# Statistics for Environmental Engineers

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FIGURE 38.3 Plot of residuals against the predicted values of the regression model у = 185.97 + 7.125t + 0.014/2 — 3.057z.

the model might be improved. The pattern of the residuals plotted against time (Figure 38.2b) suggests that adding a t2 term may be helpful. This was done to obtain:

у = 186.0 + 7.12z- 3.06t + 0.0143t2

which has R2 = 0.97 and residual mean square = 81.5. A diagnostic plot of the residuals (Figure 38.3) reveals no inadequacies. Similar plots of residuals against the independent variables also support the model. This model is adequate to describe the data.

The most complicated model, which has six parameters, is:

у = 152 + 20.9z — 2.741 1.13z2 — 0.014312 — 0.080zt

The model contains quadratic terms for time and depth and the interaction of depth and time (zt). The analysis of variance for this model is given in Table 38.2. This information is produced by computer programs that do linear regression. For now we do not need to know how to calculate this, but we should understand how it is interpreted.

Across the top, SS is sum of squares and df = degrees of freedom associated with a sum of squares quantity. MS is mean square, where MS = SS/df. The sum of squares due to regression is the regression sum of squares (RegSS): RegSS = 20,255.5. The sum of squares due to residuals is the residual sum of squares (RSS); RSS = 308.8. The total sum of squares, or Total SS, is:

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