# Multivariate Statistical Tests for Comparing Classification Algorithms

## O. T. Yildiz, O. Aslan, E. Alpaydin

Learning and Intelligent OptimizatioN LION5, Jan 17-21 2011, Rome Italy, Lecture Notes in Computer Science, Vol. 6683, pp. 1-15, Springer.
## Abstract

The misclassification error which is usually used in tests to compare classification
algorithms, does not make a distinction between the sources of error, namely, false
positives and false negatives. Instead of summing these in a single number, we
propose to collect multivariate statistics and use multivariate tests on them.
Information retrieval uses the measures of precision and recall, and signal
detection uses true positive rate (tpr) and false positive rate (fpr) and a
multivariate test can also use such two values instead of combining them in a
single value, such as error or average precision. For example, we can have bivariate
tests for (precision, recall) or (tpr, fpr). We propose to use the pairwise test
based on Hotelling's multivariate $T^2$ test to compare two algorithms or
multivariate analysis of variance (MANOVA) to compare $L>2$ algorithms. In our
experiments, we show that the multivariate tests have higher power than the
univariate error test, that is, they can detect differences that the error test
cannot, and we also discuss how the decisions made by different multivariate tests
differ, to be able to point out where to use which. We also show how multivariate or
univariate pairwise tests can be used as post-hoc tests after MANOVA to find cliques
of algorithms, or order them along separate dimensions.