Hotelling's T2 test is the multivariate generlisation of the Student's t test. A one-sample Hotelling's T2 test can be used to test if a set of vectors of data (which should be a sample of a single statistical population) has a mean equal to a hypothetical mean. A two-sample Hotelling's T2 test may be used to test for significant differences between the mean vectors (multivariate means) of two multivariate data sets are different.

`HotellingsT2Test(x, ...)`# S3 method for default
HotellingsT2Test(x, y = NULL, mu = NULL, test = "f", ...)

# S3 method for formula
HotellingsT2Test(formula, data, subset, na.action, ...)

A list with class 'htest' containing the following components:

- statistic
the value of the T2-statistic. (That is the scaled value of the statistic that has an F distribution or a chisquare distribution depending on the value of

`test`

).- parameter
the degrees of freedom for the T2-statistic.

- p.value
the p-value for the test.

- null.value
the specified hypothesized value of the mean or mean difference depending on whether it was a one-sample test or a two-sample test.

- alternative
a character string with the value 'two.sided'.

- method
a character string indicating what type of test was performed.

- data.name
a character string giving the name of the data (and grouping vector).

- x
a numeric data frame or matrix.

- y
an optional numeric data frame or matrix for the two sample test. If

`NULL`

a one sample test is performed.- mu
a vector indicating the hypothesized value of the mean (or difference in means if a two sample test is performed).

`NULL`

represents origin or no difference between the groups.- test
if

`"f"`

, the decision is based on the F-distribution, if`"chi"`

a chi-squared approximation is used.- formula
a formula of the form

`x ~ g`

where`x`

is a numeric matrix giving the data values and`g`

a factor with two levels giving the corresponding groups.- data
an optional matrix or data frame (or similar: see

`model.frame`

) containing the variables in the formula`formula`

. By default the variables are taken from`environment(formula)`

.- subset
an optional vector specifying a subset of observations to be used.

- na.action
a function which indicates what should happen when the data contain NAs. Defaults to

`getOption("na.action")`

.- ...
further arguments to be passed to or from methods.

Klaus Nordhausen, <klaus.nordhausen@uta.fi>

The classical test for testing the location of a multivariate population or for testing the mean
difference for two multivariate populations. When `test = "f"`

the F-distribution is used for
the test statistic and it is assumed that the data are normally distributed. If the chisquare
approximation is used, the normal assumption can be relaxed to existence of second moments.
In the two sample case both populations are assumed to have the same covariance matrix.

The formula interface is only applicable for the 2-sample tests.

Nordhausen K., Sirkia S., Oja H. and Tyler D. E. (2012) *ICSNP: Tools for
Multivariate Nonparametrics*. R package version 1.0-9.

https://cran.r-project.org/package=ICSNP

Anderson, T.W. (2003), *An introduction to
multivariate analysis*, New Jersey: Wiley.

```
math.teach <- data.frame(
teacher = factor(rep(1:2, c(3, 6))),
satis = c(1, 3, 2, 4, 6, 6, 5, 5, 4),
know = c(3, 7, 2, 6, 8, 8, 10, 10, 6))
with(math.teach,
HotellingsT2Test(cbind(satis, know) ~ teacher))
```

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