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STEPWISE REGRESSION METHODOLOGY
n g0 = yi i =1 n g1 = xi yi g = X y = i =1 n gk = x ki yi i =1
the normal equations can be put in matrix form A = g If the matrix A is nonsingular, the solution for the regression coef cients is written as = A 1g = ( X X ) 1 X y The regression equation is obtained by solving a set of k + 1 equations for the like number of unknowns. This involves the inversion of k + 1 by k + 1 matrix X X. Step 3 in the following section was used to calculate the regression equation for each waste group. As a visual representation, the scatter diagram for the transportation equipment manufacturing group is displayed in Fig. 16.3 for annual waste versus the number of employees.
300 Annual solid waste (tons) 250 200 150 100 50 0 0 500 1000 Number of employees 1500 2000
Figure 16.3 Scatter plot example of annual solid waste versus the number of employees for transportation equipment manufacturers waste group.
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 speci ed level of con dence Accounts for multicolinearity (correlation between variables) Selects the most ef cient 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 rst 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 independent variable now in the equations. If the partial F statistic for a variable is less than FOUT, the variable is dropped from the model. For this research, an FOUT corresponding to the 95 percent con dence level was used. Stepwise regression requires two cutoff variables, FIN and FOUT. Some analysts prefer FIN = FOUT, although this is not necessary. Frequently FIN > FOUT, making it relatively more dif cult to add an independent variable than to delete one (Montgomery, 2001). Minitab was applied to conduct 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 regression 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 SST = ( yi y )2 = ( yi y )2 + ( yi yi )2
i =1 i =1 i =1 n n n
Using a different notation for the SST identity: SST = SSR + SSE with SST = ( yi y )2 = total sum of squares
i =1 n
STEPWISE REGRESSION METHODOLOGY
and SSR = ( yi y )2 = regression sum of squares
i =1 n
There are k degrees of freedom associated with SSR and SST has n 1 degrees of freedom. Therefore, after subtraction, SSE has n k 1 degrees of freedom. The esti2 mate of is given by the error sum of squares divided by the degrees of freedom (Walpole and Myers, 1993). All three of the sums of squares appear on the printout of most multiple regression computer packages, including Minitab, which was used for this research. Minitab and SYSTAT were used to calculate these values. SYSTAT was used to verify results. The F statistic is calculated using the following equation: f = SSR/k SSR/k = SSE /(n k 1) s2
The results of the sum of squares and F statistic may be represented in an analysis of variance (ANOVA) table (Table 16.2).
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