In statistical analysis, Freedman's paradox, 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, the R2 goes to ρ, the F-statistic for the overall regression goes to 1.0, and 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,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 in addition to the accompanying issue of model selection bias,Burnham, K. P., & Anderson, D. R. (2002). Model Selection and Multimodel Inference: A Practical-Theoretic Approach, 2nd ed. Springer-Verlag. whereby estimators of predictor variables that have a weak relationship with the response variable are biased.
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