On the Comparison of Two Populations

Abstract

A comparison of two populations is an interesting and very common statistical problem. It involves finding statistically significant differences based on given samples. The most common way is to verify the hypothesis concerned the equality of certain, characteristic parameters, i.e. mean, standard deviation or fraction. The most efficient parametric tests need to fulfill assumptions about the normal distribution of examined populations. There are, however, cases where comparing "the shape" of multivariate populations could be crucial. Additionally, the distributions of tested populations are either unknown or cannot be treated as multivariate normal distributions. This paper presents the results of investigations on the comparison of two populations where the differences between eigenvectors were implemented. Test statistics, based on the differences between first eigenvectors of tested populations, make it possible to examine the differences of a shape, regardless of its mean or standard deviation. They could be used, for example, to test the variability of a given phenomenon even with the trends. It was proposed to verify the hypotheses with permutation tests, where no assumptions about the distribution must be fulfilled. Doing so would make it possible to use different test statistics as well. At the end of the paper, the characteristics of the examined tests were estimated using Monte Carlo simulation.