Statistical significance

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 ${\displaystyle �}$, 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, ${\displaystyle �}$, 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 ${\displaystyle �\le �}$. 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[edit]

Main article: History of statistics

Statistical significance dates to the 1700s, 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 ${\displaystyle �}$. They recommended that ${\displaystyle �}$ 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[edit]

The significance level ${\displaystyle �}$ is the threshold for ${\displaystyle �}$ below which the null hypothesis is rejected even though by assumption it were true, and something else is going on. This means that ${\displaystyle �}$ 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[edit]

Main articles: Statistical hypothesis testing, Null hypothesis, Alternative 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. 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.^[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 ${\displaystyle �}$.

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, ${\displaystyle �}$. ${\displaystyle �}$ 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 ${\displaystyle �}$ 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[edit]

Further information: 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 (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.