Deborah Mayo

After some general remarks about the interrelation between philosophical and statistical thinking, the discussion centres largely on significance tests. These are defined as the calculation of p-values rather than as formal procedures for ‘acceptance‘ and ‘rejection‘. A number of types of null hypothesis are described and a principle for evidential interpretation set out governing the implications of p- values in the specific circumstances of each application, as contrasted with a long-run interpretation. A number of more complicated situ- ations are discussed in which modification of the simple p-value may be essential.

Philosophers of science have long been concerned with the question of what a given scientific theory tells us about the contents of the world, but relatively little attention has been paid to how we set out to build theories and to the relevance of pre-theoretical methodology on a theory’s interpretation. In the traditional view, the form and content of a mature theory can be separated from any tentative ontological assumptions that went into its development. For this reason, the target of interpretation is taken to be the mature theory and nothing more. On this view, positions on ontology emerge only once a theory is to hand, not as part of the process of theory building. Serious attention to theory creation suggests this is too simple. In particular, data collection and experimentation are influenced both by theory and by assumptions about the entities thought to be the target of study. Initial reasoning about possible ontologies has an influence on the choice of theoretical variables as well as on the judgments of the methodology appropriate to investigate them.

I argue that the Bayesian Way of reconstructing Duhem's problem fails to advance a solution to the problem of which of a group of hypotheses ought to be rejected or "blamed" when experiment disagrees with prediction. But scientists do regularly tackle and often enough solve Duhemian problems. When they do, they employ a logic and methodology which may be called error statistics. I discuss the key properties of this approach which enable it to split off the task of testing auxiliary hypotheses from that of appraising a primary hypothesis. By discriminating patterns of error, this approach can at least block, if not also severely test, attempted explanations of an anomaly. I illustrate how this approach directs progress with Duhemian problems and explains how scientists actually grapple with them

The field of quantified risk assessment is a new field, only about 20 years old, and already it is considered to be in a crisis. As Funtowicz and J.R. Ravetz (1985) put it:The concept of risk in terms of probability has proved to be so elusive, and statistical inference so problematic, that many experts in the field have recently either lost hope of finding a scientific solution or lost faith in Risk Analysis as a tool for decisionmaking. (p.219)Thus the ‘art’ of the assessment of risks… is at an impasse. The early hopes that it could be reduced to a science are frustrated. …[O]thers are tending to introduce the ‘human’ and ‘cultural’ factors. The question now becomes, to what extent should these predominate? Would it be to the reduction or exclusion of the ‘scientific’ aspects? For, …if the perceived phenomena of ‘risks’ are interpreted as lacking all objective content or being merely a small part of some total cultural configuration, then there is no basis for dialogue between opposed positions on such problems, (pp.220-221)

In How the Laws of Physics Lie (1983)2 Cartwright argues for being a realist about theoretical entities but non-realist about theoretical laws. Her reason for this distinction is that only the former involves causal explanation, and accepting causal explanations commits us to the existence of the causal entity invoked. “What is special about explanation by theoretical entity is that it is causal explanation, and existence is an internal characteristic of causal claims. There is nothing similar for theoretical laws.” (p. 93). For, according to Cartwright, the acceptability of a theoretical explanation is a matter of its ability to satisfy such criteria as prganizing and simplifying, and in her view, “success at organizing, predicting, and classifying is never an argument for truth.” (p. 91). In contrast, Cartwright claims, “When I infer from an effect to a cause, I am asking what made the effect occur, what brought it about.

While the common procedure of statistical significance testing and its accompanying concept of p-values have long been surrounded by controversy, renewed concern has been triggered by the replication crisis in science. Many blame statistical significance tests themselves, and some regard them as sufficiently damaging to scientific practice as to warrant being abandoned. We take a contrary position, arguing that the central criticisms arise from misunderstanding and misusing the statistical tools, and that in fact the purported remedies themselves risk damaging science. We argue that banning the use of p-value thresholds in interpreting data does not diminish but rather exacerbates data-dredging and biasing selection effects. If an account cannot specify outcomes that will not be allowed to count as evidence for a claim—if all thresholds are abandoned—then there is no test of that claim. The contributions of this paper are: To explain the rival statistical philosophies underlying the ongoing controversy; To elucidate and reinterpret statistical significance tests, and explain how this reinterpretation ameliorates common misuses and misinterpretations; To argue why recent recommendations to replace, abandon, or retire statistical significance undermine a central function of statistics in science: to test whether observed patterns in the data are genuine or due to background variability.

A common intuition about evidence is that if data x have been used to construct a hypothesis H, then x should not be used again in support of H. It is no surprise that x fits H, if H was deliberately constructed to accord with x. The question of when and why we should avoid such “double-counting” continues to be debated in philosophy and statistics. It arises as a prohibition against data mining, hunting for significance, tuning on the signal, and ad hoc hypotheses, and as a preference for predesignated hypotheses and “surprising” predictions. I have argued that it is the severity or probativeness of the test—or lack of it—that should determine whether a double-use of data is admissible. I examine a number of surprising ambiguities and unexpected facts that continue to bedevil this debate

The issues of double-counting, use-constructing, and selection effects have long been the subject of debate in the philosophical as well as statistical literature. I have argued that it is the severity, stringency, or probativeness of the test—or lack of it—that should determine if a double-use of data is admissible. Hitchcock and Sober ([2004]) question whether this ‘severity criterion' can perform its intended job. I argue that their criticisms stem from a flawed interpretation of the severity criterion. Taking their criticism as a springboard, I elucidate some of the central examples that have long been controversial, and clarify how the severity criterion is properly applied to them. Severity and Use-Constructing: Four Points (and Some Clarificatory Notes) 1.1 Point 1: Getting beyond ‘all or nothing’ standpoints 1.2 Point 2: The rationale for prohibiting double-counting is the requirement that tests be severe 1.3 Point 3: Evaluate severity of a test T by its associated construction rule R 1.4 Point 4: The ease of passing vs. ease of erroneous passing: Statistical vs. ‘Definitional’ probability The False Dilemma: Hitchcock and Sober 2.1 Marsha measures her desk reliably 2.2 A false dilemma Canonical Errors of Inference 3.1 How construction rules may alter the error-probing performance of tests 3.2 Rules for accounting for anomalies 3.3 Hunting for statistically significant differences Concluding Remarks

The crisis of replication has led many to blame statistical significance tests for making it too easy to find impressive looking effects that do not replicate. However, the very fact it becomes difficult to replicate effects when features of the tests are tied down actually serves to vindicate statistical significance tests. While statistical significance tests, used correctly, serve to bound the probabilities of erroneous interpretations of data, this error control is nullified by data-dredging, multiple testing, and other biasing selection effects. Arguments claiming to vitiate statistical significance tests attack straw person variants of tests that commit well-known fallacies and misinterpretations. There is a tension between popular calls for preregistration – arguably, one of the most promising ways to boost replication – and accounts that downplay error probabilities: Bayes Factors, Bayesian posteriors, likelihood ratios. By underscoring the importance of error control for well testedness, the replication crisis points to reformulating tests so as to avoid fallacies and report the extent of discrepancies that are and are not indicated with severity.

The growing availability of computer power and statistical software has greatly increased the ease with which practitioners apply statistical methods, but this has not been accompanied by attention to checking the assumptions on which these methods are based. At the same time, disagreements about inferences based on statistical research frequently revolve around whether the assumptions are actually met in the studies available, e.g., in psychology, ecology, biology, risk assessment. Philosophical scrutiny can help disentangle 'practical' problems of model validation, and conversely, a methodology of statistical model validation can shed light on a number of issues of interest to philosophers of science

Discussions of science and values in risk management have largely focused on how values enter into arguments about risks, that is, issues of acceptable risk. Instead this volume concentrates on how values enter into collecting, interpreting, communicating, and evaluating the evidence of risks, that is, issues of the acceptability of evidence of risk. By focusing on acceptable evidence, this volume avoids two barriers to progress. One barrier assumes that evidence of risk is largely a matter of objective scientific data and therefore uncontroversial. The other assumes that evidence of risk, being "just" a matter of values, is not amenable to reasoned critique. Denying both extremes, this volume argues for a more constructive conclusion: understanding the interrelations of scientific and value issues enables a critical scrutiny of risk assessments and better public deliberation about social choices. The contributors, distinguished philosophers, policy analysts, and natural and social scientists, analyze environmental and medical controversies, and assumptions underlying views about risk assessment and the scientific and statistical models used in risk management.

The growing acceptance and success of experimental economics has increased the interest of researchers in tackling philosophical and methodological challenges to which their work increasingly gives rise. I sketch some general issues that call for the combined expertise of experimental economists and philosophers of science, of experiment, and of inductive‐statistical inference and modeling. †To contact the author, please write to: 235 Major Williams, Virginia Tech, Blacksburg, VA 24061‐0126; e‐mail: [email protected].

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

In Philosophical Problems of Statistical Inference, Seidenfeld argues that the Neyman-Pearson (NP) theory of confidence intervals is inadequate for a theory of inductive inference because, for a given situation, the 'best' NP confidence interval, [CIλ], sometimes yields intervals which are trivial (i.e., tautologous). I argue that (1) Seidenfeld's criticism of trivial intervals is based upon illegitimately interpreting confidence levels as measures of final precision; (2) for the situation which Seidenfeld considers, the 'best' NP confidence interval is not [CIλ] as Seidenfeld suggests, but rather a one-sided interval [CI0]; and since [CI0] never yields trivial intervals, NP theory escapes Seidenfeld's criticism entirely; (3) Seidenfeld's criterion of non-triviality is inadequate, for it leads him to judge an alternative confidence interval, [CI alt. ], superior to [CIλ] although [CI alt. ] results in counterintuitive inferences. I conclude that Seidenfeld has not shown that the NP theory of confidence intervals is inadequate for a theory of inductive inference

An important theme to have emerged from the new experimentalist movement is that much of actual scientific practice deals not with appraising full-blown theories but with the manifold local tasks required to arrive at data, distinguish fact from artifact, and estimate backgrounds. Still, no program for working out a philosophy of experiment based on this recognition has been demarcated. I suggest why the new experimentalism has come up short, and propose a remedy appealing to the practice of standard error statistics. I illustrate a portion of my proposal using Galison's experimental narrative on neutral currents

While orthodox (Neyman-Pearson) statistical tests enjoy widespread use in science, the philosophical controversy over their appropriateness for obtaining scientific knowledge remains unresolved. I shall suggest an explanation and a resolution of this controversy. The source of the controversy, I argue, is that orthodox tests are typically interpreted as rules for making optimal decisions as to how to behave--where optimality is measured by the frequency of errors the test would commit in a long series of trials. Most philosophers of statistics, however, view the task of statistical methods as providing appropriate measures of the evidential-strength that data affords hypotheses. Since tests appropriate for the behavioral-decision task fail to provide measures of evidential-strength, philosophers of statistics claim the use of orthodox tests in science is misleading and unjustified. What critics of orthodox tests overlook, I argue, is that the primary function of statistical tests in science is neither to decide how to behave nor to assign measures of evidential strength to hypotheses. Rather, tests provide a tool for using incomplete data to learn about the process that generated it. This they do, I show, by providing a standard for distinguishing differences (between observed and hypothesized results) due to accidental or trivial errors from those due to systematic or substantively important discrepancies. I propose a reinterpretation of a commonly used orthodox test to make this learning model of tests explicit