A Smoothing Cohort Model in Age–Period–Cohort Analysis With Applications to Homicide Arrest Rates and Lung Cancer Mortality Rates

Abstract

This article considers the effects of age, period, and cohort in social studies and chronic disease epidemiology through age–period–cohort (APC) analysis. These factors are linearly dependent; thus, the multiple classification model, a regression model that takes these factors as covariates in APC analysis, suffers from an identifiability problem with multiple estimators. A data set of homicide arrest rates is used to illustrate the problem. A smoothing cohort model is proposed that allows flexible structure of the effects for age, period, and cohort and avoids the identifiability problem. Results are provided for the consistency of estimation of model intercept and age effects as the number of periods goes to infinity under a mild bounded cohort condition. This also leads to consistent estimation for period and cohort effects. Analyses of homicide arrest rate and lung cancer mortality rate data demonstrate that the smoothing cohort model yields unique parameter estimation with sensible trend interpretation.

Keywords

consistent estimation identifiability intrinsic estimator semiparametric singular design matrix spline smoothing

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References

Cheng, H. , L. Aylward , C. Beall , T.B. Starr , R.C. Brunet , G. Carrier , and E. Delzell. 2006. ``TCDD Exposure-Response Analysis and Risk Assessment.'' Risk Analysis 26:1059-71.

Clayton, D. and E. Schifflers. 1987. ``Models for Temporal Variation in Cancer Rates. II: Age-Period-Cohort Models.'' Statistics in Medicine 6:469-81.

Enting, I.G. , C.M. Trudinger , and D.M. Etheridge. 2006. ``Propagating Data Uncertainty Through Smoothing Spine Fits'' Tellus Series B: Chemical and Physical Methodology 58:305-309.

Fienberg, S.E. and W.M. Mason. 1985. ``Specification and Implementation of Age, Period and Cohort Models'' Pp. 44-88 in Cohort Analysis in Social Research: Beyond the Identification Problem, edited by W. Mason and S. E. Fienberg. New York: Springer.

Fu, W.J. 2000. ``Ridge Estimator in Singular Design With Applications to Age-Period-Cohort Analysis of Disease Rates.'' Communications in Statistics Theory and Method 29:263-78.

Fu, W.J. and P. Hall. 2006. ``Asymptotic Properties of Estimators in Age-Period-Cohort Analysis.'' Statistics and Probability Letters 76:1925-29.

Fu, W.J. , P. Hall , and T. Rohan. 2004. ``Age-Period-Cohort Analysis: Structure of Estimators, Estimability, Sensitivity and Asymptotics.'' Technical report, Department of Epidemiology, Michigan State University, East Lansing.

Glenn, N.D. 2003. ``Distinguishing Age, Period, and Cohort Effects '' Pp. 465-76 in Handbook of the Life Course, edited by J. T. Mortimer and M. J. Shanahan. New York: Kluwer Academic/Plenum.

Hastie, T. and R. Tibshirani. 1990. Generalized Additive Models. New York: Chapman and Hall.

10.

Heuer, C. 1997. ``Modeling of Time Trends and Interactions in Vital Rates Using Restricted Regression Splines.'' Biometrics 53:161-77.

11.

Holford, T.R. 1985. ``An Alternative Approach to Statistical Age-Period-Cohort Analysis.'' Journal of Chronic Disease 38:831-36.

12.

———. 1991. ``Understanding the Effects of Age, Period and Cohort on Incidence and Mortality Rates.'' Annual Review of Public Health 12:425-57.

13.

James, I.R. and M.R. Segal. 1982. ``On a Method of Mortality Analysis Incorporating Age-Year Interaction, With Application to Prostate Cancer Mortality .'' Biometrics 38:433-43.

14.

Kupper, L.L. , J.M. Janis , A. Karmous , and B.G. Greenberg. 1985. ``Statistical Age-Period-Cohort Analysis: A Review and Critique.'' Journal of Chronic Disease 38:811-30.

15.

Lee, W.C. and R.C. Lin. 1996. ``Autoregressive Age-Period-Cohort Models .'' Statistics in Medicine 15:273-81.

16.

Mason, K.O. , W.M. Mason , H.H. Winsborough , and W.K. Poole. 1973. ``Some Methodological Issues in the Cohort Analysis of Archival Data. American Sociological Review 38:242-58.

17.

McCullagh, P. and J.A. Nelder. 1991. Generalized Linear Models. 2d ed. New York: Chapman and Hall/CRC.

18.

Moolgavkar, S.H. , R.G. Stevens , and J.A.H. Lee. 1979. ``Effect of Age on Incidence of Breast Cancer in Females.'' Journal of National Cancer Institute 62:493-501.

19.

O'Brien, R.M. 2000. ``Age Period Cohort Characteristic Models.'' Social Science Research 29:123-39.

20.

Osmond, C. and M.J. Gardner. 1982. ``Age, Period and Cohort Models Applied to Cancer Mortality Rates.'' Statistics in Medicine 1:245-59.

21.

Pearl, J. 2000. Causality: Models, Reasoning, and Inference. Cambridge, UK: Cambridge University Press .

22.

Rogers, W.L. 1982. ``Estimable Functions in Age, Period and Cohort Effects .'' American Sociological Review 47:774-87.

23.

Sasieni, P.D. and J. Adams. 2000. ``Analysis of Cervical Cancer Mortality and Incidence Data From England and Wales: Evidence of a Beneficial Effect of Screening.'' Journal of Royal Statistical Society A 163:191-209.

24.

Shao, J. 1999. Mathematical Statistics. New York: Springer.

25.

Stevens, R.G. , S.H. Moolgavkar , and J.A.H. Lee. 1982. ``Temporal Trends in Breast Cancer .'' American Journal of Epidemiology 115:759-77.

26.

Stockard, J. and R.M. O'Brien . 2002. ``Cohort Effects on Suicide Rates: International Variations.'' American Sociological Review 67:854-72.

27.

Tirosh, S. , D. Van de Ville , and M. Unse. 2006. ``Polyharmonic Smoothing Splines and the Multidimensional Wiener Filtering of Fractal-Like Signals. '' IEEE Transactions on Image Processing 15:2616-30.

28.

Wilmoth, J.R. 1990. ``Variation in Vital Rates by Age, Period, and Cohort .'' Sociological Review 20:295-335.

29.

Winship, C. , and David J. Harding . 2008. ``A Mechanism-Based Approach to the Identification of Age—Period—Cohort Models.'' Sociological Methods & Research 36:362-401.

30.

Yang, Y. , W.J. Fu , and K. Land. 2004. ``A Methodological Comparison of Age-Period-Cohort Models: The Intrinsic Estimator and Conventional Generalized Linear Models .'' Sociological Methodology 34:75-110.