# Statistics for Environmental Engineers

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After modifying a model by adding, or in this case dropping, a term, an additional test should be made to compare the regression sum of squares of the two models. Details of this test are given in texts on regression analysis (Draper and Smith, 1998) and in Chapter 40. Here, the test is illustrated by example.

The regression sum of squares for the complete model (Model A) is 20,256. Dropping the z2 term to get Model B reduced the regression sum of squares by only 54. We need to consider that a reduction of 54 in the regression sum of squares may not be a statistically significant difference.

The reduction in the regression sum of squares due to dropping z2 can be thought of as a variance associated with the z term. If this variance is small compared to the variance of the pure experimental

error, then the term z contributes no real information and it should be dropped from the model. In

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contrast, if the variance associated with the z term is large relative to the pure error variance, the term should remain in the model.

There were no repeated measurements in this experiment, so an independent estimate of the variance due to pure error variance cannot be computed. The best that can be done under the circumstances is to use the residual mean square of the complete model as an estimate of the pure error variance. The residual mean square for the complete model (Model A) is 51.5. This is compared with the difference in regression sum of squares of the two models; the difference in regression sum of squares between Models A and B is 54. The ratio of the variance due to z2 and the pure error variance is F = 54/51.5 = 1.05. This value is compared against the upper 5% point of the F distribution (1, 6 degrees of freedom). The degrees of freedom are 1 for the numerator (1 degree of freedom for the one parameter that was dropped from the model) and 6 for the denominator (the mean residual sum of squares). From Table C in the appendix, F16 = 5.99. Because 1.05 < 5.99, we conclude that removing the z2 term does not result in a significant reduction in the regression sum of squares. Therefore, the z term is not needed in the model.

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