Chapter
|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
Early Literature
Why Cohort Analysis?
Why is there an APC Identification
Problem?
Conventional Solutions
New Developments: Models, Methods,
and Substantive Research
Research design I: Age-by-Time Period
Tables of Rates/Proportions
INTRODUCTION
17
18
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18
19
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
Directions for Future Research
Continuously Evolving Cohort Effects
Model
Longitudinal Cohort Analysis of Balanced
Age-by-Cohort Data Structure
References
© 2011 Elsevier Inc. All rights reserved.
DOI: 10.1016/B978-0-12-380880-6.00002-2
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Theoretical developments combined with new methodological 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 context – 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 empirically 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 confounded. Similarly, comparing data from multiple
time periods for the same age group precludes distinguishing between period and cohort effects. The
second reason is the use of linear regression models
that suffer from either specification errors or an identification 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 empirical analyses. The remainder of this chapter is organized as follows. The next section summarizes the state
of knowledge on cohort analysis in aging research in
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THEORY AND METHODS
sociology, demography, and epidemiology. Following
on further, the next section synthesizes new developments 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 membership could be as important in determining behavior
as other social structural features such as socioeconomic status (SES). APC analysis is, in this sense, synonymous 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 developmental 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 physiological 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 biographies 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 exposures 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 ageperiod model) are subject to model specification
errors and spurious results (Mason et al., 1973). The
18
“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 factors 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 beginning 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 investigations 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 epidemiology (e.g. Clayton & Schifflers, 1987; Holford, 1991;
Kupper et al., 1985). Several recent reviews provide
useful additional material on these and related contributions 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 variables 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 example, 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, different identifying constraints (e.g. two age groups vs
two cohorts) can produce widely different estimates
of age, period, and cohort effects. And third, all justidentified 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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AGING, COHORTS, AND METHODS
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 quadratic 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) specifying 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 variable. This is a popular approach because of its substantive appeal. After all, the indicator variables of
age, period, and cohort serve as surrogates for different 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 specification. Although replacing an accounting dimension 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 original age, period, and cohort categories can be conceived 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 analysis leave us today? If a researcher has a temporally 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 contributions 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 focusing 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 identification problem is not inevitable in all settings –
only in the case of linear models that assume additive 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, disability, or mortality. Using the conventional data
structure, rates or proportions are arranged in rectangular 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 ageby-period matrix. Among the limitations of conventional solutions to the identification problem is
the lack of guidelines for estimating APC models of
rates or proportions. A three-step procedure is helpful for this purpose. Step 1 is to conduct descriptive
data analyses using graphics. The objective is to provide qualitative understanding of patterns of age,
19
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THEORY AND METHODS
300
250
1920 birth cohort
1915
Rate/100 000
200
1910
150
1905
1900
100
1895
50
1995–99
1990–94
1985–89
1890
1980–84
1885
1975–79
1880
1970–74
1965–69
1875
1870
1960–64
0
30
35
40
45
50
55
60
65
70
75
80
85
90 95+
Age
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).
period, cohort, and two-way age by period and age by
cohort variations. Step 2 is model fitting. The objective is to ascertain whether the data are sufficiently
well-described by any single factor or two-way combination 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 analyses. 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
20
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 interaction 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 explaining 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 overall 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
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AGING, COHORTS, AND METHODS
Age effect
2.0
Male
Female
1.0
–2.0
0.0
–4.0
–1.0
9
–9
4
–9
95
19
9
–8
90
19
4
85
–8
80
19
-7
9
19
4
–7
75
19
9
70
–6
19
65
19
–6
4
–2.0
19
20 25 30 35 40 45 50 55 60 65 70 75 80 85 90 95+
Age
60
Log coefficient
0.0
–6.0
Period effect
2.0
Male
Female
Year
Cohort effect
2.0
Male
Female
Log coefficient
1.0
0.0
–1.0
–2.0
75
19
65
19
55
45
19
19
35
25
15
19
19
05
19
19
95
18
85
18
75
18
18
65
–3.0
Cohort
Figure 2.2 Intrinsic estimates of age, period, and cohort effects of lung cancer mortality by sex.
Source: Yang (2008b).
models of female lung cancer mortality were produced: 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 solution to the APC model and is the unique estimable
function of both the linear and nonlinear components of the APC model. It has minimal assumptions
and is estimated using principal component regression. The reason why the IE is useful lies in the special 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 conventional equality constraints produce. For the mathematical 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 estimation of these models, but shares some of the limitations 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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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 population-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 potential solutions to the identification problem in addition 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 surveys 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 application 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 additivity of age, period, and cohort effects not only produce
the identification problem, but are poor approximations of the processes generating social changes.
Second, simple linear models do not account for
multilevel heterogeneity in the data and may underestimate standard errors. Third, the APC accounting
model cannot include explanatory variables and test
substantive hypotheses. A useful alternative is a different 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 historical time and cohort membership; and (3) can accommodate 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 leveltwo (period and/or cohort) heterogeneity, assuming
fixed period and cohort effects ignores this hetero-
22
geneity and may not be adequate. In this case, crossclassified 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 individuals within the same periods and cohorts share
unobserved random variance and (2) test explanatory
hypotheses of A, P, and C effects by including additional 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 happiness 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 estimate 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 interaction terms with age. The level-2 (period- and cohortlevel) 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 variables. 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 probabilities 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 increasing rate across age. Thus, with age comes happiness.
Everything else being equal, women are slightly happier 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 hypothesis. Longitudinal panel data are needed to sort out
whether selective survival of the happier or other
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AGING, COHORTS, AND METHODS
processes are more plausible explanations of the
stratification of aging and subjective well-being.
Estimates of variance components at level-2 indicated significant period and cohort effects controlling 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 patterns? One hypothesis is that period effects may
reflect changes in macroeconomic variables such as
gross domestic product (GDP) and levels of joblessness between the 1970s and 1990s (Di Tella et al.,
2003). Exploratory analysis suggests that GDP is concurrent 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 determine 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 environments into which cohorts are born and age. The
baby boomer pattern is consistent with the Easterlin
hypothesis that large cohort sizes have negative consequences on cohort members’ socioeconomic and
other life course outcomes, such as sense of wellbeing 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 associated with happiness and larger cohort size reduces happiness 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 diminishes 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 stratification operates over the life course and historical time
and to test the extent to which aging and cohort theories apply to subjective well-being in ways that cannot be achieved by a conventional linear regression
analysis. The HAPC model does not incur the identification 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 modeling strategy that takes advantage of the multilevel
data structure and rich covariates in repeated crosssectional 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 specification for the level-2 variables (period and cohort).
A frequently used alternative is a fixed effects specification for the level-2 variables in which dummy variables 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 representing 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 surveys (Yang & Land, 2008).
Second, the random effects of level-2 variables in
the HAPC models, particularly the variance components 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 properties. 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 hypothesis of the presence of random effects. The F statistic 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 linear 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) estimation 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 estimates 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 definition, 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 variance components, and uses the Bayes rule to yield
the joint model or posterior distribution. Gibbs sampling 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 members 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 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 accelerated longitudinal panel design –
where multiple birth cohorts are followed over multiple 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
24
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 different 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 centering 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 confounded, 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 usually cover several decades, longitudinal designs typically span much shorter time periods (e.g. a decade or
so). So, the effects of period can be assumed to be trivial and omitted from the models, especially if the theoretical focus is on aging. Second, it is challenging to
estimate a separate period effect in accelerated longitudinal 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), depending on substantive focus, but not both because within
individuals age and period are the same. The simultaneous 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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AGING, COHORTS, AND METHODS
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 analysis of life course patterns and cohort variations in sex
and race health gaps (Yang & Lee, 2009). An important 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 inconsistent (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 explanations for decreasing health gaps with age are the ageas-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 interaction effects, net of
cohort effects. Third, are there intercohort variations
in the intracohort health differentials by sex and race?
The study further tests the intercohort difference in intracohort 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 analytic sample consisted of 10 174 person-year observations 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 characteristics including birth cohort and its quadratic function, 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 illnesses, 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 personyear data set. Compared to alternative modeling techniques, 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 significantly 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 controls for the effects of attrition by including dummy
variables indicating the deceased and nonrespondents
in the level-2 models. All statistical analyses are performed 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 significant findings. First, the intercohort change hypothesis is
supported for all three health outcomes. There are significant cohort differences in health trajectories, with
more recent cohorts faring better in physical functioning 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
25
Part
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THEORY AND METHODS
shows that the black excess in depression has become
more pronounced for more recent cohorts. We further 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.
constant sex and race gaps in physical and self-rated
health trajectories within cohorts. Third, there is support for the hypothesis of intercohort variations in intracohort 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
1
White
Black
CES-D
0.8
0.6
0.4
0.2
0
–1905
1905–1914 1915–1924 1925–1934 1935–1944 1945–1954 1955–1964
Birth cohort
Figure 2.3
Predicted mean levels of CES-D score by birth cohort: race gap.
Source: Yang & Lee (2009).
1.2
1955–64 cohort
1
0.8
CES-D
1935–44 cohort
0.6
1915–24 cohort
0.4
0.2
–1905 cohort
Men
Women
0
–0.2
25
Figure 2.4
35
40
45
50
55
60 65
Age
70
75
80
Predicted age growth trajectories of CES-D score by birth cohort: sex gap.
Source: Yang & Lee (2009).
26
30
85
90 95+
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AGING, COHORTS, AND METHODS
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 provides 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 variations 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 intercohort differences in disability or self-rated health levels. 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 trajectories indicate the persistent disadvantages of women
and African Americans.
The analytic framework exemplified above facilitates 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 methods, a recent analysis reveals patterns of social stratification 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 trajectories: 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 physical well-being, QOL, and longevity, and also conditions 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 methods 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 organizations of data in research on aging. A vast literature
in the social sciences uses linear models methodology for APC analysis. Different solutions to the
model identification problem often produce ambiguous and inconsistent results and have ignited continuous 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 highlight 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 identification problem) characterizes the majority of previous 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 phenomena 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 methodological and the other data design – hold promise
for moving cohort analysis to a new era of methodological 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 phenomena. 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 people in their teens and twenties. Similarly, it assumes
that a cohort effect remains the same as long as the
27
Part
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THEORY AND METHODS
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 outcomes. 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 accumulating 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 additive model. The new model allows age profiles to
change over time and period effects to differ for people of different ages. The model also defines cohort
effects as an accumulation of age-by-period interactions 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 modeling for this more complex form of cohort effects has
been presented. This new model was applied to analyze 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 flexibility of the model, however, comes at a high computational 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 developments 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.
28
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 ageheterogeneous at the end of data collection. The
growth curve models applied to these data therefore
yield estimates of cohort differences in age trajectories 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 problems for inference, but is less than perfect for the purposes 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 criteria 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 provides 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 longitudinal 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 underlying persistent cohort differences in mortality need
to be better understood. The “cohort morbidity phenotype” 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 conditions and late-life mortality is solely based on aggregate population data from developed countries and
needs further testing using individual life histories
across multiple birth cohorts in other national populations such as those in the LMGP.
The APC problem has intrigued and frustrated social
scientists for decades. Recent developments in techniques 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 techniques, coupled with new and superior data sources,
set the stage for enhancing our understanding of the
complex interplay of human aging, cohort characteristics, and historical events and processes.
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AGING, COHORTS, AND METHODS
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