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statistical hypothesis testing A statistical hypothesis test is a method of statistical inference used to decide whether the data provide sufficient evidence to reject a particular hypothesis. A statistical hypothesis test typically involves a calculation of a test statistic. T ...
, a result has statistical significance when a result at least as "extreme" would be very infrequent if the
null hypothesis The null hypothesis (often denoted ''H''0) is the claim in scientific research that the effect being studied does not exist. The null hypothesis can also be described as the hypothesis in which no relationship exists between two sets of data o ...
were true. 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; 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. The result is said to be ''statistically significant'', by the standards of the study, when p \le \alpha. The significance level for a study is chosen before data collection, and is typically set to 5% or much lower—depending on the field of study. In any
experiment An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs whe ...
or
observation Observation in the natural sciences is an act or instance of noticing or perceiving and the acquisition of information from a primary source. In living beings, observation employs the senses. In science, observation can also involve the percep ...
that involves drawing a sample from a
population Population is a set of humans or other organisms in a given region or area. Governments conduct a census to quantify the resident population size within a given jurisdiction. The term is also applied to non-human animals, microorganisms, and pl ...
, there is always the possibility that an observed effect would have occurred due to
sampling error In statistics, sampling errors are incurred when the statistical characteristics of a population are estimated from a subset, or sample, of that population. Since the sample does not include all members of the population, statistics of the sample ...
alone. 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, thereby rejecting the null hypothesis. 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. For example, the term
clinical significance In medicine and psychology, clinical significance is the practical importance of a treatment effect—whether it has a real genuine, palpable, noticeable effect on daily life. Types of significance Statistical significance Statistical significanc ...
refers to the practical importance of a treatment effect.


History

Statistical significance dates to the 18th century, in the work of John Arbuthnot and
Pierre-Simon Laplace Pierre-Simon, Marquis de Laplace (; ; 23 March 1749 – 5 March 1827) was a French polymath, a scholar whose work has been instrumental in the fields of physics, astronomy, mathematics, engineering, statistics, and philosophy. He summariz ...
, who computed the ''p''-value for the
human sex ratio The human sex ratio is the ratio of males to females in a population in the context of anthropology and demography. In humans, the natural sex ratio at birth is slightly biased towards the male sex. It is estimated to be about 1.05 worldwide or ...
at birth, assuming a null hypothesis of equal probability of male and female births; see for details. In 1925,
Ronald Fisher Sir Ronald Aylmer Fisher (17 February 1890 – 29 July 1962) was a British polymath who was active as a mathematician, statistician, biologist, geneticist, and academic. For his work in statistics, he has been described as "a genius who a ...
advanced the idea of statistical hypothesis testing, which he called "tests of significance", in his publication ''
Statistical Methods for Research Workers ''Statistical Methods for Research Workers'' is a classic book on statistics, written by the statistician R. A. Fisher. It is considered by some to be one of the 20th century's most influential books on statistical methods, together with his '' T ...
''. Fisher suggested a probability of one in twenty (0.05) as a convenient cutoff level to reject the null hypothesis. In a 1933 paper,
Jerzy Neyman Jerzy Spława-Neyman (April 16, 1894 – August 5, 1981; ) was a Polish mathematician and statistician who first introduced the modern concept of a confidence interval into statistical hypothesis testing and, with Egon Pearson, revised Ronald Fis ...
and
Egon Pearson Egon Sharpe Pearson (11 August 1895 – 12 June 1980) was one of three children of Karl Pearson and Maria, née Sharpe, and, like his father, a British statistician. Career Pearson was educated at Winchester College and Trinity College ...
called this cutoff the ''significance level'', which they named \alpha. They recommended that \alpha 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. In his 1956 publication ''Statistical Methods and Scientific Inference,'' he recommended that significance levels be set according to specific circumstances.


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. This is also called
false positive A false positive is an error in binary classification in which a test result incorrectly indicates the presence of a condition (such as a disease when the disease is not present), while a false negative is the opposite error, where the test resu ...
and
type I error Type I error, or a false positive, is the erroneous rejection of a true null hypothesis in statistical hypothesis testing. A type II error, or a false negative, is the erroneous failure in bringing about appropriate rejection of a false null hy ...
. Sometimes researchers talk about the confidence level instead. This is the probability of not rejecting the null hypothesis given that it is true. Confidence levels and confidence intervals were introduced by Neyman in 1937.


Role in statistical hypothesis testing

Statistical significance plays a pivotal role in statistical hypothesis testing. It is used to determine whether the
null hypothesis The null hypothesis (often denoted ''H''0) is the claim in scientific research that the effect being studied does not exist. The null hypothesis can also be described as the hypothesis in which no relationship exists between two sets of data o ...
should be rejected or retained. The null hypothesis is the hypothesis that no effect exists in the phenomenon being studied. 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. 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 Type I error, or a false positive, is the erroneous rejection of a true null hypothesis in statistical hypothesis testing. A type II error, or a false negative, is the erroneous failure in bringing about appropriate rejection of a false null hy ...
). It is usually set at or below 5%. For example, when \alpha is set to 5%, the
conditional probability In probability theory, conditional probability is a measure of the probability of an Event (probability theory), event occurring, given that another event (by assumption, presumption, assertion or evidence) is already known to have occurred. This ...
of a
type I error Type I error, or a false positive, is the erroneous rejection of a true null hypothesis in statistical hypothesis testing. A type II error, or a false negative, is the erroneous failure in bringing about appropriate rejection of a false null hy ...
, ''given that the null hypothesis is true'', is 5%, and a statistically significant result is one where the observed ''p''-value is less than (or equal to) 5%. When drawing data from a sample, this means that the rejection region comprises 5% of the
sampling distribution In statistics, a sampling distribution or finite-sample distribution is the probability distribution of a given random-sample-based statistic. For an arbitrarily large number of samples where each sample, involving multiple observations (data poi ...
. 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 A research question is "a question that a research project sets out to answer". Choosing a research question is an essential element of both quantitative and qualitative research. Investigation will require data collection and analysis, and the ...
or
alternative hypothesis In statistical hypothesis testing, the alternative hypothesis is one of the proposed propositions in the hypothesis test. In general the goal of hypothesis test is to demonstrate that in the given condition, there is sufficient evidence supporting ...
specifies a direction such as whether a group of objects is ''heavier'' or the performance of students on an assessment is ''better''. 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. 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 Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the s ...
and
manufacturing Manufacturing is the creation or production of goods with the help of equipment, labor, machines, tools, and chemical or biological processing or formulation. It is the essence of the secondary sector of the economy. The term may refer ...
, statistical significance is often expressed in multiples of the
standard deviation In statistics, the standard deviation is a measure of the amount of variation of the values of a variable about its Expected value, mean. A low standard Deviation (statistics), deviation indicates that the values tend to be close to the mean ( ...
or sigma (''σ'') of a
normal distribution In probability theory and statistics, a normal distribution or Gaussian distribution is a type of continuous probability distribution for a real-valued random variable. The general form of its probability density function is f(x) = \frac ...
, with significance thresholds set at a much stricter level (for example 5''σ''). For instance, the certainty of the
Higgs boson The Higgs boson, sometimes called the Higgs particle, is an elementary particle in the Standard Model of particle physics produced by the excited state, quantum excitation of the Higgs field, one of the field (physics), fields in particl ...
particle's existence was based on the 5''σ'' 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 are not uncommon—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 and not replicable. 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.


Effect size

Effect size is a measure of a study's practical significance. 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 In statistics, an effect size is a value measuring the strength of the relationship between two variables in a population, or a sample-based estimate of that quantity. It can refer to the value of a statistic calculated from a sample of data, the ...
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 A correlation coefficient is a numerical measure of some type of linear correlation, meaning a statistical relationship between two variables. The variables may be two columns of a given data set of observations, often called a sample, or two c ...
between two variables or its square, and other measures.


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 likelihood that the result was a false positive.


Challenges


Overuse in some journals

Starting in the 2010s, some journals began questioning whether significance testing, and particularly using a threshold of =5%, was being relied on too heavily as the primary measure of validity of a hypothesis. 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, requiring authors to use other measures to evaluate hypotheses and impact. 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." Some statisticians prefer to use alternative measures of evidence, such as likelihood ratios or
Bayes factor The Bayes factor is a ratio of two competing statistical models represented by their evidence, and is used to quantify the support for one model over the other. The models in question can have a common set of parameters, such as a null hypothesis ...
s. Using
Bayesian statistics Bayesian statistics ( or ) is a theory in the field of statistics based on the Bayesian interpretation of probability, where probability expresses a ''degree of belief'' in an event. The degree of belief may be based on prior knowledge about ...
can avoid confidence levels, but also requires making additional assumptions, and may not necessarily improve practice regarding statistical testing. The widespread abuse of statistical significance represents an important topic of research in
metascience Metascience (also known as meta-research) is the use of scientific methodology to study science itself. Metascience seeks to increase the quality of scientific research while reducing inefficiency. It is also known as "research on research" and ...
.


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". 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. Other researchers responded that imposing a more stringent significance threshold would aggravate problems such as
data dredging Data dredging, also known as data snooping or ''p''-hacking is the misuse of data analysis to find patterns in data that can be presented as statistically significant, thus dramatically increasing and understating the risk of false positives. Th ...
; alternative propositions are thus to select and justify flexible ''p''-value thresholds before collecting data, or to interpret ''p''-values as continuous indices, thereby discarding thresholds and statistical significance. 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. In 2019, over 800 statisticians and scientists signed a message calling for the abandonment of the term "statistical significance" in science, and the ASA published a further official statement declaring (page 2):


See also

*
A/B testing A/B testing (also known as bucket testing, split-run testing or split testing) is a user-experience research method. A/B tests consist of a randomized experiment that usually involves two variants (A and B), although the concept can be also exte ...
,
ABX test An ABX test is a method of comparing two choices of sensory stimuli to identify detectable differences between them. A subject is presented with two known samples (sample , the first reference, and sample , the second reference) followed by one un ...
*
Estimation statistics Estimation statistics, or simply estimation, is a data analysis framework that uses a combination of effect sizes, confidence intervals, precision planning, and meta-analysis to plan experiments, analyze data and interpret results. It complement ...
* Fisher's method for combining
independent Independent or Independents may refer to: Arts, entertainment, and media Artist groups * Independents (artist group), a group of modernist painters based in Pennsylvania, United States * Independentes (English: Independents), a Portuguese artist ...
test Test(s), testing, or TEST may refer to: * Test (assessment), an educational assessment intended to measure the respondents' knowledge or other abilities Arts and entertainment * ''Test'' (2013 film), an American film * ''Test'' (2014 film) ...
s of significance * Look-elsewhere effect *
Multiple comparisons problem Multiple comparisons, multiplicity or multiple testing problem occurs in statistics when one considers a set of statistical inferences simultaneously or estimates a subset of parameters selected based on the observed values. The larger the numbe ...
*
Sample size Sample size determination or estimation is the act of choosing the number of observations or replicates to include in a statistical sample. The sample size is an important feature of any empirical study in which the goal is to make inferences abo ...
* Texas sharpshooter fallacy (gives examples of tests where the significance level was set too high)


References


Further reading

* Imbens, Guido W. 2021.
Statistical Significance, ''p''-Values, and the Reporting of Uncertainty
" ''Journal of Economic Perspectives'' 35 (3): 157–74. * Lydia Denworth, "A Significant Problem: Standard scientific methods are under fire. Will anything change?", ''
Scientific American ''Scientific American'', informally abbreviated ''SciAm'' or sometimes ''SA'', is an American popular science magazine. Many scientists, including Albert Einstein and Nikola Tesla, have contributed articles to it, with more than 150 Nobel Pri ...
'', vol. 321, no. 4 (October 2019), pp. 62–67. "The use of ''p'' values for nearly a century ince 1925to determine statistical significance of
experiment An experiment is a procedure carried out to support or refute a hypothesis, or determine the efficacy or likelihood of something previously untried. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs whe ...
al results has contributed to an illusion of
certainty Certainty (also known as epistemic certainty or objective certainty) is the epistemic property of beliefs which a person has no rational grounds for doubting. One standard way of defining epistemic certainty is that a belief is certain if and ...
and o reproducibility crises in many scientific fields. There is growing determination to reform statistical analysis... Some esearcherssuggest changing statistical methods, whereas others would do away with a threshold for defining "significant" results." (p. 63.) * 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 The University of Michigan Press is a university press that is a part of Michigan Publishing at the University of Michigan Library. It publishes 170 new titles each year in the humanities and social sciences. Titles from the press have earn ...
, 2009. . 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, – 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. 150-152 (open access). Highlights common misunderstandings about the p value. *Cohen, Joseph (1994)

. 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. *

" (February 1994): article by Bruce Thompon hosted by the ERIC Clearinghouse on Assessment and Evaluation, Washington, D.C. *

(no date): an article from the Statistical Assessment Service at George Mason University, Washington, D.C. {{DEFAULTSORT:Statistical Significance Statistical hypothesis testing