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

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 overview 1.1Behavioristic and inferential rationales for Neyman–Pearson (N–P) tests 1.2Severity rationale: induction as severe testing 1.3Severity as a meta-statistical concept: three required restrictions on the N–P paradigm2. Error statistical tests from the severity perspective 2.1N–P test T(α): type I, II error probabilities and power 2.2Specifying test T(α) using p-values3. Neyman's post-data use of power 3.1Neyman: does failure to reject H warrant confirming H?4. Severe testing as a basic concept for an adequate post-data inference 4.1The severity interpretation of acceptance (SIA) for test T(α) 4.2The fallacy of acceptance (i.e., an insignificant difference): Ms Rosy 4.3Severity and power5. Fallacy of rejection: statistical vs. substantive significance 5.1Taking a rejection of H0 as evidence for a substantive claim or theory 5.2A statistically significant difference from H0 may fail to indicate a substantively important magnitude 5.3Principle for the severity interpretation of a rejection (SIR) 5.4Comparing significant results with different sample sizes in T(α): large n problem 5.5General testing rules for T(α), using the severe testing concept6. The severe testing concept and confidence intervals 6.1Dualities between one and two-sided intervals and tests 6.2Avoiding 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? Introduction and overview 1.1Behavioristic and inferential rationales for Neyman–Pearson (N–P) tests 1.2Severity rationale: induction as severe testing 1.3Severity as a meta-statistical concept: three required restrictions on the N–P paradigm 1.1Behavioristic and inferential rationales for Neyman–Pearson (N–P) tests 1.2Severity rationale: induction as severe testing 1.3Severity as a meta-statistical concept: three required restrictions on the N–P paradigm Error statistical tests from the severity perspective 2.1N–P test T(α): type I, II error probabilities and power 2.2Specifying test T(α) using p-values 2.1N–P test T(α): type I, II error probabilities and power 2.2Specifying test T(α) using p-values Neyman's post-data use of power 3.1Neyman: does failure to reject H warrant confirming H? 3.1Neyman: does failure to reject H warrant confirming H? Severe testing as a basic concept for an adequate post-data inference 4.1The severity interpretation of acceptance (SIA) for test T(α) 4.2The fallacy of acceptance (i.e., an insignificant difference): Ms Rosy 4.3Severity and power 4.1The severity interpretation of acceptance (SIA) for test T(α) 4.2The fallacy of acceptance (i.e., an insignificant difference): Ms Rosy 4.3Severity and power Fallacy of rejection: statistical vs. substantive significance 5.1Taking a rejection of H0 as evidence for a substantive claim or theory 5.2A statistically significant difference from H0 may fail to indicate a substantively important magnitude 5.3Principle for the severity interpretation of a rejection (SIR) 5.4Comparing significant results with different sample sizes in T(α): large n problem 5.5General testing rules for T(α), using the severe testing concept 5.1Taking a rejection of H0 as evidence for a substantive claim or theory 5.2A statistically significant difference from H0 may fail to indicate a substantively important magnitude 5.3Principle for the severity interpretation of a rejection (SIR) 5.4Comparing significant results with different sample sizes in T(α): large n problem 5.5General testing rules for T(α), using the severe testing concept The severe testing concept and confidence intervals 6.1Dualities between one and two-sided intervals and tests 6.2Avoiding shortcomings of confidence intervals 6.1Dualities between one and two-sided intervals and tests 6.2Avoiding shortcomings of confidence intervals Beyond the N–P paradigm: pure significance, and misspecification tests Concluding comments: have we shown severity to be a basic concept in a N–P philosophy of induction?