, Volume 56, Issue 3, pp 1131–1159 | Cite as

A General Age-Specific Mortality Model With an Example Indexed by Child Mortality or Both Child and Adult Mortality

  • Samuel J. ClarkEmail author


The majority of countries in Africa and nearly one-third of all countries require mortality models to infer the complete age schedules of mortality that are required to conduct population estimates, projections/forecasts, and other tasks in demography and epidemiology. Models that relate child mortality to mortality at other ages are important because almost all countries have measures of child mortality. A general, parameterizable component model (SVD-Comp) of mortality is defined using the singular value decomposition and calibrated to the relationship between child or child/adult mortality and mortality at other ages in the observed mortality schedules of the Human Mortality Database. Cross-validation is used to validate the model, and the predictive performance of the model is compared with that of the log-quadratic (Log-Quad) model, which is designed to do the same thing. Prediction and cross-validation tests indicate that the child mortality–calibrated SVD-Comp is able to accurately represent the observed mortality schedules in the Human Mortality Database, is robust to the selection of mortality schedules used for calibration, and performs better than the Log-Quad model. The child mortality–calibrated SVD-Comp can be used where and when child mortality is available but mortality at other ages is unknown.


Mortality Model SVD HMD SVD-Comp 



This work was supported in part by Grants R01 HD086227 and R01 HD054511 from the Eunice Kennedy Shriver National Institute of Child Health and Human Development (NICHD). The funder had no part in the design, execution, or interpretation of the work. Tables of regression coefficients were formatted using the LaTeX package stargazer (Hlavac 2015).

Supplementary material

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ESM 1 (PDF 794 kb)
13524_2019_785_MOESM2_ESM.pdf (12.2 mb)
ESM 2 (PDF 12.1 mb)


  1. Alexander, M., Zagheni, E., & Barbieri, M. (2017). A flexible Bayesian model for estimating subnational mortality. Demography, 54, 2025–2041.CrossRefGoogle Scholar
  2. Bell, W. R. (1997). Comparing and assessing time series methods for forecasting age-specific fertility and mortality rates. Journal of Official Statistics, 13, 279–303.Google Scholar
  3. Bourgeois-Pichat, J. (1962). Factor analysis and sex-age-specific death rates: A contribution to the study of the dimensions of mortality. United Nations Population Bulletin, 6, 147–201.Google Scholar
  4. Bourgeois-Pichat, J. (1990). Application de l’analyse factorielle à l’étude de la mortalitié [Application of factor analysis to the study of mortality]. Population (French ed.), 45, 773–802.CrossRefGoogle Scholar
  5. Bozik, J. E., & Bell, W. R. (1987). Forecasting age specific fertility using principal components. In Proceedings of the American Statistical Association, Social Statistics Section (pp. 396–401). Alexandria, VA: American Statistical Association.Google Scholar
  6. Brass, W. (1971). On the scale of mortality. In W. Brass (Ed.), Biological aspects of demography (pp. 69–110). London, UK: Taylor and Francis.Google Scholar
  7. Carter, L. R., & Lee, R. D. (1986). Joint forecasts of U.S. marital fertility, nuptiality, births, and marriages using time series models. Journal of the American Statistical Association, 81, 902–911.CrossRefGoogle Scholar
  8. Clark, S. J. (2001). An investigation into the impact of HIV on population dynamics in Africa (Doctoral dissertation). University of Pennsylvania, Philadelphia, PA. Retrieved from, and available at
  9. Clark, S. J. (2015). A singular value decomposition-based factorization and parsimonious component model of demographic quantities correlated by age: Predicting complete demographic age schedules with few parameters (Working paper). Retrieved from
  10. Clark, S. J., Jasseh, M., Punpuing, S., Zulu, E., Bawah, A., & Sankoh, O. (2009, May). INDEPTH model life tables 2.0. Paper presented at the annual meeting of the Population Association of America, Detroit, MI.Google Scholar
  11. Clark, S. J., & Sharrow, D. J. (2011a, April). Contemporary model life tables for developed countries: An application of model-based clustering. Paper presented at the annual meeting of the Population Association of America. Washington, DC.Google Scholar
  12. Clark, S. J., & Sharrow, D. J. (2011b). Contemporary model life tables for developed countries: An application of model-based clustering (Working Paper No. 107). Seattle: University of Washington Center for Statistics and the Social Sciences. Retrieved from
  13. Coale, A. J., & Demeny, P. (1966). Regional model life tables and stable populations. Princeton, NJ: Princeton University Press.Google Scholar
  14. Coale, A. J., & Trussell, T. J. (1974). Model fertility schedules: Variations in the age structure of childbearing in human populations. Population Index, 40, 185–258.CrossRefGoogle Scholar
  15. Fosdick, B. K., & Hoff, P. D. (2012). Separable factor analysis with applications to mortality data. Annals of Applied Statistics, 8, 120–147.CrossRefGoogle Scholar
  16. Golub, G. H., Hoffman, A., & Stewart, G. W. (1987). A generalization of the Eckart-Young-Mirsky matrix approximation theorem. Linear Algebra and Its Applications, 88–89, 317–327.CrossRefGoogle Scholar
  17. Gompertz, B. (1825). On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Philosophical Transactions of the Royal Society of London, 115, 513–583.CrossRefGoogle Scholar
  18. Good, I. J. (1969). Some applications of the singular decomposition of a matrix. Technometrics, 11, 823–831.CrossRefGoogle Scholar
  19. Heligman, L., & Pollard, J. H. (1980). The age pattern of mortality. Journal of the Institute of Actuaries, 107, 49–80.CrossRefGoogle Scholar
  20. Hlavac, M. (2015). stargazer: Well-formatted regression and summary statistics tables [r package version 5.2]. Cambridge, MA: Harvard University. Retrieved from
  21. INDEPTH Network. (2002). INDEPTH mortality patterns for Africa. In Population and health in developing countries (Vol. 1, pp. 83–128). Ottawa, Canada: International Development Research Centre.Google Scholar
  22. Ledermann, S. (1969). Nouvelles tables-types de mortalité [New standard mortality tables] (Travaux et Documents No. 53, Institut national d’études démographiques). Paris: Presses Universitaires de France.Google Scholar
  23. Ledermann, S., & Breas, J. (1959). Les dimensions de la mortalité [The dimensions of mortality]. Population (French ed.), 14, 637–682.Google Scholar
  24. Lee, R. D. (1993). Modeling and forecasting the time series of U.S. fertility: Age distribution, range, and ultimate level. International Journal of Forecasting, 9, 187–202.CrossRefGoogle Scholar
  25. Lee, R. D., & Carter, L. R. (1992). Modeling and forecasting U.S. mortality. Journal of the American Statistical Association, 87, 659–671.Google Scholar
  26. Li, N. (2015). Estimating life tables for developing countries (Technical Paper No. 2014/4). New York, NY: United Nations, Department of Economic and Social Affairs, Population Division. Retrieved from
  27. Li, N., & Gerland, P. (2011, April). Modifying the Lee-Carter method to project mortality changes up to 2100. Paper presented at the 2011 annual meeting of the Population Association of America, Washington, DC.Google Scholar
  28. Li, T., & Anderson, J. J. (2009). The vitality model: A way to understand population survival and demographic heterogeneity. Theoretical Population Biology, 76, 118–131.CrossRefGoogle Scholar
  29. Makeham, W. M. (1860). On the law of mortality and the construction of annuity tables. Assurance Magazine, and Journal of the Institute of Actuaries, 8, 301–310.CrossRefGoogle Scholar
  30. Max Planck Institute for Demographic Research, University of California, Berkeley, & Institut d’Études Démographiques (INED). (n.d.). Human life table database [Data set]. Retrieved from
  31. Murray, C. J., Ferguson, B. D., Lopez, A. D., Guillot, M., Salomon, J. A., & Ahmad, O. (2003). Modified logit life table system: Principles, empirical validation, and application. Population Studies, 57, 165–182.CrossRefGoogle Scholar
  32. R Foundation for Statistical Computing. (2016). The R project for statistical computing. Retrieved from http://www.r-project.orgGoogle Scholar
  33. Sharrow, D., Clark, S. J., Collinson, M., Kahn, K., & Tollman, S. (2013). The age pattern of increases in mortality affected by HIV: Bayesian fit of the Heligman-Pollard model to data from the Agincourt HDSS field site in rural northeast South Africa. Demographic Research, 29, 1039–1096. CrossRefGoogle Scholar
  34. Sharrow, D. J., Clark, S. J., & Raftery, A. E. (2014). Modeling age-specific mortality for countries with generalized HIV epidemics. PloS ONE, 9(5), e96447.
  35. Stewart, G. W. (1993). On the early history of the singular value decomposition. SIAM Review, 35, 551–566.CrossRefGoogle Scholar
  36. Strang, G. (2009). Introduction to linear algebra (4th ed.). Wellesley, MA: Wellesley-Cambridge Press.Google Scholar
  37. United Nations. (1955). Age and sex patterns of mortality: Model life-tables for under-developed countries (Population Studies No. 22). New York, NY: United Nations, Department of Economic and Social Affairs, Population Division.Google Scholar
  38. United Nations. (1982). Model life tables for developing countries. (Population Studies No. 77). New York, NY: United Nations, Department of Economic and Social Affairs, Population Division.Google Scholar
  39. United Nations. (2015a). World population prospects: The 2015 revision (DVD ed.). New York, NY: United Nations, Department of Economic and Social Affairs, Population Division.Google Scholar
  40. United Nations. (2015b). World population prospects: The 2015 revision. New York, NY: United Nations, Department of Economic and Social Affairs, Population Division.Google Scholar
  41. United Nations. (2015c). World population prospects: The 2015 revision, methodology of the United Nations population estimates and projections (Working Paper No. ESA/P/WP.242). New York, NY: United Nations, Department of Economic and Social Affairs, Population Division.Google Scholar
  42. University of California, Berkeley, & Max Planck Institute for Demographic Research. (n.d.). Human Mortality Database [Data set]. Available from
  43. Wang, H., Dwyer-Lindgren, L., Lofgren, K. T., Rajaratnam, J. K., Marcus, J. R., Levin-Rector, A., . . . Murray, C. J. L. (2013). Age-specific and sex-specific mortality in 187 countries, 1970–2010: A systematic analysis for the Global Burden of Disease Study 2010. Lancet, 380, 2071–2094.Google Scholar
  44. Wilmoth, J., Vallin, J., & Caselli, G. (1989). Quand certaines générations ont une mortalité différente de celle que l’on pourrait attendre [When some generations have different mortality than expected]. Population (French ed.), 44, 335–376.Google Scholar
  45. Wilmoth, J., Zureick, S., Canudas-Romo, V., Inoue, M., & Sawyer, C. (2012). A flexible two-dimensional mortality model for use in indirect estimation. Population Studies, 66, 1–28.CrossRefGoogle Scholar
  46. Wilmoth, J. R. (1988). On the statistical analysis of large arrays of demographic rates (Doctoral dissertation). Department of Statistics, Princeton University, Princeton, NJ.Google Scholar
  47. Wilmoth, J. R. (1990). Variation in vital rates by age, period, and cohort. Sociological Methodology, 20, 295–335.CrossRefGoogle Scholar
  48. Wilmoth, J. R., & Caselli, G. (1987). A simple model for the statistical analysis of large arrays of mortality data: Rectangular vs. diagonal structure (IIASA Working Paper WP-87-058). Laxenburg, Austria: International Institute for Applied Systems Analysis.Google Scholar
  49. World Health Organization. (n.d.). Global Health Observatory data repository [Data set]. Retrieved from
  50. Zaba, B. (1979). The four-parameter logit life table system. Population Studies, 33, 79–100.CrossRefGoogle Scholar

Copyright information

© Population Association of America 2019

Authors and Affiliations

  1. 1.Department of SociologyThe Ohio State UniversityColumbusUSA
  2. 2.MRC/Wits Rural Public Health and Health Transitions Research Unit (Agincourt), School of Public Health, Faculty of Health SciencesUniversity of the WitwatersrandJohannesburgSouth Africa

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