t-statistic

In 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 hypothesis testing via Student's t-test. For example, it is used in estimating the population mean from a sampling distribution of sample means if the population standard deviation is unknown.

Definition and features

Let ${\displaystyle \scriptstyle {\widehat {\beta }}}$ be an estimator of parameter β in some statistical model. Then a t-statistic for this parameter is any quantity of the form

${\displaystyle t_{\widehat {\beta }}={\frac {{\widehat {\beta }}-\beta _{0}}{\operatorname {s.e.} ({\widehat {\beta }})}}}$

where β0 is a non-random, known constant which may or may not match the actual unknown parameter value β, and ${\displaystyle \operatorname {s.e.} ({\widehat {\beta }})}$ is the standard error of the estimator ${\displaystyle \scriptstyle {\widehat {\beta }}}$ for β.

By default, statistical packages report t-statistic with β0 = 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 H0: β = β0, then a non-zero β0 may be used.

If ${\displaystyle \scriptstyle {\widehat {\beta }}}$ is an ordinary least squares estimator in the classical linear regression model (that is, with normally distributed and homoscedastic error terms), and if the true value of the parameter β is equal to β0, then the sampling distribution of the t-statistic is the Student's t-distribution with (n − k) degrees of freedom, where n is the number of observations, and k is the number of regressors (including the intercept).

In the majority of models, the estimator ${\displaystyle \scriptstyle {\widehat {\beta }}}$ is consistent for β and is distributed asymptotically normally. If the true value of the parameter β is equal to β0 and the quantity ${\displaystyle \scriptstyle \operatorname {s.e.} ({\widehat {\beta }})}$ 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 (depending on the test setting).

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:

${\displaystyle g(x,X)={\frac {x-{\overline {X}}}{s}}}$

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 ${\displaystyle N(\mu ,\sigma ^{2})}$ with unknown mean and variance, the t-statistic of a future observation ${\displaystyle X_{n+1},}$ after one has made n observations, is an ancillary statistic – a pivotal quantity (does not depend on the values of μ and σ2) that is a statistic (computed from observations). This allows one to compute a frequentist prediction interval (a predictive confidence interval), via the following t-distribution:

${\displaystyle {\frac {X_{n+1}-{\overline {X}}_{n}}{s_{n}{\sqrt {1+n^{-1}}}}}\sim T^{n-1}}$

Solving for ${\displaystyle X_{n+1}}$ yields the prediction distribution

${\displaystyle {\overline {X}}_{n}+s_{n}{\sqrt {1+n^{-1}}}\cdot T^{n-1}}$

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 ${\displaystyle X_{n+1}}$ will fall in that interval.

History

The term "t-statistic" is abbreviated from "hypothesis test statistic", while "Student" was the pen name of William Sealy Gosset, who introduced the t-statistic and t-test in 1908, while working for the Guinness brewery in Dublin, Ireland.