Statistics for Environmental Engineers

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The simplest possible optimal design has only two observations, one at t1= 11в2= 5 days and t2= 20 days.) Two well-placed observations are better than six badly located ones (Design A). The confidence region is smaller and the parameter estimates will be independent (the confidence region will tend to be elliptic rather than elongated). Replication of the optimal design quickly reduces the joint confidence region. The design with five replicates (a total of 10 observations) is about equal to Design C with 20 observations. This shows that Design C has about half its observations made at times that contribute little information about the parameter values. The extra experimental effort has gone to confirming the mathematical form of the model


This approach to designing experiments that are efficient for estimating parameters in nonlinear models depends on the experimenter assuming that the form of the model is correct. The goal is to estimate parameters in a known model, and not to discover the correct form of the model.

The most efficient experimental strategy is to start with simple designs, even as small as n = p observations, and then to work iteratively. The first experiment provides parameter estimates that can be used to plan additional experiments that will refine the parameter estimates.

In many cases, the experimenter will not want to make measurements at only the p locations that are optimal based on the criterion of maximizing A. If setting up the experiment is costly but each measurement is inexpensive, it may preferable to use several observations at near optimal locations. If the experiment runs a long time (long-term BOD test) and it is difficult to store experimental material (wastewater), the initial design should be augmented, but still emphasizing the critical experimental regions. It may be desirable to add some observations at nonoptimal locations to check the adequacy of the model. Augmenting the optimal design is sensible. The design criterion, after all, provides advice — not orders.

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