Bayesian Ridge Estimation of Age-Period-Cohort Models

  • Minle XuEmail author
  • Daniel A. Powers
Part of the The Springer Series on Demographic Methods and Population Analysis book series (PSDE, volume 39)


Age-Period-Cohort (APC) analysis offers a framework to study trends in the three temporal dimensions underlying age by period tables. However, the perfect linear relationship among age, period, and cohort leads to a well-known identification issue due perfect colinearity from the identity Cohort = Period − Age. A number of methods have been proposed to deal with this identification issue, e.g., the intrinsic estimator (IE), which may be viewed as a limiting form of ridge regression. Bayesian regression offers an alternative approach to modeling tabular age, period, cohort data. This study views the ridge estimator from a Bayesian perspective by introducing prior distributions for the ridge parameters, which permits these parameters to be estimated jointly with the substantive parameters rather than being assigned (and fixed) a-priori. Results show that a Bayesian ridge model with a common prior for the ridge parameter yields estimated age, period, and cohort effects similar to those based on the intrinsic estimator and to those based on a conventional ridge estimator with a shrinkage penalty obtained from cross-validation. The performance of Bayesian models with distinctive priors for the ridge parameters of age, period, and cohort effects is, however, affected by the choice of prior distributions. Further investigation of the influence of the choice of prior distributions is therefore warranted.


Age Period and cohort analysis Ridge regression Bayesian analysis 


  1. Arraiz, G. A., Wigle, D. T., & Mao, Y. (1990). Is cervical cancer increasing among young women in Canada? Canadian Journal of Public Health, 81, 396–397.Google Scholar
  2. Baker, A., & Bray, I. (2005). Bayesian projections: What are the effects of excluding data from younger age groups? American Journal of Epidemiology, 162, 798–805.CrossRefGoogle Scholar
  3. Berzuini, C., Clayton, D., & Bernardinelli, L. (1994). Bayesian inference on the Lexis diagram. Bulletin of the International Statistical Institute, 55, 149–164.Google Scholar
  4. Browning, M., Crawford, I., & Knoef, M. (2012). The age-period cohort problem: Set identification and point identification (CEMMAP working paper CWP02/12). Retrieved from
  5. Casella, G., & George, E. I. (1992). Explaining the Gibbs sampler. The American Statistician, 46, 167–174.Google Scholar
  6. Congdon, P. (2006). Bayesian statistical modelling (Wiley series in probability and statistics). doi: 10.1002/9780470035948.
  7. Draper, N. R., & Smith, H. (1981). Applied regression analysis (2nd ed.). New York: Wiley.Google Scholar
  8. Efron, B., Hastie, T., Johnstone, I., & Tibshirani, R. (2004). Least angle regression. The Annals of Statistics, 32(2), 407–499.Google Scholar
  9. Fienberg, S. E., & Mason, W. M. (1979). Identification and estimation of Age-Period-Cohort models in the analysis of discrete archival data. Sociological Methodology, 10, 1–67. doi: 10.2307/270764.CrossRefGoogle Scholar
  10. Fu, W. J. (2000). Ridge estimator in singular design with application to age-period-cohort analysis of disease rates. Communications in Statistics Theory and Methods, 29, 263–278. doi: 10.1080/03610920008832483.CrossRefGoogle Scholar
  11. Fu, W. J., & Hall, P. (2006). Asymptotic properties of estimators in age-period-cohort analysis. Statistics and Probability Letters, 76, 1925–1929. doi: 10.1016/j.spl.2006.04.051.CrossRefGoogle Scholar
  12. Fu, W. J., Hall, P., & Rohan, T. (2003). Age-period-cohort analysis: Structure of estimators, estimability, sensitivity and asymptotics. Technical Report, Department of Epidemiology, Michigan State University, East Lansing.Google Scholar
  13. Gelman, A., Carlin, J. B., Stern, H. S., Runson, D. B., Vehtari, A., & Rubin, D. B. (2013). Bayesian data analysis (3rd ed.). Boca Raton: Chapman and Hall/CRC.Google Scholar
  14. Geman, S., & Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6, 721–741.CrossRefGoogle Scholar
  15. Glenn, N. D. (1976). Cohort analysts’ futile quest: Statistical attempts to separate age, period and cohort effects. American Sociological Review, 41, 900–904.CrossRefGoogle Scholar
  16. Glenn, N. D. (2005). Cohort analysis (2nd ed.). Thousand Oaks: Sage.CrossRefGoogle Scholar
  17. Golub, G. H., Heath, M., & Wahba, G. (1979). Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics, 21, 215–223.CrossRefGoogle Scholar
  18. Hoerl, A. E. (1962). Application of ridge analysis to regression problems. Chemical Engineering Progress, 58, 54–59.Google Scholar
  19. Hoerl, A. E., & Kennard, R. W. (1970). Ridge regression: Biased estimation for non-orthogonal problems. Technometrics, 12, 55–67.CrossRefGoogle Scholar
  20. Hsiang, T. C. (1975). A Bayesian view on ridge regression. The Statistician, 24, 267–268. doi: 10.2307/2987923.CrossRefGoogle Scholar
  21. Keyes, K. M., Utz, R. L., Robinson, W., & Li, G. (2010). What is a cohort effect? Comparison of three statistical methods for modeling cohort effects in obesity prevalence in the United States, 1971–2006. Social Science and Medicine, 70, 1100–1108.CrossRefGoogle Scholar
  22. Knorr-Held, L., & Rainer, E. (2001). Projections of lung cancer mortality in West Germany: A case study in Bayesian prediction. Biostatistics, 2, 109–129.CrossRefGoogle Scholar
  23. Kupper, J. J., & Janis, J. M. (1980). The multiple classification model in age, period, and cohort analysis: Theoretical considerations (Institute of Statistics Mimeo No. 1311). Chapel Hill: Department of Biostatistics University of North Carolina.Google Scholar
  24. Kupper, J. J., Janis, J. M., Karmous, A., & Greenberg, B. G. (1985). Statistical age-period-cohort analysis: A review and critique. Journal of Chronic Disease, 38, 811–830.CrossRefGoogle Scholar
  25. Marquardt, D. W. (1970). Generalized inverses, ridge regression, biased linear estimation, and nonlinear estimation. Technometrics, 12, 591–612. doi: 10.2307/1267205.CrossRefGoogle Scholar
  26. Mason, W. M., & Wolfinger, N. H. (2001). Cohort analysis. International Encyclopedia of the Social and Behavioral Sciences, 2189–2194. doi: 10.1016/b0-08-043076-7/00401-0.
  27. Mason, K. O., Mason, W. M., Winsborough, H. H., & Poole, W. K. (1973). Some methodological issues in cohort analysis of archival data. American Sociological Review, 38, 242–258. doi: 10.2307/2094398.CrossRefGoogle Scholar
  28. Mosteller, F., & Tukey, J. W. (1977). Data analysis and regression. Reading: Addison-Wesley.Google Scholar
  29. O’Brien, R. M., Hudson, K., & Stockard, J. (2008). A mixed model estimation of age, period, and cohort effects. Sociological Methods & Research, 36, 402–428.CrossRefGoogle Scholar
  30. Plummer, M. (2003). JAGS: A program for analysis of Bayesian graphical models using Gibbs sampling. Proceedings of the 3rd International Workshop on Distributed Statistical Computing. Retrieved from
  31. Plummer, M. (2014). rjags: Bayesian graphical models using MCMC. R package version 3-13.
  32. R Core Team. (2013). R: A language and environment for statistical computing. R Foundation for Statistical Computing, Vienna, Austria. URL
  33. Schmid, V. J., & Held, L. (2007). Bayesian age-period-cohort modeling and prediction – BAMP. Journal of Statistical Software, 21(8), 1–15.CrossRefGoogle Scholar
  34. Suzuki, E. (2012). Time changes, so do people. Social Science and Medicine, 75, 452–456.CrossRefGoogle Scholar
  35. Tibshirani, R. (1996). Regression shrinkage and selection via the Lasso. Journal of the Royal Statistical Society, Series B, 58(1), 267–288.Google Scholar
  36. Tu, Y. K., Smith, G. D., & Gilthorpe, M. S. (2011). A new approach to age-period-cohort analysis using partial least squares: The trend in blood pressure in Glasgow Alumni Cohort. Plos One, 6(4), e19401. 1371/journal.pone. 001901.CrossRefGoogle Scholar
  37. Tu, Y. K., Kramer, N., & Lee, W. (2013). Addressing the identification problem in age-period-cohort analysis: A tutorial on the use of partial least squares and principle components analysis. Epidemiology, 23, 583–593.CrossRefGoogle Scholar
  38. Tukey, J. W. (1977). Exploratory data analysis. Reading: Addison-Wesley.Google Scholar
  39. Vizcaino, A. P., Moreno, V., Bosch, F. X., Munoz, N., Barros-Dios, X. M., & Parkin, D. M. (1998). International trends in the incidence of cervical cancer I: Adenocarcinoma and adenosquamous cell carcinomas. International Journal of Cancer, 75, 536–545.CrossRefGoogle Scholar
  40. Yang, Y., & Land, K. C. (2008). Age-period-cohort analysis of repeated cross-section surveys: Fixed or random effects? Sociological Methods and Research, 36, 297–326. doi: 10.1177/0049124106292360.CrossRefGoogle Scholar
  41. Yang, Y., & Land, K. C. (2013). Age-period-cohort analysis. Chapman & Hall/CRC Interdisciplinary Statistics Series. doi: 10.1201/b13902.
  42. Yang, Y., Fu, W. J., & Land, K. C. (2004). A methodological comparison of age-period-cohort models: The intrinsic estimator and conventional generalized linear models. Sociological Methodology, 34, 75–110. doi: 10.1111/j.0081-1750.2004.00148.x.CrossRefGoogle Scholar
  43. Yang, Y., Schulehoffer-Wohl, S., Fu, W. J., & Land, K. C. (2008). The intrinsic estimator for age-period-cohort analysis: What it is and how to use it. American Journal of Sociology, 113, 1697–1736.CrossRefGoogle Scholar
  44. Zheng, T., Hofford, T. R., Ma, Z., Chen, Y., Liu, W., Ward, B. A., & Boyle, P. (1996). The continuing increase in adenocarcinoma of the uterine cervix: A birth cohort phenomenon. International Journal of Epidemiology, 25, 252–258.CrossRefGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Sociology and Population Research CenterUniversity of Texas at AustinAustinUSA

Personalised recommendations