the stronger the relationship among variables. One can test the
hypothesis H that the regression is not significant by forming the ratio
0

/
/
f = SSR k = SSR k
SSE n k s 2
/(
−− )1
−
and rejecting H at the α-level of significance when f > f k n k − . ) 1
(,
α
0
Another test, the t-test is the standard method used to evaluate individual coeffi-
cients in a multiple regression model. The addition of any single variable to a regres-
sion system will increase the regression sum of squares and thus reduce the error sum