The bingo model of survivorship: 1. probabilistic aspects

Human disease mortality kinetics are explored through a chain model embodying principles of extreme value theory and competing risks

The distributions for human disease-specific mortality exhibit two striking characteristics: survivorship curves that intersect near the longevity limit; and, the clustering of best-fitting Weibull shape parameter values into groups centered on integers. Correspondingly, we have hypothesized that the distribution intersections result from either competitive processes or population partitioning and the integral clustering in the shape parameter results from the occurrence of a small number of rare, rate-limiting events in disease progression. In this report we initiate a theoretical examination of these questions by exploring serial chain model dynamics and parameteric competing risks theory. The links in our chain models are composed of more than one bond, where the number of bonds in a link are denoted the link size and are the number of events necessary to break the link and, hence, the chain. We explored chains with all links of the same size or with segments of the chain composed of different size links (competition). Simulations showed that chain breakage dynamics depended on the weakest-link principle and followed kinetics of extreme-values which were very similar to human mortality kinetics. In particular, failure distributions for simple chains were Weibull-type extreme-value distributions with shape parameter values that were identifiable with the integral link size in the limit of infinite chain length. Furthermore, for chains composed of several segments of differing link size, the survival distributions for the various segments converged at a point in the S(t) tails indistinguishable from human data. This was also predicted by parameteric competing risks theory using Weibull underlying distributions. In both the competitive chain simulations and the parametric competing risks theory, however, the shape values for the intersecting distributions deviated from the integer values typical of human data. We conclude that rare events can be the source of integral shapes in human mortality, that convergence is a salient feature of multiple endpoints, but that pure competition may not be the best explanation for the exact type of convergence observable in human mortality. Finally, while the chain models were not motivated by any specific biological structures, interesting biological correlates to them may be useful in gerontological research.

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.

A geneticist's approach to psychiatric disease

The pattern of the genetics of psychiatric disease is various, in detail and in kind; one cannot always expect to find simple, trustworthy explanations of what are complex relationships. There will be some diseases, mostly of the disruptive type, that will prove to be Mendelian defects; but it would be idle to expect that of most. Many disorders will be elucidated only when the nub of the problem is better defined. Mindless application of standard genetic techniques devised for quite different purposes is no substitute for articulate speculation built on sound fact and cogent testing. In this respect genetic evidence is no different from that of physiology, pathology and pharmacology, which are perhaps less unfamiliar to readers than genetics. Finally, no answer has been found unless it deals in those terms in which the question first arose. If the topic of the genetic analysis is some psychosis, the final predicate must be a statement about the psychosis, not about some arbitrary abstraction from it.

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.

The bingo model of survivorship. III. Genetic principles

The broad relationships are explored between the genetic and the phenotypic structures of the bingo-gamma model (ie, the shortest waiting time among competing, independent, multiple-hit systems). Finite algorithms are derived to compute in closed form the joint and marginal distributions; the distribution, density, and hazard functions of time to failure; the respective total probabilities of dying from failure of each competing system; and the raw and central moments. The algorithm is the computer counterpart of a generating function. The number of competing systems and their individual orders and transition parameters may be chosen at will. Classical Galton-Fisher theory does not apply: neither means nor variances are additive nor are their effects homogeneous; rather, those systems with shorter mean survival more or less mask the impact of those with longer means. Thus even huge differences among means for alleles of any one component may be almost totally concealed phenotypically; even the maximal genetic covariation may in practice remain totally unrecognized and the heritability estimated close to zero. The proportional specific mortality is a less capricious index and is naturally additive, but, though a monotonic function of the underlying parameters, it is neither linear nor homogeneous.

Familial aggregation in Alzheimer dementia--I. A model for the age-dependent expression of an autosomal dominant gene

An autosomal dominant genetic etiology has been proposed for Alzheimer Dementia (AD), but many cases appear to be sporadic. Evaluation of the possible genetic transmission of AD from its familial aggregation requires consideration of (1) the proportion of index cases with genetic disease, and (2) the consequences of typically very late onset. To investigate these factors, a provisional biomathematical genetic model was developed from the empirical age-specific incidence of AD in relatives. Based upon the premise of an autosomal dominant AD gene in proband families, the modeling technique provides estimates of the proportion of genetic index cases (as opposed to phenocopies) and the parameters of age-dependent gene expression. With appropriate parameters the model accurately reflects the age-specific familial risk of AD, suggesting the appropriateness of its underlying assumptions. The estimated proportions of genetic index cases suggest that heritable disease constitutes a majority of AD. In cases ascertained by the presence of aphasia or apraxia the estimated proportion of genetic cases is 100%. The greatest likelihood of gene expression is in the ninth decade, however, suggesting that most genetically predisposed relatives will die from other causes before developing AD.

The bingo model of survivorship. II: statistical aspects of the bingo model of multiplicity 1 with application to hereditary polyposis of the colon

Some Mendelian disorders (Huntington chorea, hereditary polyposis coli) are not manifest at birth but show a distribution in the age of onset. Patients at risk fall into three groups. In type I, they are affected when first examined. In type II, they are not affected at one visit, but are at a later visit. Those of type III (who comprise an indistinguishable mixture of those who have, and those who have not, inherited the gene) are never found to be affected. This paper posits a model that the age of onset is logistic. (It is a degenerate bingo model in which competing causes of death may be ignored.) The statistical properties of maximum likelihood estimation (MLE) are explored by Monte Carlo simulation of this logistic function with known arbitrary parameters. Two schemes are used: point-prevalence (or synchronic) data of types I and III, and piecewise longitudinal (diachronic) data; this allows all three types to be included. Samples of various sizes between 25 and 100 are used. While estimates of the parameters are positively biased (especially with small samples), the estimate of the mean appears to be consistent, almost unbiased, and fairly precise, though somewhat larger than the estimates from the lower bound (a fact that calls for some caution in interpreting actual data). The MLE was applied to 109 patients with the Gardner syndrome (GS); measures of variability found by applying MLE to four random subsets of 25 each were compared against the asymptotic estimates. The analysis was also applied to 36 persons with familial polyposis coli (FPC). The mean age of onset in GS and FPC was similar, and since they are rather earlier than is currently believed, it is recommended that regular supervision be started at not later than 10 years of age.

Paradoxical fixation of deleterious alleles in two-locus systems with epistasis

The dynamics of four deterministic models of interaction between two Mendelian loci are explored numerically. At one locus there are two hypostatic alleles: h, the wild type and H, a deleterious mutant that in either heterozygous or homozygous state (depending on the specifics of the model) produces an abnormal phenotype. At another arbitrarily linked epistatic locus there are two alleles: e, the wild type with no effect on the expression of the H locus, and E, an epistatic mutant that in heterozygous or homozygous state (again, depending on the specifics of the model) blocks the expression of H. The parameters are the initial gamete frequencies, the recombination fraction, the genotypic viabilities, and the forward and back mutation rates at each locus. The He haplotype is eventually eliminated (unless back mutation occurs) from the population. If mutation is ignored, the evolutionary outcome is determined by the initial gamete frequencies and is either (1) an edge equilibrium comprising one pair of haplotypes only (he and He, or he and hE, or hE and HE, or He and HE, or, if there is no recombination, he and HE or hE and He); or (2) a corner equilibrium consisting of a single gametic type. Given that the forward mutation rates at both loci are greater than the back mutation rates, then the outcome is always the corner equilibrium in which HE is universal (apart from transient perturbations by mutation). In the process of fixation, the phenotypic impact of the deleterious allele H becomes neutralized by the epistatic allele E. The rate at which the initially harmful gene replaces the wild type gene depends on the recombination fraction, the genotypic viabilities, and the mutation rates.

The dynamics of quantifiable homeostasis. I. The individual

In this field of enquiry, "homeostasis" is understood in the sense of Bernard, Cannon, and Wiener, that is, with the system of continuous adjustments, in a trait not inherently stable, to meet the challenges of the environment. It is shown that the clinical fitness of a trait may depend on factors other than the mean or the variance. In particular the broad pattern of variation of the phenotype may be of importance and hence the characteristics of the homeostatic mechanisms for the control of traits subject to variation: notably the homing value (the "setting" of the homeostat), and the strength with which the organism responds to departures from this value. These parameters are related, but perhaps only remotely, to the traditional notion of the value of a phenotype. In general, where the environmental value is variable, there exist circumstances in which the optimal control would be neither extremely tight nor extremely loose. For instance, anticipatory action on the part of the body may be vitiated by too tight a control. Some illustrations are given of genetic disorders in which there is a fault in the strength of control. While neither the pattern of inheritance nor the impact on the species is explored in detail, the broad implications are indicated. Compromise adjustments are explored where two or more traits are regulated through the same homeostatic device.