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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 θ 1 = 750 and θ 2 = 0.3 while
the other reports θ 1 = 13,000 and θ 2 = 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.
An interesting variation on nonlinear least squares is fitting kinetic models that cannot be solved explicitly
for the measured response. For example, the change in substrate concentration for Michaelis-Menten biok-
S 0 
inetics in a batch reactor is K m ln   --- S 0 –( S) = v m t . There is no analytic solution for S, so the predicted

S
value of S must be calculated iteratively and then compared with the observed value. Goudar and Devlin
(2001) discuss this problem.

References
Berthouex, P. M. and J. E. Szewczyk (1984). “Discussion of Influence of Toxic Metals on the Repression of
Carbonaceous Oxygen Demand,” Water Res., 18, 385–386.
Boyle, W. C., P. M. Berthouex, and T. C. Rooney (1974). “Pitfalls in Parameter Estimation for Oxygen Transfer
Data,” J. Envir. Engr. Div., ASCE, 100, 391–408.
Draper, N. R. and H. Smith (1998). Applied Regression Analysis, 3rd ed., New York, John Wiley.
Goudar, C. T. and J. F. Devlin (2001), “Nonlinear Estimation of Microbial and Enzyme Kinetic Parameters
from Progress Curve Data,” Water Envir. Res., 73, 260–265.

Exercises
35.1 Soybean Oil. The data below are from a long-term BOD test on soybean oil, where y = g BOD
per g of soybean oil, and t = time in days. There are four replicates at each time. (a) Fit the
first-order BOD model y = θ 1 [1 − exp (−θ 2 t)] to estimate the ultimate BOD θ 1 and the rate
coefficient, θ 2 . (b) Plot the data and the fitted model. (c) Plot the residuals as a function of
2 2
the predicted values and as a function of time. (d) Estimate σ from s = S R /(n − 2) and from
the replicate observations. Compare the two values obtained. (e) Map the sum of squares
surface and indicate the approximate 95% joint confidence region.

© 2002 By CRC Press LLC