# Statistics for Environmental Engineers

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The fitted model is:

Model C:    pH = 5.82 + 1.11Z1 + 1.38Z2 — 0.0057Z3 WA

(t-ratios)    (8.43)    (12.19)    (6.68)

TABLE 40.2

Alternate Models for pH at Cosby Creek

 Model Reg SS Res SS R2 A pH = 5.77 — 0.00008 WA + 0.998Z1 + 1.65Z2 — 0.005Z1 WA — 0.008Z2WA 4.278 0.662 0.866 B pH = 5.82 + 0.95Z1 + 1.60Z2 — 0.005Z1WA — 0.008Z2WA 4.278 0.662 0.866 C pH = 5.82 + 1.11Z1 + 1.38Z2 — 0.0057Z3WA 4.229 0.712 0.856

This simplification of the model can be checked in a more formal way by comparing regression sums of squares of the simplified model with the more complicated one. The regression sum of squares is a measure of how well the model fits the data. Dropping an important term will cause the regression sum of squares to decrease by a noteworthy amount, whereas dropping an unimportant term will change the regression sum of squares very little. An example shows how we decide whether a change is “noteworthy” (i.e., statistically significant).

If two models are equivalent, the difference of their regression sums of squares will be small, within an allowance for variation due to random experimental error. The variance due to experimental error can be estimated by the mean residual sum of squares of the full model (Model A).

The variance due to the deleted term is estimated by the difference between the regression sums of squares of Model A and Model C, with an adjustment for their respective degrees of freedom. The ratio of the variance due to the deleted term is compared with the variance due to experimental error by computing the F statistic, as follows:

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