Let be an estimator of parameter β in some statistical model. Then a t-statistic for this parameter is any quantity of the form where β0 is a non-random, known constant which may or may not match the actual unknown parameter value β, and is the standard error of the estimator for β. By default, statistical packages report t-statistic with . However, when t-statistic is needed to test the hypothesis of the form, then a non-zero β0 may be used. If is an ordinary least squares estimator in the classical linear regression model, and if the true value of the parameter β is equal toβ0, then the sampling distribution of the t-statistic is the Student's t-distribution with degrees of freedom, where n is the number of observations, and k is the number of regressors. In the majority of models, the estimator is consistent forβ and is distributed asymptotically normally. If the true value of the parameter β is equal to β0 and the quantity 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 with a unit root is regressed in the augmented Dickey–Fuller test, the test t-statistic will asymptotically have one of the Dickey–Fuller distributions.
Use
Most frequently, t statistics are used in Student's t-tests, a form of statistical hypothesis testing, and in the computation of certain confidence intervals. The key property of the t statistic is that it is a 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: 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 with unknown mean and variance, the t-statistic of a future observation after one has made n observations, is an ancillary statistic – a pivotal quantity that is a statistic. This allows one to compute a frequentist prediction interval, via the following t-distribution: Solving for yields the prediction distribution from which one may compute predictive confidence intervals – given a probability p, one may compute intervals such that 100p% of the time, the next observation will fall in that interval.
History
The term "t-statistic" is abbreviated from "hypothesis test statistic". In statistics, the t-distribution was first derived as a posterior distribution in 1876 by Helmert and Lüroth. The t-distribution also appeared in a more general form as Pearson Type IV distribution in Karl Pearson's 1895 paper. However, the T-Distribution, also known as Student's T Distribution gets its name from William Sealy Gosset who first published it in English literature in his 1908 paper titled Biometrika using his pseudonym "Student" because his employer preferred staff to use pen names when publishing scientific papers instead of their real name, so he used the name "Student" to hide his identity. Gosset worked at the Guinness Brewery in 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 Ronald Fisher that the distribution became well known as "Student's distribution" and "Student's t-test"
Related concepts
z-score : 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. This is rare outside of standardized testing.
Studentized residual: In regression analysis, the standard errors of the estimators at different data points vary, and thus one must divide the different residuals by different estimates for the error, yielding what are called studentized residuals.
