Statistical Methods in Medical Research 4ed

728 Laboratory assays
Proportion positive, P
1
0
Distribution of tolerances
Dose, X
Fig. 20.4 Quantal-response curve with the corresponding tolerance distribution.
In an assay of T and S, the response curves, plotted against x ˆ log X,willbe
parallel in the sense that they differ by a constant horizontal distance, log r, and
a full analysis requires that the data be fitted by two parallel curves of this sort. A
natural approach is to linearize the response curves by applying to the response
one of the standard transformations for binary data (§§10.8 and 14.1). One can
then suppose that the expectations of the transformed values are linearly related
to the log dose, x, by the equations
)
Y ˆ aS ‡ bx for S
and
: …20:20†
Y ˆ aT ‡ bx for T
The slope parameter b is inversely proportional to the standard deviation of
the log-tolerance distribution. For the logistic transformation (14.5) with
Y ˆ ln‰P=…1 P†Š, for instance, the proportionality factor is p= 3
p ˆ 1 814.
Although (20.20) has essentially the same form as (20.3) and (20.4), the
methods described in §20.1 for parallel-line assays cannot be used, since the
random variation about the lines is not normally distributed with constant
variance. The model may be fitted iteratively by the maximum likelihood procedure
for logistic regression described in §14.2, using a dummy variable to
distinguish between S and T, as in (11.55). Writing aT ˆ aS ‡ d, the log potency
ratio is given by log r ˆ d=b. A computer program for logistic regression may be
used to obtain the estimates aS, d and b of aS, d and b, respectively, and hence log
r is estimated by M ˆ d=b. From (5.17), an approximate formula for var(M) is
var…M† ' Vd
b 2
2dCdb
b 3
‡ d2 Vb
b 4 ;