Computational Statistics

, Volume 33, Issue 2, pp 595–621 | Cite as

Model selection criteria based on cross-validatory concordance statistics

Original Paper
First Online:
Received:
Accepted:
  • 65 Downloads

Abstract

In the logistic regression framework, we present the development and investigation of three model selection criteria based on cross-validatory analogues of the traditional and adjusted c-statistics. These criteria are designed to estimate three corresponding measures of predictive error: the model misspecification prediction error, the fitting sample prediction error, and the sum of prediction errors. We aim to show that these estimators serve as suitable model selection criteria, facilitating the identification of a model that appropriately balances goodness-of-fit and parsimony, while achieving generalizability. We examine the properties of the selection criteria via an extensive simulation study designed as a factorial experiment. We then employ these measures in a practical application based on modeling the occurrence of heart disease.

Keywords

Akaike information criterion Logistic regression Prediction ROC curve Variable selection 
This is a preview of subscription content, log in to check access

Notes

Acknowledgements

We wish to thank our referees for their valuable feedback, which served to improve the original version of this manuscript.

References

  1. Akaike H (1973) Information theory and an extension of the maximum likelihood principle. In: Petrov BN, Csaki F (eds) 2nd international symposium on information theory. Akademia Kiado, Budapest, pp 267–281Google Scholar
  2. Akaike H (1974) A new look at the statistical model identification. IEEE Trans Autom Control AC–19:716–723MathSciNetCrossRefMATHGoogle Scholar
  3. Allen DM (1974) The relationship between variable selection and data augmentation and a method for prediction. Technometrics 16:125–127MathSciNetCrossRefMATHGoogle Scholar
  4. Arlot S, Celisse A (2010) A survey of cross-validation procedures for model selection. Stat Surv 4:40–79MathSciNetCrossRefMATHGoogle Scholar
  5. Bengtsson T, Cavanaugh JE (2006) An improved Akaike information criterion for state-space model selection. Comput Stat Data Anal 50:2635–2654MathSciNetCrossRefMATHGoogle Scholar
  6. Bozdogan H (1987) Model selection and Akaike’s information criterion (AIC): the general theory and its analytical extensions. Psychometrika 52:345–370MathSciNetCrossRefMATHGoogle Scholar
  7. Cavanaugh JE (1999) A large-sample model selection criterion based on Kullback’s symmetric divergence. Stat Probab Lett 44:333–344MathSciNetCrossRefMATHGoogle Scholar
  8. Cavanaugh JE, Shumway RH (1997) A bootstrap variant of AIC for state-space model selection. Stat Sin 7:473–496MathSciNetMATHGoogle Scholar
  9. Cook NR (2007) Use and misuse of the receiver operating characteristic curve in risk prediction. Circulation 115:928–935CrossRefGoogle Scholar
  10. Davies SL, Neath AA, Cavanaugh JE (2005) Cross validation model selection criteria for linear regression based on the Kullback–Leibler discrepancy. Stat Methodol 2:249–266MathSciNetCrossRefMATHGoogle Scholar
  11. Efron B (1983) Estimating the error rate of a prediction rule: improvement on cross-validation. J Am Stat Assoc 78:316–331MathSciNetCrossRefMATHGoogle Scholar
  12. Efron B (1986) How biased is the apparent error rate of a prediction rule? J Am Stat Assoc 81:461–470MathSciNetCrossRefMATHGoogle Scholar
  13. Golub GH, Heath M, Wahba G (1979) Generalized cross-validation as a method for choosing a good ridge parameter. Technometrics 21:215–223MathSciNetCrossRefMATHGoogle Scholar
  14. Gonen M, Heller G (2005) Concordance probability and discriminatory power in proportional hazards regression. Biometrika 92:965–970MathSciNetCrossRefMATHGoogle Scholar
  15. Hanley JA, McNeil BJ (1982) The meaning and use of the area under a receiver operating characteristic (ROC) curve. Radiology 143:29–36CrossRefGoogle Scholar
  16. Hastie T, Tibshirani R, Friedman J (2009) The elements of statistical learning, 2nd edn. Springer, New YorkCrossRefMATHGoogle Scholar
  17. Heagerty PJ, Zheng Y (2005) Survival model predictive accuracy and ROC curves. Biometrics 61:92–105MathSciNetCrossRefMATHGoogle Scholar
  18. Hilden J, Habbema JD, Bjerregaard B (1978) The measurement of performance in probabilistic diagnosis. II. Trustworthiness of the exact values of the diagnostic probabilities. Methods Inf Med 17:227–237CrossRefGoogle Scholar
  19. Hosmer DW, Lemeshow S (1980) A goodness-of-fit test for the multiple logistic regression model. Commun Stat A10:1043–1069CrossRefMATHGoogle Scholar
  20. Hurvich CM, Tsai CL (1989) Regression and time series model selection in small samples. Biometrika 76:297–307MathSciNetCrossRefMATHGoogle Scholar
  21. Hurvich CM, Shumway RH, Tsai CL (1990) Improved estimators of Kullback–Leibler information for autoregressive model selection in small samples. Biometrika 77:709–719MathSciNetGoogle Scholar
  22. Ishiguro M, Sakamoto Y, Kitagawa G (1997) Bootstrapping log likelihood and EIC, an extension of AIC. Ann Inst Stat Math 49:411–434MathSciNetCrossRefMATHGoogle Scholar
  23. Kullback S (1968) Information theory and statistics. Dover, New YorkMATHGoogle Scholar
  24. Kullback S, Leibler RA (1951) On information and sufficiency. Ann Math Stat 22:79–86MathSciNetCrossRefMATHGoogle Scholar
  25. Lemeshow S, Hosmer DW (1982) A review of goodness of fit statistics for use in the development of logistic regression models. Am J Epidemiol 115:92–106CrossRefGoogle Scholar
  26. Linhart H, Zucchini W (1986) Model selection. Wiley, New YorkMATHGoogle Scholar
  27. Mallows CL (1973) Some comments on \(C_p\). Technometrics 15:661–675MATHGoogle Scholar
  28. Metz CE (1986) ROC methodology in radiologic imaging. Investig Radiol 21:720–733CrossRefGoogle Scholar
  29. Metz CE (1989) Some practical issues of experimental design and data analysis in radiologic ROC studies. Investig Radiol 24:234–245CrossRefGoogle Scholar
  30. Pan W (2001) Akaike’s information criterion in generalized estimating equations. Biometrics 57:120–125MathSciNetCrossRefMATHGoogle Scholar
  31. Pencina MJ, D’Agostino RB Sr, D’Agostino RB Jr, Vasan RS (2008) Evaluating the added predictive ability of a new marker: from area under the ROC curve to reclassification and beyond. Stat Med 27:157–172MathSciNetCrossRefGoogle Scholar
  32. Royston P, Altman DG (2010) Visualizing and assessing discrimination in the logistic regression model. Stat Med 29:2508–2520MathSciNetCrossRefGoogle Scholar
  33. Schwarz G (1978) Estimating the dimension of a model. Ann Stat 6:461–464MathSciNetCrossRefMATHGoogle Scholar
  34. Shao J (1993) Linear model selection by cross-validation. J Am Stat Assoc 88:486–495MathSciNetCrossRefMATHGoogle Scholar
  35. Shibata R (1980) Asymptotically efficient selection of the order of the model for estimating parameters of a linear process. Ann Stat 8:147–164MathSciNetCrossRefMATHGoogle Scholar
  36. Shibata R (1981) An optimal selection of regression variables. Biometrika 68:45–54MathSciNetCrossRefMATHGoogle Scholar
  37. Shibata R (1997) Bootstrap estimate of Kullback–Leibler information for model selection. Stat Sin 7:375–394MathSciNetMATHGoogle Scholar
  38. Steyerberg EW, Vickers AJ, Cook NR, Gerds T, Gonen M, Obuchowski N, Pencina MJ, Kattan MW (2010) Assessing the performance of prediction models: a framework for some traditional and novel measures. Epidemiology 21:128–138CrossRefGoogle Scholar
  39. Stone M (1977) An asymptotic equivalence of choice of model by cross-validation and Akaike’s criterion. J R Stat Soc Ser B 39:44–47MathSciNetMATHGoogle Scholar
  40. Sugiura N (1978) Further analysis of the data by Akaike’s information criterion and the finite corrections. Commun Stat A7:13–26MathSciNetCrossRefMATHGoogle Scholar
  41. Takeuchi K (1976) Distribution of information statistics and criteria for adequacy of models. Math Sci 153:12–18 (in Japanese)Google Scholar
  42. Tibshirani R (1996) Regression shrinkage and selection via the Lasso. J R Stat Soc Ser B 58:267–288MathSciNetMATHGoogle Scholar
  43. Ten Eyck P, Cavanaugh JE (2015) The adjusted concordance statistic. In: Karagrigoriou A, Oliveira T, Skiadas C (eds) Statistical, stochastic and data analysis methods and applications. ISAST, Athens, pp 143–156Google Scholar
  44. Vieu P (1994) Choice of regressors in nonparametric estimation. Comput Stat Data Anal 17:575–594CrossRefMATHGoogle Scholar
  45. Zhang P (1991) Variable selection in nonparametric regression with continuous covariates. Ann Stat 19:1869–1882MathSciNetCrossRefMATHGoogle Scholar
  46. Zhou XH, Obuchowski NA, McClish DK (2002) Stat Methods Diagn Med. Wiley, New YorkCrossRefGoogle Scholar
  47. Zou H, Hastie T (2005) Regularization and variable selection via the elastic net. J R Stat Soc Ser B 67:301–320MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany 2017

Authors and Affiliations

  1. 1.Institute for Clinical and Translational ScienceThe University of IowaIowa CityUSA
  2. 2.Department of BiostatisticsThe University of IowaIowa CityUSA

Personalised recommendations