D.A. Sprott’s research while affiliated with University of Waterloo and other places

SUMMARY The concepts of sufficiency and ancillarity are extended to marginal and conditional sufficiency in the presence of nuisance parameters. It is required that under the assumption of the absence of knowledge of the nuisance parameter β no information shall be lost in basing inferences about Θ on a statistic that is marginally or conditionally sufficient for Θ. The approach is based on the likelihood function and relates the two previously studied apparently dissimilar structures of (a) the factorization of the likelihood and (b) group sufficiency. Some examples are given and the final example illustrates a possible loss of information entailed by the use of statistics that are not marginally or conditionally sufficient.

Methods of statistical analysis are presented for one or more dilution series experiments where the quantity of interest is the number of virus particles required to infect a cell. These methods are illustrated on several data sets drawn from the literature. Data from seven series, which have been used to support a two-particle model in the literature, are here shown to reject such a model decisively, whereas fifteen other experiments are found to be in excellent agreement with a one-particle model.

Ailing [1971] considers a dilution series model arising in virology and discusses the estimation of a parameter h called the hit number. His analysis is based on asymptotic properties of various estimators. The present paper develops other methods for making inferences about h. Tests for the homogeneity of data from several series and for the goodness of fit of the model are also considered. The analysis is based upon the use of sufficient and ancillary statistics to factor the joint likelihood into parts appropriate for the various types of problem considered, and is illustrative of a general approach to problems of inference.

Some logical aspects of tests of significance are illustrated using the example of equality of means of Poisson-distributed observations. Specifically, in the conventional test for the significance of a difference between two Poisson-distributed observations, the significance level is computed from the conditional distribution given their observed total. However, an unconditional test has greater power and so is sometimes advocated in place of the conditional test. The present paper argues that power considerations are not relevant in choosing between conditional and unconditional tests, and that the conditional test is the appropriate one. The example is extended to include tests of equality of means of two Poisson samples (pointing out an error in a formula that is sometimes used) and also to include tests concerning the ratio of Poisson means. There is a general discussion of significance tests with reference to the above examples.

The probability function, P(T)=Θe−Θt/ns(1−e−Θt/ns)n−[1−(1−e−Θt/ns)n]s−1 where θ is associated with the number of random “hits” per unit time, s is taken as 2000 and n as 2x, can simulate mortality curves for various species. This function is derived by assuming that critical gene sets exist which comprise the corresponding loci on each of the n strands where each strand may correspond to a double helix. The number of genes on each strand is assumed to be, on average, 2000.

Likelihood methods of dealing with some multiparameter problems are introduced and exemplified. Specifically, methods of eliminating nuisance parameters from the likelihood function so that inferences can be made about the parameters of interest are considered. In this regard integrated likelihoods, maximum relative likelihoods, conditional likelihoods, marginal likelihoods and second‐order likelihoods are introduced and their uses illustrated in examples. Marginal and conditional likelihoods are dependent upon factorings of the likelihood function. They are applied to the linear functional relationship and to related models and are found to give intuitively appealing results. These methods indicate that in many situations commonly encountered objective methods of eliminating unwanted parameters from the likelihood function can be adopted. This gives an alternative method of interpreting multiparameter likelihoods to that offered by the Bayesian approach.

The exact likelihood functions are examined for several examples of data chosen from the literature. These are compared with the likelihoods arising from the large sample approximations and with point estimates that were actually used in the literature. It is concluded that large sample approximations (application of standard maximum likelihood theory) can be misleading for inferences and should be checked against the actual likelihood functions. Similarly point estimates can be misleading or uninformative and their properties such as bias, variance, etc. are relatively unimportant. Because of the availability of high speed computers, exact methods and asymptotic comparisons are now feasible and this should be reflected where possible in the theory and application of statistical inference.