Farm Animal Metabolism and Nutrition

Appendix

Non-linear curve fitting

Useful numerical information is extracted
from gas data by means of non-linear curve
fitting. To understand the significance of
such information, it is helpful to have an
overview of the curve-fitting process and
some appreciation of the statistical tests
used to evaluate the quality of a fit
(Motulsky and Ransnas, 1987).
At least three different pieces of soft-
ware have been applied for this fitting.
France et al. (1993) used the Maximum
Likelihood Program (Ross, 1987). Cone et al.
(1996) used a program called NLREG (non-
linear regression analysis). Schofield et al.
(1994) used a commercial program called
TableCurve (published by Jandel Scientific,
now incorporated into SPSS Science,
Chicago, Illinois) that is powerful, graphic-
ally oriented and easy to use. The following
description applies to TableCurve.
After importing the data set (as an xy
curve) into the program, one first con-
structs or imports an equation, called a
user-defined function, to use for fitting.
Taking the single pool logistic equation
(Equation 10.5) as an example, the equa-
tion would be entered as:

Y = #A (1 + exp(2 
 4

#B (x
#C)))
^1 (10.9)

Here, Y is the gas volume at time x, #A is
the asymptotic gas volume (units, ml), #B

is the specific rate (units, h
^1 ) and #C is the

lag (units, h). #A, #B and #C are parameters

to be calculated by the program such that

the function represented by Equation 10.9

passes through, or close to, as many of the

data points as possible. The next step is to

assign starting values to #A, #B and #C and

to set limits on possible values. For the

logistic equation, we can specify that all

parameters must be >0. The asymptotic

volume #A is easily estimated from the

data curve. The specific rate #B usually lies

within the range 0.02–0.2 h
^1 , and the lag

#C is also readily estimated from the data

curve. The program allows one to see the

visual effects on the curve shape of

changes in these starting values. In addi-

tion, the program can perform a ‘limited’ fit

to help obtain these values. With starting

values assigned to produce a curve of

approximately the right shape, we are now

ready to do a full-scale fit. During this

process, the parameters are adjusted itera-

tively to minimize the sum of the squared

errors (the difference between the calcu-

lated and actual data points, SSE). The

fitting process is extremely fast and takes

<1 s for most of the functions and data sets

we use.

Fitting stops and convergence is

assumed when the coefficient of deter-

mination (r^2 ) did not change in the sixth

significant figure for five consecutive itera-

tions. Among the statistics reported by

TableCurve, Fstatistic and r^2 were used to

evaluate a fit. The Fstatistic (mean square

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