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In statistical hypothesis testing,^[1]^[2] a result has statistical significance when a result at least as "extreme" would be very infrequent if the null hypothesis were true.^[3] More precisely, a study's defined significance level, denoted by $\alpha$ , is the probability of the study rejecting the null hypothesis, given that the null hypothesis is true;^[4] and the p-value of a result, $p$ , is the probability of obtaining a result at least as extreme, given that the null hypothesis is true.^[5] The result is statistically significant, by the standards of the study, when $p\leq \alpha$ .^[6]^[7]^[8]^[9]^[10]^[11]^[12] The significance level for a study is chosen before data collection, and is typically set to 5%^[13] or much lower—depending on the field of study.^[14]

In any experiment or observation that involves drawing a sample from a population, there is always the possibility that an observed effect would have occurred due to sampling error alone.^[15]^[16] But if the p-value of an observed effect is less than (or equal to) the significance level, an investigator may conclude that the effect reflects the characteristics of the whole population,^[1] thereby rejecting the null hypothesis.^[17]

This technique for testing the statistical significance of results was developed in the early 20th century. The term significance does not imply importance here, and the term statistical significance is not the same as research significance, theoretical significance, or practical significance.^[1]^[2]^[18]^[19] For example, the term clinical significance refers to the practical importance of a treatment effect.^[20]

History

Statistical significance dates to the 18th century, in the work of John Arbuthnot and Pierre-Simon Laplace, who computed the p-value for the human sex ratio at birth, assuming a null hypothesis of equal probability of male and female births; see p-value § History for details.^[21]^[22]^[23]^[24]^[25]^[26]^[27]

In 1925, Ronald Fisher advanced the idea of statistical hypothesis testing, which he called "tests of significance", in his publication Statistical Methods for Research Workers.^[28]^[29]^[30] Fisher suggested a probability of one in twenty (0.05) as a convenient cutoff level to reject the null hypothesis.^[31] In a 1933 paper, Jerzy Neyman and Egon Pearson called this cutoff the significance level, which they named $\alpha$ . They recommended that $\alpha$ be set ahead of time, prior to any data collection.^[31]^[32]

Despite his initial suggestion of 0.05 as a significance level, Fisher did not intend this cutoff value to be fixed. In his 1956 publication Statistical Methods and Scientific Inference, he recommended that significance levels be set according to specific circumstances.^[31]

Related concepts

The significance level $\alpha$ is the threshold for $p$ below which the null hypothesis is rejected even though by assumption it were true, and something else is going on. This means that $\alpha$ is also the probability of mistakenly rejecting the null hypothesis, if the null hypothesis is true.^[4] This is also called false positive and type I error.

Sometimes researchers talk about the confidence level $γ = (1 - α)$ instead. This is the probability of not rejecting the null hypothesis given that it is true.^[33]^[34] Confidence levels and confidence intervals were introduced by Neyman in 1937.^[35]

Role in statistical hypothesis testing

In a two-tailed test, the rejection region for a significance level of $α = 0.05$ is partitioned to both ends of the sampling distribution and makes up 5% of the area under the curve (white areas).

Statistical significance plays a pivotal role in statistical hypothesis testing. It is used to determine whether the null hypothesis should be rejected or retained. The null hypothesis is the hypothesis that no effect exists in the phenomenon being studied.^[36] For the null hypothesis to be rejected, an observed result has to be statistically significant, i.e. the observed p-value is less than the pre-specified significance level $\alpha$ .

To determine whether a result is statistically significant, a researcher calculates a p-value, which is the probability of observing an effect of the same magnitude or more extreme given that the null hypothesis is true.^[5]^[12] The null hypothesis is rejected if the p-value is less than (or equal to) a predetermined level, $\alpha$ . $\alpha$ is also called the significance level, and is the probability of rejecting the null hypothesis given that it is true (a type I error). It is usually set at or below 5%.

For example, when $\alpha$ is set to 5%, the conditional probability of a type I error, given that the null hypothesis is true, is 5%,^[37] and a statistically significant result is one where the observed p-value is less than (or equal to) 5%.^[38] When drawing data from a sample, this means that the rejection region comprises 5% of the sampling distribution.^[39] These 5% can be allocated to one side of the sampling distribution, as in a one-tailed test, or partitioned to both sides of the distribution, as in a two-tailed test, with each tail (or rejection region) containing 2.5% of the distribution.

The use of a one-tailed test is dependent on whether the research question or alternative hypothesis specifies a direction such as whether a group of objects is heavier or the performance of students on an assessment is better.^[3] A two-tailed test may still be used but it will be less powerful than a one-tailed test, because the rejection region for a one-tailed test is concentrated on one end of the null distribution and is twice the size (5% vs. 2.5%) of each rejection region for a two-tailed test. As a result, the null hypothesis can be rejected with a less extreme result if a one-tailed test was used.^[40] The one-tailed test is only more powerful than a two-tailed test if the specified direction of the alternative hypothesis is correct. If it is wrong, however, then the one-tailed test has no power.

Significance thresholds in specific fields

In specific fields such as particle physics and manufacturing, statistical significance is often expressed in multiples of the standard deviation or sigma (σ) of a normal distribution, with significance thresholds set at a much stricter level (for example 5σ).^[41]^[42] For instance, the certainty of the Higgs boson particle's existence was based on the 5σ criterion, which corresponds to a p-value of about 1 in 3.5 million.^[42]^[43]

In other fields of scientific research such as genome-wide association studies, significance levels as low as 5×10⁻⁸ are not uncommon^[44]^[45]—as the number of tests performed is extremely large.

Limitations

Researchers focusing solely on whether their results are statistically significant might report findings that are not substantive^[46] and not replicable.^[47]^[48] There is also a difference between statistical significance and practical significance. A study that is found to be statistically significant may not necessarily be practically significant.^[49]^[19]

Effect size

Effect size is a measure of a study's practical significance.^[49] A statistically significant result may have a weak effect. To gauge the research significance of their result, researchers are encouraged to always report an effect size along with p-values. An effect size measure quantifies the strength of an effect, such as the distance between two means in units of standard deviation (cf. Cohen's d), the correlation coefficient between two variables or its square, and other measures.^[50]

Reproducibility

A statistically significant result may not be easy to reproduce.^[48] In particular, some statistically significant results will in fact be false positives. Each failed attempt to reproduce a result increases the likelihood that the result was a false positive.^[51]

Challenges

Overuse in some journals

Starting in the 2010s, some journals began questioning whether significance testing, and particularly using a threshold of $α$ =5%, was being relied on too heavily as the primary measure of validity of a hypothesis.^[52] Some journals encouraged authors to do more detailed analysis than just a statistical significance test. In social psychology, the journal Basic and Applied Social Psychology banned the use of significance testing altogether from papers it published,^[53] requiring authors to use other measures to evaluate hypotheses and impact.^[54]^[55]

Other editors, commenting on this ban have noted: "Banning the reporting of p-values, as Basic and Applied Social Psychology recently did, is not going to solve the problem because it is merely treating a symptom of the problem. There is nothing wrong with hypothesis testing and p-values per se as long as authors, reviewers, and action editors use them correctly."^[56] Some statisticians prefer to use alternative measures of evidence, such as likelihood ratios or Bayes factors.^[57] Using Bayesian statistics can avoid confidence levels, but also requires making additional assumptions,^[57] and may not necessarily improve practice regarding statistical testing.^[58]

The widespread abuse of statistical significance represents an important topic of research in metascience.^[59]

Redefining significance

In 2016, the American Statistical Association (ASA) published a statement on p-values, saying that "the widespread use of 'statistical significance' (generally interpreted as 'p ≤ 0.05') as a license for making a claim of a scientific finding (or implied truth) leads to considerable distortion of the scientific process".^[57] In 2017, a group of 72 authors proposed to enhance reproducibility by changing the p-value threshold for statistical significance from 0.05 to 0.005.^[60] Other researchers responded that imposing a more stringent significance threshold would aggravate problems such as data dredging; alternative propositions are thus to select and justify flexible p-value thresholds before collecting data,^[61] or to interpret p-values as continuous indices, thereby discarding thresholds and statistical significance.^[62] Additionally, the change to 0.005 would increase the likelihood of false negatives, whereby the effect being studied is real, but the test fails to show it.^[63]

In 2019, over 800 statisticians and scientists signed a message calling for the abandonment of the term "statistical significance" in science,^[64] and the ASA published a further official statement ^[65] declaring (page 2):

We conclude, based on our review of the articles in this special issue and the broader literature, that it is time to stop using the term "statistically significant" entirely. Nor should variants such as "significantly different," " $p\leq 0.05$ ," and "nonsignificant" survive, whether expressed in words, by asterisks in a table, or in some other way.

References

^ ^a ^b ^c Sirkin, R. Mark (2005). "Two-sample t tests". Statistics for the Social Sciences (3rd ed.). Thousand Oaks, CA: SAGE Publications, Inc. pp. 271–316. ISBN 978-1-4129-0546-6.
^ ^a ^b Borror, Connie M. (2009). "Statistical decision making". The Certified Quality Engineer Handbook (3rd ed.). Milwaukee, WI: ASQ Quality Press. pp. 418–472. ISBN 978-0-87389-745-7.
^ ^a ^b Myers, Jerome L.; Well, Arnold D.; Lorch, Robert F. Jr. (2010). "Developing fundamentals of hypothesis testing using the binomial distribution". Research design and statistical analysis (3rd ed.). New York, NY: Routledge. pp. 65–90. ISBN 978-0-8058-6431-1.
^ ^a ^b Dalgaard, Peter (2008). "Power and the computation of sample size". Introductory Statistics with R. Statistics and Computing. New York: Springer. pp. 155–56. doi:10.1007/978-0-387-79054-1_9. ISBN 978-0-387-79053-4.
^ ^a ^b "Statistical Hypothesis Testing". www.dartmouth.edu. Archived from the original on 2020-08-02. Retrieved 2019-11-11.
^ Johnson, Valen E. (October 9, 2013). "Revised standards for statistical evidence". Proceedings of the National Academy of Sciences. 110 (48): 19313–19317. Bibcode:2013PNAS..11019313J. doi:10.1073/pnas.1313476110. PMC 3845140. PMID 24218581.
^ Redmond, Carol; Colton, Theodore (2001). "Clinical significance versus statistical significance". Biostatistics in Clinical Trials. Wiley Reference Series in Biostatistics (3rd ed.). West Sussex, United Kingdom: John Wiley & Sons Ltd. pp. 35–36. ISBN 978-0-471-82211-0.
^ Cumming, Geoff (2012). Understanding The New Statistics: Effect Sizes, Confidence Intervals, and Meta-Analysis. New York, USA: Routledge. pp. 27–28.
^ Krzywinski, Martin; Altman, Naomi (30 October 2013). "Points of significance: Significance, P values and t-tests". Nature Methods. 10 (11): 1041–1042. doi:10.1038/nmeth.2698. PMID 24344377.
^ Sham, Pak C.; Purcell, Shaun M (17 April 2014). "Statistical power and significance testing in large-scale genetic studies". Nature Reviews Genetics. 15 (5): 335–346. doi:10.1038/nrg3706. PMID 24739678. S2CID 10961123.
^ Altman, Douglas G. (1999). Practical Statistics for Medical Research. New York, USA: Chapman & Hall/CRC. pp. 167. ISBN 978-0-412-27630-9.
^ ^a ^b Devore, Jay L. (2011). Probability and Statistics for Engineering and the Sciences (8th ed.). Boston, MA: Cengage Learning. pp. 300–344. ISBN 978-0-538-73352-6.
^ Craparo, Robert M. (2007). "Significance level". In Salkind, Neil J. (ed.). Encyclopedia of Measurement and Statistics. Vol. 3. Thousand Oaks, CA: SAGE Publications. pp. 889–891. ISBN 978-1-4129-1611-0.
^ Sproull, Natalie L. (2002). "Hypothesis testing". Handbook of Research Methods: A Guide for Practitioners and Students in the Social Science (2nd ed.). Lanham, MD: Scarecrow Press, Inc. pp. 49–64. ISBN 978-0-8108-4486-5.
^ Babbie, Earl R. (2013). "The logic of sampling". The Practice of Social Research (13th ed.). Belmont, CA: Cengage Learning. pp. 185–226. ISBN 978-1-133-04979-1.
^ Faherty, Vincent (2008). "Probability and statistical significance". Compassionate Statistics: Applied Quantitative Analysis for Social Services (With exercises and instructions in SPSS) (1st ed.). Thousand Oaks, CA: SAGE Publications, Inc. pp. 127–138. ISBN 978-1-4129-3982-9.
^ McKillup, Steve (2006). "Probability helps you make a decision about your results". Statistics Explained: An Introductory Guide for Life Scientists (1st ed.). Cambridge, United Kingdom: Cambridge University Press. pp. 44–56. ISBN 978-0-521-54316-3.
^ Myers, Jerome L.; Well, Arnold D.; Lorch, Robert F. Jr. (2010). "The t distribution and its applications". Research Design and Statistical Analysis (3rd ed.). New York, NY: Routledge. pp. 124–153. ISBN 978-0-8058-6431-1.
^ ^a ^b Hooper, Peter. "What is P-value?" (PDF). University of Alberta, Department of Mathematical and Statistical Sciences. Archived from the original (PDF) on March 31, 2020. Retrieved November 10, 2019.
^ Leung, W.-C. (2001-03-01). "Balancing statistical and clinical significance in evaluating treatment effects". Postgraduate Medical Journal. 77 (905): 201–204. doi:10.1136/pmj.77.905.201. ISSN 0032-5473. PMC 1741942. PMID 11222834.
^ Brian, Éric; Jaisson, Marie (2007). "Physico-Theology and Mathematics (1710–1794)". The Descent of Human Sex Ratio at Birth. Springer Science & Business Media. pp. 1–25. ISBN 978-1-4020-6036-6.
^ John Arbuthnot (1710). "An argument for Divine Providence, taken from the constant regularity observed in the births of both sexes" (PDF). Philosophical Transactions of the Royal Society of London. 27 (325–336): 186–190. doi:10.1098/rstl.1710.0011.
^ Conover, W.J. (1999), "Chapter 3.4: The Sign Test", Practical Nonparametric Statistics (Third ed.), Wiley, pp. 157–176, ISBN 978-0-471-16068-7
^ Sprent, P. (1989), Applied Nonparametric Statistical Methods (Second ed.), Chapman & Hall, ISBN 978-0-412-44980-2
^ Stigler, Stephen M. (1986). The History of Statistics: The Measurement of Uncertainty Before 1900. Harvard University Press. pp. 225–226. ISBN 978-0-674-40341-3.
^ Bellhouse, David (2001), "John Arbuthnot", in C.C. Heyde; E. Seneta (eds.), in Statisticians of the Centuries, Springer, pp. 39–42, ISBN 978-0-387-95329-8
^ Hald, Anders (1998), "Chapter 4. Chance or Design: Tests of Significance", A History of Mathematical Statistics from 1750 to 1930, Wiley, p. 65
^ Cumming, Geoff (2011). "From null hypothesis significance to testing effect sizes". Understanding The New Statistics: Effect Sizes, Confidence Intervals, and Meta-Analysis. Multivariate Applications Series. East Sussex, United Kingdom: Routledge. pp. 21–52. ISBN 978-0-415-87968-2.
^ Fisher, Ronald A. (1925). Statistical Methods for Research Workers. Edinburgh, UK: Oliver and Boyd. pp. 43. ISBN 978-0-05-002170-5.
^ Poletiek, Fenna H. (2001). "Formal theories of testing". Hypothesis-testing Behaviour. Essays in Cognitive Psychology (1st ed.). East Sussex, United Kingdom: Psychology Press. pp. 29–48. ISBN 978-1-84169-159-6.
^ ^a ^b ^c Quinn, Geoffrey R.; Keough, Michael J. (2002). Experimental Design and Data Analysis for Biologists (1st ed.). Cambridge, UK: Cambridge University Press. pp. 46–69. ISBN 978-0-521-00976-8.
^ Neyman, J.; Pearson, E.S. (1933). "The testing of statistical hypotheses in relation to probabilities a priori". Mathematical Proceedings of the Cambridge Philosophical Society. 29 (4): 492–510. Bibcode:1933PCPS...29..492N. doi:10.1017/S030500410001152X. S2CID 119855116.
^ "Conclusions about statistical significance are possible with the help of the confidence interval. If the confidence interval does not include the value of zero effect, it can be assumed that there is a statistically significant result." Prel, Jean-Baptist du; Hommel, Gerhard; Röhrig, Bernd; Blettner, Maria (2009). "Confidence Interval or P-Value?". Deutsches Ärzteblatt Online. 106 (19): 335–9. doi:10.3238/arztebl.2009.0335. PMC 2689604. PMID 19547734.
^ StatNews #73: Overlapping Confidence Intervals and Statistical Significance
^ Neyman, J. (1937). "Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability". Philosophical Transactions of the Royal Society A. 236 (767): 333–380. Bibcode:1937RSPTA.236..333N. doi:10.1098/rsta.1937.0005. JSTOR 91337. S2CID 19584450.
^ Meier, Kenneth J.; Brudney, Jeffrey L.; Bohte, John (2011). Applied Statistics for Public and Nonprofit Administration (3rd ed.). Boston, MA: Cengage Learning. pp. 189–209. ISBN 978-1-111-34280-7.
^ Healy, Joseph F. (2009). The Essentials of Statistics: A Tool for Social Research (2nd ed.). Belmont, CA: Cengage Learning. pp. 177–205. ISBN 978-0-495-60143-2.
^ McKillup, Steve (2006). Statistics Explained: An Introductory Guide for Life Scientists (1st ed.). Cambridge, UK: Cambridge University Press. pp. 32–38. ISBN 978-0-521-54316-3.
^ Health, David (1995). An Introduction To Experimental Design And Statistics For Biology (1st ed.). Boston, MA: CRC press. pp. 123–154. ISBN 978-1-85728-132-3.
^ Hinton, Perry R. (2010). "Significance, error, and power". Statistics explained (3rd ed.). New York, NY: Routledge. pp. 79–90. ISBN 978-1-84872-312-2.
^ Vaughan, Simon (2013). Scientific Inference: Learning from Data (1st ed.). Cambridge, UK: Cambridge University Press. pp. 146–152. ISBN 978-1-107-02482-3.
^ ^a ^b Bracken, Michael B. (2013). Risk, Chance, and Causation: Investigating the Origins and Treatment of Disease (1st ed.). New Haven, CT: Yale University Press. pp. 260–276. ISBN 978-0-300-18884-4.
^ Franklin, Allan (2013). "Prologue: The rise of the sigmas". Shifting Standards: Experiments in Particle Physics in the Twentieth Century (1st ed.). Pittsburgh, PA: University of Pittsburgh Press. pp. Ii–Iii. ISBN 978-0-8229-4430-0.
^ Clarke, GM; Anderson, CA; Pettersson, FH; Cardon, LR; Morris, AP; Zondervan, KT (February 6, 2011). "Basic statistical analysis in genetic case-control studies". Nature Protocols. 6 (2): 121–33. doi:10.1038/nprot.2010.182. PMC 3154648. PMID 21293453.
^ Barsh, GS; Copenhaver, GP; Gibson, G; Williams, SM (July 5, 2012). "Guidelines for Genome-Wide Association Studies". PLOS Genetics. 8 (7): e1002812. doi:10.1371/journal.pgen.1002812. PMC 3390399. PMID 22792080.
^ Carver, Ronald P. (1978). "The Case Against Statistical Significance Testing". Harvard Educational Review. 48 (3): 378–399. doi:10.17763/haer.48.3.t490261645281841. S2CID 16355113.
^ Ioannidis, John P. A. (2005). "Why most published research findings are false". PLOS Medicine. 2 (8): e124. doi:10.1371/journal.pmed.0020124. PMC 1182327. PMID 16060722.
^ ^a ^b Amrhein, Valentin; Korner-Nievergelt, Fränzi; Roth, Tobias (2017). "The earth is flat (p > 0.05): significance thresholds and the crisis of unreplicable research". PeerJ. 5: e3544. doi:10.7717/peerj.3544. PMC 5502092. PMID 28698825.
^ ^a ^b Hojat, Mohammadreza; Xu, Gang (2004). "A Visitor's Guide to Effect Sizes". Advances in Health Sciences Education. 9 (3): 241–9. doi:10.1023/B:AHSE.0000038173.00909.f6. PMID 15316274. S2CID 8045624.
^ Pedhazur, Elazar J.; Schmelkin, Liora P. (1991). Measurement, Design, and Analysis: An Integrated Approach (Student ed.). New York, NY: Psychology Press. pp. 180–210. ISBN 978-0-8058-1063-9.
^ Stahel, Werner (2016). "Statistical Issue in Reproducibility". Principles, Problems, Practices, and Prospects Reproducibility: Principles, Problems, Practices, and Prospects. pp. 87–114. doi:10.1002/9781118865064.ch5. ISBN 978-1-118-86497-5.
^ "CSSME Seminar Series: The argument over p-values and the Null Hypothesis Significance Testing (NHST) paradigm". www.education.leeds.ac.uk. School of Education, University of Leeds. Retrieved 2016-12-01.
^ Novella, Steven (February 25, 2015). "Psychology Journal Bans Significance Testing". Science-Based Medicine.
^ Woolston, Chris (2015-03-05). "Psychology journal bans P values". Nature. 519 (7541): 9. Bibcode:2015Natur.519....9W. doi:10.1038/519009f.
^ Siegfried, Tom (2015-03-17). "P value ban: small step for a journal, giant leap for science". Science News. Retrieved 2016-12-01.
^ Antonakis, John (February 2017). "On doing better science: From thrill of discovery to policy implications" (PDF). The Leadership Quarterly. 28 (1): 5–21. doi:10.1016/j.leaqua.2017.01.006.
^ ^a ^b ^c Wasserstein, Ronald L.; Lazar, Nicole A. (2016-04-02). "The ASA's Statement on p-Values: Context, Process, and Purpose". The American Statistician. 70 (2): 129–133. doi:10.1080/00031305.2016.1154108.
^ García-Pérez, Miguel A. (2016-10-05). "Thou Shalt Not Bear False Witness Against Null Hypothesis Significance Testing". Educational and Psychological Measurement. 77 (4): 631–662. doi:10.1177/0013164416668232. ISSN 0013-1644. PMC 5991793. PMID 30034024.
^ Ioannidis, John P. A.; Ware, Jennifer J.; Wagenmakers, Eric-Jan; Simonsohn, Uri; Chambers, Christopher D.; Button, Katherine S.; Bishop, Dorothy V. M.; Nosek, Brian A.; Munafò, Marcus R. (January 2017). "A manifesto for reproducible science". Nature Human Behaviour. 1 (1): 0021. doi:10.1038/s41562-016-0021. PMC 7610724. PMID 33954258.
^ Benjamin, Daniel; et al. (2018). "Redefine statistical significance". Nature Human Behaviour. 1 (1): 6–10. doi:10.1038/s41562-017-0189-z. hdl:10281/184094. PMID 30980045.
^ Chawla, Dalmeet (2017). "'One-size-fits-all' threshold for P values under fire". Nature. doi:10.1038/nature.2017.22625.
^ Amrhein, Valentin; Greenland, Sander (2017). "Remove, rather than redefine, statistical significance". Nature Human Behaviour. 2 (1): 0224. doi:10.1038/s41562-017-0224-0. PMID 30980046. S2CID 46814177.
^ Vyse, Stuart (November 2017). "Moving Science's Statistical Goalposts". csicop.org. CSI. Retrieved 10 July 2018.
^ McShane, Blake; Greenland, Sander; Amrhein, Valentin (March 2019). "Scientists rise up against statistical significance". Nature. 567 (7748): 305–307. Bibcode:2019Natur.567..305A. doi:10.1038/d41586-019-00857-9. PMID 30894741.
^ Wasserstein, Ronald L.; Schirm, Allen L.; Lazar, Nicole A. (2019-03-20). "Moving to a World Beyond "p < 0.05"". The American Statistician. 73 (sup1): 1–19. doi:10.1080/00031305.2019.1583913.

External links

The article "Earliest Known Uses of Some of the Words of Mathematics (S)" contains an entry on Significance that provides some historical information.
"The Concept of Statistical Significance Testing Archived 2022-09-07 at the Wayback Machine" (February 1994): article by Bruce Thompon hosted by the ERIC Clearinghouse on Assessment and Evaluation, Washington, D.C.
"What does it mean for a result to be "statistically significant"?" (no date): an article from the Statistical Assessment Service at George Mason University, Washington, D.C.

[Sirkin-1] Sirkin, R. Mark (2005). "Two-sample t tests". Statistics for the Social Sciences (3rd ed.). Thousand Oaks, CA: SAGE Publications, Inc. pp. 271–316. ISBN 978-1-4129-0546-6.

[Borror-2] Borror, Connie M. (2009). "Statistical decision making". The Certified Quality Engineer Handbook (3rd ed.). Milwaukee, WI: ASQ Quality Press. pp. 418–472. ISBN 978-0-87389-745-7.

[Myers_et_al-p65-3] Myers, Jerome L.; Well, Arnold D.; Lorch, Robert F. Jr. (2010). "Developing fundamentals of hypothesis testing using the binomial distribution". Research design and statistical analysis (3rd ed.). New York, NY: Routledge. pp. 65–90. ISBN 978-0-8058-6431-1.

[Dalgaard-4] Dalgaard, Peter (2008). "Power and the computation of sample size". Introductory Statistics with R. Statistics and Computing. New York: Springer. pp. 155–56. doi:10.1007/978-0-387-79054-1_9. ISBN 978-0-387-79053-4.

[:0-5] "Statistical Hypothesis Testing". www.dartmouth.edu. Archived from the original on 2020-08-02. Retrieved 2019-11-11.

[Johnson-6] Johnson, Valen E. (October 9, 2013). "Revised standards for statistical evidence". Proceedings of the National Academy of Sciences. 110 (48): 19313–19317. Bibcode:2013PNAS..11019313J. doi:10.1073/pnas.1313476110. PMC 3845140. PMID 24218581.

[Redmond_and_Colton-7] Redmond, Carol; Colton, Theodore (2001). "Clinical significance versus statistical significance". Biostatistics in Clinical Trials. Wiley Reference Series in Biostatistics (3rd ed.). West Sussex, United Kingdom: John Wiley & Sons Ltd. pp. 35–36. ISBN 978-0-471-82211-0.

[Cumming-p27-8] Cumming, Geoff (2012). Understanding The New Statistics: Effect Sizes, Confidence Intervals, and Meta-Analysis. New York, USA: Routledge. pp. 27–28.

[Krzywinski_and_Altman-9] Krzywinski, Martin; Altman, Naomi (30 October 2013). "Points of significance: Significance, P values and t-tests". Nature Methods. 10 (11): 1041–1042. doi:10.1038/nmeth.2698. PMID 24344377.

[Sham_and_Purcell-10] Sham, Pak C.; Purcell, Shaun M (17 April 2014). "Statistical power and significance testing in large-scale genetic studies". Nature Reviews Genetics. 15 (5): 335–346. doi:10.1038/nrg3706. PMID 24739678. S2CID 10961123.

[Altman-11] Altman, Douglas G. (1999). Practical Statistics for Medical Research. New York, USA: Chapman & Hall/CRC. pp. 167. ISBN 978-0-412-27630-9.

[Devore-12] Devore, Jay L. (2011). Probability and Statistics for Engineering and the Sciences (8th ed.). Boston, MA: Cengage Learning. pp. 300–344. ISBN 978-0-538-73352-6.

[Salkind-13] Craparo, Robert M. (2007). "Significance level". In Salkind, Neil J. (ed.). Encyclopedia of Measurement and Statistics. Vol. 3. Thousand Oaks, CA: SAGE Publications. pp. 889–891. ISBN 978-1-4129-1611-0.

[Sproull-14] Sproull, Natalie L. (2002). "Hypothesis testing". Handbook of Research Methods: A Guide for Practitioners and Students in the Social Science (2nd ed.). Lanham, MD: Scarecrow Press, Inc. pp. 49–64. ISBN 978-0-8108-4486-5.

[Babbie2-15] Babbie, Earl R. (2013). "The logic of sampling". The Practice of Social Research (13th ed.). Belmont, CA: Cengage Learning. pp. 185–226. ISBN 978-1-133-04979-1.

[Faherty-16] Faherty, Vincent (2008). "Probability and statistical significance". Compassionate Statistics: Applied Quantitative Analysis for Social Services (With exercises and instructions in SPSS) (1st ed.). Thousand Oaks, CA: SAGE Publications, Inc. pp. 127–138. ISBN 978-1-4129-3982-9.

[McKillup-17] McKillup, Steve (2006). "Probability helps you make a decision about your results". Statistics Explained: An Introductory Guide for Life Scientists (1st ed.). Cambridge, United Kingdom: Cambridge University Press. pp. 44–56. ISBN 978-0-521-54316-3.

[Myers_et_al-p124-18] Myers, Jerome L.; Well, Arnold D.; Lorch, Robert F. Jr. (2010). "The t distribution and its applications". Research Design and Statistical Analysis (3rd ed.). New York, NY: Routledge. pp. 124–153. ISBN 978-0-8058-6431-1.

[:1-19] Hooper, Peter. "What is P-value?" (PDF). University of Alberta, Department of Mathematical and Statistical Sciences. Archived from the original (PDF) on March 31, 2020. Retrieved November 10, 2019.

[20] Leung, W.-C. (2001-03-01). "Balancing statistical and clinical significance in evaluating treatment effects". Postgraduate Medical Journal. 77 (905): 201–204. doi:10.1136/pmj.77.905.201. ISSN 0032-5473. PMC 1741942. PMID 11222834.

[21] Brian, Éric; Jaisson, Marie (2007). "Physico-Theology and Mathematics (1710–1794)". The Descent of Human Sex Ratio at Birth. Springer Science & Business Media. pp. 1–25. ISBN 978-1-4020-6036-6.

[22] John Arbuthnot (1710). "An argument for Divine Providence, taken from the constant regularity observed in the births of both sexes" (PDF). Philosophical Transactions of the Royal Society of London. 27 (325–336): 186–190. doi:10.1098/rstl.1710.0011.

[Conover1999-23] Conover, W.J. (1999), "Chapter 3.4: The Sign Test", Practical Nonparametric Statistics (Third ed.), Wiley, pp. 157–176, ISBN 978-0-471-16068-7

[Sprent1989-24] Sprent, P. (1989), Applied Nonparametric Statistical Methods (Second ed.), Chapman & Hall, ISBN 978-0-412-44980-2

[25] Stigler, Stephen M. (1986). The History of Statistics: The Measurement of Uncertainty Before 1900. Harvard University Press. pp. 225–226. ISBN 978-0-674-40341-3.

[Bellhouse2001-26] Bellhouse, David (2001), "John Arbuthnot", in C.C. Heyde; E. Seneta (eds.), in Statisticians of the Centuries, Springer, pp. 39–42, ISBN 978-0-387-95329-8

[Hald1998-27] Hald, Anders (1998), "Chapter 4. Chance or Design: Tests of Significance", A History of Mathematical Statistics from 1750 to 1930, Wiley, p. 65

[Cumming-28] Cumming, Geoff (2011). "From null hypothesis significance to testing effect sizes". Understanding The New Statistics: Effect Sizes, Confidence Intervals, and Meta-Analysis. Multivariate Applications Series. East Sussex, United Kingdom: Routledge. pp. 21–52. ISBN 978-0-415-87968-2.

[Fisher1925-29] Fisher, Ronald A. (1925). Statistical Methods for Research Workers. Edinburgh, UK: Oliver and Boyd. pp. 43. ISBN 978-0-05-002170-5.

[Poletiek-30] Poletiek, Fenna H. (2001). "Formal theories of testing". Hypothesis-testing Behaviour. Essays in Cognitive Psychology (1st ed.). East Sussex, United Kingdom: Psychology Press. pp. 29–48. ISBN 978-1-84169-159-6.

[Quinn-31] Quinn, Geoffrey R.; Keough, Michael J. (2002). Experimental Design and Data Analysis for Biologists (1st ed.). Cambridge, UK: Cambridge University Press. pp. 46–69. ISBN 978-0-521-00976-8.

[Neyman-32] Neyman, J.; Pearson, E.S. (1933). "The testing of statistical hypotheses in relation to probabilities a priori". Mathematical Proceedings of the Cambridge Philosophical Society. 29 (4): 492–510. Bibcode:1933PCPS...29..492N. doi:10.1017/S030500410001152X. S2CID 119855116.

[33] "Conclusions about statistical significance are possible with the help of the confidence interval. If the confidence interval does not include the value of zero effect, it can be assumed that there is a statistically significant result." Prel, Jean-Baptist du; Hommel, Gerhard; Röhrig, Bernd; Blettner, Maria (2009). "Confidence Interval or P-Value?". Deutsches Ärzteblatt Online. 106 (19): 335–9. doi:10.3238/arztebl.2009.0335. PMC 2689604. PMID 19547734.

[34] StatNews #73: Overlapping Confidence Intervals and Statistical Significance

[Neyman1937-35] Neyman, J. (1937). "Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability". Philosophical Transactions of the Royal Society A. 236 (767): 333–380. Bibcode:1937RSPTA.236..333N. doi:10.1098/rsta.1937.0005. JSTOR 91337. S2CID 19584450.

[Meier-36] Meier, Kenneth J.; Brudney, Jeffrey L.; Bohte, John (2011). Applied Statistics for Public and Nonprofit Administration (3rd ed.). Boston, MA: Cengage Learning. pp. 189–209. ISBN 978-1-111-34280-7.

[Healy2009-37] Healy, Joseph F. (2009). The Essentials of Statistics: A Tool for Social Research (2nd ed.). Belmont, CA: Cengage Learning. pp. 177–205. ISBN 978-0-495-60143-2.

[Healy2006-38] McKillup, Steve (2006). Statistics Explained: An Introductory Guide for Life Scientists (1st ed.). Cambridge, UK: Cambridge University Press. pp. 32–38. ISBN 978-0-521-54316-3.

[Heath-39] Health, David (1995). An Introduction To Experimental Design And Statistics For Biology (1st ed.). Boston, MA: CRC press. pp. 123–154. ISBN 978-1-85728-132-3.

[Hinton_2014-40] Hinton, Perry R. (2010). "Significance, error, and power". Statistics explained (3rd ed.). New York, NY: Routledge. pp. 79–90. ISBN 978-1-84872-312-2.

[Vaughan-41] Vaughan, Simon (2013). Scientific Inference: Learning from Data (1st ed.). Cambridge, UK: Cambridge University Press. pp. 146–152. ISBN 978-1-107-02482-3.

[Bracken-42] Bracken, Michael B. (2013). Risk, Chance, and Causation: Investigating the Origins and Treatment of Disease (1st ed.). New Haven, CT: Yale University Press. pp. 260–276. ISBN 978-0-300-18884-4.

[franklin-43] Franklin, Allan (2013). "Prologue: The rise of the sigmas". Shifting Standards: Experiments in Particle Physics in the Twentieth Century (1st ed.). Pittsburgh, PA: University of Pittsburgh Press. pp. Ii–Iii. ISBN 978-0-8229-4430-0.

[Clarke_et_al-44] Clarke, GM; Anderson, CA; Pettersson, FH; Cardon, LR; Morris, AP; Zondervan, KT (February 6, 2011). "Basic statistical analysis in genetic case-control studies". Nature Protocols. 6 (2): 121–33. doi:10.1038/nprot.2010.182. PMC 3154648. PMID 21293453.

[Barsh_et_al-45] Barsh, GS; Copenhaver, GP; Gibson, G; Williams, SM (July 5, 2012). "Guidelines for Genome-Wide Association Studies". PLOS Genetics. 8 (7): e1002812. doi:10.1371/journal.pgen.1002812. PMC 3390399. PMID 22792080.

[Carver-46] Carver, Ronald P. (1978). "The Case Against Statistical Significance Testing". Harvard Educational Review. 48 (3): 378–399. doi:10.17763/haer.48.3.t490261645281841. S2CID 16355113.

[Ioannidis-47] Ioannidis, John P. A. (2005). "Why most published research findings are false". PLOS Medicine. 2 (8): e124. doi:10.1371/journal.pmed.0020124. PMC 1182327. PMID 16060722.

[peerj.com-48] Amrhein, Valentin; Korner-Nievergelt, Fränzi; Roth, Tobias (2017). "The earth is flat (p > 0.05): significance thresholds and the crisis of unreplicable research". PeerJ. 5: e3544. doi:10.7717/peerj.3544. PMC 5502092. PMID 28698825.

[A_Visitor’s_Guide_to_Effect_Sizes-49] Hojat, Mohammadreza; Xu, Gang (2004). "A Visitor's Guide to Effect Sizes". Advances in Health Sciences Education. 9 (3): 241–9. doi:10.1023/B:AHSE.0000038173.00909.f6. PMID 15316274. S2CID 8045624.

[Pedhazur-50] Pedhazur, Elazar J.; Schmelkin, Liora P. (1991). Measurement, Design, and Analysis: An Integrated Approach (Student ed.). New York, NY: Psychology Press. pp. 180–210. ISBN 978-0-8058-1063-9.

[51] Stahel, Werner (2016). "Statistical Issue in Reproducibility". Principles, Problems, Practices, and Prospects Reproducibility: Principles, Problems, Practices, and Prospects. pp. 87–114. doi:10.1002/9781118865064.ch5. ISBN 978-1-118-86497-5.

[52] "CSSME Seminar Series: The argument over p-values and the Null Hypothesis Significance Testing (NHST) paradigm". www.education.leeds.ac.uk. School of Education, University of Leeds. Retrieved 2016-12-01.

[53] Novella, Steven (February 25, 2015). "Psychology Journal Bans Significance Testing". Science-Based Medicine.

[54] Woolston, Chris (2015-03-05). "Psychology journal bans P values". Nature. 519 (7541): 9. Bibcode:2015Natur.519....9W. doi:10.1038/519009f.

[55] Siegfried, Tom (2015-03-17). "P value ban: small step for a journal, giant leap for science". Science News. Retrieved 2016-12-01.

[56] Antonakis, John (February 2017). "On doing better science: From thrill of discovery to policy implications" (PDF). The Leadership Quarterly. 28 (1): 5–21. doi:10.1016/j.leaqua.2017.01.006.

[Wasserstein_129–133-57] Wasserstein, Ronald L.; Lazar, Nicole A. (2016-04-02). "The ASA's Statement on p-Values: Context, Process, and Purpose". The American Statistician. 70 (2): 129–133. doi:10.1080/00031305.2016.1154108.

[58] García-Pérez, Miguel A. (2016-10-05). "Thou Shalt Not Bear False Witness Against Null Hypothesis Significance Testing". Educational and Psychological Measurement. 77 (4): 631–662. doi:10.1177/0013164416668232. ISSN 0013-1644. PMC 5991793. PMID 30034024.

[59] Ioannidis, John P. A.; Ware, Jennifer J.; Wagenmakers, Eric-Jan; Simonsohn, Uri; Chambers, Christopher D.; Button, Katherine S.; Bishop, Dorothy V. M.; Nosek, Brian A.; Munafò, Marcus R. (January 2017). "A manifesto for reproducible science". Nature Human Behaviour. 1 (1): 0021. doi:10.1038/s41562-016-0021. PMC 7610724. PMID 33954258.

[60] Benjamin, Daniel; et al. (2018). "Redefine statistical significance". Nature Human Behaviour. 1 (1): 6–10. doi:10.1038/s41562-017-0189-z. hdl:10281/184094. PMID 30980045.

[61] Chawla, Dalmeet (2017). "'One-size-fits-all' threshold for P values under fire". Nature. doi:10.1038/nature.2017.22625.

[62] Amrhein, Valentin; Greenland, Sander (2017). "Remove, rather than redefine, statistical significance". Nature Human Behaviour. 2 (1): 0224. doi:10.1038/s41562-017-0224-0. PMID 30980046. S2CID 46814177.

[63] Vyse, Stuart (November 2017). "Moving Science's Statistical Goalposts". csicop.org. CSI. Retrieved 10 July 2018.

[64] McShane, Blake; Greenland, Sander; Amrhein, Valentin (March 2019). "Scientists rise up against statistical significance". Nature. 567 (7748): 305–307. Bibcode:2019Natur.567..305A. doi:10.1038/d41586-019-00857-9. PMID 30894741.

[Wasserstein2_129–133-65] Wasserstein, Ronald L.; Schirm, Allen L.; Lazar, Nicole A. (2019-03-20). "Moving to a World Beyond "p < 0.05"". The American Statistician. 73 (sup1): 1–19. doi:10.1080/00031305.2019.1583913.

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@@ Line 1: / Line 1: @@
-{{Short description|Concept in inferential statistics}}
+{{Short description | Concept in inferential statistics}}
-In [[statistical hypothesis testing]],<ref name=Sirkin>{{cite book |last1 = Sirkin|first1 = R. Mark |chapter= Two-sample t tests | title = Statistics for the Social Sciences| edition=3rd |publisher = SAGE Publications, Inc | location = Thousand Oaks, CA | year = 2005 |isbn =978-1-412-90546-6 |pages=271–316}}</ref><ref name=Borror>{{cite book |last1 = Borror |first1 = Connie M. |chapter= Statistical decision making |title = The Certified Quality Engineer Handbook | edition=3rd |publisher = ASQ Quality Press |location = Milwaukee, WI | year = 2009 |isbn =978-0-873-89745-7 |pages=418–472}}</ref> a result has '''statistical significance''' when it is very unlikely to have occurred given the [[null hypothesis]].{{clarify|date=March 2022|reason=This definition is linked to as an explanation of "statistically significant" in many articles despite being incomprehensible to almost all readers.}}<ref name="Myers et al-p65" /> More precisely, a study's defined '''significance level''', denoted by <math>\alpha</math>, is the probability of the study rejecting the null hypothesis, given that the null hypothesis is true;<ref name="Dalgaard">{{cite book |last=Dalgaard |first=Peter |title=Introductory Statistics with R |location=New York |publisher=Springer |year=2008 |pages=155–56 |isbn=978-0-387-79053-4 |doi=10.1007/978-0-387-79054-1_9 |chapter=Power and the computation of sample size |series=Statistics and Computing }}</ref> and the [[p-value|''p''-value]] of a result, ''<math>p</math>'', is the probability of obtaining a result at least as extreme, given that the null hypothesis is true.<ref name=":0">{{Cite web|url=http://www.dartmouth.edu/~matc/X10/Show.htm|title=Statistical Hypothesis Testing|website=www.dartmouth.edu|access-date=2019-11-11|archive-date=2020-08-02|archive-url=https://web.archive.org/web/20200802050104/http://www.dartmouth.edu/~matc/X10/Show.htm|url-status=dead}}</ref> The result is '''statistically significant,''' by the standards of the study, when <math>p \le \alpha</math>.<ref
+In [[statistical hypothesis testing]],<ref name=Sirkin>{{cite book |last1 = Sirkin|first1 = R. Mark |chapter= Two-sample t tests | title = Statistics for the Social Sciences| edition=3rd |publisher = SAGE Publications, Inc | location = Thousand Oaks, CA | year = 2005 |isbn =978-1-4129-0546-6 |pages=271–316}}</ref><ref name=Borror>{{cite book |last1 = Borror |first1 = Connie M. |chapter= Statistical decision making |title = The Certified Quality Engineer Handbook | edition=3rd |publisher = ASQ Quality Press |location = Milwaukee, WI | year = 2009 |isbn =978-0-87389-745-7 |pages=418–472}}</ref> a result has '''statistical significance''' when a result at least as "extreme" would be very infrequent if the [[null hypothesis]] were true.<ref name="Myers et al-p65" /> More precisely, a study's defined '''significance level''', denoted by <math>\alpha</math>, is the probability of the study rejecting the null hypothesis, given that the null hypothesis is true;<ref name="Dalgaard">{{cite book |last=Dalgaard |first=Peter |title=Introductory Statistics with R |location=New York |publisher=Springer |year=2008 |pages=155–56 |isbn=978-0-387-79053-4 |doi=10.1007/978-0-387-79054-1_9 |chapter=Power and the computation of sample size |series=Statistics and Computing }}</ref> and the [[p-value|''p''-value]] of a result, ''<math>p</math>'', is the probability of obtaining a result at least as extreme, given that the null hypothesis is true.<ref name=":0">{{Cite web|url=http://www.dartmouth.edu/~matc/X10/Show.htm|title=Statistical Hypothesis Testing|website=www.dartmouth.edu|access-date=2019-11-11|archive-date=2020-08-02|archive-url=https://web.archive.org/web/20200802050104/http://www.dartmouth.edu/~matc/X10/Show.htm|url-status=dead}}</ref> The result is '''statistically significant,''' by the standards of the study, when <math>p \le \alpha</math>.<ref
-name="Johnson">{{cite journal |last= Johnson| first= Valen E. |date= October 9, 2013 |title= Revised standards for statistical evidence |journal= Proceedings of the National Academy of Sciences |doi= 10.1073/pnas.1313476110 | pmid= 24218581 |volume=110 | issue= 48 |pages=19313–19317|pmc=3845140 | bibcode= 2013PNAS..11019313J | doi-access= free }}</ref><ref name="Redmond and Colton">{{cite book | last1 = Redmond|first1 = Carol | last2 = Colton | first2 = Theodore | chapter = Clinical significance versus statistical significance | title = Biostatistics in Clinical Trials | series = Wiley Reference Series in Biostatistics | edition=3rd |publisher = John Wiley & Sons Ltd  |location = West Sussex, United Kingdom | year = 2001 |isbn = 978-0-471-82211-0 |pages = 35–36}}</ref><ref name="Cumming-p27">{{cite book | last1 = Cumming|first1 = Geoff | title = Understanding The New Statistics: Effect Sizes, Confidence Intervals, and Meta-Analysis |publisher = Routledge |location = New York, USA | year = 2012|pages = 27–28}}</ref><ref name="Krzywinski and Altman">{{cite journal |last1= Krzywinski |first1= Martin |last2= Altman |first2= Naomi |date= 30 October 2013   |title= Points of significance: Significance, P values and t-tests |journal= Nature Methods |volume= 10 |issue= 11 |pages= 1041–1042 |doi= 10.1038/nmeth.2698 |pmid= 24344377 |doi-access= free }}</ref><ref name="Sham and Purcell">{{cite journal |last1= Sham |first1= Pak C.|last2= Purcell |first2= Shaun M |date= 17 April 2014 |title= Statistical power and significance testing in large-scale genetic studies |journal= Nature Reviews Genetics |volume= 15 |issue= 5 |pages= 335–346 |doi= 10.1038/nrg3706 |pmid= 24739678|s2cid= 10961123}}</ref><ref name="Altman">{{cite book | last1 = Altman|first1 = Douglas G. | title = Practical Statistics for Medical Research | url = https://archive.org/details/isbn_9780412276309| url-access = registration|publisher = Chapman & Hall/CRC |location = New York, USA | year = 1999 | isbn = 978-0412276309  |pages = [https://archive.org/details/isbn_9780412276309/page/167 167]}}</ref><ref name=Devore>{{cite book |last1 = Devore|first1 = Jay L.|title = Probability and Statistics for Engineering and the Sciences| edition=8th |publisher = Cengage Learning |location = Boston, MA | year = 2011 |isbn =978-0-538-73352-6 |pages=300–344}}</ref> The significance level for a study is chosen before data collection, and is typically set to 5%<ref name="Salkind">{{cite encyclopedia|year=2007|title=Significance level|encyclopedia=Encyclopedia of Measurement and Statistics|publisher=SAGE Publications|location=Thousand Oaks, CA|editor-last1=Salkind|editor-first1=Neil J.|volume=3|pages=889–891|isbn=978-1-412-91611-0|last1=Craparo|first1=Robert M.}}</ref> or much lower—depending on the field of study.<ref name="Sproull">{{cite book|title=Handbook of Research Methods: A Guide for Practitioners and Students in the Social Science|last1=Sproull|first1=Natalie L.|publisher=Scarecrow Press, Inc.|year=2002|isbn=978-0-810-84486-5|edition=2nd|location=Lanham, MD|pages=[https://archive.org/details/handbookofresear00spro/page/49 49–64]|chapter=Hypothesis testing|chapter-url=https://archive.org/details/handbookofresear00spro/page/49}}</ref>
+name="Johnson">{{cite journal |last= Johnson| first= Valen E. |date= October 9, 2013 |title= Revised standards for statistical evidence |journal= Proceedings of the National Academy of Sciences |doi= 10.1073/pnas.1313476110 | pmid= 24218581 |volume=110 | issue= 48 |pages=19313–19317|pmc=3845140 | bibcode= 2013PNAS..11019313J | doi-access= free }}</ref><ref name="Redmond and Colton">{{cite book | last1 = Redmond|first1 = Carol | last2 = Colton | first2 = Theodore | chapter = Clinical significance versus statistical significance | title = Biostatistics in Clinical Trials | series = Wiley Reference Series in Biostatistics | edition=3rd |publisher = John Wiley & Sons Ltd  |location = West Sussex, United Kingdom | year = 2001 |isbn = 978-0-471-82211-0 |pages = 35–36}}</ref><ref name="Cumming-p27">{{cite book | last1 = Cumming|first1 = Geoff | title = Understanding The New Statistics: Effect Sizes, Confidence Intervals, and Meta-Analysis |publisher = Routledge |location = New York, USA | year = 2012|pages = 27–28}}</ref><ref name="Krzywinski and Altman">{{cite journal |last1= Krzywinski |first1= Martin |last2= Altman |first2= Naomi |date= 30 October 2013   |title= Points of significance: Significance, P values and t-tests |journal= Nature Methods |volume= 10 |issue= 11 |pages= 1041–1042 |doi= 10.1038/nmeth.2698 |pmid= 24344377 |doi-access= free }}</ref><ref name="Sham and Purcell">{{cite journal |last1= Sham |first1= Pak C.|last2= Purcell |first2= Shaun M |date= 17 April 2014 |title= Statistical power and significance testing in large-scale genetic studies |journal= Nature Reviews Genetics |volume= 15 |issue= 5 |pages= 335–346 |doi= 10.1038/nrg3706 |pmid= 24739678|s2cid= 10961123}}</ref><ref name="Altman">{{cite book | last1 = Altman|first1 = Douglas G. | title = Practical Statistics for Medical Research | url = https://archive.org/details/isbn_9780412276309| url-access = registration|publisher = Chapman & Hall/CRC |location = New York, USA | year = 1999 | isbn = 978-0-412-27630-9  |pages = [https://archive.org/details/isbn_9780412276309/page/167 167]}}</ref><ref name=Devore>{{cite book |last1 = Devore|first1 = Jay L.|title = Probability and Statistics for Engineering and the Sciences| edition=8th |publisher = Cengage Learning |location = Boston, MA | year = 2011 |isbn =978-0-538-73352-6 |pages=300–344}}</ref> The significance level for a study is chosen before data collection, and is typically set to 5%<ref name="Salkind">{{cite encyclopedia|year=2007|title=Significance level|encyclopedia=Encyclopedia of Measurement and Statistics|publisher=SAGE Publications|location=Thousand Oaks, CA|editor-last1=Salkind|editor-first1=Neil J.|volume=3|pages=889–891|isbn=978-1-4129-1611-0|last1=Craparo|first1=Robert M.}}</ref> or much lower—depending on the field of study.<ref name="Sproull">{{cite book|title=Handbook of Research Methods: A Guide for Practitioners and Students in the Social Science|last1=Sproull|first1=Natalie L.|publisher=Scarecrow Press, Inc.|year=2002|isbn=978-0-8108-4486-5|edition=2nd|location=Lanham, MD|pages=[https://archive.org/details/handbookofresear00spro/page/49 49–64]|chapter=Hypothesis testing|chapter-url=https://archive.org/details/handbookofresear00spro/page/49}}</ref>
-In any [[experiment]] or [[Observational study|observation]] that involves drawing a [[Sample (statistics)|sample]] from a [[Statistical population|population]], there is always the possibility that an observed effect would have occurred due to [[sampling error]] alone.<ref name=Babbie2>{{cite book |last1 = Babbie|first1 = Earl R. |chapter= The logic of sampling | title = The Practice of Social Research| edition=13th |publisher = Cengage Learning |location = Belmont, CA | year = 2013|isbn =978-1-133-04979-1 |pages=185–226}}</ref><ref name=Faherty>{{cite book |last1 = Faherty | first1 = Vincent | chapter= Probability and statistical significance | title = Compassionate Statistics: Applied Quantitative Analysis for Social Services (With exercises and instructions in SPSS) | edition=1st |publisher = SAGE Publications, Inc |location = Thousand Oaks, CA | year = 2008 |isbn =978-1-412-93982-9 |pages=127–138}}</ref> But if the ''p''-value of an observed effect is less than (or equal to) the significance level, an investigator may conclude that the effect reflects the characteristics of the whole population,<ref name=Sirkin/> thereby rejecting the null hypothesis.<ref name=McKillup>{{cite book |last1 = McKillup|first1 = Steve |title = Statistics Explained: An Introductory Guide for Life Scientists|chapter-url = https://archive.org/details/statisticsexplai0000mcki|chapter-url-access = registration| edition = 1st | publisher = Cambridge University Press|location = Cambridge, United Kingdom | year = 2006 |chapter=Probability helps you make a decision about your results | isbn = 978-0-521-54316-3 |pages=[https://archive.org/details/statisticsexplai0000mcki/page/44 44–56]}}</ref>
+In any [[experiment]] or [[Observational study|observation]] that involves drawing a [[Sampling (statistics)|sample]] from a [[Statistical population|population]], there is always the possibility that an observed effect would have occurred due to [[sampling error]] alone.<ref name=Babbie2>{{cite book |last1 = Babbie|first1 = Earl R. |chapter= The logic of sampling | title = The Practice of Social Research| edition=13th |publisher = Cengage Learning |location = Belmont, CA | year = 2013|isbn =978-1-133-04979-1 |pages=185–226}}</ref><ref name=Faherty>{{cite book |last1 = Faherty | first1 = Vincent | chapter= Probability and statistical significance | title = Compassionate Statistics: Applied Quantitative Analysis for Social Services (With exercises and instructions in SPSS) | edition=1st |publisher = SAGE Publications, Inc |location = Thousand Oaks, CA | year = 2008 |isbn =978-1-4129-3982-9 |pages=127–138}}</ref> But if the ''p''-value of an observed effect is less than (or equal to) the significance level, an investigator may conclude that the effect reflects the characteristics of the whole population,<ref name=Sirkin/> thereby rejecting the null hypothesis.<ref name=McKillup>{{cite book |last1 = McKillup|first1 = Steve |title = Statistics Explained: An Introductory Guide for Life Scientists|chapter-url = https://archive.org/details/statisticsexplai0000mcki|chapter-url-access = registration| edition = 1st | publisher = Cambridge University Press|location = Cambridge, United Kingdom | year = 2006 |chapter=Probability helps you make a decision about your results | isbn = 978-0-521-54316-3 |pages=[https://archive.org/details/statisticsexplai0000mcki/page/44 44–56]}}</ref>
-This technique for testing the statistical significance of results was developed in the early 20th century. The term ''significance'' does not imply importance here, and the term ''statistical significance'' is not the same as research significance, theoretical significance, or practical significance.<ref name=Sirkin /><ref name=Borror /><ref name="Myers et al-p124">{{cite book |last1 = Myers|first1 = Jerome L. |last2 = Well|first2 = Arnold D. |last3 = Lorch Jr |first3 = Robert F. |chapter= The t distribution and its applications |title = Research Design and Statistical Analysis | edition=3rd |publisher = Routledge |location = New York, NY | year = 2010 |isbn =978-0-805-86431-1 |pages=124–153}}</ref><ref name=":1">{{Cite web|url=http://www.stat.ualberta.ca/~hooper/teaching/misc/Pvalue.pdf|title=What is P-value?|last=Hooper|first=Peter|website=University of Alberta, Department of Mathematical and Statistical Sciences|access-date=November 10, 2019}}</ref> For example, the term [[clinical significance]] refers to the practical importance of a treatment effect.<ref>{{Cite journal|last=Leung|first=W.-C.|date=2001-03-01|title=Balancing statistical and clinical significance in evaluating treatment effects|url= |journal=Postgraduate Medical Journal|language=en|volume=77|issue=905|pages=201–204|doi=10.1136/pmj.77.905.201|issn=0032-5473|pmid=11222834|pmc=1741942}}</ref>
+This technique for testing the statistical significance of results was developed in the early 20th century. The term ''significance'' does not imply importance here, and the term ''statistical significance'' is not the same as research significance, theoretical significance, or practical significance.<ref name=Sirkin /><ref name=Borror /><ref name="Myers et al-p124">{{cite book |last1 = Myers|first1 = Jerome L. |last2 = Well|first2 = Arnold D. |last3 = Lorch |first3 = Robert F. Jr. |chapter= The t distribution and its applications |title = Research Design and Statistical Analysis | edition=3rd |publisher = Routledge |location = New York, NY | year = 2010 |isbn =978-0-8058-6431-1 |pages=124–153}}</ref><ref name=":1">{{Cite web|url=http://www.stat.ualberta.ca/~hooper/teaching/misc/Pvalue.pdf|title=What is P-value?|last=Hooper|first=Peter|website=University of Alberta, Department of Mathematical and Statistical Sciences|access-date=November 10, 2019|archive-date=March 31, 2020|archive-url=https://web.archive.org/web/20200331153714/http://www.stat.ualberta.ca/~hooper/teaching/misc/Pvalue.pdf|url-status=dead}}</ref> For example, the term [[clinical significance]] refers to the practical importance of a treatment effect.<ref>{{Cite journal|last=Leung|first=W.-C.|date=2001-03-01|title=Balancing statistical and clinical significance in evaluating treatment effects|url= |journal=Postgraduate Medical Journal|language=en|volume=77|issue=905|pages=201–204|doi=10.1136/pmj.77.905.201|issn=0032-5473|pmid=11222834|pmc=1741942}}</ref>
 ==History==
 {{Main|History of statistics}}
-Statistical significance dates to the 1700s, in the work of [[John Arbuthnot]] and [[Pierre-Simon Laplace]], who computed the [[p-value|''p''-value]] for the [[human sex ratio]] at birth, assuming a null hypothesis of equal probability of male and female births; see {{slink|p-value|History|display=''p''-value}} for details.<ref>{{cite book |title=The Descent of Human Sex Ratio at Birth |first1=Éric |last1=Brian |first2=Marie |last2=Jaisson |chapter=Physico-Theology and Mathematics (1710–1794) |pages=1–25 |year=2007 |publisher=Springer Science & Business Media |isbn=978-1-4020-6036-6}}</ref><ref>{{cite journal|author=John Arbuthnot |title=An argument for Divine Providence, taken from the constant regularity observed in the births of both sexes|journal=[[Philosophical Transactions of the Royal Society of London]] | volume=27| pages=186–190 | year=1710 | url=http://www.york.ac.uk/depts/maths/histstat/arbuthnot.pdf|doi=10.1098/rstl.1710.0011|issue=325–336|doi-access=free}}</ref><ref name="Conover1999">{{Citation
+Statistical significance dates to the 18th century, in the work of [[John Arbuthnot]] and [[Pierre-Simon Laplace]], who computed the [[p-value|''p''-value]] for the [[human sex ratio]] at birth, assuming a null hypothesis of equal probability of male and female births; see {{slink|p-value|History|display=''p''-value}} for details.<ref>{{cite book |title=The Descent of Human Sex Ratio at Birth |first1=Éric |last1=Brian |first2=Marie |last2=Jaisson |chapter=Physico-Theology and Mathematics (1710–1794) |pages=1–25 |year=2007 |publisher=Springer Science & Business Media |isbn=978-1-4020-6036-6}}</ref><ref>{{cite journal|author=John Arbuthnot |title=An argument for Divine Providence, taken from the constant regularity observed in the births of both sexes|journal=[[Philosophical Transactions of the Royal Society of London]] | volume=27| pages=186–190 | year=1710 | url=http://www.york.ac.uk/depts/maths/histstat/arbuthnot.pdf|doi=10.1098/rstl.1710.0011|issue=325–336|doi-access=free}}</ref><ref name="Conover1999">{{Citation
 |last=Conover
 |first=W.J.
@@ Line 27: / Line 27: @@
 |publisher=Chapman & Hall
 |isbn=978-0-412-44980-2
-}}</ref><ref>{{cite book |title=The History of Statistics: The Measurement of Uncertainty Before 1900 |first=Stephen M. |last=Stigler |publisher=Harvard University Press |year=1986 |isbn=978-0-67440341-3 |pages=[https://archive.org/details/historyofstatist00stig/page/225 225–226]}}</ref><ref name="Bellhouse2001">{{Citation
+}}</ref><ref>{{cite book |title=The History of Statistics: The Measurement of Uncertainty Before 1900 |first=Stephen M. |last=Stigler |publisher=Harvard University Press |year=1986 |isbn=978-0-674-40341-3 |pages=[https://archive.org/details/historyofstatist00stig/page/225 225–226]}}</ref><ref name="Bellhouse2001">{{Citation
 |last=Bellhouse
-|first=P.
+|first=David
-|title=in Statisticians of the Centuries by C.C. Heyde and E. Seneta
+|title=in Statisticians of the Centuries
+|editor1-link=Chris Heyde
+|editor1=C.C. Heyde
+|editor2-link=Eugene Seneta
+|editor2=E. Seneta
 |year=2001
 |publisher=Springer
@@ Line 46: / Line 50: @@
 </ref>
-In 1925, [[Ronald Fisher]] advanced the idea of statistical hypothesis testing, which he called "tests of significance", in his publication ''[[Statistical Methods for Research Workers]]''.<ref name="Cumming">{{cite book|title=Understanding The New Statistics: Effect Sizes, Confidence Intervals, and Meta-Analysis|publisher=Routledge|year=2011|isbn=978-0-415-87968-2|series=Multivariate Applications Series|location=East Sussex, United Kingdom|pages=21–52|chapter=From null hypothesis significance to testing effect sizes|last1=Cumming|first1=Geoff}}</ref><ref name="Fisher1925">{{cite book|title=Statistical Methods for Research Workers|publisher=Oliver and Boyd|year=1925|location=Edinburgh, UK|pages=[https://archive.org/details/statisticalmethoe7fish/page/43 43]|last1=Fisher|first1=Ronald A.|isbn=978-0-050-02170-5|url=https://archive.org/details/statisticalmethoe7fish/page/43}}</ref><ref name="Poletiek">{{cite book|title=Hypothesis-testing Behaviour|publisher=Psychology Press|year=2001|isbn=978-1-841-69159-6|edition=1st|series=Essays in Cognitive Psychology|location=East Sussex, United Kingdom|pages=29–48|chapter=Formal theories of testing|last1=Poletiek|first1=Fenna H.}}</ref> Fisher suggested a probability of one in twenty (0.05) as a convenient cutoff level to reject the null hypothesis.<ref name=Quinn>{{cite book |last1 = Quinn |first1 = Geoffrey R. |last2 = Keough |first2 = Michael J. |title = Experimental Design and Data Analysis for Biologists |edition = 1st |publisher = Cambridge University Press |location = Cambridge, UK |year = 2002 |isbn = 978-0-521-00976-8 |pages = [https://archive.org/details/experimentaldesi0000quin/page/46 46–69] |url = https://archive.org/details/experimentaldesi0000quin/page/46 }}</ref> In a 1933 paper, [[Jerzy Neyman]] and [[Egon Pearson]] called this cutoff the ''significance level'', which they named <math>\alpha</math>. They recommended that <math>\alpha</math> be set ahead of time, prior to any data collection.<ref name=Quinn /><ref name="Neyman">{{Cite journal|last2=Pearson|first2=E.S.|year=1933|title=The testing of statistical hypotheses in relation to probabilities a priori|journal=Mathematical Proceedings of the Cambridge Philosophical Society|volume=29|issue=4|pages=492–510|doi=10.1017/S030500410001152X|last1=Neyman|first1=J.|bibcode=1933PCPS...29..492N }}</ref>
+In 1925, [[Ronald Fisher]] advanced the idea of statistical hypothesis testing, which he called "tests of significance", in his publication ''[[Statistical Methods for Research Workers]]''.<ref name="Cumming">{{cite book|title=Understanding The New Statistics: Effect Sizes, Confidence Intervals, and Meta-Analysis|publisher=Routledge|year=2011|isbn=978-0-415-87968-2|series=Multivariate Applications Series|location=East Sussex, United Kingdom|pages=21–52|chapter=From null hypothesis significance to testing effect sizes|last1=Cumming|first1=Geoff}}</ref><ref name="Fisher1925">{{cite book|title=Statistical Methods for Research Workers|publisher=Oliver and Boyd|year=1925|location=Edinburgh, UK|pages=[https://archive.org/details/statisticalmethoe7fish/page/43 43]|last1=Fisher|first1=Ronald A.|isbn=978-0-05-002170-5|url=https://archive.org/details/statisticalmethoe7fish/page/43}}</ref><ref name="Poletiek">{{cite book|title=Hypothesis-testing Behaviour|publisher=Psychology Press|year=2001|isbn=978-1-84169-159-6|edition=1st|series=Essays in Cognitive Psychology|location=East Sussex, United Kingdom|pages=29–48|chapter=Formal theories of testing|last1=Poletiek|first1=Fenna H.}}</ref> Fisher suggested a probability of one in twenty (0.05) as a convenient cutoff level to reject the null hypothesis.<ref name=Quinn>{{cite book |last1 = Quinn |first1 = Geoffrey R. |last2 = Keough |first2 = Michael J. |title = Experimental Design and Data Analysis for Biologists |edition = 1st |publisher = Cambridge University Press |location = Cambridge, UK |year = 2002 |isbn = 978-0-521-00976-8 |pages = [https://archive.org/details/experimentaldesi0000quin/page/46 46–69] |url = https://archive.org/details/experimentaldesi0000quin/page/46 }}</ref> In a 1933 paper, [[Jerzy Neyman]] and [[Egon Pearson]] called this cutoff the ''significance level'', which they named <math>\alpha</math>. They recommended that <math>\alpha</math> be set ahead of time, prior to any data collection.<ref name=Quinn /><ref name="Neyman">{{Cite journal|last2=Pearson|first2=E.S.|year=1933|title=The testing of statistical hypotheses in relation to probabilities a priori|journal=Mathematical Proceedings of the Cambridge Philosophical Society|volume=29|issue=4|pages=492–510|doi=10.1017/S030500410001152X|last1=Neyman|first1=J.|bibcode=1933PCPS...29..492N |s2cid=119855116 }}</ref>
 Despite his initial suggestion of 0.05 as a significance level, Fisher did not intend this cutoff value to be fixed. In his 1956 publication ''Statistical Methods and Scientific Inference,'' he recommended that significance levels be set according to specific circumstances.<ref name=Quinn />
@@ Line 53: / Line 57: @@
 The significance level <math>\alpha</math> is the threshold for <math>p</math> below which the null hypothesis is rejected even though by assumption it were true, and something else is going on. This means that <math>\alpha</math> is also the probability of mistakenly rejecting the null hypothesis, if the null hypothesis is true.<ref name="Dalgaard" /> This is also called [[False positives and false negatives#False positive error|false positive]] and [[Type I and type II errors#Type I error|type I error]].
-Sometimes researchers talk about the [[confidence level]] {{math|''γ'' {{=}} (1 − ''α'')}} instead. This is the probability of not rejecting the null hypothesis given that it is true.<ref>"Conclusions about statistical significance are possible with the help of the confidence interval. If the confidence interval does not include the value of zero effect, it can be assumed that there is a statistically significant result." {{cite journal|title=Confidence Interval or P-Value?|journal=Deutsches Ärzteblatt Online|volume=106|issue=19|pages=335–9|doi=10.3238/arztebl.2009.0335|pmid=19547734|pmc=2689604|year=2009|last1=Prel|first1=Jean-Baptist du|last2=Hommel|first2=Gerhard|last3=Röhrig|first3=Bernd|last4=Blettner|first4=Maria}}</ref><ref>[https://www.cscu.cornell.edu/news/statnews/stnews73.pdf StatNews #73: Overlapping Confidence Intervals and Statistical Significance]</ref> Confidence levels and confidence intervals were introduced by Neyman in 1937.<ref name="Neyman1937">{{cite journal|year=1937|title=Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability|jstor=91337|journal=[[Philosophical Transactions of the Royal Society A]]|volume=236|issue=767|pages=333–380|doi=10.1098/rsta.1937.0005|last1=Neyman|first1=J.|bibcode=1937RSPTA.236..333N |author-link=Jerzy Neyman|doi-access=free}}</ref>
+Sometimes researchers talk about the [[confidence level]] {{math|''γ'' {{=}} (1 − ''α'')}} instead. This is the probability of not rejecting the null hypothesis given that it is true.<ref>"Conclusions about statistical significance are possible with the help of the confidence interval. If the confidence interval does not include the value of zero effect, it can be assumed that there is a statistically significant result." {{cite journal|title=Confidence Interval or P-Value?|journal=Deutsches Ärzteblatt Online|volume=106|issue=19|pages=335–9|doi=10.3238/arztebl.2009.0335|pmid=19547734|pmc=2689604|year=2009|last1=Prel|first1=Jean-Baptist du|last2=Hommel|first2=Gerhard|last3=Röhrig|first3=Bernd|last4=Blettner|first4=Maria}}</ref><ref>[https://www.cscu.cornell.edu/news/statnews/stnews73.pdf StatNews #73: Overlapping Confidence Intervals and Statistical Significance]</ref> Confidence levels and confidence intervals were introduced by Neyman in 1937.<ref name="Neyman1937">{{cite journal|year=1937|title=Outline of a Theory of Statistical Estimation Based on the Classical Theory of Probability|jstor=91337|journal=[[Philosophical Transactions of the Royal Society A]]|volume=236|issue=767|pages=333–380|doi=10.1098/rsta.1937.0005|last1=Neyman|first1=J.|bibcode=1937RSPTA.236..333N |author-link=Jerzy Neyman|doi-access=|s2cid=19584450 }}</ref>
 ==Role in statistical hypothesis testing==
@@ Line 60: / Line 64: @@
 [[File:NormalDist1.96.png|250px|thumb|In a [[one- and two-tailed tests|two-tailed test]], the rejection region for a significance level of {{math|''α'' {{=}} 0.05}} is partitioned to both ends of the [[sampling distribution]] and makes up 5% of the area under the curve (white areas).]]
-Statistical significance plays a pivotal role in statistical hypothesis testing. It is used to determine whether the [[null hypothesis]] should be rejected or retained. The null hypothesis is the default assumption that nothing happened or changed.<ref name=Meier>{{cite book |last1 = Meier|first1 = Kenneth J.|last2=Brudney |first2=Jeffrey L. |last3=Bohte|first3=John |title = Applied Statistics for Public and Nonprofit Administration| edition=3rd |publisher = Cengage Learning |location = Boston, MA | year = 2011 |isbn =978-1-111-34280-7 |pages=189–209}}</ref> For the null hypothesis to be rejected, an observed result has to be statistically significant, i.e. the observed ''p''-value is less than the pre-specified significance level <math>\alpha</math>.
+Statistical significance plays a pivotal role in statistical hypothesis testing. It is used to determine whether the [[null hypothesis]] should be rejected or retained. The null hypothesis is the hypothesis that no effect exists in the phenomenon being studied.<ref name=Meier>{{cite book |last1 = Meier|first1 = Kenneth J.|last2=Brudney |first2=Jeffrey L. |last3=Bohte|first3=John |title = Applied Statistics for Public and Nonprofit Administration| edition=3rd |publisher = Cengage Learning |location = Boston, MA | year = 2011 |isbn =978-1-111-34280-7 |pages=189–209}}</ref> For the null hypothesis to be rejected, an observed result has to be statistically significant, i.e. the observed ''p''-value is less than the pre-specified significance level <math>\alpha</math>.
 To determine whether a result is statistically significant, a researcher calculates a ''p''-value, which is the probability of observing an effect of the same magnitude or more extreme given that the null hypothesis is true.<ref name=":0" /><ref name="Devore"/> The null hypothesis is rejected if the ''p''-value is less than (or equal to) a predetermined level, <math>\alpha</math>.  <math>\alpha</math> is also called the ''significance level'', and is the probability of rejecting the null hypothesis given that it is true (a [[Type I error#Type I error|type I error]]). It is usually set at or below 5%.
-For example, when <math>\alpha</math> is set to 5%, the [[conditional probability]] of a [[Type I error#Type I error|type I error]], ''given that the null hypothesis is true'', is 5%,<ref name="Healy2009">{{cite book |last1 = Healy|first1 = Joseph F. |title = The Essentials of Statistics: A Tool for Social Research | edition=2nd |publisher = Cengage Learning |location = Belmont, CA | year = 2009 | isbn =978-0-495-60143-2 |pages=177–205}}</ref> and a statistically significant result is one where the observed ''p''-value is less than (or equal to) 5%.<ref name="Healy2006">{{cite book |last1 =  McKillup|first1 = Steve |title = Statistics Explained: An Introductory Guide for Life Scientists |url =  https://archive.org/details/statisticsexplai0000mcki|url-access =  registration| edition=1st |publisher =  Cambridge University Press  |location = Cambridge, UK | year = 2006 |isbn =978-0-521-54316-3 |pages=[https://archive.org/details/statisticsexplai0000mcki/page/32 32–38]}}</ref> When drawing data from a sample, this means that the rejection region comprises 5% of the [[sampling distribution]].<ref name=Heath>{{cite book |last1 = Health|first1 = David |title = An Introduction To Experimental Design And Statistics For Biology| edition=1st |publisher = CRC press |location = Boston, MA | year = 1995 |isbn =978-1-857-28132-3 |pages=123–154}}</ref> These 5% can be allocated to one side of the sampling distribution, as in a [[one-tailed test]], or partitioned to both sides of the distribution, as in a [[two-tailed test]], with each tail (or rejection region) containing 2.5% of the distribution.
+For example, when <math>\alpha</math> is set to 5%, the [[conditional probability]] of a [[Type I error#Type I error|type I error]], ''given that the null hypothesis is true'', is 5%,<ref name="Healy2009">{{cite book |last1 = Healy|first1 = Joseph F. |title = The Essentials of Statistics: A Tool for Social Research | edition=2nd |publisher = Cengage Learning |location = Belmont, CA | year = 2009 | isbn =978-0-495-60143-2 |pages=177–205}}</ref> and a statistically significant result is one where the observed ''p''-value is less than (or equal to) 5%.<ref name="Healy2006">{{cite book |last1 =  McKillup|first1 = Steve |title = Statistics Explained: An Introductory Guide for Life Scientists |url =  https://archive.org/details/statisticsexplai0000mcki|url-access =  registration| edition=1st |publisher =  Cambridge University Press  |location = Cambridge, UK | year = 2006 |isbn =978-0-521-54316-3 |pages=[https://archive.org/details/statisticsexplai0000mcki/page/32 32–38]}}</ref> When drawing data from a sample, this means that the rejection region comprises 5% of the [[sampling distribution]].<ref name=Heath>{{cite book |last1 = Health|first1 = David |title = An Introduction To Experimental Design And Statistics For Biology| edition=1st |publisher = CRC press |location = Boston, MA | year = 1995 |isbn =978-1-85728-132-3 |pages=123–154}}</ref> These 5% can be allocated to one side of the sampling distribution, as in a [[one-tailed test]], or partitioned to both sides of the distribution, as in a [[two-tailed test]], with each tail (or rejection region) containing 2.5% of the distribution.
-The use of a one-tailed test is dependent on whether the [[research question]] or [[alternative hypothesis]] specifies a direction such as whether a group of objects is ''heavier'' or the performance of students on an assessment is ''better''.<ref name="Myers et al-p65">{{cite book |last1 = Myers | first1 = Jerome L. | last2 = Well | first2 = Arnold D. | last3 = Lorch Jr. | first3 = Robert F. | chapter = Developing fundamentals of hypothesis testing using the binomial distribution | title = Research design and statistical analysis | edition=3rd | publisher = Routledge |location = New York, NY | year = 2010 | isbn = 978-0-805-86431-1  | pages=65–90}}</ref> A two-tailed test may still be used but it will be less [[Statistical power|powerful]] than a one-tailed test, because the rejection region for a one-tailed test is concentrated on one end of the null distribution and is twice the size (5% vs. 2.5%) of each rejection region for a two-tailed test. As a result, the null hypothesis can be rejected with a less extreme result if a one-tailed test was used.<ref name="Hinton 2014">{{cite book |last1 = Hinton | first1 = Perry R. | chapter = Significance, error, and power | title = Statistics explained | edition=3rd | publisher = Routledge |location = New York, NY | year = 2010 | isbn = 978-1-848-72312-2  | pages=79–90}}</ref> The one-tailed test is only more powerful than a two-tailed test if the specified direction of the alternative hypothesis is correct. If it is wrong, however, then the one-tailed test has no power.
+The use of a one-tailed test is dependent on whether the [[research question]] or [[alternative hypothesis]] specifies a direction such as whether a group of objects is ''heavier'' or the performance of students on an assessment is ''better''.<ref name="Myers et al-p65">{{cite book |last1 = Myers | first1 = Jerome L. | last2 = Well | first2 = Arnold D. | last3 = Lorch | first3 = Robert F. Jr. | chapter = Developing fundamentals of hypothesis testing using the binomial distribution | title = Research design and statistical analysis | edition=3rd | publisher = Routledge |location = New York, NY | year = 2010 | isbn = 978-0-8058-6431-1  | pages=65–90}}</ref> A two-tailed test may still be used but it will be less [[Statistical power|powerful]] than a one-tailed test, because the rejection region for a one-tailed test is concentrated on one end of the null distribution and is twice the size (5% vs. 2.5%) of each rejection region for a two-tailed test. As a result, the null hypothesis can be rejected with a less extreme result if a one-tailed test was used.<ref name="Hinton 2014">{{cite book |last1 = Hinton | first1 = Perry R. | chapter = Significance, error, and power | title = Statistics explained | edition=3rd | publisher = Routledge |location = New York, NY | year = 2010 | isbn = 978-1-84872-312-2  | pages=79–90}}</ref> The one-tailed test is only more powerful than a two-tailed test if the specified direction of the alternative hypothesis is correct. If it is wrong, however, then the one-tailed test has no power.
 === Significance thresholds in specific fields ===
 {{Further|Standard deviation|Normal distribution}}
-In specific fields such as [[particle physics]] and [[manufacturing]], statistical significance is often expressed in multiples of the [[standard deviation]] or sigma (''σ'') of a [[normal distribution]], with significance thresholds set at a much stricter level (e.g. 5''σ'').<ref name=Vaughan>{{cite book |last1 = Vaughan|first1 = Simon |title = Scientific Inference: Learning from Data | edition=1st |publisher = Cambridge University Press|location = Cambridge, UK | year = 2013 |isbn = 978-1-107-02482-3 |pages=146–152}}</ref><ref name=Bracken>{{cite book |last1 = Bracken|first1 = Michael B.|title = Risk, Chance, and Causation: Investigating the Origins and Treatment of Disease |url = https://archive.org/details/riskchancecausat0000brac|url-access = registration| edition=1st |publisher =Yale University Press|location = New Haven, CT | year = 2013 |isbn = 978-0-300-18884-4 |pages=[https://archive.org/details/riskchancecausat0000brac/page/260 260–276]}}</ref> For instance, the certainty of the [[Higgs boson]] particle's existence was based on the 5''σ'' criterion, which corresponds to a ''p''-value of about 1 in 3.5 million.<ref name="Bracken"/><ref name=franklin>{{cite book |last1 = Franklin|first1 = Allan|chapter= Prologue: The rise of the sigmas |title=Shifting Standards: Experiments in Particle Physics in the Twentieth Century|edition=1st |publisher = University of Pittsburgh Press|location = Pittsburgh, PA | year = 2013 |isbn = 978-0-822-94430-0 |pages=Ii–Iii}}</ref>
+In specific fields such as [[particle physics]] and [[manufacturing]], statistical significance is often expressed in multiples of the [[standard deviation]] or sigma (''σ'') of a [[normal distribution]], with significance thresholds set at a much stricter level (for example 5''σ'').<ref name=Vaughan>{{cite book |last1 = Vaughan|first1 = Simon |title = Scientific Inference: Learning from Data | edition=1st |publisher = Cambridge University Press|location = Cambridge, UK | year = 2013 |isbn = 978-1-107-02482-3 |pages=146–152}}</ref><ref name=Bracken>{{cite book |last1 = Bracken|first1 = Michael B.|title = Risk, Chance, and Causation: Investigating the Origins and Treatment of Disease |url = https://archive.org/details/riskchancecausat0000brac|url-access = registration| edition=1st |publisher =Yale University Press|location = New Haven, CT | year = 2013 |isbn = 978-0-300-18884-4 |pages=[https://archive.org/details/riskchancecausat0000brac/page/260 260–276]}}</ref> For instance, the certainty of the [[Higgs boson]] particle's existence was based on the 5''σ'' criterion, which corresponds to a ''p''-value of about 1 in 3.5 million.<ref name="Bracken"/><ref name=franklin>{{cite book |last1 = Franklin|first1 = Allan|chapter= Prologue: The rise of the sigmas |title=Shifting Standards: Experiments in Particle Physics in the Twentieth Century|edition=1st |publisher = University of Pittsburgh Press|location = Pittsburgh, PA | year = 2013 |isbn = 978-0-8229-4430-0 |pages=Ii–Iii}}</ref>
-In other fields of scientific research such as [[Genome-wide association study|genome-wide association studies]], significance levels as low as {{val|5|e=-8}} are not uncommon<ref name="Clarke et al">{{cite journal | last1 =  Clarke | first1 = GM | last2 = Anderson | first2 = CA | last3 = Pettersson | first3 = FH | last4 = Cardon | first4 = LR | last5 = Morris | first5 = AP | last6 = Zondervan | first6= KT | title = Basic statistical analysis in genetic case-control studies | journal = Nature Protocols | volume = 6 | issue = 2 | pages = 121–33 | date = February 6, 2011 | pmid = 21293453 | doi = 10.1038/nprot.2010.182 | pmc=3154648}}</ref><ref name="Barsh et al">{{cite journal | last1 = Barsh | first1 = GS | last2 = Copenhaver | first2 = GP | last3 = Gibson | first3 = G | last4 = Williams | first4 = SM | title = Guidelines for Genome-Wide Association Studies | journal = PLOS Genetics | volume = 8 | issue = 7 | pages = e1002812 | date = July 5, 2012 | pmid = 22792080 | doi = 10.1371/journal.pgen.1002812 | pmc=3390399}}</ref>—as the number of tests performed is extremely large.
+In other fields of scientific research such as [[Genome-wide association study|genome-wide association studies]], significance levels as low as {{val|5|e=-8}} are not uncommon<ref name="Clarke et al">{{cite journal | last1 =  Clarke | first1 = GM | last2 = Anderson | first2 = CA | last3 = Pettersson | first3 = FH | last4 = Cardon | first4 = LR | last5 = Morris | first5 = AP | last6 = Zondervan | first6= KT | title = Basic statistical analysis in genetic case-control studies | journal = Nature Protocols | volume = 6 | issue = 2 | pages = 121–33 | date = February 6, 2011 | pmid = 21293453 | doi = 10.1038/nprot.2010.182 | pmc=3154648}}</ref><ref name="Barsh et al">{{cite journal | last1 = Barsh | first1 = GS | last2 = Copenhaver | first2 = GP | last3 = Gibson | first3 = G | last4 = Williams | first4 = SM | title = Guidelines for Genome-Wide Association Studies | journal = PLOS Genetics | volume = 8 | issue = 7 | pages = e1002812 | date = July 5, 2012 | pmid = 22792080 | doi = 10.1371/journal.pgen.1002812 | pmc=3390399 | doi-access = free }}</ref>—as the number of tests performed is extremely large.
 == Limitations ==
-Researchers focusing solely on whether their results are statistically significant might report findings that are not substantive<ref name="Carver">{{Cite journal | last1 = Carver| first1 = Ronald P. | title = The Case Against Statistical Significance Testing | journal = Harvard Educational Review | volume = 48| issue = 3 | pages = 378–399 | year = 1978| doi = 10.17763/haer.48.3.t490261645281841 | s2cid = 16355113 | url = https://semanticscholar.org/paper/cb9adb96be34b2652fce8c2a3e8324a0f1ce0048 }}</ref> and not replicable.<ref name="Ioannidis">{{cite journal | last1 = Ioannidis | first1 = John P. A. | title = Why most published research findings are false | journal = PLOS Medicine | volume = 2 | issue = 8 | pages = e124 | year = 2005 | doi=10.1371/journal.pmed.0020124 | pmid=16060722 | pmc=1182327}}</ref><ref name="peerj.com">{{cite journal|last1= Amrhein|first1=Valentin|last2=Korner-Nievergelt|first2=Fränzi|last3=Roth|first3=Tobias|title=The earth is flat (p > 0.05): significance thresholds and the crisis of unreplicable research|journal=PeerJ|date=2017|volume=5|page=e3544|doi=10.7717/peerj.3544|pmid=28698825|pmc=5502092}}</ref> There is also a difference between statistical significance and practical significance. A study that is found to be statistically significant may not necessarily be practically significant.<ref name="A Visitor’s Guide to Effect Sizes">{{cite journal|last1=Hojat|first1=Mohammadreza|last2=Xu|first2=Gang|title=A Visitor's Guide to Effect Sizes|journal=Advances in Health Sciences Education|volume=9|issue=3|pages=241–9|date=2004|doi=10.1023/B:AHSE.0000038173.00909.f6|pmid=15316274|s2cid=8045624}}</ref><ref name=":1">{{Cite web|url=http://www.stat.ualberta.ca/~hooper/teaching/misc/Pvalue.pdf|title=What is P-value?|last=Hooper|first=Peter|website=University of Alberta, Department of Mathematical and Statistical Sciences|access-date=November 10, 2019}}</ref>
+Researchers focusing solely on whether their results are statistically significant might report findings that are not substantive<ref name="Carver">{{Cite journal | last1 = Carver| first1 = Ronald P. | title = The Case Against Statistical Significance Testing | journal = Harvard Educational Review | volume = 48| issue = 3 | pages = 378–399 | year = 1978| doi = 10.17763/haer.48.3.t490261645281841 | s2cid = 16355113 }}</ref> and not replicable.<ref name="Ioannidis">{{cite journal | last1 = Ioannidis | first1 = John P. A. | title = Why most published research findings are false | journal = PLOS Medicine | volume = 2 | issue = 8 | pages = e124 | year = 2005 | doi=10.1371/journal.pmed.0020124 | pmid=16060722 | pmc=1182327 | doi-access = free }}</ref><ref name="peerj.com">{{cite journal|last1= Amrhein|first1=Valentin|last2=Korner-Nievergelt|first2=Fränzi|last3=Roth|first3=Tobias|title=The earth is flat (p > 0.05): significance thresholds and the crisis of unreplicable research|journal=PeerJ|date=2017|volume=5|page=e3544|doi=10.7717/peerj.3544|pmid=28698825|pmc=5502092 |doi-access=free }}</ref> There is also a difference between statistical significance and practical significance. A study that is found to be statistically significant may not necessarily be practically significant.<ref name="A Visitor’s Guide to Effect Sizes">{{cite journal|last1=Hojat|first1=Mohammadreza|last2=Xu|first2=Gang|title=A Visitor's Guide to Effect Sizes|journal=Advances in Health Sciences Education|volume=9|issue=3|pages=241–9|date=2004|doi=10.1023/B:AHSE.0000038173.00909.f6|pmid=15316274|s2cid=8045624}}</ref><ref name=":1"/>
 === Effect size ===
 {{Main|Effect size}}
-Effect size is a measure of a study's practical significance.<ref name="A Visitor’s Guide to Effect Sizes"/> A statistically significant result may have a weak effect. To gauge the research significance of their result, researchers are encouraged to always report an [[effect size]] along with ''p''-values. An effect size measure quantifies the strength of an effect, such as the distance between two means in units of standard deviation (cf. [[Cohen's d]]), the [[Pearson product-moment correlation coefficient|correlation coefficient]] between two variables or [[Coefficient of determination|its square]], and other measures.<ref name=Pedhazur>{{cite book | last1 = Pedhazur | first1 = Elazar J. | last2=Schmelkin|first2=Liora P. | title = Measurement, Design, and Analysis: An Integrated Approach| edition=Student|publisher = Psychology Press |location = New York, NY | year = 1991|isbn =978-0-805-81063-9 |pages=180–210}}</ref>
+Effect size is a measure of a study's practical significance.<ref name="A Visitor’s Guide to Effect Sizes"/> A statistically significant result may have a weak effect. To gauge the research significance of their result, researchers are encouraged to always report an [[effect size]] along with ''p''-values. An effect size measure quantifies the strength of an effect, such as the distance between two means in units of standard deviation (cf. [[Cohen's d]]), the [[Pearson product-moment correlation coefficient|correlation coefficient]] between two variables or [[Coefficient of determination|its square]], and other measures.<ref name=Pedhazur>{{cite book | last1 = Pedhazur | first1 = Elazar J. | last2=Schmelkin|first2=Liora P. | title = Measurement, Design, and Analysis: An Integrated Approach| edition=Student|publisher = Psychology Press |location = New York, NY | year = 1991|isbn =978-0-8058-1063-9 |pages=180–210}}</ref>
 === Reproducibility ===
 {{Main|Reproducibility}}
-A statistically significant result may not be easy to reproduce.<ref name="peerj.com"/> In particular, some statistically significant results will in fact be false positives. Each failed attempt to reproduce a result increases the likelihood that the result was a false positive.<ref>{{cite journal|last1=Stahel|first1=Werner|title=Statistical Issue in Reproducibility|journal=Principles, Problems, Practices, and Prospects Reproducibility: Principles, Problems, Practices, and Prospects|date=2016|pages=87–114|doi=10.1002/9781118865064.ch5|isbn=9781118864975}}</ref>
+A statistically significant result may not be easy to reproduce.<ref name="peerj.com"/> In particular, some statistically significant results will in fact be false positives. Each failed attempt to reproduce a result increases the likelihood that the result was a false positive.<ref>{{cite book|last1=Stahel|first1=Werner|chapter=Statistical Issue in Reproducibility|title=Principles, Problems, Practices, and Prospects Reproducibility: Principles, Problems, Practices, and Prospects|date=2016|pages=87–114|doi=10.1002/9781118865064.ch5|isbn=978-1-118-86497-5}}</ref>
 == Challenges ==
@@ Line 96: / Line 100: @@
 Other editors, commenting on this ban have noted: "Banning the reporting of ''p''-values, as Basic and Applied Social Psychology recently did, is not going to solve the problem because it is merely treating a symptom of the problem. There is nothing wrong with hypothesis testing and ''p''-values per se as long as authors, reviewers, and action editors use them correctly."<ref>{{Cite journal|last=Antonakis|first=John|date=February 2017|title=On doing better science: From thrill of discovery to policy implications|journal=The Leadership Quarterly|volume=28|issue=1|pages=5–21|doi=10.1016/j.leaqua.2017.01.006|url=https://serval.unil.ch/resource/serval:BIB_8EF01CB80A64.P001/REF.pdf}}</ref> Some statisticians prefer to use alternative measures of evidence, such as [[likelihood ratio]]s or [[Bayes factor]]s.<ref name="Wasserstein 129–133"/> Using [[Bayesian statistics]] can avoid confidence levels, but also requires making additional assumptions,<ref name="Wasserstein 129–133">{{Cite journal|last1=Wasserstein|first1=Ronald L.|last2=Lazar|first2=Nicole A.|date=2016-04-02|title=The ASA's Statement on p-Values: Context, Process, and Purpose|journal= [[The American Statistician]]|volume=70|issue=2|pages=129–133| doi=10.1080/00031305.2016.1154108|url=http://revistas.ucm.es/index.php/TEKN/article/view/57194|doi-access=free}}</ref> and may not necessarily improve practice regarding statistical testing.<ref>{{Cite journal|last=García-Pérez|first=Miguel A.|date=2016-10-05|title=Thou Shalt Not Bear False Witness Against Null Hypothesis Significance Testing|journal=Educational and Psychological Measurement|volume=77|issue=4|language=en|pages=631–662|doi=10.1177/0013164416668232|pmid=30034024|pmc=5991793|issn=0013-1644}}</ref>
-The widespread abuse of statistical significance represents an important topic of research in [[metascience (research)|metascience]].<ref>{{cite journal |last1=Ioannidis |first1=John P. A. |last2=Ware |first2=Jennifer J. |last3=Wagenmakers |first3=Eric-Jan |last4=Simonsohn |first4=Uri |last5=Chambers |first5=Christopher D. |last6=Button |first6=Katherine S. |last7=Bishop |first7=Dorothy V. M. |last8=Nosek |first8=Brian A. |last9=Munafò |first9=Marcus R. |title=A manifesto for reproducible science |journal=Nature Human Behaviour |volume=1 |pages=0021 |language=en |doi=10.1038/s41562-016-0021 |date=January 2017|pmid=33954258 |pmc=7610724 |doi-access=free }}</ref>
+The widespread abuse of statistical significance represents an important topic of research in [[metascience (research)|metascience]].<ref>{{cite journal |last1=Ioannidis |first1=John P. A. |last2=Ware |first2=Jennifer J. |last3=Wagenmakers |first3=Eric-Jan |last4=Simonsohn |first4=Uri |last5=Chambers |first5=Christopher D. |last6=Button |first6=Katherine S. |last7=Bishop |first7=Dorothy V. M. |last8=Nosek |first8=Brian A. |last9=Munafò |first9=Marcus R. |title=A manifesto for reproducible science |journal=Nature Human Behaviour |volume=1 |pages=0021 |language=en |doi=10.1038/s41562-016-0021 |date=January 2017|issue=1 |pmid=33954258 |pmc=7610724 |doi-access=free }}</ref>
 === Redefining significance ===
-In 2016, the [[American Statistical Association]] (ASA) published a statement on ''p''-values, saying that "the widespread use of 'statistical significance' (generally interpreted as '''p''&nbsp;≤ 0.05') as a license for making a claim of a scientific finding (or implied truth) leads to considerable distortion of the scientific process".<ref name="Wasserstein 129–133"/> In 2017, a group of 72 authors proposed to enhance reproducibility by changing the ''p''-value threshold for statistical significance from 0.05 to 0.005.<ref>{{Cite journal|last=Benjamin|first=Daniel |display-authors=etal |title=Redefine statistical significance|journal=Nature Human Behaviour|volume=1|issue=1 |pages=6–10|doi=10.1038/s41562-017-0189-z|pmid=30980045 |year=2018 |doi-access=free}}</ref> Other researchers responded that imposing a more stringent significance threshold would aggravate problems such as [[data dredging]]; alternative propositions are thus to select and justify flexible ''p''-value thresholds before collecting data,<ref>{{Cite journal|last=Chawla|first=Dalmeet|date=2017|title='One-size-fits-all' threshold for P values under fire|url=https://www.nature.com/news/one-size-fits-all-threshold-for-p-values-under-fire-1.22625|journal=Nature|doi=10.1038/nature.2017.22625}}</ref> or to interpret ''p''-values as continuous indices, thereby discarding thresholds and statistical significance.<ref>{{cite journal|last1= Amrhein|first1 = Valentin|last2=Greenland |first2=Sander|title=Remove, rather than redefine, statistical significance|journal=Nature Human Behaviour|date=2017|volume=2|issue = 1|page=0224|doi=10.1038/s41562-017-0224-0|pmid = 30980046|s2cid = 46814177}}</ref> Additionally, the change to 0.005 would increase the likelihood of false negatives, whereby the effect being studied is real, but the test fails to show it.<ref>{{cite web |last1=Vyse |first1=Stuart |title=Moving Science's Statistical Goalposts |url=https://www.csicop.org/si/show/moving_sciences_statistical_goal_posts |website=csicop.org |date=November 2017 |publisher=CSI |access-date=10 July 2018}}</ref>
+In 2016, the [[American Statistical Association]] (ASA) published a statement on ''p''-values, saying that "the widespread use of 'statistical significance' (generally interpreted as '''p''&nbsp;≤ 0.05') as a license for making a claim of a scientific finding (or implied truth) leads to considerable distortion of the scientific process".<ref name="Wasserstein 129–133"/> In 2017, a group of 72 authors proposed to enhance reproducibility by changing the ''p''-value threshold for statistical significance from 0.05 to 0.005.<ref>{{Cite journal|last=Benjamin|first=Daniel |display-authors=etal |title=Redefine statistical significance|journal=Nature Human Behaviour|volume=1|issue=1 |pages=6–10|doi=10.1038/s41562-017-0189-z|pmid=30980045 |year=2018 |doi-access=free|hdl=10281/184094|hdl-access=free}}</ref> Other researchers responded that imposing a more stringent significance threshold would aggravate problems such as [[data dredging]]; alternative propositions are thus to select and justify flexible ''p''-value thresholds before collecting data,<ref>{{Cite journal|last=Chawla|first=Dalmeet|date=2017|title='One-size-fits-all' threshold for P values under fire|url=https://www.nature.com/news/one-size-fits-all-threshold-for-p-values-under-fire-1.22625|journal=Nature|doi=10.1038/nature.2017.22625}}</ref> or to interpret ''p''-values as continuous indices, thereby discarding thresholds and statistical significance.<ref>{{cite journal|last1= Amrhein|first1 = Valentin|last2=Greenland |first2=Sander|title=Remove, rather than redefine, statistical significance|journal=Nature Human Behaviour|date=2017|volume=2|issue = 1|page=0224|doi=10.1038/s41562-017-0224-0|pmid = 30980046|s2cid = 46814177}}</ref> Additionally, the change to 0.005 would increase the likelihood of false negatives, whereby the effect being studied is real, but the test fails to show it.<ref>{{cite web |last1=Vyse |first1=Stuart |title=Moving Science's Statistical Goalposts |url=https://www.csicop.org/si/show/moving_sciences_statistical_goal_posts |website=csicop.org |date=November 2017 |publisher=CSI |access-date=10 July 2018}}</ref>
-In 2019, over 800 statisticians and scientists signed a message calling for the abandonment of the term "statistical significance" in science,<ref>{{Cite journal|last1=McShane|first1=Blake|last2=Greenland|first2=Sander|last3=Amrhein|first3=Valentin|date=March 2019|title=Scientists rise up against statistical significance|journal=Nature|language=EN|volume=567|issue=7748|pages=305–307|doi=10.1038/d41586-019-00857-9|pmid=30894741|bibcode=2019Natur.567..305A |doi-access=free}}</ref> and the American Statistical Association published a further official statement <ref name="Wasserstein2 129–133">{{Cite journal|last1=Wasserstein|first1=Ronald L.|last2=Schirm|first2=Allen L.|last3=Lazar|first3=Nicole A.|date=2019-03-20|title=Moving to a World Beyond "p < 0.05"|journal= [[The American Statistician]]|volume=73|issue=sup1|pages=1–19| doi=10.1080/00031305.2019.1583913|doi-access=free}}</ref> declaring (page 2): {{Blockquote
+In 2019, over 800 statisticians and scientists signed a message calling for the abandonment of the term "statistical significance" in science,<ref>{{Cite journal|last1=McShane|first1=Blake|last2=Greenland|first2=Sander|last3=Amrhein|first3=Valentin|date=March 2019|title=Scientists rise up against statistical significance|journal=Nature|language=EN|volume=567|issue=7748|pages=305–307|doi=10.1038/d41586-019-00857-9|pmid=30894741|bibcode=2019Natur.567..305A |doi-access=free}}</ref> and the ASA published a further official statement <ref name="Wasserstein2 129–133">{{Cite journal|last1=Wasserstein|first1=Ronald L.|last2=Schirm|first2=Allen L.|last3=Lazar|first3=Nicole A.|date=2019-03-20|title=Moving to a World Beyond "p < 0.05"|journal= [[The American Statistician]]|volume=73|issue=sup1|pages=1–19| doi=10.1080/00031305.2019.1583913|doi-access=free}}</ref> declaring (page 2): {{Blockquote
 |text=We conclude, based on our review of the articles in this special issue and the broader literature, that it is time to stop using the term "statistically significant" entirely. Nor should variants such as "significantly different," "<math>p \le 0.05</math>," and "nonsignificant" survive, whether expressed in words, by asterisks in a table, or in some other way.}}
@@ Line 119: / Line 123: @@
 ==Further reading==
 * Lydia Denworth, "A Significant Problem: Standard scientific methods are under fire. Will anything change?", ''[[Scientific American]]'', vol. 321, no. 4 (October 2019), pp.&nbsp;62–67. "The use of [[p value|''p'' values]] for nearly a century [since 1925] to determine statistical significance of [[experiment]]al results has contributed to an illusion of [[certainty]] and [to] [[reproducibility|reproducibility crises]] in many [[science|scientific fields]]. There is growing determination to reform statistical analysis... Some [researchers] suggest changing statistical methods, whereas others would do away with a threshold for defining "significant" results." (p. 63.)
-*[[Stephen Ziliak|Ziliak, Stephen]] and [[Deirdre McCloskey]] (2008), ''[http://www.press.umich.edu/titleDetailDesc.do?id=186351 The Cult of Statistical Significance: How the Standard Error Costs Us Jobs, Justice, and Lives]''. Ann Arbor, [[University of Michigan Press]], 2009. {{isbn|978-0-472-07007-7}}. Reviews and reception: [http://blogs.roosevelt.edu/sziliak/cult-of-statistical-significance/ (compiled by Ziliak)]
+*[[Stephen Ziliak|Ziliak, Stephen]] and [[Deirdre McCloskey]] (2008), ''[http://www.press.umich.edu/titleDetailDesc.do?id=186351 The Cult of Statistical Significance: How the Standard Error Costs Us Jobs, Justice, and Lives] {{Webarchive|url=https://web.archive.org/web/20100608082545/http://www.press.umich.edu/titleDetailDesc.do?id=186351 |date=2010-06-08 }}''. Ann Arbor, [[University of Michigan Press]], 2009. {{isbn|978-0-472-07007-7}}. Reviews and reception: [http://blogs.roosevelt.edu/sziliak/cult-of-statistical-significance/ (compiled by Ziliak)]
 *{{cite journal | last1 = Thompson | first1 = Bruce | year = 2004 | title = The "significance" crisis in psychology and education | journal = Journal of Socio-Economics | volume = 33 | issue = 5| pages = 607–613 | doi=10.1016/j.socec.2004.09.034}}
-*Chow, Siu L., (1996). ''[http://websites.psychology.uwa.edu.au/labs/cogscience/Publications/Lewandowsky-Mayberry%20(1996)%20-%20Critics%20Rebuttted.pdf Statistical Significance: Rationale, Validity and Utility],'' Volume 1 of series ''Introducing Statistical Methods,'' Sage Publications Ltd, {{isbn|978-0-7619-5205-3}} – argues that statistical significance is useful in certain circumstances.
+*Chow, Siu L., (1996). ''[http://websites.psychology.uwa.edu.au/labs/cogscience/Publications/Lewandowsky-Mayberry%20(1996)%20-%20Critics%20Rebuttted.pdf Statistical Significance: Rationale, Validity and Utility] {{Webarchive|url=https://web.archive.org/web/20131203024039/http://websites.psychology.uwa.edu.au/labs/cogscience/Publications/Lewandowsky-Mayberry%20(1996)%20-%20Critics%20Rebuttted.pdf |date=2013-12-03 }},'' Volume 1 of series ''Introducing Statistical Methods,'' Sage Publications Ltd, {{isbn|978-0-7619-5205-3}} – argues that statistical significance is useful in certain circumstances.
-*Kline, Rex, (2004). ''[http://www.worldcat.org/title/beyond-significance-testing-reforming-data-analysis-methods-in-behavioral-research/oclc/53288011&referer=brief_results Beyond Significance Testing: Reforming Data Analysis Methods in Behavioral Research]'' Washington, DC: American Psychological Association.
+*Kline, Rex, (2004). ''[https://www.worldcat.org/oclc/53288011 Beyond Significance Testing: Reforming Data Analysis Methods in Behavioral Research]'' Washington, DC: American Psychological Association.
 *[[Regina Nuzzo|Nuzzo, Regina]] (2014). [https://web.archive.org/web/20140213062055/http://www.nature.com/news/scientific-method-statistical-errors-1.14700#/b5 Scientific method: Statistical errors]. ''Nature'' Vol. 506, p.&nbsp;150-152 (open access). Highlights common misunderstandings about the p value.
 *Cohen, Joseph (1994). [http://ist-socrates.berkeley.edu/~maccoun/PP279_Cohen1.pdf] {{Webarchive|url=https://web.archive.org/web/20170713081635/http://ist-socrates.berkeley.edu/~maccoun/PP279_Cohen1.pdf |date=2017-07-13 }}. The earth is round (p<.05). American Psychologist. Vol 49, p.&nbsp;997-1003. Reviews problems with null hypothesis statistical testing.
@@ Line 129: / Line 133: @@
 ==External links==
 {{Wikiversity}}
-* The article "[http://jeff560.tripod.com/s.html Earliest Known Uses of Some of the Words of Mathematics (S)]" contains an entry on Significance that provides some historical information.
+* The article "[https://jeff560.tripod.com/s.html Earliest Known Uses of Some of the Words of Mathematics (S)]" contains an entry on Significance that provides some historical information.
-* "[http://www.ericdigests.org/1995-1/testing.htm The Concept of Statistical Significance Testing]" (February 1994): article by Bruce Thompon hosted by the ERIC Clearinghouse on Assessment and Evaluation, Washington, D.C.
+* "[https://www.ericdigests.org/1995-1/testing.htm The Concept of Statistical Significance Testing] {{Webarchive|url=https://web.archive.org/web/20220907064024/https://www.ericdigests.org/1995-1/testing.htm |date=2022-09-07 }}" (February 1994): article by Bruce Thompon hosted by the ERIC Clearinghouse on Assessment and Evaluation, Washington, D.C.
 * "[https://web.archive.org/web/20120419105227/http://stats.org/faq_significance.htm What does it mean for a result to be "statistically significant"?]" (no date): an article from the Statistical Assessment Service at George Mason University, Washington, D.C.