The bingo model of survivorship. III. Genetic principles

Angular homeostasis. IX. Polygonal orbits with a moving target: implications for anomalous numbers of digits in congenital heart disease

This paper explores properties of discrete processes in which a pursuer seeks a target that is moving at constant velocity r that is a fixed proportion of the speed of the pursuer. The pursuer is subjected to proportional angular homeostasis, so chosen that the number of steps per circuit is small. The orbits relative to the target may assume any of four forms: polygons that reverse their sense an infinite number of times; or polygons that after a finite number of reversals ultimately come to have an integer numbers of sides; or have a rational numbers of sides; or have an irrational number of sides that densely fill an annulus. None of the polygons is regular. In the parameter space, the boundary line between the first of these sets and the other three has a somewhat bizarre pattern and may possibly be fractal, but no proof is forthcoming. Unlike the pattern with a stationary target, there may be a set or catchment of diverse values of the speed ratio, r, and the correction coefficient, b that all result in figures of some specified number, n, of sides (although with vertices in differing locations). Catchments have been found for only those polygons that have the winding number of 1. The implications are discussed that this property has for the genetic coding of biological traits that are countable. Some attention is also paid to the relevance of polygons with few sides to ontogenic growth when the correction coefficient is cyclically arc- or time-dependent.

Threshold model in the genetics of age-dependent disease in twins: I. General principles as applied to Alzheimer disease

In the context of the etiology and pathogenesis of Alzheimer disease (AD), we discuss assumptions under which categorical data on the phenotypes of twin pairs may be used to estimate standardized characteristics (correlation and the critical threshold) for an age-dependent multinomial process. Important topics include Erlangian, gamma, and Poisson processes, tetrachoric and trichoric functions, and degrees of freedom and how they relate to estimation from both an abstract and a practical standpoint. Under plausible assumptions about the age-dependence of a heritable trait, it is possible to generate sufficient degrees of freedom to test the genetic model and to explore age-dependence and the impact of environmental factors that may influence it systematically. Though general in scope, the model is focused on data from twin pairs. The statistical strategy is briefly outlined, but its properties are not examined in detail.

Angular homeostasis. VIII. Pursuit of a slowly moving target in a plane: relevance to lateralization in cardiovascular ontogeny

We explore the pursuit in a plane of a target moving at constant slow speed in a straight line. Two models of the pursuit are given. In the continuous case, the pursuer is moving at constant speed and is subject to proportionate angular homeostasis with correction constant b. In the discrete version movement occurs at a constant speed in a sequence of straight line segments of constant length (called the step size, s) the end of the segments being called the vertices. The pattern considered is not the absolute position of the pursuer, but its distance and orientation relative to the target. Both the transients and the asymptotic orbit are addressed. A key quantity is r, the speed of the target expressed as a fraction of that of the pursuer. If the speed of the pursuer is defined as unity, r is also the ratio of the speeds. There exists a critical speed fraction, R(b,s), a function of b and s, that defines what the term slow designates. R(b,s), which has to be found numerically, has the following property. For r less than R(b,s), the asymptotic path is a simple closed curve. In the discrete case the vertices converge to a simple closed curve. The larger r, the more the path (or in the discrete analogue its set of vertices) departs from a circle, and the more eccentric the target is with respect to it. Interest centers on two issues. First we address the transient patterns of the path, notably whether or not the sense of any particular path (clockwise or counterclockwise) is the same throughout, or changes at some stage.(ABSTRACT TRUNCATED AT 250 WORDS)

Estimating genetic parameters of survival distributions: a multifactorial model

For a number of genetically influenced illnesses, age of onset is correlated between relatives. In some cases, age of onset also serves as an index of an individual's inherited liability to an illness. Typically, survival models for age of onset distributions do not allow for these effects. We have used the gamma distribution as the basis of a model which specifies genetic variability in age of onset. Specifically, we allow the hazard frequency parameter of the gamma distribution to be a monotonic function of an individual's inherited liability to an illness. Simulations under various gamma models indicate that age of onset correlations may be strikingly different from correlations in liability. Maximum likelihood estimation procedures were used to obtain parameter estimates from simulated data sets and to determine whether it is possible to discriminate between gamma processes of different order. The results of 1,000 pairs of twins (500 monozygotic and 500 dizygotic) gave parameter estimates that were fairly accurate but that decreased in precision as the order of the gamma process increased. We discuss the effect this may have on inferences made about age of onset.

The bingo model. IV. The statistics of survivorship in the bingo-gamma model

The survivorship (time to death or failure) of a bingo-gamma (BG) model is defined as the minimum among the waiting times for completion among k independent gamma processes. The ith process is of order ni, with a mean rate for the occurrence of hits of ai. In this paper we address the case where, for all competing processes, the order and the rate at which hits occur are the same but both they and k are unknown. We denote by k the multiplicity, by n the order or the number of hits to failure, and by a the transition parameter. The joint maximum likelihood estimator (MLE) of the three parameters of this BG process is developed. An algorithm for calculating it has been devised and a computer program in BASIC has been written. The properties of the MLE have been explored systematically, mainly by Monte Carlo simulation. The distributions, means, variances, covariances, and correlation coefficients of the three parameters are explored for samples of size 25 and samples of size 100. Also, the simple average of the observed survival times (which gives a method of moments estimator of the mean survival) is compared with the MLE of the mean survival; the two estimators seem to be unbiased and about equally efficient.

The bingo model of survivorship. V. The problems of conformation to the empirical evidence

We discuss the statistical and biological problems of adapting the theoretical bingo model to the analysis of empirical data. A distinction is made between an idealized pathogenetic model, which aims to represent the disease in as much authentic detail as the present state of knowledge allows and in components that have literal interpretation, and an empirical model, which deals with those effects of the pathogenetic model that one may hope to observe clinically. We review a variety of empirical models distinguishable by the amount of data available on intermediate degrees of damage short of total destruction. The relationship of damage to time is explored, and we consider the criteria and usefulness of linearization of this relationship where the diachronic ("longitudinal") data are few and extend over a comparatively short time. Every time a patient is examined, the degree of cumulative damage is assessed in each of the body systems of interest. Thus the examination will furnish a set of measurements, which is obtained on each of several examinations, taken over a period that for preference is long relative to the survival of the system. Specific disorders discussed include dentition and enlargement of the aorta with age in the Marfan syndrome.