Robert C. Busby

An eight-parameter model of human mortality—The single decrement case

Developed in this paper is an eight-parameter model of human mortality. A step-wise nonlinear least-squares procedure for estimating the parameters from abridged life tables is also described and implemented. Used for purposes of illustration were nine period life tables, ranging from 1900 to 1977, for the United States white male population. The agreement between the observed and calculated survival functions in the nine life tables was very good. Apart from its phenomenological interest, the model provides an effective means for calculating interpolations and extrapolations of abridged life tables, which are useful making population projections and in computer graphics.

On the analysis of straight line data in pharmacology and biochemistry

Abstract The determination of drug potency, efficacy and receptor affinity constants, common calculations in pharmacology and biochemistry, require estimates of the slope and intercept of a linear regression line obtained from measured points (xi, yi). Three questions not explicitly covered in the basic statistical literature and, consequently, answered differently by scientists, arise often. (1) When there exist several ys corresponding to each value of the controlled parameter x, should all of these be used as points or only their average value at that x in constructing the line? (2) What are precise confidence limits of an abscissa corresponding to a specified ordinale? (3) What are confidence limits for the intercepts of a regression line whose slope is constrained to +1 or -1? In this paper we show that all the data should be used (question 1) and provide derivations for the formulas used to get the confidence limits mentioned in questions (2) and (3).

Theory and applications of a computationally efficient cohort simulation model of human reproduction

Abstract Ease of implementation and computational efficiency are two necessary criteria if a simulation system is to be run repeatedly. Described in this paper is a cohort simulation model, based on the theory of terminating renewal processes, which satisfies these two criteria. There are two versions of the model. In one version, waiting times till pregnancy and times spent in the postpartum sterile state, as well as parity progression ratios reflecting hypothetical birth intentions, are taken into account. Unlike simulation systems described in earlier papers, pregnancy wastage is not accommodated in this version of the model. A second version is a model of birth intervals in which parity progression ratios and distributions of waiting times among live births, both of which may reflect pregnancy wastage when based on birth history data, serve as computer input. Female mortality, expressed as a survival function, and a distribution of age at marriage in a cohort are essential parts of both versions of the system. High efficiency in computing the many required convolutions has been obtained by use of a fast Fourier transform algorithm. After an overview of computer software design is given, the computer input for twelve simulation runs is described. These twelve runs are designed to test the impact of various combinations of levels of mortality, age of marriage, and fertility on population growth. One of the interesting substantive conclusions stemming from the simulation runs was that in populations of low mortality and fertility, late age at marriage, as observed in some historical populations, can be a significant factor in increasing the population doubling time.

Quadratic operator equations and periodic operator continued fractions

Abstract Conditions are given that assure convergence of an operator-valued periodic continued fraction of period two. These results and techniques are applied to get a solution of the quadratic operator equation in a complex Hilbert space. Special attention is then given to the important case of the quadratic matrix equation connected with the steady-state solution of the matrix Riccati equation from control theory. It is shown that a modification of the traditional matrix power approximation technique leads to a new, efficient and highly simplified method of approximating the unique nonnegative definite solution that exists in many important special cases.

Algorithms for a projection model of demographic indicators based on generalized age-dependent branching processes

Abstract Demographic indicators are not only used extensively to describe the current state of human populations, but they also enter into the formulation and implementation of population policies. Successes and⧸or failures of policies are frequently judged in terms of changes in demographic indicators over time. Using indicators in this way gives rise to a need for a methodology for projecting them over time in terms of basic determinants underlying the demographic evolution of human populations so that prospective population policies may be studied by computer simulation. The basic determinants of population dynamics considered in this paper were mortality, age at marriage, and fertility. The mathematical machinery used to incorporate these determinants into a workable one-sex mathematical system was that of generalized age-dependent branching processes. Given an initial age distribution and stationary law of evolution, algorithms for projecting three classes of indicators in time were developed. Included in the first class were the age distribution and rate of population growth. A second class included indicators of mortality, with specific attention being given to age-specific, crude, and infant death rates. Indicators of fertility made up a third class. Singled out for study were: age-specific birth rates, the total and net fertility rates, and the crude birth rate. Renewal theory was used to develop asymptotic formulas for all indicators considered; many of these formulas will be familiar to students of stable population theory. Some connections between the Leslie matrix and the projection system developed in this paper were also pointed out.

Population momentum—A formulation based on a stochastic population process

Abstract The momentum of population growth is studied within a unifying framework based on a stochastic population process with time homogeneous laws of evolution. After setting down some general asymptotic formulas for mean functions in Section 2, which involve the Fisherian reproductive value, it is shown in Section 3 that a stable initial age structure leads to formulas describing exponential growth when the time variable t is sufficiently larger. An alternative derivation of Keyfitzs formula for mean asymptotic population size, under a regime of replacement fertility and a stable initial age structure, is given in Section 4. Described in Section 5 are six computer simulation runs designed to study the momentum of population growth under various conditions. An example is provided whereby a population would continue to grow for about 30 years even if there were an abrupt change to a fertility regime in which mean family size was one offspring. Among the intellectual lines of descent upon which this paper rests is one initiated by Richard Bellman, during the early fifties, in a stochastic process that subsequently became known as the Bellman-Harris process.

Computerization of a population projection model based on generalized age-dependent branching processes: a connection with the Leslie matrix

Abstract The population projection model based on generalized age-dependent branching processes developed by Mode and Busby (1981) involves the solution of a large number of renewal type equations. It is shown that these equations may be solved recursively. Such a solution has two implications. One is that the projection model may be very efficiently computerized. Second, the recursive algorithm developed has striking similarities to two traditional methods of population projection used by demographers: the Leslie matrix and cohort component methods. The results presented here associate traditional projection techniques with the theory of age-dependent branching processes.

Convergence of ‘Periodic in the Limit’ Operator Continued Fractions

By straightforward analysis, we prove a convergence theorem regarding continued fractions whose entries are members of a Banach algebra. If the entries converge fast enough to a certain fixed element, convergence is assured. Under suitable other restrictions, a stronger theorem results.