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16.3.4 STEP 4: EVALUATE RESULTS AND DETERMINE FINAL MODELS
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The goals of this step is to develop the nal models and determine the variables that ef ciently predict solid waste generation for each waste group. The following procedure is used:
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Apply stepwise regression method to the 20 waste groups (data matrices) Remove outliers and recalculate Evaluate the results statistically with: ANOVA F-test (strength of entire models) t-test (strength of each independent variable) Coef cient of determination Validate regression model assumptions Report nal results
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TABLE 16.2 SOURCE OF VARIATION
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MULTIVARIABLE ANOVA TABLE FORMAT SUM OF SQUARES DEGREES OF FREEDOM MEAN SQUARE
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Regression Error Total
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SSR SSE SST
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k n k 1 n 1
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SSR SSE/(n k 1)
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(SSR/k)/s 2
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 +
2 an unbiased estimator of is given by the error or residual mean square
s2 = where SSE =
i =1 n
SSE n k 1
= ( yi yi )2
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 tted regression line is the coef cient of multiple determination (R2) (Walpole and Myers, 1993): ( yi y )2
R2 =
SSR = SST
( yi y )2
i =1
i =1 n
The coef cient of determination indicates what proportion of the total variation in the response is explained by the tted model. The regression sum of squares can be used to give some indication concerning whether or not the model is an adequate explanation of the true situation (Walpole and Myers, 1993). The R2 value is the percent of variation explained by each independent variable. The higher an R2 for a dependent and independent variable is, the stronger the relationship among variables. One can test the hypothesis H0 that the regression is not signi cant by forming the ratio f = SSR/k SSR/k = SSE /(n k 1) s2
and rejecting H0 at the -level of signi cance when f > f ( k , n k 1) . Another test, the t-test is the standard method used to evaluate individual coef cients in a multiple regression model. The addition of any single variable to a regression system will increase the regression sum of squares and thus reduce the error sum
STEPWISE REGRESSION METHODOLOGY
of squares (Walpole and Myers, 1993). The decision is whether the increase in the regression is suf cient to warrant using the variable in the model. The use of unimportant variables can reduce the effectiveness of the prediction equation by increasing the variance of the estimated response (Walpole and Myers, 1993). The following mathematics illustrate the method used to evaluate individual coef cients and determine if each variable effectively aids in predicting total annual waste. A t-test, which is a statistical test on a sample from a normally distributed population, was conducted at the 95 percent con dence level to determine if there was signi cant correlation between the variables. The null hypothesis (H0) was de ned such that the slope of the population regression line ( i) is zero, in other words variables are not correlated. This would mean that there is no linear relationship between the independent variables (xi) and dependent variables (yi). The alternate hypothesis (H1) states that the slope of the population regression is not equal to zero, in other words, the tested variables are correlated and do have a relationship. The t-tests were conducted as follows: Hypothesis test H 0 : 1 = 0 H1 : 1 0 Decision rule Reject H0 if t > t /2, n 1 or t < t /2, n 1 The t value was calculated at the signi cance level and n 1 degrees of freedom. The decision rule is based upon a two-tail test, where /2 and + /2 de ne the critical region. Calculated test statistic values (see equation below) with a value less than /2 or greater than + /2 will indicate a relationship exists between the dependent and independent variable. This rule is based upon the t distribution. Test statistic t= b s/ S xx
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