On a stochastic integral of a branching process

This paper is concerned with the properties of a stochastic integral which arises in the study of a modified Markov branching process. Explicit expressions are found for the mean and the limit distribution of the integral.     

Testing for the Equality of the Variance-Covariance Matrices of Two Jointly Normal Vector Variables

Assume a joint 2P-variate normal distribution for the p-component vectors x and y with unknown mean and unknown variance-covariance matrices Σ_xx and Σ_yy respectively, with Cov (x , y )=Σ_xy. No assumptions are made about the nature of Σ_xy. The likelihood ratio method is investigated to test the hypothesis that Σ_xx = Σ_yy. A method of numerical solution to the likelihood equations in the restricted parameter case is given when p=2, and approximate solutions are suggested for p>2.     

The form of the talus in some higher primates: a multivariate study

Sixteen measurements of the talus have been taken on 334 tali of a total of eleven primate groups and several additional single individual specimens. Multivariate morphometric (canonical and generalized distance) analyses of these data in extant higher primates are presented and used to define the relative morphological positions of fossils of the genera Proconsul and Limnopithecus, of individual specimens from Kromdraai, Olduvai and Kiik-Koba (Homo neanderthalensis), and a group of specimens of Bronze Age man from Jericho. Following preliminary studies the ultimate analysis suggests that the various extant arboreal primates examined fall within an envelope that is defined by Macaca together with various other Old World monkeys and extending in different directions to the extreme genera (a) Pongo, (b) Hylobates and (c) Ateles. This separation is thus one which defines generally quadrupedal monkeys and separates the various extreme arboreal locomotor modes of (a) acrobatic climbing and hanging, (b) richochetal brachiation and (c) prehensile-tailed arm-swinging and hanging, respectively. Man and the African apes are well separated both from each other and from this spectrum of arboreally adapted genera. Bronze Age man from Jericho and Neandertal man from Kiik-Koba lie relatively close to the position for modern man although significantly separated from him. Limnopithecus, Proconsul, and the specimens from Kromdraai and Olduvai all lie within the envelope of arboreal species and specifically rather close to, although significantly different from, the orang-utan; they differ markedly from both man and the African apes. The possibility exists that the resemblances of Proconsul and Limnopithecus relate to arboreal habitus in these species. The findings for the specimens from Kromdraai and Olduvai suggest either that the morphological resemblances to arboreal forms relate to a previous arboreal history for these species, or that bipedality is much less advanced or uniquely different from that displayed by Homo. It is not inconceivable that both conditions might apply.     

Effects of dominance on the probability of fixation of a mutant allele

We consider whether the fixation probability of an allele in a two-allele diploid system is always a monotonic function of the selective advantage of the allele. We show that while this conjecture is correct for intermediate dominance, it is not correct in general for either overdominant or underdominant alleles, and that for some parameter ranges the fixation probability can initially decrease and then increase as a function of the amount of selection. We have partial results that characterize the ranges of parameters for which this happens.      

On the probability of loss of new mutations in the presence of linkage disequilibrium

A new selectively neutral mutation occurs in a multilocus genetic background that has achieved a stable equilibrium at which there is a linkage disequilibrium. Perturbation techniques are applied to an extension of the branching process formulation of Fisher in order to address the question of extinction probabilities. We show that under appropriate conditions the probability of extinction of the new mutant is increased by the existence of linkage disequilibrium in the genetic background.Research supported in part by NIH grant GM 28016     

Tests for multivariate recurrent events in the presence of a terminal event

In studies involving diseases associated with high rates of mortality, trials are frequently conducted to evaluate the effects of therapeutic interventions on recurrent event processes terminated by death. In this setting, cumulative mean functions form a natural basis for inference for questions of a health economic nature, and Ghosh and Lin (2000) recently proposed a relevant class of test statistics. Trials of patients with cancer metastatic to bone, however, involve multiple types of skeletal complications, each of which may be repeatedly experienced by patients over their lifetime. Traditionally the distinction between the various types of events is ignored and univariate analyses are conducted based on a composite recurrent event. However, when the events have different impacts on patients' quality of life, or when they incur different costs, it can be important to gain insight into the relative frequency of the specific types of events and treatment effects thereon. This may be achieved by conducting separate marginal analyses with each analysis focusing on one type of recurrent event. Global inferences regarding treatment benefit can then be achieved by carrying out multiplicity adjusted marginal tests, more formal multiple testing procedures, or by constructing global test statistics. We describe methods for testing for differences in mean functions between treatment groups which accommodate the fact that each particular event process is ultimately terminated by death. The methods are illustrated by application to a motivating study designed to examine the effect of bisphosphonate therapy on the incidence of skeletal complications among patients with breast cancer metastatic to bone. We find that there is a consistent trend towards a reduction in the cumulative mean for all four types of skeletal complications with bisphosphonate therapy; there is a significant reduction in the need for radiation therapy for the treatment of bone. The global test suggests that bisphosphonate therapy significantly reduces the overall number of skeletal complications.