View source for T-statistic

{{Short description|Ratio in statistics}}
{{DISPLAYTITLE:''t''-statistic}}
In [[Statistics|statistics]], the '''''t''-statistic''' is the ratio of the departure of the estimated value of a parameter from its hypothesized value to its standard error. It is used in [[Statistical hypothesis testing|hypothesis testing]] via [[Student's t-test|Student's ''t''-test]]. The ''t''-statistic is used in a ''t''-test to determine whether to support or reject the null hypothesis. It is very similar to the [[Standard score|z-score]] but with the difference that ''t''-statistic is used when the sample size is small or the population standard deviation is unknown. For example, the ''t''-statistic is used in estimating the population mean from a [[Sampling distribution|sampling distribution]] of [[Sample mean|sample mean]]s if the population [[Standard deviation|standard deviation]] is unknown. It is also used along with [[P-value|p-value]] when running hypothesis tests where the p-value tells us what the odds are of the results to have happened.

==Definition and features==
Let <math>\hat\beta</math> be an [[Estimator|estimator]] of parameter ''β'' in some [[Statistical model|statistical model]]. Then a ''t''-statistic for this parameter is any quantity of the form
: <math>
    t_{\hat{\beta}} = \frac{\hat\beta - \beta_0}{\operatorname{s.e.}(\hat\beta)},
  </math>
where ''β''<sub>0</sub> is a non-random, known constant, which may or may not match the actual unknown parameter value ''β'', and <math>\operatorname{s.e.}(\hat\beta)</math> is the standard error of the estimator <math>\hat\beta</math> for ''β''.

By default, statistical packages report ''t''-statistic with {{nowrap|''β''<sub>0</sub> {{=}} 0}} (these ''t''-statistics are used to test the significance of corresponding regressor). However, when ''t''-statistic is needed to test the hypothesis of the form {{nowrap|''H''<sub>0</sub>: ''β'' {{=}} ''β''<sub>0</sub>}}, then a non-zero ''β''<sub>0</sub> may be used.

If <math>\hat\beta</math> is an [[Ordinary least squares|ordinary least squares]] estimator in the classical linear regression model (that is, with [[Normal distribution|normally distributed]] and [[Homoscedasticity|homoscedastic]] error terms), and if the true value of the parameter ''β'' is equal to ''β''<sub>0</sub>, then the [[Sampling distribution|sampling distribution]] of the ''t''-statistic is the [[Student's t-distribution|Student's ''t''-distribution]] with {{nowrap|(''n'' − ''k'')}} degrees of freedom, where ''n'' is the number of observations, and ''k'' is the number of regressors (including the intercept){{Citation needed|date=November 2018}}.

In the majority of models, the estimator <math>\hat\beta</math> is [[Consistent estimator|consistent]] for ''β'' and is distributed asymptotically normally. If the true value of the parameter ''β'' is equal to ''β''<sub>0</sub>, and the quantity <math>\operatorname{s.e.}(\hat\beta)</math> correctly estimates the asymptotic variance of this estimator, then the ''t''-statistic will asymptotically have the standard normal distribution.

In some models the distribution of the ''t''-statistic is different from the normal distribution, even asymptotically. For example, when a [[Time series|time series]] with a [[Unit root|unit root]] is regressed in the [[Augmented Dickey–Fuller test|augmented Dickey–Fuller test]], the test ''t''-statistic will asymptotically have one of the Dickey–Fuller distributions (depending on the test setting).

==Use==

Most frequently, ''t'' statistics are used in Student's ''t''-tests, a form of [[Statistical hypothesis testing|statistical hypothesis testing]], and in the computation of certain [[Confidence interval|confidence interval]]s.

The key property of the ''t'' statistic is that it is a [[Pivotal quantity|pivotal quantity]] – while defined in terms of the sample mean, its sampling distribution does not depend on the population parameters, and thus it can be used regardless of what these may be.

One can also divide a residual by the sample [[Standard deviation|standard deviation]]:
:<math> g(x,X) = \frac{x - \overline{X}}{s} </math>
to compute an estimate for the number of standard deviations a given sample is from the mean, as a sample version of a z-score, the z-score requiring the population parameters.

===Prediction===
Given a normal distribution <math>N(\mu, \sigma^2)</math> with unknown mean and variance, the ''t''-statistic of a future observation <math>X_{n+1},</math> after one has made ''n'' observations, is an [[Ancillary statistic|ancillary statistic]] – a pivotal quantity (does not depend on the values of ''μ'' and ''σ''<sup>2</sup>) that is a statistic (computed from observations). This allows one to compute a frequentist [[Prediction interval|prediction interval]] (a predictive [[Confidence interval|confidence interval]]), via the following t-distribution:
: <math>\frac{X_{n+1} - \overline{X}_n}{s_n \sqrt{1 + n^{-1}}} \sim T^{n-1}.</math>
Solving for <math>X_{n+1}</math> yields the prediction distribution
: <math>\overline{X}_n + s_n \sqrt{1 + n^{-1}} \cdot T^{n-1},</math>
from which one may compute predictive confidence intervals – given a probability ''p'', one may compute intervals such that 100''p''% of the time, the next observation <math>X_{n+1}</math> will fall in that interval.

==History==
{{further|Student's t-test}}
The term "''t''-statistic" is abbreviated from "hypothesis test statistic".<ref>{{Cite book|url=https://books.google.com/books?id=kiToDwAAQBAJ&pg=PA397|title=The Microbiome in Health and Disease|date=2020-05-29|publisher=Academic Press|isbn=978-0-12-820001-8|pages=397|language=en}}</ref>{{Citation needed|date=February 2011}} In statistics, the t-distribution was first derived as a [[Posterior probability|posterior distribution]] in 1876 by Helmert<ref>{{Citation|last=Szabó|first=István|chapter=Systeme aus einer endlichen Anzahl starrer Körper|date=2003|title=Einführung in die Technische Mechanik|pages=196–199|publisher=Springer Berlin Heidelberg|doi=10.1007/978-3-642-61925-0_16|isbn=978-3-540-13293-6}}</ref><ref>{{Cite journal|last=Schlyvitch|first=B.|s2cid=27311567|date=October 1937|title=Untersuchungen über den anastomotischen Kanal zwischen der Arteria coeliaca und mesenterica superior und damit in Zusammenhang stehende Fragen|journal=Zeitschrift für Anatomie und Entwicklungsgeschichte|volume=107|issue=6|pages=709–737|doi=10.1007/bf02118337|issn=0340-2061}}</ref><ref>{{Cite journal|last=Helmert|date=1876|title=Die Genauigkeit der Formel von Peters zur Berechnung des wahrscheinlichen Beobachtungsfehlers directer Beobachtungen gleicher Genauigkeit|journal=Astronomische Nachrichten|language=de|volume=88|issue=8–9|pages=113–131|doi=10.1002/asna.18760880802|bibcode=1876AN.....88..113H|url=https://zenodo.org/record/1424695}}</ref> and Lüroth.<ref>{{Cite journal|last=Lüroth|first=J.|date=1876|title=Vergleichung von zwei Werthen des wahrscheinlichen Fehlers|journal=Astronomische Nachrichten|language=de|volume=87|issue=14|pages=209–220|doi=10.1002/asna.18760871402|bibcode=1876AN.....87..209L|url=https://zenodo.org/record/1424693}}</ref><ref>{{cite journal | journal = [[Biometrika]]|doi=10.1093/biomet/83.4.891|mr=1766040|title=Studies in the history of probability and statistics XLIV. A forerunner of the t-distribution|year=1996|last1=Pfanzagl|first1=J.|volume=83|issue=4|pages=891–898}}</ref><ref>{{Cite journal|last=Sheynin|first=Oscar|s2cid=121241599|date=1995|title=Helmert's work in the theory of errors|journal=Archive for History of Exact Sciences|language=en|volume=49|issue=1|pages=73–104|doi=10.1007/BF00374700|issn=0003-9519}}</ref> The t-distribution also appeared in a more general form as Pearson Type [[Pearson distribution|IV]] distribution in [[Biography:Karl Pearson|Karl Pearson]]'s 1895 paper.<ref>{{cite journal|doi=10.1098/rsta.1895.0010|issn=1364-503X|title=X. Contributions to the mathematical theory of evolution.—II. Skew variation in homogeneous material|journal=Philosophical Transactions of the Royal Society of London A|year=1895|volume=186|pages=343–414|bibcode=1895RSPTA.186..343P |doi-access=free|last1=Pearson |first1=Karl }}</ref> However, the T-Distribution, also known as [[Student's t-distribution|Student's T Distribution]] gets its name from [[Biography:William Sealy Gosset|William Sealy Gosset]] who was first to publish the result in English in his 1908 paper titled "The Probable Error of a Mean" (in [[Biometrika]]) using his pseudonym "Student"<ref>{{cite journal|author="Student" ([[Biography:William Sealy Gosset|William Sealy Gosset]]) |title=The Probable Error of a Mean|journal=[[Biometrika]]|year=1908|volume=6|issue=1|pages=1–25|doi=10.1093/biomet/6.1.1|hdl=10338.dmlcz/143545|jstor=2331554}}</ref><ref>{{Cite web|url=https://www.tdistributiontable.com|title=T Table {{!}} History of T Table, Etymology, one-tail T Table, two-tail T Table and T-statistic}}</ref> because his employer preferred their staff to use pen names when publishing scientific papers instead of their real name, so he used the name "Student" to hide his identity.<ref>{{cite journal|doi=10.1126/science.351.6280.1406|pmid=27013722|title=Pseudonymous fame|year=2016|last1=Wendl|first1=M. C.|journal=Science|volume=351|issue=6280|page=1406}}</ref> Gosset worked at the Guinness Brewery in [[Place:Dublin|Dublin]], Ireland, and was interested in the problems of small samples – for example, the chemical properties of barley where sample sizes might be as few as 3. Hence a second version of the etymology of the term Student is that Guinness did not want their competitors to know that they were using the t-test to determine the quality of raw material. Although it was William Gosset after whom the term "Student" is penned, it was actually through the work of [[Biography:Ronald Fisher|Ronald Fisher]] that the distribution became well known as "Student's distribution"<ref>{{Cite journal |doi=10.15438/rr.v4i2.72 |title=Subgroup Analysis of Topical Tranexamic Acid in Total Knee Arthroplasty |year=2014 |last1=Tuttle |first1=Md |last2=Anazonwu, Bs |first2=Walter |last3=Rubin, Md |first3=Lee |journal=Reconstructive Review |volume=4 |issue=2 |pages=37–41 |doi-access=free }}</ref><ref>{{Cite book|last=Walpole, Ronald E.|title=Probability & statistics for engineers & scientists|date=2006|publisher=Pearson|others=Myers, H. Raymond.|isbn=81-7758-404-9|edition= 7th|location=New Delhi|oclc=818811849}}</ref> and "[[Student's t-test]]"

==Related concepts==
*''z''-score (standardization): If the population parameters are known, then rather than computing the t-statistic, one can compute the z-score; analogously, rather than using a ''t''-test, one uses a [[Z-test|''z''-test]]. This is rare outside of standardized testing.
*[[Studentized residual]]: In [[Regression analysis|regression analysis]], the standard errors of the estimators at different data points vary (compare the middle versus endpoints of a [[Simple linear regression|simple linear regression]]), and thus one must divide the different residuals by different estimates for the error, yielding what are called [[Studentized residual|studentized residual]]s.

==See also==
* [[F-test|''F''-test]]
* [[Hotelling's t-squared statistic|''t''<sup>2</sup>-statistic]]
*[[Student's t-distribution|Student's T-Distribution]]
*[[Student's t-test]]
*[[Statistical hypothesis testing|Hypothesis testing]]
*[[Folded-t and half-t distributions]]
*[[Chi-squared distribution]]

==References==
{{Reflist}}


{{DEFAULTSORT:T-statistic}}
[[Category:Statistical ratios]]
[[Category:Parametric statistics]]
[[Category:Normal distribution]]

{{Sourceattribution|T-statistic}}