# hotelling.stat: Calculate Hotelling's two sample T-squared test statistic In Hotelling: Hotelling's T^2 Test and Variants

## Description

Calculate Hotelling's T-squared test statistic for the difference in two multivariate means.

## Usage

 `1` ```hotelling.stat(x, y, shrinkage = FALSE, var.equal = TRUE) ```

## Arguments

 `x` a nx by p matrix containing the data points from sample 1 or a list containing elements `mean`, `cov`, and `n` where `mean` is a mean vector of length p, `cov` is a variance-covariance matrix of dimension p by p, and `n` is the sample size `y` a ny by p matrix containg the data points from sample 2 or a list containing elements `mean`, `cov`, and `n` where `mean` is a mean vector of length p, `cov` is a variance-covariance matrix of dimension p by p, and `n` is the sample size `shrinkage` set to `TRUE` if the covariance matrices are to be estimated using Schaefer and Strimmer's James-Stein shrinkage estimator. Note this only works when raw data is supplied, and will not work if summary statistics are supplied. `var.equal` set to `TRUE` if the covariance matrices are (assumed to be) equal

## Details

Note, the sample size requirements are that nx + ny - 1 > p. The procedure will stop if this is not met and the shrinkage estimator is not being used. The shrinkage estimator has not been rigorously tested for this application (small p, smaller n).

## Value

A list containing the following components:

 `statistic` Hotelling's (unscaled) T-squared statistic `m` The scaling factor - this can be used by by multiplying it with the test statistic, or dividing the critical F value `df` a vector of length containing the numerator and denominator degrees of freedom `nx` The sample size of sample 1 `ny` The sample size of sample 2 `p` The number of variables to be used in the comparison

James M. Curran

Taylor Hersh

## References

Hotelling, H. (1931). “The generalization of Student's ratio.” Annals of Mathematical Statistics 2 (3): 360–378.

Schaefer, J., and K. Strimmer (2005). “A shrinkage approach to large-scale covariance matrix estimation and implications for functional genomics.” Statist. Appl. Genet. Mol. Biol. 4: 32.

Opgen-Rhein, R., and K. Strimmer (2007). “Accurate ranking of differentially expressed genes by a distribution-free shrinkage approach.” Statist. Appl. Genet. Mol. Biol. 6: 9.

NEL, D.G. and VAN DER MERWE, C.A. (1986). “A solution to the - multivariate Behrens-Fisher problem.” Comm. Statist. Theor.- Meth., A15, 12, 3719-3736.

## Examples

 ```1 2 3 4 5 6``` ```data(container.df) split.data = split(container.df[,-1],container.df\$gp) x = split.data[] y = split.data[] hotelling.stat(x, y) hotelling.stat(x, y, TRUE) ```

Hotelling documentation built on Sept. 9, 2021, 9:09 a.m.