The package includes functions for age-period-cohort analysis. The statistical model is a generalized linear model (GLM) allowing for age, period and cohort factors, or a sub-set of the factors. The canonical parametrisation of Kuang, Nielsen and Nielsen (2008a) is used. The outline of an analysis is described below.
Bent Nielsen <bent.nielsen@nuffield.ox.ac.uk> 29 Jan 2015 updated 7 July 2025.
| Package: | apc |
| Type: | Package |
| Version: | 3.0.0 |
| Date: | 2025-07-07 |
| License: | GPL-3 |
The apc package uses the canonical parameters suggested by Kuang, Nielsen and Nielsen (2008a) and generalized by Nielsen (2014). These evolve around the second differences of age, period and cohort factors as well as an three parameters (level and two slopes) for a linear plane. The age, period and cohort factors themselves are not identifiable. They could be ad hoc identified by associating the levels and two slopes to the age, period and cohort factors in a particular way. This should be done with great care as such ad hoc identification easily masks which information is coming from the data and which information is coming from the choice of ad hoc identification scheme. An illustration is given below. A short description of the package can be found in Nielsen (2015).
A formal analysis of the identification of the age-period-cohort model can be found in Nielsen and Nielsen (2014). Forecasting is discussed in Kuang, Nielsen and Nielsen (2008b, 2011) and Martinez Miranda, Nielsen and Nielsen (2015). Methods for cross section data are introduced in Fannon, Monden and Nielsen (2019). Methods for panel data are introduced in Fannon (2020). For a recent overview see Fannon and Nielsen (2019). Methods for 2-sample analysis and mixed frequency age, period times scales are available for aggregate data, see Nielsen (2022a,b).
Inference. When analyzing aggregate data using an over-dispersed Poisson model or a log-normal model, inference is based on a Central Limit Theorem for infinitely divisible distributions developed in Harnau and Nielsen (2018) and Kuang and Nielsen (2020). This supports the situation where the data array has fixed dimensions but the information content in each cell is thought to be large.
The package covers age-period-cohort models for three types of data.
Tables of aggregate data.
Repeated cross sectional data.
Panel data.
Vignettes showing how to use the package and reproduce existing results are available on vignettes.
The apc package can be used as follows.
Aggregate data. For a vignette with an introduction to analysis of aggregate data, see see IntroductionAggregateData.pdf, IntroductionAggregateData.R. on vignettes.
Organize the data in as an apc.data.list.
Data are included in matrix format. Information needs to be given about the original data format.
Optionally, information can be given about the labels for the time scales.
Data in age-period format with mixed frequency is possible. That is age and period can be grouped differently. Choose options data.format and unit accordingly.
Construct descriptive plots using apc.plot.data.all.
This gives a series of descriptive plots. The plots can be called individually through
Plot data sums using apc.plot.data.sums.
Numerical values can be obtained through apc.data.sums.
Sparsity plots of data using apc.plot.data.sparsity.
Plot data using all combinations of two time scales using apc.plot.data.within.
Get an deviance table for the age-period-cohort model through
apc.fit.table. For two-sample models choose apc.fit.table.2s.
Estimate a particular (sub-model of) age-period-cohort model through
apc.fit.model. For two-sample models choose apc.fit.model.2s.
Plot probability transforms of observed responses given fit using
apc.plot.fit.pt.
Plot estimated parameters through
apc.plot.fit. For two-sample models choose apc.plot.fit.2s.
Numerical values of certain transformations of the canonical parameter can be obtained through
apc.identify. For mixed frequency data use apc.identify.mixed.
Recursive analysis can be done by selecting a subset of the observations through
apc.data.list.subset and then repeating analysis. This will reveal how sensitive
the results are to particular age, period and cohort groups.
Forecasting. Some functions have been been added for forecasting in from a Poisson response-only model
with an age-cohort parametrization
apc.forecast.ac
and with an age-period parametrization
apc.forecast.ap.
See also the overview on
apc.forecast
Repeated cross section and Panel Data. For a vignette with an introduction to analysis of repeated cross section data and panel data, see IntroductionIndividualData.pdf, IntroductionIndividualData.R on Vignettes. Further examples can be found in a second vignette, see IntroductionIndividualDataFurtherExample.pdf, IntroductionIndividualDataFurtherExample.R.
Data examples include
Aggregate data: 1-sample
data.asbestos includes counts of deaths from mesothelioma in the UK. This dataset has no measure for exposure. It can be analysed using a Poisson model with an "APC" or an "AC" design. Source: Martinez Miranda, Nielsen and Nielsen (2015). Also used in Nielsen (2015).
data.Italian.bladder.cancer includes counts of deaths from bladder cancer in the Italy. This dataset includes a measure for exposure. It can be analysed using a Poisson model with an "APC" or an "AC" design. Source: Clayton and Schifflers (1987a).
data.Belgian.lung.cancer includes counts of deaths from lung cancer in the Belgium. This dataset includes a measure for exposure. It can be analysed using a Poisson model with an "APC", "AC", "AP" or "Ad" design. Source: Clayton and Schifflers (1987a).
data.Japanese.breast.cancer includes counts of deaths from breast cancer in the Japan. This dataset includes a measure for exposure. It can be analysed using a Poisson model with an "APC" design. Source: Clayton and Schifflers (1987b).
Aggregate data: 2-sample & mixed frequency
data.Swiss.suicides includes mixed-frequency counts of suicides for women and men in Switzerland. This is used as illustration in Nielsen (2022b), see vignettes. Source: Riebler, Held, Rue and Bopp (2012).
Repeated cross section data
Wage data from the package ISLR
Panel data
PSID7682 data from the package AER. These are panel data on earnings for 595 individuals for the years 1976-1982.
Clayton, D. and Schifflers, E. (1987a) Models for temperoral variation in cancer rates. I: age-period and age-cohort models. Statistics in Medicine 6, 449-467.
Clayton, D. and Schifflers, E. (1987b) Models for temperoral variation in cancer rates. II: age-period-cohort models. Statistics in Medicine 6, 469-481.
Fannon, Z. (2020). D.Phil. thesis. University of Oxford.
Fannon, Z., Monden, C. and Nielsen, B. (2018) Age-period cohort modelling and covariates, with an application to obesity in England 2001-2014. Nuffield Discussion Paper: https://www.nuffield.ox.ac.uk/economics/Papers/2018/2018W05_obesity.pdf. Supplement Code for replication: https://www.nuffield.ox.ac.uk/economics/Papers/2018/2018W05_obesityReplication.zip.
Fannon, Z. and Nielsen, B. (2019) Age-period-cohort models. Oxford Research Encyclopedia of Economics and Finance. Oxford University Press. Download: tools:::Rd_expr_doi("https://doi.org/10.1093/acrefore/9780190625979.013.495"). Nuffield Discussion Paper: https://www.nuffield.ox.ac.uk/economics/Papers/2018/2018W04_age_period_cohort_models.pdf.
Harnau, J. and Nielsen, B. (2018) Over-dispersed age-period-cohort models. Journal of the American Statistical Association 113, 1722-1732. Download: Article, tools:::Rd_expr_doi("https://doi.org/10.1080/01621459.2017.1366908"). Nuffield Discussion Paper: https://www.nuffield.ox.ac.uk/economics/Papers/2017/HarnauNielsen2017apcDP.pdf.
Kuang, D. and Nielsen, B. (2020) Generalized Log-Normal Chain-Ladder. Scandinavian Actuarial Journal 2020, 553--576. Download: Open access: tools:::Rd_expr_doi("https://doi.org/10.1080/03461238.2019.1696885"). Nuffield Discussion Paper: https://www.nuffield.ox.ac.uk/economics/Papers/2018/2018W02_KuangNielsen2018GLNCL.pdf.
Kuang, D., Nielsen, B. and Nielsen, J.P. (2008a) Identification of the age-period-cohort model and the extended chain ladder model. Biometrika 95, 979-986. Download: tools:::Rd_expr_doi("https://doi.org/10.1093/biomet/asn026"). Nuffield Discussion Paper: http://www.nuffield.ox.ac.uk/economics/papers/2007/w5/KuangNielsenNielsen07.pdf.
Kuang, D., Nielsen, B. and Nielsen, J.P. (2008b) Forecasting with the age-period-cohort model and the extended chain-ladder model. Biometrika 95, 987-991. Download: tools:::Rd_expr_doi("https://doi.org/10.1093/biomet/asn038"). Nuffield Discussion Paper: http://www.nuffield.ox.ac.uk/economics/papers/2008/w9/KuangNielsenNielsen_Forecast.pdf.
Kuang, D., Nielsen, B. and Nielsen, J.P. (2011) Forecasting in an extended chain-ladder-type model. Journal of Risk and Insurance 78, 345-359. Download: tools:::Rd_expr_doi("https://doi.org/10.1111/j.1539-6975.2010.01395.x"). Nuffield Discussion Paper: http://www.nuffield.ox.ac.uk/economics/papers/2010/w5/Forecast24jun10.pdf.
Martinez Miranda, M.D., Nielsen, B. and Nielsen, J.P. (2015) Inference and forecasting in the age-period-cohort model with unknown exposure with an application to mesothelioma mortality. Journal of the Royal Statistical Society A 178, 29-55. Download: tools:::Rd_expr_doi("https://doi.org/10.1111/rssa.12051"). Nuffield Discussion paper: http://www.nuffield.ox.ac.uk/economics/papers/2013/Asbestos8mar13.pdf.
Nielsen, B. (2015) apc: An R package for age-period-cohort analysis. R Journal 7, 52-64. Download: Open access.
Nielsen, B. (2014) Deviance analysis of age-period-cohort models. Download: Nuffield Discussion paper: http://www.nuffield.ox.ac.uk/economics/papers/2014/apc_deviance.pdf.
Nielsen, B. (2022a) Age-period-cohort analysis of mixed frequency data. Download: Nuffield Discussion Paper: https://www.nuffield.ox.ac.uk/economics/Papers/2022/2022-W02apc_mixed.pdf.
Nielsen, B. (2022b) Two-sample age-period-cohort models with an application to Swiss suicide rates. Download: Nuffield Discussion Paper: https://www.nuffield.ox.ac.uk/economics/Papers/2022/2022-W03apc_2sample.pdf.
Nielsen, B. and Nielsen, J.P. (2014) Identification and forecasting in mortality models. The Scientific World Journal. vol. 2014, Article ID 347043, 24 pages. Download: tools:::Rd_expr_doi("https://doi.org/10.1155/2014/347043").
Riebler, A. and Held, L. and Rue, H. and Bopp, M. (2012) Gender-specific differences and the impact of family integration on time trends in age-stratified Swiss suicide rates. Journal of the Royal Statistical Society, Series A, 175, 479-490.
Vignettes are available on Vignettes.
Further information, including minor upgrades and a python version can be found on apc development web page.