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Valid post-selection inference. (English) Zbl 1267.62080
Summary: It is common practice in statistical data analysis to perform data-driven variable selection and derive statistical inference from the resulting model. Such inference enjoys none of the guarantees that classical statistical theory provides for tests and confidence intervals when the model has been chosen a priori. We propose to produce valid “post-selection inference” by reducing the problem to one of simultaneous inference and hence suitably widening conventional confidence and retention intervals. Simultaneity is required for all linear functions that arise as coefficient estimates in all submodels. By purchasing “simultaneity insurance” for all possible submodels, the resulting post-selection inference is rendered universally valid under all possible model selection procedures. This inference is therefore generally conservative for particular selection procedures, but it is always less conservative than the full H. Scheffé protection [The analysis of variance. (1959; Zbl 0086.34603)]. Importantly it does not depend on the truth of the selected submodel, and hence it produces valid inference even in wrong models. We describe the structure of the simultaneous inference problem and give some asymptotic results.

62J05 Linear regression; mixed models
62J10 Analysis of variance and covariance (ANOVA)
62J15 Paired and multiple comparisons; multiple testing
62H12 Estimation in multivariate analysis
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