Modeling Human Fertility Using Variance-Adjusted Logistic Family of Distributions

One of the important demographic features of any population is the fertility rate which has a direct relationship with both the social and the biological environment. Demographic factors like age at marriage, present family size, gender preference (Mahadevan [27], Bhasin [3], Asari and John [2], Chachra and Bhasin [18], Bhasin and Nag [19]) and socio economic factors like education, occupation, religion, contraceptive practice, etc. (Bhatia [20], Asari and John [2]) are the determinants of desired family size and all these are considered as the cause of the variation of fertility. So far researchers have proposed a variety of reproductivity measures and mathematical models to describe the reproductivity pattern of human population (Islam and Ali [21], Peristera and Kostaki [22] and Nasir et al. [23]). The objective of this study is to examine the current pattern of Age Specific Fertility Rate(ASFR) and to study the trend in fertile age groups by fitting non linear models to the ASFR data for all the states in India obtained from the sample registration system [1], of India. Cross Validity Prediction Power (CVPP), Shrinkage, and R2 are used to identify the best model for the states. Model identification for Forward Age Specific Fertility Rate (FASFR) and Backward Age Specific Fertility Rate (BASFR) along with validity measures are also presented in this paper.

As a continent with 54 independent states Africa’s diversity is often highlighted but frequently forgotten when fertility is discussed. Fifty and more years ago to consider that all African countries and societies had a single fertility pattern (large numbers of children) and single trend (unchanging over time) was a valid characterization. Since the 1960s however that uniformity has disappeared replaced by substantial inter- and intra-country differences in fertility patterns and trends that render previous perceptions of continent-wide homogeneity obsolete. In this chapter we consider two African countries—Ghana and Kenya—whose fertility patterns and trends and their determinants have been well documented (Bongaarts 2008; Garenne 2008; Machiyama 2010; Shapiro and Gebreselassie 2008; Sneeringer 2009). Both countries have benefited from regularWorld Fertility Surveys (WFS) and Demographic and Health Surveys (DHS) that record trends in fertility family planning (FP) and other relevant indicators. The recently introduced Performance Monitoring and Accountability 2020 (PMA2020) surveys monitor progress since 2012 for the FP2020 initiative and occasional Situation Analysis and Service Provision Assessment surveys have also detailed the readiness of the health system in both countries to make quality FP services available. Ghana and Kenya share some common history: both have relatively strong health system legacies from the period of British colonialization; both were among the earliest countries to achieve independence; they were the first two African countries that developed policies to address population growth in the 1960s; and both have received substantial and sustained resources over several decades from many external donors and technical assistance organizations explicitly intended to increase the availability and quality of family planning services. However they are composed of cultures that are both diverse within each country and markedly different in many ways between the two countries. The two countries demonstrate remarkably different pathways in fertility and family planning patterns and trends from the 1970s to the present. We highlight some of the key differences and similarities explain why they have occurred and identify insights that could inform a wider understanding of fertility transitions and the role of family planning in other African countries. (excerpt)

BackgroundPakistan has an inadequate vital event registration system, leading to fewer than half of all births being registered, and this issue is further exacerbated by systematic recall errors and omission of births. This study aims to evaluate direct and indirect methods of fertility estimation to analyze the trends and patterns of fertility rates in Pakistan from 1990 to 2018.Design/methodology/approachIndirect methods are utilized in this study to evaluate the direction and extent of changes in total and age-specific fertility rates, and these findings are compared to direct estimates. The study draws data on livebirths from four waves of the Pakistan Demographic and Health Survey that took place between 1990 and 2018. To ensure the quality of data, graphical methods and Whipple and Myers indices are employed. Additionally, the Brass Relational Gompertz model was used to analyze the data.ResultsThe Relational Gompertz model revealed that total fertility rates (TFRs) were higher than direct estimates by 0.4 children and age-specific fertility rates (ASFR) were higher for all age groups except the oldest. The difference was more significant among younger women aged 15–24, and less so for age groups 29 and above. The gap in estimated fertility between direct and indirect methods decreased with age.ConclusionThe indirect method is an invaluable tool in situations where direct measurement of fertility rates is challenging or impossible. By utilizing this method, policymakers can gain important insights into the fertility patterns and trends of a population, which is crucial for making informed decisions on fertility planning.

These analyses do not demonstrate how changes in mortality which lead to differences in widowhood affect birth rates nor do they show how a fall in mortality will affect the birth rate when other factors such as marriage and age-specific marital fertility rates are held constant. These aspects were not considered by Coale and Tye in their recent study of the effects of changes in age patterns of fertility on growth and birth rates. They were concerned with the effects of changes from a younger to an older pattern of age-specific fertility rates on the growth and birth rates at the same level of mortality and on the total fertility rate or gross reproduction rate (G.R.R.). Hence in this present study we propose to examine the effects of changes in mortality on the birth rate and related measures such as growth rate mean length of generation mean duration of potentially fertile married life etc. when (i) the pattern and magnitude of age-specific marital fertility rates are held constant so that at a different level of mortality the age-specific fertility rates for all women change and hence the G.R.R. changes and (ii) the marriage is held constant to a large extent at both levels of mortality. (excerpt)

Previous authors have used simple models to investigate the relative importance to population increase of variations in the total and age-specific reproductive rates. But while acknowledging that the latter were the product of the age specific birth and death rates, they have used their models only to investigate changes in total or age-specific birth rates and have not been concerned with variations in death rates. This paper extends the use of Lewontin's (1965) model, to a wide range of values of r, the exponential rate of population increase. It shows how the relative importance of changes in certain life-history features can change with r and be reversed when r is near to zero. It is also shown that variations in mortality rate are not necessarily best expressed in analogous terms to variations in birth rate. If more suitable terms are used it is seen that changes in mortality rate can be of varying importance depending on the existing mortality rate. They can be overwhelmingly important when the mortality rate is high.

The modeling of fertility patterns is an essential method researchers use to understand world-wide population patterns. Various types of fertility models have been reported in the literature to capture the patterns specific to developed countries. While much effort has been put into reducing fertility rates in Africa, models which describe the fertility patterns have not been adequately described. This article presents a flexible parametric model that can adequately capture the varying patterns of the age-specific fertility curves of African countries. The model has parameters that are interpretable in terms of demographic indices. The performance of this model was compared with other commonly used models and Akaike's Information Criterion was used for selecting the model with best fit. The presented model was able to reproduce the empirical fertility data of 11 out of 15 countries better than the other models considered. (ProQuest: ... denotes formulae omitted.) 1. Introduction Parametric and non parametric models have been reported to have useful applications in demographic research. Apart from being useful when creating hypothetical rate schedules in forecasting and projection, they also serve to condense complex data into smaller indices (Schmertmann 2003; Peristera and Kostaki 2007). Several models have; therefore, been proposed to model fertility as the major determinant (of the three demographic variables namely, fertility, mortality, and migration,); of the size and structure of any population. These models have been commonly created for the developed countries of the world and usually fit excellently the population they are intended to model (Hoem et al. 1981). It is pertinent, however, to mention that though there are many fertility models in the literature, few have been specifically generated to describe age-specific fertility patterns in Africa; despite the fact that most governments of the sub Saharan African countries are targeting lower total fertility rates to meet the Millennium Development Goals (MDGs) (United Nations 2000). To make reaching these targets possible, a better understanding of the current pattern of age specific fertility rate (ASFR) of African countries is required. Mathematical models, when well constructed, can aid in this understanding as they provide better insight into some characteristics of the distributional pattern of fertility in Africa. The goal of any modeling exercise is to extract as much information as possible from available data and to provide an accurate representation of both the known and unknown aspects of the phenomenon being studied (Salomon and Murray 2001). Modeling fertility in Africa has also become necessary to enable a meaningful comparison of fertility across the countries in the region in the face of the current fertility transition. Already, fertility can be compared using a wide variety of existing conventional measures, summary indices, or averages that are commonly reported for fertility data. These include total fertility rate (TFR), general fertility rate, and the crude birth rate. Few comparisons, however, are made based upon the detailed distribution of the age-specific fertility curve. Not all information in the curve can be conveyed by these summary indices. There is still much to be described in terms of the variance, skew, kurtosis, and symmetry of fertility distributions for individual countries on the continent. In this article, we propose a mathematical model for ASFR using the complementary error function (defined below). The proposed model is a flexible one that can capture various shapes of ASFR. It also provides a mathematical description of some fertility indices through its interpretable parameters. The efficacy of the model was determined by comparing its performance with other fertility models. The age pattern of fertility in Africa is described in the next section. In Section 3, we provide a brief review of some existing models for fertility patterns and then propose our model in Section 4. …

In this chapter, we examine the sub-regional trends and patterns in the fertility transition in the continent by exploring the levels and patterns in period fertility as well as two proximate determinants of fertility. We develop period age-specific fertility rates (ASFRs), Total Fertility Rates (TFRs), median ages at first marriage, and contraceptive prevalence rates for sub-Saharan African countries. Correlation analysis is conducted between the two proximate determinants and corresponding level of fertility. Data come from the Demographic and Health Surveys (DHS) conducted between 1986 and 2018. Prior, a literature review is undertaken. It shows that other explanations for the fertility transition apart from the classical, including contingent lives as well as social institutions, are also important. Two fertility patterns characterise the African fertility transition – an early childbearing peak and to some extent one slightly later. Age at first marriage and contraceptive prevalence are negatively correlated with fertility levels. The fertility transition is most advanced in Southern Africa. The author recommends that: 1) adolescent, teenage, and youth reproductive health, as well as family planning programmes remain core intervention areas; 2) from a research perspective, other models predicting fertility schedules more pertinently and accurately need to be considered.

Projection of individual age-specific fertility rates is a forecasting problem of high dimension. We solve this dimensionality problem by using parametric curves to approximate the annual age-specific rates and a multivariate time series model to forecast the curve parameters. These yield forecasts of future fertility curves, which are then used to compute age-specific fertility rate forecasts. This reduces the dimensionality of the forecasting problem and also guarantees that long-run projections of age-specific fertility rates will exhibit a smooth shape across age similar to historical data. Short-term projections are improved by also using simple techniques to forecast the deviations of the fitted curves from the actual rates. The article applies this approach to age-specific fertility data for U.S. white women from 1921–1984. The resulting forecasts are examined, and the multivariate model is used to investigate possible relations between the curve parameters, expressed as the total fertility rate, the mean age of childbearing, and the standard deviation of age at childbearing. The only strong relationship found is the contemporaneous relationship between the mean and standard deviation of age at childbearing. A variation of this approach, in conjunction with traditional demographic judgment, was used in a recent set of U.S. Census Bureau population projections. We discuss this implementation and compare the Census Bureau projections with those produced directly from the model presented here.

Ethiopia has experienced a considerable decline in fertility rates in urban and rural areas across various regions over the past few decades, largely due to government initiatives. Ethiopian Demographic and Health Surveys (EDHS) from 2000, 2005, 2011, and 2016 provide substantial evidence of this trend in terms of total fertility rates (TFRs). Due to the lack of reliable vital registration systems and recent census data, research heavily relies on these EDHS surveys. A review of the literature reveals a gap in understanding the age patterns of fertility and the fertility transition in Ethiopia and its regions. To address this, this study proposes a “model fertility table” for Ethiopia, providing reliable estimates of age-specific fertility rates (ASFRs) based on TFR data. Using TFRs from multiple EDHS surveys, ASFRs are derived for the years 2000, 2005, 2011, and 2016 to study regional fertility transition in Ethiopia. The results show a typical uni-modal distribution of ASFRs, with a broad peak in fertility rates among women aged 20 – 24, 25 – 29, and 30 – 34 during the early stages of the transition. As TFRs decline, the peak shifts toward older age groups. In addition, the fertility pattern becomes more concentrated in older age groups. Significant fertility differences were observed between regional and rural-urban areas. This study has both theoretical and practical implications. It introduces a new methodology for population studies and offers detailed ASFR data, aiding policymakers in designing targeted fertility and health policies and addressing regional fertility differences for more effective interventions.

연령별 출산율의 분석 및 예측은 인구추계를 위한 중요한 요소 중 하나로, 연령별 출산율을 적합하고, 예측하기 위해 다양한 모수적 모형들이 이용되었다. 기존의 방법들은 특정 시점에서의 연령별 출산율을 모수적 모형 등을 이용하여 추정하고, 전 시점에서의 모형의 모수 추정치를 시계열 모형을 통해 적합한 후 예측하는 방법을 이용하여 왔다. 그러나 이 방법은 고정된 시점에서의 변동성과 모수 추정치의 시계열의 변동성을 함께 고려하지 못하여, 인구추계 시 변동성이과소평가되는 문제점이 지적되어 왔다. 이러한 문제를 해결하기 위한 방안으로 베이지안 통계학이 이용될 수 있다. 최근 UN(The United Nation)의 세계인구추계(world population prospects)에서는 합계출산율과 기대수명을 베이지안 방법을 이용하여 예측하였다. 그러나 이들의 연구는 단지 합계 출산율과 기대수명에 한정되어 있다. 본 논문에서는 연령별 출산율의 예측하기 위한 베이지안 방법을 제안하고, 이를 우리나라의 연령별 출산율 예측에 이용하고자 한다. 각 시점에서 연령별 출산율이 모수적 모형을 따름을 가정하고, 모수적 모형의 모수가 시계열 모형을 따름을 가정한 후, 이를 MCMC(Markov chain Monte Carlo)방법을 이용하여 미래 연령별 출산율을 예측하고 신뢰구간을 구하고자 한다.Analysis and prediction of age specific fertility rates is one of important components for population projection. To predict age specific fertility rates, parametric models were fit to data at each time point, and parameter estimates from the model were forecasted by using time series models. However, this approach cannot take into account the uncertainty from model fitting at fixed time and the uncertainty from modeling time series of parameter estimates at the same time. Since it is well known that Bayesian approach can overcome this problem, UN (the United Nations) forecasted total fertility rates and life expectancy at birth at world population prospects by using Bayesian approach. However, this work was limited to only total fertility rates and life expectancy at birth. In this paper, Bayesian method to predict age specific fertility rates are proposed, and applied to forecast of Korean age specific fertility rates. By using MCMC (Markov chain Monte Carlo) method, the uncertainty from model fitting at each time and from modeling time series of parameters can be considered at the same time. Finally, forecasts of age specific fertility rates and their confidence intervals are presented by using the proposed method.

The importance of projection in national and state level planning and policy formulation is quite well recognized in any country which attempts at achieving sustainability in human development and improving the quality of life. Population projection exercises are basically a part of forecasting growth of human population in the future years over a time horizon. There are various means of extrapolating past trend of change in human population over the future years. These means are determined by the assumptions that are made on the determinants of population change such as time, pattern of changes in fertility, mortality, and migration and other associated factors. The success of population projection depends not only on the technique of projection but also on the proximity of the assumptions to reality so that changes in the future years get estimated with least possible errors. Projections, however, might not suffice when there are significant deviations from the assumption that prevailing conditions would continue unchanged in the future. Also, the projection might not be satisfactory due to failure to incorporate adequately the changes in the policy parameters, technological changes, changes in the migration pattern, etc. Forecasting attempts at overcoming these drawbacks by incorporating the elements of judgment in the projection exercise. Forecasting enjoys the advantage of being based upon one or more assumptions that are likely to be realized in the future years. Thus, forecasts give more realistic picture of the future.KeywordsRoot Mean Square ErrorTotal Fertility RatePopulation ProjectionExponential SmoothingPolynomial Regression ModelThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Modelling is a well-established concept for understanding the typical shape and pattern of age-specific fertility. The distribution of India’s age-specific fertility rate (ASFR) is unimodal and positively skewed and is distinct from the ASFR of the developed countries. The existing models (P-K model, Gompertz model, Skew-normal model and G-P model considered here) that were developed, based on the experiences of the developed countries, failed to fit the single-year age-specific fertility pattern for India as a whole and for the six selected states. Our study has proposed four flexible models, to capture the diverse age pattern of fertility, observed in the Indian states. The proposed models were compared in three ways; among themselves, with the original models and with the popular Hadwiger model. The parameters of these proposed models were estimated through the Non-Linear Least Squares Method. To find the model with best fit, we used the corrected version of Akaike’s Information Criterion (AICc). Optimization of the four original models was successfully done. When the model was fitted to the empirical data of the 4th round of the National Family Health Survey conducted in 2015–2016, the results of this study showed that all the four proposed models outperform their corresponding original models and the Hadwiger model. When comparison among the proposed models was done, the Modified Gompertz Model provided the best fit for India, Uttar Pradesh and Gujarat. Whereas, the Modified P-K model gave the best fit for West Bengal, Tripura and Karnataka. The Modified G-P model is the most suitable model for Punjab. Although our proposed models illustrated the fitting of ASFR for India as a whole and the selected six states only, it provides an important tool for the policymakers and the government authorities to project fertility rates and to understand the fertility transitions in India and various other states.

On the sensitivity of the intrinsic growth rate to changes in the age-specific birth and death rates

The Effects of State-Level In Vitro Fertilization (IVF) Insurance Mandates on the Fertility of American Women

This study aims to understand trends in global fertility from 1950-2010 though the analysis of age-specific fertility rates. This approach incorporates both the overall level, as when the total fertility rate is modeled, and different patterns of age-specific fertility to examine the relationship between changes in age-specific fertility and fertility decline. Singular value decomposition is used to capture the variation in age-specific fertility curves while reducing the number of dimensions, allowing curves to be described nearly fully with three parameters. Regional patterns and trends over time are evident in parameter values, suggesting this method provides a useful tool for considering fertility decline globally. The second and third parameters were analyzed using model-based clustering to examine patterns of age-specific fertility over time and place; four clusters were obtained. A country’s demographic transition can be traced through time by membership in the different clusters, and regional patterns in the trajectories through time and with fertility decline are identified.