Severe Testing as a Basic Concept in a Neyman–Pearson Philosophy of Induction

Abstract

Despite the widespread use of key concepts of the Neyman–Pearson (N–P) statistical paradigm—type I and II errors, significance levels, power, confidence levels—they have been the subject of philosophical controversy and debate for over 60 years. Both current and long-standing problems of N–P tests stem from unclarity and confusion, even among N–P adherents, as to how a test's (pre-data) error probabilities are to be used for (post-data) inductive inference as opposed to inductive behavior. We argue that the relevance of error probabilities is to ensure that only statistical hypotheses that have passed severe or probative tests are inferred from the data. The severity criterion supplies a meta-statistical principle for evaluating proposed statistical inferences, avoiding classic fallacies from tests that are overly sensitive, as well as those not sensitive enough to particular errors and discrepancies. 1. Introduction and overview1.1Behavioristic and inferential rationales for Neyman–Pearson (N–P) tests1.2Severity rationale: induction as severe testing1.3Severity as a meta-statistical concept: three required restrictions on the N–P paradigm2. Error statistical tests from the severity perspective2.1N–P test T(α): type I, II error probabilities and power2.2Specifying test T(α) using p-values3. Neyman's post-data use of power3.1Neyman: does failure to reject H warrant confirming H?4. Severe testing as a basic concept for an adequate post-data inference4.1The severity interpretation of acceptance (SIA) for test T(α)4.2The fallacy of acceptance (i.e., an insignificant difference): Ms Rosy4.3Severity and power5. Fallacy of rejection: statistical vs. substantive significance5.1Taking a rejection of H0 as evidence for a substantive claim or theory5.2A statistically significant difference from H0 may fail to indicate a substantively important magnitude5.3Principle for the severity interpretation of a rejection (SIR)5.4Comparing significant results with different sample sizes in T(α): large n problem5.5General testing rules for T(α), using the severe testing concept6. The severe testing concept and confidence intervals6.1Dualities between one and two-sided intervals and tests6.2Avoiding shortcomings of confidence intervals7. Beyond the N–P paradigm: pure significance, and misspecification tests8. Concluding comments: have we shown severity to be a basic concept in a N–P philosophy of induction?