Chin Long Chiang

A fertility table for the analysis of human reproduction.

Abstract Birth order instead of age of women is considered the basic variable in the study of human reproduction. In the proposed table, parity of women is the main criterion used to summarize the reproductive experience of a study population. The elements of the table serve to describe the fertility pattern of a female population; to determine the population distribution by completed family size; and to estimate the parity progression ratios, the parity specific fertility rates, the mean waiting time between the i th birth and the completion of family, and others. The table can be used to test the temporal difference and difference between countries in fertility and to evaluate the effectiveness of family planning programs. A fertility table based on current U.S. data has been constructed for illustration.

A transition-probability model for the study of chronic diseases

Abstract Presented in this paper is a modification of the general illness-death model for the study of chronic diseases. Coronary heart diseases (CHD) are used as an example. The model contains two illness (transient) states: S 1 , the “healthy” state; S 2 , state of having CHD; R 1 , state of death from other causes; R 2 , death from CHD. Transitions between states are governed by intensity functions. The study population consists of two groups of people: a group of n 1 people who are initially healthy (in S 1 ), and a group of n 2 people who are affected with CHD (in S 2 ) at the time of entering the study. The n 1 + n 2 people are subject to various lengths of time of observation. The time of first symptom of having CHD and the time of death are treated as random variables. Likelihood function of these random variables has been derived and maximum likelihood estimators of the intensity functions have been obtained. Formulas for the varianves-covariances of the ML estimators have been obtained by using the information matrix. Application to the model to an empirical data set is also discussed.

Survival and stages of disease

Abstract Development of many chronic conditions is characterized by stages. Diseases advance with time from mild through intermediate stages to severe to death. Generally, the process is irreversible, but a patient may die while being in any one of the stages. Suppose there are k stages in a disease process S 1 ,…, S k , and a death state R . Given an individual in stage S i at time τ, for 0⩽τ d τ) possible transitions are S i → S i +1 with intensity v i,i +1 ϑ(τ) and S i → R with intensity μ i ϑ(τ). The density function of survival time T of an individual who is initially in stage S 1 at time 0 is given by f T (t) = ∑ j=1 k v 12… v j−1 , j μ j θ(t) ∑ i=1 j 1 ∏ l=1 l≠1 (v ii −v ll e v ii ∫ t O θ (τ) dτ where v ii =-( v i,i +1 +μ i ). Particular functions for θ(τ), moments and estimation of parameters are discussed.

A survival model and estimation of time to tumor

In this paper we introduce a stochastic model of survival distribution, where the mortality intensity is a function of the accumulated effect of an individuals continuous exposure to toxic material in the environment (absorbing coefficient) and his biological reaction to the toxin absorbed (discharging coefficient). Formulas for the density function, the distribution function, and the expectation of lifetime are presented. The paper also includes special cases where there is a change in exposure level or exposure is discontinued or exposure is discrete in time. The model is then applied to the NCTRs serial sacrifice experimental study on mice fed 2-AAF, including some mice whose feeding was discontinued. The random variable here is the time to tumor. The chi-square test shows a good fit of the model to the data (P = 0.365). In addition to the parameters and their standard errors, estimates are computed for the expectation, variance, and percentiles of time to tumor, and for the age-specific cancer incidence rates. Confidence intervals for the parameters are also given.

Parity progression ratio and fertility rate

A formula that expresses the relationship between the maximum likelihood estimators of the parity progression ratio and the parity specific fertility rate has been derived [by C. L. Chiang and B. J. van den Berg] based on the assumption that fertility is independent of a womans age. This assumption is unrealistic and severely limits theoretical development of fertility analysis. The purpose of the present note is to derive a similar formula for the corresponding unknown parameters without the independence assumption. (EXCERPT)

The Theory of Multistage Carcinogenesis

Abstract Cancerous cells in many tissues develop by stages. A normal cell must undergo several mutations to become a neoplastic cell. This paper presents a time-dependent stochastic model to describe the k -stage carcinogenic process. The distribution of the time required for a given number of mutations and the probability of developing neoplastic cells in a given time interval are derived. The problem of estimation for a special case is also discussed.

An algebraic treatment of finite Markov chains

Abstract Simple and explicit formulas are presented for the n -step transition probabilities p ij ( n ) in finite Markov chains. Formulas are also obtained for the limiting probability distribution and the mean return time for irreducible ergodic finite chains. An example is given for illustration.

A true rate of population growth—Lotka's intrinsic rate of natural increase revisited

In this paper, Lotkas intrinsic rate of current population growth is evaluated. A new method of computing the net reproduction rate and a new rate of population growth are proposed. The proposed rate is the rate of growth of the female population per woman per year. The rate is positive, equal to zero, or negative as a population is increasing, remaining stationary, or decreasing. The rate for the 1987 U.S. white female population was R = -0.0037. This means that the white population was decreasing in 1987 and was losing 3.7 females for every 1000 women per year.

Mathematical modeling of early diabetes mellitus

Abstract A new stochastic model is presented for the study of chronic diseases in which individuals, who are identified as at high risk, are not necessarily expected to progress to develop the specific disease. The model considers the manifestation of the disease as a function of time as documented in prospective epidemiological studies of individual patients. An illustration is given with empirical data from the Boston Gestational Diabetes Study of 615 women who experienced glucose intolerance initially confined to gestation (gestational diabetes) and were followed longitudinally for 16 years for incidence of diabetes mellitus.

On multiple transition time in a simple illness death process— a fix-neyman model

Abstract This paper deals with a time homogeneous stochastic system which consists of two transient states: S 1 and S 2 , and r absorbing states: R 1 ,…, R r . An individual who is in a transient state S α at time t = 0 may move from one transient state to another within any finite length of time, and he may enter an absorbing state. Let T ( m ) αα be the length of time needed for making m transitions from S α and S β and back to S α , and T ( m ) α β , β ≠ α , be the length of time needed for making m transitions from S α to S β . Formulas have been derived for the density functions and the distributions functions of the multiple transition times. Various properties of the distributions of T ( m ) αα and T ( m ) α β have been investigated. It has been shown that, for any given α, β , and m , T ( m ) αα and T ( m ) α β are improper random variables. The properized density functions of the multiple transition times and corresponding expectations and variances have been derived. The density functions of the survival time (time before entering an absorbing state) have also been discussed.