Biology Reference
In-Depth Information
r
WSSR
N
Þ
1
ii
¼
ð
P
T
P
:
Asymptotic Standard Error
i
(8-51)
The asymptotic standard error for the i-th estimated parameter is related
to the weighted sum of squared residual of the fit, WSSR, the number
of data points, N, and the ii-th element of the inverse of the P
T
P
matrix that was already evaluated by the Gauss-Newton procedure and
used in Eq. (8-45). It is commonly used because it requires almost no
additional computer time to evaluate. There are, however, three
assumptions required to utilize these asymptotic standard errors as
realistic estimates of the precision of the estimated parameters: (1) The
fitting equation must be linear; (2) a large number of data points are
required; and (3) the parameter correlation should be near zero. The
consequence of these required assumptions is that the asymptotic
standard errors usually significantly underestimate the actual precision
of the estimated parameters. This means the significance of the results
will be overestimated, and conclusions not justified by the data will be
made. More sophisticated methods, beyond the scope of this text, such
as the support plane method (see Johnson and Frasier [1985]) and the
bootstrap approach (see Efron and Tibshirani [1993] for the details), can be
used for better precision.
D. Goodnes s -o f - F i t
Parameter estimation procedures, such as weighted least-squares, can
find an optimal fit of almost any equation to almost any data set. This
does not mean, however, that the fitted curve accurately describes the
experimental data. For example, the hemoglobin-oxygen binding data
shown in Figure 7-4 of Chapter 7 could be least-squares fit to a straight
line, but it would not provide a realistic description of the data points.
Likewise, the slope and intercept of this optimal straight line would
provide no information about the molecular mechanism of hemoglobin
function. Goodness-of-fit tests provide rigorous statistical criteria to
decide if the fitted equation actually provides a good description of the
experimental data. Furthermore, if the form of the fitting equation is
based upon mechanistic hypotheses about what is being measured, then
goodness-of-fit tests also provide rigorous statistical criteria to test the
mechanistic hypotheses. Because of this, the choice of the fitting equation
should always be dictated by the mechanistic hypotheses under study.
Most goodness-of-fit criteria are based on the distribution of the
residuals—the weighted differences r
i
between the data points and the
fitted curve in Eq. (8-37). If the justifying assumptions for the least-
squares approach from Section A above are satisfied, then the residuals
should follow a Gaussian distribution; if they do not, then one or more of
these assumptions is not valid. Assumptions 1, 2, and 4 are within the
control of the experimental protocol and thus can be verified
independently. If, in a carefully controlled and performed experiment,
