Statistics for Environmental Engineers

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Many published papers show estimates of 01 and 02 derived from measurements at just a few early times, that is, at days 1 through 5. The experimenters may have reasoned something like this: “I am interested mainly in an estimate of the rate constant, and I want to get the data in the shortest time possible. If 02 is known, I can compute 01. Because the reaction rate is high on days 1 through 5, data over this range will give a good estimate of 02 and the experiment can be finished within a few days.”

This plausible argument is wrong. The experiment gives poor estimates of both 01 and 02. The rate constant can only be precisely estimated if the asymptotic level is well estimated, and this requires having measurements at 15 to 20 days no matter how many measurements are made at early times. The problem is that data at early times only allow estimation of the initial slope of the curve, which is:

00 = constant


dt t=0

Thus, any values of 01 and 02 that have a product nearly equal to the slope of the curve over the first few days will reasonably fit the data collected on the first few days. The shape of the joint confidence region will be similar to Figure 34.1d. The hyperbolic shape reflects the parameter correlation and shows that neither parameter is well estimated. Well-designed experiments will yield precise, uncorrelated parameter estimates and the joint confidence region will tend toward being elliptical.

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