# American Institute of Mathematical Sciences

February  2021, 4(1): 45-59. doi: 10.3934/mfc.2021001

## The uses and abuses of an age-period-cohort method: On the linear algebra and statistical properties of intrinsic and related estimators

 1 Department of Sociology, The University of British Columbia, Vancouver, BC V6T 1Z1, Canada 2 Department of Applied Mathematics, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong 3 School of Mathematics and Physics, The University of Queensland, Brisbane, QLD 4072, Australia 4 School of Medicine, University of California, San Francisco, CA 94121, USA 5 Department of Sociology, Social Work, and Anthropology, Utah State University, Logan, UT 84322, USA 6 Department of Sociology, Yale University, New Haven, CT 06511, USA 7 Department of Sociology and Social Science Research Institute, Duke University, Durham, NC 27708, USA

* Corresponding author: Qiang Fu

Received  September 2020 Published  December 2020

Fund Project: This research was partially supported by the Research Grants Council of Hong Kong [Project No. PolyU 15334616] and partially based on class notes provided by Qiang Fu during the course "Age-Period-Cohort Analysis: Principles, Models and Application", given in the Institute for Empirical Social Science Research (IESSR) at Xi'an Jiaotong University (July 2015) and in the School of Public Administration at Zhongnan University of Economics and Law (July 2018)

As a sophisticated and popular age-period-cohort method, the Intrinsic Estimator (IE) and related estimators have evoked intense debate in demography, sociology, epidemiology and statistics. This study aims to provide a more holistic review and critical assessment of the overall methodological significance of the IE and related estimators in age-period-cohort analysis. We derive the statistical properties of the IE from a linear algebraic perspective, provide more precise mathematical proofs relevant to the current debate, and demonstrate the essential, yet overlooked, link between the IE and classical statistical tools that have been employed by scholars for decades. This study offers guidelines for the future use of the IE and related estimators in demographic research. The exposition of the IE and related estimators may help redirect, if not settle, the logic of the debate.

Citation: Qiang Fu, Xin Guo, Sun Young Jeon, Eric N. Reither, Emma Zang, Kenneth C. Land. The uses and abuses of an age-period-cohort method: On the linear algebra and statistical properties of intrinsic and related estimators. Mathematical Foundations of Computing, 2021, 4 (1) : 45-59. doi: 10.3934/mfc.2021001
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