Statistical Methods in Medical Research 4ed

where Vd ˆ var(d ), Cdb ˆ cov(d, b) andVbˆvar(b). These variances and the
covariance should be obtainable from the computer output. Approximate
100…1 a)% confidence limits for log r are then
p
M za var…M† ,
where za is the usual standardized normal deviate. Limits for r are obtained by
taking antilogs.
By analogy with (20.8), Fieller's theorem can be used to give more reliable
approximations. From (5.13) the limits for log r then become
M gCdb
Vb
za
b Vd 2MCdb ‡ M 2 Vb g Vd
,
1 g
where g ˆ za 2Vb=b2 .
Approximate x2 tests for (i) parallelism and (ii) linearity may be obtained by
the analysis of deviance, by taking: (i) the difference between the deviances of the
fitted model and a model with separate slopes for S and T; and (ii) the deviance
of the latter model.
An alternative approach to the analysis of quantal-response data, probit
analysis, is widely used. This method, which is fully described by Finney (1971),
provides the maximum likelihood solution for the probit or normal equivalent
deviate (NED) transformation (§14.1) rather than for the logit. Here P is the
distribution function for a standard normal deviate Y (a constant 5 being added
to the NED to give the probit). This transformation thus implies that the tolerance
distribution for the log dose is normal, and the slope parameter b in (20.20) is the
reciprocal of the standard deviation. There is no evidence to suggest that either
model is more consistent with real data than the other, and curves fitted by the two
methods are invariably very similar.
Before the advent of computer programs for the efficient analysis of quantalresponse
data, a number of simple methods were in common use. These mainly
aimed to estimate the location of the tolerance distribution as summarized by its
median. This is the dose at which P ˆ 0 5, and is called the median effective dose
(ED50). In the comparison of two response curves, the ED50 for T and S are in the
ratio 1 : r. If they are estimated by X0T and X0S, the ratio X0S=X0T will provide an
estimate of r. The ED50 is sometimes given slightly different names according to
the type of response: e.g. LD50 or median lethal dose if the response is lethal. A full
account of these short-cut methods is given by Finney (1978, §§18.6±18.8).
The up-and-down (or staircase) method
20.4 Quantal-response assays 729
In experiments or assays in which the main purpose is to estimate the ED50, the
choice of dose levels is important. Doses yielding very low or very high values of
C 2 db
Vb
1
2