Page 290 - Solid Waste Analysis and Minimization a Systems Approach

268 SOLID WASTE ESTIMATION AND PREDICTION

16.3.3 STEP 3: CONDUCT REGRESSION PROCEDURE AND
SELECT INDEPENDENT VARIABLES (STEPWISE REGRESSION
METHOD)

The stepwise regression method was used to select the independent variables for each
of the 20 remaining waste groups using the developed matrices. The stepwise method
was chosen for several reasons:

■ Statistically evaluates the addition or removal of every selected potential independent
variable at the specified level of confidence
■ Accounts for multicolinearity (correlation between variables)
■ Selects the most efficient variables for the model, considering redundancy
■ The method is programmable

Stepwise regression is an algorithm that applies an iterative process to determine the
optimal independent variables. For the first step, all independent variables are entered
into the model and are represented by their partial F statistic. A partial F statistic is the
ratio of variance explained by the independent variable divided by total variation
between all observations of the sample. An independent variable added at an earlier
step may now be redundant because of the relationships between it and the independ-
ent variable now in the equations. If the partial F statistic for a variable is less than
F OUT , the variable is dropped from the model. For this research, an F OUT correspon-
ding to the 95 percent confidence level was used. Stepwise regression requires two
cutoff variables, F and F OUT . Some analysts prefer F = F OUT , although this is not
IN
IN
necessary. Frequently F IN > F OUT , making it relatively more difficult to add an inde-
pendent variable than to delete one (Montgomery, 2001). Minitab was applied to con-
duct this analysis.
The F statistic is calculated based on the sum of squares from the regression results
for each variable (the method discussed in the previous section determines the regres-
sion results). The mathematics for the F statistic is shown below.
For multiple linear regression, the error and regression sum of squares take the same
form as in the simple linear regression case (Walpole and Myers, 1993). The total sum
of squares (SST) identity is

n n
n
n
SST = ∑ ( y − y) 2 = ∑ ( ˆ i y) 2 + ∑ ( y − ˆ ) 2
y −
y
i
i
i
i=1 i=1 i=1
Using a different notation for the SST identity:
SST = SSR + SSE
with

n
SST = ∑ ( y − y) 2 = total sum of squares
i
i=1