# Statistics for Environmental Engineers

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The Optimal Design and Design A have confidence regions that are about the same size. What is not indicated by the A-1/2 values is that the region for optimal design will be elliptical while Design A will be elongated. Design A has been used very often in kinetic studies. It is inefficient. It does not give independent estimates of the parameters because all the observations are in the early part of the experiment and none are on the asymptote. It tends to estimate the product в1в 2 rather than 61 and в2 independently. In fact, в1 and в2 are estimated so poorly as to be virtually useless; the joint confidence region is hyperbolic and elongated (banana-shaped) instead of the more elliptical shape we would like. This weakness in the design cannot be overcome merely by putting more observations into the same region. To improve, the observations must be made at times that yield more information.

Design B, with 10 observations, is similar to Design A, but the observations cover a wider range of time and four points are duplicated. This almost doubles the experimental work but it reduces the size of the confidence region by a factor of four. This reduction is due mainly to putting some observations on the asymptote. Adding five observations will do nothing if they are made in the wrong places. For example, duplicating four of the five points in Design A will not do much to improve the shape of the confidence region or to reduce its size.

Design C has 20 observations and yields a well-shaped confidence region that is half the size obtained by Design B. Also, it has the advantage of providing a check on the model over the range of the experimental space. If the first-order model is wrong, this design should reveal it. On the other hand, if the model is correct, the precise parameter estimates can be attained with less work by replicating the optimal design.

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