Will the ASA's Efforts to Improve Statistical Practice be Successful? Some Evidence to the Contrary

ABSTRACT

Recent efforts by the American Statistical Association to improve statistical practice, especially in countering the misuse and abuse of null hypothesis significance testing (NHST) and p-values, are to be welcomed. But will they be successful? The present study offers compelling evidence that this will be an extraordinarily difficult task. Dramatic citation-count data on 25 articles and books severely critical of NHST's negative impact on good science, underlining that this issue was/is well known, did nothing to stem its usage over the period 1960–2007. On the contrary, employment of NHST increased during this time. To be successful in this endeavor, as well as restoring the relevance of the statistics profession to the scientific community in the 21st century, the ASA must be prepared to dispense detailed advice. This includes specifying those situations, if they can be identified, in which the p-value plays a clearly valuable role in data analysis and interpretation. The ASA might also consider a statement that recommends abandoning the use of p-values.     

The adaptive calibration of testing p-values

Abstract

In statistical hypothesis testing, a p-value is expected to be distributed as the uniform distribution on the interval (0, 1) under the null hypothesis. However, some p-values, such as the generalized p-value and the posterior predictive p-value, cannot be assured of this property. In this paper, we propose an adaptive p-value calibration approach, and show that the calibrated p-value is asymptotically distributed as the uniform distribution. For Behrens–Fisher problem and goodness-of-fit test under a normal model, the calibrated p-values are constructed and their behavior is evaluated numerically. Simulations show that the calibrated p-values are superior than original ones.     

The widespread misinterpretation of p-values as error probabilities

The anonymous mixing of Fisherian (p-values) and Neyman–Pearsonian (α levels) ideas about testing, distilled in the customary but misleading p < α criterion of statistical significance, has led researchers in the social and management sciences (and elsewhere) to commonly misinterpret the p-value as a ‘data-adjusted’ Type I error rate. Evidence substantiating this claim is provided from a number of fronts, including comments by statisticians, articles judging the value of significance testing, textbooks, surveys of scholars, and the statistical reporting behaviours of applied researchers. That many investigators do not know the difference between p’s and α’s indicates much bewilderment over what those most ardently sought research outcomes—statistically significant results—means. Statisticians can play a leading role in clearing this confusion. A good starting point would be to abolish the p < α criterion of statistical significance.     

A Proposed Hybrid Effect Size Plus p-Value Criterion: Empirical Evidence Supporting its Use

ABSTRACT

When the editors of Basic and Applied Social Psychology effectively banned the use of null hypothesis significance testing (NHST) from articles published in their journal, it set off a fire-storm of discussions both supporting the decision and defending the utility of NHST in scientific research. At the heart of NHST is the p-value which is the probability of obtaining an effect equal to or more extreme than the one observed in the sample data, given the null hypothesis and other model assumptions. Although this is conceptually different from the probability of the null hypothesis being true, given the sample, p-values nonetheless can provide evidential information, toward making an inference about a parameter. Applying a 10,000-case simulation described in this article, the authors found that p-values’ inferential signals to either reject or not reject a null hypothesis about the mean (α?=?0.05) were consistent for almost 70% of the cases with the parameter’s true location for the sampled-from population. Success increases if a hybrid decision criterion, minimum effect size plus p-value (MESP), is used. Here, rejecting the null also requires the difference of the observed statistic from the exact null to be meaningfully large or practically significant, in the researcher’s judgment and experience. The simulation compares performances of several methods: from p-value and/or effect size-based, to confidence-interval based, under various conditions of true location of the mean, test power, and comparative sizes of the meaningful distance and population variability. For any inference procedure that outputs a binary indicator, like flagging whether a p-value is significant, the output of one single experiment is not sufficient evidence for a definitive conclusion. Yet, if a tool like MESP generates a relatively reliable signal and is used knowledgeably as part of a research process, it can provide useful information.     

P-Value Precision and Reproducibility

P-values are useful statistical measures of evidence against a null hypothesis. In contrast to other statistical estimates, however, their sample-to-sample variability is usually not considered or estimated, and therefore not fully appreciated. Via a systematic study of log-scale p-value standard errors, bootstrap prediction bounds, and reproducibility probabilities for future replicate p-values, we show that p-values exhibit surprisingly large variability in typical data situations. In addition to providing context to discussions about the failure of statistical results to replicate, our findings shed light on the relative value of exact p-values vis-a-vis approximate p-values, and indicate that the use of *, **, and *** to denote levels 0.05, 0.01, and 0.001 of statistical significance in subject-matter journals is about the right level of precision for reporting p-values when judged by widely accepted rules for rounding statistical estimates.     

Why are p-Values Controversial?

While it is often argued that a p-value is a probability; see Wasserstein and Lazar, we argue that a p-value is not defined as a probability. A p-value is a bijection of the sufficient statistic for a given test which maps to the same scale as the Type I error probability. As such, the use of p-values in a test should be no more a source of controversy than the use of a sufficient statistic. It is demonstrated that there is, in fact, no ambiguity about what a p-value is, contrary to what has been claimed in recent public debates in the applied statistics community. We give a simple example to illustrate that rejecting the use of p-values in testing for a normal mean parameter is conceptually no different from rejecting the use of a sample mean. The p-value is innocent; the problem arises from its misuse and misinterpretation. The way that p-values have been informally defined and interpreted appears to have led to tremendous confusion and controversy regarding their place in statistical analysis.     

Three Recommendations for Improving the Use of p-Values

ABSTRACT

Researchers commonly use p-values to answer the question: How strongly does the evidence favor the alternative hypothesis relative to the null hypothesis? p-Values themselves do not directly answer this question and are often misinterpreted in ways that lead to overstating the evidence against the null hypothesis. Even in the “post p?<?0.05 era,” however, it is quite possible that p-values will continue to be widely reported and used to assess the strength of evidence (if for no other reason than the widespread availability and use of statistical software that routinely produces p-values and thereby implicitly advocates for their use). If so, the potential for misinterpretation will persist. In this article, we recommend three practices that would help researchers more accurately interpret p-values. Each of the three recommended practices involves interpreting p-values in light of their corresponding “Bayes factor bound,” which is the largest odds in favor of the alternative hypothesis relative to the null hypothesis that is consistent with the observed data. The Bayes factor bound generally indicates that a given p-value provides weaker evidence against the null hypothesis than typically assumed. We therefore believe that our recommendations can guard against some of the most harmful p-value misinterpretations. In research communities that are deeply attached to reliance on “p?<?0.05,” our recommendations will serve as initial steps away from this attachment. We emphasize that our recommendations are intended merely as initial, temporary steps and that many further steps will need to be taken to reach the ultimate destination: a holistic interpretation of statistical evidence that fully conforms to the principles laid out in the ASA statement on statistical significance and p-values.     

The p-Value Requires Context,Not a Threshold

Abstract

It is widely recognized by statisticians, though not as widely by other researchers, that the p-value cannot be interpreted in isolation, but rather must be considered in the context of certain features of the design and substantive application, such as sample size and meaningful effect size. I consider the setting of the normal mean and highlight the information contained in the p-value in conjunction with the sample size and meaningful effect size. The p-value and sample size jointly yield 95% confidence bounds for the effect of interest, which can be compared to the predetermined meaningful effect size to make inferences about the true effect. I provide simple examples to demonstrate that although the p-value is calculated under the null hypothesis, and thus seemingly may be divorced from the features of the study from which it arises, its interpretation as a measure of evidence requires its contextualization within the study. This implies that any proposal for improved use of the p-value as a measure of the strength of evidence cannot simply be a change to the threshold for significance.     

The p-value Function and Statistical Inference

ABSTRACT

This article has two objectives. The first and narrower is to formalize the p-value function, which records all possible p-values, each corresponding to a value for whatever the scalar parameter of interest is for the problem at hand, and to show how this p-value function directly provides full inference information for any corresponding user or scientist. The p-value function provides familiar inference objects: significance levels, confidence intervals, critical values for fixed-level tests, and the power function at all values of the parameter of interest. It thus gives an immediate accurate and visual summary of inference information for the parameter of interest. We show that the p-value function of the key scalar interest parameter records the statistical position of the observed data relative to that parameter, and we then describe an accurate approximation to that p-value function which is readily constructed.     

What Have We (Not) Learnt from Millions of Scientific Papers with P Values?

ABSTRACT

P values linked to null hypothesis significance testing (NHST) is the most widely (mis)used method of statistical inference. Empirical data suggest that across the biomedical literature (1990–2015), when abstracts use P values 96% of them have P values of 0.05 or less. The same percentage (96%) applies for full-text articles. Among 100 articles in PubMed, 55 report P values, while only 4 present confidence intervals for all the reported effect sizes, none use Bayesian methods and none use false-discovery rate. Over 25 years (1990–2015), use of P values in abstracts has doubled for all PubMed, and tripled for meta-analyses, while for some types of designs such as randomized trials the majority of abstracts report P values. There is major selective reporting for P values. Abstracts tend to highlight most favorable P values and inferences use even further spin to reach exaggerated, unreliable conclusions. The availability of large-scale data on P values from many papers has allowed the development and applications of methods that try to detect and model selection biases, for example, p-hacking, that cause patterns of excess significance. Inferences need to be cautious as they depend on the assumptions made by these models and can be affected by the presence of other biases (e.g., confounding in observational studies). While much of the unreliability of past and present research is driven by small, underpowered studies, NHST with P values may be also particularly problematic in the era of overpowered big data. NHST and P values are optimal only in a minority of current research. Using a more stringent threshold, as in the recently proposed shift from P < 0.05 to P < 0.005, is a temporizing measure to contain the flood and death-by-significance. NHST and P values may be replaced in many fields by other, more fit-for-purpose, inferential methods. However, curtailing selection biases requires additional measures, beyond changes in inferential methods, and in particular reproducible research practices.     

The False Positive Risk: A Proposal Concerning What to Do About p-Values

Abstract

It is widely acknowledged that the biomedical literature suffers from a surfeit of false positive results. Part of the reason for this is the persistence of the myth that observation of p?<?0.05 is sufficient justification to claim that you have made a discovery. It is hopeless to expect users to change their reliance on p-values unless they are offered an alternative way of judging the reliability of their conclusions. If the alternative method is to have a chance of being adopted widely, it will have to be easy to understand and to calculate. One such proposal is based on calculation of false positive risk(FPR). It is suggested that p-values and confidence intervals should continue to be given, but that they should be supplemented by a single additional number that conveys the strength of the evidence better than the p-value. This number could be the minimum FPR (that calculated on the assumption of a prior probability of 0.5, the largest value that can be assumed in the absence of hard prior data). Alternatively one could specify the prior probability that it would be necessary to believe in order to achieve an FPR of, say, 0.05.     

A method for combining p-values in meta-analysis by gamma distributions

Combining p-values from statistical tests across different studies is the most commonly used approach in meta-analysis for evolutionary biology. The most commonly used p-value combination methods mainly incorporate the z-transform tests (e.g., the un-weighted z-test and the weighted z-test) and the gamma-transform tests (e.g., the CZ method [Z. Chen, W. Yang, Q. Liu, J.Y. Yang, J. Li, and M.Q. Yang, A new statistical approach to combining p-values using gamma distribution and its application to genomewide association study, Bioinformatics 15 (2014), p. S3]). However, among these existing p-value combination methods, no method is uniformly most powerful in all situations [Chen et al. 2014]. In this paper, we propose a meta-analysis method based on the gamma distribution, MAGD, by pooling the p-values from independent studies. The newly proposed test, MAGD, allows for flexible accommodating of the different levels of heterogeneity of effect sizes across individual studies. The MAGD simultaneously retains all the characters of the z-transform tests and the gamma-transform tests. We also propose an easy-to-implement resampling approach for estimating the empirical p-values of MAGD for the finite sample size. Simulation studies and two data applications show that the proposed method MAGD is essentially as powerful as the z-transform tests (the gamma-transform tests) under the circumstance with the homogeneous (heterogeneous) effect sizes across studies.     

Content Audit for p-value Principles in Introductory Statistics

ABSTRACT

Longstanding concerns with the role and interpretation of p-values in statistical practice prompted the American Statistical Association (ASA) to make a statement on p-values. The ASA statement spurred a flurry of responses and discussions by statisticians, with many wondering about the steps necessary to expand the adoption of these principles. Introductory statistics classrooms are key locations to introduce and emphasize the nuance related to p-values; in part because they engrain appropriate analysis choices at the earliest stages of statistics education, and also because they reach the broadest group of students. We propose a framework for statistics departments to conduct a content audit for p-value principles in their introductory curriculum. We then discuss the process and results from applying this course audit framework within our own statistics department. We also recommend meeting with client departments as a complement to the course audit. Discussions about analyses and practices common to particular fields can help to evaluate if our service courses are meeting the needs of client departments and to identify what is needed in our introductory courses to combat the misunderstanding and future misuse of p-values.     

The Minimaxity of the Mid P-value under Linear and Squared Loss Functions

The mid-p is defined as the sum of the probabilities of all outcomes more extreme than an observed value, plus half of the probabilities of all outcomes exactly as extreme. On the one hand, it offers greater power than the standard p-value, but on the other, tests based on the mid-p statistic may have greater Type I error than their nominal level. This article investigates the mid p-value's properties under the estimated truth paradigm, which views p-values as estimators of the truth. The mid-p is shown to minimize the maximum risk for one-sided and two-sided tests.     

Putting the P-Value in its Place

ABSTRACT

As the debate over best statistical practices continues in academic journals, conferences, and the blogosphere, working researchers (e.g., psychologists) need to figure out how much time and effort to invest in attending to experts' arguments, how to design their next project, and how to craft a sustainable long-term strategy for data analysis and inference. The present special issue of The American Statistician promises help. In this article, we offer a modest proposal for a continued and informed use of the conventional p-value without the pitfalls of statistical rituals. Other statistical indices should complement reporting, and extra-statistical (e.g., theoretical) judgments ought to be made with care and clarity.     

Editorial Collaborators

Abstract

The present note explores sources of misplaced criticisms of P-values, such as conflicting definitions of “significance levels” and “P-values” in authoritative sources, and the consequent misinterpretation of P-values as error probabilities. It then discusses several properties of P-values that have been presented as fatal flaws: That P-values exhibit extreme variation across samples (and thus are “unreliable”), confound effect size with sample size, are sensitive to sample size, and depend on investigator sampling intentions. These properties are often criticized from a likelihood or Bayesian framework, yet they are exactly the properties P-values should exhibit when they are constructed and interpreted correctly within their originating framework. Other common criticisms are that P-values force users to focus on irrelevant hypotheses and overstate evidence against those hypotheses. These problems are not however properties of P-values but are faults of researchers who focus on null hypotheses and overstate evidence based on misperceptions that p?=?0.05 represents enough evidence to reject hypotheses. Those problems are easily seen without use of Bayesian concepts by translating the observed P-value p into the Shannon information (S-value or surprisal) –log₂(p).     

Significance test for linear regression: how to test without P-values?

The discussion on the use and misuse of p-values in 2016 by the American Statistician Association was a timely assertion that statistical concept should be properly used in science. Some researchers, especially the economists, who adopt significance testing and p-values to report their results, may felt confused by the statement, leading to misinterpretations of the statement. In this study, we aim to re-examine the accuracy of the p-value and introduce an alternative way for testing the hypothesis. We conduct a simulation study to investigate the reliability of the p-value. Apart from investigating the performance of p-value, we also introduce some existing approaches, Minimum Bayes Factors and Belief functions, for replacing p-value. Results from the simulation study confirm unreliable p-value in some cases and that our proposed approaches seem to be useful as the substituted tool in the statistical inference. Moreover, our results show that the plausibility approach is more accurate for making decisions about the null hypothesis than the traditionally used p-values when the null hypothesis is true. However, the MBFs of Edwards et al. [Bayesian statistical inference for psychological research. Psychol. Rev. 70(3) (1963), pp. 193–242]; Vovk [A logic of probability, with application to the foundations of statistics. J. Royal Statistical Soc. Series B (Methodological) 55 (1993), pp. 317–351] and Sellke et al. [Calibration of p values for testing precise null hypotheses. Am. Stat. 55(1) (2001), pp. 62–71] provide more reliable results compared to all other methods when the null hypothesis is false.KEYWORDS: Ban of P-value, Minimum Bayes Factors, belief functions     

Divergence versus decision P-values: A distinction worth making in theory and keeping in practice: Or,how divergence P-values measure evidence even when decision P-values do not

There are two distinct definitions of “P-value” for evaluating a proposed hypothesis or model for the process generating an observed dataset. The original definition starts with a measure of the divergence of the dataset from what was expected under the model, such as a sum of squares or a deviance statistic. A P-value is then the ordinal location of the measure in a reference distribution computed from the model and the data, and is treated as a unit-scaled index of compatibility between the data and the model. In the other definition, a P-value is a random variable on the unit interval whose realizations can be compared to a cutoff α to generate a decision rule with known error rates under the model and specific alternatives. It is commonly assumed that realizations of such decision P-values always correspond to divergence P-values. But this need not be so: Decision P-values can violate intuitive single-sample coherence criteria where divergence P-values do not. It is thus argued that divergence and decision P-values should be carefully distinguished in teaching, and that divergence P-values are the relevant choice when the analysis goal is to summarize evidence rather than implement a decision rule.     

A generalized p-value approach to inference on the performance measures of an M/Ek/1 queueing system

Abstract

The hypothesis tests of performance measures for an M/E_k/1 queueing system are considered. With pivotal models deduced from sufficient statistics for the unknown parameters, a generalized p-value approach to derive tests about parametric functions are proposed. The focus is on derivation of the p-values of hypothesis testing for five popular performance measures of the system in the steady state. Given a sample T, let p(T) be the p values we developed. We derive a closed form expression to show that, for small samples, the probability P(p(T) ? γ) is approximately equal to γ, for 0 ? γ ? 1.