Abstract
Cohort Change Ratios (CCRs) appear to have been overlooked in regard to a major canon of formal demography, stable population theory. CCRs are explored here as a tool for examining the transient dynamics of a population as it moves toward the stable equivalent that is captured in most formal demographic models based on asymptotic population dynamics. We employ simulation and a regression-based approach to model trajectories toward this stability. This examination is done in conjunction with the Leslie Matrix and data for 62 countries selected from the US Census Bureau’s International Data Base. We use an Index of Stability (S), which defines stability as the point when S is equal zero (operationalized as S = 0.000000). The Index also is used to define initial stability for a given population and four subsequent “quasi-stable” points on the temporal path to stability (S = .01, S = .05, S = .001, and S = .0005). The regression-based analysis reveals that the initial conditions as defined by the initial Stability Index along with fertility and migration play a role in determining time to stability up until the quasi-stable point of S = .0005 is reached. After this point, the initial conditions are no longer a factor and mortality joins the fertility and migration components in determining the remaining time to stability. Overall all, we find that fertility and mortality have an inverse relationship with time to stability while migration has a positive relationship. The initial Stability Index has an inverse relationship with time to quasi-stability at S = .01, S = .005, S = .001, and S = .0005. We also find that a regression model works very well in estimating the intrinsic rate of increase from the initial rate of increase, but that this model can be improved by adding the components of change. We also compare time to stability and intrinsic r as estimated using the CCR Leslie Matrix approach to, respectively, estimates of time to stability and intrinsic r found using analytic methods and find that the former are consistent with the latter.
The authors are grateful to a number of people for comments and suggestions, including Hiram Beltran-Sanchez, Stan Drezek, Barry Edmonston, Victor M. Garcia-Guerrero, David Hamiter, Richard Verdugo, Robert Schoen, Webb Sprague, and Jeff Tayman.
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Notes
- 1.
The input data used to generate cohort change ratios need to be separated by a time interval that is consistent with the age groups used in the input data. For example, if the data are in 5 year age groups (up to the terminal, open-ended age group), the time interval should be either 5 years or 10 years. If the data are in 10-year age groups then the time interval will need to be 10 years. If the data are in single-year age groups, then the time interval should be 1 year. Fortunately, most population data are provided in 5-year age groups.
Although we do not provide a proof here, it is easy to show that using CCRs to move a population through time is consistent with the fundamental demographic equation. This consistency is important for two reasons. First, as noted by Land (1986) any quantitative approach to forecasting is constrained to satisfy various mathematical identities, and a demographic approach should ideally satisfy demographic accounting identities, which are summarized in the fundamental demographic equation. The second reason is based on the argument by Vaupel and Yashin (1985) that a demographic forecasting method needs to be consistent with the fundamental demographic equation in order to minimize the potential errors associated with hidden heterogeneity.
- 2.
If one has a life table, the CCRs for a given population could be compared to their corresponding survival rates and the effects of migration could be separated from the effects of mortality . This would be similar to using a life table to estimate net migration by age using the Forward Life Table Survival Method. Again, an important assumption is that differential net undercount by age is absent or at least very minimal.
- 3.
Often, the Index of Dissimilarity is expressed as a percentage, whereby the formula shown in Eq. (12.5) is multiplied by 100. In our use of this Index, we define “zero” to six significant digits. That is, when S is equal to “0.000000,” we define this stability. If fewer or more significant digits were used, the point at which stability is reached would, of course, be different.
It is worthwhile to note here that Keyfitz and Flieger (1968: 23 and 24–41) display a “dissimilarity” score between a current population age distribution and the age distribution for the corresponding stable population. The index is the sum of positive differences between the two distributions. This index is only one simple step from the Index of Dissimilarity . However, even so, it is neither employed by Keyfitz and Flieger (1968) to define a stable population nor used to estimate time to stability . However, Keyfitz (1968: 47) does use it to define the distance to stability and other measures of this distance are found in Caswell (2001), Cohen (1979b), Schoen (2006), Schoen and Kim (1991) and Tuljapurkar (1982).
Also, as noted in the text and described by Keyfitz (1968: 47), the Index of Dissimilarity could be used in conjunction with the relative age distribution at stability and at the initial launch point. As an example of this use, the highest value found in the 62 country data set is for Hong Kong, which has a Dissimilarity Index of .399073; the lowest is found for Guatemala, with a Dissimilarity Index of .05001. Thus, Hong Kong’s age distribution at origin is furthest from its stable age distribution while Guatemala’s is closest. As would be expected, Hong Kong’s time to stability (740 years) is much longer than Guatemala’s (250 years). These two respective indices also provide an easy-to-interpret measure of how different the initial population age structure is from the age structure at stability. For Hong Kong, 39.91 % of the initial population needs to be re-allocated to match its relative age distribution at stability while for Guatemala only 5 % needs to be reallocated. Due to the specific dynamics underlying a country’s path to stability, the Index is not the sole determinant of time to stability, however. For example, Hong Kong does not take the longest time to reach stability of the 62 countries (Singapore does, at 890 years) and Guatemala does not take the shortest time to reach stability (El Salvador does, at 225 years).
- 4.
The Leslie Matrix was implemented as a “macro” in Excel using Excel’s coding language, VBA. The code as well as a “template” excel file with instructions on how to implement the Leslie Matrix are available on request from the authors. Also available from the authors are the files for all 62 countries as well as the summary file containing life expectancy at birth, the total fertility rate, and the mean CCR for ages 20–24, 25–29, and 30–34.
There are different ways in which the Leslie Matrix could be implemented in terms of the constant ASFRs and CCRs. For example, once one developed the ASFRs for both sexes (as we have done) and set them up in a forecast cycle (which in our case here is for a 5 year period), the ASFRs could then be adjusted for infant mortality . One also could determine the mid-cycle populations of child-bearing ages (15–19, 20–24,…, 45–49) and then apply the either the unadjusted ASFRS or mortality-adjusted ASFRs to them. We implemented the ASFRs without an adjustment for mortality and applied them to the population at the beginning of the forecast cycle. In the long run to stability, the different implementations are not likely to create substantial differences in the time to stability , but they could make a difference if one were attempting to develop realistic forecasts with much shorter horizons (e.g., 10 years, 20 years and even 50 years).
- 5.
Given that the path to stability is non-linear, we also explored regression models in which the time (number of years) to stability was transformed using natural logarithms. However, we found that other than the change in the regression coefficients to accommodate the transformation, these models were not substantially different than their non-transformed counterparts. For example, the model that corresponds to the provided in Eq. (12.6) has an R2 of .61 and an adjusted R2 of .59 and the rank-order of the standardized coefficients is the same as found for the model shown in Exhibit 12.3 for time to S = zero, which are those associated with Eq. (12.5). It is useful to note here that the NCSS regression procedure employs Huber’s method when skewed residuals are encountered. As such, it is a robust approach and its results will vary from those found using OLS methods which do not employ this method when skewed residuals are encountered.
It is worthwhile to note that some of the effects of the predictor variables found in Eq. (12.6), may also be non-linear on their own and interactive. We have not explored these possibilities here, but they may prove useful in future work.
It also is worthwhile to mention work by Preston (1986) in which he found that there is a close approximation between the intrinsic growth rate of a population and the mean of age-specific growth rates below age T, the mean length of a generation. He concluded therefore that where a disparity exists between the intrinsic growth rate and the actual growth rate of a population (whether or not net migration is included in both rates), it must be attributable to an unusual growth rate of the population block above age T.
- 6.
While it appears that regression analysis has not been used to estimate intrinsic r from an initial r, Bourgeois-Pichat employed it to estimate intrinsic r from the proportional age distribution of a given population (see Keyfitz and Flieger 1968: 40).
- 7.
Based on comments by Barry Edmonston, we also constructed models for estimating r using the initial Stability Index (S). In the model in which all three components of change were included along with initial r, we found that e0 was not statistically significant. We then eliminated this predictor variable and re-ran the model with the other two components of change, Initial S, and initial r, and found a model with an adjusted R2 of .948 and predictor variables that were all statistically significant. These results suggest that the difference between the initial age distribution and the stable age distribution may be a factor in the difference between initial r and r, a suggestion provided by Barry Edmonston. This idea may also account for the difference between the regression model for estimating r from initial r that was constructed using the Keyfitz and Flieger (1968) data and the model for estimating r from initial r that was constructed using the Census Bureau’s International Data Base. That is, the differences found between initial population age structures and the stable ones for each of the 67 populations taken from Keyfitz and Flieger (1968), on the one hand, may vary from the differences found for each of the 62 populations taken from the US Census Bureau’s International Data Base, on the other. This is a topic for future research.
- 8.
There may be approaches other than the one we employ (Caswell 2001: 95–97) to compare with the estimate of time to stability generated from our CCR approach using regression and the Stability Index (S). For example, it may be possible to substitute a variation of the Kullback Distance (Nair and Nair 2010; Schoen 2006: 29–33) for the Stability Index as an independent variable in a regression model such as we employed. With appropriate modifications, the Kullback Distance potentially could be used with cohort change ratios and its results compared with both those generated by the CCR method and the analytic approach we used. It is useful to note that the Kullback Distance declines monotonically during the process of convergence (Schoen 2006: 31), which is similar to the behavior of S, where the initial decline may not be monotonic, but becomes so at some point and overall, is monotonic or nearly so. Also, like S, the Kullback Distance possesses a number of desirable properties (Schoen 2006: 31). However, the Kullback Distance also may generate different values than the method we used and, as such, yield different summary statistics in a comparison with the CCR approach. Similarly, there variations on the analytic approach we used to estimate r, which is taken from Caswell (2001: 74–75). Descriptions of variations that potentially could be used can be found in Barclay (1958: 216–222), Coale (1957, 1972), Dublin and Lotka (1925), Keyfitz and Flieger (1968), Lotka (1907), Pressat (2009: 318–328), Preston et al. (2001:138–170), and United Nations (1968). Again, as we noted in regard to time to stability, these approaches may generate different values of r than the method we used and, as such, yield different summary statistics in a comparison with the values of r generated by the CCR approach.
- 9.
Caswell (2001: 572) notes that demographers have addressed the “two-sex” problem since the 1940s, but that much of the literature focuses on the “consistency” problem: how to make estimates of intrinsic r based on male and female life tables agree. Although he notes that those studies that deal with demographic dynamics in any detail have focused on models lacking age structure, examples of studies using age can be found in Schoen (1988).
- 10.
References
Alho, J. (2008). Migration, fertility, and aging in stable populations. Demography, 45(3), 641–650.
Alho, J., & Spencer, B. (2005). Statistical demography and forecasting. New York: Springer.
Arthur, W. B. (1981). Why a population converges to stability. The American Mathematical Monthly, 86(8), 557–563.
Arthur, W. B., & Vaupel, J. (1984). Some relationships in population dynamics. Population Index, 50(2), 214–226.
Bacaër, N. (2011). A short history of population dynamics. Dordrecht: Springer.
Barclay, R. (1958). Techniques of population analysis. New York: Wiley.
Bennett, N., & Horuchi, S. (1984). Mortality estimation from registered deaths in less developed countries. Demography, 21(2), 217–233.
Caswell, H. (2001). Matrix population models: Construction, analysis, and interpretation (2nd ed.). Sunderland: Sinauer Associates, Inc.
Coale, A. J. (1957). A new method for calculating Lotka’s r – The intrinsic rate of growth in a stable population. Population Studies, 11, 92–94.
Coale, A. J. (1972). The growth and structure of human populations: A mathematical investigation. Princeton: Princeton University Press.
Coale, A. J., & Demeny, P. (1966). Regional model life tables and stable populations. Princeton: Princeton University Press.
Coale, A. J., & Trussell, J. (1974). Model fertility schedules: Variations in the age structure of childbearing in human populations. Population Index, 40(2), 185–258.
Cohen, J. (1979a). Ergodic theorems in demography. Bulletin of the American Mathematical Society, 1(2), 275–295.
Cohen, J. (1979b). The cumulative distance from an observed to a stable population age structure. SIAM Journal of Applied Mathematics, 36, 169–175.
Dublin, L., & Lotka, A. (1925). On the true rate of natural increase: As exemplified by the population of the United States, 1920. Journal of the American Statistical Association, 20, 305–339.
Espenshade, T. (1986). Population dynamics with immigration and low fertility. Population and Development Review, 12, 248–261.
Espenshade, T., Bouvier, L., & Arthur, W. B. (1982). Immigration and the stable population model. Demography, 19, 125–133.
Hamilton, C. H., & Perry, J. (1962). A short method for projecting population by age from one decennial census to another. Social Forces, 41, 163–170.
Hardy, G. F., & Wyatt, F. B. (1911). Report of the actuaries in relation to the scheme of insurance against sickness, disablement, &c., embodied in the national insurance bill. Journal of the Institute of Actuaries XLV, 406–443.
Hobbs, F. (2004). Age and sex composition. In J. Siegel, & D. A. Swanson (Eds.), The methods and materials of demography (2nd ed., pp. 125–173). San Diego: Elsevier Academic Press.
Keyfitz, N. (1968). Introduction to the mathematics of population. Reading: Addison-Wesley.
Keyfitz, N. (1974). A general condition for stability in demographic processes. Canadian Studies in Population, 1, 29–35.
Keyfitz, N. (1977). Introduction to the mathematics of population. New York: Addison-Wesley.
Keyfitz, N. (1980). Multistate demography and its data: A comment. Environment and Planning A, 12, 615–622.
Keyfitz, N., & Flieger, W. (1968). World population: An analysis of vital data. Chicago: University of Chicago Press.
Kim, Y., & Schoen, R. (1993a). On the intrinsic force of convergence to stability. Mathematical Population Studies, 4(2), 89–102.
Kim, Y., & Schoen, R. (1993b). Crossovers that link populations with the same vital rates. Mathematical Population Studies, 4(1), 1–19.
Kim, Y., & Sykes, Z. (1976). An experimental study of weak ergodicity in human populations. Theoretical Population Biology, 10, 150–172.
Land, K. (1986). Methods for national population forecasts: A review. Journal of the American Statistical Association, 81, 888–901.
Le Bras, H. (2008). The nature of demography. Princeton: Princeton University Press.
Liaw, K. L. (1980). Multistate dynamics: The convergence of an age-by-region population system. Environment and Planning A, 12, 589–613.
Lotka, A. J. (1907). Relation between birth rates and death rates. Science (New Series), 26(653), 21–22.
McCann, J. (1973). A more accurate short method of approximating Lotka’s r. Demography, 10(4), 567–570.
Mitra, S., & Cerone, P. (1986). Migration and stability. Genus, 42(1-2), 1–12.
Nair, S. B., & Nair, P. S. (2010). Momentum of population growth in India. In S. Nangia, N. C. Jana, & R. B. Bhagat, (Eds.), State of natural and human resources in India, Part 2 (pp. 399–424). New Delhi: Concept Publishing Company, Ltd.
Pollak, R. (1986). A re-formulation of the two-sex problem. Demography, 23, 247–259.
Pollard, A., Yusuf, F., & Pollard, G. (1974). Demographic techniques (3rd ed.). New York: Pergamon Press.
Popoff, C., & Judson, D. (2004). Some methods of estimation for statistically underdeveloped areas. In J. Siegel & D. A. Swanson (Eds.), The methods and materials of demography (2nd ed., pp. 603–641). San Diego: Elsevier Academic Press.
Pressat, R. (2009). Demographic analysis: Projections on natality, fertility and replacement (2nd Paperback Printing). New Brunswick: Aldine Transaction.
Preston, S. (1986). The relation between actual and intrinsic growth rates. Population Studies, 40(3), 343–351.
Preston, S., & Coale, A. J. (1982). Age structure, growth, attrition, and accession: A new synthesis. Population Index, 48, 217–259.
Preston, S., Heuveline, P., & Guillot, M. (2001). Demography: Measuring and modeling population processes. Malden: Blackwell Publishing.
Rogers, A. (1985). Regional population projection models (Vol. 4, Scientific Geography Series). Beverly Hills: Sage Publications.
Rogers, A. (1995). Multiregional demography: Principles, methods, and extensions. New York: Wiley.
Rogers, A., Little, J., & Raymer, J. (2010). The indirect estimation of migration. Dordrecht: Springer.
Rogers-Bennett, L., & Leaf, R. (2006). Elasticity analysis of size-based red and white abalone matrix models: Management and conservation. Ecological Applications, 16(1), 213–224.
Schoen, R. (1988). Modeling multigroup populations. New York: Plenum Press.
Schoen, R. (2006). Dynamic population models. Dordrecht: Springer.
Schoen, R., & Kim, Y. (1991). Movement toward stability as a fundamental principle of population dynamics. Demography, 28(3), 455–466.
Sivamurthy, M. (1982). Growth and structure of human population in the presence of migration. London: Academic.
Smith, S., Tayman, J., & Swanson, D. A. (2013). A practitioner’s guide to state and local population projections. Dordrecht: Springer.
Stubben, C., & Milligan, B. (2007). Estimating and analyzing demographic models using the popbio package in R. Journal of Statistical Software, 22(11), 1–23.
Swanson, D. A., & Tayman, J. (2013). Subnational population estimates. Dordrecht: Springer.
Swanson, D. A., & Tayman, J. (2014). Measuring uncertainty in population forecasts: A new approach (pp. 203–215). In M. Marsili & G. Capacci (Eds.), Proceedings of the 6th EUROSTAT/UNECE work session on demographic projections National Institute of Statistics, Rome, Italy.
Swanson, D. A., & Tedrow, L. (2012). Using cohort change ratios to estimate life expectancy in populations with negligible migration: A new approach. Canadian Studies in Population, 39, 83–90.
Swanson, D. A., & Tedrow, L. (2013). Exploring stable population concepts from the perspective of cohort change ratios. The Open Demography Journal, 6, 1–17.
Swanson, D. A., Schlottmann, A., & Schmidt, R. (2010). Forecasting the population of census tracts by age and sex: An example of the Hamilton-Perry method in action. Population Research and Policy Review, 29(1), 47–63.
Sykes, Z. M. (1969). Stochastic versions of the matrix model of population dynamics. Journal of the American Statistical Association, 64(325), 111–130.
Tuljapurkar, S. (1982). Why use population entropy? It determines the rate of convergence. Journal of Mathematical Biology, 13, 325–337.
United Nations. (1968). The concept of a stable population: Applications to the study of populations with incomplete demographic statistics (Population Studies No. 39). New York: United Nations, Department of Economic and Social Affairs.
United Nations. (2002). Methods for estimating adult mortality. New York: Population Division, United Nations.
United Nations. (2008). 2006 Demographic yearbook. New York: Department of Economic and Social Affairs, United Nations.
Vaupel, J., & Yashin, A. (1985). Heterogeneity’s ruses: Some surprising effects of selection on population dynamics. The American Statistician, 39(August), 176–185.
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Swanson, D.A., Tedrow, L.M., Baker, J. (2016). Exploring Stable Population Concepts from the Perspective of Cohort Change Ratios: Estimating the Time to Stability and Intrinsic r from Initial Information and Components of Change. In: Schoen, R. (eds) Dynamic Demographic Analysis. The Springer Series on Demographic Methods and Population Analysis, vol 39. Springer, Cham. https://doi.org/10.1007/978-3-319-26603-9_12
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