Английская Википедия:Freedman's paradox
Шаблон:Short description 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,
- 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,[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
- ↑ Шаблон:Cite journal
- ↑ Шаблон:Cite journal
- ↑ 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 Шаблон:Doi
- ↑ Burnham, K. P., & Anderson, D. R. (2002). Model Selection and Multimodel Inference: A Practical-Theoretic Approach, 2nd ed. Springer-Verlag.
