# Age, Period and Cohort Processes in Longitudinal and Life Course Analysis: A Multilevel Perspective

## Abstract

This chapter considers age, period and cohort (APC) as different sources of health-related change. Age (or, life course) effects are individual, often biological, sources of change, whilst periods and cohorts can be thought of as social contexts affecting individuals that reside within them. Due to the mathematical confounding of age, period and cohort, careful consideration of each is important – otherwise what appears to be, for example, a period (year) effect could in fact be a mixture of age and cohort processes. Naive life course approaches could thus produce misleading results when APC effects are not all considered. However, the mathematical confounding also often makes modelling all three effects together impossible, and the dangers of attempting to do so, or of ignoring one effect without critical forethought, is illustrated through the example of the obesity epidemic. This example uses Yang and Land’s Hierarchical APC model which it is claimed (incorrectly) solves the identification problem. Finally, we suggest a flexible multilevel framework that extends Yang and Land’s model, and by making relatively strong assumptions (in this case that there are no long-run period trends) can model age, period and cohort effects robustly and explicitly, so long as those assumptions are correct. This is illustrated using health data from the British Household Panel Survey. We argue that this theory driven approach is often the most appropriate for conceptualising APC effects, and producing valid empirical inference about both individual life courses and the spatial and temporal contexts in which they exist.

## Keywords

Monte Carlo Markov Chain Cohort Effect General Health Questionnaire Deviance Information Criterion Micro Model## Introduction

*age*, meaning that they change as they progress through their life course. Second, change can occur over time due to differences between

*cohort*groups, whereby as new cohorts replace old cohorts, the social composition (and thus the health) of society as a whole can change. Third, and finally, change can occur as a result of

*period*effects, whereby passage through time results in a change in health, regardless of the age of the individual. Suzuki (2012, p. 452) demonstrates the difference between these with the following fictional dialogue:

- A:
I can’t seem to shake off this tired feeling. Guess I’m just getting old. [Age effect]

- B:
Do you think it’s stress? Business is down this year, and you’ve let your fatigue build up. [Period effect]

- A:
Maybe. What about you?

- B:
Actually, I’m exhausted too! My body feels really heavy.

- A:
You’re kidding. You’re still young. I could work all day long when I was your age.

- B:
Oh, really?

- A:
Yeah, young people these days are quick to whine. We were not like that. [Cohort effect]

Understanding what combination of APC causes changes in health is of importance to many researchers, especially since different combinations of APC can have different public health policy implications. Unfortunately, meaningfully partitioning change into these three dimensions with statistical methods is far from straightforward, because age, period and cohort are exactly linearly dependent. This chapter considers the very serious implications of this ‘identification problem’ for longitudinal and life course research. Whilst the focus will be on health, the methodological and conceptual issues apply across the social sciences and beyond.

The chapter is structured as followed. We first outline the APC identification problem, and why simply controlling for age, period and cohort, as you might for imperfectly co-linear variables, does not work in the case of age, period and cohort. We show that the identification problem needs to be carefully considered whenever life course or longitudinal change is modelled, and that naïve models can radically reassign effects between age, period and cohort, producing misleading results. Next, we outline some proposed solutions to the identification problem, focusing on Yang and Land’s Hierarchical Age-Period-Cohort (HAPC) model (Yang and Land 2006, 2013), and, using the example of the obesity epidemic, show how they often do not work. Finally, we outline what we consider to be best practice when considering APC effects, by extending the HAPC model, and demonstrate this with an example examining APC effects on mental health (measured by the General Health Questionnaire) using the British Household Panel Survey (BHPS). We argue that whilst there is no method that can mechanically separate APC effects in all scenarios, when good theory is used to make robust assumptions regarding APC effects, researchers can often make useful and non-arbitrary inference.

## The Age-Period-Cohort Identification Problem

^{1}:

*Period*with

*Age + Cohort*, we get

*Age + Cohort*for

*Period*gives us

The continued search for a statistical technique that can be mechanically applied always to correctly estimate the effects is one of the most bizarre instances in the history of science of repeated attempts to do the logically impossible (Glenn 2005, p. 6).

## ‘Solutions’ to the APC Identification Problem

The most common ‘solution’, and that suggested first by Mason et al. (1973), is to constrain certain parameters in a model to be equal.^{2} Thus, each age, period and cohort group is entered into a regression model as a dummy variable, but two groups are combined as if they were a single group. This means that the dependency in Eq. 10.1 no longer applies (that is, it is no longer possible to always be sure of the value of one of the APC variables if you know the value of the other two). However, as Mason et al. recognised (but unfortunately many who use the Mason et al. method do not), solving the dependency in the model does not solve the dependency in the real world (Glenn 1976, 2005; Osmond and Gardner 1989). Whilst the model will produce an answer, there is no way of knowing whether that answer is correct unless we know that the constraint imposed is exactly correct. Thus whilst saying that individuals born in 1960 are substantively the same as those born in 1961 may seem innocuous, such an assumption could have a profound effect on the estimated results, and produce very different results from models using other apparently innocuous assumptions. Crucially, all of these models will have identical model fit statistics, meaning there is no way of choosing one constraint over another without strong prior knowledge. Other models use similar constraints, for example using aggregated groups for one of APC similarly constrains the parameters within those groups, for example see Page et al. (2013). These models are subject to the same problem – the identification problem is merely hidden beneath coarser data. Unless there is very good theory to believe that the groupings imposed are exactly valid, the model will generally fail to produce correct inference.

In recent years more solutions to the identification problem have been proposed.^{3} This section now focusses on one of these – Yang and Land’s Hierarchical APC (HAPC) model (Yang and Land 2006, 2013).

_{1}and cohort j

_{2}. The ‘micro’ model has linear and quadratic age terms, with coefficients \( {\beta}_1 \) and \( {\beta}_2 \) respectively, a constant that varies across both periods and cohorts, and a level-1 residual error term. The ‘macro’ model defines the intercept in the micro model by a non-varying constant \( {\beta}_0 \), and a residual term for each period and cohort. The period, cohort and level-1 residuals are all assumed to follow Normal distributions, each with variances that are estimated.

the underidentification problem of the classical APC accounting model has been resolved by the specification of the quadratic function for the age effects (Yang and Land 2006, p. 84)

An HAPC framework does not incur the identification problem because the three effects are not assumed to be linear and additive at the same level of analysis (Yang and Land 2013, p. 191)

This contextual approach … helps to deal with (actually completely avoids) the identification problem (Yang and Land 2013, p. 71)

## How to Model APC Effects Robustly

Whilst the HAPC model does not work as its authors intended, it does offer a compelling conceptual framework which is useful looking forward to ways one might model age, period and cohort effects together in a single model without falling foul of the identification problem. We have argued from the beginning that discerning APC effects mechanically is impossible. However, if we are willing to make certain assumptions about the nature of those APC effects, then inference is possible, and the HAPC model provides us with a useful framework in which to do so.

^{4}in the model. Thus, Eq. 10.5 is extended to:

^{5}If this assumption is justified, such a model will produce correct inference both about the linear age and cohort trends, and about the period and cohort random deviations from those trends (Bell and Jones 2014a). We would argue that often constraining the period trend to zero is a reasonable course of action. For us, the mechanism for long-run change is more easily conceptualised through cohorts than periods – change occurring by influencing people in their formative years rather than ‘something in the air’ that influences all age groups equally and simultaneously. However, this is of course dependent on the research question and subject area, and the researchers own understanding of the process at hand.

Having made the above assumption, and thus (assuming the assumption is valid) dealt with the identification problem, the model can now be extended in a number of ways. First, using the multilevel framework, additional levels can be added to fit the structure of the data being used. The HAPC model was originally designed for repeated cross-sectional data (such as the ONS Longitudinal Study (Office of National Statistics 2008)), where a cross-sectional sample of individuals is measured on multiple occasions, but individuals are not followed through time across these occasions. Where panel data (such as the BHPS) is used, that is data that does follow individuals over time, an individual level should be included to account for dependency within individuals between occasions. For other data designs, the HAPC model does not work so well: cross-sectional studies control for periods by design, but therefore cannot differentiate between age and cohort effects; whilst single cohort studies (such as the Millennium Cohort Study (Hansen 2014)) control for cohorts by design but cannot differentiate age and period effects.

Another extension would be to include an interaction between the age and cohort variable in the fixed part of the model. This is particularly useful for panel data, which effectively takes the form of an accelerated longitudinal design (for example see Freitas and Jones 2012). The age-by-cohort interaction allows for the possibility that the life course effect varies by generation – i.e. that there is not a single life course pattern that applies across all cohorts. In our view it is not appropriate to interpret this interaction as a period effect as others have done – for examples see Bell and Jones (2014c); the model still assumes that period effects are absent. The presence of an age-by-cohort interaction term is often thought of as a threat to inference about the life course, that is, a problem that needs to be corrected for (Miyazaki and Raudenbush 2000). However it seems to us that the interaction term can itself be of substantive interest, in understanding how life course trajectories have changed with changing cohort groups. Such an approach is increasingly common in the social medical sciences (Yang 2007; Shaw et al. 2014; Chen et al. 2010; Yang and Lee 2009), and in sociology more generally (McCulloch 2014). However, such designs are usually not combined with the cross-classified structure that characterises the HAPC model.

The model can be further elaborated by adding covariates at any level, or by allowing the effect of variables to vary at certain levels. For example, one could allow the life course (age) effects to vary between individuals, as is regularly done in simpler multilevel life course studies. One could also include control variables of various types, and interact these with the age and cohort variables to test whether the effects of these variables is constant over various dimensions of time.

The next section of this chapter puts this methodology into practice using the BHPS data to consider the life course and longitudinal effects on mental health.

## Example: APC Effects on Mental Health with the BHPS

This example^{6} uses data from the British Household Panel Survey (BHPS) to consider the age, period and cohort effects on mental health. The BHPS surveyed individuals from approximately 5,000 households (around ~10,000 individuals) from across the United Kingdom (UK), every year between 1991 and 2008 (Taylor et al. 2010). These individuals are measured on a wide range of social, demographic, economic and medical characteristics. Here, our outcome of interest is mental health and to that end, we use the General Health Questionnaire (GHQ) (Goldberg 1972) to form our dependent variable. For the GHQ, respondents are asked 12 questions, and asked how far they agree with those questions on a four point scale. Each question is thus assigned a score from 0 to 3 on that scale, which are summed to create a single 36 point scale which can be treated as a continuous variable.^{7} It is argued that the GHQ is a measure of psychiatric illness, both in terms of the severity of that illness, or as a probability of that individual being a psychiatric case (Goldberg and Williams 1988; Weich and Lewis 1998, p. 9), with high scores indicating a higher degree of psychiatric disorder. It should also be noted, however, that the GHQ is “sensitive to recent change in psychological well-being” (Weich and Lewis 1998, p. 12) and “transient disorders, which may remit without treatment” (Goldberg and Williams 1988, p. 5), and as such also encompasses a subjective understanding of mental health that complicates an understanding of individuals being ‘cases’ or ‘non-cases’.

Parameter estimates for the three models presented here. (1) A two-level model with age, cohort and gender specified in the fixed part of the model; (2) a six-level version of model 1; (3) as model 2, but with the age coefficient allowed to vary at the person level

1. 2 levels | 2. 6 levels | 3. Age random slopes | ||||
---|---|---|---|---|---|---|

Mean estimate | SE | Mean estimate | SE | Mean estimate | SE | |

| ||||||

Constant | 10.934 | 0.046*** | 10.882 | 0.100*** | 10.922 | 0.111*** |

| −0.009 | 0.040 | 0.030 | 0.067 | 0.044 | 0.070 |

Age | −0.176 | 0.043*** | −0.069 | 0.008*** | −0.057 | 0.009*** |

Age | 0.044 | 0.002*** | 0.044 | 0.003*** | 0.046 | 0.003*** |

| 0.008 | 0.004* | 0.011 | 0.007 + | 0.012 | 0.007* |

Birth year | −0.001 | 0.000*** | −0.000 | 0.000** | −0.000 | 0.000** |

Female | 1.322 | 0.052*** | 1.317 | 0.051*** | 1.279 | 0.055*** |

Female * age | 0.051 | 0.042 | 0.053 | 0.043 | 0.043 | 0.047 |

Female * birth year | 0.011 | 0.005** | 0.012 | 0.005** | 0.011 | 0.005* |

Birth year * age | −0.021 | 0.008** | ||||

| ||||||

Local authority | 0.160 | 0.142 | ||||

Household-year | 2.458 | 2.425 | ||||

Period | 0.015 | 0.015 | ||||

Cohort group (5 year interval) | 0.057 | 0.059 | ||||

Person (intercept) | 12.505 | 12.174 | 11.000 | |||

Person (covariance) | 0.077 | |||||

Person (age slope) | 0.008 | |||||

Occasion | 16.996 | 14.564 | 14.268 | |||

DIC | 1,120,612 | 1,113,079 | 1,110,445 |

Having established the significance of terms in the fixed part of the model, we then built up the random part of the model, to create the cross-classified structure portrayed in Fig. 10.3. A single level was added at a time, with the significance of that term assessed on the basis of a reduction in the Deviance Information Criterion (DIC) (Spiegelhalter et al. 2002). Our dataset contains 405 local authorities, 113,907 household years,^{8} 18 years, nineteen 5-year cohort groups,^{9} 25,889 people and 194,217 measured occasions, which form the structure of the random part of our models. In this case, it was found that all 6 levels were significant (the variances at these levels are different from zero) and were retained in the model (model 2 in Table 10.1). Finally we tested whether there were differential effects in the age effect between individuals, by allowing the linear age effect to vary at the person level; we also tested whether the random cohort variation was different for different genders, by allowing the gender effect to vary at the cohort level. We found the former to be significant and the latter insignificant (see model 3).^{10}

The results from three models – a two-level model, and two six-level models (one with random intercepts only and one with random slopes at the individual level on the age coefficient) – can be found in Table 10.1. As can be seen, model 1 (the two-level model) shows evidence of a significant age-cohort interaction with a negative coefficient estimate; this suggests that the life course effect is larger for earlier cohorts than for later cohorts. However this became insignificant when new levels were added to the model, and so in models 2 and 3 this term was removed.

^{11}in Fig. 10.8. It can be seen that those with higher GHQ scores will experience a greater increase in GHQ scores over their lifetime than those with lower GHQ scores, and so the variance between individuals is greater amongst older people than younger people – this is a result of the positive covariance term (0.077) in model 3 of Table 10.1.

This model is for illustration only – one would normally add additional time varying and time invariant control variables (for example employment status, social position, income, wealth and ethnicity) in an attempt to account for the unexplained variation in the random part of the model. One could also further extend the model in any of the other ways mentioned above. It is also worth noting that, whilst this model presented here uses a continuous outcome, other outcomes could be used with different link functions (for example, if you wanted to analyse a binary health outcome, a logit or probit version of this model could be used).

## Conclusions

The aim of this chapter is to highlight the perils in modelling age, period and cohort effects, and to provide some ways in which these perils can be overcome in longitudinal and life course research, in health and beyond. We have shown that researchers must put serious thought into which of age period and cohort they believe are behind changes that occur in society, and these must be appropriately specified in their model for accurate, policy-relevant inference to be made. We have highlighted a number of attempts to disentangle APC effects, and shown the shortcomings of these. Finally, we have presented a framework by which APC effects can be robustly measured, so long as certain assumptions (in our case the assumption of an absence of period trends) can be made. So long as this is the case, both long run polynomial trends and discrete random fluctuations can be modelled effectively, within a multilevel framework that can incorporate further variables and levels. We do not claim that our most complex model is always necessary (indeed in our example the simpler two-level model did a good job of accurately partitioning APC effects); however undoubtedly the extendibility of the model we present here is one of its strengths. Overall, we hope the chapter will encourage people to take the APC identification problem seriously and, when investigating life course and longitudinal effects, bear it in mind when constructing their statistical model.

## Footnotes

- 1.
This and the subsequent section are adapted from Bell and Jones (2014a), section “The Age-Period-Cohort Identification Problem”.

- 2.
This section is in part adapted from Bell and Jones (2013).

- 3.
- 4.
Here we only include a linear cohort trend, but if we find it to be necessary we could additionally include polynomials, as we do in the subsequent example.

- 5.
Where we are unwilling to constrain a trend to zero, but are willing to constrain it to an alternative value, informative priors can be used in a Bayesian framework. See Bell and Jones (2015).

- 6.
This analysis is a simplified version of that done by Bell (2014), which engages in more detail in the substantive debates about mental health, and uses further control variables and interaction terms.

- 7.
The GHQ is often assessed as a dichotomous outcome, where each question is scored as a ‘case’ or ‘non-case’ and respondents who are cases for 3 or more questions are considered cases overall. However, as Goldberg (1972, p. 1) states, “the distribution of psychiatric symptoms in the general population does not correspond to a sharp dichotomy between ‘cases’ and ‘normals’. Psychiatric disturbance may be thought of as being evenly distributed throughout the population in varying degrees of severity.”

- 8.
The BHPS does not provide data on households that is linked across time – that is, households are conceptualised here as transitory, changing each year.

- 9.
Cohorts were grouped into 5-year intervals in the random part of the model, to account for the autocorrelation between cohort years. However, single year groups were used to define the fixed part cohort trends.

- 10.
All models were run using Bayesian Monte Carlo Markov Chain (MCMC) estimation using MLwiN version 2.29 (Rasbash et al. 2014; Browne 2009), with a 500 iteration burn-in and 50,000 iteration chain length. For hierarchical models, starting values were obtained from Iterative Generalised Least Sqaures (IGLS) estimation (Goldstein 1989), whilst for the cross-classified models (which cannot be estimated in IGLS in MLwiN) the previous model’s estimates were used as starting values, with small (relatively non informative) values used for any new parameters. To speed up convergence, hierarchical centering was used, which reduces the correlation between the parameter chains and so improves the mixing of the MCMC algorithms (Browne 2009, p. 401). All parameter chains were visually inspected for convergence, and the Effective Sample Size (ESS) was used to assess whether the model had been run for long enough. It was found that 50,000 iterations are sufficient to produce ESS scores of over 400 for all parameters. For practical advice on MCMC estimation see Jones and Subramanian (2013).

- 11.
Coverage intervals are not to be confused with confidence intervals. The latter gives the uncertainty around a parameter, the former gives the expected variation for the data.

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