Statistics for Environmental Engineers

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Computing estimates of parameters values is easy. Hand-held calculators can do it for a straight line, and inexpensive commercial software for personal computers can do it for large linear and nonlinear models. Unfortunately, most of these computing resources do not compute or plot the joint confidence region for the parameters.

For linear models, exact confidence regions can be calculated. If the model is nonlinear, we construct an approximate joint confidence region. Although the confidence level (i.e., 95%) associated with the region is approximate, the size and shape are correct. The effort required to determine these confidence regions is rewarded with useful information.

The highly correlated parameter estimates that result from weak experimental designs can lead to serious disputes. Suppose that two laboratories made independent experiments to estimate the parameters in a model and, furthermore, that both used a poor design. One might report 61 = 750 and 62 = 0.3 while the other reports 61 = 13,000 and в2 = 0.1. Each might suspect the other of doing careless work, or of making a mistake in calculating the parameter values. If the experimental design produced an elongated confidence region with a high degree of correlation, both sets of parameter values could be within the confidence region. In a sense, both could be considered correct. And both could be useless. If the joint confidence region were presented along with the estimated parameter values, it would be clear that the disagreement arises from flawed experimental designs and not from flawed analytical procedures.

The examples presented here have, for graphical convenience, been based on two-parameter models. We can obtain a picture of the joint confidence region when there are three parameters by making contour maps of slices through the volume that is bounded by the critical sum of squares value.

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