# eqCorrMatTest: equality of two correlation matrices test In ldstatsHD: Linear Dependence Statistics for High-Dimensional Data

## Description

tests the equality of two correlation matrices coming from two independent (or paired) datasets, that can possibly be high dimensional.

## Usage

 1 2 3 4 5 eqCorrMatTest(D1, D2 = NULL, testStatistic = c("AS", "max", "exc"), testNullDist = c("asyIndep","asyDep", "np"), nite= 500, paired = FALSE, threshold = 2.3, excAdj = TRUE, exact = FALSE, conf.level = 0.95, saddlePoint = FALSE, MINint =2, MAXint=100, ...) 

## Arguments

 D1 first population dataset in matrix n_1\times p form. D2 second population dataset in matrix n_2\times p form. If D2 = NULL, an hypothesis testing for non-identity correlation matrix is performed instead. testStatistic test statistic used for the hypothesis testing: name that uniquely identifies "AS" for average of squares based test, "max" for an extreme values test and "exc" for an exceedances-based test. If "all", all tests are performed. testNullDist Null distribution approximation. Either assuming independence between sample coefficients ("asyIndep"), accounting for dependence in a parametric distribution ("asyDep"), or accounting for dependence by a non-parametric distribution ("np"). nite number of iterations used to generate the permuted samples (only if testNullDist = "asyDep" or testNullDist = "np"). It is also used to approximate a null distribution by Monte Carlo when testNullDist = "asyIndep" and testStatistic = "exc". paired if TRUE, observations in D1 and D2 are assumed to be matched (n_1 must be equal to n_2). threshold exceedance threshold used if testStatistic = "exc". excAdj weight for the exceedances test. If excAdj = FALSE the test statistic is given by the squared exceedances. In contrast, If excAdj = TRUE the test statistic is given by the squared of the exceedances minus the threshold. exact permuted samples method: if TRUE it forces to have the exact same number of observations in the two conditions in the samples exchanging process. If FALSE, permutations are made exchanging matched observations from the two datasets randomly with probability equal to 0.5. conf.level confidence level of the interval. saddlePoint saddle-point approximation is used to find the p-value for exceedances-based test under asymptotic independence null distribution (only for excAdj = FALSE). MINint minimum expected number of exceedances for saddle point approximation. MAXint maximum expected number of exceedances for saddle point approximation. ... arguments passed to or from other methods to the low level.

## Details

The extreme value test is the most powerful test against very sparse alternatives whereas sum of squares test is the most powerful when the true differential correlation matrix is dense. Otherwise, for a reasonable selection of the exceedance threshold, the exceedances test overperforms the power of the other two tests.

Paired structures can be used in this function. For instance, if paired = TRUE then the correlation between sample correlation coefficients in the two matrices is estimated to adjust the test statistic.

Asymptotic independence tests are fast since they do not compute permuted samples and can be used, even for paired data, under weakly dependent assumptions (very sparse correlation matrices) when sample sizes are large. If these assumptions do not hold, wrong representations of the p-values under H_0 could be found, in which case, permuted based null distributions should be used instead.

Testing if a correlation matrix is the identity matrix can also be performed when D2 = NULL. Note that the same type of test statistics and null distributions are available in this setting. Nevertheless, Monte Carlo simulations are used instead of permuted samples. Here null distributions are approximated by computing sample correlation matrices of generated data following a N_p(0,I) under the assumption of normality.

## Value

An object of class eqCorrMatTest containing the test statistics, p-values and confidence intervals for the selected tests.

## Author(s)

Caballe, Adria <[email protected]>, Natalia Bochkina, Claus Mayer and Ioannis Papastathopoulos.

## References

To come

eqCorTestByRows for testing linear independence and equality of sames rows in two correlation matrices.
  1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 #### data EX2 <- pcorSimulatorJoint(nobs = 50, nclusters = 3, nnodesxcluster = c(40,40,40), pattern = "pow", diffType = "cluster", dataDepend = "diag", pdiff=0.5) #### eq. corr. mat. test ## not run #test1 <- eqCorrMatTest(EX2$D1, EX2$D2, testStatistic = c("AS", "exc", "max"), # paired = TRUE, nite = 400) #print(test1) ## not run #test2 <- eqCorrMatTest(EX2\$D1, NULL, testStatistic = c("AS", "exc", "max"), # paired = TRUE, nite = 400) #print(test2)