The multistate life table with duration‐dependence

A method for generalizing the multistate, or increment‐decrement, life table to include rates which depend upon duration of exposure to risk, as well as upon age, is proposed. The method is built upon the linear approximation, called the linear integration hypothesis, developed mainly by Rogers and his colleagues. Although the use of rates indexed by duration categories leads to a substantial increase in the state space of the model, it is possible to arrange the rates in such a way that matrices to be inverted are no larger than those encountered in the usual multistate life table. In the more general approach it is possible to derive several new summary indices of the life‐table cohort's history, such as the mean and median time in current status, at any age. The method is illustrated using a simple four‐state marital‐status model which has appeared often in the literature; here, rates of divorce and widowhood vary by duration of marriage as well as age.     

Home ownership,life cycle stage,and residential mobility

Previous research has shown that mobility rates decline with increasing age and duration of residence. These relationships are investigated further for the case of residential mobility using residence histories obtained in interviews with 2264 Rhode Island residents. Three methods of classifying segments of a person’s life into life cycle stages are compared: age, life cycle stages based on marital status and child-rearing periods, and a combined age-marital status classification. These classifications were not found to be equivalent in that there was considerable variation in mobility rates by life cycle stage within age categories and by age within life cycle categories. The age-marital status classification was selected for use in the remainder of the analysis because it had the least variation in mobility rates within categories and required far less data for computation than the life cycle stages. When mobility rates were examined by home ownership, age-marital status, and duration of previous residence, it was observed that there was little variation in mobility rates by duration for home owners while the mobility rates for renters declined with duration.     

A fertility analysis of New Zealand marriage cohorts

New Zealand annual returns of non-Maori live legitimate births 1913 to 1955 were tabulated longitudinally by marriage duration and birth order. The figures were used as numerators of fertility rates of forty marriage cohorts, specific for duration and birth order ; the twenty-four oldest cohorts had virtually completed fertility. For the denominators of the rates, the cohorts of initial size (estimated by a method similar to that described by P. H. Karmel) were survived by observing changes over time through divorce, death, widowhood, war-widowhood, and net external migration. The aim was to provide a set of data of the best possible approximation on variations of cohort fertility.

Total cohort rates, cumulated for twenty years’ duration of marriage, and segmental rates of relevant sub-periods are given for parities o to 7+. Total rates of the cohorts of completed fertility yielded values of family size, and were also used for parity progression ratios (by L. Henry’s formula). Segmental rates permitted a study of changes in timing maternities.

The analysis, which needs as a corollary a nuptiality analysis, is concerned, so far with past experience. It confirms the decline in number and proportion of large families. Both the parity progression ratios, and segmental rates of cohorts of as yet incomplete fertility, suggest some recent recovery in three- and fourchild families. For total fertility, such recovery might even be more significant than the observed decline in large families. On the other hand, birth-order specific changes in timing suggest lengthening of intervals among older cohorts as a trend upon which is superimposed the fluctuation due to postponement of, and recovery of a portion of postponed maternities. A construction of hypothetical timing patterns for incomplete cohorts by a simplified method of projection does not lead very far without support of observations on spacing that differentiate for family size.     

Recovering event histories by cubic spline interpolation

If event data are recorded in discrete intervals of time, errors are introduced when the data are converted from the unit in which they were recorded, such as date, to another unit such as age or duration. The problem is illustrated by the inconsistent age at marriage schedules published by 2 recent US censuses. This paper develops a general method for treating problems of this type using cubic spline interpolation. The method is used to adjust US age at marriage schedules, explaining a substantial part of the discrepancy in the 1960 and 1970 censuses.     

A new method for estimating the level of natural fertility in populations practicing birth control

The natural fertility schedule of a population is the schedule of age-specific marital fertility we would observe if no birth control were being practiced. In natural fertility (no birth control) populations we can observe the natural fertility schedule directly, but in populations practicing birth control the natural fertility schedule is disguised by the marital fertility rates in those age intervals in which control is exercised, the marital fertility rates being below the natural rates. This paper elaborates a method for estimating the natural fertility schedules of populations practicing birth control. Two alternative models are presented, one nonlinear and one linear. The use of these models is then illustrated with reference to the Old Order Amish population of Lancaster County, Pennsylvania. It is shown that for Amish women born between 1860 and 1900 both the nonlinear and linear models yield an estimate of .843 as the ratio between Amish and Hutterite natural fertility schedules.     

A modification for use in destabilized populations of brass's technique for estimating completeness of death registration

Summary Brass has developed a method of estimating completeness of death registration using only data on deaths and population by age and sex. In this paper, his method is briefly outlined and the assumptions upon which it is based are discussed. In particular, the implications of the failure of the assumption of stability of the population are investigated. It is found that in populations where mortality has been declining, use of the technique leads to underestimation of completeness. A modification of the technique based on knowledge of the duration and rate of mortality change is proposed for use in such populations. Using simulated destabilized populations, the modification is tested and found to yield more accurate estimates of completeness of death registration than the unmodified technique. The usefulness of the modified technique is further illustrated by applying it to data for Costa Rican females in 1963.     

Farm Household Lifecycles and Land Use in the Ecuadorian Amazon

This paper describes trends in population, household formation, fragmentation of landholdings, and changes in land use between 1990 and 1999 in the Northern Ecuadorian Amazon (NEA)—one of the areas of highest biodiversity in the world. It also shows how changes in land use are associated with the duration (or age) of the farm settlement, which is also linked to the stage in the farm household lifecycle and household composition. The study is based upon data from two detailed household surveys, which result in a cohort of 246 farm households interviewed in 1990 and 1999 as well as 383 new farm households constituted after 1990. Distinct patterns of land use are linked with duration of settlement, independent of when settlement occurred in the region.     

Factors influencing contraception continuation rates: The oral and the intrauterine methods

The technique of multi-sort analysis was used to compare the relative duration of use of oral and intrauterine contraception among low socio-economic groups in Baltimore City. The clinical records of over 12,000 women who had received contraception through the Baltimore public services were analyzed. The type of method a woman used was found to be the most important factor affecting her probability of continuing contraception. Her age, race, and how long she had been using the method also influenced this probability. The strictness of the criteria used for the definition of continuation rates and the fact that experience was incomplete for many intrauterine contraceptors probably in part explain the somewhat low continuation rates obtained for both methods. It is suggested that the intrinsic characteristics of each method may lead to a clinical impression favoring the oral even when statistical evidence, as presented here, favors the intrauterine method.     

Recovering event histories by cubic spline interpolation

If event history data are recorded in discrete intervals of time, errors are introduced when the data are converted from the unit in which they were recorded, such as date, to another unit such as age or duration. The problem is illustrated by the inconsistent age at marriage schedules published by two recent U.S. censuses. This paper develops a general method for treating problems of this type using cubic spline interpolation. The method is used to adjust U.S. age at marriage schedules, explaining a substantial part of the discrepancy in the 1960 and 1970 censuses.     

Duration of residence and prospective migration: Further evidence

A basic assumption of the Cornell Mobility Model insofar as it is relevant to inter-community migration is that an individual’s propensity to move is a function of, among other things, his length of residence in the community. Parameters of the functional relationship of migration probabilities to duration status are estimated from migration histories for a sample of residents in Monterrey, Mexico, and compared with estimates reported by P. A. Morrison using data from Amsterdam, The Netherlands. In both data sets a negative nonlinear relation of the probability of migrating to duration status, as well as an interaction between age and duration status, is found. Values of parameters describing the relation within age groups differ sufficiently between the two data sets, however, to suggest that further specification of conditions under which a particular form of functional relation will obtain is necessary if the model is to be useful in migration research.     

Census-derived estimates of fertility by duration since first marriage in the Republic of Korea

This paper estimates ever-married birth rates by age and duration since first marriage and ever-married total fertility rates for the Republic of Korea, derived by applying an extension of the own-children method of fertility estimation to the 1975 and 1980 censuses. Since each census provides annual estimates for the 15-year period previous to enumeration, there is a ten-year period of overlapping estimates that facilitates checks for consistency and accuracy. Comparisons are also made with estimates derived from the 1974 Korea National Fertility Survey, which was part of the World Fertility Survey. The method works well, except in its application to the 1975 Census where the evidence suggests considerable misreporting of age at first marriage because of the way the question was asked and coded. Results confirm that ever-married fertility fell substantially in Korea between 1961 and 1980, with a temporary resurgence in the late 1960s and early 1970s. Ever-married fertility rose at younger ages and shorter durations and fell at older ages and longer durations. Ever-married fertility differentials by urban-rural residence and by education were usually in the expected direction, with urban fertility generally lower than rural fertility and the fertility of those with more education usually lower than the fertility of those with less education. Differential ever-married fertility by urban-rural residence and education declined over the estimation period.     

A model to estimate births averted due to IUCDs and sterilizations

The computation of births averted due to a specific family planning programme requires special treatment instead of application of the usual population projection technique using conventional fertility rates. The reason for this is that adopters of a family planning method are a selective group from the general population with respect to a number of demographic and biological characteristics, including the initial susceptibility status at the time of adoption. In the present paper a matrix of annual birth probabilities specific to age at adoption of a method and duration since adoption is obtained. One matrix is presented for adopters of IUCD, another for salpingectomy, and a third for vasectomy. These matrices are used to obtain estimates of births averted for India due to IUCDs and sterilizations performed during 1956–69. The results are compared with those that could have been obtained by using the usual age-specific marital fertility rates under identical conditions, except for initial susceptibility. The entire work is programmed on a high speed electronic computer.     

Measuring marital fertility control with CPA

Cohort Parity Analysis (CPA) is an indirect method for measuring the extent and timing of the adoption of fertility control within marriage. Cohort Parity Analysis uses 2 sets of parity distributions, 1 from the population under study (the "target" population) and 1 from a similar population in which fertility control is absent (the "model" population). Fertility control is defined as any set of behaviors in the target population that cause its parity progression rates to differ from those of a similar but noncontrolling population. The target population contains an unidentified mixture of controlling and noncontrolling couples for a given age range and marriage duration. The model population is similar in all respects except that it contains no controllers. Examples are taken from urban and rural Irish populations in the 1911 census data. Cohort parity analysis rests on 2 basic assumptions: 1) For every specified marriage duration and marriage age, there exists some parity above which controllers are never found; and 2) Parity progression rates of those in the mixed population who have never initiated fertility control are identical to those in the model population. Further, one must identify a cutoff parity beyond which controllers are presumed not to be found. In these urban Irish examples cutoff parity for each marriage age and duration is set at 20% of that of the noncontrolling rural Irish population. The upper and lower bounds on the percentage of controllers are derived by obtaining the lower and upper bounds on the size of the group at that parity who never controlled and subtracting these estimates from the observed percentage of the target population observed at that parity. Flow charts are used to present the methodology of cohort parity analysis in a simple format. the method is illustrated in the case of Ireland in 1911. Aggregate parity distributions for the county boroughs of Dublin, Belfast, Cork, LOndonderry, Limerick, and Waterford are taken as the target distribution; and parity distributions for those living outside the county boroughs are taken as the model distributions. The analysis shows that 28% of urban-dwelling Irish women who married before age 30 and were married between 4 and 29 years in 1911 used some effective means of fertility control. The parity distributions also show that women who married later controlled their fertility to a greater extent and that between 20.9 and 25.5% of women married 4 years in 1911 had already practiced fertility control by 1911, but only 15.5% of those married 15-19 years had used fertility control by 1911. Cohort Parity Analysis is thus a useful method of examining historical populations in the process of fertility transition.     

Age at marriage as a factor in state divorce rate differentials

The prominent East-to-West gradient in state divorce rates has frequently been partially attributed to an ecological factor called "frontier atmosphere," which incorporates the effects of mobility on social integration. This paper employs linear statistical procedures to show that, after adjustment of state divorce rates for age distribution, the percent ever married among teenage females is the best statistical predictor of 1960 and 1970 divorce rates, whereas "frontierness," conventionally measured by population mobility, is shown to have negligible effect, controlling for other selected legal and nonlegal factors. This finding suggests that previous ecological studies of divorce have overemphasized the effect of mobility and social stability on geographic divorce rate differentials, perhaps due to the neglect of demographic variables such as age distribution and especially age at marriage.     

Estimating detailed distributions from grouped sociodemographic data: ‘get me started in’ curve fitting using nonlinear models

In much demographic analysis, it is important to know how occurrence-exposure rates or transition probabilities vary continuously by age or by time. Often we have coarse or fluctuating data so there can be a need for estimation and smoothing. Since the distributions of rates or counts across age or another variable are often curved, a nonlinear model is likely to be appropriate. The main focus of this paper is on the estimation of detailed information from grouped data such as age and income bands; however, the methods we outline could also be applied to other settings such as smoothing rates where the original data are ragged. The ability to carry out curve fitting is a very useful skill for population geographers and demographers. Curve fitting is not well covered in statistics textbooks, and whilst there is a large literature in journals thoroughly discussing the detail of functions which define curves, these texts are likely to be inaccessible to researchers who are not specialists in mathematics. We aim here to make nonlinear modelling as accessible as possible. We demonstrate how to carry out nonlinear regression using SPSS, giving stepped-through hypothetical and research examples. We note other software in which nonlinear regression can be carried out, and outline alternative methods of curve fitting.     

Age-parity-specific birth rates and birth probabilities from census or survey data on own children

The ‘own-children’ method is a census- or survey-based reverse-survival technique for estimating age-specific birth rates for years previous to enumeration. In this paper it is extended to estimate birth rates and birth probabilities by parity as well as by age. The refined method is tested for accuracy by comparing ‘own-children’ estimates of age-parity-specific birth probabilities for the United States in 1969 with parallel estimates obtained by the method used in Vital Statistics of the United States. The ‘own-children’ estimates give relatively lower values at ages 15-19 and higher values at ages above 40, but compare well in between. The discrepancies are probably due mainly to adoption of illegitimate children of young mothers by older women. The ‘own-children’ estimate of the total fertility rate is very close to the published vital statistics estimate.     

An analytic simulation model of human reproduction with demographic and biological components

Abstract This Monte Carlo model for simulating the reproductive history of a cohort of women is described in detail. The model provides for patterns of survival, sterility, formation and dissolution of sexual unions, fecundability, lactation, foetal wastage, family planning practices etc. Natality indices specific for marital status, for duration of marriage and for age, as well as analyses of birth spacing patterns are among the results that may be obtained. In the model, the experimental unit is an individual woman. The complete life history of a woman is generated and recorded before the history of the next woman is generated. The data for the whole cohort are analyzed at the end of the programme. The model includes two kinds of states into which a woman may pass, namely: (1) permanent changes of status such as death, sterility, or becoming a family planner, and (2) temporary states, each with a probability distribution of length of stay. The probabilities of the various events or changes of state may vary from age, parity, and other features of a woman's status or history. Natural fecundability at any age may also vary from woman to woman. In this programme natality patterns and specific indices such as age-specific fertility rates are produced, in a quasi-realistic fashion, by the interplay of the demographic and biological parameters postulated for any cohort. Consequently, the effect of changes in anyone factor can be studied, as well as the interaction resulting from changes in several factors. The purposes and potentials of the model are both substantive and methodological. As an illustration, a series of computer runs attempting to simulate the reproductive patterns of Indian women is presented. These results, as well as some additional ones, indicate some effects of changes in marital patterns, levels of fecundability, duration of post-partum non-susceptibility, age incidence of sterility and foetal wastage. In the final section of the paper, the advantages and possible applications of the model are discussed together with the limitations encountered to date in the efforts to apply the model.     

Patterns underlying fertility schedules: A decomposition by both age and marriage duration

Summary Model fertility schedules based on the proportions married and the age pattern of marital fertility are unsatisfactory to the extent that marital fertility does not depend on age alone but also on other factors. Most notably, models based just on age fail to allow for differences between populations in their composition by marriage duration. Examination of the major series of fertility rates specific by both age and duration of marriage (Sweden 1911-70, England and Wales, 1941-70) reveals striking underlying regularities. The marital fertility rates observed at any given point in time can be factored into three independent components - an overall level, a vector of age effects common to all marriage durations and a vector of duration effects common to all age groups. A simple product of these three components is shown to approximate the data very closely over the entire series, despite major changes in the aggregate levels of fertility and nuptiality during the periods concerned. Not only are the data tightly structured, conforming very closely to this simple multiplicative model, but the vectors of age and duration effects themselves are shown to exhibit clear and meaningful regularities.     

Child mortality in rural India

This paper focuses on infant and child mortality in rural areas of India. We construct a flexible duration model, which allows for frailty at multiple levels and interactions between the child’s age and individual, socioeconomic, and environmental characteristics. The model is estimated using the Indian National Family and Health Survey 1998/1999. The estimation results show that socioeconomic and environmental characteristics have significantly different impacts on mortality rates at different ages. These are particularly important immediately after birth. The parameter estimates indicate that child mortality can be reduced substantially, particularly by improving the education of women, providing safe water, and reducing indoor air pollution caused by dirty cooking fuels. Finally, we still found substantial differences in mortality rates between states, which are associated with differences in schooling expenditures, female immunization, and poverty rates.     

Duration dependence and heterogeneity in French youth unemployment durations

This paper investigates the degree to which the individual exit rate out of unemployment for young job seekers changes as a function of the elapsed unemployment duration. We use a nonparametric estimation method for population data on outflows from different duration classes. The method also provides estimates of the amount of unobserved heterogeneity in these data. We explicitly take into account that individual exit rates are affected by the business cycle. The method is applied to population data on young French unemployed job seekers. The results are used for policy recommendations. Received: 17 October 1996/Accepted: 23 July 1998