Student's t-test

Student's t-test is a statistical test used to test whether the difference between the response of two groups is statistically significant or not. It is any statistical hypothesis test in which the test statistic follows a Student's t-distribution under the null hypothesis. It is most commonly applied when the test statistic would follow a normal distribution if the value of a scaling term in the test statistic were known (typically, the scaling term is unknown and is therefore a nuisance parameter). When the scaling term is estimated based on the data, the test statistic, under certain conditions, follows a Student's t distribution. The t-test's most common application is to test whether the means of two populations are significantly different. In many cases, a Z-test will yield very similar results to a t-test because the latter converges to the former as the size of the dataset increases.

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 in 1908 in the scientific journal Biometrika using the pseudonym "Student" because his employer preferred staff to use pen names when publishing scientific papers. 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 with small sample sizes. 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" was coined, it was actually through the work of Ronald Fisher that the distribution became well known as "Student's distribution" and "Student's t-test".

Gosset devised the t-test as an economical way to monitor the quality of stout. The t-test work was submitted to and accepted in the journal Biometrika and published in 1908.

Guinness had a policy of allowing technical staff leave for study (so-called "study leave"), which Gosset used during the first two terms of the 1906-1907 academic year in Professor Karl Pearson's Biometric Laboratory at University College London. Gosset's identity was then known to fellow statisticians and to editor-in-chief Karl Pearson.

A one-sample Student's t-test is a location test of whether the mean of a population has a value specified in a null hypothesis. In testing the null hypothesis that the population mean is equal to a specified value μ₀, one uses the statistic

\(t = \frac{\bar{x} - \mu_0}{s/\sqrt{n}},\)

where \(\bar x\) is the sample mean, s is the sample standard deviation and n is the sample size. The degrees of freedom used in this test are n − 1. Although the parent population does not need to be normally distributed, the distribution of the population of sample means \(\bar x\) is assumed to be normal.

By the central limit theorem, if the observations are independent and the second moment exists, then \(t\) will be approximately normal \(\mathcal{N}(0, 1)\). This is only an approximation as the central limit theorem would apply to t if s was the actual standard deviation of x, while it is the sample standard deviation as the actual standard deviation is not generally known. Therefore, t asymptotically follows a Student's t-distribution.

A two-sample location test of the null hypothesis that the means of two populations are equal. All such tests are usually called Student's t-tests, though strictly speaking that name should only be used if the variances of the two populations are also assumed to be equal; the form of the test used when this assumption is dropped is sometimes called Welch's t-test. These tests are often referred to as unpaired or independent samples t-tests, as they are typically applied when the statistical units underlying the two samples being compared are non-overlapping.

Two-sample t-tests for a difference in means involve independent samples (unpaired samples) or paired samples. Paired t-tests are a form of blocking, and have greater power (probability of avoiding a type II error, also known as a false negative) than unpaired tests when the paired units are similar with respect to "noise factors" (see confounder) that are independent of membership in the two groups being compared. In a different context, paired t-tests can be used to reduce the effects of confounding factors in an observational study.

Most test statistics have the form t = Z/s, where Z and s are functions of the data.

Z may be sensitive to the alternative hypothesis (i.e., its magnitude tends to be larger when the alternative hypothesis is true), whereas s is a scaling parameter that allows the distribution of t to be determined.

As an example, in the one-sample t-test

\(t = \frac{Z}{s} = \frac{\bar{X} - \mu}{\hat\sigma / \sqrt{n}},\)

where \(\bar{X}\) is the sample mean from a sample X₁, X₂, …, X_n, of size n, s is the standard error of the mean, \(\hat\sigma = \sqrt{\frac{1}{n-1} \sum_i (X_i - \bar X)^2}\) is the estimate of the standard deviation of the population, and μ is the population mean.

The assumptions underlying a t-test in the simplest form above are that:

• X follows a normal distribution with mean μ and variance σ/n.
• s(n − 1)/σ follows a χ distribution with n − 1 degrees of freedom. This assumption is met when the observations used for estimating s come from a normal distribution (and i.i.d. for each group).
• Z and s are independent.

In the t-test comparing the means of two independent samples, the following assumptions should be met:

• The means of the two populations being compared should approximately follow normal distributions. Under weak assumptions, this follows in large samples from the central limit theorem, even when the distribution of observations in each group is non-normal.
• If using Student's original definition of the t-test, the two populations being compared should have the same variance (testable using F-test, Levene's test, Bartlett's test, or the Brown-Forsythe test; or assessable graphically using a Q-Q plot). If the sample sizes in the two groups being compared are equal, Student's original t-test is highly robust to the presence of unequal variances. Welch's t-test is insensitive to equality of the variances regardless of whether the sample sizes are similar.
• The data used to carry out the test should either be sampled independently from the two populations being compared or be fully paired. This is in general not testable from the data, but if the data are known to be dependent (e.g. paired by test design), a dependent test has to be applied. For partially paired data, the classical independent t-tests may give invalid results as the test statistic might not follow a t distribution, while the dependent t-test is sub-optimal as it discards the unpaired data.

Most two-sample t-tests are robust to all but large deviations from the assumptions.

\(\sqrt{n}(\bar{X} - \mu) \xrightarrow{d} N(0, \sigma^2)\) as per the Central limit theorem,

\(s^2 \xrightarrow{p} \sigma^2\) as per the law of large numbers,

\(\therefore \frac{\sqrt{n}(\bar{X} - \mu)}{s} \xrightarrow{d} N(0, 1)\).

Condensed: the full section is in Wikipedia.

Explicit expressions that can be used to carry out various t-tests are given below. In each case, the formula for a test statistic that either exactly follows or closely approximates a t-distribution under the null hypothesis is given. Also, the appropriate degrees of freedom are given in each case. Each of these statistics can be used to carry out either a one-tailed or two-tailed test.

Once the t value and degrees of freedom are determined, a p-value can be found using a table of values from Student's t-distribution. If the calculated p-value is below the threshold chosen for statistical significance (usually the 0.10, the 0.05, or 0.01 level), then the null hypothesis is rejected in favor of the alternative hypothesis.

Suppose one is fitting the model

\(Y = \alpha + \beta x + \varepsilon,\)

where x is known, α and β are unknown, ε is a normally distributed random variable with mean 0 and unknown variance σ, and Y is the outcome of interest. We want to test the null hypothesis that the slope β is equal to some specified value β₀ (often taken to be 0, in which case the null hypothesis is that x and y are uncorrelated).

Let

\(\begin{align} \hat\alpha, \hat\beta &= \text{least-squares estimators}, \\ SE_{\hat\alpha}, SE_{\hat\beta} &= \text{the standard errors of least-squares estimators}. \end{align}\)

Then

\(t_\text{score} = \frac{\hat\beta - \beta_0}{ SE_{\hat\beta} } \sim \mathcal{T}_{n-2}\)

has a t-distribution with n − 2 degrees of freedom if the null hypothesis is true. The standard error of the slope coefficient:

\(SE_{\hat\beta} = \frac{\sqrt\frac{1}{n - 2}\sum_{i=1}^n (y_i - \hat y_i)^2}}{\sqrt{\displaystyle \sum_{i=1}^n (x_i - \bar{x})^2}\)

can be written in terms of the residuals. Let

\(\begin{align} \hat\varepsilon_i &= y_i - \hat y_i = y_i - (\hat\alpha + \hat\beta x_i) = \text{residuals} = \text{estimated errors}, \\ \text{SSR} &= \sum_{i=1}^n {\hat\varepsilon_i}^2 = \text{sum of squares of residuals}. \end{align}\)

Then t_score is given by

\(t_\text{score} = \frac{(\hat\beta - \beta_0) \sqrt{n-2}}{\sqrt{\frac{SSR}{\sum_{i=1}^n (x_i - \bar{x})^2}}}.\)

\(t_\text{score} = \frac{r\sqrt{n - 2}}{\sqrt{1 - r^2}},\)

\(t_\text{score,intercept} = \frac{\alpha}{\beta} \frac{t_\text{score,slope}}{\sqrt{s_\text{x}^2 + \bar{x}^2}},\)

Condensed: the full section is in Wikipedia.

The test deals with the famous Behrens-Fisher problem, i.e., comparing the difference between the means of two normally distributed populations when the variances of the two populations are not assumed to be equal, based on two independent samples.

The test is developed as an exact test that allows for unequal sample sizes and unequal variances of two populations. The exact property still holds even with extremely small and unbalanced sample sizes (e.g. \(\ m \equiv n_\mathsf{X} = 50\\) vs. \(\ n \equiv n_\mathsf{Y} = 5\\)).

The statistic to test whether the means are different can be calculated as follows:

Let \(\ X = \left[\ X_1, X_2, \ldots, X_m\ \right]^\top\\) and \(\ Y = \left[\ Y_1, Y_2, \ldots, Y_n\ \right]^\top\\) be the i.i.d. sample vectors (for \(\ m \ge n\\)) from \(\ \mathsf{Norm}\left(\ \mu_\mathsf{X},\ \sigma_\mathsf{X}^2\ \right)\\) and \(\ \mathsf{Norm}\left(\ \mu_\mathsf{Y},\ \sigma_\mathsf{Y}^2\ \right)\\) separately.

Let \(\ (P^\top)_{n\times n}\\) be an \(n\times n\) orthogonal matrix whose elements of the first row are all \(\ \tfrac{ 1 }{ \sqrt{ n\ } }\ ,\) similarly, let \(\ (Q^\top)_{n\times m}\\) be the first \(\ n\\) rows of an \(\ m\times m\\) orthogonal matrix (whose elements of the first row are all \(\ \tfrac{ 1 }{ \sqrt{ m\ } }\\)).

Then \(\ Z \equiv \frac{\ \left( Q^\top \right)_{n\times m}\ X\ }{ \sqrt{ m\ } }\ -\ \frac{\ \left( P^\top \right)_{n\times n}\ Y\ }{ \sqrt{ n\ } }\\) is an n-dimensional normal random vector:

\(Z ~\sim~ \mathsf{Norm}\left(\ \left[\ \mu_\mathsf{X} - \mu_\mathsf{Y},\ 0,\ 0,\ \ldots,\ 0\ \right]^\top\ ,\ \left( \frac{\ \sigma_\mathsf{X}^2\ }{ m } + \frac{\ \sigma_\mathsf{Y}^2\ }{ n }\right)\ I_n\ \right) ~.\)

From the above distribution we see that the first element of the vector Z is

\(Z_1 = \bar X - \bar Y = \frac{ 1 }{\ m\ } \sum_{i=1}^m\ X_i - \frac{ 1 }{\ n\ } \sum_{j=1}^n\ Y_j\ ,\)

\(Z_1 - \left( \mu_\mathsf{X} - \mu_\mathsf{Y} \right) ~\sim~ \mathsf{Norm}\left(\ 0,\ \frac{\ \sigma_\mathsf{X}^2\ }{ m } + \frac{\ \sigma_\mathsf{Y}^2\ }{ n }\ \right)\ ,\)

\(\frac{\ \sum_{i=2}^n Z^2_i\ }{\ n - 1\ } ~\sim~ \frac{\ \chi^2_{n - 1}\ }{\ n - 1\ } \times\left( \frac{\ \sigma_\mathsf{X}^2\ }{ m }+\frac{\ \sigma_\mathsf{Y}^2\ }{ n } \right)\)

\(Z_1 - \left( \mu_\mathsf{X} - \mu_\mathsf{Y} \right) \quad \perp \quad \sum_{i=2}^n Z^2_i\ ,\)

\(T_\mathsf{e} ~\equiv~ \frac{\ Z_1 - \left( \mu_\mathsf{X} - \mu_\mathsf{Y} \right)\ }{\ \sqrt{ \left( \sum_{i=2}^{n} Z^2_i \right) /\left( n - 1 \right)\ }\ } ~\sim~ t_{n - 1} ~.\)

Condensed: the full section is in Wikipedia.

This test is used when the samples are dependent; that is, when there is only one sample that has been tested twice (repeated measures) or when there are two samples that have been matched or "paired". This is an example of a paired difference test. The t statistic is calculated as

\(t = \frac{\bar{X}_D - \mu_0}{s_D/\sqrt n},\)

where \(\bar{X}_D\) and \(s_D\) are the average and standard deviation of the differences between all pairs. The pairs are e.g. either one person's pre-test and post-test scores or between-pairs of persons matched into meaningful groups (for instance, drawn from the same family or age group: see table). The constant μ₀ is zero if we want to test whether the average of the difference is significantly different. The degree of freedom used is n − 1, where n represents the number of pairs.

Let A₁ denote a set obtained by drawing a random sample of six measurements:

\(A_1=\{30.02,\ 29.99,\ 30.11,\ 29.97,\ 30.01,\ 29.99\}\)

and let A₂ denote a second set obtained similarly:

\(A_2=\{29.89,\ 29.93,\ 29.72,\ 29.98,\ 30.02,\ 29.98\}\)

These could be, for example, the weights of screws that were manufactured by two different machines.

We will carry out tests of the null hypothesis that the means of the populations from which the two samples were taken are equal.

The difference between the two sample means, each denoted by X_i, which appears in the numerator for all the two-sample testing approaches discussed above, is

\(\bar{X}_1 - \bar{X}_2 = 0.095.\)

The sample standard deviations for the two samples are approximately 0.05 and 0.11, respectively. For such small samples, a test of equality between the two population variances would not be very powerful. Since the sample sizes are equal, the two forms of the two-sample t-test will perform similarly in this example.

If the approach for unequal variances (discussed above) is followed, the results are

\(\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}} \approx 0.04849\)

and the degrees of freedom

\(\text{d.f.} \approx 7.031.\)

The test statistic is approximately 1.959, which gives a two-tailed test p-value of 0.09077.

If the approach for equal variances (discussed above) is followed, the results are

\(s_p \approx 0.08399\)

and the degrees of freedom

\(\text{d.f.} = 10.\)

The test statistic is approximately equal to 1.959, which gives a two-tailed p-value of 0.07857.

The t-test provides an exact test for the equality of the means of two i.i.d. normal populations with unknown, but equal, variances. (Welch's t-test is a nearly exact test for the case where the data are normal but the variances may differ.) For moderately large samples and a one tailed test, the t-test is relatively robust to moderate violations of the normality assumption. In large enough samples, the t-test asymptotically approaches the z-test, and becomes robust even to large deviations from normality.

If the data are substantially non-normal and the sample size is small, the t-test can give misleading results. See Location test for Gaussian scale mixture distributions for some theory related to one particular family of non-normal distributions.

When the normality assumption does not hold, a non-parametric alternative to the t-test may have better statistical power. However, when data are non-normal with differing variances between groups, a t-test may have better type-1 error control than some non-parametric alternatives. Furthermore, non-parametric methods, such as the Mann-Whitney U test discussed below, typically do not test for a difference of means, so should be used carefully if a difference of means is of primary scientific interest. For example, Mann-Whitney U test will keep the type 1 error at the desired level alpha if both groups have the same distribution. It will also have power in detecting an alternative by which group B has the same distribution as A but after some shift by a constant (in which case there would indeed be a difference in the means of the two groups). However, there could be cases where group A and B will have different distributions but with the same means (such as two distributions, one with positive skewness and the other with a negative one, but shifted so to have the same means). In such cases, MW could have more than alpha level power in rejecting the Null hypothesis but attributing the interpretation of difference in means to such a result would be incorrect.

In the presence of an outlier, the t-test is not robust. For example, for two independent samples when the data distributions are asymmetric (that is, the distributions are skewed) or the distributions have large tails, then the Wilcoxon rank-sum test (also known as the Mann-Whitney U test) can have three to four times higher power than the t-test. The nonparametric counterpart to the paired samples t-test is the Wilcoxon signed-rank test for paired samples. For a discussion on choosing between the t-test and nonparametric alternatives, see Lumley, et al. (2002).

One-way analysis of variance (ANOVA) generalizes the two-sample t-test when the data belong to more than two groups.

When both paired observations and independent observations are present in the two sample design, assuming data are missing completely at random (MCAR), the paired observations or independent observations may be discarded in order to proceed with the standard tests above. Alternatively making use of all of the available data, assuming normality and MCAR, the generalized partially overlapping samples t-test could be used.