Statistical Methods in Medical Research 4ed

(1989) propose an approach which tries to identify appropriate error structures
from the data themselves. These authors assume that the data obey the model
y …l†
i
ˆ Vmaxxi
KM ‡ xi
…l†
‡ x u i ei,
where ei are independent normal random variables with mean 0 and variance s 2
and z …l† denotes the Box±Cox transformation (see §10.8). The parameters l and
u, as well as KM, Vmax and s, are estimated by maximum likelihood. The form of
error model induced by certain pairs of values for l and u are described in
Table 1 of Ruppert et al. (1989).
An alternative approach to estimating the form of the error from the data
was discussed by Nelder (1991) in a rejoinder to Ruppert et al. Nelder advocated
using a generalized linear model, with mean given by (20.21) and variance
function V…m† ˆm u ; a reciprocal link applied to (20.21) gives a form that is
linear in KM=Vmax and 1=Vmax. Parameters are estimated by extended quasilikelihood
(see McCullagh & Nelder, 1989, p. 349): values of u close to zero will
correspond to the model that has constant variance on the scale of (20.21) and
values close to 2 correspond to errors with a constant coefficient of variation.
The relative merits of the approaches advocated by Nelder and by Ruppert et al.
are discussed by the authors at the end of Nelder's article.
Example 20.1
20.5 Some special assays 733
The data from Forsyth et al. (1993) considered in Example 12.5 are now reconsidered. The
Michaelis±Menten equation is fitted in six ways. Figure 20.5(a) shows (20.21) fitted
directly using non-linear methods and assuming a constant error variance on the scale
of the reaction velocities: Fig. 20.5 (b, c and d) shows, respectively, the data transformed
by (20.22), (20.23) and (20.24), together with the ordinary least squares regression line.
From Fig. 20.5(a) it appears that the data conform closely to the Michaelis±Menten
equation but this impression is not fully sustained in the other plots. In each of the other
plots there is a discrepant point which, in each case, corresponds to the observation with
the smallest substrate concentration. In Fig. 20.5(b) the discrepancy is perhaps less
obvious because the discrepant point is so influential that it forces the fitted line to depart
from the obvious line of the remaining points.
Figure 20.6 is the direct linear plot for these data: three of the 10 intersections between
the five lines result in intersections with negative values for both KM and Vmax. These
points all involve intersection with the line corresponding to the point which appeared
discrepant in Fig. 20.5. The one intersection with this line that is in the first quadrant is
clearly distant from the other intersections, which are relatively closely clustered.
The estimates of KM and Vmax from these methods are shown in Table 20.1.
The estimated values are clearly highly dependent on the method of fitting. Admittedly
this is what might be expected when only five points are analysed, but this number is not
atypical in applications. The negative values from the Lineweaver±Burk transformation
emphasize the sensitivity of this method to even slightly unusual observations. As