Biochemical Reactions - A First Look
85
Example 3.11 Application of the least-squares estimate
Using Eq. (3.51) on the data analyzed in Example 3.10, i.e., R, and qm
are given respectively by Eqs. (3) and
(4) of Example 3.10, we find
^-0.142^
-0.060
q" “
0.064
(1)
^ 0.077;
which is almost identical with the estimates for the rates found in Example 3.10, where information
concerning the measurement errors of the rates was included.______________________________________
If any components in the residual vector are significantly different from zero, either there must be a
significant error in at least one of the measurements or the applied model is not correct. To quantify
what is meant by residuals significantly different from zero, we introduce the test function
h
given
by the sum of weighted squares of the residuals, i.e., the residuals are weighted according to their
accuracy:
h
=
eTP ]e
(3.52)
When the raw measurements are uncorrelated, the test function
h
is chi-square distributed [see, e.g.,
Wang and Stephanopoulos (1983)], and Heijden
ei al.
(1994b) proved that this is also the case of
correlated raw data. The degrees of freedom of the distribution is equal to rank(P) = rank(R).
The calculated value of
h
is compared with values of the ^distribution at the given value of
rank(R). If at a high enough confidence level
\-0
one obtains that A
is larger than
X
2, then there is
something wrong with the data or the model. Table 3.5 is an extract from a table of
X1
values.
Normally a confidence level of at least 95% should be used.
Table 3.5 Values of the X2 distribution.
Confidence level П - ft)
Degrees of freedom
0.500
0.750
0.900
0.950
0.975
0.990
1
0.46
1.32
2.71
3.84
5.02
6.63
2
1.39
2.77
4.61
5.99
7.38
9.21
3
2.37
4.11
6.25
7.81
9.35
11.30
4
3.36
5.39
7.78
9.49
11.10
13.30
Л
_________________
4.35
6.63
9.24
11.10
12.80
15.10