Degrees of Freedom and Model Selection in Linear Models with Repeated Measurement Errors

Abstract

Measurement errors are always inevitable in many data collection procedures. Despite the fundamental role of degrees of freedom in statistics, its behavior when measurement errors exist is not well understood. In this paper, we propose an unbiased and consistent estimator of the degrees of freedom to measure the model complexity when the data has repeated covariate measurement errors. Based on the estimated degrees of freedom, we propose a model selection criterion. The selection procedure based on the proposed criterion can achieve asymptotic loss efficiency and selection consistency with a more relaxed condition. Comprehensive simulation studies demonstrate the asymptotic properties of the proposed approach. We apply the proposed selection procedure to obesity data. The proposed approach achieves good prediction performance with more parsimonious models.

Keywords

Degrees of freedom / Model selection / Repeated measurement error / Selection consistency / 62F07 / 62J05

References

[1] Biswas S, Hansda CK, Singh N, Malla VR, Pal AK. Determinants of nutritional status among scheduled tribe women in india. Clinical Epidemiology and Global Health, 2022, 17 101119

[2] Blackwell M, Honaker J, King G. A unified approach to measurement error and missing data: overview and applications. Sociological Methods & Research, 2017, 46(3): 303-341

[3] Bode ED, Mathias KC, Stewart DF, Moffatt SM, Jack K, Smith DL. Cardiovascular disease risk factors by bmi and age in united states firefighters. Obesity, 2021, 29(7): 1186-1194

[4] Carroll RJ, Ruppert D, Stefanski LA, Crainiceanu CM. Measurement Error in Nonlinear Models: a Modern Perspective, 2006, Boca Raton, Chapman and Hall/CRC

[5] Casas-Agustench P, Arnett DK, Smith CE, Lai C-Q, Parnell LD, Borecki IB, Frazier-Wood AC, Allison M, Chen Y-DI, Taylor KD, et al.. Saturated fat intake modulates the association between an obesity genetic risk score and body mass index in two us populations. J. Acad. Nutr. Diet., 2014, 114(12): 1954-1966

[6] Celis-Morales CA, Lyall DM, Gray S, Steell L, Anderson J, Iliodromiti S, Welsh P, Guo Y, Petermann F, Mackay DF, et al.. Dietary fat and total energy intake modifies the association of genetic profile risk score on obesity: evidence from 48 170 uk biobank participants. Int. J. Obes., 2017, 41(12): 1761-1768

[7] Dandapat B, Biswas S, Patra B. Religion, nutrition and birth weight among currently married women (15–49) in india: a study based on nfhs-5. Clinical Epidemiology and Global Health, 2023, 20 101218

[8] Efron B. How biased is the apparent error rate of a prediction rule?. J. Am. Stat. Assoc., 1986, 81(394): 461-470

[9] Efron B. The estimation of prediction error: covariance penalties and cross-validation. J. Am. Stat. Assoc., 2004, 99(467): 619-632

[10] Flynn CJ, Hurvich CM, Simonoff JS. Efficiency for regularization parameter selection in penalized likelihood estimation of misspecified models. J. Am. Stat. Assoc., 2013, 108(503): 1031-1043

[11] Hales C, Carroll M, Fryar C, Ogden C. Prevalence of obesity and severe obesity among adults: United states, 2017–2018. NCHS Data Brief, 2020, 360: 1-8

[12] Jennings, J., Kim, J.M., Lee, J., Taylor, D.: Measurement error, fixed effects, and false positives in accounting research. Review of Accounting Studies, 1–37 (2023)

[13] Li K-C. Asymptotic optimality for cp, cl, cross-validation and generalized cross-validation: discrete index set. Ann. Stat., 1987, 15(3): 958-975

[14] Liang H, Li R. Variable selection for partially linear models with measurement errors. J. Am. Stat. Assoc., 2009, 104(485): 234-248

[15] Ma Y, Li R. Variable selection in measurement error models. Bernoulli, 2010, 16(1): 274

[16] Magnus JR, Neudecker H. Matrix Differential Calculus with Applications in Statistics and Econometrics, 2019, New York, John Wiley & Sons

[17] Memisoglu A, Hu FB, Hankinson SE, Manson JE, De Vivo I, Willett WC, Hunter DJ. Interaction between a peroxisome proliferator-activated receptor $\gamma $ gene polymorphism and dietary fat intake in relation to body mass. Hum. Mol. Genet., 2003, 12(222923-2929

[18] Mukherjee A, Chen K, Wang N, Zhu J. On the degrees of freedom of reduced-rank estimators in multivariate regression. Biometrika, 2015, 102(2): 457-477

[19] Murillo AL, Affuso O, Peterson CM, Li P, Wiener HW, Tekwe CD, Allison DB. Illustration of measurement error models for reducing bias in nutrition and obesity research using 2-d body composition data. Obesity, 2019, 27(3): 489-495

[20] Nakamura T. Proportional hazards model with covariates subject to measurement error. Biometrics, 1992, 48(3): 829-838

[21] Qin G, Zhang J, Zhu Z. Simultaneous mean and covariance estimation of partially linear models for longitudinal data with missing responses and covariate measurement error. Comput. Stat. Data Anal., 2016, 96: 24-39

[22] Schennach SM. Recent advances in the measurement error literature. Annual Review of Economics, 2016, 8: 341-377

[23] Schott JR. Matrix Analysis for Statistics, 2016, New York, John Wiley & Sons

[24] Shao J. An asymptotic theory for linear model selection. Stat. Sin., 1997, 7(2): 221-242

[25] Stein CM. Estimation of the mean of a multivariate normal distribution. Ann. Stat., 1981, 9(6): 1135-1151

[26] Tibshirani RJ. Degrees of freedom and model search. Stat. Sin., 2015, 25(3): 1265-1296

[27] Zhang X, Wang H, Ma Y, Carroll RJ. Linear model selection when covariates contain errors. J. Am. Stat. Assoc., 2017, 112(520): 1553-1561

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School of Mathematical Sciences, University of Science and Technology of China and Springer-Verlag GmbH Germany, part of Springer Nature