The main idea...
Hotelling's T^{2} test (Hotelling, 1931) is the multivariate generlisation of the Student's t test; however, objects subject to a Hotelling's T^{2} should be described by multiple response variables. A onesample Hotelling's T^{2} test can be used to test if a set of objects (which should be a sample of a single statistical population) has a mean equal to a hypothetical mean (Figure 1a). A twosample Hotelling's T^{2} test may be used to test for significant differences between the mean vectors (multivariate means) of two multivariate data sets (Figure 1b).
Null hypothesis (onesample) 
The (multivariate) vector of means of a group of objects is equal to a hypothetical vector of means. 
Null hypothesis (twosample) 
The (multivariate) vectors of means of two groups of objects are equal. 
For testing more than two groups, consider multivariate analysis of variance (MANOVA).


Figure 1: Schematic illustrating the logic of a one and twoway Hotelling's T^{2} test in a simple, twodimensional space.
Linear combinations of the original variables are used to build a synthetic variable that best separates either a group from a hypothetical mean (μ_{0}; a), or two groups of multivariatenormal data (b). In other words, the maximum possible T^{2} value is found. Points indicate multivariate means of each population and circles indicate multivariate dispersion. The significance of this separation may be tested by comparison of transformed T^{2} values to an Fdistribution.

Assumptions
 The variables of each data set follow a multivariate normal distribution. Each variable may be tested for univariate normality.
 The objects have been independently sampled.
 In a twosampled test, the two data sets being tested have (near) equivalent variancecovariance matrices.Bartlett's test may be used to evaluate if this assumption holds.
 Each data set describes one population with one multivariate mean. No subpopulations exist within each data set.
Warnings
 Hotelling's T^{2} test is sensitive to violations of the assumption of independently sampled objects. Any interdependence, and hence redundancy, will reduce the power of the test by reducing the effect sample size. Both timeseries data and data sampled along some nonrandom spatial range may be autocorrelated which may mean objects are not independent. Test for temporal autocorrelation prior to conducting Hotelling's T^{2} test.
 Twosample Hotelling's T^{2} tests are sensitive to violations of the assumption of equal variances and covariances. This is especially true if sample sizes differ between the two data sets being tested.
Implementations
 R
 The "Hotelling" package includes Hotelling's T^{2} test and a number of useful variants. Further, JamesStein shrinkage estimators may be used in computing Hotelling's T^{2} test. Permutationbased tests are also available along with plotting functions.
References
