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35

Precision of Parameter Estimates

in Nonlinear Models

KEY WORDS biokinetics, BOD, confidence interval, confidence region, joint confidence region, crit-
ical sum of squares, Monod model, nonlinear least squares, nonlinear regression, parameter correlation,
parameter estimates, precision, residuals sum of squares.

The precision of parameter estimates in nonlinear models is defined by a boundary on the sum of squares
surface. For linear models this boundary traces symmetric shapes (parabolas, ellipses, or ellipsoids). For
nonlinear models the shapes are not symmetric and they are not defined by any simple geometric equation.
The critical sum of squares value:

p


p

S c = S R + S R ------------F p,n− p,α = S R 1 + ------------F p,n− p,α  



p
p
n –
n –
bounds an exact (1 − α)100% joint confidence region for a linear model. In this, p is the number of
parameters estimated, n is the number of observations, F p,n−p,α is the upper α percent value of the F
distribution with p and n – p degrees of freedom, and S R is the residual sum of squares. We are using
2
2
s = S R /(n − p) as an estimate of σ .
Joint confidence regions for nonlinear models can be defined by this expression but the confidence
level will not be exactly 1 – α. In general, the exact confidence level is not known, so the defined region
2
is called an approximate 1 – α confidence region (Draper and Smith, 1998). This is because s = S R /
2
(n − p) is no longer an unbiased estimate of σ .
Case Study: A Bacterial Growth Model
Some data obtained by operating a continuous flow biological reactor at steady-state conditions are:
x (mg// //L COD) 28 55 83 110 138
y (1// //h) 0.053 0.060 0.112 0.105 0.099
The Monod model has been proposed to fit the data:

y i = --------------- +
θ 1 x i
θ 2 + x i e i
−1 −1
where y i = growth rate (h ) obtained at substrate concentration x, θ 1 = maximum growth rate (h ), and
θ 2 = saturation constant (in units of the substrate concentration).
The parameters θ 1 and θ 2 were estimated by minimizing the sum of squares function:

θ 1 x
minimize S = ∑   y i – -------------- x  2
θ 2 + 

© 2002 By CRC Press LLC