el.test2: Empirical likelihood hypothesis testing for two mean vectors

In Compositional: Compositional Data Analysis

Empirical likelihood hypothesis testing for two mean vectors

Description

Empirical likelihood hypothesis testing for two mean vectors.

Usage

el.test2(y1, y2, R = 0, ncores = 1, graph = FALSE)

Arguments

y1	A matrix containing the Euclidean data of the first group.
y2	A matrix containing the Euclidean data of the second group.
R	If R is 0, the classical chi-square distribution is used, if R = 1, the corrected chi-square distribution (James, 1954) is used and if R = 2, the modified F distribution (Krishnamoorthy and Yanping, 2006) is used. If R is greater than 3 bootstrap calibration is performed.
ncores	How many to cores to use.
graph	A boolean variable which is taken into consideration only when bootstrap calibration is performed. IF TRUE the histogram of the bootstrap test statistic values is plotted.

Details

The H_0 is that \pmb{\mu}_1 = \pmb{\mu}_2 and the two constraints imposed by EL are

\frac{1}{n_j}\sum_{i=1}^{n_j}\left\lbrace\left[1+\pmb{\lambda}_j^T\left({\bf x}_{ji}-\pmb{\mu} \right)\right]^{-1}\left({\bf x}_{ij}-\pmb{\mu}\right)\right\rbrace={\bf 0},

where j=1,2 and the \pmb{\lambda}_js are Lagrangian parameters introduced to maximize the above expression. Note that the maximization of is with respect to the \pmb{\lambda}_js. The probabilities of the j-th sample have the following form

p_{ji}=\frac{1}{n_j} \left[1+\pmb{\lambda}_j^T \left({\bf x}_{ji}-\pmb{\mu} \right)\right]^{-1}

. The log-likelihood ratio test statistic can be written as

\Lambda=\sum_{j=1}^2\sum_{i=1}^{n_j}\log{n_jp_{ij}}.

The test is implemented by searching for the mean vector that minimizes the sum of the two one sample EL test statistics.

Value

A list including:

test	The empirical likelihood test statistic value.
modif.test	The modified test statistic, either via the chi-square or the F distribution.
dof	Thre degrees of freedom of the chi-square or the F distribution.
pvalue	The asymptotic or the bootstrap p-value.
mu	The estimated common mean vector.
runtime	The runtime of the bootstrap calibration.

Author(s)

Michail Tsagris.

R implementation and documentation: Michail Tsagris [e-mail].

References

Amaral G.J.A., Dryden I.L. and Wood A.T.A. (2007). Pivotal bootstrap methods for k-sample problems in directional statistics and shape analysis. Journal of the American Statistical Association, 102(478): 695–707.

Owen A. B. (2001). Empirical likelihood. Chapman and Hall/CRC Press.

Owen A.B. (1988). Empirical likelihood ratio confidence intervals for a single functional. Biometrika, 75(2): 237–249.

Preston S.P. and Wood A.T.A. (2010). Two-Sample Bootstrap Hypothesis Tests for Three-Dimensional Labelled Landmark Data. Scandinavian Journal of Statistics, 37(4): 568–587.

See Also

eel.test2, maovjames, hotel2T2, james

Examples

el.test2( y1 = as.matrix(iris[1:25, 1:4]), y2 = as.matrix(iris[26:50, 1:4]), R = 0 )
el.test2( y1 = as.matrix(iris[1:25, 1:4]), y2 = as.matrix(iris[26:50, 1:4]), R = 1 )
el.test2( y1 =as.matrix(iris[1:25, 1:4]), y2 = as.matrix(iris[26:50, 1:4]), R = 2 )

Compositional documentation built on April 4, 2025, 1:46 a.m.