Another test for heterogeneity of host resistance in dilution assays

Abstract

Suppose that A is the average density per unit volume in a suspension of infective particles such as virus particles. To estimate A the usual method is to make up a series of inocula of various dilutions containing expected numbers of particles .. .Ai-1, Ai, Ai+4,... which are known multiples of A. Each of these is then tested by inoculation in a host such as an egg. We consider only the case where the dilution series is twofold (Ai = A2i, say) and the same number of eggs, N, is tested at each dilution. Then if we are sure that an egg is infected if and only if the inoculum contains at least one infective particle, the probability that the egg remains sterile is Pi = exp {-A2i}. If each particle is not certainly infective but has a probability p of infecting the egg, the probability of sterility of an egg chosen at random is exp {-Ap2i} provided that p does not vary from egg to egg. It is then only possible to estimate Ap from the results.If, however, p varies from egg to egg with a probability distribution f(p), the probability of an egg, chosen at random, being sterile is e-Ap2if(p) dp and when plotted against i, this gives a flatter curve. Thus any test of the goodnessof-fit of the original hypothesis provides us with a method of testing for variation in host resistance.Instead of fitting by maximum likelihood and then using x2 for this purpose, a more effective test has been proposed (Moran (1954a,b)). Provided that the series is sufficiently long to range from almost certainly sterile to almost certainly fertile levels we calculate a quantity T = Efm(N -fm). Here fm is the m number of fertile eggs at the dilution level 2m and N is the number of eggs tested at each level. The mean and variance of T have been calculated, thus enabling a rapid test to be made on the assumption that T is approximately normally distributed. This test has the advantage of being very much faster than the %2-test and also appears, in most cases, to be substantially more powerful (Armitage & Spicer (1956)), since it is so constructed that T is large for one particular kind of divergence from the theoretically expected numbers, namely, that in the direction of a flattening of the graph of Pi against i.

The Microbial Degradation of 2,4-Dichlorophenoxyacetic Acid in Soil

“…These tests by Moran and Stevens are worthwhile only if several dilutions have been tested and if positives occur at a concentration well below that at which all tubes would be expected to be positive. The tests are unsatisfactory for lO-fold dilutions (Moran 1958 andStevens 1958) and cannot be used if only three dilutions are tested (Taylor 1962). The test by Moran (1954a) was used as a test for homogeneity for twofold dilutions by Russek and Colwell (1983) in a computational program of the MPN.…”

Infectivity Titration

“…Stevens [22] proposes the statistic R, equivalent to Moran's [18] D + 1, defined as the number of dilutions between (and including) the first at which not all hosts are infected, and the last at which at least one is infected. As an example, in the following series of increasing dilutions with single observations (+ representing infection), the value of R is 5.…”

Testing and Adjusting for Departures from Nominal Dispersion in Generalized Linear Models

“…Much early work studied overdispersion in Poisson models; examples are papers by Fisher (1950) and David and Johnson (1952). Papers by Moran (1954aMoran ( , b, 1958, Armitage and Spicer (1956), Armitage (1959) and Neyman and Scott (1966) have considered overdispersion in bioassay models. Still others have considered overdispersion relative to Poisson regression and log-linear models; some papers are by Anderson (1988), Breslow (1984Breslow ( , 1990b, Dean and Lawless (1989) and Wilson (1989).…”

“…One approach consists of modelling overdispersion by including additional variance parameters in the model, as advocated by Efron (1986), Albert and Peeple (1989) and Smyth (1989). A second approach is to consider the parameter of interest to be random and to test for overdispersion, as proposed by Buhler et al (1965), Potthoff and Whittinghill (1966), Moran (1973), Lindsay (1980Lindsay ( , 1982Lindsay ( , 1983, Cox (1983) and Zelterman and Chen (1988). Other approaches are given by Moore (1986) and Barnwal and Paul (1988) in related problems.…”