Statistical Methods in Medical Research 4ed

attempt to model the processes that lead to this downturn; see, for example,
Margolin et al. (1981) and Breslow (1984a). A simpler, but perhaps less satisfying
approach is to identify the dose at which the downturn occurs and to test for a
monotonic dose response across all lower doses; see, for example, Simpson and
Margolin (1986).
It is also quite common for the conditions relating to the assumptions of a
Poisson distribution not to be met. This usually seems to arise because the
variation between plates within a replicate exceeds what you would expect if
the counts on the plates all came from the same Poisson distribution. A test of
the hypothesis that all counts in the ith dose group come from the same Poisson
distribution can be made by referring
P ri
jˆ1 …Yij Y i† 2
Y i
20.5 Some special assays 739
to a x2 distribution with ri 1 degrees of freedom. A test across all dose groups
can be made by adding these quantities from i ˆ 1toDand referring the sum to
a x2 distribution with P ri D degrees of freedom.
A plausible mechanism by which the assumptions for a Poisson distribution
are violated is for the number of microbes put on each plate within a replicate to
vary. Suppose that the mutation rate at the ith dose is li and the number of
microbes placed on the jth plate at this dose is Nij. If experimental technique
is sufficiently rigorous, then it may be possible to claim that the count Nij is
constant from plate to plate. If the environments of the plates are sufficiently
similar for the same mutation rate to apply to all plates in the ith dose
group, then it is likely that Yij is Poisson, with mean liNij. However, it may be
more realistic to assume that the Nij vary about their target value, and variation
in environments for the plates, perhaps small variations in incubation
temperatures, leads to mutation rates that also vary slightly about their
expected values. Conditional on these values, the counts from a plate will
still be Poisson, but unconditionally the counts will exhibit extra-Poisson variation.
In this area of application, extra-Poisson variation is often encountered. It is
then quite common to assume that the counts follow a negative binomial distribution.
If the mean of this distribution is m, then the variance is m ‡ am2 , for
some non-negative constant a …a ˆ 0 corresponds to Poisson variation). A crude
justification for this, albeit based in part on mathematical tractability, is that the
negative binomial distribution would be obtained if the liNij varied about their
expected values according to a gamma distribution.
If extra-Poisson variation is present, the denominator of the test statistic
(20.27) will tend to be too small and the test will be too sensitive. An amended
version is obtained by changing the denominator to ‰
p Y…1 ‡ ^aY†S 2 xŠ, with ^a an