Hotelling's T2 Test for Comparing Multivariate Mean Vectors

The classical test for the location of a multivariate population (one-sample) or for the difference between the mean vectors of two multivariate populations (two-sample). It is the multivariate generalisation of Student's t-test.

Usage

hotellingsT2Test(x, ...)

# S3 method for class 'formula'
hotellingsT2Test(formula, data, subset, na.action = na.pass, ...)

# Default S3 method
hotellingsT2Test(x, y = NULL, mu = NULL, test = c("f", "chi"), ...)

Arguments

x

a numeric matrix or data frame (observations in rows, variables in columns).

...

further arguments passed to or from methods.

formula

a formula of the form cbind(v1, v2, ...) ~ g where the left-hand side is a numeric matrix of response variables and g is a factor with exactly two levels.

data

an optional data frame (or similar, see model.frame()) containing the variables in formula. Defaults to the environment of formula.

subset

an optional vector specifying a subset of observations.

na.action

a function indicating what should happen when the data contain NAs. Defaults to getOption("na.action").

y

an optional numeric matrix or data frame for the two-sample test. If NULL (default) a one-sample test is performed.

mu

a numeric vector of length \(p\) giving the hypothesised mean (one-sample) or mean difference (two-sample). NULL is interpreted as the zero vector.

test

a character string selecting the reference distribution: "f" (exact F-distribution, default) or "chi" (chi-squared approximation).

Value

An object of class "htest" containing:

statistic

the value of the T2-statistic (scaled to follow an F- or chi-squared distribution depending on test).

parameter

degrees of freedom of the reference distribution.

p.value

the p-value of the test.

estimate

the sample mean vector (one-sample) or the difference of the sample mean vectors (two-sample).

null.value

the hypothesised mean or mean difference.

alternative

always "two.sided".

method

a character string describing the test variant performed.

data.name

a character string giving the name(s) of the input data.

Details

When test = "f" the test statistic follows an exact F-distribution under the assumption of multivariate normality. When test = "chi" a chi-squared approximation is used; it relies on large-sample asymptotic theory and is less sensitive to departures from multivariate normality than the F-test, but remains only asymptotically correct.

In the two-sample case both populations are assumed to share the same covariance matrix; a pooled within-group estimate is used.

The formula interface (cbind(v1, v2) ~ g) is available for the two-sample case only.

Note

Based on code by Klaus Nordhausen, adapted to conform to package standards.

References

Anderson, T. W. (2003). An Introduction to Multivariate Statistical Analysis (3rd ed.). Wiley.

Nordhausen, K., Sirkia, S., Oja, H., & Tyler, D. E. (2012). ICSNP: Tools for Multivariate Nonparametrics. R package version 1.0-9. https://cran.r-project.org/package=ICSNP

See also

Other test.location: brunnerMunzelTest(), moodMedianTest(), signTest(), tTestA(), vanWaerdenTest(), yuenTTest(), zTest()

Examples

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))

hotellingsT2Test(cbind(satis, know) ~ teacher, data = math.teach)
#> 
#> 	Hotelling's two-sample T-squared test
#> 
#> data:  cbind(satis, know) ~ teacher
#> T.2 = 9, df1 = 2, df2 = 6, p-value = 0.01562
#> alternative hypothesis: two.sided
#> null values:
#> location difference location difference 
#>                   0                   0 
#> sample estimates:
#> mean difference of satis  mean difference of know 
#>                       -3                       -4 
#> 

# chi-squared approximation
hotellingsT2Test(cbind(satis, know) ~ teacher, data = math.teach,
                 test = "chi")
#> 
#> 	Hotelling's two-sample T-squared test
#> 
#> data:  cbind(satis, know) ~ teacher
#> T.2 = 21, df = 2, p-value = 2.754e-05
#> alternative hypothesis: two.sided
#> null values:
#> location difference location difference 
#>                   0                   0 
#> sample estimates:
#> mean difference of satis  mean difference of know 
#>                       -3                       -4 
#>