Executive Summary Project title: Listeria monocytogenes

17© Elsevier Inc. All rights reserved.DOI:

2011 10.1016/B978-0-12-380880-6.00002-2

Aging, Cohorts, and Methods Yang Yang Department of Sociology & Lineberger Comprehensive Cancer Center, University of North Carolina, Chapel Hill, North Carolina

CHAPTER CONTENTS

Introduction 17 Early Literature 18

Why Cohort Analysis? 18 Why is there an APC Identification Problem? 18 Conventional Solutions 18

New Developments: Models, Methods, and Substantive Research 19

Research design I: Age-by-Time Period Tables of Rates/Proportions 19

Guidelines for Estimating APC Models of Rates 19

Temporal Changes of US Adult Lung Cancer Mortality 20

Conclusion 21

Research Design II: Repeated Cross-Sectional Surveys 21

Hierarchical Age-Period-Cohort (HAPC) Models 22

Social Inequality of Happiness 22

Conclusion and Extensions 23

Research Design III: Accelerated Longitudinal Panels 24

Growth Curve Models of Individual and Cohort Change 24

Sex and Race Health Disparities Across the Life Course 25

Conclusion 27

Directions for Future Research 27 Continuously Evolving Cohort Effects Model 27 Longitudinal Cohort Analysis of Balanced Age-by-Cohort Data Structure 28

References 29

INTRODUCTION

Theoretical developments combined with new meth-odological tools and data sources have contributed to considerable growth in aging research in the social sciences over the past few decades ( Elder, 1985 ). To address key questions of aging and the life course paradigm – how individual lives unfold with age and are shaped by historical time and social con-text – researchers often need to compare age-specific data recorded at different points in time and from different birth cohorts. A systematic study of such data, termed the age-period-cohort (APC), or simply “cohort analysis,” is one of the most useful means to gain a greater understanding of the process of aging within and across human populations.

The challenges posed by cohort analysis are well known. Whether observed time-related differences are related to aging or cohort variation is a question usually deemed conceptually important but empiri-cally intractable for two reasons. The first is data limitations. Using cross-sectional data at one point in time, for example, aging and cohort effects are con-founded. Similarly, comparing data from multiple time periods for the same age group precludes dis-tinguishing between period and cohort effects. The second reason is the use of linear regression models that suffer from either specification errors or an iden-tification problem and consequently are incapable of distinguishing age, period, and cohort effects. What should one do to model these effects in empirical research to understand the true social and biological mechanisms generating the data? This chapter aims to provide some useful guidelines on how to conduct cohort analysis. The expository strategy is to provide those guidelines in the context of illustrative empiri-cal analyses. The remainder of this chapter is organ-ized as follows. The next section summarizes the state of knowledge on cohort analysis in aging research in

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Part THEORY AND METHODS| 1 |

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sociology, demography, and epidemiology. Following on further, the next section synthesizes new develop-ments in cohort analysis, including models, methods, and substantive applications. Finally, the last section outlines new directions for which methods and data are promising, but not yet completely developed.

EARLY LITERATURE

Why Cohort Analysis? In a classic article, Norman Ryder (1965) made an extended argument for the conceptual relevance of cohorts to an extraordinary range of substantive issues in social research. He argued that cohort member-ship could be as important in determining behavior as other social structural features such as socioeco-nomic status (SES). APC analysis is, in this sense, syn-onymous with cohort analysis ( Smith, 2008 ). Among various cohorts defined by different initial events (such as marriage and college/university entrance), birth cohorts are the most commonly examined unit of analysis in demographic and aging research. A birth cohort shares the same birth year and ages together. Birth cohorts born in different time periods that encounter different historical and social conditions as they age would conceivably have diverse develop-mental paths. Cohort analysis concerns the precise depiction of such time-related phenomena, is central to the study of aging and the life course, and is crucial for inference. Age effects represent aging-related physi-ological or developmental changes and offer clues to etiology in epidemiologic studies. Period effects reflect changes in contemporaneous social, historical, and epidemiologic conditions that affect all living cohorts. Cohort effects reflect different formative experiences resulting from the intersection of individual biogra-phies and macrosocial influences. In the absence of period and cohort changes, age effects are broadly applicable across historical eras and different cohorts. Presence of either or both of these changes, however, indicates the existence of exogenous forces or expo-sures that are period- and/or cohort-specific.

Why is there an APC Identification Problem? The essence of cohort analysis is the identification and quantification of different sources of variation that are associated with age, period, and/or cohort effects in an outcome of interest. Early investigators developed models for situations in which all three factors account for a substantive phenomenon and in which simpler two-factor models (such as an age-period model) are subject to model specification errors and spurious results ( Mason et al., 1973 ). The

“conventional linear regression model,” also known as the APC multiple classification/accounting model ( Mason et al., 1973 ), has been widely adopted as the general methodology for cohort analysis, but it also suffers from a well-known model identification problem. Specifically, any two of the constituent fac-tors perfectly predict the third. For example, if date of birth (i.e. birth cohort) and time of measurement are known, age also is known. This produces multiple instead of unique estimates of the three effects.

The Mason et al. (1973) article spawned a large methodological literature in the social sciences begin-ning with Glenn’s critique (1976) and Mason et al.’s reply (1976), followed by Fienberg and Mason’s work (1978). Similar debates occurred between Rodgers (1982) and Smith et al. (1982) . These early investi-gations culminated in an edited volume on cohort analysis by Mason and Fienberg in 1985. Limitations of those analytic strategies propelled the search for new approaches. For example, a Bayesian approach was proposed by Saski and Suzuki (1987) , which was again critiqued by Glenn, who cautioned against “mechanical solutions” to the problem (1987). APC analysis also took root in biostatistics and epidemi-ology (e.g. Clayton & Schifflers, 1987; Holford, 1991; Kupper et al., 1985 ). Several recent reviews provide useful additional material on these and related con-tributions to cohort analysis ( Mason & Wolfinger, 2002; Yang, 2007a, 2009 ).

Conventional Solutions Within the framework of linear regression models, three conventional solutions to the identification problem have been used. The purpose of each is to break the linear dependency between the three vari-ables and, thus, to identify the model, but each has limitations.

The constrained coefficients approach imposes one or more equality constraints on the coefficients of one parameter vector to just-identify (one constraint) or over-identify (two or more constraints) the model ( Mason et al., 1973; Yang et al., 2004 ). For exam-ple, in an APC analysis of US tuberculosis mortality, Mason and Smith (1985) placed the constraint by equating the contrasts for ages 0–9 and 10–19. This yielded unique estimates for age, period, and cohort effects. This is the most widely used approach, but is not immune to problems. First, the analyst must rely on prior or external information that rarely exists or cannot be verified to select constraints. Second, dif-ferent identifying constraints (e.g. two age groups vs two cohorts) can produce widely different estimates of age, period, and cohort effects. And third, all just-identified constrained coefficients models produce the same levels of goodness-of-fit to the data. Thus, model fit cannot be used as the criterion for selecting the best model ( Yang et al., 2008 ).

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The nonlinear parametric (algebraic) transformation approach defines a nonlinear parametric function for at least one of the age, period, and cohort variables so that its relationship to others is nonlinear (Mason & Fienberg, 1985). For example, one can specify a quad-ratic or cubic function of age to model the nonlinear rate of change in happiness with age ( Yang, 2008a ). The drawbacks of this approach are that: (1) specify-ing a structure of the parameters makes the model less flexible and (2) it may not be clear what nonlinear function should be defined.

The proxy variables approach uses one or more proxy variables to replace the age, period, or cohort vari-able. This is a popular approach because of its sub-stantive appeal. After all, the indicator variables of age, period, and cohort serve as surrogates for differ-ent sets of unmeasured structural correlates ( Hobcraft et al., 1982 ). Examples of proxy variables for cohort effects include relative cohort size ( O’Brien et al., 1999 ) and cohort mean years of smoking before age 40 ( Preston & Wang, 2006 ). Unemployment rate, labor force size, and gender role attitudes have been used as proxy variables for period effects ( Pavalko et al., 2007 ). Smith and colleagues (1982, p. 792) , however, cautioned that replacing cohort or period with a measured variable “leaves open the question of whether all of the right measured variables have been included in an appropriately wrought speci-fication. Although replacing an accounting dimen-sion with measured variables solves an identification problem, it makes room for others.” Thus, assuming that all of the variation associated with the A, P, or C dimension is fully accounted for by the chosen proxy, variable(s) may be unwarranted. Winship and Harding (2008) proposed a mechanism-based approach that accommodates a more general set of models. Using the framework of causal modeling described by Pearl (2000) 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 are available and the orig-inal age, period, and cohort categories can be con-ceived as the exogenous elements of a causal chain ( Smith, 2008 ), this is an enriched and sophisticated alternative to the proxy variable approach.

Where does the early literature on cohort analy-sis leave us today? If a researcher has a tempo-rally ordered data set and wants to tease out its age, period, and cohort components, how should he/she proceed? Can any methodological guidelines be recommended? A problem with much of the extant literature is that there is a lack of useful guidelines on how to conduct an APC analysis. Instead, one is left with the impression that either it is impossible to obtain meaningful estimates of the distinct con-tributions of age, cohort, and time period, or that

the conduct of an APC analysis is an esoteric art that is best left to a few skilled methodologists ( Yang, 2009 ). I now seek to redress this situation by focus-ing on recent developments in APC analysis for three research designs commonly used in research on aging.

NEW DEVELOPMENTS: MODELS, METHODS, AND SUBSTANTIVE RESEARCH

In this section, I introduce a new estimation method to solve the identification problem within the context of conventional linear models. Note that the iden-tification problem is not inevitable in all settings – only in the case of linear models that assume addi-tive effects of age, period, and cohort. The text then describes other families of models that are not subject to this problem and are suitable for many research questions. Empirical studies of aging, health, and well-being for each of the three research designs will be use as illustrative examples. Sample codes using standard statistical software packages are available at http://home.uchicago.edu/~yangy/apc .

Research Design I: Age-by-Time Period Tables of Rates/Proportions In demographic and epidemiologic investigations, researchers are typically interested in aggregate population-level data such as rates of morbidity, dis-ability, or mortality. Using the conventional data structure, rates or proportions are arranged in rectan-gular arrays with age intervals defining the rows and time periods defining the columns. Age and period are usually of equal interval length (e.g. 5 or 10 years), so the diagonal elements of the matrix correspond to birth cohorts. An example of this data structure is shown in Figure 2.1 , where age-specific lung cancer death rates for 14 five-year age groups (30 to 95�) of women are plotted by eight time periods between 1960 and 1999 and for 21 cohorts born between 1870 and 1920 (mid-year).

Guidelines for Estimating APC Models of Rates Many previous applications of cohort analysis using linear regression models are based on the age-by-period matrix. Among the limitations of con-ventional solutions to the identification problem is the lack of guidelines for estimating APC models of rates or proportions. A three-step procedure is help-ful for this purpose. Step 1 is to conduct descriptive data analyses using graphics. The objective is to pro-vide qualitative understanding of patterns of age,


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period, cohort, and two-way age by period and age by cohort variations. Step 2 is model fitting. The objec-tive is to ascertain whether the data are sufficiently well-described by any single factor or two-way com-bination of age, period, and cohort. If these analyses suggest that only one or two of the three effects are operative, the analysis can proceed with a reduced model that omits one or two groups of variables and there is no identification problem. If, however, these analyses suggest that all three dimensions are at work, Yang and colleagues (2004, 2008) recommend Step 3: applying a new method of estimation of the linear model, the intrinsic estimator (IE). We will now look at these steps using examples from recent analyses of US adult chronic disease mortality change ( Yang, 2008b ). The example below focuses on lung cancer mortality (see Yang, 2008b for the complete set of analyses of other causes of death).

Temporal Changes of US Adult Lung Cancer Mortality Figure 2.1 is a graphical presentation of the female lung cancer rate data for the Step 1 descriptive anal-yses. The age pattern shows much higher rates of deaths due to lung cancers in older ages than young adulthood. Rates of lung cancer deaths in women

aged 40 and older also increased dramatically. Large increases in successive cohorts are also evident: more recent cohorts of women had higher death rates than their predecessors at the same ages. Because cohort effects can be interpreted as a special form of interac-tion effect between age and period, cohort effects can be detected by non-parallel age-specific curves by time period – a pattern evident in Figure 2.1 . Peak ages shifted to the left over time, indicating higher death rates at younger ages for more recent female cohorts. The graphical analyses present all available rates, but provide no summary or quantitative assessment of the source of mortality change ( Kupper et al., 1985 ). The curve of age-specific death rates for women in any given time period cuts across multiple birth cohort curves. Therefore, the period curve is affected by both varying age effects is cohort effects. The question of which of these effects is more important in explain-ing the mortality change can only be addressed using regression modeling.

In Step 2, the relative importance of the A, P, and C dimensions can be determined by comparing over-all measures of model fit (e.g. R-squared for linear regression models or deviance statistics/penalized likelihood functions such as Bayes the Information Criterion for generalized linear models). For this analysis, goodness-of-fit statistics for four log-linear

Figure 2.1 US female lung cancer mortality rates per 100 000 by age at death, period of death, and birth cohort. Source: Yang (2008b) .

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models of female lung cancer mortality were pro-duced: age effects only (A), age and period effects (AP), age and cohort effects (AC), and age, period, and cohort effects (APC). The three-factor model best fit the data, regardless of the statistics used. Thus, all three sources of variation contribute to the mortality change and need to be included in the model.

In Step 3, one estimates the APC model coefficients to reveal the age, period, and cohort effects. The IE is a new method of estimation that yields a unique solu-tion to the APC model and is the unique estimable function of both the linear and nonlinear compo-nents of the APC model. It has minimal assumptions and is estimated using principal component regres-sion. The reason why the IE is useful lies in the spe-cial constraint it imposes – removing the influence of the number of age and period groups (which is not related to the outcome) on coefficient estimates. This constraint allows the IE to avoid the bias that the con-ventional equality constraints produce. For the math-ematical foundations of the IE see Yang et al. (2004) and for additional information about its statistical properties see Yang et al. (2008) .

The log coefficient estimates using the IE on the lung cancer mortality data by sex are shown in Figure 2.2 . Age effects are concave: the mortality

risk increased rapidly from early adulthood, peaked around ages 80 to 85, and then leveled off. Cohort mortality curves increased first and decreased for more recent cohorts, following an inverse-U shape. Period effects are modest compared to age and cohort effects.

Conclusion Is the IE a “final” or “universal” solution to the APC “conundrum”? No. There will never be a solution within the confine of linear models . Researchers should never naively apply any estimator to APC data and expect to get meaningful results. In all cases, APC analysis should be approached with great caution and awareness of its many pitfalls. The IE appears to be the most useful approach to the identification and estima-tion of these models, but shares some of the limita-tions of conventional linear models.

Research Design II: Repeated Cross-Sectional Surveys Repeated cross-sectional surveys, such as the Current Population Survey (CPS) and General Social Survey (GSS), provide unique opportunities for cohort analysis. Pooling data of all survey years yields the

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Figure 2.2 Intrinsic estimates of age, period, and cohort effects of lung cancer mortality by sex. Source: Yang (2008b) .


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age-by-year data structure needed for cohort analysis. Many researchers previously assumed that the APC identification problem for repeated cross-sectional surveys is the same as that for population rates. But sample surveys are different from aggregate popula-tion-level data in two important ways. First, repeated cross-sectional surveys allow age intervals to differ from period intervals. More importantly, the sample size for each age-by-period cell exceeds rather than equals one. This data structure suggests a multi-level data structure. In this design, respondents are nested in and cross-classified simultaneously by the two higher-level social contexts defined by time period and birth cohort. That is, individual members of any birth cohort are interviewed in multiple replications of the survey, and individual respondents in any particular wave of the survey can be drawn from multiple birth cohorts.

Hierarchical Age-Period-Cohort (HAPC) Models Given these data characteristics, there are two poten-tial solutions to the identification problem in addi-tion to those described in the first section. First, within the framework of the conventional linear regression model, different temporal groupings for the A, P, and C dimensions can be used to break the linear dependency. For example, one can use single year of age , time periods corresponding to years in which sur-veys are conducted (which may be several years apart), and cohorts defined either by five- or ten-year intervals, which are conventional in demography, or by applica-tion of substantive classification (e.g. War babies, baby boomers, Baby Busters, etc.). Despite the appeal of this approach, it has several problems.

First and foremost, linear models assuming additiv-ity of age, period, and cohort effects not only produce the identification problem, but are poor approxi-mations of the processes generating social changes. Second, simple linear models do not account for multilevel heterogeneity in the data and may under-estimate standard errors. Third, the APC accounting model cannot include explanatory variables and test substantive hypotheses. A useful alternative is a dif-ferent family of models that (1) do not assume fixed age, period, and cohort effects that are additive and therefore avoid the identification problem; (2) can statistically characterize contextual effects of histori-cal time and cohort membership; and (3) can accom-modate covariates to aid better conceptualization of specific social processes generating observed patterns in the data.

The Hierarchical APC (HAPC) models approach ( Yang, 2006; Yang & Land, 2006 ) is appropriate for these purposes and can address questions unique to repeated cross-sectional surveys. If there is level-two (period and/or cohort) heterogeneity, assuming fixed period and cohort effects ignores this hetero-

geneity and may not be adequate. In this case, cross-classified random effects models (CCREMs), mixed (fixed and random) effects models, can be used to quantify the level-2 heterogeneity. The objectives of CCREM are to (1) assess the possibility that indi-viduals within the same periods and cohorts share unobserved random variance and (2) test explanatory hypotheses of A, P, and C effects by including addi-tional variables at all levels.

Social Inequality of Happiness To illustrate the application of HAPC models, I will use the example of subjective well-being from a recent study ( Yang, 2008a ). This study is based on time series data on happiness in the US as reported by people 18 years of age and older from the GSS 1972–2004 (N � 28 869). This study addressed the following questions: Do people get happier with age and over time? How do social inequalities in happi-ness vary over the life course and by time? Are some people born to be happy? Are there any birth cohort differences in happiness?

The HAPC-CCREM method was applied to esti-mate fixed effects of age and other individual-level covariates and random effects of period and cohort. Hypotheses about differential sex, race, and SES effects over the life course were tested by entering their inter-action terms with age. The level-2 (period- and cohort-level) model was further specified to take into account error terms associated with period and cohort. The simplest specification of a HAPC model, as introduced by Yang and Land (2006) , is a linear mixed effects model for normally distributed outcomes. Generalized linear mixed model specifications need to be used for dichotomous and multiple categorical outcome vari-ables. In this case, the ordinal logit HAPC models of happiness were estimated. These estimates were obtained using the SAS PROC GLIMMIX.

Next came a calculation of the predicted prob-abilities of being very happy by age, sex, and race, adjusting for all other factors and net of period and cohort effects. The age curves of happiness are quadratic and convex. The J-shaped net age effects suggest that average happiness levels bottom out in early adulthood and then increase at an increas-ing rate across age. Thus, with age comes happiness. Everything else being equal, women are slightly hap-pier than men at all ages, and the gap converges at older ages. And when other factors are held constant, the race gap shows stronger trends of convergence with reversals at around the age of 70, with blacks becoming slightly happier than whites afterwards. Thus, life course variations in social inequalities in happiness are more consistent with the age-as-leveler hypothesis than the cumulative advantage hypoth-esis. Longitudinal panel data are needed to sort out whether selective survival of the happier or other


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processes are more plausible explanations of the stratification of aging and subjective well-being.

Estimates of variance components at level-2 indi-cated significant period and cohort effects control-ling for age and other individual-level variables. The results on period and cohort effects as estimated by the random effect coefficients showed variations in levels of happiness over time for all groups and that sex and race disparities decreased during the last 30 years. The most interesting cohort effect was a dip in happiness levels for cohorts born between 1945 and 1960, i.e. the baby boomers.

What accounts for these period and cohort pat-terns? One hypothesis is that period effects may reflect changes in macroeconomic variables such as gross domestic product (GDP) and levels of jobless-ness between the 1970s and 1990s ( Di Tella et al., 2003 ). Exploratory analysis suggests that GDP is con-current with the period changes in happiness and that the unemployment rate has a lagged effect. The two period-level covariates (GDP per capita and changes in unemployment rate) can be added to the models for the overall mean, sex, and race effects to deter-mine whether the economic environment explains period variations in happiness and sex and race gaps therein. In addition, the cohort analysis proposition emphasizes the exogenous social demographic envi-ronments into which cohorts are born and age. The baby boomer pattern is consistent with the Easterlin hypothesis that large cohort sizes have negative con-sequences on cohort members’ socioeconomic and other life course outcomes, such as sense of well-being and mental stress ( Easterlin, 1987 ). Relative cohort size (adopted from O’Brien et al., 1999 ) can be added to the model for the overall mean to account for cohort-level variance.

Results of the Yang (2008a,b) study indicated that GDP per capita was significantly and positively associ-ated with happiness and larger cohort size reduces hap-piness significantly, although the residual period effect remains significant. Economic conditions also affect sex and race differences over time. Higher GDP per capita reduces the sex and race differences by 83% and 38%, respectively. Higher unemployment rates, interestingly, decrease the race gap. Relative cohort size not only reduces the cohort variance by 73%, but also dimin-ishes its statistical significance. So the lower levels of happiness in baby boomers are largely due to their large size compared to their predecessors and successors. More competition to go to school and find jobs meant more pressure and strains as the boomers came of age – something that can decrease their positive affect and assessment of quality of life (QOL).

Conclusion and Extensions The empirical example above suggests that HAPC analysis moves far beyond simply identifying age and

cohort trends. It allows us to see how social stratifica-tion operates over the life course and historical time and to test the extent to which aging and cohort the-ories apply to subjective well-being in ways that can-not be achieved by a conventional linear regression analysis. The HAPC model does not incur the identi-fication problem because the age, period, and cohort effects are not assumed to be linear and additive. This analytic technique opens new doors to research on aging by offering a systematic and flexible mod-eling strategy that takes advantage of the multilevel data structure and rich covariates in repeated cross-sectional surveys.

HAPC models can be extended in various ways to address other research questions. We focus on three extensions here: fixed vs random effects models, an F -test for the presence of random effects, and full Bayesian HAPC models. First, the HAPC-CCREM approach illustrated above uses a mixed (fixed and random) effects model with a random effects specifica-tion for the level-2 variables (period and cohort). A frequently used alternative is a fixed effects specifi-cation for the level-2 variables in which dummy vari-ables are used to represent cohorts and survey years. This approach may be appropriate when the number of survey replications is relatively small (three to five). In general, however, the random effects specification is preferable to the fixed effects specification for the level-2 variables because (1) it avoids an assumption of fixed effects models that the dummy variables rep-resenting the fixed cohort and periods effects fully account for the group effects; (2) it allows group-level covariates to be incorporated into the model and explicitly models cohort characteristics and period events to test explanatory hypotheses; and (3) it is generally more statistically efficient for unbalanced research designs (designs in which there are unequal numbers of respondents in the period-by-cohort cells), which is typical in repeated cross-section sur-veys ( Yang & Land, 2008 ).

Second, the random effects of level-2 variables in the HAPC models, particularly the variance compo-nents of cohort and period, are obtained by using the restricted maximum likelihood (REML) method, which assumes that the level-2 residual or error terms are asymptotic normally distributed, and yields variance estimators with good large sample proper-ties. When the number of level-2 units, in this case cohorts and periods, is not large, this assumption may not be appropriate. Further, the z -scores for the REML estimates of the variance components are only proximate. To test more exactly whether the birth cohort and period effects make statistically significant contributions to explained variance in an outcome variable, a general linear hypothesis may be applied. Specifically, one can use an F test to test the hypo-thesis of the presence of random effects. The F statis-tic is preferred over the z -score when the sample sizes


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for random effects are small ( Littell et al., 2006 ) and can be constructed using sum of squares for the lin-ear mixed effects model and log-likelihood functions for some generalized linear mixed effects models. Interested readers should see Yang et al. (2009) for detail on the formulation of the F -test.

Third, the standard REML-EB (empirical Bayes) esti-mation algorithm for mixed effects models has other limitations when applied to APC analyses of finite time period survey data. As noted above, the numbers of periods and birth cohorts in surveys usually are too small to satisfy the large sample criterion required by the maximum likelihood estimation of variance components. In addition, the sample sizes within each cohort are highly unbalanced. These result in inaccurate REML estimates of variance-covariance components that further result in inaccurate EB esti-mates of fixed effects. This is because errors in the REML estimates may produce increased uncertainty in the EB estimates of fixed effects coefficients that will not be reflected in the standard errors. A remedy is to use the full Bayesian approach that, by defini-tion, ensures that inferences about every parameter fully account for the uncertainty associated with all others. To this end, one specifies the level-1 model as the likelihood function, assigns prior probability distributions for all fixed effect parameters and vari-ance components, and uses the Bayes rule to yield the joint model or posterior distribution. Gibbs sam-pling has been widely used to derive the marginal posterior distributions for each model parameter and can be implemented using the software WinBUGS (freely available at http://www.mrc-bsu.cam.ac.uk/bugs/ ). For an example of Bayesian HAPC analysis, see Yang (2006) .

Research Design III: Accelerated Longitudinal Panels The former two designs rely on information from synthetic cohorts that contain different cohort mem-bers at each point in time. Inferences drawn from such designs, therefore, assume that synthetic cohorts mimic true cohorts, and 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 migra-tion 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 accelerated longitudinal panel design – where multiple birth cohorts are followed over multi-ple points in time – is an important advance in aging and cohort research. The primary advantage of this design is that it provides cross-time linkages within

individuals and hence information pertaining to true birth cohorts.

Growth Curve Models of Individual and Cohort Change Hierarchical models are a useful tool for modeling longitudinal data. Because repeated observations over time (level-1 units) can be viewed as nested within individuals (level-2 units), one can specify growth curve models to assess simultaneously the intracohort age changes and intercohort differences. Although the application of two-level hierarchical regression models to standard longitudinal data is relatively straightforward, growth curve analysis of multi-cohort multi-wave data is complicated by two issues.

First, because the observable age trajectories of dif-ferent cohorts initiate and end at different ages, cohort comparisons are based on different segments of cohort members’ life courses. This raises the question of the potential confounding of the age and cohort effects. Two analytic strategies help to resolve this problem. The first is the use of centered age variables (e.g. centered around cohort median). Age center-ing eases the interpretation of the intercept, stabilizes estimation, and prevents the bias in the estimate that arises from systematic variation in mean age across the cohorts, hence eliminating the confounding of age and cohort variables. Second, the models yield tests of significance of overlapping segments of the life course of adjacent cohorts. As waves of data accumulate, the number of overlapping ages of adjacent cohorts increases, which increases statistical power. In this case, age and cohort will become less and less con-founded, making it increasingly possible to compare cohort differences in age trajectories.

The second issue concerns period effects. This model does not explicitly incorporate period effects for both substantive and methodological reasons. First, in contrast to synthetic cohort designs that usu-ally cover several decades, longitudinal designs typi-cally span much shorter time periods (e.g. a decade or so). So, the effects of period can be assumed to be triv-ial and omitted from the models, especially if the the-oretical focus is on aging. Second, it is challenging to estimate a separate period effect in accelerated longi-tudinal designs. In the framework of the growth curve models, the level-1 analysis reflects within-individual change by modeling the outcome as a function of the time indicator ( Singer & Willett, 2003 ). This means that one can include age or wave (period), depend-ing on substantive focus, but not both because within individuals age and period are the same. The simulta-neous estimation of period effects creates the model identification problem, which requires different data designs and mixed model specifications to resolve ( Yang & Land, 2006, 2008 ). Third, one does not need


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to estimate period effect per se and can focus instead on the age by cohort interaction effects.

Sex and Race Health Disparities Across the Life Course This illustrative example comes from a recent analy-sis of life course patterns and cohort variations in sex and race health gaps ( Yang & Lee, 2009 ). An impor-tant question in recent aging research is whether, and how, social status affects health dynamics in later life. Evidence about whether social disparities in health grow or diminish over the life course has been incon-sistent (e.g. House et al., 1994; Lynch, 2003; Wilson et al., 2007 ). Another key unresolved issue concerns the distinct roles of aging and cohort succession. Whether recent birth cohorts show more or less sex and race inequalities in health due to cohort changes in social attainment, health capital, and the roles of women and blacks merits further study.

This study systematically examines intercohort variations and intracohort heterogeneity in health trajectories to shed light on the social mechanisms generating sex and race inequalities in health over the life course. It addresses three questions. First, are there cohort differences in age trajectories of health? It tests the intercohort change hypothesis by modeling main cohort and age by cohort interaction effects. Second, how do sex and race health disparities change over the life course within cohorts? The prevailing explana-tions for decreasing health gaps with age are the age-as-leveler and selective survival processes. The primary explanation for increasing health gaps with age is the cumulative advantage process. The double jeopardy hypothesis has also been proposed as a secondary explanation. Because these theoretical perspectives all emphasize intracohort differentiation in aging, a proper test should control for cohort differences that might otherwise be confounded with aging effects, but this was not previously tested. The current analysis tests the intracohort inequality hypothesis by modeling age by sex and age by race inter action effects, net of cohort effects. Third, are there inter cohort variations in the intracohort health differentials by sex and race? The study further tests the intercohort difference in intra-cohort inequality hypothesis by incorporating three-way interactions between age, cohort, and social status.

The data are from the Americans’ Changing Lives (ACL) study, which uses an accelerated longitudinal design. An initial sample of 3617 adults aged 25 and older from seven 10-year birth cohorts (before 1905 to 1964) were interviewed in 1986, 1989, 1994, and 2001/2002. Three health outcomes were of interest: depressive symptoms as measured by an 11-item CES-D (Center for Epidemiologic Studies – Depression) scale, physical disability as measured by a summary index indicating levels of functional disability (range 1–4),

and self-rated health as measured by a scale indicating levels of perceived general health (range 1–5). The ana-lytic sample consisted of 10 174 person-year observa-tions from black and white respondents at all waves for which data on all variables were available.

Growth curve models are the method of analysis. The level-1 model characterizes within-individual change with age. In this model, the response variable is modeled as a function of linear and quadratic terms of age, where age is centered around the cohort median. The level-2 model assesses individual differences in change with age and determines the associations between the change with age and person-level charac-teristics including birth cohort and its quadratic func-tion, sex, race, and the interaction effects of sex and cohort and race and cohort. A similar model was tested for the quadratic growth rate and was not significant. Control variables are entered at level-1 for time-varying covariates (family income, marital status, chronic ill-nesses, body mass index (BMI), and smoking).

The hierarchical linear modeling (HLM) growth curve methodology has the advantage of allowing data that are unbalanced in time ( Raudenbush & Bryk, 2002 ). That is, it incorporates all individuals with data for the estimation of trajectories, regardless of the number of waves s/he contributes to the person-year data set. Compared to alternative modeling tech-niques, this substantially reduces the number of cases lost to follow-up due to mortality or nonresponse. However, this method does not completely resolve the sample selection problem unless it distinguishes those lost to follow up from those with complete data for all waves. If mortality and nonresponse are signifi-cantly correlated with worse health and key covariates of health, such as age and SES, parameter estimates of the health trajectories may be biased if they are not controlled. To account for this, the analysis con-trols for the effects of attrition by including dummy variables indicating the deceased and nonrespondents in the level-2 models. All statistical analyses are per-formed using SAS PROC MIXED.

The analysis tested the cohort-based models (full model) against the models combining all cohorts (reduced model with no cohort effect) to determine whether cohort-specific trajectories can actually be represented by a single mean-age trajectory. The hypothesis that there are no cohort differences in age trajectories was rejected.

Results from the growth models yield several signifi-cant findings. First, the intercohort change hypothesis is supported for all three health outcomes. There are sig-nificant cohort differences in health trajectories, with more recent cohorts faring better in physical function-ing and self-rated health, but worse in mental health. Second, the intracohort inequality hypothesis is supported only for depression. Specifically, there is evidence of a converging sex gap in depression trajectories, but


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constant sex and race gaps in physical and self-rated health trajectories within cohorts. Third, there is sup-port for the hypothesis of intercohort variations in intrac-ohort sex and race differences in mean levels of all health outcomes, but not in the growth rates other than depression. Sex and race gaps in mean levels of health narrowed across cohorts for disability, but widened for depression and perceived health. The predicted mean levels of CES-D scores are plotted in Figure 2.3 by cohort for blacks and whites based on the model estimates, adjusting for age and all other factors. It

shows that the black excess in depression has become more pronounced for more recent cohorts. We fur-ther find significant intercohort differences in age changes of intracohort sex gaps in depression. Figure 2.4 shows the predicted age trajectories of depressive symptoms by sex for selected cohorts. It suggests that the sex gaps in growth rates of depression are strongly contingent upon cohort membership. We see less convergence in the male–female gap in depression with age in more recent cohorts, suggesting a weaker age-as-leveler process in these cohorts.

Figure 2.3 Predicted mean levels of CES-D score by birth cohort: race gap. Source: Yang & Lee (2009) .

–0.2

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Figure 2.4 Predicted age growth trajectories of CES-D score by birth cohort: sex gap. Source: Yang & Lee (2009) .

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Conclusion This analysis provides an example of how to address some long-standing questions in the stratification of aging from the perspective of cohort analysis. It pro-vides substantial evidence that the process of cohort change is important for the theory, measurement, and analysis of social inequalities in health over the life course. Controlling for cohort effects substantially alters existing explanations of sex and race gaps in health trajectories over the life course. For example, because we do not find any significant changes in the intracohort sex or race gaps in disability or self-rated health with age, but find substantial intercohort vari-ations in sex and race gaps in mean levels of these outcomes, we conclude that any changes with age observed in previous studies are actually due to inter-cohort differences in disability or self-rated health lev-els. Furthermore, there is no meaningful cumulative advantage process or age-as-leveler effect at work in later or earlier cohorts’ physical and self-rated health when the confounding effects of cohort and other social factors are controlled. The constant intracohort disparities in disability and self-rated health trajecto-ries indicate the persistent disadvantages of women and African Americans.

The analytic framework exemplified above facili-tates studies of patterns of inequality within and between birth cohorts that can provide a particularly useful line of inquiry into how individual lives evolve with social change. For example, analyzing five waves of data from the China Health and Nutrition Survey (CHNS) from 1991 to 2004 with growth curve meth-ods, a recent analysis reveals patterns of social stratifi-cation in health trajectories for multiple cohorts in the context of an ever-changing macro-social environment in China ( Chen et al., 2010 ). The study found strong cohort variations in SES disparities on health trajecto-ries: the effect of education on mean level of health is smaller for more recent cohorts; the income gap in age trajectories of health increases for earlier cohorts but decreases for most recent cohorts. These cohort effects are in an opposite direction to recent US studies (e.g. Lynch, 2003; Wilson et al., 2007 ). Thus, this study highlights the uniqueness of China’s social, economic, and political structures; China’s stage of epidemiologic transition; and the changing impact of the government on health care.

In sum, cohort membership contextualizes aging-related outcomes such as mental and physi-cal well-being, QOL, and longevity, and also condi-tions social inequalities therein. The significance of cohort change in shaping the individual life suggests that new patterns may emerge for future cohorts as a result of their new life circumstances. The meth-ods and findings illustrated here should prompt future examinations of aging-related hypotheses in a cohort-specific context.

DIRECTIONS FOR FUTURE RESEARCH

Demographic rates and social indicators arrayed over time by age or cohort are canonical organiza-tions of data in research on aging. A vast literature in the social sciences uses linear models method-ology for APC analysis. Different solutions to the model identification problem often produce ambig-uous and inconsistent results and have ignited con-tinuous debates on whether any solutions exist or which solutions are better. Researchers do not agree on methodological solutions to these problems and have concluded that APC analysis is still in its infancy ( Mason & Wolfinger, 2002 ).

The developments introduced in this chapter high-light new solutions to this problem – or simply new ways of thinking about the problem. They provide an important insight that the identification problem is model-specific rather than data-specific ( Fu, 2008 ). Because the use of linear models (and hence the iden-tification problem) characterizes the majority of pre-vious studies, investigators erroneously concluded that the problem is inherent in APC analysis. In fact, it is the linear models, not the APC data structure, that incur the identification problem. Thus, the most effective way of accounting for aging-related phenom-ena and social and demographic change is to develop alternative approaches that do not treat age, period, and cohort as independent covariates in an additive fixed effects model. In addition to the new estimator for the conventional models and hierarchical models already discussed above, two approaches – one meth-odological and the other data design – hold promise for moving cohort analysis to a new era of methodo-logical development and substantive research.

Continuously Evolving Cohort Effects Model It is well known that the linear model suffers from identification problems. Less appreciated is that the model also suffers from a conceptual problem. The model rests on key assumptions that do not always accurately describe most APC-related phe-nomena. It assumes that the effect of age is the same for all periods and cohorts ( Hobcraft et al., 1982 ). But the influence of age may change over time and across cohorts. Consider, for instance, the dramatic declines in infant mortality over the past century. It also assumes that the effect of period is the same for people of all ages. But period effects are often age-specific. For example, the influenza epidemic of 1918 caused especially high mortality among peo-ple in their teens and twenties. Similarly, it assumes that a cohort effect remains the same as long as the


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cohort lives. But cohorts must change ( Ryder, 1965 ). That is, cohorts are continuously exposed to events whose influences accumulate over the life course. Wars and epidemics are examples of events that may occur in the middle of a cohort’s life and leave an imprint on all of its subsequent behaviors and out-comes. New events constantly occur. A model with unchanging cohort effects is appropriate only if all relevant events occur before the initial observation and only if these events’ impacts stay fixed as the cohort ages ( Hobcraft et al., 1982 ). To capture the process of changing cohort effects over time, however, one needs a more general model – a framework that Hobcraft et al. (1982) labeled “continuously accumu-lating cohort effects.”

Schulhofer-Wohl and Yang (2009) addressed this gap in a recent paper. They developed a general model that relaxes the assumption of the conventional addi-tive model. The new model allows age profiles to change over time and period effects to differ for peo-ple of different ages. The model also defines cohort effects as an accumulation of age-by-period interac-tions over all events across the life course. Although a longstanding literature on theories of social change conceptualizes cohort effects in exactly this way, this is the first time that a method of statistically mod-eling for this more complex form of cohort effects has been presented. This new model was applied to ana-lyze changes in age-specific mortality rates in Sweden over the past 150 years and found that the model fits the data dramatically better than the additive model. The analyses also yield interesting results that show the utility of this model in testing competing theories about the evolution of human mortality. The flexibil-ity of the model, however, comes at a high computa-tional cost because it involves the estimation of a large number of parameters. The inclusion of additional covariates also has not been analytically attempted. So, this new approach presents both opportunities and challenges for future analysts. When further improved, it may find applications in many areas of research.

Longitudinal Cohort Analysis of Balanced Age-by-Cohort Data Structure The multi-cohort multi-wave data design is especially important for aging and cohort analysis. Although more longitudinal surveys using this design are available, such as the Health and Retirement Survey (HRS) and the National Long Term Care Survey, cohort studies would benefit from further develop-ments in data collection.

The usual accelerated longitudinal design has two limitations: (1) some ages cannot be observed for all cohorts and (2) coverage of the individual life course and historical time is extremely restricted.

The imbalance of the age-by-cohort structure arises from the fact that the baseline survey consists of cohorts of different ages and follow-up surveys occur at the same times. As a result, cohorts age for exactly the same number of years but will remain age-heterogeneous at the end of data collection. The growth curve models applied to these data therefore yield estimates of cohort differences in age trajecto-ries based only on the overlapping age groups of adjacent cohorts rather than the entire possible range of ages. As mentioned above, this could affect the accuracy of the estimates when the number of waves is small or the overlapping age intervals are few. Increasing the number of follow-ups alleviates prob-lems for inference, but is less than perfect for the pur-poses of disentangling aging effects from birth cohort differences and observing period effects.

A better design is one in which the age-by-cohort data structure is balanced and extends for a long time. Extant secondary data that meet these cri-teria are exceedingly rare, however. The Liaoning Multigenerational Panel (LMGP) data ( Campbell & Lee, 2009 ) may be an exceptional resource for cohort studies. The LMGP is a database of at least one million observations of 200 000 individuals from eighteenth to early-twentieth Chinese population registers. It pro-vides entire life histories for men and nearly complete life histories for women. The length of the historical period spanned and the sheer number of observations included make it a great candidate for future longi-tudinal research on aging. It holds the potential for vastly enhancing our ability to estimate various age and cohort models.

Data like these are difficult to collect and compile, but can be highly useful for a variety of substantive investigations. For example, the mechanisms under-lying persistent cohort differences in mortality need to be better understood. The “cohort morbidity phe-notype” hypothesis has been proposed to link large cohort improvement in survival to reductions in exposures to infections, inflammation, and increased nutrition in early life ( Finch & Crimmins, 2004 ). But evidence of an association between early life condi-tions and late-life mortality is solely based on aggre-gate population data from developed countries and needs further testing using individual life histories across multiple birth cohorts in other national popu-lations such as those in the LMGP.

The APC problem has intrigued and frustrated social scientists for decades. Recent developments in tech-niques for modeling APC data – techniques that avoid the identification problem that has long compromised previous APC analyses – are available and have already generated important substantive findings. These tech-niques, coupled with new and superior data sources, set the stage for enhancing our understanding of the complex interplay of human aging, cohort characteris-tics, and historical events and processes.


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