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Bounding Analyses of Age-Period-Cohort Effects

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Demography

Abstract

For more than a century, researchers from a wide range of disciplines have sought to estimate the unique contributions of age, period, and cohort (APC) effects on a variety of outcomes. A key obstacle to these efforts is the linear dependence among the three time scales. Various methods have been proposed to address this issue, but they have suffered from either ad hoc assumptions or extreme sensitivity to small differences in model specification. After briefly reviewing past work, we outline a new approach for identifying temporal effects in population-level data. Fundamental to our framework is the recognition that it is only the slopes of an APC model that are unidentified, not the nonlinearities or particular combinations of the linear effects. One can thus use constraints implied by the data along with explicit theoretical claims to bound one or more of the APC effects. Bounds on these parameters may be nearly as informative as point estimates, even with relatively weak assumptions. To demonstrate the usefulness of our approach, we examine temporal effects in prostate cancer incidence and homicide rates. We conclude with a discussion of guidelines for further research on APC effects.

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Notes

  1. Following the convention in the APC literature, we use the shorthand of “effects” when referring to age, period, and cohort processes (e.g., Fienberg and Mason 1979; Glenn 1981; Mason et al. 1973; O’Brien 2015a; Yang and Land 2013a:). We discuss the issue of interpreting the coefficients from an APC model in the online appendix.

  2. For example, researchers have examined verbal ability (Alwin 1991; Hauser and Huang 1997; Wilson and Gove 1999; Yang and Land 2006), social trust (Clark and Eisenstein 2013; Putnam 1995; Robinson and Jackson 2001; Schwadel and Stout 2012), party identification (Bartels and Jackman 2014; Ghitza and Gelman 2014; Hout and Knoke 1975; Tilley and Evans 2014), religious affiliation (Chaves 1989; Firebaugh and Harley 1991), drug use (Chen et al. 2003; Kerr et al. 2004; O’Malley et al. 1984; Vedøy 2014), obesity (Diouf et al. 2010; Fu and Land 2015; Reither et al. 2009), cancer (Clayton and Schifflers 1987; Liu et al. 2001), and mental health (Lavori et al. 1987; Lewinsohn et al. 1993; Yang 2008).

  3. Researchers have used various terms in the literature to refer to the linear and nonlinear effects of an APC model. In this article, we refer to the linear effects as “slopes” or “linear effects”; conversely, we refer to the nonlinear effects as “nonlinearities” or “deviations.” By “effects” or “overall effects,” we refer to the combination of the linear and nonlinear effects.

  4. However, APC analysis arguably dates back to at least the 1860s, predating the eponymous diagrams of Wilhelm Lexis (see Keiding 2011).

  5. As Rodgers (1982a:785) cautioned, “Although a constraint of the type described by Mason et al. (1973; 1979) seems trivial, in fact it is exquisitely precise and has effects that are multiplied so that even a slight inconsistency between the constraint and reality, or small measurement errors, can have very large effects on estimates.” However, see also the reply by Smith et al. (1982) as well as the rejoinder by Rodgers (1982b).

  6. For notable examples of researchers identifying APC effects using explicit, theoretically motivated equality constraints, see Mason and Smith (1985) and O’Brien (2015a).

  7. For alternative perspectives on these techniques, see Fu (2016) and Land et al. (2016).

  8. For example, with reference to the IE, Yang and Land (2013a:119) noted that “the objective of the IE is not to estimate the unidentifiable regression coefficient vector.” That is, the IE finds the point on the solution line closest to the origin in terms of Euclidean distance, but it does not necessarily recover the actual age, period, and cohort effects.

  9. For simplicity of exposition we assume that age and period are aggregated into intervals of equal width. Additional complications arise when the age and period intervals are not equally spaced, because this can generate artifactual cyclical patterns. For approaches to estimating temporal effects when age and period intervals are unequal, see Holford (2006).

  10. Alternatively, one could fix the parameters at one of the levels to 0. By convention, researchers typically fix to 0 the first set of levels (e.g., αi = 1 = πj = 1 = γk = 1 = 0) or the last set (e.g., αi = I= πj = J = γk = K = 0), although other sets could be used.

  11. They are equivalent in the sense that as basis vectors they span the same space.

  12. The null vector is unique up to multiplication by a scalar.

  13. A simple linear transformation can be used to convert agei to i – i* because i – i* = (agei – age*) / (Δage), where age* is the midpoint for all age groups, and ∆age is the fixed difference between the midpoints. For example, suppose we have age1 = 32, age2 = 37, age3 = 42, age4 = 47, and age5 = 52. The midpoint across all age groups is 42, and the fixed difference between the groups is 5. Thus, we can calculate that age1 = 32 equals (32 – 42) / 5 = –2, which is equivalent to i – i* = 1 – 3 = –2.

  14. There is considerable disagreement in the social science and statistics literature on the causal status of nonmanipulable variables. As Rubin (1986) and Holland (1986) argued, such variables do not themselves have well-defined causal effects. However, in Pearl’s (2009) framework, these variables may be ascribed a causal status, with corresponding counterfactuals, even though they are not manipulable.

  15. One could also estimate the θs using Yijk = μ + (periodj)(θ1) + (cohortk)(θ2 – θ1) + εijk or Yijk = μ + (agei)(θ1 – θ2) + (periodj)(θ2) + εijk.

  16. The estimates when the age slope is constrained to equal 0 are α* = 0, π* = θ1, and γ* = θ2 – θ1, and the corresponding estimates when the cohort slope is fixed to 0 are α* = θ1 – θ2, π* = θ2, and γ* = 0.

  17. Figure 1 is a graphical representation of the APC identification problem. If there were no linear dependency, we would have three planes intersecting at a single point in the parameter space (O’Brien 2015a).

  18. The age-period-cohort origin is (0, 0, 0), but this is not directly visible on the 2D-APC graph unless θ2 − θ1 = 0.

  19. The –1 slope relating period to age and cohort as well as the differing direction of the period axis relative to the age and cohort axes in the 2D-APC graph are reflected in the opposing sign of ν in Eq. (4) for the period slope compared with that for the age and cohort slopes.

  20. Likewise, each plane in Fig. 1 can be thought of as a function of linear equations based on θ1 and θ2. For example, the age-period plane is defined by θ1 = α + π, so it is equivalent to the linear equation π = θ1 − α. Similarly, the period-cohort plane can be thought of as a function of θ2 = π + γ, with π = θ2 − γ.

  21. As a reviewer noted, some of these nonlinearities could be noise. One way to address possible noise is to smooth out the nonlinearities by setting the parameters for the higher-order nonlinearities to 0. Alternatively, one could use natural cubic splines, treating age, period, and cohort as continuous rather than categorical variables (see Heuer 1997).

  22. The raw data are yearly, ranging from 1973 to 2013.

  23. Nearly identical results were obtained with Poisson regression, but for ease of exposition, we present our findings using a classical linear regression with a logged rate outcome.

  24. Estimated values of both θ1 and θ2 are statistically significant at the conventional threshold of .05.

  25. F tests indicate that all three time scales have statistically significant nonlinearities at the conventional threshold of .05.

  26. However, this interpretation is not strictly correct because we can estimate the values of θ1 and θ2. For example, if we assume that the period slope is 0, then the age slope must be 7.344 rather than 0.

  27. Estimated values of both θ1 and θ2 are statistically significant at the conventional threshold of .05.

  28. F tests show that at the conventional threshold of .05, all three time scales exhibit statistically significant nonlinearities.

  29. The 2D-APC graph of the solution line is available in the online appendix.

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  1. Department of Sociology, University of Toronto, Toronto, ON, M5S 2J4, Canada

    Ethan Fosse

  2. Department of Sociology, Harvard University, Cambridge, MA, 02138, USA

    Christopher Winship

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Fosse, E., Winship, C. Bounding Analyses of Age-Period-Cohort Effects. Demography 56, 1975–2004 (2019). https://doi.org/10.1007/s13524-019-00801-6

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