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Title: Redefine Statistical Significance
Authors: Daniel J. Benjamin1*, James O. Berger2, Magnus Johannesson3*, Brian A.
Nosek4,5,
E.J.
Wagenmakers6,
Richard
Berk7,
10,
Kenneth
A.
Bollen8,
Björn
Brembs9,
Lawrence Brown10, Colin Camerer11, David Cesarini12, 13, Christopher D. Chambers14,
Merlise Clyde2, Thomas D. Cook15,16, Paul De Boeck17, Zoltan Dienes18, Anna Dreber3,
Kenny Easwaran19, Charles Efferson20, Ernst Fehr21, Fiona Fidler22, Andy P. Field18,
Malcolm Forster23, Edward I. George10, Richard Gonzalez24, Steven Goodman25, Edwin
Green26, Donald P. Green27, Anthony Greenwald28, Jarrod D. Hadfield29, Larry V.
Hedges30, Leonhard Held31, Teck Hua Ho32, Herbert Hoijtink33, James Holland
Jones39,40, Daniel J. Hruschka34, Kosuke Imai35, Guido Imbens36, John P.A. Ioannidis37,
Minjeong Jeon38, Michael Kirchler41, David Laibson42, John List43, Roderick Little44,
Arthur Lupia45, Edouard Machery46, Scott E. Maxwell47, Michael McCarthy48, Don
Moore49, Stephen L. Morgan50, Marcus Munafó51, 52, Shinichi Nakagawa53, Brendan
Nyhan54, Timothy H. Parker55, Luis Pericchi56, Marco Perugini57, Jeff Rouder58, Judith
Rousseau59, Victoria Savalei60, Felix D. Schönbrodt61, Thomas Sellke62, Betsy
Sinclair63, Dustin Tingley64, Trisha Van Zandt65, Simine Vazire66, Duncan J. Watts67,
Christopher Winship68, Robert L. Wolpert2, Yu Xie69, Cristobal Young70, Jonathan
Zinman71, Valen E. Johnson72*
Affiliations:
1Center for Economic and Social Research and Department of Economics, University of
Southern California, Los Angeles, CA 900893332, USA.
2Department of Statistical Science, Duke University, Durham, NC 277080251, USA.
3Department of Economics, Stockholm School of Economics, SE113 83 Stockholm,
Sweden.
4University of Virginia, Charlottesville, VA 22908, USA.
5Center for Open Science, Charlottesville, VA 22903, USA.
6University of Amsterdam, Department of Psychology, 1018 VZ Amsterdam, The
Netherlands.
7University of Pennsylvania, School of Arts and Sciences and Department of
Criminology, Philadelphia, PA 191046286, USA.
8University of North Carolina Chapel Hill, Department of Psychology and Neuroscience,
Department of Sociology, Chapel Hill, NC 275993270, USA.
9
Institute
of
Zoology

Neurogenetics,
Universität
Regensburg,
Universitätsstrasse 31
93040 Regensburg, Germany.
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10Department of Statistics, The Wharton School, University of Pennsylvania,
Philadelphia, PA 19104, USA.
11Division of the Humanities and Social Sciences, California Institute of Technology,
Pasadena, CA 91125, USA.
12Department of Economics, New York University, New York, NY 10012, USA.
13The Research Institute of Industrial Economics (IFN), SE 102 15 Stockholm, Sweden.
14Cardiff University Brain Research Imaging Centre (CUBRIC), CF24 4HQ, UK.
15Northwestern University, Evanston, IL 60208, USA.
16Mathematica Policy Research, Washington, DC, 200024221, USA.
17Department of Psychology, Quantitative Program, Ohio State University, Columbus,
OH 43210, USA.
18School of Psychology, University of Sussex, Brighton BN1 9QH, UK.
19Department of Philosophy, Texas A&M University, College Station, TX 778434237,
USA.
20Department of Psychology, Royal Holloway University of London, Egham Surrey
TW20 0EX, UK.
21Department of Economics, University of Zurich, 8006 Zurich, Switzerland.
22School of BioSciences and School of Historical & Philosophical Studies, University of
Melbourne, Vic 3010, Australia.
23Department of Philosophy, University of Wisconsin  Madison, Madison, WI 53706,
USA.
24Department of Psychology, University of Michigan, Ann Arbor, MI 481091043, USA.
25Stanford University, General Medical Disciplines, Stanford, CA 94305, USA.
26Department of Ecology, Evolution and Natural Resources SEBS, Rutgers University,
New Brunswick, NJ 089018551, USA.
27Department of Political Science, Columbia University in the City of New York, New
York, NY 10027, USA.
28Department of Psychology, University of Washington, Seattle, WA 981951525, USA.
29Institute of Evolutionary Biology School of Biological Sciences, The University of
Edinburgh, Edinburgh EH9 3JT, UK.
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30Weinberg College of Arts & Sciences Department of Statistics, Northwestern
University, Evanston, IL 60208, USA.
31Epidemiology, Biostatistics and Prevention Institute (EBPI), University of Zurich,
8001 Zurich, Switzerland.
32National University of Singapore, Singapore 119077.
33Department of Methods and Statistics, Universiteit Utrecht, 3584 CH Utrecht, The
Netherlands.
34School of Human Evolution and Social Change, Arizona State University, Tempe, AZ
852872402, USA.
35Department of Politics and Center for Statistics and Machine Learning, Princeton
University, Princeton NJ 08544, USA.
36Stanford University, Stanford, CA 943055015, USA.
37Departments of Medicine, of Health Research and Policy, of Biomedical Data
Science, and of Statistics and MetaResearch Innovation Center at Stanford (METRICS),
Stanford University, Stanford, CA 94305, USA.
38 Advanced Quantitative Methods, Social Research Methodology, Department of
Education, Graduate School of Education & Information Studies, University of
California, Los Angeles, CA 900951521, USA.
39Department of Life Sciences, Imperial College London, Ascot SL5 7PY, UK.
40Department of Earth System Science, Stanford, CA 94305 4216, USA.
41Department of Banking and Finance, University of Innsbruck and University of
Gothenburg, A6020 Innsbruck, Austria.
42Department of Economics, Harvard University, Cambridge, MA 02138, USA.
43Department of Economics, University of Chicago, Chicago, IL 60637, USA.
44Department of Biostatistics, University of Michigan, Ann Arbor, MI 481092029, USA.
45Department of Political Science, University of Michigan, Ann Arbor, MI 481091045,
USA.
46Department of History and Philosophy of Science, University of Pittsburgh, Pittsburgh
PA 15260, USA.
47Department of Psychology, University of Notre Dame, Notre Dame, IN 46556, USA.
48School of BioSciences, University of Melbourne, Vic 3010, Australia.
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49Haas School of Business, University of California at Berkeley, Berkeley, CA 94720
1900A, USA.
50Johns Hopkins University, Baltimore, MD 21218, USA.
51MRC Integrative Epidemiology Unit, University of Bristol, Bristol BS8 1TU, UK.
52UK Centre for Tobacco and Alcohol Studies, School of Experimental Psychology,
University of Bristol, Bristol BS8 1TU, UK.
53Evolution & Ecology Research Centre and School of Biological, Earth and
Environmental Sciences, University of New South Wales, Sydney, NSW 2052, Australia.
54Department of Government, Dartmouth College, Hanover, NH 03755, USA.
55Department of Biology, Whitman College, Walla Walla, WA 99362, USA.
56Department of Mathematics, University of Puerto Rico, Rio Piedras Campus, San
Juan, PR 009368377.
57Department of Psychology, University of Milan  Bicocca, 20126 Milan, Italy.
58Department of Psychological Sciences, University of Missouri, Columbia, MO 65211,
USA.
59Université Paris Dauphine, 75016 Paris, France.
60Department of Psychology, The University of British Columbia, Vancouver, BC Canada
V6T 1Z4.
61Department Psychology, LudwigMaximiliansUniversity Munich, Leopoldstraße 13,
80802 Munich, Germany.
62Department of Statistics, Purdue University, West Lafayette, IN 479072067, USA.
63Department of Political Science, Washington University in St. Louis, St. Louis, MO
631304899, USA.
64Government Department, Harvard University, Cambridge, MA 02138, USA.
65Department of Psychology, Ohio State University, Columbus, OH 43210, USA.
66Department of Psychology, University of California, Davis, CA, 95616, USA.
67Microsoft Research. 641 Avenue of the Americas, 7th Floor, New York, NY 10011,
USA.
68Department of Sociology, Harvard University, Cambridge, MA 02138, USA.
69Department of Sociology, Princeton University, Princeton NJ 08544, USA.
70Department of Sociology, Stanford University, Stanford, CA 943052047, USA.
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71Department of Economics, Dartmouth College, Hanover, NH 037553514, USA.
72Department of Statistics, Texas A&M University, College Station, TX 77843, USA.
*Correspondence to: Daniel J. Benjamin, daniel.benjamin@gmail.com; Magnus
Johannesson, magnus.johannesson@hhs.se; Valen E. Johnson,
vejohnson@exchange.tamu.edu.
One Sentence Summary: We propose to change the default Pvalue threshold for
statistical significance for claims of new discoveries from 0.05 to 0.005.
Main Text:
The lack of reproducibility of scientific studies has caused growing concern over
the credibility of claims of new discoveries based on “statistically significant” findings.
There has been much progress toward documenting and addressing several causes of this
lack of reproducibility (e.g., multiple testing, Phacking, publication bias, and under
powered studies). However, we believe that a leading cause of nonreproducibility has
not yet been adequately addressed: Statistical standards of evidence for claiming new
discoveries in many fields of science are simply too low. Associating “statistically
significant” findings with P < 0.05 results in a high rate of false positives even in the
absence of other experimental, procedural and reporting problems.
For fields where the threshold for defining statistical significance for new
discoveries is 𝑃<0.05, we propose a change to 𝑃<0.005. This simple step would
immediately improve the reproducibility of scientific research in many fields. Results that
would currently be called “significant” but do not meet the new threshold should instead
be called “suggestive.” While statisticians have known the relative weakness of using
𝑃≈0.05 as a threshold for discovery and the proposal to lower it to 0.005 is not new (1,
2), a critical mass of researchers now endorse this change.
We restrict our recommendation to claims of discovery of new effects. We do not
address the appropriate threshold for confirmatory or contradictory replications of
existing claims. We also do not advocate changes to discovery thresholds in fields that
have already adopted more stringent standards (e.g., genomics and highenergy physics
research; see Potential Objections below).
We also restrict our recommendation to studies that conduct null hypothesis
significance tests. We have diverse views about how best to improve reproducibility, and
many of us believe that other ways of summarizing the data, such as Bayes factors or
other posterior summaries based on clearly articulated model assumptions, are preferable
to Pvalues. However, changing the Pvalue threshold is simple, aligns with the training
undertaken by many researchers, and might quickly achieve broad acceptance.
Strength of evidence from Pvalues
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In testing a point null hypothesis 𝐻! against an alternative hypothesis 𝐻! based on
data 𝑥obs, the Pvalue is defined as the probability, calculated under the null hypothesis,
that a test statistic is as extreme or more extreme than its observed value. The null
hypothesis is typically rejected—and the finding is declared “statistically significant”—if
the Pvalue falls below the (current) Type I error threshold α = 0.05.
!From a Bayesian perspective,!a more direct measure of the strength of evidence
for 𝐻! relative to 𝐻! is the ratio of their probabilities. By Bayes’ rule, this ratio may be
written as:
!
!
Pr 𝐻!𝑥obs
Pr 𝐻!𝑥obs
=𝑓𝑥obs𝐻!
𝑓𝑥obs𝐻!
×
Pr 𝐻!
Pr 𝐻!
≡𝐵𝐹 × prior odds ,!
(1)!
!
where 𝐵𝐹 is the Bayes factor that represents the evidence from the data, and the prior
odds can be informed by researchers’ beliefs, scientific consensus, and validated
evidence from similar research questions in the same field. Multiple hypothesis testing,
Phacking, and publication bias all reduce the credibility of evidence. Some of these
practices reduce the prior odds of 𝐻! relative to 𝐻! by changing the population of
hypothesis tests that are reported. Prediction markets (3) and analyses of replication
results (4) both suggest that for psychology experiments, the prior odds of 𝐻! relative to
𝐻! may be only about 1:10. A similar number has been suggested in cancer clinical trials,
and the number is likely to be much lower in preclinical biomedical research (5).
There is no unique mapping between the Pvalue and the Bayes factor since the
Bayes factor depends on 𝐻!. However, the connection between the two quantities can be
evaluated for particular test statistics under certain classes of plausible alternatives (Fig.
1).
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Fig. 1. Relationship between the Pvalue and the Bayes Factor. The Bayes factor (BF)
is defined as !!obs!!
!!obs!!
. The figure assumes that observations are drawn i.i.d. according to
𝑥 ~ 𝑁𝜇,𝜎!, where the mean 𝜇 is unknown and the variance 𝜎! is known. The Pvalue
is from a twosided z test (or equivalently a onesided 𝜒!
! test) of the null hypothesis
𝐻!:𝜇=0.
“Power”: BF obtained by defining 𝐻! as putting ½ probability on 𝜇=±𝑚 for the value
of 𝑚 that gives 75% power for the test of size α = 0.05. This 𝐻! represents an effect size
typical of that which is implicitly assumed by researchers during experimental design.
“Likelihood Ratio Bound”: BF obtained by defining 𝐻! as putting ½ probability on
𝜇=±𝑥, where 𝑥 is approximately equal to the mean of the observations. These BFs are
upper bounds among the class of all 𝐻!’s that are symmetric around the null, but they are
improper because the data are used to define 𝐻!. “UMPBT”: BF obtained by defining 𝐻!
according to the uniformly most powerful Bayesian test (5) that places ½ probability on
𝜇=±𝑤, where 𝑤 is the alternative hypothesis that corresponds to a onesided test of size
0.0025. This curve is indistinguishable from the “Power” curve that would be obtained if
the power used in its definition was 80% rather than 75%. “Local𝐻! Bound”: BF =
!
!!" !" !, where 𝑝 is the Pvalue, is a largesample upper bound on the BF from among all
unimodal alternative hypotheses that have a mode at the null and satisfy certain regularity
conditions (15). For more details, see the Supplementary Online Materials (SOM).
A twosided Pvalue of 0.05 corresponds to Bayes factors in favor of 𝐻! that range from
about 2.5 to 3.4 under reasonable assumptions about 𝐻! (Fig. 1). This is weak evidence
from at least three perspectives. First, conventional Bayes factor categorizations (6)
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characterize this range as “weak” or “very weak.” Second, we suspect many scientists
would guess that 𝑃≈0.05 implies stronger support for 𝐻! than a Bayes factor of 2.5 to
3.4. Third, using equation (1) and prior odds of 1:10, a Pvalue of 0.05 corresponds to at
least 3:1 odds (i.e., the reciprocal of the product !
!"
× 3.4) in favor of the null hypothesis!
Why 0.005?
The choice of any particular threshold is arbitrary and involves a tradeoff
between Type I and II errors. We propose 0.005 for two reasons. First, a twosided P
value of 0.005 corresponds to Bayes factors between approximately 14 and 26 in favor of
𝐻!. This range represents “substantial” to “strong” evidence according to conventional
Bayes factor classifications (6).
Second, in many fields the 𝑃<0.005 standard would reduce the false positive
rate to levels we judge to be reasonable. If we let 𝜙 denote the proportion of null
hypotheses that are true, (1−𝛽) the power of tests in rejecting false null hypotheses, and
𝛼 the Type I error/significance threshold, then as the population of tested hypotheses
becomes large, the false positive rate (i.e., the proportion of true null effects among the
total number of statistically significant findings) can be approximated by
!
!
false positive rate ≈ 𝛼𝜙
𝛼𝜙 +(1−𝛽)(1−𝜙).!
(2)!
For different levels of the prior odds that there is a true effect, !!!
!, and for significance
thresholds 𝛼=0.05 and 𝛼=0.005, Figure 2 shows the false positive rate as a function
of power 1−𝛽.
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Fig. 2. Relationship between the Pvalue threshold, power, and the false positive
rate. Calculated according to Equation (2), with prior odds defined as !!!
!
=!" !!
!"(!!). For
more details, see the Supplementary Online Materials (SOM).
In many studies, statistical power is low (e.g., ref. 7). Fig. 2 demonstrates that low
statistical power and 𝛼=0.05 combine to produce high false positive rates.
For many, the calculations illustrated by Fig. 2 may be unsettling. For example,
the false positive rate is greater than 33% with prior odds of 1:10 and a Pvalue threshold
of 0.05, regardless of the level of statistical power. Reducing the threshold to 0.005
would reduce this minimum false positive rate to 5%. Similar reductions in false positive
rates would occur over a wide range of statistical powers.
Empirical evidence from recent replication projects in psychology and
experimental economics provide insights into the prior odds in favor of 𝐻!. In both
projects, the rate of replication (i.e., significance at P < 0.05 in the replication in a
consistent direction) was roughly double for initial studies with P < 0.005 relative to
initial studies with 0.005 < P < 0.05: 50% versus 24% for psychology (8), and 85%
versus 44% for experimental economics (9). Although based on relatively small samples
of studies (93 in psychology, 16 in experimental economics, after excluding initial studies
with P > 0.05), these numbers are suggestive of the potential gains in reproducibility that
would accrue from the new threshold of P < 0.005 in these fields. In biomedical research,
96% of a sample of recent papers claim statistically significant results with the P < 0.05
threshold (10). However, replication rates were very low (5) for these studies, suggesting
a potential for gains by adopting this new standard in these fields as well.
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Potential Objections
We now address the most compelling arguments against adopting this higher
standard of evidence.
The false negative rate would become unacceptably high. Evidence that does not
reach the new significance threshold should be treated as suggestive, and where possible
further evidence should be accumulated; indeed, the combined results from several
studies may be compelling even if any particular study is not. Failing to reject the null
hypothesis does not mean accepting the null hypothesis. Moreover, the false negative rate
will not increase if sample sizes are increased so that statistical power is held constant.
For a wide range of common statistical tests, transitioning from a Pvalue
threshold of 𝛼=0.05 to 𝛼=0.005 while maintaining 80% power would require an
increase in sample sizes of about 70%. Such an increase means that fewer studies can be
conducted using current experimental designs and budgets. But Figure 2 shows the
benefit: false positive rates would typically fall by factors greater than two. Hence,
considerable resources would be saved by not performing future studies based on false
premises. Increasing sample sizes is also desirable because studies with small sample
sizes tend to yield inflated effect size estimates (11), and publication and other biases
may be more likely in an environment of small studies (12). We believe that efficiency
gains would far outweigh losses.
The proposal does not address multiple hypothesis testing, Phacking, publication
bias, low power, or other biases (e.g., confounding, selective reporting, measurement
error), which are arguably the bigger problems. We agree. Reducing the Pvalue
threshold complements—but does not substitute for—solutions to these other problems,
which include good study design, ex ante power calculations, preregistration of planned
analyses, replications, and transparent reporting of procedures and all statistical analyses
conducted.
The appropriate threshold for statistical significance should be different for
different research communities. We agree that the significance threshold selected for
claiming a new discovery should depend on the prior odds that the null hypothesis is true,
the number of hypotheses tested, the study design, the relative cost of Type I versus Type
II errors, and other factors that vary by research topic. For exploratory research with very
low prior odds (well outside the range in Figure 2), even lower significance thresholds
than 0.005 are needed. Recognition of this issue led the genetics research community to
move to a “genomewide significance threshold” of 5×108 over a decade ago. And in
highenergy physics, the tradition has long been to define significance by a “5sigma”
rule (roughly a Pvalue threshold of 3×107). We are essentially suggesting a move from a
2sigma rule to a 3sigma rule.
Our recommendation applies to disciplines with prior odds broadly in the range
depicted in Figure 2, where use of P < 0.05 as a default is widespread. Within those
disciplines, it is helpful for consumers of research to have a consistent benchmark. We
feel the default should be shifted.
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Changing the significance threshold is a distraction from the real solution, which
is to replace null hypothesis significance testing (and brightline thresholds) with more
focus on effect sizes and confidence intervals, treating the Pvalue as a continuous
measure, and/or a Bayesian method. Many of us agree that there are better approaches to
statistical analyses than null hypothesis significance testing, but as yet there is no
consensus regarding the appropriate choice of replacement. For example, a recent
statement by the American Statistical Association addressed numerous issues regarding
the misinterpretation and misuse of Pvalues (as well as the related concept of statistical
significance), but failed to make explicit policy recommendations to address these
shortcomings (13) . Even after the significance threshold is changed, many of us will
continue to advocate for alternatives to null hypothesis significance testing.
Concluding remarks
Ronald Fisher understood that the choice of 0.05 was arbitrary when he
introduced it (14). Since then, theory and empirical evidence have demonstrated that a
lower threshold is needed. A much larger pool of scientists are now asking a much larger
number of questions, possibly with much lower prior odds of success.
For research communities that continue to rely on null hypothesis significance
testing, reducing the Pvalue threshold for claims of new discoveries to 0.005 is an
actionable step that will immediately improve reproducibility. We emphasize that this
proposal is about standards of evidence, not standards for policy action nor standards for
publication. Results that do not reach the threshold for statistical significance (whatever
it is) can still be important and merit publication in leading journals if they address
important research questions with rigorous methods. This proposal should not be used to
reject publications of novel findings with 0.005 < P < 0.05 properly labeled as suggestive
evidence. We should reward quality and transparency of research as we impose these
more stringent standards, and we should monitor how researchers’ behaviors are affected
by this change. Otherwise, science runs the risk that the more demanding threshold for
statistical significance will be met to the detriment of quality and transparency.
Journals can help transition to the new statistical significance threshold. Authors
and readers can themselves take the initiative by describing and interpreting results more
appropriately in light of the new proposed definition of “statistical significance.” The
new significance threshold will help researchers and readers to understand and
communicate evidence more accurately.
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References and Notes:
1. A. G. Greenwald et al., Effect sizes and p values: What should be reported and
what should be replicated? Psychophysiology 33, 175183 (1996).
2. V. E. Johnson, Revised standards for statistical evidence. Proc. Natl. Acad. Sci.
U.S.A. 110, 1931319317 (2013).
3. A. Dreber et al., Using prediction markets to estimate the reproducibility of
scientific research. Proc. Natl. Acad. Sci. U.S.A. 112, 1534315347 (2015).
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4. V. E. Johnson et al., On the reproducibility of psychological science. J. Am. Stat.
Assoc. 112, 110 (2016).
5. G. C. Begley, J. P. A. Ioannidis, Reproducibility in science: Improving the
standard for basic and preclinical research. Circ. Res. 116, 116126 (2015).
6. R. E. Kass, A. E. Raftery, Bayes Factors. J. Am. Stat. Assoc. 90, 773795 (1995).
7. D. Szucs, J. P. A. Ioannidis, Empirical assessment of published effect sizes and
power in the recent cognitive neuroscience and psychology literature. PLoS Biol.
15, (2017).
8. Open Science Collaboration, Estimating the reproducibility of psychological
science. Science 349, (2015).
9. C. Camerer et al., Evaluating replicability of laboratory experiments in
economics. Science 351, 14331436 (2016).
10. D. Chavalarias et al., Evolution of reporting p values in the biomedical literature,
19902015. JAMA 315, 11411148 (2016).
11. A. Gelman, J. Carlin, Beyond power calculations: Assessing Type S (Sign) and
Type M (Magnitude) errors. Perspect. Psychol. Sci. 9, 641651 (2014).
12. D. Fanelli, R. Costas, J. P. A. Ioannidis, Metaassessment of bias in science. Proc.
Natl. Acad. Sci. U.S.A. 114, 37143719 (2017).
13. R. L. Wasserstein, N. A. Lazar, The ASA’s statement on pvalues: Context,
process, and purpose. Am. Stat. 70 (and online comments), 129133 (2016).
14. R. A. Fisher, Statistical Methods for Research Workers (Oliver & Boyd,
Edinburgh, 1925).
15. T. Sellke, M. J. Bayarri, J. O. Berger, Calibration of pvalues for testing precise
null hypotheses. Am. Stat. 55, 6271 (2001).
Acknowledgements: We thank Deanna L. Lormand, Rebecca Royer and Anh Tuan
Nguyen Viet for excellent research assistance.
Supplementary Materials:
Supplementary Text
R code used to generate Figures 1 and 2
Supplementary Materials:
Supplementary Text
Figure 1
All four curves in Figure 1 describe the relationship between (i) a Pvalue based
on a twosided normal test and (ii) a Bayes factor or a bound on a Bayes factor. The P
values are based on a twosided test that the mean 𝜇 of an independent and identically
distributed sample of normally distributed random variables is 0. The variance of the
observations is known. Without loss of generality, we assume that the variance is 1 and
the sample size is also 1. The curves in the figure differ according to the alternative
hypotheses that they assume for calculating (ii).
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Because these curves involve twosided tests, all alternative hypotheses are restricted to
be symmetric around 0. That is, the density assumed for the value of 𝜇 under the
alternative hypothesis is always assumed to satisfy 𝑓𝜇=𝑓−𝜇.
The curve labeled “Power” corresponds to defining the alternative hypothesis so that
power is 75% in a twosided 5% test. This is achieved by assuming that 𝜇 under the
alternative hypothesis is equal to ±𝑧!.!"# +𝑧!.!" =±2.63. That is, the alternative
hypothesis places ½ its prior mass on 2.63 and ½ its mass on 2.63.
The curve labeled UMPBT corresponds to the uniformly most powerful Bayesian test (2)
that corresponds to a classical, twosided test of size 𝛼=0.005. The alternative
hypothesis for this Bayesian test places ½ mass at 2.81 and ½ mass at 2.81. The null
hypothesis for this test is rejected if the Bayes factor exceeds 25.7. Note that this curve is
nearly identical to the “Power” curve if that curve had been defined using 80% power,
rather than 75% power. The Power curve for 80% power would place ½ its mass at
±2.80.
The Likelihood Ratio Bound curve represents an approximate upper bound on the Bayes
factor obtained by defining the alternative hypothesis as putting ½ its mass on ±𝑥, where
𝑥 is the observed sample mean. Over the range of Pvalues displayed in the figure, this
alternative hypothesis very closely approximates the maximum Bayes factor that can be
attained from among the set of alternative hypotheses constrained to be of the form 0.5×
[𝑓𝜇+𝑓−𝜇] for some density function f.
The LocalH1 curve is described fully in the figure caption.!A!fuller!explanation!and!
discussion!of!this!bound!can!be!found!in!ref.!15.
Equation 2 and Figure 2
This equation defines the largesample relationship between the false positive
rate, power 1−𝛽, type I error rate 𝛼, and the probability that the null hypothesis is true
when a large number of independent experiments have been conducted. More
specifically, suppose that n independent hypothesis tests are conducted, and suppose that
in each test the probability that the null hypothesis is true is 𝜙. If the null hypothesis is
true, assume that the probability that it is falsely rejected (i.e., a false positive occurs) is
𝛼. For the test 𝑗=1,…,𝑛, define the random variable 𝑋
!=1 if the null hypothesis is
true and the null hypothesis is rejected, and 𝑋
!=0 if either the alternative hypothesis is
true or the null hypothesis is not rejected. Note that the 𝑋
! are independent Bernoulli
random variables with Pr 𝑋
!=1=𝛼𝜙. Also for test j, define another random variable
𝑌
!=1 if the alternative hypothesis is true and the null hypothesis is rejected, and 0
otherwise. It follows that the 𝑌
! are independent Bernoulli random variables with
Pr 𝑌
!=1=1−𝜙1−𝛽. Note that 𝑌
! is independent of 𝑌
! for 𝑗≠𝑘, but 𝑌
! is not
independent of 𝑋
!. For the n experiments, the false positive rate can then be written as:
𝐹𝑃𝑅 =
𝑋
!
!
!!!
𝑋
!+𝑌
!
!
!!!
!
!!!
=
𝑋
!/𝑛
!
!!!
𝑋
!/𝑛+𝑌
!/𝑛
!
!!!
!
!!!
.
!
14!
By the strong law of large numbers, 𝑋
!/𝑛
!
!!! converges almost surely to 𝛼𝜙, and
𝑌
!/𝑛
!
!!! converges almost surely to 1−𝜙1−𝛽. Application of the continuous
mapping theorem yields
𝐹𝑃𝑅
a.s.
𝛼𝜙
𝛼𝜙 +(1−𝜙)(1−𝛽).
Figure 2 illustrates this relationship for various values of 𝛼 and prior odds for the
alternative, !!!
!
.
!
15!
R code used to generate Figure 1:
type1=.005
type1Power=0.05
type2=0.25
p=1c(9000:9990)/10000
xbar = qnorm(1p/2)
# alternative based on 80% POWER IN 5% TEST
muPower = qnorm(1type2)+qnorm(1type1Power/2)
bfPow = 0.5*(dnorm(xbar,muPower,1)+dnorm(xbar,
muPower,1))/dnorm(xbar,0,1)
muUMPBT = qnorm(0.9975)
bfUMPBT = 0.5*(dnorm(xbar,muUMPBT,1)+dnorm(xbar,
muUMPBT,1))/dnorm(xbar,0,1)
# twosided "LR" bound
bfLR = 0.5/exp(0.5*xbar^2)
bfLocal = 1/(2.71*p*log(p))
#coordinates for dashed lines
data = data.frame(p,bfLocal,bfLR,bfPow,bfUMPBT)
U_005 = max(data$bfLR[data$p=="0.005"])
L_005 = min(data$bfLocal[data$p=="0.005"])
U_05 = max(data$bfLR[data$p=="0.05"])
L_05 = min(data$bfUMPBT[data$p=="0.05"])
# Local bound; no need for twosided adjustment
#plot margins
par(mai=c(0.8,0.8,.1,0.4))
par(mgp=c(2,1,0))
matplot(p,cbind(bfLR,1/(2.71*p*log(p))),type='n',log='xy',
xlab=expression(paste(italic(P) ,"value")),
ylab="Bayes Factor",
ylim = c(0.3,100),
bty="n",xaxt="n",yaxt="n")
lines(p,bfPow,col="red",lwd=2.5)
lines(p,bfLR,col="black",lwd=2.5)
lines(p,bfUMPBT,col="blue",lwd=2.5)
lines(p,bfLocal,col="green",lwd=2.5)
legend(0.015,100,c(expression(paste("Power")),"Likelihood Ratio
Bound","UMPBT",expression(paste("Local",italic(H)[1],"
Bound"))),lty=c(1,1,1,1),
lwd=c(2.5,2.5,2.5,2.5),col=c("red","black","blue","green"),
cex = 0.8)
#text(0.062,65, "\u03B1", font =3, cex = 0.9)
#customizing axes
#x axis
!
16!
axis(side=1,at=c(
2,0.001,0.0025,0.005,0.010,0.025,0.050,0.100,0.14),
labels =
c("","0.0010","0.0025","0.0050","0.0100","0.0250","0.0500","0.1000",
""),lwd=1,
tck = 0.01, padj = 1.1, cex.axis = .8)
#y axis on the left  main
axis(side=2,at=c(0.2, 0.3,0.5,1,2,5,10,20,50,100),labels =
c("","0.3","0.5","1.0","2.0","5.0","10.0","20.0","50.0","100.0"),lwd
=1,las= 1,
tck = 0.01, hadj = 0.6, cex.axis = .8)
#y axis on the left  secondary (red labels)
axis(side=2,at=c(L_005,U_005),labels = c(13.9,25.7),lwd=1,las= 1,
tck = 0.01, hadj = 0.6, cex.axis = .6,col.axis="red")
#y axis on the right  main
axis(side=4,at=c(0.2, 0.3,0.5,1,2,5,10,20,50,100),labels =
c("","0.3","0.5","1.0","2.0","5.0","10.0","20.0","50.0","100.0"),lwd
=1,las= 1,
tck = 0.01, hadj = 0.4, cex.axis = .8)
#y axis on the right  secondary (red labels)
axis(side=4,at=c(L_05,U_05),labels = c(2.4,3.4),lwd=1,las= 1,
tck = 0.01, hadj = 0.4, cex.axis = .6,col.axis="red")
###dashed lines
segments(x0 = 0.000011, y0= U_005, x1 = 0.005, y1 = U_005, col =
"gray40", lty = 2)
segments(x0 = 0.000011, y0= L_005, x1 = 0.005, y1 = L_005, col =
"gray40", lty = 2)
segments(x0 = 0.005, y0= 0.00000001, x1 = 0.005, y1 = U_005, col =
"gray40", lty = 2)
segments(x0 = 0.05, y0= U_05, x1 = 0.14, y1 = U_05, col = "gray40",
lty = 2)
segments(x0 = 0.05, y0= L_05, x1 = 0.14, y1 = L_05, col = "gray40",
lty = 2)
segments(x0 = 0.05, y0= 0.00000001, x1 = 0.05, y1 = U_05, col =
"gray40", lty = 2)
!
17!
R code used to generate Figure 2:
pow1=c(5:999)/1000 # power range for 0.005 tests
pow2=c(50:999)/1000 # power range for 0.05 tests
alpha=0.005 # test size
pi0=5/6 # prior probability
N=10^6 # doesn't matter
#graph margins
par(mai=c(0.8,0.8,0.1,0.1))
par(mgp=c(2,1,0))
plot(pow1,alpha*N*pi0/(alpha*N*pi0+pow1*(1pi0)*N),type='n',ylim =
c(0,1), xlim = c(0,1.5),
xlab='Power ',
ylab='False positive rate', bty="n", xaxt="n", yaxt="n")
#grid lines
segments(x0 = 0.058, y0 = 0, x1 = 1, y1 = 0,lty=1,col = "gray92")
segments(x0 = 0.058, y0 = 0.2, x1 = 1, y1 = 0.2,lty=1,col =
"gray92")
segments(x0 = 0.058, y0 = 0.4, x1 = 1, y1 = 0.4,lty=1,col =
"gray92")
segments(x0 = 0.058, y0 = 0.6, x1 = 1, y1 = 0.6,lty=1,col =
"gray92")
segments(x0 = 0.058, y0 = 0.8, x1 = 1, y1 = 0.8,lty=1,col =
"gray92")
segments(x0 = 0.058, y0 = 1, x1 = 1, y1 = 1,lty=1,col = "gray92")
lines(pow1,alpha*N*pi0/(alpha*N*pi0+pow1*(1
pi0)*N),lty=1,col="blue",lwd=2)
odd_1_5_1 = alpha*N*pi0/(alpha*N*pi0+pow1[995]*(1pi0)*N)
alpha=0.05
pi0=5/6
lines(pow2,alpha*N*pi0/(alpha*N*pi0+pow2*(1
pi0)*N),lty=2,col="blue",lwd=2)
odd_1_5_2 = alpha*N*pi0/(alpha*N*pi0+pow2[950]*(1pi0)*N)
alpha=0.05
pi0=10/11
lines(pow2,alpha*N*pi0/(alpha*N*pi0+pow2*(1
pi0)*N),lty=2,col="red",lwd=2)
odd_1_10_2 = alpha*N*pi0/(alpha*N*pi0+pow2[950]*(1pi0)*N)
alpha=0.005
pi0=10/11
lines(pow1,alpha*N*pi0/(alpha*N*pi0+pow1*(1
pi0)*N),lty=1,col="red",lwd=2)
odd_1_10_1 = alpha*N*pi0/(alpha*N*pi0+pow1[995]*(1pi0)*N)
alpha=0.05
pi0=40/41
!
18!
lines(pow2,alpha*N*pi0/(alpha*N*pi0+pow2*(1
pi0)*N),lty=2,col="green",lwd=2)
odd_1_40_2 = alpha*N*pi0/(alpha*N*pi0+pow2[950]*(1pi0)*N)
alpha=0.005
pi0=40/41
lines(pow1,alpha*N*pi0/(alpha*N*pi0+pow1*(1
pi0)*N),lty=1,col="green",lwd=2)
odd_1_40_1 = alpha*N*pi0/(alpha*N*pi0+pow1[995]*(1pi0)*N)
#customizing axes
axis(side=2,at=c(0.5,0,0.2,0.4,0.6,0.8,1.0),labels =
c("","0.0","0.2","0.4","0.6","0.8","1.0"),
lwd=1,las= 1,tck = 0.01, hadj = 0.4, cex.axis = .8)
axis(side=1,at=c(0.5,0,0.2,0.4,0.6,0.8,1.0),labels =
c("","0.0","0.2","0.4","0.6","0.8","1.0"),
lwd=1,las= 1, tck = 0.01, padj = 1.1, cex.axis = .8)
legend(1.05,1,c("Prior odds = 1:40","Prior odds = 1:10","Prior odds
= 1:5"),pch=c(15,15,15),
col=c("green","red","blue"), cex = 1)
############### Use these commands to add brackets in Figure 2
library(pBrackets)
#add text and brackets
text(1.11,(odd_1_5_2+odd_1_40_2)/2, expression(paste(italic(P)," <
0.05 threshold")), cex = 0.9,adj=0)
text(1.11,(odd_1_5_1+odd_1_40_1)/2, expression(paste(italic(P)," <
0.005 threshold")), cex = 0.9,adj=0)
brackets(1.03, odd_1_40_1, 1.03, odd_1_5_1, h = NULL, ticks = 0.5,
curvature = 0.7, type = 1,
col = 1, lwd = 1, lty = 1, xpd = FALSE)
brackets(1.03, odd_1_40_2, 1.03, odd_1_5_2, h = NULL, ticks = 0.5,
curvature = 0.7, type = 1,
col = 1, lwd = 1, lty = 1, xpd = FALSE)
!