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  February 2021 Early access  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
References:
[1]

A. Agresti, An Introduction to Categorical Data Analysis, Wiley Series in Probability and Statistics, John Wiley & Sons, Inc., Hoboken, NJ, 2019.

[2]

A.-L. Boulesteix and K. Strimmer, Partial least squares: A versatile tool for the analysis of high-dimensional genomic data, Briefings in Bioinformatics, 8 (2007), 32-44.  doi: 10.1093/bib/bbl016.

[3]

T. L. Boullion and P. L. Odell, Generalized Inverse Matrices, Wiley-Interscience, New York-London-Sydney, 1971.

[4]

M. Browning, I. Crawford and M. Knoef, The Age-Period Cohort Problem: Set Identification and Point Identification, Technical report, Cemmap working paper, 2012.

[5]

D. Clayton and E. Schifflers, Models for temporal variation in cancer rates. II: Age-period-cohort models, Statistics in Medicine, 6 (1987), 469-481.  doi: 10.1002/sim.4780060406.

[6]

S. E. Fienberg and W. M. Mason, Identification and estimation of age-period-cohort models in the analysis of discrete archival data, Sociological Methodology, 10 (1979), 1-67.  doi: 10.2307/270764.

[7]

S. E. Fienberg and W. M. Mason, Specification and implementation of age, period and cohort models, in Cohort Analysis in Social Research, Springer-Verlag, New York, 1985, 45–88. doi: 10.1007/978-1-4613-8536-3_3.

[8]

E. Fosse and C. Winship, Analyzing age-period-cohort data: A review and critique, Ann. Rev. Sociology, 45 (2019), 467-492.  doi: 10.1146/annurev-soc-073018-022616.

[9]

E. Fosse and C. Winship, Moore-Penrose estimators of age-period-cohort effects: Their interrelationship and properties, Sociological Science, 5 (2018), 304-334.  doi: 10.15195/v5.a14.

[10]

Q. Fu, The persistence of power despite the changing meaning of homeownership: An age-period-cohort analysis of urban housing tenure in China, 1989-2011, Urban Studies, 53 (2016), 1225-1243.  doi: 10.1177/0042098015571240.

[11]

Q. Fu and K. C. Land, Does urbanization matter? A temporal analysis of the socio-demographic gradient in the rising adulthood overweight epidemic in China, 1989-2009, Population, Space and Place, 23 (2017), 1-17.  doi: 10.1002/psp.1970.

[12]

Q. Fu and K. C. Land, The increasing prevalence of overweight and obesity of children and youth in China, 1989-2009: An age-period-cohort analysis, Population Res. Policy Rev., 34 (2015), 901-921.  doi: 10.1007/s11113-015-9372-y.

[13]

Q. FuK. C. Land and V. L. Lamb, Violent physical bullying victimization at school: Has there been a recent increase in exposure or intensity? An age-period-cohort analysis in the United States, 1991 to 2012, Child Indicators Research, 9 (2016), 485-513.  doi: 10.1007/s12187-015-9317-3.

[14]

W. Fu, Constrained estimators and consistency of a regression model on a Lexis diagram, J. Amer. Statist. Assoc., 111 (2016), 180-199.  doi: 10.1080/01621459.2014.998761.

[15]

W. J. Fu, Ridge estimator in singular design with application to age-period-cohort analysis of disease rates, Comm. Stat. Theory Methods, 29 (2000), 263-278.  doi: 10.1080/03610920008832483.

[16]

W. J. Fu and P. Hall, Asymptotic properties of estimators in age-period-cohort analysis, Statist. Probab. Lett., 76 (2006), 1925-1929.  doi: 10.1016/j.spl.2006.04.051.

[17]

F. Girosi and G. King, Demographic Forecasting, Princeton University Press, Princeton, NJ, 2008.

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N. D. Glenn, Cohort Analysis, SAGE Publications, Inc., Thousand Oaks, CA, 2005. doi: 10.4135/9781412983662.

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N. D. Glenn, Cohort analysts' futile quest: Statistical attempts to separate age, period and cohort effects, Amer. Sociological Rev., 41 (1976), 900-904.  doi: 10.2307/2094738.

[20]

M. H. Graham, Confronting multicollinearity in ecological multiple regression, Ecology, 84 (2003), 2809-2815.  doi: 10.1890/02-3114.

[21]

T. Hastie, R. Tibshirani and J. Friedman, The Elements of Statistical Learning. Data Mining, Inference, and Prediction, Springer Series in Statistics, Springer, NY, 2009. doi: 10.1007/978-0-387-84858-7.

[22]

J. HobcraftJ. Menken and S. Preston, Age, period, and cohort effects in demography: A review, Population Index, 48 (1982), 4-43.  doi: 10.2307/2736356.

[23]

T. R. Holford, The estimation of age, period and cohort effects for vital rates, Biometrics, 39 (1983), 311-324.  doi: 10.2307/2531004.

[24]

S. Y. Jeon, Do Data Structures Matter? A Simulation Study for Testing the Validity of Age-Period-Cohort Models, Ph.D thesis, Utah State University, 2017.

[25]

L. L. KupperJ. M. JanisA. Karmous and B. G. Greenberg, Statistical age-period-cohort analysis: A review and critique, J. Chronic Diseases, 38 (1985), 811-830.  doi: 10.1016/0021-9681(85)90105-5.

[26]

L. L. KupperJ. M. JanisI. A. SalamaC. N. YoshizawaB. G. Greenberg and H. H. Winsborough, Age-period-cohort analysis: An illustration of the problems in assessing interaction in one observation per cell data, Comm. Stat. - Theory and Methods, 12 (1983), 201-217.  doi: 10.1080/03610928308828640.

[27]

K. C. LandQ. FuX. GuoS. Y. JeonE. N. Reither and E. Zang, Playing with the rules and making misleading statements: A response to Luo, Hodges, Winship, and Powers, Amer. J. Sociology, 122 (2016), 962-973. 

[28]

L. Luo, Assessing validity and application scope of the intrinsic estimator approach to the age-period-cohort problem, Demography, 50 (2013), 1945-1967.  doi: 10.1007/s13524-013-0243-z.

[29]

L. LuoJ. HodgesC. Winship and D. Powers, The sensitivity of the intrinsic estimator to coding schemes: Comment on Yang, Schulhofer-Wohl, Fu, and Land, Amer. J. Sociology, 122 (2016), 930-961.  doi: 10.1086/689830.

[30]

K. O. MasonW. M. MasonH. H. Winsborough and W. K. Poole, Some methodological issues in cohort analysis of archival data, Amer. Sociological Rev., 38 (1973), 242-258.  doi: 10.2307/2094398.

[31]

W. M. Mason and H. L. Smith, Age-period-cohort analysis and the study of deaths from pulmonary tuberculosis, in Cohort Analysis in Social Research, Springer, NY, 1985,151–227. doi: 10.1007/978-1-4613-8536-3_6.

[32]

W. M. Mason and N. H. Wolfinger, Cohort analysis, in International Encyclopedia of the Social & Behavioral Sciences, Pergamon, Oxford, 2001, 2189–2194. doi: 10.1016/B0-08-043076-7/00401-0.

[33]

P. McCullagh and J. A. Nelder, Generalized Linear Models, Monographs on Statistics and Applied Probability, Chapman & Hall, London, 1989.

[34]

C. E. McCulloch and S. R. Searle, Generalized, Linear, and Mixed Models, Wiley Series in Probability and Statistics: Texts, References, and Pocketbooks Section, Wiley-Interscience [John Wiley & Sons], New York, 2001. doi: 10.1002/0471722073.

[35]

T. Næs and H. Martens, Principal component regression in NIR analysis: Viewpoints, background details and selection of components, J. Chemometrics, 2 (1988), 155-167.  doi: 10.1002/cem.1180020207.

[36]

R. M. O'Brien, Age period cohort characteristic models, Social Sci. Res., 29 (2000), 123-139.  doi: 10.1006/ssre.1999.0656.

[37]

R. M. O'Brien, Age–period–cohort models and the perpendicular solution, Epidemiologic Methods, 4 (2015), 87-99.  doi: 10.1515/em-2014-0006.

[38]

R. M. O'Brien, Constrained estimators and age-period-cohort models, Sociol. Methods Res., 40 (2011), 419-452.  doi: 10.1177/0049124111415367.

[39]

C. Osmond and M. J. Gardner, Age, period and cohort models applied to cancer mortality rates, Statistics in Medicine, 1 (1982), 245-259.  doi: 10.1002/sim.4780010306.

[40] W. H. PressS. A. TeukolskyW. T. Vetterling and B. P. Flannery, Numerical Recipes. The Art of Scientific Computing, Cambridge University Press, Cambridge, 2007. 
[41]

C. Robertson and P. Boyle, Age-period-cohort analysis of chronic disease rates. I: Modelling approach, Statistics in Medicine, 17 (1998), 1305-1323.  doi: 10.1002/(SICI)1097-0258(19980630)17:12<1305::AID-SIM853>3.0.CO;2-W.

[42]

C. RobertsonS. Gandini and P. Boyle, Age-period-cohort models: A comparative study of available methodologies, J. Clinical Epidemiology, 52 (1999), 569-583.  doi: 10.1016/s0895-4356(99)00033-5.

[43]

W. L. Rodgers, Estimable functions of age, period, and cohort effects, Amer. Sociological Rev., 47 (1982), 774-787.  doi: 10.2307/2095213.

[44]

W. L. Rodgers, Reply to comment by Smith, Mason, and Fienberg, Amer. Sociological Rev., 47 (1982), 793-796.  doi: 10.2307/2095215.

[45]

S. R. Searle, Linear Models, John Wiley & Sons, Inc., New York-London-Sydney, 1971. doi: 10.1002/9781118491782.

[46]

A. Sen and M. Srivastava, Regression Analysis. Theory, Methods, and Applications, Springer Texts in Statistics, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-4470-7.

[47]

H. L. SmithW. M. Mason and S. E. Fienberg, Estimable Functions of Age, Period, and Cohort Effects: More chimeras of the age-period-cohort accounting framework: Comment on Rodgers, Amer. Sociological Rev., 47 (1982), 787-793.  doi: 10.2307/2095214.

[48] G. Strang, Introduction to Linear Algebra, Wellesley-Cambridge Press, Wellesley, MA, 2016. 
[49]

R. Tarone and K. Chu, Age-period-cohort analyses of breast-, ovarian-, endometrial- and cervical-cancer mortality rates for caucasian women in the USA, J. Epidemiology and Biostatistics, 5 (2000), 221-231. 

[50]

R. E. Tarone and K. C. Chu, Implications of birth cohort patterns in interpreting trends in breast cancer rates, JNCI: J. National Cancer Institute, 84 (1992), 1402-1410.  doi: 10.1093/jnci/84.18.1402.

[51]

Y.-K. TuN. Krämer and W.-C. Lee, Addressing the identification problem in age-period-cohort analysis: A tutorial on the use of partial least squares and principal components analysis, Epidemiology, 23 (2012), 583-593.  doi: 10.1097/EDE.0b013e31824d57a9.

[52]

A. W. van der Vaart, Asymptotic Statistics, Cambridge Series in Statistical and Probabilistic Mathematics, 3, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9780511802256.

[53]

J. R. Wilmoth, Variation in vital rates by age, period, and cohort, Sociological Methodology, 20 (1990), 295-335.  doi: 10.2307/271089.

[54]

M. Xu and D. A. Powers, Bayesian ridge estimation of age-period-cohort models, in Dynamic Demographic Analysis, Springer Ser. Demogr. Methods Popul. Anal., 39, Springer, Cham, 2016,337–359. doi: 10.1007/978-3-319-26603-9_17.

[55]

Y. YangW. J. Fu and K. C. Land, A methodological comparison of age-period-cohort models: The intrinsic estimator and conventional generalized linear models, Sociological Methodology, 34 (2004), 75-110.  doi: 10.1111/j.0081-1750.2004.00148.x.

[56]

Y. Yang and K. C. Land, Age-Period-Cohort Analysis. New Models, Methods, and Empirical Applications, Chapman & Hall/CRC Interdisciplinary Statistics Series, CRC Press, Boca Raton, FL, 2013. doi: 10.1201/b13902.

[57]

Y. YangS. Schulhofer-WohlW. J. Fu and K. C. Land, The intrinsic estimator for age-period-cohort analysis: What it is and how to use it, Amer. J. Sociology, 113 (2008), 1697-1736.  doi: 10.1086/587154.

show all references

References:
[1]

A. Agresti, An Introduction to Categorical Data Analysis, Wiley Series in Probability and Statistics, John Wiley & Sons, Inc., Hoboken, NJ, 2019.

[2]

A.-L. Boulesteix and K. Strimmer, Partial least squares: A versatile tool for the analysis of high-dimensional genomic data, Briefings in Bioinformatics, 8 (2007), 32-44.  doi: 10.1093/bib/bbl016.

[3]

T. L. Boullion and P. L. Odell, Generalized Inverse Matrices, Wiley-Interscience, New York-London-Sydney, 1971.

[4]

M. Browning, I. Crawford and M. Knoef, The Age-Period Cohort Problem: Set Identification and Point Identification, Technical report, Cemmap working paper, 2012.

[5]

D. Clayton and E. Schifflers, Models for temporal variation in cancer rates. II: Age-period-cohort models, Statistics in Medicine, 6 (1987), 469-481.  doi: 10.1002/sim.4780060406.

[6]

S. E. Fienberg and W. M. Mason, Identification and estimation of age-period-cohort models in the analysis of discrete archival data, Sociological Methodology, 10 (1979), 1-67.  doi: 10.2307/270764.

[7]

S. E. Fienberg and W. M. Mason, Specification and implementation of age, period and cohort models, in Cohort Analysis in Social Research, Springer-Verlag, New York, 1985, 45–88. doi: 10.1007/978-1-4613-8536-3_3.

[8]

E. Fosse and C. Winship, Analyzing age-period-cohort data: A review and critique, Ann. Rev. Sociology, 45 (2019), 467-492.  doi: 10.1146/annurev-soc-073018-022616.

[9]

E. Fosse and C. Winship, Moore-Penrose estimators of age-period-cohort effects: Their interrelationship and properties, Sociological Science, 5 (2018), 304-334.  doi: 10.15195/v5.a14.

[10]

Q. Fu, The persistence of power despite the changing meaning of homeownership: An age-period-cohort analysis of urban housing tenure in China, 1989-2011, Urban Studies, 53 (2016), 1225-1243.  doi: 10.1177/0042098015571240.

[11]

Q. Fu and K. C. Land, Does urbanization matter? A temporal analysis of the socio-demographic gradient in the rising adulthood overweight epidemic in China, 1989-2009, Population, Space and Place, 23 (2017), 1-17.  doi: 10.1002/psp.1970.

[12]

Q. Fu and K. C. Land, The increasing prevalence of overweight and obesity of children and youth in China, 1989-2009: An age-period-cohort analysis, Population Res. Policy Rev., 34 (2015), 901-921.  doi: 10.1007/s11113-015-9372-y.

[13]

Q. FuK. C. Land and V. L. Lamb, Violent physical bullying victimization at school: Has there been a recent increase in exposure or intensity? An age-period-cohort analysis in the United States, 1991 to 2012, Child Indicators Research, 9 (2016), 485-513.  doi: 10.1007/s12187-015-9317-3.

[14]

W. Fu, Constrained estimators and consistency of a regression model on a Lexis diagram, J. Amer. Statist. Assoc., 111 (2016), 180-199.  doi: 10.1080/01621459.2014.998761.

[15]

W. J. Fu, Ridge estimator in singular design with application to age-period-cohort analysis of disease rates, Comm. Stat. Theory Methods, 29 (2000), 263-278.  doi: 10.1080/03610920008832483.

[16]

W. J. Fu and P. Hall, Asymptotic properties of estimators in age-period-cohort analysis, Statist. Probab. Lett., 76 (2006), 1925-1929.  doi: 10.1016/j.spl.2006.04.051.

[17]

F. Girosi and G. King, Demographic Forecasting, Princeton University Press, Princeton, NJ, 2008.

[18]

N. D. Glenn, Cohort Analysis, SAGE Publications, Inc., Thousand Oaks, CA, 2005. doi: 10.4135/9781412983662.

[19]

N. D. Glenn, Cohort analysts' futile quest: Statistical attempts to separate age, period and cohort effects, Amer. Sociological Rev., 41 (1976), 900-904.  doi: 10.2307/2094738.

[20]

M. H. Graham, Confronting multicollinearity in ecological multiple regression, Ecology, 84 (2003), 2809-2815.  doi: 10.1890/02-3114.

[21]

T. Hastie, R. Tibshirani and J. Friedman, The Elements of Statistical Learning. Data Mining, Inference, and Prediction, Springer Series in Statistics, Springer, NY, 2009. doi: 10.1007/978-0-387-84858-7.

[22]

J. HobcraftJ. Menken and S. Preston, Age, period, and cohort effects in demography: A review, Population Index, 48 (1982), 4-43.  doi: 10.2307/2736356.

[23]

T. R. Holford, The estimation of age, period and cohort effects for vital rates, Biometrics, 39 (1983), 311-324.  doi: 10.2307/2531004.

[24]

S. Y. Jeon, Do Data Structures Matter? A Simulation Study for Testing the Validity of Age-Period-Cohort Models, Ph.D thesis, Utah State University, 2017.

[25]

L. L. KupperJ. M. JanisA. Karmous and B. G. Greenberg, Statistical age-period-cohort analysis: A review and critique, J. Chronic Diseases, 38 (1985), 811-830.  doi: 10.1016/0021-9681(85)90105-5.

[26]

L. L. KupperJ. M. JanisI. A. SalamaC. N. YoshizawaB. G. Greenberg and H. H. Winsborough, Age-period-cohort analysis: An illustration of the problems in assessing interaction in one observation per cell data, Comm. Stat. - Theory and Methods, 12 (1983), 201-217.  doi: 10.1080/03610928308828640.

[27]

K. C. LandQ. FuX. GuoS. Y. JeonE. N. Reither and E. Zang, Playing with the rules and making misleading statements: A response to Luo, Hodges, Winship, and Powers, Amer. J. Sociology, 122 (2016), 962-973. 

[28]

L. Luo, Assessing validity and application scope of the intrinsic estimator approach to the age-period-cohort problem, Demography, 50 (2013), 1945-1967.  doi: 10.1007/s13524-013-0243-z.

[29]

L. LuoJ. HodgesC. Winship and D. Powers, The sensitivity of the intrinsic estimator to coding schemes: Comment on Yang, Schulhofer-Wohl, Fu, and Land, Amer. J. Sociology, 122 (2016), 930-961.  doi: 10.1086/689830.

[30]

K. O. MasonW. M. MasonH. H. Winsborough and W. K. Poole, Some methodological issues in cohort analysis of archival data, Amer. Sociological Rev., 38 (1973), 242-258.  doi: 10.2307/2094398.

[31]

W. M. Mason and H. L. Smith, Age-period-cohort analysis and the study of deaths from pulmonary tuberculosis, in Cohort Analysis in Social Research, Springer, NY, 1985,151–227. doi: 10.1007/978-1-4613-8536-3_6.

[32]

W. M. Mason and N. H. Wolfinger, Cohort analysis, in International Encyclopedia of the Social & Behavioral Sciences, Pergamon, Oxford, 2001, 2189–2194. doi: 10.1016/B0-08-043076-7/00401-0.

[33]

P. McCullagh and J. A. Nelder, Generalized Linear Models, Monographs on Statistics and Applied Probability, Chapman & Hall, London, 1989.

[34]

C. E. McCulloch and S. R. Searle, Generalized, Linear, and Mixed Models, Wiley Series in Probability and Statistics: Texts, References, and Pocketbooks Section, Wiley-Interscience [John Wiley & Sons], New York, 2001. doi: 10.1002/0471722073.

[35]

T. Næs and H. Martens, Principal component regression in NIR analysis: Viewpoints, background details and selection of components, J. Chemometrics, 2 (1988), 155-167.  doi: 10.1002/cem.1180020207.

[36]

R. M. O'Brien, Age period cohort characteristic models, Social Sci. Res., 29 (2000), 123-139.  doi: 10.1006/ssre.1999.0656.

[37]

R. M. O'Brien, Age–period–cohort models and the perpendicular solution, Epidemiologic Methods, 4 (2015), 87-99.  doi: 10.1515/em-2014-0006.

[38]

R. M. O'Brien, Constrained estimators and age-period-cohort models, Sociol. Methods Res., 40 (2011), 419-452.  doi: 10.1177/0049124111415367.

[39]

C. Osmond and M. J. Gardner, Age, period and cohort models applied to cancer mortality rates, Statistics in Medicine, 1 (1982), 245-259.  doi: 10.1002/sim.4780010306.

[40] W. H. PressS. A. TeukolskyW. T. Vetterling and B. P. Flannery, Numerical Recipes. The Art of Scientific Computing, Cambridge University Press, Cambridge, 2007. 
[41]

C. Robertson and P. Boyle, Age-period-cohort analysis of chronic disease rates. I: Modelling approach, Statistics in Medicine, 17 (1998), 1305-1323.  doi: 10.1002/(SICI)1097-0258(19980630)17:12<1305::AID-SIM853>3.0.CO;2-W.

[42]

C. RobertsonS. Gandini and P. Boyle, Age-period-cohort models: A comparative study of available methodologies, J. Clinical Epidemiology, 52 (1999), 569-583.  doi: 10.1016/s0895-4356(99)00033-5.

[43]

W. L. Rodgers, Estimable functions of age, period, and cohort effects, Amer. Sociological Rev., 47 (1982), 774-787.  doi: 10.2307/2095213.

[44]

W. L. Rodgers, Reply to comment by Smith, Mason, and Fienberg, Amer. Sociological Rev., 47 (1982), 793-796.  doi: 10.2307/2095215.

[45]

S. R. Searle, Linear Models, John Wiley & Sons, Inc., New York-London-Sydney, 1971. doi: 10.1002/9781118491782.

[46]

A. Sen and M. Srivastava, Regression Analysis. Theory, Methods, and Applications, Springer Texts in Statistics, Springer-Verlag, New York, 1990. doi: 10.1007/978-1-4612-4470-7.

[47]

H. L. SmithW. M. Mason and S. E. Fienberg, Estimable Functions of Age, Period, and Cohort Effects: More chimeras of the age-period-cohort accounting framework: Comment on Rodgers, Amer. Sociological Rev., 47 (1982), 787-793.  doi: 10.2307/2095214.

[48] G. Strang, Introduction to Linear Algebra, Wellesley-Cambridge Press, Wellesley, MA, 2016. 
[49]

R. Tarone and K. Chu, Age-period-cohort analyses of breast-, ovarian-, endometrial- and cervical-cancer mortality rates for caucasian women in the USA, J. Epidemiology and Biostatistics, 5 (2000), 221-231. 

[50]

R. E. Tarone and K. C. Chu, Implications of birth cohort patterns in interpreting trends in breast cancer rates, JNCI: J. National Cancer Institute, 84 (1992), 1402-1410.  doi: 10.1093/jnci/84.18.1402.

[51]

Y.-K. TuN. Krämer and W.-C. Lee, Addressing the identification problem in age-period-cohort analysis: A tutorial on the use of partial least squares and principal components analysis, Epidemiology, 23 (2012), 583-593.  doi: 10.1097/EDE.0b013e31824d57a9.

[52]

A. W. van der Vaart, Asymptotic Statistics, Cambridge Series in Statistical and Probabilistic Mathematics, 3, Cambridge University Press, Cambridge, 1998. doi: 10.1017/CBO9780511802256.

[53]

J. R. Wilmoth, Variation in vital rates by age, period, and cohort, Sociological Methodology, 20 (1990), 295-335.  doi: 10.2307/271089.

[54]

M. Xu and D. A. Powers, Bayesian ridge estimation of age-period-cohort models, in Dynamic Demographic Analysis, Springer Ser. Demogr. Methods Popul. Anal., 39, Springer, Cham, 2016,337–359. doi: 10.1007/978-3-319-26603-9_17.

[55]

Y. YangW. J. Fu and K. C. Land, A methodological comparison of age-period-cohort models: The intrinsic estimator and conventional generalized linear models, Sociological Methodology, 34 (2004), 75-110.  doi: 10.1111/j.0081-1750.2004.00148.x.

[56]

Y. Yang and K. C. Land, Age-Period-Cohort Analysis. New Models, Methods, and Empirical Applications, Chapman & Hall/CRC Interdisciplinary Statistics Series, CRC Press, Boca Raton, FL, 2013. doi: 10.1201/b13902.

[57]

Y. YangS. Schulhofer-WohlW. J. Fu and K. C. Land, The intrinsic estimator for age-period-cohort analysis: What it is and how to use it, Amer. J. Sociology, 113 (2008), 1697-1736.  doi: 10.1086/587154.

[1]

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