Comparison of Independent Samples of High‐Dimensional Data by Pairwise Distance Measures

Abstract

Pairwise distance or association measures of sample elements are often used as a basis for hierarchical cluster analyses. They can also be used in tests for the comparison of pre-defined subgroups of the total sample. Usually this is done with permutation tests In this paper, we compare such a procedure with alternative tests for high-dimensional data based on spherically distributed scores in simulation experiments and with real data. The tests based on the pairwise distance or similarity measures perform quite well in this comparison. As the number of possible permutations is small in very small samples, this might restrict the use of the test. Therefore, we propose an exact parametric small sample version of the test using randomly rotated samples.

A new proposal for a principal component‐based test for high‐dimensional data applied to the analysis of PhyloChip data

“…Just as in the case of its application for tests based on pairwise distance or similarity measures (Kropf et al, 2007;Kropf and Adolf, 2009), this reflects the potential of this tool to support ''unconventional'' test statistics with emphasis to special high-dimensional applications, where otherwise only approximations would have been used. In the above principal component-based tests one can even omit the estimation of the number q of relevant principal components and use the maximum number of minðn À r 1 ; pÞ nonzero eigenvalues instead.…”

“…The second alternative test included here for comparisons is a representative of the class of tests based on pairwise distance or similarity measures between the sample elements (Anderson, 2001;Kropf et al, 2004Kropf et al, , 2007Kropf et al, , 2009), which has been successful applied to the analysis of high-dimensional microbial fingerprint data. Particularly, we use Pearson correlation coefficients to assess the similarity in each pair of high-dimensional observation vectors.…”

“…This can be checked in permutation tests if the groups are large enough. Otherwise parametric rotation tests (Langsrud, 2005;La¨uter et al, 2005;Kropf et al, 2007;Dørum et al, 2009) may help. In this paper, such distance-based methods are only used for comparisons.…”

Rotation test with pairwise distance measures of sample vectors in a GLM

“…Furthermore, it is not possible to compare very small samples because the number of possible permutations is too low to yield p-values below the pre-defined threshold (as = 0.05). Therefore, in Kropf et al (2007), parametric rotation tests have been applied in the comparison of independent samples. Here we consider a generalization of this approach to general linear models.…”

“…In Kropf et al (2007), the special situation of the comparison of independent samples based on pairwise measures is considered and an exact parametric test version is derived as rotation test (cf. Langsrud, 2005;Läuter et al, 2005) based on pairwise similarity or distance measures.…”