Statistical significance

In statistical hypothesis testing,[1][2] statistical significance (or a statistically significant result) is attained whenever the observed p-value of a test statistic is less than the significance level defined for the study.[3][4][5][6][7][8][9] The p-value is the probability of obtaining results at least as extreme as those observed, given that the null hypothesis is true. The significance level, α, is the probability of rejecting the null hypothesis, given that it is true.[10] This statistical technique for testing the significance of results was developed in the early 20th century.

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.[11][12] But if the p-value of an observed effect is less than the significance level, an investigator may conclude that that effect reflects the characteristics of the whole population,[1] thereby rejecting the null hypothesis.[13] A significance level is chosen before data collection, and typically set to 5%[14] or much lower, depending on the field of study.[15]

The term significance does not imply importance and the term statistical significance is not the same as research, theoretical, or practical significance.[1][2][16] For example, the term clinical significance refers to the practical importance of a treatment effect.

History

Main article: History of statistics

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.[17][18][19] Fisher suggested a probability of one in twenty (0.05) as a convenient cutoff level to reject the null hypothesis.[20] In a 1933 paper, Jerzy Neyman and Egon Pearson called this cutoff the significance level, which they named α. They recommended that α be set ahead of time, prior to any data collection.[20][21]

Despite his initial suggestion of 0.05 as a significance level, Fisher did not intend this cutoff value to be fixed. For instance, in his 1956 publication Statistical methods and scientific inference he recommended that significant levels be set according to specific circumstances.[20]

The significance level α is the threshhold for p below which the experimenter assumes the null hypothesis is false, and something else is going on. This means α is also the probability of mistakenly rejecting the null hypothesis, if the null hypothesis is true.[22]

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. [23][24] Confidence levels and confidence intervals were introduced by Neyman in 1937.[25]

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 default assumption that nothing happened or changed.[26] 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.

To determine whether a result is statistically significant, a researcher calculates a p-value, which is the probability of observing an effect given that the null hypothesis is true.[9] The null hypothesis is rejected if the p-value is less than a predetermined level, α. α is 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 α is set to 5%, the conditional probability 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 data from a sample, this means that the rejection region comprises 5% of the sampling distribution.[29] 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.[30] 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.[31] 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.

Stringent 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 (e.g. 5σ).[32][33] 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.[33][34]

In other fields of scientific research such as genome-wide association studies significance levels as low as 5×10−8 are not uncommon.[35][36]

Limitations

Researchers focusing solely on whether their results are statistically significant might report findings that are not substantive[37] and not replicable.[38] 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. [39]

Effect size

Main article: Effect size

Effect size is a measure of a study's practical significance. [40] 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 between two variables or its square, and other measures.[41]

Reproducibility

Main article: Reproducibility

A statistically significant result may not be easy to reproduce. In particular, some statistically significant results will in fact be false positives. Each failed attempt to reproduce a result increases the belief that the result was a false positive. [42]

Controversy around 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.[43] Some journals encouraged authors to do more detailed analysis than just a statistical significance test. In social psychology, the Journal of Basic and Applied Social Psychology banned the use of significance testing altogether from papers it published, requiring authors to use other measure to evaluate hypotheses and impact.[44][45]

See also

References

  1. 1 2 3 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 1-412-90546-X.
  2. 1 2 Borror, Connie M. (2009). "Statistical decision making". The Certified Quality Engineer Handbook (3rd ed.). Milwaukee, WI: ASQ Quality Press. pp. 418–472. ISBN 0-873-89745-5.
  3. ↑ 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 0-471-82211-6.
  4. ↑ Cumming, Geoff (2012). Understanding The New Statistics: Effect Sizes, Confidence Intervals, and Meta-Analysis. New York, USA: Routledge. pp. 27–28.
  5. ↑ Krzywinski, Martin; Altman, Naomi (30 October 2013). "Points of significance: Significance, P values and t-tests". Nature Methods. Nature Publishing Group. 10 (11): 1041–1042. doi:10.1038/nmeth.2698. Retrieved 3 July 2014.
  6. ↑ Sham, Pak C.; Purcell, Shaun M (17 April 2014). "Statistical power and significance testing in large-scale genetic studies". Nature Reviews Genetics. Nature Publishing Group. 15 (5): 335–346. doi:10.1038/nrg3706. Retrieved 3 July 2014.
  7. ↑ Johnson, Valen E. (October 9, 2013). "Revised standards for statistical evidence". Proceedings of the National Academy of Sciences. National Academies of Science. 110: 19313–19317. doi:10.1073/pnas.1313476110. Retrieved 3 July 2014.
  8. ↑ Altman, Douglas G. (1999). Practical Statistics for Medical Research. New York, USA: Chapman & Hall/CRC. p. 167. ISBN 978-0412276309.
  9. 1 2 Devore, Jay L. (2011). Probability and Statistics for Engineering and the Sciences (8th ed.). Boston, MA: Cengage Learning. pp. 300–344. ISBN 0-538-73352-7.
  10. ↑ Schlotzhauer, Sandra (2007). Elementary Statistics Using JMP (SAS Press) (PAP/CDR ed.). Cary, NC: SAS Institute. pp. 166–169. ISBN 1-599-94375-1.
  11. ↑ Babbie, Earl R. (2013). "The logic of sampling". The Practice of Social Research (13th ed.). Belmont, CA: Cengage Learning. pp. 185–226. ISBN 1-133-04979-6.
  12. ↑ 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 1-412-93982-8.
  13. ↑ 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 0-521-54316-9.
  14. ↑ Craparo, Robert M. (2007). "Significance level". In Salkind, Neil J. Encyclopedia of Measurement and Statistics. 3. Thousand Oaks, CA: SAGE Publications. pp. 889–891. ISBN 1-412-91611-9.
  15. ↑ 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 0-810-84486-9.
  16. ↑ Myers, Jerome L.; Well, Arnold D.; Lorch Jr, Robert F. (2010). "The t distribution and its applications". Research Design and Statistical Analysis: Third Edition (3rd ed.). New York, NY: Routledge. pp. 124–153. ISBN 0-805-86431-8.
  17. ↑ 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 0-415-87968-X.
  18. ↑ Fisher, Ronald A. (1925). Statistical Methods for Research Workers. Edinburgh, UK: Oliver and Boyd. p. 43. ISBN 0-050-02170-2.
  19. ↑ 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 1-841-69159-3.
  20. 1 2 3 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 0-521-00976-6.
  21. ↑ 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: 492–510. doi:10.1017/S030500410001152X.
  22. ↑ Schlotzhauer, Sandra (2007). Elementary Statistics Using JMP (SAS Press) (PAP/CDR ed.). Cary, NC: SAS Institute. pp. 166–169. ISBN 1-599-94375-1.
  23. ↑ "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." "Confidence Interval or P-Value?". doi:10.3238/arztebl.2009.0335.
  24. ↑ StatNews #73: Overlapping Confidence Intervals and Statistical Significance
  25. ↑ 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: 333–380. doi:10.1098/rsta.1937.0005.
  26. ↑ 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 1-111-34280-6.
  27. ↑ Healy, Joseph F. (2009). The Essentials of Statistics: A Tool for Social Research (2nd ed.). Belmont, CA: Cengage Learning. pp. 177–205. ISBN 0-495-60143-8.
  28. ↑ McKillup, Steve (2006). Statistics Explained: An Introductory Guide for Life Scientists (1st ed.). Cambridge, UK: Cambridge University Press. pp. 32–38. ISBN 0-521-54316-9.
  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.
  30. ↑ Myers, Jerome L.; Well, Arnold D.; Lorch, Jr., Robert F. (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 0-805-86431-8.
  31. ↑ Hinton, Perry R. (2010). "Significance, error, and power". Statistics explained (3rd ed.). New York, NY: Routledge. pp. 79–90. ISBN 1-848-72312-1.
  32. ↑ Vaughan, Simon (2013). Scientific Inference: Learning from Data (1st ed.). Cambridge, UK: Cambridge University Press. pp. 146–152. ISBN 1-107-02482-X.
  33. 1 2 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 0-300-18884-6.
  34. ↑ 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 0-822-94430-8.
  35. ↑ 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 3154648Freely accessible. PMID 21293453.
  36. ↑ 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 3390399Freely accessible. PMID 22792080.
  37. ↑ Carver, Ronald P. (1978). "The Case Against Statistical Significance Testing". Harvard Educational Review. 48: 378–399.
  38. ↑ Ioannidis, John P. A. (2005). "Why most published research findings are false". PLoS Medicine. 2: e124. doi:10.1371/journal.pmed.0020124. PMC 1182327Freely accessible. PMID 16060722.
  39. ↑ Hojat, Mohammadreza; Xu, Gang (2004). "A Visitor's Guide to Effect Sizes". Advances in Health Sciences Education.
  40. ↑ Hojat, Mohammadreza; Xu, Gang (2004). "A Visitor's Guide to Effect Sizes". Advances in Health Sciences Education.
  41. ↑ 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 0-805-81063-3.
  42. ↑ Stahel, Werner (2016). "Statistical Issue in Reproducibility". Principles, Problems, Practices, and Prospects Reproducibility: Principles, Problems, Practices, and Prospects: 87-114.
  43. ↑ "CSSME Seminar Series: The argument over p-values and the Null Hypothesis Significance Testing (NHST) paradigm » School of Education » University of Leeds". www.education.leeds.ac.uk. Retrieved 2016-12-01.
  44. ↑ Woolston, Chris (2015-03-05). "Psychology journal bans P values". Nature. 519 (7541): 9–9. doi:10.1038/519009f.
  45. ↑ Siegfried, Tom (2015-03-17). "P value ban: small step for a journal, giant leap for science". Science News. Retrieved 2016-12-01.

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