# Statistics for Environmental Engineers

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For linear models, the elements Xj do not involve the parameters of the model. They are functions only of the independent variables (xj) or combinations of them. (This is the characteristic that defines a model as being linear in the parameters.) It is easily shown that the minimum variance design for a linear model spaces observations as far apart as possible. This result is intuitive in the case of fitting n = во + в1х; the estimate of во is enhanced by making an observation near the origin and the estimate of вi is enhanced by making the second observation at the largest feasible value of x. This simple example also points out the importance of the qualifier “if the model is assumed to be correct.” Making measurements at two widely spaced settings of x is ideal for fitting a straight line, but it has terrible deficiencies if the correct model is quadratic. Obviously the design strategy is different when we know the form of the model compared to when we are seeking to discover the form of the model. In this chapter, the correct form of the model is assumed to be known.

Returning now to the design of experiments to estimate parameters in nonlinear models, we see a difficulty in going forward. To find the settings of Xj that maximize X’X, the values of the elements X, must be expressed in terms of numerical values for the parameters. The experimenter’s problem is to provide these numerical values.

At first this seems an insurmountable problem because we are planning the experiment because the values of в are unknown. Is it not necessary, however, to know in advance the answer that those experiments will give in order to design efficient experiments. The experimenter always has some prior knowledge (experienced judgment or previous similar experiments) from which to “guess” parameter values that are not too remote from the true values. These a priori estimates, being the best available information about the parameter values, are used to evaluate the elements of the derivative matrix and design the first experiments.

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