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Scanning EAN13 Supplement 5 In Visual C# Using Barcode reader for .NET framework Control to read, scan read, scan image in .NET applications. Bar Code Reader In None Using Barcode scanner for Software Control to read, scan read, scan image in Software applications. SOLID WASTE ESTIMATION AND PREDICTION
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S S yy S xy n 2
In the preceding equations, is the slope of the regression line and s is an unbiased estimator of 2 (the population standard deviation). The variables Sxx, Syy, and Sxy are the sum of squares for x and y (Walpole and Myers, 1993). 16.3.5 STEP 5: VALIDATE REGRESSION ASSUMPTIONS
There are ve major assumptions or ideal conditions for the estimation and inference in multiple regression models (Dielman, 1996). The ve assumptions are 1 2 3 4 5 The relationship is statistically signi cant. 2 The residuals, ei , have constant variance e . The residuals are independent. The residuals are normally distributed. The explanatory variables are not highly correlated. The rst assumption was tested and validated using the Ftest. The Ftest was discussed in the previous section at the 95 percent con dence level. To access the assumptions of constant variance around the regression line, that residuals are randomly distributed, and residuals are normally distributed, residual plot analysis was conducted. A residual describes the error in the t of the model at the ith data point (Walpole and Myers, 1993) and is described in the following equation: ei = yi yi In a residual plot of ei versus an explanatory variable x, the residuals should appear scattered randomly about the zero line with no difference in the amount of variation in the residuals, regardless of the value of x (Dielman, 1996). If there appears to be a difference in variation (for example, if the residuals are more spread out for larger values of x than for small values), then the assumption of constant variance may be violated. STEPWISE REGRESSION METHODOLOGY
EMP residual plot 600 400 200 Residuals 0 200 0 400 600 800 EMP 100 200 300
Example residual plot.
Figure 16.4 is an example of a residual plot for number of employees for the transportation equipment manufacturer s waste group. As shown by the residual plot, the residuals appear to be random and of equal variance, except for two outliers, which were removed and the model was recalculated. No patterns appear to be present. This validates the constant variance and random error assumptions of the regression model. The same tests were conducted for the other 19 waste groups as well. The nal assumption in a multiple regression model is that explanatory (independent) variables are not correlated with one another. When explanatory variables are correlated with one another, the problem of multicolinearity is said to exist. The presence of a high degree of multicolinearity among the explanatory variables will result in the following problems (Dielman, 1996): The standard deviation of the regression coef cients will be disproportionately large. The regression coef cient estimates will be unstable, and the accuracy will vary signi cantly for different independent variables. To detect multicolinearity, several methods have been developed. One method involves computing the pairwise correlations between explanatory variables. One rule of thumb suggested by some researchers is that mutlicolinearity may be a serious problem if any pairwise correlation is greater than 0.5 (Dielman, 1996). Multicolinearity was examined using this method and all pairwise correlation was below 0.5. The correlations between random variables are denoted by r and are calculated as: r = rx1x2 = S12 S11S22 S11, S12, and S22 may be found in the (X X) matrix in the off diagonals. All correlations between independent variables were less than 0.001. SOLID WASTE ESTIMATION AND PREDICTION
16.4 Analysis of Results and Summary of Findings
This section discusses the results and ndings from the regression analysis conducted on the 20 waste groups. Included in this discussion are lists of the independent variables that are signi cant in predicting annual solid waste for each group, the full regression results (Ftest, ttest, and R2coef cient of determination), ANOVA, and correlation analysis. Table 16.3 and Fig. 16.5 display the signi cant variables that are TABLE 16.3 SIGNIFICANT VARIABLES THAT INFLUENCE SOLID WASTE FOR THE 20 WASTE GROUPS LANDFILL DISPOSAL COST ($/TON)

