Freedman's paradox

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In statistical analysis, Freedman's paradox,[1][2] named after David Freedman, is a problem in model selection whereby predictor variables with no relationship to the dependent variable can pass tests of significance – both individually via a t-test, and jointly via an F-test for the significance of the regression. Freedman demonstrated (through simulation and asymptotic calculation) that this is a common occurrence when the number of variables is similar to the number of data points.

Specifically, if the dependent variable and k regressors are independent normal variables, and there are n observations, then as k and n jointly go to infinity in the ratio k/n=ρ, (1) the R2 goes to ρ, (2) the F-statistic for the overall regression goes to 1.0, and (3) the number of spuriously significant regressors goes to αk where α is the chosen critical probability (probability of Type I error for a regressor). This third result is intuitive because it says that the number of Type I errors equals the probability of a Type I error on an individual parameter times the number of parameters for which significance is tested.

More recently, new information-theoretic estimators have been developed in an attempt to reduce this problem,[3] in addition to the accompanying issue of model selection bias,[4] whereby estimators of predictor variables that have a weak relationship with the response variable are biased.

References[edit]

  1. ^ Freedman, D. A. (1983) "A note on screening regression equations." The American Statistician, 37, 152–155.
  2. ^ Freedman, Laurence S.; Pee, David (November 1989). "Return to a Note on Screening Regression Equations". The American Statistician. 43 (4): 279–282. doi:10.2307/2685389. JSTOR 2685389. 
  3. ^ Lukacs, P. M., Burnham, K. P. & Anderson, D. R. (2010) "Model selection bias and Freedman's paradox." Annals of the Institute of Statistical Mathematics, 62(1), 117–125 doi:10.1007/s10463-009-0234-4
  4. ^ Burnham, K. P., & Anderson, D. R. (2002). Model Selection and Multimodel Inference: A Practical-Theoretic Approach, 2nd ed. Springer-Verlag.