the handbook of food engineering practice crc press chapter 10 ...

S = SS {1+ N p
n-Np F[N p, n-N p , (1-q)]} (27)
where f is the fitted nonlinear model, SS is the nonlinear least square estimate
of the fitted model, i.e. SS= Σ[ln(A i -f)] 2 for i=1 to n , n is the number of data points, N p
the number of parameters derived from the nonlinear least squares, 100(1-q)% the
confidence level and F the F -statistics.
The values used for deriving the confidence contour for the nonlinear
regression of the nonenzymatic browning data were as follows (Table 8):
SS=1.331 E-3; E A /R=15,796; A o =99.32 and F(3,34,90%)=2.27.
The fitted model, f, is replaced with the appropriate model based on the reaction order:
zero-order
⎡
⎤
f = A o + t exp ⎣ - E A ⎛1
⎞
R ⎝ T - 1
T ref ⎠⎦
first order
f = exp⎨ ⎧
⎡
⎤
ln(A ⎩ o ) + t exp ⎣ - E A ⎛1
⎞
R ⎝ T - 1
T ref ⎠⎦
⎭ ⎬⎫
n - order (n not equal to 1)
f = ⎨ ⎧
A ⎩ o
(n-1) ⎡
⎤
+ (1-n) t exp ⎣ - E A ⎛1
⎞
R ⎝ T - 1 (1/(1-n))
T ref ⎠⎦
The appropriate sign +/- in the above equations should be chosen. For a
reaction where concentration increases a positive should be used. For a depletion reactrion
the negative sign should be utilized.
⎭ ⎬⎫
The algorithm implemented to derive the confidence region is as follows:
a. Initial concentration is assumed constant and the estimated value derived by the
nonlinear regression is utilized.
b. The confidence contour is derived by choosing values of E A /R and k ref which fulfill
the equality expressed in Eq. (27). Obviously, the value of E A /R and k ref are varied within
the range of values that satisfies the inequality listed in Eq. (27). This trial and error
procedure is normally carried out on a computer.
The derived confidence contour is depicted in Fig. 11. It shows the span in the
calculated values of E A /R and k ref . When comparing the confidence regions derived by the
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