OBJECTIVE To demonstrate the analytical value of a cubic parameterization of the age curve of fertility and to explore its features, especially its usefulness in separating fertility level and fertility timing. METHODS Using mathematical analysis, the cubic fertility curve is derived and examined in both continuous and discrete forms. RESULTS The cubic curve for replacement level fertility is found and expressed in terms of the mean age of fertility. That baseline cubic birth rate density, proportionately adjusted for the level of fertility, is shown to plausibly fit observed birth rates and imply a new approximation for their implicit stable growth rate. Because the proposed cubic model separates the effects of fertility level (quantum) and fertility timing (tempo), it leads to new period/cohort and population momentum relationships and provides a structure for relating fertility trajectories to birth sequences in changing rate models. CONTRIBUTION The cubic parameterization can simplify the representation of age curves of fertility rates while capturing their essential features. With a proportional adjustment at all ages to reflect fertility level, the cubic model can separate level and timing effects and permit numerous analytical applications. Of note, those applications include a new and superior approach to how changes in period tempo with constant quantum affect cohort fertility.

Supplementary Figures S1-S2 from Evaluation of Colon Cancer–Specific Antigen 2 as a Potential Serum Marker for Colorectal Cancer

The family transformations associated with the Second Demographic Transition can be seen as the consequence of three macro factors–economic, ideological, and demographic–acting through two intermediate factors–the rise of gender competition and the fall in the social capital value of marriage and children. This theoretical perspective provides the background for a dozen focused analyses that explore theoretical issues, develop methodological innovations, and provide detailed examinations of actual populations in transformation.

OBJECTIVE Few methods are available for analyzing populations with changing rates. Here hyperstable models are presented and substantially extended to facilitate such analyses. METHODS Hyperstable models, where a known birth trajectory yields a consistent set of age-specific birth rates, are set out in both discrete and continuous form. Mathematical analysis is used to find new relationships between model functions for a range of birth trajectories. RESULTS Hyperstable population projection matrices can create bridges that project any given initial population to any given ending population. New, explicit relationships are found between period and cohort births for exponential, polynomial, and sinusoidal birth trajectories. In quadratic and cubic models, the number of cohort births equals the number of period births a generation later, with a modest adjustment. In sinusoidal models, cohort births equal the number of period births a generation later, modified by a factor related to cycle length. CONTRIBUTION Because of their adaptability, structure, and internal relationships, hyperstable birth models afford a valuable platform for analyzing populations with changing fertility. The new relationships found provide insight into dynamic models and period-cohort connections and offer useful applications to analysts.

From a population perspective, the trajectories of both the total fertility at successive time periods and the total fertility of successive birth cohorts are derived from the same array of age-specific fertility rates. This analysis uses the assumption of constant age-specific fertility proportions to derive new explicit relationships between period and cohort fertility. In short, period total fertility is approximately equal to the total fertility of the cohort born a generation earlier, with a modest additive adjustment. A simple relationship also links both period and cohort total fertility to ACF, the average fertility of the childbearing cohorts in a given year. Assuming that fertility levels follow a cubic curve, cohort values from the derived relationships are then compared to observed cohort fertility values for the United States in 1917-2019. Despite substantial violations of the constant proportional fertility assumption, the calculated values deviate from the observed values by an average of only 7-8%. Short-term projections suggest that U.S. cohort fertility will continue to decline.

The risk of many demographic events varies by both current state and duration in that state. However, the use of such semi-Markov models has been substantially constrained by data limitations. Here, a new specification of the semi-Markov transition probability matrix in terms of the underlying rates is provided, and a general procedure is developed to estimate semi-Markov probabilities and rates from adjacent population data. Multistate models recognizing marriage and divorce by duration in state are constructed for United States Females, 1995. The results show that recognizing duration in the married and divorced states adds significantly to the model’s analytical value. Extending the constant-α method to semi-Markov models, 2000–2005 U.S. population data and 1995 cross-product ratios are employed to estimate 2000–2005 duration-dependent transfer probabilities and rates. The present analyses provide new relationships between probabilities and rates in semi-Markov models. Extending the constant cross-product ratio estimation approach opens new sources of data and expands the range of data susceptible to state-duration analyses.

Cross-product ratios (αs), which are structurally analogous to odds ratios, are statistically sound and demographically meaningful measures. Assuming constant cross-product ratios in the elements of a matrix of multistate transition probabilities provides a new basis both for calculating probabilities from minimal data and for modeling populations with changing demographic rates. Constant-α estimation parallels log linear modeling, in which the αs are the fixed interactions, and the main effects are calculated from relevant data. Procedures are presented showing how an N state model’s matrix of transition probabilities can be found from the constant αs and (1) the state composition of adjacent populations, (2) (N – 1) known probabilities, (3) (N – 1) known transfer rates, or (4) (2N – 1) known numbers of transfers. The scope and flexibility of constant-α models makes them applicable to a broad range of demographic subjects, including marital/union status, political affiliation, residential status, and labor force status. Here, an application is provided to the important but understudied topic of poverty status. Census data, separately for men and women, provide age-specific numbers of persons in three poverty statuses for the years 2009 and 2014. Using an estimated transition matrix that furnishes a set of cross-product ratios, the constant-α approach allows the calculation of male and female poverty status life tables for the 2009–2014 period. The results describe the time spent in each poverty state and the transitions between states over the entire life course.

As fertility in much of the developed world remains far below replacement level, it is appropriate to look closely at trends and patterns in childlessness. The rationale for doing so is reinforced by a multidimensional scaling analysis of 80 countries, which finds that contemporary fertility patterns are largely determined by two factors: the overall level and the proportion childless. Parity status life tables for 24 low fertility nations for periods since 2000 show that 15 of them have period parity progression rates implying that over 20% of women will never have any children. Commonly used figures on proportions at parity zero in cohorts completing their reproductive years have understated the level of childlessness inherent in recent data by ignoring the behavior of younger cohorts. Still, even those cohort data reveal an upward trend in childlessness. The likelihood of a resurgence in childlessness is bolstered by steady increases in the mean ages at first birth observed in all 24 study populations. Looking ahead, high proportions childless can be consistent with stable, egalitarian unions, as children now bring few resources to parents while making great demands upon them.

The prevalence of multipartner fertility (W) depends on both the level of fertility (TFR) and the likelihood (s) that a woman has a birth of parity 2 or higher by a different partner. The relationship between those factors is analyzed, and the joint variability of W with TFR and s is shown for a range of contemporary values. Similar calculations are made for the fraction of children with a maternal half sibling. Index S is then proposed as a measure that can approximate the probability that a woman has a birth by a different partner based on the TFR and the observed prevalence of multipartner fertility or children with half siblings.

A number of contemporary populations are exhibiting sustained fertility at levels substantially below long-term replacement. Nonetheless, relatively few populations are actually diminishing in size. Here, we approach that apparent paradox by analyzing the time before the number in a birth cohort, and its descendants, falls below the initial number in the cohort. First, models are examined with constant below replacement fertility, cohort extinction at age 75 or 85, and no mortality below the highest age attained. For a net reproduction rate (NRR) of 0.75, it takes 150 years for the cohort’s descendants to be fewer than the cohort’s original size if persons live to age 85, and over 130 years if persons live to age 75. If the NRR is at least 0.60, it takes a century before the descendants are fewer in number than the original cohort. Second, projections are done for the USA 2012, Italy 2012, and Hong Kong 2011 assuming that fertility and mortality remain constant. The results resemble the projections. For example, in Italy, with actual mortality and an NRR of 0.70, it takes over 125 years before the descendants of a cohort are fewer in number than the initial cohort. A relatively simple equation for the long term “time to decline” is presented, showing that it depends primarily on the level of fertility, secondarily on longevity, and only modestly on the mean age of fertility.

Demographic analyses of fertility customarily focus on age-specific birth rates, implicitly assuming that, at every age, fertility is the same for women of every parity. This chapter looks at the implications of that common assumption, and finds that they are far-reaching. From a set of age-specific birth rates, one can determine the likelihood that sibships of any size and gender composition arise in a cohort. Calculations for several fertility levels and a variety of sibship constellations are given. Further, overall cohort experience can be divided into subcohorts based on their ultimate number of children, and those subcohorts can be further broken down into the pathways by which that ultimate parity is reached. Such an analysis allows a detailed examination of fertility timing and birth spacing. Illustrative calculations indicate that higher parity subcohorts have children of every birth order earlier and move to higher parities faster than lower parity subcohorts. Because of the sequential nature of fertility, parity homogeneity casts a long shadow.

Demographers have long been interested in parity, the number of children a person has ever produced. Explicit recognition of parity opens the door to analyzing the kinship network. This chapter first examines kinship ties in models that vary both the level of fertility and the pattern of parity progression in order to quantify the influence of parity progression on the number of kin. We then re-examine the American experience over the 1917 to 2005–2010 period to see how fertility levels, family sizes, and number of kin have changed over time. The end of the Baby Boom in the mid-1960s marked a watershed, a Sibsize Transition, ending the era of large families and reshaping the American kinship network.

Demographic analyses of multistate populations are commonplace, as are situations where population stocks are known but population flows are not. Still, demographic models for multistate populations with changing rates remain at an early stage of development, limiting dynamic analyses and analytical projections. Here, a new approach, the Intrinsic Linkage-Rate Ratio (IL-RR) model, is presented and explored. The key IL parameter, w, is a simple weight for projecting populations. Using the ultimate state composition implied by the prevailing rates, the IL-RR model provides new relationships that connect multistate populations over time and allow analytical population projections. Parameter w reflects population metabolism and scales the level of the transfer rates. Compositional change is driven by the sequence of implicit stable population compositions. The IL-RR approach also provides a new method for estimating transfer rates within an interval from population numbers at the beginning and end of the interval. The new relationships developed advance the ability of demographers to model multistate populations with changing rates and to relate population stocks and flows.

In multistate populations, the rates of interstate transfer cannot generally be determined from the size and composition of the populations at the beginning and end of a time interval. With N living states, the population data give only N equations to determine the N 2 possible rates. Here, the QERT (quadratic estimation of rates of transfer) approach is advanced that allows the transfer rates to be estimated when the products of selected pairs of rates can be assumed constant. The solution can be written in closed form and, for N living states, involves no more than N −1 quadratic equations. Compared to the leading alternative approaches, QERT provides very similar numerical estimates, while yielding the underlying behavioral rates, having flexible input requirements, accommodating all structural zeros, and reproducing the exact solution when interstate transfers are strictly hierarchical. The QERT approach is applied to construct labor force life tables for U.S. men and women for 2005–2010. The results show that labor force participation differences between men and women have continued to narrow, and that the QERT approach can generate robust worklife estimates. QERT thus provides new opportunities for demographic analysis in the absence of direct data on behavioral rates.

Hierarchical models are characterized by having N living states connected by N–1 rates of transfer. Demographic measures for such models can be calculated directly from counts of the number of persons in each state at two nearby points in time. Exploiting the ability of population stocks to determine the flows in hierarchical models expands the range of demographic analysis. The value of such analyses is illustrated by an application to childbearing, where the states of interest reflect the number of children a woman has born. Using Census data on the distribution of women by age and parity, a parity status life table for U.S. Women, 2005-2010, is constructed. That analysis shows that nearly a quarter of American women are likely to remain childless, with a 0-3 child pattern replacing the 2-4 child pattern of the past.

The lack of vital statistics data on American marriage and divorce has made it difficult to follow post-1995 changes in marriage behavior. Here, a new approach, Rate Estimation from Adjacent Populations (REAP), is used in conjunction with vital statistics mortality data and recently released divorce data from the American Community Survey to construct marital status life tables that reflect the lifetime implications of observed or inferred rates of marriage, divorce, and mortality. Methodologically, the chapter sets forth the features of the REAP approach. Substantively, the analysis shows that the retreat from marriage is continuing, but unevenly. The probability that a woman ever marries has fallen to 80 %, and the average age at first marriage has risen, slightly, to 27 years. At the same time, the probability of divorce appears to be holding steady at about 43–46 %. The results suggest that the great transformation of the American family has not yet run its course.

Demographic models of marriage and fertility that incorporate the behavior of both males and females encounter the so-called 'two-sex problem.' That problem arises because observed one-sex male and female rates of marriage and birth are influenced by the total age–sex composition. The leading two-sex solution that has been advanced is the harmonic mean approach, which assumes that the sum of male and female marriage or birth rates is independent of such compositional effects. Although other serious alternatives have been proposed, the harmonic mean approach is based on the most plausible demographic foundation. The marriage squeeze is the best known manifestation of two-sex population dynamics. Using the harmonic mean approach, analyses indicate that over the past 50 years marriage squeezes have had significant impacts on the timing of marriage. Nonetheless, marriage squeezes have had less influence on the probability of ever marrying, and are unlikely to have played a major role in large-scale social change.

BACKGROUNDDynamic population models, or models with changing vital rates, are only beginning to receive serious attention from mathematical demographers. Despite considerable progress, there is still no general analytical solution for the size or composition of a population generated by an arbitrary sequence of vital rates.OBJECTIVEThe paper introduces a new approach, Intrinsic Linkage, that in many cases can analytically determine the birth trajectory of a dynamic birth-death population.METHODSIntrinsic Linkage assumes a weighted linear relationship between (i) the time trajectory of proportional increases in births in a population and (ii) the trajectory of the intrinsic rates of growth of the projection matrices that move the population forward in time. Flexibility is provided through choice of the weighting parameter, w, that links these two trajectories.RESULTSNew relationships are found linking implied intrinsic and observed population patterns of growth. Past experience is "forgotten" through a process of simple exponential decay. When the intrinsic growth rate trajectory follows a polynomial, exponential, or cyclical pattern, the population birth trajectory can be expressed analytically in closed form. Numerical illustrations provide population values and relationships in metastable and cyclically stable models. Plausible projection matrices are typically found for a broad range of values of w, although w appears to vary greatly over time in actual populations.CONCLUSIONSThe Intrinsic Linkage approach extends current techniques for dynamic modeling, revealing new relationships between population structures and the changing vital rates that generate them.