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Should age-period-cohort analysts accept innovation without scrutiny? A response to Reither, Masters, Yang, Powers, Zheng, and Land

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Post-print version available at http://eprints.whiterose.ac.uk/87648/. This commentary clarifies our original commentary (Bell & Jones, 2014c) and illustrates some concerns we have regarding the response article in this issue (Reither et al., 2015). In particular, we argue that (a) linear effects do not have to be produced by exact linear mathematical functions to behave as if they were linear, (b) linear effects by this wider definition are extremely common in real life social processes, and (c) in the presence of these effects, the Hierarchical Age Period Cohort (HAPC) model will often not work. Although Reither et al. do not define what a ‘non-linear monotonic trend’ is (instead, only stating that it isn’t a linear effect) we show that the model often doesn’t work in the presence of such effects, by using data generated as a ‘non-linear monotonic trend’ by Reither et al. themselves. We then question their discussion of fixed and random effects before finishing with a discussion of how we argue that theory should be used, in the context of the obesity epidemic.
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Should age-period-cohort analysts accept
innovation without scrutiny? A response
to Reither, Masters, Yang, Powers, Zheng
and Land
Andrew Bell a b and Kelvyn Jones a b
aSchool of Geographical Sciences
University of Bristol
University Road
Bristol
BS8 1SS
UK
bCentre for Multilevel Modelling
University of Bristol
2 Priory Road
Bristol
BS8 1TX
UK
Corresponding Author:
Andrew Bell
School of Geographical Sciences
University of Bristol
University Road
Bristol
BS8 1SS
UK
Email: andrew.bell@bristol.ac.uk
Acknowledgements: Thanks to Ron Johnston for his helpful advice.
2
Abstract
This commentary clarifies our original commentary (Bell & Jones, 2014c) and illustrates some concerns
we have regarding the response article in this issue (Reither et al., 2015). In particular, we argue that
(a) linear effects do not have to be produced by exact linear mathematical functions to behave as if
they were linear, (b) linear effects by this wider definition are extremely common in real life social
processes, and (c) in the presence of these effects, the Hierarchical Age Period Cohort (HAPC) model
will often not work. Although Reither et al. do not define what a ‘non-linear monotonic trend’ is
(instead, only stating that it isn’t a linear effect) we show that the model often doesn’t work in the
presence of such effects, by using data generated as a non-linear monotonic trend by Reither et al.
themselves. We then question their discussion of fixed and random effects before finishing with a
discussion of how we argue that theory should be used, in the context of the obesity epidemic.
Research Highlights
We clarify the nature of the identification problem in all APC analysis
The Hierarchical APC model will sometimes work, but sometimes is not enough
Simulations using plausible data structures show the model often does not work
Relying in theory is problematic, but this is often all researchers can do
Keywords
Age-period-cohort models, obesity, collinearity, model identification, cohort effects, multilevel
modelling
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We thank the Social Science & Medicine editors for allowing us to respond to the above article and
allowing the debate regarding age-period-cohort (APC) identification to be furthered. In their article,
Reither et al. (2015, henceforth RMYPZL) argue the following:
Only when period and cohort effects are exactly linear does the Hierarchical APC (HAPC)
model give fallacious results.
In the real world, period and cohort effects are never exactly linear.
Thus, in the real world, the HAPC model will work.
The HAPC model should only be used when goodness-of-fit statistics (such as AIC and BIC)
suggest a simpler model (including only one or two APC dimensions) would be insufficient.
We address each of these arguments in turn below, showing that each of them is flawed, and that our
original critique of the HAPC model (Bell & Jones, 2014c) remains justified.
What is a linear trend, and what is a non-linear monotonic trend?
RMYPZL argue that the HAPC model will only fail to produce accurate results in the presence of linear
effects, and we agree with this. But what is a linear effect? One answer, and that suggested by
RMYPZL, is that it is a process produced by an exact linear algebraic association: y=mx+c. However, in
the real world data are generated by social processes, not equations, meaning RMYPZL are right to
claim that such effects never occur exactly in real life. However, our definition of a linear trend is wider
than this: we argue that a linear trend exists when, if an algebraically linear expression is removed
from the data at hand, this would have the effect of flattening that trend. It would be difficult to argue
that social processes never produce data that fills these criteria. Furthermore, there need only be a
linear component to the data generating process to fulfil this definition. Other effects (stochastic,
quadratic, or whatever else) can also be present, so long as there is also a linear component, defined
as above. Whilst the HAPC model will sometimes work under these circumstances, as shown by the
simulations in RMYPZL, we argue that it will often not work. We also argue that a model that only
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works some of the time is not a particularly useful model to social scientists, and at least needs to be
applied with care and awareness of its limitations.
As for what a ‘non-linear monotonic trend’ is, the answer is less clear. RMYPZL state what it is not, but
do not say what it is. This is convenient for their argument: it means that there is no possibility of
questioning the model with simulations because any simulated DGP which the HAPC model fails to
replicate can be dismissed as being unrealistically linear. All that we have to go on are the six trends
(period and cohort trends from each of equations 2-4 in RMYPZL) which we must therefore assume
are examples that are consistent with ‘real life’ situations.
Some counter-simulations
However, RMYPZL’s argument does not even stand up for these six trends. We generated data where
the period trend was the same as that in RMYPZL’s equation 2, the age trend was the same as that in
RMYPZL’s equation 4, and the cohort trend was based on the period trend in figure 4 (we took every
other period effect so the numbers matched the numbers necessary for 7-year cohort groupings).
Thus, the DGP used was as follows:
Logit[Pr(Y=1)] = -1.988 + (0.059*Age-gm) + (-0.001*(Age-gm)2) + (-.474*C1) + (-.423*C2) + (-.362*C3)
+ (-.314*C4) + (-.214*C5) + (-.135*C6) + (-.054*C7) + (.029*C8) + (.172*C9) + (.256*C10) + (.410*C11)
+ (0.529*C12) + (0.605*C13) + (-.02*P1) + (.03*P2) + (.04*P3) + (.04*P4) + (.02*P5) + (-.03*P6) + (-
.03*P7) + (-.05*P8) + (-.05*P9) + (-.05*P10) + (-.05*P11) + (-.05*P12) + (-.06*P13) + (-.05*P14) + (-
.05*P15) + (-.02*P16) + (0*P17) + (.02*P18) + (-.02*P19) + (-.02*P20) + (0*P21) + (-.02*P22) +
(.02*P23) + (.05*P24) + (.08*P25) + (.1*P26) + (.1*P27) + Uc + Up
Uc ~ N(0,.01), Up ~ N(0,.01)
(1)
We fitted these data to the HAPC model 100 times, specifying 5-year groupings in the model. If
RMYPZL are correct, unbiased results should be produced because the DGP contains, by their own
definition, no linear effects and only ‘non-linear monotonic trends’ for periods and cohorts.
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The results are shown in figure 1. As can be seen, the model fails to pick up the cohort trend, finds a
period trend where there is none, and underestimates the strength of the age trend; in sum, the
HAPC gets it radically wrong in not identifying the true patterns.
[Figure 1 about here]
The use of fit statistics
RMYPZL argue that fit statistics should be used to check that all of the elements of APC are required.
Whilst some of the authors have stated this in the past with regard to the Intrinsic Estimator (e.g. Yang
& Land, 2013:126), in neither their articles nor their book (as far as we are aware) have they stated
that this is necessary before using the HAPC model. Indeed they have regularly claimed that the HAPC
approach “completely avoids the identification problem(Yang & Land, 2013:70) without any such ifs
or buts. Thus, many researchers will (and have) used the HAPC model without taking this step.
Consequently, we welcome this important clarification.
However, there is a problem with this. In each of the models presented in table 1 of RMYPZL, there
are age, period and cohort effects present in the DGPs, even if those effects are only random variation
(generated by the Uc and Up coefficients). As such, in all four of the simulated cases, the model fit
statistics find the incorrect answer. Moreover, in two cases, different model fit statistics give different
answers. This is unsurprising given our previous arguments about the APC identification problem (Bell
& Jones, 2013, 2014a): model fit statistics will never be able to solve the identification problem,
because they cannot tell the difference between DGPs with different linear (by our definition) APC
effects. Model fit statistics showing the full APC model is preferable only suggests that there is
significant non-linear variation present in each of the three dimensions it does not make it possible
to assign linear (by our definition) trends correctly.
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Fixed and Random effects
A theme that runs through RMYPZL’s article is the question of whether one should use fixed (FE) or
random (RE) effects models. We want to emphasise that this is a separate issue to the identification
problem and RMYPZL conflate the two in their article, which distracts from the issue at hand. We
agree with the conceptual treatment of cohorts and periods as random effects. But an appropriate
conceptual treatment does not mean the model works in practice. RMYPZL claim that we “assume”
the classical APC accounting/linear regression model where “the effects of all three temporal
dimensions are fixed [effects]” (RMYPZL:6). This is not the case; indeed this claim does not really make
sense. Statistical models are not assumed; they are used to appropriately represent the social
processes that produced the data at hand. Our argument is simply that in many situations the model
used by RMYPZL fails to accurately represent these processes. Should we use FE or RE? Whilst in other
scenarios we have actually argued strongly for the latter (Bell & Jones, 2015b), if we are looking for an
automatic, general solution to the identification problem, the answer is that we should use neither.
The problem is that RMYPZL are completely wrong when they say that the identification problem “is
not data specific, but model specific” (RMYPZL:4) if linear trends (by our definition) exist in the
dataset, you will encounter problems regardless of what model you use, whether a RE HAPC, a FE
accounting model, or whatever else.
Solid Theory
Finally, we want to thank RMYPZL for their discussion of solid theory which, in general, we agree with.
In our article we did not give an opinion one way or another regarding which of period or cohort
effects are more likely to be responsible for the obesity epidemic. Our point was simply that this
question cannot be answered using the data and methods of Reither et al. (2009). We therefore
welcome RMYPZL’s theoretically informed review of previous studies of obesity. Indeed that is exactly
what we called for in our original commentary (Bell & Jones, 2014c).
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In particular, RMYPZL cite two papers (Flegal et al., 2002; Lee et al., 2011) which find a non-linear trend
in periods: obesity/BMI appears relatively flat until about 1980 and then increases from there. It is
unlikely that such a nonlinearity could be the result of cohort effects. We do not dispute this logic
except to make two points. First, another study, albeit in a different cultural context, have used a
similar analysis of non-linearities, and found that cohorts fit best (Olsen et al., 2006). Second, the non-
linearities were not, and could not, be found by Reither et al’s study, because their data only went as
far back as 1976 and so no such non-linearities were present.
Our point about strong theory was not that it is a solution to the APC identification problem. We agree
that “compelling speculation can never replace evidence in any field of scientific inquiry (RMYPZL:22).
However this doesn’t mean that any evidence will do, and where the choice is between compelling
speculation and misleading evidence dressed up as science, we would always choose the former.
Conclusions
RMYPZL ask the question “Should APC studies return to the methodologies of the 1970s?” The answer
to this rather loaded question is clearly no. However that previous methods don’t work does not mean
that new innovations do, and this discussion should guide readers in making their own minds up about
whether the HAPC model is appropriate for their purposes. We have argued before that there are
some situations where the HAPC model could be used (e.g. when periods and cohorts have no
continuous trends) and it can easily be adapted to incorporate theory where appropriate (Bell, 2014;
Bell & Jones, 2014b, 2015a). However the model does not work as a general purpose APC model; no
model does. Our concerns mirror those of others regarding another the Intrinsic Estimator (Luo, 2013;
Pelzer et al., 2014) and we agree with Fienberg (2013:1983) that “the search for methodological
solutions to the APC identity is an endless and fruitless quest. It is surely time to move onto
substantively focused considerations of the meaning of the three components in settings of interest.”
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sectional and panel data. Polit Sci Res Methods, 3, 133-153.
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US adults, 1999-2000. JAMA, 288, 1723-1727.
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Period-Cohort Problem. Demography, 50, 1945-1967.
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in Denmark. Epidemiology, 17, 292-295.
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of the Intrinsic Estimator in APC Models. Demography, online.
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the obesity epidemic in the United States. Soc Sci Med, 69, 1439-1448.
Reither, E.N., Masters, R.K., Yang, Y.C., Powers, D.A., et al. (2015). Should age-period-cohort studies
return to the methodologies of the 1970s? Soc Sci Med, online.
Yang, Y., & Land, K.C. (2013). Age-Period-Cohort Analysis: New models, methods, and empirical
applications. Boca Raton, FL: CRC Press.
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Figure 1: The DGP (black) and results (grey) from fitting the HAPC model to 100 datasets generated as
in equation 1.
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Reither et al. (2009) use a Hierarchical Age-Period-Cohort model (HAPC - Yang & Land, 2006) to assess changes in obesity in the USA population. Their results suggest that there is only a minimal effect of cohorts, and that it is periods which have driven the increase in obesity over time. We use simulations to show that this result may be incorrect. Using simulated data in which it is cohorts, rather than periods, that are responsible for the rise in obesity, we are able to replicate the period-trending results of Reither et al. In this instance, the HAPC model misses the true cohort trend entirely, erroneously finds a period trend, and underestimates the age trend. Reither et al.’s results may be correct, but because age, period and cohort are confounded there is no way to tell. This is typical of age-period-cohort models, and shows the importance of caution when any APC model is used. We finish with a discussion of ways forward for researchers wishing to model age, period and cohort in a robust and non-arbitrary manner.
Book
Age-Period-Cohort Analysis: New Models, Methods, and Empirical Applications is based on a decade of the authors’ collaborative work in age-period-cohort (APC) analysis. Within a single, consistent HAPC-GLMM statistical modeling framework, the authors synthesize APC models and methods for three research designs: age-by-time period tables of population rates or proportions, repeated cross-section sample surveys, and accelerated longitudinal panel studies. The authors show how the empirical application of the models to various problems leads to many fascinating findings on how outcome variables develop along the age, period, and cohort dimensions. The book makes two essential contributions to quantitative studies of time-related change. Through the introduction of the GLMM framework, it shows how innovative estimation methods and new model specifications can be used to tackle the "model identification problem" that has hampered the development and empirical application of APC analysis. The book also addresses the major criticism against APC analysis by explaining the use of new models within the GLMM framework to uncover mechanisms underlying age patterns and temporal trends. Encompassing both methodological expositions and empirical studies, this book explores the ways in which statistical models, methods, and research designs can be used to open new possibilities for APC analysis. It compares new and existing models and methods and provides useful guidelines on how to conduct APC analysis. For empirical illustrations, the text incorporates examples from a variety of disciplines, such as sociology, demography, and epidemiology. Along with details on empirical analyses, software and programs to estimate the models are available on the book’s web page.
Article
Social scientists have recognized the importance of age-period-cohort (APC) models for half a century, but have spent much of this time mired in debates about the feasibility of APC methods. Recently, a new class of APC methods based on modern statistical knowledge has emerged, offering potential solutions. In 2009, Reither, Hauser and Yang used one of these new methods – hierarchical APC (HAPC) modeling – to study how birth cohorts may have contributed to the U.S. obesity epidemic. They found that recent birth cohorts experience higher odds of obesity than their predecessors, but that ubiquitous period-based changes are primarily responsible for the rising prevalence of obesity. Although these findings have been replicated elsewhere, recent commentaries by Bell and Jones call them into question – along with the new class of APC methods. Specifically, Bell and Jones claim that new APC methods do not adequately address model identification and suggest that “solid theory” is often sufficient to remove one of the three temporal dimensions from empirical consideration. They also present a series of simulation models that purportedly show how the HAPC models estimated by Reither et al. (2009) could have produced misleading results. However, these simulation models rest on assumptions that there were no period effects, and associations between period and cohort variables and the outcome were perfectly linear. Those are conditions under which APC models should never be used. Under more tenable assumptions, our own simulations show that HAPC methods perform well, both in recovering the main findings presented by Reither et al. (2009) and the results reported by Bell and Jones. We also respond to critiques about model identification and theoretically-imposed constraints, finding little pragmatic support for such arguments. We conclude by encouraging social scientists to move beyond the debates of the 1970s and toward a deeper appreciation for modern APC methodologies.
Article
This article explores an important property of the intrinsic estimator that has received no attention in literature: the age, period, and cohort estimates of the intrinsic estimator are not unique but vary with the parameterization and reference categories chosen for these variables. We give a formal proof of the non-uniqueness property for effect coding and dummy variable coding. Using data on female mortality in the United States over the years 1960-1999, we show that the variation in the results obtained for different parameterizations and reference categories is substantial and leads to contradictory conclusions. We conclude that the non-uniqueness property is a new argument for not routinely applying the intrinsic estimator.
Article
There is ongoing debate regarding the shape of life-course trajectories in mental health. Many argue the relationship is U-shaped, with mental health declining with age to mid-life, then improving. However, I argue that these models are beset by the age-period-cohort (APC) identification problem, whereby age, cohort and year of measurement are exactly collinear and their effects cannot be meaningfully separated. This means an apparent life-course effect could be explained by cohorts. This paper critiques two sets of literature: the substantive literature regarding life-course trajectories in mental health, and the methodological literature that claims erroneously to have 'solved' the APC identification problem statistically (e.g. using Yang and Land's Hierarchical APC-HAPC-model). I then use a variant of the HAPC model, making strong but justified assumptions that allow the modelling of life-course trajectories in mental health (measured by the General Health Questionnaire) net of any cohort effects, using data from the British Household Panel Survey, 1991-2008. The model additionally employs a complex multilevel structure that allows the relative importance of spatial (households, local authority districts) and temporal (periods, cohorts) levels to be assessed. Mental health is found to increase throughout the life-course; this slows at mid-life before worsening again into old age, but there is no evidence of a U-shape--I argue that such findings result from confounding with cohort processes (whereby more recent cohorts have generally worse mental health). Other covariates were also evaluated; income, smoking, education, social class, urbanity, ethnicity, gender and marriage were all related to mental health, with the latter two in particular affecting life-course and cohort trajectories. The paper shows the importance of understanding APC in life-course research generally, and mental health research in particular.
Article
Post-print version available at http://eprints.whiterose.ac.uk/87660/. This commentary discusses the age-period-cohort identification problem. It shows that, despite a plethora of proposed solutions in the literature, no model is able to solve the identification problem because the identification problem is inherent to the real-world processes being modelled. As such, we cast doubt on the conclusions of a number of papers, including one presented here (Page et al., this issue). We conclude with some recommendations for those wanting to model age, period and cohort in a compelling way.