Statistics for Environmental Engineers

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A fundamental concept in model discrimination is that rival models often diverge noticeably only at extreme conditions. It follows that extremes must be studied or discrimination on a statistical basis is impossible. This gives the researcher the choice of restricting the experiment to a limited range of conditions and accepting any plausible model that fits the data over this range, or testing at more stressful conditions to obtain evidence in favor of one among several rival models. To discriminate between models (and between mechanisms), the models must be put in jeopardy of failing.


Figure 47.1 shows two examples of this. The two first-order reactions in series, with a reversible second step, cannot be discriminated from the irreversible series A ^ B ^ C unless observations are made at long times. The Haldane inhibition model cannot be distinguished from the simpler Monod model unless studies are made at high concentrations. One might expect the problem of model discrimination to be fairly obvious with models such as these, but examples of weak experimental designs do exist even for these models.


Given models that seem adequate on statistical grounds, we might try to select the best model on the basis of (1) minimum sum of squares, (2) lack of fit tests (F tests), (3) fewest parameters (parsimony), (4) simplest functional form, and (5) estimated parameter values consistent with the mechanistic premise of the model. In practice, all of these criteria come into consideration.

Case Study: Two Rival Biological Models


We consider two models for bacterial growth in industrial biotechnology and biological wastewater treatment. Both assume that the specific growth rate of bacterial solid is limited by the concentration of a single substrate. The two models are by Monod (1942) and Tiessier (1936):

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