Statistical significance

In statistical hypothesis testing, statistical significance (or a statistically significant result) is attained when a p-value is less than the significance level (denoted α, alpha). The p-value is the probability of obtaining at least as extreme results given that the null hypothesis is true whereas the significance level α is the probability of rejecting the null hypothesis given that it is true. Equivalently, when the null hypothesis specifies the value of a parameter, the data are said to be statistically significant at given confidence level γ = 1 − α when the computed confidence interval for that parameter fails to contain the value specified by the null hypothesis.

As a matter of good scientific practice, a significance level is chosen before data collection and is often set to 0.05 (5%). Other significance levels (e.g., 0.01) may be used, depending on the field of study. 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. But if the p-value is less than the significance level (e.g., p < 0.05), then an investigator may conclude that the observed effect actually reflects the characteristics of the population rather than just sampling error. Investigators may then report that the result attains statistical significance, thereby rejecting the null hypothesis.

The present-day concept of statistical significance originated with Ronald Fisher when he developed statistical hypothesis testing based on p-values in the early 20th century. It was Jerzy Neyman and Egon Pearson who later recommended that the significance level be set ahead of time, prior to any data collection.

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

History

Main article: History of statistics

The concept of statistical significance was originated by Ronald Fisher when he developed statistical hypothesis testing, which he described as "tests of significance", in his 1925 publication Statistical Methods for Research Workers. Fisher suggested a probability of one in twenty (0.05) as a convenient cutoff level to reject the null hypothesis. In their 1933 paper, Jerzy Neyman and Egon Pearson recommended that the significance level (e.g. 0.05), which they called α, be set ahead of time, prior to any data collection.

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

Role in statistical hypothesis testing

Main articles: Statistical hypothesis testing, Null hypothesis, p-value, and Type I and type II errors

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, where it is used to determine whether a null hypothesis should be rejected or retained. A null hypothesis is the general or default statement that nothing happened or changed. For a null hypothesis to be rejected as false, the result has to be identified as being statistically significant, i.e. unlikely to have occurred due to sampling error alone.

To determine whether a result is statistically significant, a researcher would have to calculate a p-value, which is the probability of observing an effect given that the null hypothesis is true. The null hypothesis is rejected if the p-value is less than the significance or α level. The α level is the probability of rejecting the null hypothesis given that it is true (type I error) and is most often set at 0.05 (5%). If the α level is 0.05, then the conditional probability of a type I error, given that the null hypothesis is true, is 5%. Then a statistically significant result is one in which the observed p-value is less than 5%, which is formally written as p < 0.05.

If the α level is set at 0.05, it means that the rejection region comprises 5% of the sampling distribution. 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. One-tailed tests are more powerful than two-tailed tests, as a null hypothesis can be rejected with a less extreme result. Nevertheless, the use of a one-tailed test is dependent on whether the research question specifies a direction such as whether a group of objects is heavier or the performance of students on an assessment is better.

Stringent significance thresholds in specific fields

Main articles: Standard deviation and 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. 5σ). 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.

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

Effect size

Main article: Effect size

Researchers focusing solely on whether their results are statistically significant might report findings that are not substantive and not replicable. To gauge the research significance of their result, researchers are therefore 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.

See also

• A/B testing
• ABX test
• Confidence level, the complement of the significance level
• Effect size
• Fisher's method for combining independent tests of significance
• Look-elsewhere effect
• Multiple comparisons problem
• Texas sharpshooter fallacy (gives examples of tests where the significance level was set too high)
• Reasonable doubt
• Sample size
• Statistical hypothesis testing

Further reading

• Ziliak, Stephen and Deirdre McCloskey (2008), 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: (compiled by Ziliak)
• Chow, Siu L., (1996). 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.
• Kline, Rex, (2004). Beyond Significance Testing: Reforming Data Analysis Methods in Behavioral Research Washington, DC: American Psychological Association.
• Nuzzo, Regina (2014). Scientific method: Statistical errors. Nature Vol. 506, p. 105-152 (open access). Highlights common misunderstandings about the p value.
• Cohen, Joseph (1994). [1]. The earth is round (p<.05). American Psychologist. Vol 49, p. 997-1003. Reviews problems with null hypothesis statistical testing.

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" (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.