Biomedical subjects
Publications and source records attributed to C Cannings.
This theoretical approach to multiple markers considers the problem of a trait locus flanked by an arbitrary number of linked, ordered markers. Using the criterion of the Expected LOD score (ELOD) for mating, an algorithm is developed to specify the subset of markers required to provide the full linkage information available from a given parental genotype.
Iterated conflicts allow the possibility of co-operative-like behaviour in games such as the Prisoner's Dilemma. The present paper is an attempt to initiate the study of iterated conflicts when, (a) the number of iterations is fixed and finite and (b) the underlying payoff matrix is general, e.g. a mixed Evolutionary Stable Strategy (ESS) could occur in the non-iterated coflict. These assumptions are in contrast to the Iterated Prisoner's Dilemma. We consider a somewhat special case which none the less produces results of an interesting nature. For those cases where there is no internal ESS in the one trial case the two-trial case is easily resolved. When the former has an internal ESS then the two-trial case yields two ESSs whose supports are a partition of the space of strategies.
A matrix may have several evolutionarily stable strategies (ESS's). It is thus possible for different populations of a species to adopt a different ESS even when the pay-offs for the populations are the same. The occurrence of different strategies does not imply different circumstances. However, there are constraints upon the collection of supports of the ESS's (i.e. pattern) that any matrix can have. The best-known of these is that the support of one ESS cannot be contained in that of another and this gives bounds on the number of different patterns possible for n x n matrices. Other general constraints are presented here. The enumeration of the patterns for 3 x 3 and 4 x 4 matrices is completed and considerable progress made on 5 x 5 matrices where the number of (permutationally distinct, maximal) patterns exceeds 16.
For symmetric matrix conflicts with aij = +/- 1 and aii = 0, an ESS corresponds to a clique in an associated graph. This result is proved and exploited to yield results on the attainable patterns for this class of conflicts and bounds for the number of ESS's which may coexist. Randomly generated matrices in this class are considered, and some results on the size of the support of a typical ESS given.
The paper presents the derivation of conditions for the initial increase of a new allele in a fairly general model of a one locus genetic system. Incorporated within the model are frequency dependent selection coefficients which allow the study of viability differences, fertility differences, segregation distortion and certain forms of non-random mating. Since in many models it is necessary to go beyond linear terms, quadratic, cubic and quartic terms are also derived explicitly and details given of their extension to higher degrees.
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A large Mormon pedigree of a proband with hemochromatosis was studied, using transferrin saturation as the quantitative phenotypic trait. The analysis indicated that the inheritance of hemochromatosis was recessive, with partial expression in some heterozygotes. The lod score of 6.88 (theta = .0) was strongly indicative of linkage between the hemochromatosis locus and the human major histocompatibility (HLA) loci.
Previous approaches to the ascertainment problem have been in terms of a registry-type situation in which the sampled structure is a random variable. However, with the exception of Bailey (1951), previous authors have conditioned on this structure. In extending the theory to pedigrees, a registry situation may also be considered, providing neither the structure nor any part of it (e.g., ages of individuals) are conditioned upon, but it may often be neither practicable nor relevant to do so. In many studies the sampling scheme may fix the number of pedigrees, analyzed, and classical "ascertainment probabilities" are no longer applicable. In an infinite population, in which the ascertainment of and phenotypic observations on, affect neither the distribution of other ascertainments or phenotypic distributions on other pedigrees and chance multiple ascertainments cannot occur, ascertainment corrections are relatively straightforward. However, effective population sizes with regard to particular rare traits may often be small, and the problem of ascertainment corrections in this case has not been fully analyzed, although some preliminary results are presented.
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A method for inferring the ancestral genotypes for the founders of a population is developed. This method uses the algorithms for the computation of probabilities on pedigrees of arbitrary complexity, developed by Cannings et al. (1978) and implemented by Thompson (1977b). When characteristics are simply determined by underlying genotypes the inference problem is simplified, and larger and more complex pedigrees may therefore be analysed. The problem of estimating the allele frequencies to be used in computing prior genotype probabilities for those founders on whom a likelihood function is not required is discussed. The same method allows us to compute extinction probabilities for any combination of original founder genes; these probabilities are interesting parameters of pedigree structure, which, since they relate to the actual genes present in a population, help to provide a clearer understanding of observed distributions of autosomal traits.
One aim in the analysis of pedigree data may be to infer the mode of inheritance of a characteristic. If only "interesting" pedigrees are analysed, the ascertainment bias may lead to some modes of inheritance being unintentionally preferred. Also, it is clearly most efficient in attempting to make such inferences, if a decision on whether to continue sampling a pedigree is made conditional on the types of individuals who have been observed; an a priori decision to examine 500 members of a pedigree may lead to much wasted effort, since the pedigree may prove to be largely uninformative. The present paper shows that provided all observed families are included in the analysis, even those which appeared "uninteresting" or "sporadic" and were not sampled further, and provided a decision to continue sampling is made conditional on types observed up to that point, the correct likelihood for the mode of inheritance may be easily computed. This opens the way for a more detailed study of the wider problem of optimal samplings rules on pedigrees.
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