Statistical significance

In statistical hypothesis testing,[1][2] statistical signif- 1.1 Related concepts icance (or a statistically significant result) is at- tained whenever the observed p-value of a test statis- The significance level α is the threshhold for p below tic is less than the significance level defined for the which the experimenter assumes the null hypothesis is study.[3][4][5][6][7][8][9] The p-value is the probability of false, and something else is going on. This means α is obtaining results at least as extreme as those observed, also the probability of mistakenly rejecting the null hy- given that the null hypothesis is true. The significance pothesis, if the null hypothesis is true.[22] level, α, is the probability of rejecting the null hypothe- Sometimes researchers talk about the confidence level γ sis, given that it is true.[10] This statistical technique for = (1 − α) instead. This is the probability of not rejecting testing the significance of results was developed in the the null hypothesis given that it is true. [23][24] Confidence early 20th century. levels and confidence intervals were introduced by Ney- In any experiment or observation that involves drawing man in 1937.[25] a sample from a population, there is always the possibil- ity that an observed effect would have occurred due to sampling error alone.[11][12] But if the p-value of an ob- 2 Role in statistical hypothesis test- served effect is less than the significance level, an inves- tigator may conclude that that effect reflects the charac- ing teristics of the whole population,[1] thereby rejecting the null hypothesis.[13] A significance level is chosen before Main articles: Statistical hypothesis testing, Null hypoth- data collection, and typically set to 5%[14] or much lower, esis, Alternative hypothesis, p-value, and Type I and type depending on the field of study.[15] II errors The term significance does not imply importance and the Statistical significance plays a pivotal role in statistical term statistical significance is not the same as research, theoretical, or practical significance.[1][2][16] For exam- ple, the term clinical significance refers to the practical importance of a treatment effect.

1 History

Main article: History of statistics

In 1925, Ronald Fisher advanced the idea of statisti- In a two-tailed test, the rejection region for a significance level of cal hypothesis testing, which he called “tests of signifi- α=0.05 is partitioned to both ends of the sampling distribution cance”, in his publication Statistical Methods for Research and makes up 5% of the area under the curve (white areas). Workers.[17][18][19] Fisher suggested a probability of one in twenty (0.05) as a convenient cutoff level to reject hypothesis testing. It is used to determine whether the the null hypothesis.[20] In a 1933 paper, Jerzy Neyman null hypothesis should be rejected or retained. The null and Egon Pearson called this cutoff the significance level, hypothesis is the default assumption that nothing hap- which they named α. They recommended that α be set pened or changed.[26] For the null hypothesis to be re- ahead of time, prior to any data collection.[20][21] jected, an observed result has to be statistically signif- Despite his initial suggestion of 0.05 as a significance icant, i.e. the observed p-value is less than the pre- level, Fisher did not intend this cutoff value to be fixed. specified significance level. For instance, in his 1956 publication Statistical methods To determine whether a result is statistically significant, and scientific inference he recommended that significant a researcher calculates a p-value, which is the probabil- levels be set according to specific circumstances.[20] ity of observing an effect given that the null hypothesis

1 2 4 SEE ALSO

is true.[9] The null hypothesis is rejected if the p-value nificance. A study that is found to be statistically signif- is less than a predetermined level, α. α is called the sig- icant, may not necessarily be practically significant. [39] nificance 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%. 3.1 Effect size

For example, when α is set to 5%, the conditional proba- Main article: Effect size bility of a type I error, given that the null hypothesis is true, is 5%,[27] and a statistically significant result is one where the observed p-value is less than 5%.[28] When drawing Effect size is a measure of a study’s practical significance. [40] data from a sample, this means that the rejection region A statistically significant result may have a weak ef- comprises 5% of the sampling distribution.[29] These 5% fect. To gauge the research significance of their result, can be allocated to one side of the sampling distribution, researchers are encouraged to always report an effect size as in a one-tailed test, or partitioned to both sides of the along with p-values. An effect size measure quantifies the distribution as in a two-tailed test, with each tail (or re- strength of an effect, such as the distance between two jection region) containing 2.5% of the distribution. means in units of standard deviation (cf. Cohen’s d), the correlation between two variables or its square, and other The use of a one-tailed test is dependent on whether the measures.[41] research question or alternative hypothesis specifies a di- rection such as whether a group of objects is heavier or the performance of students on an assessment is better.[30] 3.2 Reproducibility A two-tailed test may still be used but it will be less powerful than a one-tailed test because the rejection re- Main article: Reproducibility gion for a one-tailed test is concentrated on one end of the null distribution and is twice the size (5% vs. 2.5%) A statistically significant result may not be easy to repro- of each rejection region for a two-tailed test. As a result, duce. In particular, some statistically significant results the null hypothesis can be rejected with a less extreme re- will in fact be false positives. Each failed attempt to re- sult if a one-tailed test was used.[31] The one-tailed test is produce a result increases the belief that the result was a only more powerful than a two-tailed test if the specified false positive. [42] direction of the alternative hypothesis is correct. If it is wrong, however, then the one-tailed test has no power. 3.3 Controversy around overuse in some journals 2.1 Stringent significance thresholds in specific fields Starting in the 2010s, some journals began question- ing whether significance testing, and particularly using Main articles: Standard deviation and Normal distribu- a threshold of α=5%, was being relied on too heavily as tion the primary measure of validity of a hypothesis.[43] Some journals encouraged authors to do more detailed analysis In specific fields such as particle physics and than just a statistical significance test. In social psychol- manufacturing, statistical significance is often ex- ogy, the Journal of Basic and Applied Social Psychology pressed in multiples of the standard deviation or sigma banned the use of significance testing altogether from pa- (σ) of a normal distribution, with significance thresholds pers it published, requiring authors to use other measure set at a much stricter level (e.g. 5σ).[32][33] For instance, to evaluate hypotheses and impact.[44][45] 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.[33][34] 4 See also In other fields of scientific research such as genome-wide association studies significance levels as low as 5×10−8 are • A/B testing, ABX test [35][36] not uncommon. • Fisher’s method for combining independent tests of significance 3 Limitations • Look-elsewhere effect • Multiple comparisons problem Researchers focusing solely on whether their results are • Sample size statistically significant might report findings that are not substantive[37] and not replicable.[38] There is also a dif- • Texas sharpshooter fallacy (gives examples of tests ference between statistical significance and practical sig- where the significance level was set too high) 3

5 References [14] Craparo, Robert M. (2007). “Significance level”. In Salkind, Neil J. Encyclopedia of Measurement and Statis- [1] Sirkin, R. Mark (2005). “Two-sample t tests”. Statistics tics. 3. Thousand Oaks, CA: SAGE Publications. pp. for the Social Sciences (3rd ed.). Thousand Oaks, CA: 889–891. ISBN 1-412-91611-9. SAGE Publications, Inc. pp. 271–316. ISBN 1-412- 90546-X. [15] Sproull, Natalie L. (2002). “Hypothesis testing”. Hand- book of Research Methods: A Guide for Practitioners and [2] Borror, Connie M. (2009). “Statistical decision making”. Students in the Social Science (2nd ed.). Lanham, MD: The Certified Quality Engineer Handbook (3rd ed.). Mil- Scarecrow Press, Inc. pp. 49–64. ISBN 0-810-84486-9. waukee, WI: ASQ Quality Press. pp. 418–472. ISBN 0-873-89745-5. [16] Myers, Jerome L.; Well, Arnold D.; Lorch Jr, Robert F. (2010). “The t distribution and its applications”. Research [3] Redmond, Carol; Colton, Theodore (2001). “Clinical sig- Design and Statistical Analysis: Third Edition (3rd ed.). nificance versus statistical significance”. Biostatistics in New York, NY: Routledge. pp. 124–153. ISBN 0-805- Clinical Trials. Wiley Reference Series in Biostatistics 86431-8. (3rd ed.). West Sussex, United Kingdom: John Wiley & Sons Ltd. pp. 35–36. ISBN 0-471-82211-6. [17] Cumming, Geoff (2011). “From null hypothesis signif- icance to testing effect sizes”. Understanding The New [4] Cumming, Geoff (2012). Understanding The New Statis- Statistics: Effect Sizes, Confidence Intervals, and Meta- tics: Effect Sizes, Confidence Intervals, and Meta-Analysis. Analysis. Multivariate Applications Series. East Sussex, New York, USA: Routledge. pp. 27–28. United Kingdom: Routledge. pp. 21–52. ISBN 0-415- 87968-X. [5] Krzywinski, Martin; Altman, Naomi (30 October 2013). “Points of significance: Significance, P values and t- [18] Fisher, Ronald A. (1925). Statistical Methods for Research tests”. Nature Methods. Nature Publishing Group. 10 Workers. Edinburgh, UK: Oliver and Boyd. p. 43. ISBN (11): 1041–1042. doi:10.1038/nmeth.2698. Retrieved 0-050-02170-2. 3 July 2014.

[6] Sham, Pak C.; Purcell, Shaun M (17 April 2014). [19] Poletiek, Fenna H. (2001). “Formal theories of testing”. “Statistical power and significance testing in large-scale Hypothesis-testing Behaviour. Essays in Cognitive Psy- genetic studies”. Nature Reviews Genetics. Nature Pub- chology (1st ed.). East Sussex, United Kingdom: Psy- lishing Group. 15 (5): 335–346. doi:10.1038/nrg3706. chology Press. pp. 29–48. ISBN 1-841-69159-3. Retrieved 3 July 2014. [20] Quinn, Geoffrey R.; Keough, Michael J. (2002). Experi- [7] Johnson, Valen E. (October 9, 2013). “Revised stan- mental Design and Data Analysis for Biologists (1st ed.). dards for statistical evidence”. Proceedings of the National Cambridge, UK: Cambridge University Press. pp. 46–69. Academy of Sciences. National Academies of Science. ISBN 0-521-00976-6. 110: 19313–19317. doi:10.1073/pnas.1313476110. Re- trieved 3 July 2014. [21] Neyman, J.; Pearson, E.S. (1933). “The testing of statisti- cal hypotheses in relation to probabilities a priori”. Math- [8] Altman, Douglas G. (1999). Practical Statistics for Med- ematical Proceedings of the Cambridge Philosophical So- ical Research. New York, USA: Chapman & Hall/CRC. ciety. 29: 492–510. doi:10.1017/S030500410001152X. p. 167. ISBN 978-0412276309. [22] Schlotzhauer, Sandra (2007). Elementary Statistics Using [9] Devore, Jay L. (2011). Probability and Statistics for Engi- JMP (SAS Press) (PAP/CDR ed.). Cary, NC: SAS Insti- neering and the Sciences (8th ed.). Boston, MA: Cengage tute. pp. 166–169. ISBN 1-599-94375-1. Learning. pp. 300–344. ISBN 0-538-73352-7. [23] “Conclusions about statistical significance are possible [10] Schlotzhauer, Sandra (2007). Elementary Statistics Using with the help of the confidence interval. If the con- JMP (SAS Press) (PAP/CDR ed.). Cary, NC: SAS Insti- fidence interval does not include the value of zero ef- tute. pp. 166–169. ISBN 1-599-94375-1. fect, it can be assumed that there is a statistically [11] Babbie, Earl R. (2013). “The logic of sampling”. The significant result.” “Confidence Interval or P-Value?". Practice of Social Research (13th ed.). Belmont, CA: Cen- doi:10.3238/arztebl.2009.0335. gage Learning. pp. 185–226. ISBN 1-133-04979-6. [24] StatNews #73: Overlapping Confidence Intervals and Sta- [12] Faherty, Vincent (2008). “Probability and statistical sig- tistical Significance nificance”. Compassionate Statistics: Applied Quantitative Analysis for Social Services (With exercises and instruc- [25] Neyman, J. (1937). “Outline of a Theory of Statistical tions in SPSS) (1st ed.). Thousand Oaks, CA: SAGE Pub- Estimation Based on the Classical Theory of Probabil- lications, Inc. pp. 127–138. ISBN 1-412-93982-8. ity”. Philosophical Transactions of the Royal Society A. 236: 333–380. doi:10.1098/rsta.1937.0005. [13] McKillup, Steve (2006). “Probability helps you make a decision about your results”. Statistics Explained: An In- [26] Meier, Kenneth J.; Brudney, Jeffrey L.; Bohte, John troductory Guide for Life Scientists (1st ed.). Cambridge, (2011). Applied Statistics for Public and Nonprofit Admin- United Kingdom: Cambridge University Press. pp. 44– istration (3rd ed.). Boston, MA: Cengage Learning. pp. 56. ISBN 0-521-54316-9. 189–209. ISBN 1-111-34280-6. 4 6 FURTHER READING

[27] Healy, Joseph F. (2009). The Essentials of Statistics: A [41] Pedhazur, Elazar J.; Schmelkin, Liora P. (1991). Mea- Tool for Social Research (2nd ed.). Belmont, CA: Cen- surement, Design, and Analysis: An Integrated Approach gage Learning. pp. 177–205. ISBN 0-495-60143-8. (Student ed.). New York, NY: Psychology Press. pp. 180–210. ISBN 0-805-81063-3. [28] McKillup, Steve (2006). Statistics Explained: An Intro- ductory Guide for Life Scientists (1st ed.). Cambridge, [42] Stahel, Werner (2016). “Statistical Issue in Reproducibil- UK: Cambridge University Press. pp. 32–38. ISBN 0- ity”. Principles, Problems, Practices, and Prospects Repro- 521-54316-9. ducibility: Principles, Problems, Practices, and Prospects: 87-114. [29] Health, David (1995). An Introduction To Experimental Design And Statistics For Biology (1st ed.). Boston, MA: CRC press. pp. 123–154. ISBN 1-857-28132-2. [43] “CSSME Seminar Series: The argument over p-values and the Null Hypothesis Significance Testing (NHST) [30] Myers, Jerome L.; Well, Arnold D.; Lorch, Jr., Robert F. paradigm » School of Education » University of Leeds”. (2010). “Developing fundamentals of hypothesis testing www.education.leeds.ac.uk. Retrieved 2016-12-01. using the binomial distribution”. Research design and sta- tistical analysis (3rd ed.). New York, NY: Routledge. pp. [44] Woolston, Chris (2015-03-05). “Psychology jour- 65–90. ISBN 0-805-86431-8. nal bans P values”. Nature. 519 (7541): 9–9. doi:10.1038/519009f. [31] Hinton, Perry R. (2010). “Significance, error, and power”. Statistics explained (3rd ed.). New York, NY: Routledge. [45] Siegfried, Tom (2015-03-17). “P value ban: small step for pp. 79–90. ISBN 1-848-72312-1. a journal, giant leap for science”. Science News. Retrieved 2016-12-01. [32] Vaughan, Simon (2013). Scientific Inference: Learning from Data (1st ed.). Cambridge, UK: Cambridge Uni- versity Press. pp. 146–152. ISBN 1-107-02482-X.

[33] Bracken, Michael B. (2013). Risk, Chance, and Causa- 6 Further reading tion: Investigating the Origins and Treatment of Disease (1st ed.). New Haven, CT: Yale University Press. pp. • Ziliak, Stephen and Deirdre McCloskey (2008), The 260–276. ISBN 0-300-18884-6. Cult of Statistical Significance: How the Standard Er- [34] Franklin, Allan (2013). “Prologue: The rise of the sig- ror Costs Us Jobs, Justice, and Lives. Ann Arbor, mas”. Shifting Standards: Experiments in Particle Physics University of Michigan Press, 2009. ISBN 978-0- in the Twentieth Century (1st ed.). Pittsburgh, PA: Univer- 472-07007-7. Reviews and reception: (compiled by sity of Pittsburgh Press. pp. Ii–Iii. ISBN 0-822-94430-8. Ziliak)

[35] Clarke, GM; Anderson, CA; Pettersson, FH; Cardon, LR; • Thompson, Bruce (2004). “The “signifi- Morris, AP; Zondervan, KT (February 6, 2011). “Basic cance” crisis in psychology and education”. statistical analysis in genetic case-control studies”. Nature Journal of Socio-Economics. 33: 607–613. Protocols. 6 (2): 121–33. doi:10.1038/nprot.2010.182. doi:10.1016/j.socec.2004.09.034. PMC 3154648 . PMID 21293453.

[36] Barsh, GS; Copenhaver, GP; Gibson, G; Williams, SM • Chow, Siu L., (1996). Statistical Significance: Ra- (July 5, 2012). “Guidelines for Genome-Wide Asso- tionale, Validity and Utility, Volume 1 of series In- ciation Studies”. PLoS Genetics. 8 (7): e1002812. troducing Statistical Methods, Sage Publications Ltd, doi:10.1371/journal.pgen.1002812. PMC 3390399 . ISBN 978-0-7619-5205-3 – argues that statistical PMID 22792080. significance is useful in certain circumstances.

[37] Carver, Ronald P. (1978). “The Case Against Statistical • Kline, Rex, (2004). Beyond Significance Testing: Significance Testing”. Harvard Educational Review. 48: Reforming Data Analysis Methods in Behavioral Re- 378–399. search Washington, DC: American Psychological [38] Ioannidis, John P. A. (2005). “Why most published re- Association. search findings are false”. PLoS Medicine. 2: e124. doi:10.1371/journal.pmed.0020124. PMC 1182327 . • Nuzzo, Regina (2014). Scientific method: Statisti- PMID 16060722. cal errors. Nature Vol. 506, p. 150-152 (open ac- cess). Highlights common misunderstandings about [39] Hojat, Mohammadreza; Xu, Gang (2004). “A Visitor’s the p value. Guide to Effect Sizes”. Advances in Health Sciences Edu- cation. • Cohen, Joseph (1994). . The earth is round (p<.05). [40] Hojat, Mohammadreza; Xu, Gang (2004). “A Visitor’s American Psychologist. Vol 49, p. 997-1003. Re- Guide to Effect Sizes”. Advances in Health Sciences Edu- views problems with null hypothesis statistical test- cation. ing. 5

7 External links

• The article "Earliest Known Uses of Some of the Words of Mathematics (S)" contains an entry on Sig- nificance that provides some historical information.

• "The Concept of Statistical Significance Testing" (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 Statis- tical Assessment Service at George Mason Univer- sity, Washington, D.C. 6 8 TEXT AND IMAGE SOURCES, CONTRIBUTORS, AND LICENSES

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