Equaldis.TStest.HD: A two-sample test for the equality of distributions for...

In sidoruvigo/Equaldis.TStest.HD: A Two-Sample Test for the Equality of Distributions for High-Dimensional Data

Description

Performs the four tests of equality of the p marginal distributions for two groups proposed in Cousido- Rocha et al. (2018). The methods have been designed for the low sample size and high dimensional setting. Furthermore, the possibility that the p variables in each data set can be weakly dependent is considered. The function also reports a set of p permutation p-values, each of them is derived from testing the equality of distributions in the two groups for each of the variables separately. These p-values are useful when the proposed tests rejects the global null hypothesis since it is possible to identify which variables have been contributed to this significance.

Usage

Equaldis.TStest.HD(X, Y, method = c("spect", "spect_ind", "boot", "us",
  "us_ind", "perm"), I.permutation.p.values = FALSE,
  b_I.permutation.p.values = c("global", "individual"))

Arguments

X	A matrix where each row is one of the p-samples in the first group.
Y	A matrix where each row is one of the p-samples in the second group.
method	the two-sample test. By default the “us” method is computed. See details.
I.permutation.p.values	Logical. Default is FALSE. A variable indicating whether to compute the permutation p-values or not when the selected method is not “perm”. See details.
b_I.permutation.p.values	The method used to compute the individual statistics on which are based the permutation p-values. Default is “global”. See details.

Details

The function implements the two-sample tests proposed by Cousido-Rocha, et al. (2018). The methods “spect”,“boot” and “us” are based on a global statistic which is the average of p individual statistics corresponding to each of the p variables. Each of these individual statistics measure the difference between the empirical characteristic functions computed from the two samples. An alternative expression of them show that it can be interpreted as a difference between the variability in each of the two samples and the variability in the combined sample. The global statistic (average) is standarized using different variance estimators given place to the three different methods. The method “spect” uses a variance estimator based on the spectral analysis theory, the method “boot” implements the block bootstrap to estimate the variance and the method “us” employs a variance estimator derived from U-statistic theory (more details in Cousido-Rocha et al., 2018). The methods “spect” and “boot” are suitable under some theoretical assumptions which include that the sequence of individual statistics that defined the global statistic is strictly stationary whereas the method “us” avoids such assumption. However the methods “spect” and “boot” have been checked in simulations and they perform well even when such assumption is violated. The methods “spect” and “us” have their corresponding version for independent data (“spect ind” and “us ind”), for which the variance estimator is simplified taking into acount the independence of the variables. The asymptotic normality (when p tends to infinity) of the standardized version of the statistic is used to compute the corresponding p-value. On the other hand, Cousido-Rocha et al. (2018) also proposed the method “perm” whose global statistic is the average of the permutation p-values corresponding to the individual statistics mentioned above. This method assumes that the sequence of p-values is strictly stationary, however in simulations it seems that it performs well where this assumption does not hold. Furthermore than defining a new global test these p-values can be also used when the global null hypothesis is rejected and we need to identify which of the p variables have been contributed to that rejection. The global statistic depends on a parameter which plays a similar role of a smoothing parameter or bandwidth in kernel density estimation. For the four global tests this parameter is estimated using the information of all the variables or features. For the individual statistics on based of which the permutation p-values are computed, we have two possibilities: (a) use the value employed in the global test (b I.permutation.p.values=“global”). (b) estimate this parameter for each variable independently using only its sample information (b I.permutation.p.values=“individual”).

Value

A list containing the following components:

standarized statistic:	the value of the standarized statistic.
p.value:	the p-value for the test.
statistic:	the value of the statistic.
variance:	the value of the variance estimator.
p:	number of samples or populations.
n:	sample size in the first group.
m:	sample size in the second group.
method:	a character string indicating which two sample test is performed.
I.statistics:	the p individual statistics.
I.permutation.p.values:	the p individual permutation p-values.
data.name:	a character string giving the name of the data.

Author(s)

• Marta Cousido-Rocha

• José Carlos Soage González

• Jacobo de Uña-Álvarez

• D. Hart, Jeffrey.

References

Cousido-Rocha, M., de Uña-Álvarez J., and Hart, J. (2018). A two-sample test for the equality of distributions for high-dimensional data. Preprint.

Examples

# We consider the microarray study of hereditary breast cancer in Hedenfalk et
#al. (2001). The data set consists of p = 3226 logged gene expression levels
# measured on n = 7 patients with breast tumors having BRCA1 mutations and on
# n = 8 patients with breast tumors having BRCA2 mutations. Our interested is
# to test the null hypothesis that the distribution of each of 500 selected
#genes is the same for the two types of tumor, BRCA1 tumor and BRCA2 tumor.
# We use the test for this purpose the four methods proposed in Cousido-Rocha et al. (2018).
#library(Equalden.HD)
#data(Hedenfalk)
### First group
X <- Equalden.HD::Hedenfalk[1:500, 1:7]
p <- dim(X)[1]
n <- dim(X)[2]
### Second group
Y <- Equalden.HD::Hedenfalk[1:500, 8:15]
m <- dim(X)[2]

res1 <- Equaldis.TStest.HD(X, Y, method = "spect")
res1
res2 <- Equaldis.TStest.HD(X, Y, method="boot")
res2
res3 <- Equaldis.TStest.HD(X, Y, method = "us")
res3

res4 <- Equaldis.TStest.HD(X, Y, method = "perm")
res4

### The four method reject the global null hypothesis.
### Hence, we use the individual permutation p-values
### to identify which genes are not equally distributed under the two tumor types.
pv <- res4$I.permutation.p.values
### We correct the multiplicity of tests using Benjamini and Hochberg (1995) method.
#library(sgof)
sgof::BH(pv)

sidoruvigo/Equaldis.TStest.HD documentation built on May 7, 2019, 7:42 a.m.