Statistical Methods in Medical Research 4ed

726 Laboratory assays
This relationship can be fitted by standard multiple regression methods (§11.6),
and the potency r (ˆ b T=b S) estimated by
R ˆ bT=bS: …20:17†
The residual variance of y is estimated by the usual residual MSq, s 2 , and the
variances and covariance of bS and bT are obtained from (11.43) and (11.44):
var…bS† ˆc11s 2 , var…bT† ˆc22s 2 , cov…bS, bT† ˆc12s 2 : …20:18†
Approximate confidence limits for r are obtained by the following formula
derived from (5.17):
var…R† ˆ s2
b2 …c22 2Rc12 ‡ R
S
2 c11†: …20:19†
A more exact solution, using Fieller's theorem, is available, as in (5.15), but is not
normally required with assays of this type.
For numerical examples of the calculations, see Finney (1978, Chapter 7).
The adequacy of the model in a slope-ratio assay can be tested by a rather
elegant analysis of variance procedure. Suppose the design is of a 1 ‡ kS ‡ kT
type; that is, there is one group of `blanks' and there are kS dose groups of S and
kT dose groups of T. Suppose also that there is replication at some or all of the
dose levels, giving n observations in all. A standard analysis (as described on
p. 360) leads to the following subdivision of DF:
Between doses kS ‡ kT
Regression 2
Deviations from model kS ‡ kT 2
Within doses n kS kT 1
n 1
The SSq for deviations from the model can be subdivided into the following parts:
Blanks 1
Intersection 1
Non-linearity for non-zero doses kS ‡ kT 4
kS ‡ kT 2
The SSq for `blanks' indicates whether the `blanks' observations are
sufficiently consistent with the remainder. It can be obtained by refitting the
multiple regression with an extra dummy variable (1 for `blanks', 0 otherwise),
and noting the reduction in the deviations SSq. A significant variance-ratio test
for `blanks' might indicate non-linearity for very low doses; if the remaining tests
were satisfactory, the assay could still be analysed adequately by omitting the
`blanks'.