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Cohort Analysis
- LAST MODIFIED: 26 August 2013
- DOI: 10.1093/obo/9780199756384-0104

- LAST MODIFIED: 26 August 2013
- DOI: 10.1093/obo/9780199756384-0104

## Introduction

Cohort analysis deals with the conceptualization, estimation, and interpretation of the differential contributions of three dimensions of temporal changes in phenomena of interest to demographers, epidemiologists, and social scientists. Researchers in these disciplines often deal with population or sample survey data in the form of observations or measurements on individuals or other units of analysis, such as groups/populations of individuals that are repeated or ordered temporally. The focus of interest is on the effects of the three temporal dimensions: (1) the ages of the individuals at the time of observation on an outcome of interest, termed *age* (A) *effects*, (2) the time periods of observation or measurement of the outcome, termed *period* (P) *effects*, and (3) the year of birth or some other shared life events for a set of individuals, termed *cohort* (C) *effects*. To address this problem, researchers compare age-specific data recorded at different points in time and from different cohorts. A systematic study of such data is termed *age-period-cohort (APC) analysis*. APC analysis has the unique ability to depict parsimoniously the entire complex of social, historical, and environmental factors that simultaneously affect individuals and populations of individuals. It has thus been widely used to address questions of enduring importance to studies of social change, aging, the etiology of diseases, and population processes and dynamics.

## Reference Works

Relatively few books devoted solely to cohort analysis have been published; rather, most of the published works on this topic are peer-reviewed journal articles, entries in encyclopedias, or chapters in books. Mason and Fienberg 1985 is an edited volume containing fourteen chapters on conceptual and methodological approaches to APC (age-period-cohort) analysis and empirical applications thereof in demography and social research. It represents the state of the art of cohort analysis as of the mid-1980s. Glenn 2005 is a short monograph that reviews the APC literature in the social sciences up to the early 2000s and illustrates dilemmas in the identification and estimation of what will be described in this article as the age-period-cohort multiple classification/accounting model (see Age-Period-Cohort Multiple Classification/Accounting Model). Smith 2008 is the introductory essay to a special issue of a research methods journal containing four articles on new approaches to the identification and estimation of APC models that go beyond the classic methods described in Mason and Fienberg 1985 and Glenn 2005. Yang and Land 2013 is a systematic exposition of a unifying generalized linear mixed models (GLMM) statistical modeling framework for APC analysis, associated estimation methods, and empirical applications of cohort analysis.

Glenn, N. D. 2005.

*Cohort analysis*. 2d ed. SAGE University Papers Series: Quantitative Applications in the Social Sciences. Thousand Oaks, CA: SAGE.Save Citation »Export Citation »E-mail Citation »

This monograph describes identification and estimation problems in the classical APC multiple classification/accounting model as studied in the period from 1973 to the early 2000s.

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Mason, W. M., and S. E. Fienberg, eds. 1985.

*Cohort analysis in social research: Beyond the identification problem*. New York: Springer-Verlag.DOI: 10.1007/978-1-4613-8536-3Save Citation »Export Citation »E-mail Citation »

An authoritative collection of articles on the conceptualization of cohort effects, problems of identification in APC models, and empirical APC analysis in the period from 1965 to 1985.

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Smith, H. L. 2008. Advances in age-period-cohort analysis: Introduction. In

*Special issue: Age–period–cohort models revisited*.*Sociological Methods & Research*36.3: 287–296.DOI: 10.1177/0049124107310636Save Citation »Export Citation »E-mail Citation »

This article introduces a special issue of the journal on new approaches to APC analysis and places its four articles in the context of the broader literature on cohort analysis. Available online for purchase or by subscription.

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Yang, Y., and K. C. Land. 2013.

*Age-period-cohort analysis: New models, methods, and empirical applications*. Chapman and Hall/CRC Interdisciplinary Statistics Series. Boca Raton, FL: Chapman and Hall.DOI: 10.1201/b13902Save Citation »Export Citation »E-mail Citation »

Presents a unifying statistical framework for APC analysis, expounds the associated methods, and describes several empirical applications.

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## Foundational Works

Three articles stand out as foundational works on cohort analysis. APC (age-period-cohort) analysis has been a popular tool in epidemiology since employed in Frost 1940 in this classic study of tuberculosis. The article is primarily descriptive, using graphs to examine changing patterns of disease rates over time, and the tradition of graphical analysis of such data has continued to be used in epidemiology to the present. In sociology and demography, Ryder 1965 is the classic statement of the case for the use of cohorts in the study of change. Ryder argues persuasively that cohorts defined by birth and other major life events or transitions should be considered as important as socioeconomic categories/social class in structuring the life circumstances and behaviors of individuals. Mason, et al. 1973 defines the algebraic structure of the classical regression model for the APC analysis of age-by-time period tables of population rates or proportions (see Age-Period-Cohort Multiple Classification/Accounting Model) and argues that meaningful three-way cohort analysis is possible under some circumstances. This model has dominated the conceptual universe of cohort analysis for four decades and remains applicable today.

Frost, W. H. 1940. The age selection of mortality from tuberculosis in successive decades.

*Milbank Memorial Fund Quarterly*18.1: 61–66.DOI: 10.2307/3347652Save Citation »Export Citation »E-mail Citation »

An innovative descriptive analysis of changes over time in tuberculosis disease rates using graphical techniques. Available online for purchase or by subscription.

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Mason, K. O., W. M. Mason, H. H. Winsborough, and W. Kenneth Poole. 1973. Some methodological issues in cohort analysis of archival data.

*American Sociological Review*38.2: 242–258.DOI: 10.2307/2094398Save Citation »Export Citation »E-mail Citation »

Lays out the algebra of the APC identification problem in analyses of age-by-time period tables of population rates or proportions and argues that a meaningful three-way analysis of the A, P, and C effects often is possible. Available online for purchase or by subscription.

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Ryder, N. B. 1965. The cohort as a concept in the study of social change.

*American Sociological Review*30.6: 843–861.DOI: 10.2307/2090964Save Citation »Export Citation »E-mail Citation »

Emphasizes the persistence of societies through processes of demographic metabolism (births and deaths), the differentiation of cohorts through their unique life and idiosyncratic historical experiences, and the utility of cohorts as an analytic concept. Available online for purchase or by subscription.

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## The Age-Period-Cohort Multiple Classification/Accounting Model

The regression model described in Mason, et al. 1973 served for over three decades as a general methodology for estimating A, P, and C (age, period, and cohort) effects in demographic and social research. A similar three-factor analysis of variance model is proposed in Greenberg, et al. 1950 that also has been the dominant model for APC analysis in biostatistics and epidemiology for decades (see Holford 2005). This methodology uses conventional linear regression models or their generalized linear model extensions as an apparatus for the APC analysis of age-by-time period tables of population (occurrence/exposure) rates or proportions with one observation on an outcome variable per cell. In such a table, the age categories under study define the rows, the time periods define the columns, and the cohorts are arrayed along the diagonals. The APC multiple classification model correspondingly explains the variation in the outcome rates as a function of an effect coefficient vector that contains an overall mean or constant term plus a set of coefficients corresponding to the effects of each of the age categories plus a set of coefficients corresponding to the effects of each of the time periods plus a set of coefficients corresponding to each of the time periods. It was termed the *APC accounting model* in Mason, et al. 1973. Building on the tradition of descriptive graphical analysis in epidemiology and the three-factor analysis of variance model of Greenberg, et al. 1950 and Kupper, et al. 1983, it emphasizes that the cohort effects in the APC accounting model characterize a specific form of interaction between the categorical age and period variables that shows up visually as a lack of “parallelism” in period-specific graphs of population rates by age.

Greenberg, B. G., J. J. Wright, and C. G. Sheps. 1950. A technique for analyzing some factors affecting the incidence of syphilis.

*Journal of the American Statistical Association*45.251: 373–399.DOI: 10.1080/01621459.1950.10501131Save Citation »Export Citation »E-mail Citation »

Proposes a three-factor analysis of variance-type model to quantify the effects of the categorized A, P, and C variables and operationalizes A as a type III Pearson curve. Available online by subscription.

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Holford, T. R. 2005. Age-period-cohort analysis. In

*Encyclopedia of biostatistics*. Vol. 1. Edited by P. Armitage and T. Colton, 82–99. Hoboken, NJ: Wiley.DOI: 10.1002/0470011815Save Citation »Export Citation »E-mail Citation »

Reviews the APC analysis problem, three-factor APC models (in the form of the accounting/multiple classification models) to assess it, and empirical applications in biostatistics and epidemiology.

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Kupper, L. L., J. M. Janis, I. A. Salama, C. N. Yoshizawa, and B. G. Greenberg. 1983. Age-period-cohort analysis: An illustration of the problems in assessing interaction in one observation per cell data.

*Communications in Statistics: Theories and Measures*12.23: 201–217.DOI: 10.1080/03610928308828640Save Citation »Export Citation »E-mail Citation »

Studies the algebra of the three-factor APC analysis of variance model with numerical illustrations of applications to both hypothetical and empirical data. Available online for purchase or by subscription.

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Mason, K. O., W. M. Mason, H. H. Winsborough, and W. Kenneth Poole. 1973. Some methodological issues in cohort analysis of archival data.

*American Sociological Review*38.2: 242–258.DOI: 10.2307/2094398Save Citation »Export Citation »E-mail Citation »

Defines the APC multiple classification/accounting model and the associated identification problem, assesses the constrained coefficient approach to the estimation of the model, and describes problems of statistical inference with this approach. Available online for purchase or by subscription.

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### The Identification Problem

The APC multiple classification model offers an elegant and appealing framework within which to conduct cohort analysis. It is, however, an underidentified model. This is due to the fact that the design matrix of the model is singular with one less-than-full column rank. This is a result of the exact linear relationship between the A, P, and C variables (age-period-cohort), in an age-by-time period table of rates with the time intervals of the A, P, and C categories fixed and equal to each other, for example, in five-year intervals. This means that the ordinary least squares estimator for a conventional linear regression specification of the model cannot be calculated because the inverse of the matrix involved in the calculation of this estimator does not exist, and maximum likelihood estimators of generalized linear model specifications have a similar problem. Because the order of the underidentification is one, placing one additional constraint on the regression coefficients of the model, for example, an equality constraint that sets the effect coefficients for two adjacent age categories, two adjacent time periods, or two adjacent cohorts equal to each other, will result in a just-identified model that can be estimated. The problem, as Mason, et al. 1973 (cited under Age-Period-Cohort Multiple Classification/Accounting Model) illustrates, is that two different such equality constraints can produce very different estimates of the A, P, and C coefficients, thus imparting an uncertainty to inferences about the corresponding temporal trends across the three dimensions. This uncertainty led to vigorous debates (see Glenn 1976 and the Mason, et al. 1976 reply). Clayton and Schifflers 1987 gives an extensive review of the identification problem and various approaches to estimation of the APC accounting model in epidemiology.

Clayton, D., and E. Schifflers. 1987. Models for temporal variation in cancer rates, II: Age-period-cohort models.

*Statistics in Medicine*6.4: 469–481.DOI: 10.1002/sim.4780060406Save Citation »Export Citation »E-mail Citation »

Contains an exposition of the APC accounting model, its identification problem, and various approaches to estimation of the model in epidemiological applications. Available online for purchase or by subscription.

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Glenn, N. D. 1976. Cohort analysts’ futile quest: Statistical attempts to separate age, period and cohort effects.

*American Sociological Review*41.5: 900–904.DOI: 10.2307/2094738Save Citation »Export Citation »E-mail Citation »

Critiques the APC accounting model. Available online for purchase or by subscription.

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Mason, W. M., K. O. Mason, and H. H. Winsborough. 1976. Reply to Glenn.

*American Sociological Review*41.5: 904–905.DOI: 10.2307/2094739Save Citation »Export Citation »E-mail Citation »

Replies to the Glenn 1976 critique.

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### Reduced APC Accounting Models

Ryder 1965 (see Foundational Works) emphasizes that the processes of births (entries) and deaths (exits) of members of a population and the resulting personnel replacements have enormous potential for social change. One methodological response to the underidentification problem of the APC (age-period-cohort) accounting model is to adapt and/or develop decomposition methods for estimating these cohort replacement effects. Firebaugh 1989 describes six possible ways to estimate cohort (personnel) replacement and time-period effects and compares the numerical performance of the methods when applied to the same data set. Three of the methods are based on algebraic decomposition: the two-component demographic standardization method (Kitagawa 1955); the forward partitioning method (Das Gupta 1978); and the backward partitioning method (Glenn 1977). The remaining three methods are based on regression: the Clogg and Eliason 1988 regression standardization approach, the Firebaugh 1989 linear regression decomposition, and the Davis 1976 survey metric analysis. All of these methods use an aggregate population-level measure of some characteristic of individuals in the population (e.g., the percentage of adults who are literate) as an outcome variable. The APC accounting model underidentification problem thus is avoided by restricting the analysis to two temporal dimensions, time period and cohort. That is, the methods are based on reduced APC specifications that do not attempt to model age effects on an outcome variable and implicitly assume that there is no significant change over time in the age distribution of the outcome under study that interacts with either of the period or cohort temporal dimensions, a point of criticism in Glenn 2005 (see Reference Works). Lee, et al. 2007 and Lee, et al. 2010 are empirical applications of the Firebaugh 1989 linear regression decomposition approach.

Clogg, C. C., and S. R. Eliason. 1988. A flexible procedure for adjusting rates and proportions, including statistical methods for group comparisons.

*American Sociological Review*53.2: 267–283.DOI: 10.2307/2095692Save Citation »Export Citation »E-mail Citation »

A systematic approach to demographic standardization based on regression methods. Available online for purchase or by subscription.

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Das Gupta, P. 1978. A general method of decomposing a difference between two rates into several components.

*Demography*15.1: 99–112.DOI: 10.2307/2060493Save Citation »Export Citation »E-mail Citation »

Develops the forward partitioning method of demographic standardization of changes in an outcome variable that is based on holding group differences in rates or in population composition at their initial levels. Available online for purchase or by subscription.

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Davis, J. A. 1976. Analyzing contingency tables with linear flow graphs: D systems.

*Sociological Methodology*7:111–145.DOI: 10.2307/270706Save Citation »Export Citation »E-mail Citation »

Develops a decomposition method for analyzing contingency table data that focuses on the difference between a category (e.g., cohort) mean and the mean of a reference category.

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Firebaugh, G. 1989. Methods for estimating cohort replacement effects.

*Sociological Methodology*19:243–262.DOI: 10.2307/270954Save Citation »Export Citation »E-mail Citation »

Reviews methods for estimating cohort replacement effects and presents a linear regression decomposition method. Available online for purchase or by subscription.

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Glenn, N. D. 1977.

*Cohort analysis*. SAGE University Papers Series: Quantitative Applications in the Social Sciences. Beverly Hills, CA: SAGE.Save Citation »Export Citation »E-mail Citation »

An early monograph that reviews concepts and methods of APC analysis.

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Kitagawa, E. M. 1955. Components of a difference between two rates.

*Journal of the American Statistical Association*30.272: 1168–1194.Save Citation »Export Citation »E-mail Citation »

A classic article on demographic decomposition of the difference between two crude rates (e.g., crude death rates) into two components, one reflecting group differences in specific rates and the other reflecting group differences in population composition. Available online for purchase or by subscription.

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Lee, K. S., D. F. Alwin, and P. A. Tufis. 2007. Beliefs about women’s labour in the reunified Germany, 1990 to 2004.

*European Sociological Review*23.4: 487–503.Save Citation »Export Citation »E-mail Citation »

Applies the Firebaugh 1989 linear regression decomposition method for estimation of cohort replacement effects to changes in beliefs about women’s labor-force participation. Available online for purchase or by subscription.

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Lee, K. S., P. A. Tufis, and D. F. Alwin. 2010. Separate spheres or increasing equality: The transformation of gender beliefs in postwar Japan.

*Journal of Marriage and the Family*72.1: 184–201.DOI: 10.1111/j.1741-3737.2009.00691.xSave Citation »Export Citation »E-mail Citation »

Applies the Firebaugh 1989 linear regression decomposition method for estimation of cohort replacement effect to changes in beliefs about gender and gender roles. Available online for purchase or by subscription.

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### Estimable Functions

Among the methodological discussions of the underidentification problem of the APC (age-period-cohort) accounting model, Rodgers 1982 was early to propose that estimable functions of the A, P, and C effect coefficients should be the object of analysis (see also Smith, et al. 1982). Estimable functions are linear combinations of the effect coefficients that are invariant to whatever solution to the estimation problem is found, for example, invariant to whatever solution to the normal equations is chosen in a conventional linear regression model specification of the APC accounting model. Searle 1997 is a classic reference for exposition of estimable functions in linear models. Rodgers made the point that the underidentification of the accounting model is the same thing as stating that the main effects of A and P and the interactive effects of C in this model are not estimable without additional constraints on the model. It also argued that certain combinations of the A, P, and C effects and nonlinear specifications thereof are estimable. Estimable functions in the context of the APC accounting model also receive intensive study and extensive analysis in epidemiology in Holford 1983; Clayton and Schifflers 1987 (cited under Identification Problem); and Kupper, et al. 1983 (cited under Age-Period-Cohort Multiple Classification/Accounting Model). In particular, these works derived an estimability condition for identifying constraints on the A, P, and C coefficient vector and discuss a principal components regression estimator of this coefficient vector that satisfies the estimability condition.

Holford, T. R. 1983. The estimation of age, period and cohort effects for vital rates.

*Biometrics*39.2: 311–324.DOI: 10.2307/2531004Save Citation »Export Citation »E-mail Citation »

Presents an exposition of the identification problem in the APC accounting model, the problem of aliasing due to the linear dependence of the age, period, and cohort dimensions, and various approaches to the specification of estimable functions. Available online for purchase or by subscription.

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Rodgers, W. L. 1982. Estimable functions of age, period, and cohort effects.

*American Sociological Review*47.6: 774–787.DOI: 10.2307/2095213Save Citation »Export Citation »E-mail Citation »

Argues for the use of estimable functions in the APC accounting model. Available online for purchase or by subscription.

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Searle, S. R. 1997.

*Linear models*. New York: Wiley.Save Citation »Export Citation »E-mail Citation »

A systematic exposition of the algebra of linear models and their statistical estimation, including estimable functions.

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Smith, H. L., W. M. Mason, and S. E. Fienberg. 1982. Estimable functions of age, period, and cohort effects: More chimeras of the age-period-cohort accounting framework—Comment on Rodgers.

*American Sociological Review*47.6: 787–793.DOI: 10.2307/2095214Save Citation »Export Citation »E-mail Citation »

Responds to Rodgers 1982 and argues that structural underidentification problems in social science models are not unique to APC analysis. Available online for purchase or by subscription.

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### The Intrinsic Estimator

Fu 2000 defines what is termed the *intrinsic estimator* (IE) for the APC (age-period-cohort) accounting model. Algebraically, the intrinsic estimator is an orthonormal transformation of the principal components regression estimator of Kupper, et al. 1983 (see Age-Period-Cohort Multiple Classification/Accounting Model) back to the coordinates of the A, P, and C categories of the accounting model specification. This allows the coefficients to be interpreted in terms of these categories. Fu 2000 specifically studied the classical ridge estimator (Hoerl and Kennard 1970a, Hoerl and Kennard 1970b) in the singular design case, where the design matrix has one less-than-full rank, of which the design matrix for the APC accounting model is an example. The model shows that (1) the ridge estimator lies in a subparameter space orthogonal to the null space of the design matrix generated by the eigenvector of the zero eigenvalue, and (2) the ridge estimator converges to the IE as the shrinkage parameter of the ridge estimator tends to zero. In other words, the intrinsic estimator can be interpreted as the limit of the ridge estimator as its shrinkage penalty goes to zero. Further studies of the properties of the intrinsic estimator in Yang, et al. 2004 and Yang, et al. 2008 show that it satisfies the estimability condition of Kupper, et al. 1983, and report empirical applications and robustness analyses of this estimator. In particular, these articles show that the IE estimates an estimable function of the coefficient vector of the APC accounting model that corresponds to the Moore–Penrose generalized inverse solution to the estimation equations of the model. An add-on program for calculating the intrinsic estimator in Stata is contributed by Yang, et al. 2008 and can be downloaded from the Statistical Software Components Archive. O’Brien 2011a discusses constrained estimators in the APC accounting model and the intrinsic estimator in particular, expressing various concerns about the Moore–Penrose solution to the estimation problem of the accounting model. Fu, et al. 2011 comments on O’Brien 2011a and similarly discusses these topics to which O’Brien 2011b responds. This exchange establishes that the constrained coefficient vector estimated by the intrinsic estimator is estimable in the Kupper, et al. 1983 sense, but, due to the particulars of the specification of the APC accounting model and the constraint applied to obtain the intrinsic estimator, it is not possible to directly show that it satisfies the necessary and sufficient conditions for estimable functions given in Searle 1997 (see Estimable Functions). Tu, et al. 2012 studies the application of partial least squares (PLS) to the APC accounting model. Whereas principal components regression extracts the components independently of the outcome variable, PLS maximizes the covariance of the components with the outcome variable *Y*, extracting the components by order of this covariance from the highest to the lowest. Tu, et al. 2012 shows with an empirical application that, as the number of components extracted by PLS approaches the maximum number possible for a design matrix, the numerical values of the PLS estimates of the age, period, and cohort effect coefficients approach, and are within sampling error of, the corresponding coefficients estimated by the IE (which uses the maximum possible number of components). The research also shows that an estimator based on the first three PLS components is a numerically reasonable approximation to the PLS effect coefficients estimated by using the maximum possible number of components.

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*29.2: 263–278.DOI: 10.1080/03610920008832483Save Citation »Export Citation »E-mail Citation »

Defines the intrinsic estimator and shows that it can be viewed as a ridge regression estimator in the case of a singular design matrix. Available online for purchase or by subscription.

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Fu, W. J., K. C. Land, and Y. Yang. 2011. On the intrinsic estimator and constrained estimators in age-period-cohort models.

*Sociological Methods & Research*40.3: 453–466.DOI: 10.1177/0049124111415355Save Citation »Export Citation »E-mail Citation »

Discusses properties of the intrinsic estimator of the APC accounting model as a constrained estimator with empirical analyses. Available online for purchase or by subscription.

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Hoerl, A. E., and R. W. Kennard. 1970a. Ridge regression: Applications to nonorthogonal problems.

*Technometrics*12.1: 69–82.DOI: 10.1080/00401706.1970.10488635Save Citation »Export Citation »E-mail Citation »

Describes empirical applications of the ridge regression estimator. Available online for purchase or by subscription.

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Hoerl, A. E., and R. W. Kennard. 1970b. Ridge regression: Biased estimation for nonorthogonal problems.

*Technometrics*12.1: 55–67.DOI: 10.1080/00401706.1970.10488634Save Citation »Export Citation »E-mail Citation »

Defines the ridge regression estimator for regression models with highly collinear regressors. Available online for purchase or by subscription.

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O’Brien, R. M. 2011a. Constrained estimators and age-period-cohort models.

*Sociological Methods & Research*40.3: 419–452.DOI: 10.1177/0049124111415367Save Citation »Export Citation »E-mail Citation »

Discusses the algebra of estimators of the APC accounting model identified by placing various constraints on the coefficient vector of the model. Available online for purchase or by subscription.

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O’Brien, R. M. 2011b. Intrinsic estimators as constrained estimators in age-period-cohort accounting models.

*Sociological Methods & Research*40.3: 467–470.DOI: 10.1177/0049124111415369Save Citation »Export Citation »E-mail Citation »

Cites methodological and substantive points of agreement and disagreement in the exchange with Fu, et al. 2011. Available online for purchase or by subscription.

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Tu, Y.-K., N. Kramer, and W.-C. Lee. 2012. Addressing the identification problem in age-period-cohort analysis: A tutorial on the use of partial least squares and principal components analysis.

*Epidemiology*23.4: 583–593.DOI: 10.1097/EDE.0b013e31824d57a9Save Citation »Export Citation »E-mail Citation »

Studies the application of partial least squares (PLS) to the APC accounting model and makes comparisons with principal components regression and the intrinsic estimator. Available online for purchase or by subscription.

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Yang, Y., W. J. Fu, and K. C. Land. 2004. A methodological comparison of age-period-cohort models: The intrinsic estimator and conventional generalized linear models.

*Sociological Methodology*34:75–110.DOI: 10.1111/j.0081-1750.2004.00148.xSave Citation »Export Citation »E-mail Citation »

Compares properties of the intrinsic estimator of APC accounting models with those of estimator obtained by arbitrary constraints on such models, with empirical applications. Available online for purchase or by subscription.

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Yang, Y., S. Schulhofer-Wohl, W. J. Fu, and K. C. Land. 2008. The intrinsic estimator for age-period-cohort analysis: What it is and how to use it.

*American Journal of Sociology*113.6: 1697–1736.DOI: 10.1086/587154Save Citation »Export Citation »E-mail Citation »

Presents an exposition of the intrinsic estimator and its statistical properties, including empirical and Monte Carlo simulation studies. Available online for purchase or by subscription.

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### Bayesian, Smoothing Cohort, Accelerating Trends, and Mixed Model Estimators

The identification and estimation problem of the APC (age-period-cohort) accounting model has been quite challenging to statistical methodologists. The response has been the specification and development of estimators for the accounting model other than those derived from equality constraints or estimable functions approaches. Nakamura 1982 and Nakamura 1986 develop a Bayesian approach introduced to American social scientists in Sasaki and Suzuki 1987. The Nakamura model achieves identification by assuming that the A, P, and C coefficients change gradually from age category to age category, time period to time period, and cohort to cohort. It then selects model parameters that maximize a likelihood function of the model and minimize a quadratic function of successive changes in the A, P, and C effect coefficients. The model is Bayesian in the sense that it selects coefficient values that minimize the Akaike 1980 ”information criterion.” Sasaki and Suzuki 1987 is critiqued in Glenn (1989) who cautions against “mechanical solutions” (p. 754) to the APC identification problem. More recently, Fu 2008 specifies a smoothing cohort approach to estimation of the APC accounting model in which the cohort coefficients are replaced by smoothed effects that are estimated by smoothing splines. The smoothing cohort model takes advantage of the fact that adjacent cohorts usually overlap in time to provide a rationale for the estimation of cohort effects with contributions from nearest neighbors. The smoothing cohort model does not suffer from the identifiability problem and yields unique estimates of the A, P, and C effects. Relatedly, Kuang, et al. 2008 proposes a canonical parameterization of the APC accounting model based on the accelerations of trends in the three factors that is exactly identified and eases interpretation, estimation, and forecasting. Using a different statistical modeling framework, O’Brien, et al. 2008 specifies a mixed (fixed and random) effects APC accounting model in which cohort effects are random while age and period effects are fixed.

Akaike, H. 1980. Likelihood and the Bayes procedure. In

*Bayesian statistics: Proceedings of the first international meeting held in Valencia, May 28 to June 2, 1979*. Edited by J. M. Bernardo, M. H. DeGroot, D. V. Lindley, and A. F. M. Smith, 143–166. Valencia, Spain: Univ. Press.Save Citation »Export Citation »E-mail Citation »

Studies the relationship of likelihood functions to Bayesian inference and defines an “information criterion.”

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Fu, W. J. 2008. A smoothing cohort model in age–period–cohort analysis with applications to homicide arrest rates and lung cancer mortality rates.

*Sociological Methods & Research*36.3: 327–361.DOI: 10.1177/0049124107310637Save Citation »Export Citation »E-mail Citation »

Develops an approach to estimation of the APC multiple classification model that can accommodate a flexible structure for the age, period, and cohort effects and avoids the identifiability problem of this model. Available online for purchase or by subscription.

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Glenn, N. D. 1989. A caution about mechanical solutions to the identification problem in cohort analysis: Comment on Sasaki and Suzuki.

*American Journal of Sociology*95.3: 754–761.DOI: 10.1086/229332Save Citation »Export Citation »E-mail Citation »

Critiques the Sasaki and Suzuki 1987 article. Available online for purchase or by subscription.

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Kuang, D., B. Nielsen, and J. P. Nielsen. 2008. Forecasting with the age-period-cohort model and the extended chain-ladder model.

*Biometrika*95.4: 987–991.DOI: 10.1093/biomet/asn038Save Citation »Export Citation »E-mail Citation »

Develops the accelerating trends in the A, P, and C factors parameterization to exactly identify the APC accounting model. Available online for purchase or by subscription.

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Nakamura, T. 1982. A Bayesian cohort model for standard cohort table analysis.

*Proceedings of the Institute of Statistical Mathematics*29.2: 77–97.Save Citation »Export Citation »E-mail Citation »

The first of author’s Bayesian cohort model analyses.

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Nakamura, T. 1986. Bayesian cohort models for general cohort table analyses.

*Annals of the Institute of Statistical Mathematics*38B.2: 353–370.DOI: 10.1007/BF02482523Save Citation »Export Citation »E-mail Citation »

The second of the author’s Bayesian cohort models.

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O’Brien, R. M., K. Hudson, and J. Stockard. 2008. A mixed model estimation of age, period, and cohort effects.

*Sociological Methods & Research*36.3: 402–428.DOI: 10.1177/0049124106290392Save Citation »Export Citation »E-mail Citation »

In the context of the APC multiple classification/accounting model, this article specifies a mixed models approach that treats cohort effects as random and period effects as fixed with empirical applications to homicide and suicide population rate data. Available online for purchase or by subscription.

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Sasaki, M., and T. Suzuki. 1987. Changes in religious commitment in the United States, Holland, and Japan.

*American Journal of Sociology*92.5: 1055–1076.DOI: 10.1086/228627Save Citation »Export Citation »E-mail Citation »

Applies the Nakamura Bayesian estimator of the APC accounting model. Available online for purchase or by subscription.

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### Proxy Variables and Causal Models

Another approach to identification and estimation of the APC (age-period-cohort) accounting model uses one or more proxy variables to replace the A, P, or C variable. This is a popular approach because of its substantive appeal. As Hobcraft, et al. 1982 argues, the indicator variables of A, P, and C often serve as surrogates for different sets of unmeasured structural correlates. Examples of proxy variables for cohort effects include the use of relative cohort size in O’Brien, et al. 1999 and cohort mean years of smoking before age 40 in Preston and Wang 2006. Unemployment rates, labor-force size, and gender-role attitudes are used as proxy variables for period effects in Pavalko, et al. 2007. The use of cohort characteristics to replace cohort effects in APC accounting models has been termed the *age-period-cohort characteristics* (APCC) model in O’Brien 2000. Although replacing an accounting dimension with measured variables solves an identification problem, it makes room for others. In particular, the proxy-variables approach assumes that all of the variation associated with the A, P, or C dimension is fully accounted for by the chosen proxy variable(s). Winship and Harding 2008 proposes a mechanism-based approach that accommodates a more general set of models. Using the front-door criterion for the identification of causal effects in the nonparametric framework of causal modeling described in Pearl 2009 to achieve identification, the approach allows any given measured variable to be associated with more than one of the age, period, and cohort dimensions and provides statistical tests for the plausibility of alternative restrictions. If a rich set of mechanism variables is available and the original age, period, and cohort categories can be conceived as the exogenous elements of a causal chain, this is an enriched and sophisticated alternative to the proxy-variable approach.

Hobcraft, J., J. Menken, and S. Preston. 1982. Age, period, and cohort effects in demography: A review.

*Population Index*48.1: 4–43.DOI: 10.2307/2736356Save Citation »Export Citation »E-mail Citation »

A thorough review and analysis of the use of APC concepts and analysis in the study of fertility, mortality, and migration processes in demography up to the date of publication. Available online for purchase or by subscription.

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O’Brien, R. M. 2000. Age period cohort characteristic models.

*Social Science Research*29.1: 123–139.DOI: 10.1006/ssre.1999.0656Save Citation »Export Citation »E-mail Citation »

Articulates a proxy-variable approach to APC analysis that uses enduring measured features of cohorts to replace the C categories in the APC accounting model.

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O’Brien, R. M., J. Stockard, and L. Isaacson. 1999. The enduring effects of cohort characteristics on age-specific homicide rates, 1960–1995.

*American Journal of Sociology*104.4: 1061–1095.DOI: 10.1086/210136Save Citation »Export Citation »E-mail Citation »

Applies a cohort characteristic proxy-variable approach for APC analysis. Available online for purchase.

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Pavalko, E. K., F. Gong, and J. S. Long. 2007. Women’s work, cohort change, and health.

*Journal of Health and Social Behavior*48.4: 352–368.DOI: 10.1177/002214650704800402Save Citation »Export Citation »E-mail Citation »

Uses proxy variables for period effects to identify and estimate an APC model. Available online for purchase or by subscription.

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Pearl, J. 2009.

*Causality: Models, reasoning, and inference*. New York: Cambridge Univ. Press.DOI: 10.1017/CBO9780511803161Save Citation »Export Citation »E-mail Citation »

A systematic analysis of causal models in scientific research.

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Preston, S. H., and H. Wang. 2006. Sex mortality differences in the United States: The role of cohort smoking patterns.

*Demography*43.4: 631–646.DOI: 10.1353/dem.2006.0037Save Citation »Export Citation »E-mail Citation »

Uses cohort mean years of smoking before age 40 as a proxy variable for cohort effects in an APC accounting model analysis of sex differences in mortality in the age range 50–84. Available online for purchase or by subscription.

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Winship, C., and D. J. Harding. 2008. A mechanism-based approach to the identification of age–period–cohort models.

*Sociological Methods & Research*36.3: 362–401.DOI: 10.1177/0049124107310635Save Citation »Export Citation »E-mail Citation »

Applies the causal mechanism modeling approach of Pearl 2009 to identify and estimate an APC accounting model. Available online for purchase or by subscription.

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## Hierarchical Age-Period-Cohort Models

The analytic models and associated identification problems of age-period-cohort (APC) analysis originally were developed for a very limited research design in the form of an age-by-time period table of population rates or proportions (see Age-Period-Cohort Multiple Classification/Accounting Model) with one observation per cell. In recent decades, new research designs have produced data sets with more information content. When combined with recently developed statistical models, these new research designs open up new opportunities for APC analysis.

### Repeated Cross-Section Surveys

One research design for which data sets are increasingly available is a repeated cross-section survey in which representative samples of a population are taken repeatedly over time but with little likelihood that the same respondents will be present in more than one cross section. Yang and Land 2006 was the first to recognize that the microdata on the individual members in repeated surveys facilitates the use of different temporal units for the ages of the respondents, the time periods in which the surveys are administered, and the birth cohorts to which members of the surveys belong. This led to a cross-classified matrix in which the individual respondents from the repeated surveys could be nested by the birth cohort and survey time period to which they belong. To allow for the possibility that sample members from a specific cohort–survey period grouping could share correlated errors or variances in an outcome variable, Yang and Land 2006 specifies hierarchical age-period-cohort (HAPC) mixed (fixed and random) effects statistical models. These HAPC models can be specified in either linear or generalized linear form. In their simplest form, the effects on an outcome variable of age (and functions thereof) and other individual-level covariates such as sex, race/ethnicity, and socioeconomic indicators are specified as fixed, and the effects of the contextual variables—time periods and cohorts—within which the individual-level observations are nested then are specified as random. On the basis of either hypotheses founded on theory or prior research, or of statistical evidence from the data, cross-level interactions of the effects of individual-level covariates with the random contextual effects then can be examined. Because the resulting HAPC models are not single-level regression models of the type used in the APC accounting model, they do not have an underidentification problem. Details on the statistical methodology and empirical applications are presented in Yang and Land 2006 and in several related publications. Yang 2006 presents a full Bayesian approach to estimation of HAPC models. Yang 2008 applies HAPC models to the study of the A, P, and C components of changes in happiness across four decades, Yang and Land 2008 compares pure fixed-effects and mixed-effects specification of HAPC models. Zheng, et al. 2011 intersects HAPC models with heteroscedastic regression models to further extend the power of the models to study the A, P, and C components of health disparities. Yang and Land 2013 presents a unifying generalized linear mixed models (GLMM) statistical modeling framework for HAPC analysis, associated estimation methods, and empirical applications of cohort analysis.

Yang, Y. 2006. Bayesian inference for hierarchical age-period-cohort models of repeated cross-section survey data.

*Sociological Methodology*36:39–74.DOI: 10.1111/j.1467-9531.2006.00174.xSave Citation »Export Citation »E-mail Citation »

Presents a full Bayesian statistical estimation and inference approach to HAPC models with empirical applications. Available online for purchase or by subscription.

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Yang, Y. 2008. Social inequalities in happiness in the United States, 1972 to 2004: An age-period-cohort analysis.

*American Sociological Review*73.2: 204–226.DOI: 10.1177/000312240807300202Save Citation »Export Citation »E-mail Citation »

Applies HAPC models to the estimation of the A, P, and C components of changes in happiness across time. Available online for purchase or by subscription.

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Yang, Y., and K. 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*36:75–97.DOI: 10.1111/j.1467-9531.2006.00175.xSave Citation »Export Citation »E-mail Citation »

Develops the HAPC framework for modeling the time period and cohort contextual effects of individual-level observations from repeated cross-section surveys. Available online for purchase.

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Yang, Y., and K. C. Land. 2008. Age–period–cohort analysis of repeated cross-section surveys: Fixed or random effects?

*Sociological Methods & Research*36.3: 297–326.DOI: 10.1177/0049124106292360Save Citation »Export Citation »E-mail Citation »

Describes and compares pure fixed effects models for repeated cross-section microdata sets with mixed (fixed and random) effects models in which cohort and period effects are specified as random. Available online for purchase or by subscription.

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Yang, Y., and K. C. Land. 2013.

*Age-period-cohort analysis: New models, methods, and empirical applications*. Chapman and Hall/CRC Interdisciplinary Statistics Series. Boca Raton, FL: Chapman and Hall.DOI: 10.1201/b13902Save Citation »Export Citation »E-mail Citation »

Presents a unifying statistical framework for APC analysis, exposits the associated methods, and describes several empirical applications.

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Zheng, H., Y. Yang, and K. C. Land. 2011. Variance function regression in hierarchical age-period-cohort models: Applications to the study of self-reported health.

*American Sociological Review*76.6: 955–983.DOI: 10.1177/0003122411430940Save Citation »Export Citation »E-mail Citation »

Synthesizes the HAPC framework with heteroscedastic regression models to facilitate the analysis of the A, P, and C components of changes in both conditional means and conditional variances of an outcome variable. Available online for purchase or by subscription.

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### Accelerated Longitudinal Panel Studies

Both the classic age-by-time period tables of rates and the repeated cross-sectional surveys research designs adopt a synthetic cohort approach. These designs generally do not contain data on the same individuals across time. Inferences drawn from such designs, therefore, assume that synthetic cohorts obtained from concatenating data from different individuals from the same cohorts mimic true cohorts, and that changes over time across synthetic cohort members mimic the age trajectories of change within true cohorts. If the composition of cohorts does not change over time due to migration or other factors and sample sizes are large, these assumptions are generally met. However, longitudinal data obtained from the same persons followed over time are increasingly available. The primary advantage of longitudinal panel data designs is that they provide cross-time linkages within individuals and hence information pertaining to true birth cohorts. The simplest longitudinal design follows one birth cohort for a period of time but cannot be used for cohort analysis due to lack of data on different cohorts. An accelerated cohort design follows multiple cohorts forward over multiple points in time and is an important advance in aging and cohort research. This design allows a more rapid accumulation of information on age for multiple cohorts than does a single longitudinal cohort design and makes cohort analysis possible. Differences among individuals within different cohorts (and/or time periods, if the study extends sufficiently long in time that potential period effects could be present in the data) can be modeled within the hierarchical age-period-cohort framework of Yang and Land 2013. Several empirical applications have been published. Using an accelerated longitudinal panel data set on depression and this approach to APC modeling, Yang 2007 discovers independent age and cohort effects: birth cohorts with different formative experiences have distinct trajectories of change in mental health with age. This highlights the relevance of social historical context represented by cohort membership to health outcomes above and beyond socioeconomic and more proximate behavioral correlates. Yang and Lee 2009 and Yang and Lee 2010 further analyze intercohort variations and intracohort disparities in health over the life course using multiple indicators of physical and mental health and both individual and cumulative health indicators. These studies jointly provide strong tests of the proposition that consideration of the process of cohort change is important for the theory, measurement, and analysis of social inequalities in health over the life course. See also Yang and Land 2013.

Yang, Y. 2007. Is old age depressing? Growth trajectories and cohort variations in late-life depression.

*Journal of Health and Social Behavior*48.1: 16–32.DOI: 10.1177/002214650704800102Save Citation »Export Citation »E-mail Citation »

Studies age-by-cohort interactions within a multilevel model of the age dependence of depressive symptoms. Available online for purchase or by subscription.

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Yang, Y., and K. C. Land. 2013.

*Age-period-cohort analysis: New models, methods, and empirical applications*. Chapman and Hall/CRC Interdisciplinary Statistics Series. Boca Raton, FL: Chapman and Hall.DOI: 10.1201/b13902Save Citation »Export Citation »E-mail Citation »

Presents a unifying statistical framework for APC analysis, including data from accelerated longitudinal cohort designs.

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Yang, Y., and L. C. Lee. 2009. Sex and race disparities in health: Cohort variations in life course patterns.

*Social Forces*87.4: 2093–2124.DOI: 10.1353/sof.0.0183Save Citation »Export Citation »E-mail Citation »

Applies the HAPC model to the study of intercohort and intracohort sex and race health disparities. Available online purchase or by subscription.

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Yang, Y., and L. C. Lee. 2010. Dynamics and heterogeneity in the process of human frailty and aging: Evidence from the US older adult population.

*Journal of Gerontology: Social Sciences*65B.2: 246–255.DOI: 10.1093/geronb/gbp102Save Citation »Export Citation »E-mail Citation »

Applies the HAPC model to examine intercohort and intracohort differences in biological frailty trajectories in older adults.

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