# Statistics for Environmental Engineers

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Nonlinear least squares was used to estimate the parameters with the following results:

RSS = 0.00243 (v =13) RMS = 0.000187 RSS = 0.00499 (v =13) RMS = 0.000384

FIGURE 47.1 Examples of two models where discrimination is impossible unless observations are made at high levels of the independent variable.

Monod model: a =    -—-

90.4 + S

Tiessier model: /2 = 0.83( 1 — exp(-0.012S))

The hats (a) indicate the predicted values. RSS is the residual sum of squares of the fitted model, v = 13 is the degrees of freedom, and RMS is the residual mean square (RMS = RSS/v). (The estimated parameter values have been rounded so the values shown will not give exactly the RSS shown.)

Having fitted two models, can we determine that one is better? Some useful diagnostic checks are to plot the data against the predictions, plot the residuals, plot joint confidence regions (Chapter 35), make a lack-of-fit test, and examine the physical meaning of the parameters.

The fitted models are plotted against the experimental data in Figure 47.2. Both appear to fit the data. Plots of the residuals (Figure 47.3) show a slightly better pattern for the Monod model, but neither model is inadequate. Figure 47.4 shows joint confidence regions that are small and well-conditioned for both models.

These statistical criteria do not disqualify either model, so we next consider the physical significance of the parameters. (In practice, an engineer would probably consider this before anything else.) If 61 represents

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