# Freedman's paradox

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 *R*^{2} 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

**^**Freedman, D. A. (1983) "A note on screening regression equations."*The American Statistician*,**37**, 152–155.**^**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.**^**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**^**Burnham, K. P., & Anderson, D. R. (2002).*Model Selection and Multimodel Inference: A Practical-Theoretic Approach,*2nd ed. Springer-Verlag.