Disentangling Age-Period-Cohort Effects: New Models, Methods, and Empirical Applications

1. Disentangling Age-Period-Cohort Effects: New Models, Methods, and Empirical Applications Kenneth C. Land, Duke University PRI Summer Methodology Workshop Presentation Pennsylvania State University June 16, 2008

2. Objectives of the Presentation • Briefly Review the Early Literature on Cohort Analysis and the Age-Period-Cohort (APC) Identification Problem • Describe Models, Methods, and Empirical Applications Recently Developed for APC Analysis in Three Research Designs: 1) APC Analysis of Age-by-Time Period Tables of Rates 2) APC Analysis of Microdata from Repeated Cross-Section Surveys 3) Cohort Analysis of Accelerated Longitudinal Panel Designs

3. Part I: The Early Literature on Cohort Analysis and the Age-Period-Cohort (APC) Identification Problem • Why cohort analysis? • See the abstract from Norman Ryder’s classic article: Ryder, Norman B. 1965. “The Cohort as A Concept in the Study of Social Change.” American Sociological Review 30:843-861.

4. Part I: The Early Literature on Cohort Analysis and the Age-Period-Cohort (APC) Identification Problem

5. Part I: The Early Literature on Cohort Analysis and the Age-Period-Cohort (APC) Identification Problem • And what is the APC identification problem? • See the abstract from the classic Mason et al. article: Mason, Karen Oppenheim, William M. Mason, H. H. Winsborough, W. Kenneth Poole. 1973. “Some Methodological Issues in Cohort Analysis of Archival Data.” American Sociological Review 38:242-258.

6. Part I: The Early Literature on Cohort Analysis and the Age-Period-Cohort (APC) Identification Problem

7. Part I: The Early Literature on Cohort Analysis and the Age-Period-Cohort (APC) Identification Problem • These two articles were particularly important in framing the literature on cohort analysis in sociology, demography, and the social sciences over the past five decades: • Ryder (1965) argued that cohort membership could be as important in determining behavior as other social structural features such as socioeconomic status. • Mason et al. (1973) specified the APC multiple classification /accounting model and defined the identification problem therein.

8. Part I: The Early Literature on Cohort Analysis and the Age-Period-Cohort (APC) Identification Problem • The Mason et al. (1973) article, in particular, spawned a large methodological literature, beginning with Norval Glenn’s (1976) critique: • Glenn, N. D. 1976. “Cohort Analysts’ Futile Quest: Statistical Attempts to Separate Age, Period, and Cohort Effects.” American Sociological Review, 41:900–905. and Mason et al.’s (1976) reply: • Mason, W. M., K. O. Mason, and H. H. Winsborough. 1976. “Reply to Glenn.” American Sociological Review, 41:904-905.

9. Part I: The Early Literature on Cohort Analysis and the Age-Period-Cohort (APC) Identification Problem • The Mason et al. reply continued with Bill Mason’s work with Stephen Fienberg: • Fienberg, Stephen E. and William M. Mason. 1978. "Identification and Estimation of Age-Period-Cohort Models in the Analysis of Discrete Archival Data." Sociological Methodology 8:1-67, which culminated in their 1985 edited volume: • Fienberg, Stephen E. and William M. Mason, Eds. 1985. Cohort Analysis in Social Research. New York: Springer-Verlag, a volume of the methodological literature on APC analysis in the social sciences as of about 25 years ago.

10. Part I: The Early Literature on Cohort Analysis and the Age-Period-Cohort (APC) Identification Problem • The critiques of new approaches also continued; see, e.g., the article applying a Bayesian statistics approach: • Saski, M., & Suzuki, T. 1987. “Changes in Religious Commitment in the United States, Holland, and Japan.” American Journal of Sociology, 92:1055–1076, and the critique: • Glenn, N. D. 1987. “A Caution About Mechanical Solutions to the Identification Problem in Cohort Analysis: A Comment on Sasaki and Suzuki.” American Journal of Sociology, 95:754–761.

11. Part I: The Early Literature on Cohort Analysis and the Age-Period-Cohort (APC) Identification Problem • Another approach, developed by Firebaugh (1989), is based on a decomposition of change over time into the relative contributions of intracohort aging and cohort replacement; see Danigelis, Hardy, and Cutler (2007) for a recent application. • Firebaugh, Glenn. 1989. “Methods for Estimating Cohort Replacement Effects.” Sociological Methodology 19:243-262. • Danigelis, Nicholas, Melissa Hardy, and Stephen J. Cutler. 2007. “Population Aging, Intracohort Aging, and Sociopolitical Attitudes.” American Sociological Review72:812-830.

12. Part I: The Early Literature on Cohort Analysis and the Age-Period-Cohort (APC) Identification Problem • This decomposition method, called for by Glenn (1977) and developed by Firebaugh, was critiqued by Rodgers (1990; with reply by Firebaugh (1990). And now Glenn (2005: 36) thinks neither this nor any similar approach to decomposition “is very helpful for understanding change.” • Firebaugh, Glenn. 1990. “Replacement Effects, Cohort and Otherwise: Response to Rodgers.” Sociological Methodology 20:439-446. • Glenn, Norval D. 1977 [2005] Cohort Analysis, [2nd edition]. Thousand Oaks, CA: Sage. • Rodgers, Willard L. 1990. “Interpreting the Components of Time Trends.” Sociological Methodology 20:421-438.

13. Part I: The Early Literature on Cohort Analysis and the Age-Period-Cohort (APC) Identification Problem • For additional material on these and related contributions to the literature on cohort analysis, see the following three recent reviews: • Mason, William M. and N. H. Wolfinger. 2002. “Cohort Analysis.” Pp. 151-228 in International Encyclopedia of the Social and Behavioral Sciences. New York: Elsevier. • Yang, Yang. 2006. “Age/Period/Cohort Distinctions.” Encyclopedia of Health and Aging, K.S. Markides (ed). Thousand Oaks, CA: Sage Publications.

14. Part I: The Early Literature on Cohort Analysis and the Age-Period-Cohort (APC) Identification Problem • Where does this literature on cohort analysis leave us today? • If a researcher has a temporally-ordered dataset and wants to tease out its age, period, and cohort components, how should he/she proceed? • Are there any methodological guidelines that can be recommended?

15. Part I: The Early Literature on Cohort Analysis and the Age-Period-Cohort (APC) Identification Problem • The problem with much of the extant literature is that there is a deficiency of useful guidelines on how to conduct an APC analysis. Rather, the literature often leads to the conclusion either that: • it is impossible to obtain meaningful estimates of the distinct contributions of age, time period, and cohort to the study of social change, or that: • the conduct of an APC analysis is an esoteric art that is best left to a few skilled methodologists.

16. Part I: The Early Literature on Cohort Analysis and the Age-Period-Cohort (APC) Identification Problem • My collaborators (Wenjiang Fu, Sam Schulhofer-Wohl, and Yang Yang) and I seek to redress this situation by focusing on recent methodological contributions to APC analysis that we and others have made for three relatively common research designs. • We think that: • developments in statistics over the past three decades (e.g., mixed (fixed and random) effects models, MCMC estimation of Bayesian models) can lead to better methods for APC analysis that can be applied by ordinary social scientists, and • this, in turn, can lead to the accumulation of more reliable knowledge about age, period, and cohort dynamics.

17. Part II: First Research Design: APC Analysis of Age-by-Time Period Tables of Rates or Proportions • Major References for Part II: • Fu, W. J. 2000. “Ridge Estimator in Singular Design with Application to Age-Period-Cohort Analysis of Disease Rates.” Communications in Statistics--Theory and Method 29:263-278. • Yang Yang, Wenjiang J. Fu, and Kenneth C. Land. 2004. “A Methodological Comparison of Age-Period-Cohort Models: The Intrinsic Estimator and Conventional Generalized Linear Models.” Sociological Methodology, 34:75-110. • Yang Yang, Sam Schulhofer-Wohl, Wenjiang J. Fu, and Kenneth C. Land. 2008. “The Intrinsic Estimator for Age-Period-Cohort Analysis: What It Is and How To Use It.” American Journal of Sociology,113(May). • Yang Yang. 2008. “Trends in U.S. Adult Chronic Disease Mortality, 1960-1999: Age, Period, and Cohort Variations.” Demography 45(May).

18. Part II: First Research Design: APC Analysis of Age-by-Time Period Tables of Rates or Proportions • Data Structure: Tabular Rate Data

19. Part II: First Research Design: APC Analysis of Age-by-Time Period Tables of Rates or Proportions • Example: Lung Cancer Death Rates for U.S. Adult Females: 1960 - 1999 Source: CDC/NCHS Multiple Cause of Death File

20. Part II: First Research Design: APC Accounting/Multiple Classification Model • The Algebra of the APC Identification Problem • Model Specification: (1) • Mijdenotes the observed occurrence/exposure rate of deaths for the i-th age group for i = 1,…,a age groups at the j-th time period for j = 1,…, p time periods of observed data • Dij denotes the number of deaths in the ij-th group, Pij denotes the size of the estimated population in the ij-th group • μdenotes the intercept or adjusted mean • αi denotes the i-th row age effect or the coefficient for the i-th age group • βj denotes the j-th column period effect or the coefficient for the j-th time period • γkdenotes the k-th cohort effect or the coefficient for the k-th cohort for k = 1,…,(a+p-1) cohorts, with k=a-i+j • εij denotes the random errors with expectation E(εij) = 0 • Fixed effect GLIM reparameterization: , or setting one of each of the categories as the reference group.

21. Part II: First Research Design: APC Accounting/Multiple Classification Model • The Algebra of the APC Identification Problem • Generalized Linear Models (GLIM): • Simple Linear Models where Yijis the expected outcome in cell (i, j) that is assumed to be normally distributed or equivalently the error term is assumed to be normally distributed with a mean of 0 and variance σ2; • Log-Linear Models log(Eij) = log(Pij) + μ + αi + βj + γk where Eij denotes the expected number of events in cell (i,j) that is assumed to be distributed as a Poisson variate, and log(Pij) is the log of the exposure Pij • Logistic Models where θijis the log odds of event and mij is the probability of event in cell (i,j).

22. Part II: First Research Design: APC Accounting/Multiple Classification Model • The Algebra of APC Identification Problem • Least-squares regression in matrix form: (2) • Identification Problem: or the solution to normal equation does not exist because the design matrix X is singular with 1-less than full column rank and (XTX)-1 does not exist due to: Period = Age + Cohort

23. Part II: First Research Design: APC Accounting/Multiple Classification Model • Conventional Solutions to APC Identification Problem • Constrained Coefficients GLIM estimator (CGLIM) • Impose one or more equality constraints on the coefficients of the coefficient vector in (2) in order to just-identify (one equality constraint) or over-identify (two or more constraints) the model; • Proxy variables approach • Use one or more proxy variables as surrogates for the age, period, or cohort coefficients (see O'Brien, R.M. 2000. "Age Period Cohort Characteristic Models." Social Science Research 29:123-139); • Nonlinear parametric (algebraic) transformation approach • Define a nonlinear parametric function of one of the age, period, or cohort variables so that its relationship to others is nonlinear. • References: • Fienberg and Mason (1985) • Yang, Yang. 2005. New Avenues for Cohort Analysis: Chapter 2. Ph.D. Dissertation. Duke University. [Proquest]

24. Part II: First Research Design: APC Accounting/Multiple Classification Model • Limitations of Conventional Solutions to APC Identification Problem • Proxy variables approach • the analyst does not want to assume that all of the variation associated with the A, P, or C dimensions is fully accounted for by a proxy variable; • Nonlinear parametric (algebraic) transformation approach • it may not be evident what nonlinear function should be defined for the effects of age, period, or cohort; • Constrained Coefficients GLIM estimator (CGLIM) • it is the most widely used of the three approaches, but suffers from some major problems summarized below.

25. Part II: First Research Design: APC Accounting/Multiple Classification Model • Limitations of Conventional Solutions to APC Identification Problem • Constrained Coefficients GLIM estimator (CGLIM) • the analyst desires to employ the flexibility of the APC accounting model with its individual effect coefficients for each of the A, P, or C categories; • the analyst needs to rely on prior or external information to find constraints that hardly exists or can be well verified; • different choices of identifying constraints can produce widely different estimates of patterns of change across the A, P, and C categories of the analysis; • all just-identified CGLIM models will produce the same levels of goodness-of-fit to the data, making it impossible to use model fit as the criterion for selecting the best constrained model. See, e.g., Yang et al. (2004) and Yang et al. (2006), for details.

26. Part II: First Research Design: APC Accounting/Multiple Classification Model • Guidelines for Estimating APC Models of Rates • Step 1: Descriptive data analyses using graphics • Step 2: Model fitting procedures • Objectives: • to provide qualitative understanding of patterns of age, or period, or cohort variations, or two-way age by period and age by cohort variations; • to ascertain whether the data are sufficiently well described by any single factor or two-way combination of the A, P, and C dimensions or if it is necessary to include all three.

27. Part II: First Research Design: APC Accounting/Multiple Classification Model • Step 1:Graphical analyses: example from Yang (2008)

28. Part II: First Research Design: APC Accounting/Multiple Classification Model • Step 1: Graphical analyses • As a first step in the analysis of a table of age-period-specific rates or age-cohort-specific, we recommend a graphical representation of the data such as the U.S. female lung cancer mortality rates shown in Figure 3 from Yang (2008). • If there are no cohort effects, then the curves of the age-specific rates should show parallel curvatures. But it can be seen that the curves of age-specific rates show substantial departure from this condition. • For example, the curve of age-specific rates for 1995-99 cuts cross a number of birth cohort curves, such as 1900, 1905, 1910, and 1920. Therefore, the shape of the period curve is affected by both varying age effects and cohort effects. The question of how these effects operate simultaneously to shift period curve motivates the use of statistical regression modeling.

29. Part II: First Research Design: APC Accounting/Multiple Classification Model • Step 2: Model fitting procedures • Examples from Yang et al. (2004) and Yang (2008)

30. Part II: First Research Design: APC Accounting/Multiple Classification Model • Step 2: Model fitting procedures • As a second step in model specification/estimation, we recommend the estimation of a sequence of nested log-linear models as illustrated in Tables 1 and 4 for analyses reported in Yang (2008). • These tables show goodness-of-fit statistics for six reduced log linear models: three gross effects models, namely, model A for age effects, model P for period effects, and model C for cohort effects; and three two-factor models, one for each of three possible pairs of effects, namely, AP, AC, and PC effects models. All of these models then are nested within a full APC model where all three factors are simultaneously controlled. • Goodness-of-fit statistics were calculated and used to select the best fitting models for male and female mortality data. Because likelihood ratio tests (Table 4) tend to favor models with a larger number of parameters, two most commonly used penalized-likelihood model selection criteria are reported in Table 1, namely, the Akaike information criterion (AIC) and the Bayesian information criterion (BIC), each of which adjust the impact of model dimensions on model deviances. • For the female lung cancer data, both the AIC and BIC statistics imply that the full APC models fit the data significantly better than any of the reduced models.

31. Part II: First Research Design: APC Accounting/Multiple Classification Model • Guidelines for Estimating APC Models of Rates • If the foregoing descriptive analyses suggest that only one or two of the A, P, and C dimensions is operative, then the analysis can proceed with a reduced model (2) that omits one or two dimensions and there is no identification problem. • If, however, these analyses suggest that all three dimensions are at work, thenYang et al. (2004, 2008) recommend: • Step 3: Apply the Intrinsic Estimator (IE).

32. Part II: First Research Design: APC Accounting/Multiple Classification Model • What is the Intrinsic Estimator (IE)? • It is a new method of estimation that yields a unique solution to the model (2) and is the unique estimable function of both the linear and nonlinear components of the APC model determined by the Moore-Penrose generalized inverse. It achieves model identification with minimal assumptions. • Why is the IE useful? • The basic idea of the IE is to remove the influence of the design matrix (which is fixed by the number of age and period groups and not related to Yij) on coefficient estimates. This produces estimates that have desirable statistical properties.

33. Part II: First Research Design: APC Accounting/Multiple Classification Model • The Intrinsic Estimator (IE): Algebraic Definition • The linear dependency between A, P, and C is mathematically equivalent to: (3) • The eigenvector B0 of eigenvalue of 0 is fixed by X:

34. Part II: First Research Design: APC Accounting/Multiple Classification Model • The Intrinsic Estimator (IE): Algebraic Definition/Geometric Representation • Parameter vector orthogonal decomposition: (4) (5) • , projection of b to the non-null space of X • t is a real number, tB0 is in the null space of X and represents trends of linear constraints – Different equality constraints used by CGLIM estimators, such as b1 and b2, yield different values of t.

35. Part II: First Research Design: APC Accounting/Multiple Classification Model • The Intrinsic Estimator (IE) Method: Algebraic Definition • From the infinite number of estimator of b in model (2): (6) • the IE estimates the parameter vector b0 corresponding to t = 0: (7) • The IE is the special estimator that uniquely determines the age, period, and cohort effects in the parameter subspace defined by b0 : (8)

36. Part II: First Research Design: APC Accounting/Multiple Classification Model The IE also can be viewed as a special form of principal components regression estimator that removes the influence of the null space of the design matrix X on the estimator: • (a)    the analyst computes the eigenvalues and eigenvectors (principal components) of the matrix XTX, • (b)   normalizes them to have unit length; • (c)    identifies the eigenvector B0 corresponding to the unique eigenvalue 0; • (d)   estimates a regression model with response vector Y and design matrix U whose column vectors are the principal components determined by the eigenvectors of non-zero eigenvalues, i.e., estimates a principal components regression model; and • (e)    then uses the orthonormal matrix of alleigenvectorsto transform the coefficients of the principal components regression model to the regression coefficients of the intrinsic estimator B.

37. Part II: First Research Design: APC Accounting/Multiple Classification Model • The Intrinsic Estimator (IE) Method • Desirable statistical properties (Yang et al. 2004): • Estimability The IE is an estimable function in the sense that it is invariable to the choice of linear constraints on b. • Unbiasedness For a fixed number of time periods of data, it is an unbiased estimator of the special parameterization (or linear function) b0 of b. • Relative efficiency For a fixed number of time periods of data, it has a smaller variance than any CGLIM estimators. • Asymptotic consistency Under suitable regularity conditions on the error term process and a fixed set of age categories, the IE will converge asymptotically to the “true” parameters. • Monte Carlo Simulation Analysis • Demonstrated numerically the foregoing finite-time-period and asymptotic properties of the IE – Presented at 2007 Annual Meetings of ASA: Sociological Methodology Paper Session (Yang, Schulhofer-Wohl, and Land).

38. Part II: First Research Design: APC Accounting/Multiple Classification Model

39. Part II: First Research Design: APC Accounting/Multiple Classification Model

40. Part II: First Research Design: APC Accounting/Multiple Classification Model • The Intrinsic Estimator (IE) Method: Computation Software • The S-Plus/R program can be obtained by writing Wenjiang J. Fu at fuw@epi.msu.edu • Stata Ado Files • Typing “ssc install apc” or “net install apc” on the Stata 9.2 command line on any computer connected to the Internet • Download from the Statistical Software Components archive at http://ideas.repec.org/s/boc/bocode.html • Uses much the same syntax as Stata’s glm command for generalized linear models. For example, to fit a log-linear model, use command: > apc_iey, exposure(exp) family(poisson) link(log) age(a) period(t) cohort(c) for a dependent variable “y”, an exposure variable “exp”, an age variable “a”, a period variable “t”, and a cohort variable “c”. • See “help apc_ie” and “helpapc_cglim” for more detail. • An example of model estimates in Yang et al. (2004) is available at: http://home.uchicago.edu/~yangy/apc_sectionC

41. Part II: First Research Design: APC Accounting/Multiple Classification Model • Example: Intrinsic Estimates of Age, Period, and Cohort Effects of Lung Cancer Mortality by Sex (Yang 2008)

42. Part II: First Research Design: APC Accounting/Multiple Classification Model • The Intrinsic Estimator (IE): Conclusion • Is the Intrinsic Estimator a “final” or “universal” solution to the APC “conundrum” in the context of age-by-time period tables of rates? • No. There will never be such a solution. • But the IE has been shown to be a useful approach to the identification and estimation of the APC accounting model that • has desirable mathematical and statistical properties; and • has passed both case studies and simulation tests of model validation.

43. Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys • Major References for Part III: • Yang, Yang. 2006. “Bayesian Inference for Hierarchical Age-Period-Cohort Models of Repeated Cross-Section Survey Data.” Sociological Methodology 36:39-74. • Yang Yang and Kenneth C. Land. 2006. “A Mixed Models Approach to the Age-Period-Cohort Analysis of Repeated Cross-Section Surveys, With an Application to Data on Trends in Verbal Test Scores.” Sociological Methodology 36:75-98. • Yang Yang and Kenneth C. Land. 2008. ”Age-Period-Cohort Analysis of Repeated Cross-Section Surveys: Fixed or Random Effects?” Sociological Methods and Research 36(February):297-326. • Yang, Yang. 2008. “Social Inequalities in Happiness in the United States, 1972 to 2004: An Age-Period-Cohort Analysis.” American Sociological Review 73(April):204-226.

44. Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys • Data Structure: Individual-level Data in the Age-by-Period Array Period j Age i

45. Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys • Solution to the Identification Problem • Many researchers previously have assumed that the APC identification problem for age-by-time period tables of rates transfers over directly to this research design. • But note that this research design yields individual-level data, i.e., microdata on the ages and other characteristics of individuals in the samples. • Solution: Use of different temporal groupings for the A, P, and C dimensions breaks the linear dependency: • Single year of age • Time periods correspond to years in which the surveys are conducted • Cohorts can be defined either by five- or ten-year intervals that are conventional in demography or by application of a substantive classification (e.g., War babies, Baby Boomers, Baby Busters, etc.).

46. Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys • Two-way Cross-Classified Data Structure in the GSS: Number of Observations by Cohort and Period in the Verbal Ability Data (Yang and Land 2006)

47. Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys • This Data Structure illustrates that: • respondents are nested in and cross-classified simultaneously by the two higher-level social contexts defined by time period and birth cohort >>> so the basic idea here is to treat time periods and birth cohorts as contexts; • individual members of any birth cohort can be interviewed in multiples replications of the survey; and • individual respondents in any particular wave of the survey can be drawn from multiple birth cohorts.

48. Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys • Further Questions: • Is there evidence for clustering (correlation) of random errors, due to the fact that: • individuals surveyed in the same year may be subject to similar unmeasured events that influence their outcomes? • members of the same birth cohort may be subject to similar unmeasured events that influence their outcomes? • How can this random variability be modeled and explained?

49. Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys • Method • Hierarchical Age-Period-Cohort (HAPC) Models • Mixed (fixed and random) effects models or hierarchical linear models (HLM) • Cross-classified random effects model (CCREM) • Objective: Model the level-two heterogeneity to: • Assess the possibility that individuals within the same periods and cohorts could share unobserved random variance; • Explain the level-two variance by contextual characteristics of time periods and birth cohorts.

50. Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys • Illustrative Application • APC Analysis of General Social Survey (GSS) Data on Verbal Test Scores: 1974 – 2000 • The Initial Papers • Alwin, D. 1991. “Family of Origin and Cohort Differences in Verbal Ability.” American Sociological Review 56:625-38. • Glenn, N.D. 1994 “Television Watching, Newspaper Reading, and Cohort Differences in Verbal Ability.” Sociology of Education 67:216-30. • The debate in the American Sociological Review • Wilson, J.A. and W.R. Gove. 1999. "The Intercohort Decline in Verbal Ability: Does It Exist?" and reply to Glenn and Alwin & McCammon. ASR 64:253-266, 287-302. • Glenn, N.D. 1999. “Further Discussion of the Evidence for An Intercohort Decline in Education-Adjusted Vocabulary.” ASR 64:267-71. • Alwin, D.F. and R.J. McCammon. 1999. “Aging Versus Cohort Interpretations of Intercohort Differences in GSS Vocabulary Scores.” ASR 64:272-86.