Age-Period-Cohort Analysis: New Models, Methods, and Empirical Analyses Kenneth C. Land, Ph.D. John Franklin Crowell Professor of Sociology and Demography Duke University Presentation Indiana University April 15, 2011 1 GUIDING PRINCIPLE FOR THIS WORK Famous quote from George E. P. Box, Emeritus Professor of Statistics, University of Wisconsin at Madison: “All statistical models are wrong, but some are useful.” Ken Land’s Version: “All statistical models are wrong, but some have better statistical properties than others – which may make them useful.” 2 Organization Briefly Review the Early Literature on Cohort Analysis and the Age-Period-Cohort (APC) Identification Problem Describe Models & Methods Developed Recently for APC Analysis for Three Research Designs, with Empirical Applications: 1) APC Analysis of Age-by-Time Period Tables of Rates 2) APC Analysis of Microdata from Repeated CrossSection Surveys 3) Cohort Analysis of Accelerated Longitudinal Panel Designs Conclusion 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 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 defining volume on the methodological literature on APC analysis in the social sciences as of about 25 10 years ago. Part I: The Early Literature on Cohort Analysis and the Age-Period-Cohort (APC) Identification Problem New approaches and critiques thereof continued over the years; see, e.g., an 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 For additional material on these and related contributions to the literature on cohort analysis, see the following three 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. Glenn, Norval D. 2005. Cohort Analysis. 2nd edition. Thousand Oaks: Sage. Yang, Yang. 2007. “Age/Period/Cohort Distinctions.” Pp. 20-22 in Encyclopedia of Health and Aging. Kyriakos S. Markides (ed). Sage Publications. 12 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? 13 Part I: The Early Literature on Cohort Analysis and the Age-Period-Cohort (APC) Identification Problem There are some guidelines – and cautions, e.g., in Glenn (2005). But can more be done with new statistical models and methods? Perhaps, but any new method must meet the criteria laid down by Glenn (2005: 20) that it may prove useful: “if it yields approximately correct estimates ‘more often than not,’ if researchers carefully assess the credibility of the estimates by using theory and side information, and if they keep their conclusions about the effects 14 tentative.” Part I: The Early Literature on Cohort Analysis and the Age-Period-Cohort (APC) Identification Problem Generally, however, the problem with much of the extant literature is a deficiency of useful guidelines on how to conduct an APC analysis. Rather, the literature often leads a researcher to conclude 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. 15 Part I: The Early Literature on Cohort Analysis and the Age-Period-Cohort (APC) Identification Problem Yang and Land and co-authors have bravely taken on Glenn’s challenge and have developed new approaches for APC analysis that are less esoteric and can be used by researchers. These new approaches are bound together as members of the class of Generalized Linear Mixed Models (GLMMs), models that allow linear and nonlinear exponential family links and mixed (both fixed and random) effects. 16 Part II: First Research Design: APC Analysis of Ageby-Time Period Tables of Rates or Proportions 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 Methods 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 114(May): 16971736. Yang Yang. 2008. “Trends in U.S. Adult Chronic Disease Mortality, 19601999: Age, Period, and Cohort Variations.” Demography 45(May):387416. 17 Part II: First Research Design: APC Analysis of Ageby-Time Period Tables of Rates or Proportions Data Structure: Tabular Rate Data 18 Part II: First Research Design: APC Analysis of Ageby-Time Period Tables of Rates or Proportions Example: Lung Cancer Death Rates for U.S. Adult Females, 1960 – 1999 Analyzed in Yang (2008) Age 20 - 24 25 - 29 30 - 34 35 - 39 40 - 44 45 - 49 50 - 54 55 - 59 60 - 64 65 - 69 70 - 74 75 - 79 80 - 84 85 - 89 90 - 94 95 - 125+ All 1960 - 64 0.1 0.2 0.8 2.3 5.1 8.6 12.5 16.1 19.9 24.5 29.2 34.0 36.9 39.8 34.2 26.5 10.3 1965 - 69 0.1 0.2 0.9 3.0 7.1 12.9 19.4 25.5 28.8 33.9 38.4 41.8 45.8 48.6 43.1 44.2 14.7 Deaths per 100,000 Population Period 1970 - 74 1975 - 79 1980 - 84 1985 - 89 0.1 0.1 0.1 0.1 0.2 0.2 0.2 0.2 1.0 0.9 0.8 0.8 3.6 3.5 3.3 2.7 9.1 10.5 9.9 8.9 18.1 22.2 23.9 23.1 28.9 36.9 44.2 47.2 40.1 53.1 69.0 78.3 46.6 69.2 92.6 115.2 51.3 78.6 114.7 145.5 52.8 77.7 120.2 168.3 56.1 76.1 111.4 162.0 57.3 75.3 102.5 141.1 59.7 75.2 96.9 120.9 60.6 73.6 91.8 108.8 51.0 68.9 82.7 104.1 21.3 28.9 38.3 47.9 Source: CDC/NCHS Multiple Cause of Death File 1990 - 94 0.1 0.2 0.9 2.9 7.5 20.9 44.9 81.8 127.3 172.6 208.2 219.4 199.8 164.5 136.3 120.0 57.0 1995 - 99 0.1 0.2 0.8 3.0 7.7 17.0 39.0 74.2 125.1 180.0 233.5 251.6 249.6 214.8 166.2 132.8 61.7 19 Part II: First Research Design: APC Accounting/Multiple Classification Model The Algebra of the APC Identification Problem Linear Model Specification: M ij Dij / Pij i j k ij (1) – Mij denotes 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 – μ denotes 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 – γk denotes 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: as the reference group. i i j j k k 0, or setting one of each of the categories 20 Part II: First Research Design: APC Accounting/Multiple Classification Model The Algebra of the APC Identification Problem Alternative Specifications In the Generalized Linear Models (GLM) Class: Simple Linear Models Yij i j k ij where Yij is the expected outcome in cell (i, j) that is assumed to be normally distributed or equivalently the error term ij 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 mij i j k 1 mij ij log where θij is the log odds of event and mij is the probability of event in cell (i,j). 21 Part II: First Research Design: APC Accounting/Multiple Classification Model The Algebra of APC Identification Problem Least-squares regression in matrix form: Y Xb (2) b (,1 ,... a1 , 1 ,..., p1 , 1 ,..., a p2 )T Identification Problem: bˆ ( X T X )1 X T Y (3) The solution to these normal equations does not exist because the Design matrix X is singular with 1 less than full rank (one column can be written as a linear combination of the others); this is due to the identity: Period = Age + Cohort thus, (XTX)-1 does not exist 22 Part II: First Research Design: APC Accounting/Multiple Classification Model Conventional Solutions to APC Identification Problem Constrained Coefficients GLIM (CGLIM) Estimator 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 mod Proxy Variables/Age-Period-Cohort Characteristic (APCC) 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. 23 Part II: First Research Design: APC Accounting/Multiple Classification Model Limitations of Conventional Solutions to APC Identification Problem Proxy Variables Approach the analyst may 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 (CGLIM) Estimator it is the most widely used of the three approaches, but suffers from some major problems summarized below. 24 Part II: First Research Design: APC Accounting/Multiple Classification Model Limitations of Conventional Solutions to APC Identification Problem Constrained Coefficients GLIM (CGLIM) Estimator: 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. 25 Part II: First Research Design: APC Accounting/Multiple Classification Model So, what can be done? Some Guidelines for Estimating APC Models for Tables of Rates or Proportions Step 1: Descriptive data analyses using graphics Step 2: Model specification tests 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. 26 Part II: First Research Design: APC Accounting/Multiple Classification Model Step 1: Graphical analyses: Female Lung Cancer Example from Yang (2008) 27 Part II: First Research Design: APC Accounting/Multiple Classification Model Step 2: Model selection procedures Examples from Yang et al. (2004) and Yang (2008) Table 1. Goodness-of-Fit Statistics for Age-Period-Cohort Log Linear Models of U.S. Adult Mortality Cause of Death Models DF Deviance AIC BIC A 112 695527 695751 695763 Heart Disease Deviance AIC BIC 782210 782434 782446 Stroke Deviance AIC BIC Lung Cancer Breast Cancer Total Female AP 105 40443 40653 40664 AC 90 72089 72269 72279 APC* 84 18903 19071 19080 52225 52435 52446 18638 18818 18827 9243 9411 9420 655622 655846 655858 12660 12870 12881 25967 26147 26157 1480 1648 1657 Deviance AIC BIC 320050 320274 320286 42126 42336 42347 5296 5476 5486 245 413 422 Deviance AIC BIC 9748 9972 9984 7403 7613 7625 1553 1733 1743 512 680 689 28 Part II: First Research Design: APC Accounting/Multiple Classification Model Guidelines for Estimating APC Models of Rates or Proportions 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, then Yang et al. (2004, 2008) recommend: Step 3: Apply the Intrinsic Estimator (IE). 29 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 the outcome observations Yij) on coefficient estimates. This constraint produces estimates that have desirable statistical properties. 30 Part II: First Research Design: APC Accounting/Multiple Classification Model Some preliminary matrix algebra concepts: Let A be a matrix of dimension q by d (q rows and d columns), let x be a column vector of dimension d, and y a column vector of dimension q. For a set of linear equations Ax = y, the set of vectors x0 of (real) numbers such that Ax0 = 0 is called the null space of the matrix A. When a matrix A is rank deficient (has linearly dependent columns), the dimension of the null space is at least one. In this case, if we have Ax = y, then we also have A(x + x0) = y. When A is rank deficient, the equation Ax = y has an infinite set of solutions, which differ by an element of the null space (if vectors x1 and x2 are solutions, then A(x1 – x2) = 0 and the vector x1 – x2 is in the null space). When A is rank deficient, there always is a well-defined solution whose projection on the null space is zero; this solution corresponds to the generalized inverse of A. 31 Part II: First Research Design: APC Accounting/Multiple Classification Model The Intrinsic Estimator (IE): Algebraic Definition The linear dependency between A, P, and C in model (2) is mathematically equivalent to: XB0 0 (4) which defines the null space for model (2) where the eigenvector B0 of eigenvalue of 0 is fixed by the design matrix X: ~ B0 B0 ~ B0 ~ B0 (0, A, P, C)T a 1 a 1 A 1 ,, (a 1) 2 2 p 1 p 1 C 1 a p ,, (a p 2) a p P 1,, ( p 1) 2 2 2 2 32 Part II: First Research Design: APC Accounting/Multiple Classification Model The Intrinsic Estimator (IE): Algebraic Definition Parameter vector orthogonal decomposition: (5) (6) b0 ( I B0 B0T )b where b0 Pproj b is the projection of b to the non-null space of X and 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. b b0 tB0 b2 b0 b1 33 0 B0 tB0 Part II: First Research Design: APC Accounting/Multiple Classification Model The Intrinsic Estimator (IE) Method: Algebraic Definition From the infinite number of estimators of b in model (2): bˆ B tB0 (7) the IE B estimates the parameter vector b0 corresponding to t = 0: B (I B0 B0T )bˆ (8) The IE is the special estimator that uniquely determines the age, period, and cohort effects in the parameter subspace defined by b0 : Xbˆ X (B tB0 ) XB tXB0 XB 0 XB (9) 34 Part II: First Research Design: APC Accounting/Multiple Classification Model The Intrinsic Estimator (IE) Method: Desirable statistical properties (Yang et al. 2004, 2008): 1) Estimability: Yang et al. (2004) established that the IE satisfies the Kupper et al. (1985) condition for estimability, namely l T B0 0 where where lT is a constraint vector (of appropriate dimension) that defines a linear function lTb of b. Reference: 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 38:811-830. 35 Part II: First Research Design: APC Accounting/Multiple Classification Model Proof: Note that l B0 (I B0 B0 )B0 B0 B0 B0 B0 B0 B0 0 T T T Estimable functions are desirable as statistical estimators because they are linear functions of the unidentified parameter vector that can be estimated without bias, i.e., they have unbiased estimators. 36 Part II: First Research Design: APC Accounting/Multiple Classification Model Yang et al. (2004) also proved independently of the Kupper et al. (1985) estimability condition that the IE has the following two properties: 2) 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. 3) Relative efficiency: For a fixed number of time periods of data, it has a smaller variance than any CGLIM estimators. 37 Part II: First Research Design: APC Accounting/Multiple Classification Model 3) Asymptotic consistency: This properties derive largely from the fact that the length of the eigenvector B0 decreases with increasing numbers of time periods of data, and, in fact, converges to zero as the number of periods of data increases without bound. Therefore, for any two estimators: bˆ2 B t 2 B0 and bˆ1 B t1 B0 where t1 and t2 are nonzero and correspond to different identifying constraints, as the number of time periods in an APC analysis increases, the difference between these two estimators decreases towards zero, and, in fact, that the estimators converge toward the IE B. 38 Part II: First Research Design: APC Accounting/Multiple Classification Model 4) Monte Carlo Simulation: Numerical simulation demonstrations of the foregoing statistical properties were given in Yang et al. (2008); one example is reproduced on the following slide. 39 Simulation Results of the IE and CGLIM Estimators: True Cohort Effects = 0 Age Effe ct: M e an Es tim ate s 2.0 True ef f ect CGLIM_p 1.5 IE CGLIM_c 25.0 20.0 0.5 15.0 0.0 10.0 -0.5 5.0 Log coef 1.0 0.0 -1.0 a1 a2 a3 a4 a5 Age a6 a7 a8 a1 a9 Pe riod Effe ct: M e an Es tim ate s 0.5 a2 a3 a4 a5 Age a6 a7 a8 a9 Pe riod Effe ct: M SE 8.0 0.3 Log coef Age Effe ct: MSE 30.0 6.0 0.1 4.0 -0.1 2.0 -0.3 -0.5 0.0 p1 p2 p3 Pe riod p4 p5 p1 Cohort Effe ct: M e an Es tim ate s 2.0 p2 p4 p5 Cohort Effe ct: M SE 400 300 Log coef 1.0 p3 Pe riod 0.0 200 -1.0 100 -2.0 40 0 c1 c3 c5 c7 Cohort c9 c11 c13 c1 c3 c5 c7 Cohort c9 c11 c13 Part II: First Research Design: APC Accounting/Multiple Classification Model Based on these statistical properties, Yang et al. (2008) also showed how the IE can be used in an asymptotic ttest to evaluate a substantively informed equality constraint on the APC accounting model with respect to whether the estimated coefficient vector that results therefrom is (statistically) estimable, that is, within sampling error of meeting the Kupper et al. condition for estimability. 41 Part II: First Research Design: APC Accounting/Multiple Classification Model The Intrinsic Estimator (IE) Method: Computation Software Two programs for calculating the IE are available for use in popular statistical packages: 1) a S-Plus/R program and 2) a Stata Ado File (both referenced in Yang et al., 2008) 42 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) Period Effect Age Effect 2.0 2.0 Male Male Female Female 1.0 -2.0 0.0 Log coefficient 0.0 -1.0 -4.0 -9 9 19 95 -9 4 19 90 -8 9 19 85 -8 4 19 80 -7 9 19 75 -7 4 19 70 -6 9 19 65 19 60 85 80 75 70 65 60 55 50 45 90 95 + Year Cohort Effect 2.0 Male Female 1.0 0.0 -1.0 -2.0 43 19 75 19 65 19 55 19 45 Cohort 19 35 19 25 19 15 19 05 18 95 18 85 18 75 -3.0 18 65 Log coefficient 35 30 25 20 40 Age -6 4 -2.0 -6.0 Some Recent Empirical Applications of the Intrinsic Estimator: Schwadel, P. 2011. “Age, period, and cohort effects on religious activities and beliefs”, Social Science Research 40:181-192. Unknown Author. 2011. “Age, Period, and Cohort Effects on Social Capital and Voting.” Social Forces 90:forthcoming. Winkler, Richelle L., Jennifer Huck, and Keith Warnke. 2009. “Deer hunter demography: An age-period-cohort approach to population projections.” Paper presented at the Population Association of America Annual Meeting, Detroit, MI, April 30, 2009. 44 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”? No. There will never be such a solution. The APC identification problem is one of structural under-identification in linear or generalized linear models for which there can only be partial solutions. 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. 45 Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys 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. 46 Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys References for Part III, Continued: Yang Yang, Steven M. Frenk, and Kenneth C. Land. 2010. “Assessing the Significance of Cohort and Period Effects in Hierarchical Age-Period-Cohort Models.” Revision of a paper presented at the American Sociological Association Annual Meeting, San Francisco, CA, August 2009. Zheng, Hui, Yang Yang, and Kenneth C. Land. 2011. “Heteroscedastic Regression in Hierarchical Age-Period-Cohort Models, With Applications to the Study of Self-Reported Health. Revision of a paper presented at the American Sociological Association Annual Meeting, Atlanta, GA, August 2010. Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys Data Structure: Individual-level Data in an Age-by-Period Array Period j nij >1 Age i 48 Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys Approach 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. Proposal: Use different temporal groupings for the A, P, and C dimensions to break 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, 49 Baby Busters, etc.). Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys Example: 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) Cohort (J) 1974 1976 1978 1982 1984 1987 12 18 8 0 0 0 1890 31 25 19 19 6 0 1895 62 52 49 27 18 17 1900 88 69 68 43 38 23 1905 77 89 69 75 50 48 1910 109 111 84 100 81 81 1915 115 104 112 110 73 97 1920 113 108 106 131 99 92 1925 129 92 90 111 81 95 1930 130 106 108 112 80 101 1935 119 140 130 127 100 142 1940 179 161 184 163 133 143 1945 179 180 197 199 170 185 1950 89 151 180 260 162 219 1955 0 8 59 175 186 190 1960 0 0 0 38 75 161 1965 0 0 0 0 0 29 1970 0 0 0 0 0 0 1975 0 0 0 0 0 0 1980 1432 1414 1463 1690 1352 1623 Total Year (K) 1988 1989 1990 1991 1993 1994 1996 1998 2000 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 13 11 5 2 0 0 0 0 0 11 12 11 11 15 15 10 0 0 34 27 25 29 13 31 27 18 8 42 36 37 41 37 60 39 24 27 60 53 40 56 55 85 59 32 37 52 53 53 40 50 84 81 68 52 47 54 43 62 43 86 72 45 64 39 59 44 37 58 101 100 61 64 49 74 49 65 58 134 117 65 78 98 84 85 74 85 168 161 104 85 101 94 95 111 99 173 169 101 111 102 117 106 118 127 198 213 149 145 109 121 102 118 103 231 208 161 147 101 86 76 91 111 182 188 157 111 32 48 55 77 81 157 188 116 145 0 0 0 1 23 59 128 84 107 0 0 0 0 0 0 4 34 62 890 929 826 933 958 1764 1764 1219 1243 Total 38 100 256 414 620 909 1088 1182 1114 1200 1447 1907 2164 2336 1918 1377 928 402 100 19500 50 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, 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. Key Points: 1) this approach builds on the recognition that age is an intrinsically individual-level property that individuals carry with them and that varies from period to period; 2) by comparison an individual’s cohort is fixed, as is the time period of a particular survey, and both cohort and period are contexts within which individuals mature and age and experience certain events. 51 Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys Further Questions: Is there evidence for clustering effects of random errors, due to the facts that: • individuals surveyed in the same year may be subject to similar unmeasured events that influence their outcomes, and • 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? 52 Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys Method: Apply Hierarchical Age-Period-Cohort (HAPC) Models These models generally are members of what statisticians call mixed (fixed and random) effects models; in the social sciences, these models typically are called hierarchical linear models (HLM). The mixed models may be linear mixed effects (LMM) models or, more generally, allow for nonlinear link functions, in which case they are generalized linear mixed models (GLMM). A form of HLMs applicable to cross-classified data of the form shown above is the class of cross-classified random effects models (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. 53 Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys Application 1 – A HAPC-LMM of General Social Survey (GSS) Data on Verbal Test Scores: 1974 – 2006 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, 287302. 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. 54 Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys Research Questions What are the distinct age, period, and cohort components of change in verbal ability in the U.S.? How can period and/or cohort level heterogeneity be explained by period and/or cohort characteristics? Analytic Method Apply the HAPC-CCREM to estimate • fixed effects of age and other individual level and level-two covariates, • random effects of period and cohort and variance components 55 Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys Because the WORDSUM outcome variable has a relatively bell-shaped sample frequency distribution, it is reasonable to use a HAPC model specification that includes a conventional normal-errors regression model. Specifically, Yang and Land (2006: 87) specified the cross-classified random effects model (CCREM): Level-1 or “Within-Cell” Model: WORDSUMijk = β0jk + β1AGEijk + β2AGE2ijk + β3EDUCATIONijk + β4FEMALEijk + β5BLACKijk + eijk , eijk ~ N(0, σ2 ) (1) Level-2 or “Between-Cell” Model: β0jk = γ0 + u0j + v0k, u0j ~ N(0, τu), v0k ~ N(0, τv) (2) Combined Model: WORDSUMijk = γ0 + β1AGEijk + β2AGE2ijk + β3EDUCATIONijk + β4FEMALEijk + β5BLACKijk + u0j + v0k + eijk (3) for i = 1, 2, …, njk individuals cross-classified within cohort j and period k; j = 1, …, 20 birth cohorts; k = 1, …, 17 time periods (survey years) 56 Table 2. HAPC Models of the GSS WORDSUM Data, 1974-2006 Fixed Effects INTERCEPT AGE AGE2 FEMALE BLACK EDUCATION Random Effects Cohort 1894 1895 1900 1905 1910 1915 1920 1925 1930 1935 1940 1945 1950 1955 1960 1965 1970 1975 1980 1985 coefficient 6.175 0.026 -0.057 0.229 -1.030 0.366 se 0.055 0.015 0.005 0.024 0.034 0.004 t ratio 112.50 1.71 -11.87 9.49 -30.07 86.57 p value < .000 .087 < .000 < .000 < .000 < .000 coefficient -0.210 -0.114 -0.051 -0.294 0.021 0.163 -0.079 0.083 0.001 0.068 0.240 0.447 0.184 -0.035 0.002 -0.157 -0.135 -0.001 0.062 -0.195 se 0.142 0.123 0.104 0.090 0.081 0.073 0.068 0.068 0.067 0.064 0.061 0.060 0.059 0.061 0.065 0.071 0.080 0.092 0.112 0.146 t ratio -1.48 -0.93 -0.49 -3.27 0.26 2.22 -1.15 1.23 0.01 1.06 3.91 7.50 3.10 -0.57 0.04 -2.20 -1.70 -0.01 0.55 -1.34 p value 0.140 0.353 0.625 0.001 0.797 0.027 0.249 0.220 0.990 0.289 < .000 < .000 0.002 0.568 0.970 0.028 0.090 0.990 0.583 0.180 57 Period 1974 1976 1978 1982 1984 1987 1988 1989 1990 1991 1993 1994 1996 1998 2000 2004 2006 Variance Components Cohort Period Individual Deviance AIC 0.033 0.060 -0.002 -0.014 0.016 -0.061 -0.128 -0.061 0.020 0.042 -0.004 0.019 -0.060 0.044 0.005 0.038 0.052 variance 0.034 0.005 3.116 87707.2 87713.2 0.043 0.043 0.042 0.040 0.042 0.040 0.046 0.046 0.047 0.046 0.045 0.039 0.039 0.043 0.043 0.043 0.045 se 0.013 0.003 0.030 0.77 1.41 -0.04 -0.36 0.37 -1.52 -2.76 -1.34 0.43 0.92 -0.09 0.49 -1.52 1.02 0.11 0.88 1.16 z value 2.56 1.49 104.87 0.442 0.158 0.967 0.718 0.709 0.129 0.006 0.182 0.670 0.358 0.926 0.623 0.128 0.306 0.915 0.381 0.247 p value .010 .135 < .000 58 Figure 1. Estimated Cohort and Period Effects and 95 Percent Confidence Bounds for GSS Verbal Ability Model Cohort Effect ˆ0 j Period Effect 7.00 Verbal Test Score 7.00 Verbal Test Score ˆ0k 6.00 5.00 6.00 5.00 Cohort Period 59 To further test whether the birth cohort and time period effects – as a whole – make statistically significant contributions to explained variance in an outcome variable, a general linear hypothesis may be applied. Specifically, one can either: 1) examine the statistical significance of the variance components (an asymptotic t-test for LMMs), or 2) use an F test to test the hypothesis of the presence of random effect. The sampling distribution of F statistic is exact in LMMs when the random effects are independently distributed as normal random variables. This F-test statistic is preferred over the z-score when the sample sizes for random effects are small. The statistical theory for such tests has been developed in a very general LMM context by E. Demidenko (Mixed Models: Theory and Applications. Wiley, 2004). 60 In the present case, for the CCREM-HAPC model of Equations (1)-(3), there are only two sets of random effect coefficient that are estimated, namely, the set of residual random effects of cohort j, u0j, and the set of residual random effects of period k, v0k. Each of these sets of random coefficients is assumed to be independently, normally distributed with mean 0 and variances τu and τv, respectively. Thus, for a CCREM-HAPC model with random intercepts of the form of Equations (1)-(3), the exact F-test amounts to testing null hypotheses for the relevance either of the birth cohort random effects: H0: τu = 0, vs. Ha: τu > 0 or the time period effects: H0: τv = 0, vs. Ha: τv > 0. Alternatively, one can test for the joint relevance of both the cohort and period effects: H0: τu = τv = 0, vs. Ha: τu > 0 or τv > 0 61 Table 3. F-tests for the Presence of Random Effects, GSS WORDSUM Data Cohort Effects SOLS Smin R M NT (SOLS Smin ) /(r m) S min /( NT r) F f0.95(r – m, NT – r) τu = 0 vs. τu > 0 69,377 68,696 25 Period Effects Cohort and Period Effects τv = 0 vs. τv > 0 τu = τv = 0 vs. τu or τv > 0 69,377 69,377 69,268 68,558 22 42 5 22,042 34.05 5 22,042 6.41 5 22,042 22.14 3.12 10.9 1.571 3.15 2.03 1.623 3.12 7.096 1.411 62 Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys BACK TO THE DEBATE ON TRENDS IN VERBAL ABILITY: So, who is right, Alwin and Glenn or Wilson and Gove? The results of the HAPC analyses show: significant random variance components that reside in all three levels of the APC data: individuals nested within cohorts and periods; quadratic age effects that are not explained away by controlling for the effects of key individual characteristics, namely, education, sex and race, and for period and cohort effects; significant contextual effects of cohorts and periods on verbal ability, but this is mainly a cohort story; and strong effects of cohort characteristics: cohorts that have a larger proportion of daily newspaper readers are better off in their verbal ability; more hours of TV watching per day tend to undermine average cohort verbal ability. Bottom Line: Alwin and Glenn are more right than Wilson and Gove. 63 Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys Extensions of HAPC Modeling: – Fixed Effects vs. Mixed Effects Model – A Full Bayesian HAPC Model – Generalized Linear Mixed Models (GLMM) 64 Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys Fixed Effects vs. Mixed Effects Model: The HAPC-CCREM approach illustrated above uses a mixed (fixed and random) effects model with a random effects specification for the level-2 (time period and cohort) contextual variables. Alternative: fixed effects specification for the level-2 variables in which ones uses dummy (indicator) variables to record the cohort and the time period of the survey. The comparison seems especially pertinent when the number of replications of the survey is relatively small— say 3 to 5. 65 Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys Fixed Effects vs. Mixed Effects Model: The estimates of cohort and time period effects from a fixed effects model for the GSS data are quite similar in pattern to those from the random effects model (Yang and Land 2008). The mixed effects model is preferred to the fixed effect model: It avoids potential model specification error by not using the assumption of the fixed effect model that the indicator/dummy variables representing the fixed cohort and periods effects fully account for all of the group effects; It allows group level covariates to be incorporated into the model and explicitly models cohort characteristics and period events to test explanatory hypotheses; For unbalanced research designs (designs in which there are unequal numbers of respondents in the cells), such as one typically has in repeated cross-section survey designs, a random effect model for the level-2 variables generally is more statistically efficient. 66 Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys A Full Bayesian HAPC Model: Limitations of HAPC Modeling Using REML-EB Estimation • • • • Small numbers of cohorts (J) and periods (K) Unbalanced data Inaccurate REML estimates of variance-covariance components Inaccurate EB estimates of fixed effects regression coefficients A Remedy: Bayesian Model Estimation (Yang 2006) • A full Bayesian approach, by definition, ensures that inferences about every parameter fully account for the uncertainty associated with all others. 67 Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys Application 2: A HAPC-GLMM of American National Election Survey (ANES) Data on Voting Turnout in U.S. Presidential Elections, 1952-2004 (Yang, Frenk, and Land 2010) The GLMM Family of Models: • Normal outcome: Linear mixed models using Gaussian link • Binomial outcome: Logistic mixed models using logit link • Ordinal or nominal outcome: Ordinal logistic mixed models • Count outcome: Poisson mixed models using log link • Count outcome with dispersion: Negative Binomial mixed models REML-EB Estimation: Use, e.g., SAS PROC GLIMMIXED 68 Application 2: A HAPC-GLMM of Voting Turnout in U.S. Presidential Elections Table 4. Descriptive Statistics for ANES Voter Turnout Data, 1952 to 2004 Variables Description Mean SD Min Max 0.76 0.43 0 1 0.45 0.1 45.62 0 0.5 0 1 0.3 0 1 16.53 18 95 16.53 -27.46 49.54 Religion PROTESTANT CATHOLIC JEW Respondent's sex: 1 = male; 0 = female Respondent's race: 1 = black; 0 = nonblack Respondent's age at survey year Centered around grand mean Respondents' religious preference 1 = Protestant; 0 = otherwise 1 = Catholic; 0 = otherwise 1 = Jew; 0 = otherwise 0.66 0.24 0.02 0.47 0.42 0.15 0 0 0 1 1 1 OTHER 1 = Other/None; 0 = otherwise 0.08 0.27 0 1 Dependent Variable VOTE Level-1 Variables MALE BLACK AGE 1= Voted in U.S. presidential elections; 0 = Did not vote in U.S. presidential elections 69 Respondent's marital status: 1 = Currently married; 0 = otherwise MARRIED 0.67 0.47 0 1 Occupational Class PROFESSIONAL 1 = Professional; 0 = otherwise 0.25 0.43 0 1 CLERICAL 1 = Clerical; 0 = otherwise 0.18 0.38 0 1 SKILLED 1 = Skilled; 0 = otherwise 0.31 0.46 0 1 LABORER 1 = Laborer; 0 = otherwise 0.03 0.17 0 1 FARMER 1 = Farmer; 0 = otherwise 0.04 0.19 0 1 NOT WORKING 1 = Not working; 0 = otherwise 0.2 0.4 0 1 Political affiliation Respondent's political affiliation DEMOCRATIC 1 = Democratic; 0 = otherwise 0.52 0.5 0 1 INDEPENDENT 1 = Independent; 0 = otherwise 0.1 0.3 0 1 REPUBLICAN 1 = Republican; 0 = otherwise 0.38 0.48 0 1 POLITICAL SOUTH¹ 1 = Political south; 0 = otherwise 0.27 0.45 0 1 Level-2 Variables N Min Max PERIOD Survey year 14 1952 2004 COHORT Five-year birth cohort 23 1859 1986 ¹Includes the eleven session states: Alabama, Arkansas, Florida, Georgia, Louisiana, Mississippi, North Carolina, South Carolina, Tennessee, Texas, Virginia Note: N=19,766 70 To model the likelihood of voter turnout in U.S. Presidential Elections, we apply the HAPC-CCREM approach and specify the following model: Level 1 or “Within-Cell” Model: Pr (VOTEijk = 1) = β0jk + β1AGEijk + β2AGE2ijk + β3MALEijk + β4BLACKijk + β5PROTESTANTijk + β6CATHOLICijk + β7JEWijk + β8PROFESSIONALijk + β9CLERICALijk + β10SKILLEDijk + β11FARMERijk + β12NOWORKijk + β13PSOUTHijk + β14CMARRIEDijk + β15DEMOCRATICijk + β16REPUBLICANijk Level 2 or “Between-Cell” Model: β0jk = γ0 + u0j + ν0k , u0j ~ N(0, τu), ν0k ~ N(0, τv) COMBINED MODEL: Pr (VOTEijk = 1) = β0jk + β1AGEijk + β2AGE2ijk + β3MALEijk + β4BLACKijk + β5PROTESTANTijk + β6CATHOLICijk + β7JEWijk + β8PROFESSIONALijk + β9CLERICALijk + β10SKILLEDijk + β11FARMERijk + β12NOWORKijk + β13PSOUTHijk + β14CMARRIEDijk + β15DEMOCRATICijk + β16REPUBLICANijk + u0j + ν0k + eijk (12) for i = 1, 2, …, njk individual within cohort j and period k; j = 1, …23 birth cohorts; k = 1, …, 14 time periods (presidential elections). 71 Table 5. HAPC Models of the ANES Voter Turnout Data, 1952-2004 Fixed Effects coefficient se t ratio p value INTERCEPT -0.49 0.134 -3.66 0.003 AGE 0.025 0.001 19.22 <.000 AGE² -0.001 0.0001 -13.85 <.000 MALE 0.21 0.044 4.73 <.000 BLACK -0.02 0.058 -0.34 0.734 PROTESTANT 0.335 0.065 5.2 <.000 CATHOLIC 0.591 0.071 8.3 <.000 JEW 1.211 0.184 6.6 <.000 PROFESSIONAL 1.414 0.105 13.45 <.000 CLERICAL 1.037 0.107 9.73 <.000 SKILLED 0.285 0.098 2.9 0.004 FARMER 0.301 0.128 2.35 0.019 NOT WORKING 0.386 0.107 3.6 <.000 POLITICAL SOUTH CURRENTLY MARRIED -0.607 0.04 -15.23 <.000 0.446 0.041 10.78 <.000 DEMOCRATIC 0.736 0.055 13.33 <.000 REPUBLICAN 0.963 0.059 16.4 <.000 72 Random Effects Cohort 1859-1875 coefficient 0.011 se 0.066 t ratio 0.17 p value 0.867 1876-1880 -0.01 0.064 -0.16 0.871 1881-1885 -0.022 0.063 -0.35 0.723 1886-1890 0.007 0.061 0.12 0.906 1891-1895 -0.026 0.059 -0.45 0.655 1896-1900 0.01 0.058 0.17 0.864 1901-1905 -0.028 0.055 -0.51 0.613 1906-1910 -0.047 0.053 -0.89 0.376 1911-1915 0.061 0.051 1.19 0.236 1916-1920 0.016 0.049 0.32 0.747 1921-1925 0.05 0.048 1.03 0.303 1926-1930 0.054 0.048 1.12 0.262 1931-1935 -0.003 0.049 -0.06 0.953 1936-1940 -0.033 0.05 -0.66 0.511 1941-1945 0.024 0.049 0.49 0.625 1946-1950 0.035 0.048 0.73 0.468 1951-1955 0.027 0.049 0.55 0.581 1956-1960 -0.106 0.05 -2.11 0.035 1961-1965 -0.032 0.053 -0.61 0.54 1966-1970 -0.014 0.057 -0.24 0.811 1971-1975 -0.003 0.06 -0.05 0.958 1976-1980 0.016 0.063 0.26 0.795 1981-1986 0.013 0.065 0.2 0.838 73 Period 1952 -0.029 0.076 -0.39 0.7 1956 -0.102 0.068 -1.50 0.134 1960 0.274 0.081 3.4 0.001 1964 0.066 0.072 0.91 0.361 1968 0.02 0.073 0.28 0.783 1972 -0.066 0.063 -1.04 0.298 1976 -0.049 0.067 -0.74 0.459 1980 -0.055 0.073 -.75 0.453 1984 0.004 0.067 0.05 0.956 1988 -0.259 0.067 -3.85 <.000 1992 0.108 0.066 1.63 0.104 1996 -0.011 0.073 -0.16 0.876 2000 -0.074 0.075 -0.99 0.323 2004 0.174 0.087 2.01 0.045 Variance Components Cohort variance 0.004 se 0.003 z value 1.33 p value 0.16 Period 0.021 0.01 2.1 0.04 Deviance 94358.54 AIC 94305.54 Source: 1952-2004 American National Election Study (N = 19,766) #p<.1; *p<.05; **p<.01; ***p<.001 74 Figure 2. Estimated Cohort and Period Effects and 95 Percent Confidence Bounds for NES Voter Turnout Model Cohort Effect ˆ 0 j Period Effect ˆ0 k 0.75 0.7 0.65 0.6 0.55 0.5 -1 8 18 75 7 18 6 8 18 1 8 18 6 1891 9 19 6 0 19 1 0 19 6 1911 1 19 6 2 19 1 2 19 6 3 19 1 1936 4 19 1 4 19 6 5 19 1 1956 6 19 1 6 19 6 7 19 1 7 19 6 81 Predicted Probability of Voting 0.8 Cohort 75 Table 6. F-Tests for the Presence of Random Effects, ANES Data l0 lmax r m NT Cohort Effects Period Effects Cohort and Period Effects τu = 0 vs. τu > 0 τv = 0 vs. τv > 0 τu = τv = 0 vs. τu or τv > 0 9,695 9,695 9,695 9,679 9,662 9,646 39 30 53 16 16 16 F 19,766 0.6957 0.4915 1.42 19,766 2.357 0.4912 4.8 19,766 1.32 0.4918 2.69 f0.95(r – m, NT – r) 1.53 1.69 1.7 (lmax – l0)/(r – m) l0/( NT – r ) 76 Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys As in the case of trends in GSS verbal ability, this analysis of Presidential voting turnout finds: significant random variance components that reside in all three levels of the APC data: individuals nested within cohorts and periods; quadratic age effects that are not explained away by controlling for the effects of individual characteristics, and for period and cohort effects; significant contextual effects of cohorts and periods on voting in Presidential elections; but Presidential voting turnout is mainly a period story. 77 Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys Application 3: A HAPC-GLMM Analysis of GSS Data on Happiness, 1972-2004 (Yang 2008) Research Questions: Who is happier? – Social stratification of subjective wellbeing Do people get happier with age and over time? How do social inequalities in happiness vary over the life course and by time? Born to be happy? Are there any birth cohort differences in happiness? 78 Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys Level 1 (Individual-Level) Model: where yijk denotes the ordinal response happiness variable in the GSS data (very happy, pretty happy, not too happy) modeled with an ordinal logit HAPC-CCREM specification, and Xp denotes a vector of other individual-level variables such as age by sex, age by race, and age by education interaction variables. 79 Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys Level 2 Model: 80 Some Findings: 81 Some Findings: 82 Some Findings: 83 Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys As in the case of trends in GSS verbal ability and NES Presidential election voting probabilities, this analysis of the GSS happiness data finds: significant random variance components that reside in all three levels of the APC data: individuals nested within cohorts and periods; quadratic age effects that are not explained away by controlling for the effects of individual characteristics, and for period and cohort effects; significant contextual effects of both cohorts and periods on voting in Presidential elections, i.e., interesting stories both for cohorts and periods. 84 Part III: Second Research Design: APC Analysis of Repeated Cross-Section Surveys Application 4: An Integration of the Hierarchical AgePeriod-Cohort Model with Heteroscedastic Regression to Develop the HAPC-HR Model, Applied to Study Variations in Self-Reported Health Disparities in the U.S., 1984-2007 (Zheng, Yang, and Land 2011) There are three standard approaches to the study of changes in health disparities: (1) across the life course (e.g., House et al. 1994; Dannefer 2003), (2) across cohorts (e.g., Lynch 2003; Warren and Hernandez 2007), and (3) across time periods (e.g., Pappas et al. 1993; Goesling 2007). All of these approaches have one thing in common: They focus on changes in health disparities as estimated by conditional expectation functions (regressions) estimated on the basis of measured demographic and socioeconomic covariates. This facilitates the estimation of between-group disparities, i.e., variations in health across groups or between-cell variation and temporal variations therein, but it ignores possible within-group disparities – variations in health inside groups or within-cell variation – and variations therein over time. To examine Age-Period-Cohort variations in both health and health disparities, we: intersect the HAPC model with a Heteroscedastic Regression (HR) model. This allows us to both: (1) disentangle age, period, and cohort effects, and (2) separate within-group health disparities from between-group health disparities. The result is a Hierarchical-Age-Period-Cohort-HeteroscedasticRegression Model (HAPC-HR) model. Application to National Health Interview Survey (NHIS) data on self-reported health, 1984-2007: With individuallevel demographic and socioeconomic that are established covariates of health used to define the cells in the Level-1 regression model: sex (1 = male, 0 = female), race (1 = white, 0 = non-white), marital status (1 = married, 0 = unmarried), work status (1 = full/part time job and 0 = not employed), education (years of formal education), and income (in 2007 dollars), here are some results. Figure 1. Observed Means of Self-Rated Health, NHIS, 1984 to 2007. 4 The whole sample Men Mean of Self-Rated Health 3.9 Women 3.8 3.7 3.6 3.5 3.4 07 06 20 20 05 20 04 03 20 02 20 01 20 20 00 20 99 19 98 97 19 96 19 95 19 19 94 19 93 19 92 91 19 90 19 89 19 19 88 19 87 19 86 85 19 19 19 84 3.3 Year * The trends are adjusted for sample weights and smoothed by a three-point moving average. 89 Figure 2. Observed Variances in Self-Rated Health, NHIS, 1984 to 2007. 1.5 The whole sample Men Women Variance in Self-Rated Health 1.4 1.3 1.2 1.1 19 84 19 85 19 86 19 87 19 88 19 89 19 90 19 91 19 92 19 93 19 94 19 95 19 96 19 97 19 98 19 99 20 00 20 01 20 02 20 03 20 04 20 05 20 06 20 07 1 Year * The trends are adjusted for sample weights and smoothed by a three-point moving average 90 Cohort 3.8 Women 3.75 3.7 3.65 3.6 3.55 3.5 3.45 Conditional Expected Value of Self-Rated Health 3.85 19 84 19 85 19 86 19 87 19 88 19 89 19 90 19 91 19 92 19 93 19 94 19 95 19 96 19 97 19 98 19 99 20 00 20 01 20 02 20 03 20 04 20 05 20 06 20 07 19 85 19 80 19 75 19 70 19 65 19 60 19 55 19 50 19 45 19 40 19 35 19 30 19 25 19 20 19 15 19 10 19 05 18 99 Conditional Expected Value of Self-Rated Health 84 81 78 75 72 69 66 63 60 57 54 51 48 45 42 39 36 33 30 27 24 21 18 Conditional Expected Value of Self-Rated Health Figure 5. Variations in Conditional Expected Values of Gender-Specific Self-Rated Health across Age, Cohort and Period, with 95% Confidence Intervals. 4.7 Men 4.5 Women 4.3 4.1 3.9 3.7 3.5 3.3 3.1 Age Men 3.85 Men 3.8 Women 3.75 3.7 3.65 3.6 3.55 3.5 3.45 Period 91 Cohort 1.2 1.1 1 Predicted Dispersion of Self-Rated Health 1.3 0.9 0.8 19 84 19 85 19 86 19 87 19 88 19 89 19 90 19 91 19 92 19 93 19 94 19 95 19 96 19 97 19 98 19 99 20 00 20 01 20 02 20 03 20 04 20 05 20 06 20 07 19 85 19 80 19 75 19 70 19 65 19 60 19 55 19 50 19 45 19 40 19 35 19 30 19 25 19 20 19 15 19 10 19 05 18 99 Predicted Dispersion of Self-Rated Health Period 84 81 78 75 72 69 66 63 60 57 54 51 48 45 42 39 36 33 30 27 24 21 18 Predicted Dispersion of Self-Rated Health Figure 6. Variations in Predicted Dispersion of Gender-Specific Self-Rated Health across Age, Cohort and Period, with 95% Confidence Intervals. 1.25 Men 1.15 Women 1.05 0.95 0.85 0.75 0.65 0.55 Age Men 1.3 Men Women Women 1.2 1.1 1 0.9 0.8 92 Part IV: Third Research Design: Cohort Analysis of Accelerated Longitudinal Panels References for Part IV: Miyazaki, Yasuo and Stephen W. Raudenbush. 2000. "Tests for Linkage of Multiple Cohorts in an Accelerated Longitudinal Design." Psychological Methods 5:4463. Yang, Yang. 2007. “Is Old Age Depressing? Growth Trajectories and Cohort Variations in Late Life Depression.” Journal of Health and Social Behavior 48:16-32. 93 Part IV: Third Research Design: Cohort Analysis of Accelerated Longitudinal Panels Accelerated Longitudinal Panel Design Definition: A longitudinal panel study of an initial sample of individuals from a broad array of ages (and thus birth cohorts) interviewed or monitored with three or more follow-up waves. The design allows a more rapid accumulation of information on age and cohort effects than a single cohort follow-up study. 94 Part IV: Third Research Design: Cohort Analysis of Accelerated Longitudinal Panels Data Structure: Accelerated Longitudinal Panel Design Age (Time) Cohort 95 Part IV: Third Research Design: Cohort Analysis of Accelerated Longitudinal Panels For this research design, the HAPC Model becomes a Growth Curve Model of Individual Change with cohort interactions: Assess the intra-individual age changes and birth cohort differences simultaneously; Assess differential cohort patterns in age changes: ageby-cohort interaction effects; Period effects? • The time period for an accelerated longitudinal panel study often is short (e.g., a decade or so), so the effects of period usually can be ignored; • In growth curve models, age and time are the same variable, so the effects of period need not be estimated; and • can be focused on the age-by-cohort interactions. • If period effects are of concern, estimate the HAPC-CCREM. 96 Part IV: Third Research Design: Cohort Analysis of Accelerated Longitudinal Panels Application: Cohort Variations in Age Trajectories of Depression in the Elderly (Yang 2007) Research Questions • Does the age growth trajectory show an increase in depressive symptoms in late life? • Is there cohort heterogeneity in levels of depressive symptoms and age growth trajectories of depressive symptoms? • What social risk factors are associated with these effects? Data • Established Populations for Epidemiologic Studies of the Elderly (EPESE) in North Carolina: A fourwave panel study of older adults aged 65+ from 1986 to 1996 97 Part IV: Third Research Design: Cohort Analysis of Accelerated Longitudinal Panels Model Specification Level-1 Repeated Observation Model Yti 0i 1i Ageti pi X pti eti (11) p Yti = CES-D for person i at time t, for i =1, …, n and t = 1, …, Ti Xpti = (marital status, economic status, health status, stress and coping resources) = expected CES-D for person i 0i = expected growth rate per year of age in CES-D for person i 1i = regression coefficient associated with X pti pi iid eti ~ N (0, 2 ) 98 Part IV: Third Research Design: Cohort Analysis of Accelerated Longitudinal Panels Model Specification Level-2 Individual Model 0i 00 01Cohorti 0q Z qi r0i q 1i 10 11Cohorti r1i (12) Zqi = (Female, Black, Education) 00 = expected CES-D for person i for the reference group (at median age in Cohort 1 at T1) 01 = main cohort effect coefficient: mean difference in CES-D between cohorts 0 q = regression coefficient associated wit Zqi 10 = age effect coefficient: expected rate of change in CES-D 11 = age*cohort coefficient: mean difference in rate of change between cohorts r0i iid 0 0 r ~ N 0, 1i 10 01 1 99 Part IV: Third Research Design: Cohort Analysis of Accelerated Longitudinal Panels Model Estimates Fixed Effect Model 1 (Total) Model 7 (Net) 00 2.856*** 2.525*** 1i 10 Growth Rate: Age, 0.048*** -0.018 Cohort 01 0.244*** -0.213** Age * Cohort 11 -0.019# -0.040*** 0i Intercept, Random Effect Variance Component Level-1: Within person 2 % Reduction 36.987*** 35.109*** 5% 0 6.170*** 3.763*** 39% In growth rate 1 0.057*** 0.051*** 11% AIC (smaller is better) 51190.5 48167.4 BIC (smaller is better) 51215.6 48192.5 Level-2: In intercept Goodness-of-fit # p < .10; * p < .05; ** p < .01; *** p < .001. 100 Part IV: Third Research Design: Cohort Analysis of Accelerated Longitudinal Panels Expected Growth Trajectories and Cohort Variations in Depression b. Model 7- Net Age and Cohort Effects 4 4 Age 95 93 91 89 87 85 83 81 79 77 75 73 65 95 93 91 89 87 85 83 81 79 77 75 73 1 71 1 69 2 67 2 71 3 69 3 67 CES-D 5 65 CES-D a. Model 1-Gross Age and Cohort Effects 5 Age All cohort 1 cohort 2 cohort 3 cohort 4 cohort 5 101 Part IV: Third Research Design: Cohort Analysis of Accelerated Longitudinal Panels Summary of Findings: The gross age trajectory of depressive symptoms during late life is positive and linear; There is substantial cohort heterogeneity in both average levels of depressive symptoms and age growth trajectories of depressive symptoms; The age growth trajectories of depressive symptoms are not significant after adjusting for cohort effects and risk factors associated with historical trends in education, life course stages, survival, health decline, stress and coping resources; Net of all the factors considered, more recent birth cohorts have higher levels of depression. 102 Conclusion A Webpage has been developed that contains copies of our papers referenced in this presentation as well as others: http://www.unc.edu/~yangy819/apc/index.html Happy Hunting for Age, Period, and Cohort Effects! 103