Statistical Methods in Medical Research 4ed

736 Laboratory assays
P m
iˆ1
ri log‰p…u, di†Š ‡ …ni ri† log‰1 p…u, di†Š
as a function of u. In order to do this, it is necessary to specify how the
probability that the agent is present varies with the dilution.
Most limiting dilution assays use a formula for p…u, di† based on the Poisson
distribution (see §3.7); Ridout (1995) discusses an alternative. Suppose the application
is to measure the concentration of a certain type of bacteria in a watersupply.
Suppose also that the experimenter has the means to detect the presence
of one or more bacteria in a sample. It is natural to suppose that the number of
bacteria per unit volume of the water-supply is Poisson, with mean u. If the
original sample is diluted with an equal volume of uncontaminated water, then
the number of bacteria in the diluted sample will have a Poisson distribution with
mean 1
2
u. Similarly a more general dilution would give a Poisson distribution
with mean ud. It follows that the probability that a sample at this dilution will
contain no bacteria is exp… ud ) and consequently
p…u, d† ˆ1 exp… ud†: …20:26†
In immunological applications this is known as the single-hit Poisson model,
because it corresponds to assuming that the number of T cells in a sample is
Poisson and a positive result will be obtained in the presence of only one T cell.
Lefkovits and Waldmann (1999) discuss other forms for p…u, di†, such as multihit
models and models representing different types of interaction between cell subtypes,
such as B cells and T cells.
The analysis of a limiting dilution assay by maximum likelihood is generally
satisfactory. It can be implemented as a generalized linear model, using a
complementary log-log link and binomial errors (see Healy, 1988, p. 93). Problems
arise if almost all results are positive or almost all are negative, but it is clear
that such data sets are intrinsically unsatisfactory and no method of analysis
should be expected to redeem them.
Several alternative methods of analysis have been suggested. Lefkovits and
Waldmann (1999) proposed ordinary linear regression through the origin with
loge‰…ni ri†=niŠ as the response variable and dilution di as the independent
variable. This is plausible because loge‰…ni ri†=niŠ estimates the log of the
proportion of assays that contain no agent which, from (20.26), is expected
to be udi. Lefkovits and Waldmann indicate that the form of the plot of
ri†=niŠ against di not only allows u to be estimated but allows system-
log e‰…ni
atic deviations from (20.26) to be discerned. Nevertheless, the method has serious
drawbacks; it takes no account of the different variances of each point, and
dilutions with ri ˆ ni are illegitimately excluded from the analysis.
Taswell (1981) compares several methods of estimation and advocates minimum-x
2 estimation. This method appears to do well but this is largely because it