Statistics for Environmental Engineers

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The data (if they are useful for model building) will restrict the plausible parameter values to lie within a certain region. The intercept and slope of a straight line, for example, must be within certain limits or the line will not pass through the data, let alone fit it reasonably well. Furthermore, if the slope is decreased somewhat in an effort to better fit the data, inevitably the intercept will increase slightly to preserve a good fit of the line. Thus, low values of slope paired with high values of intercept are plausible, but high slopes paired with high intercepts are not. This relationship between the parameter values is called parameter correlation. It may be strong or weak, depending primarily on the settings of the x variables at which experimental trials are run.


Figure 34.1 shows some joint confidence regions that might be observed for a two-parameter model. Panels (a) and (b) show typical elliptical confidence regions of linear models; (c) and (d) are for nonlinear models that may have confidence regions of irregular shape. A small joint confidence region indicates precise parameter estimates. The orientation and shape of the confidence region are also important. It may show that one parameter is estimated precisely while another is only known roughly, as in (b) where в2 is estimated more precisely than в1. In general, the size of the confidence region decreases as the number of observations increases, but it also depends on the actual choice of levels at which measurements are made. This is especially important for nonlinear models. The elongated region in (d) could result from placing the experimental runs in locations that are not informative.

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