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Statistical tests that compare classification algorithms are univariate anduse a single performance measure, e.g., misclassification error, $F$ measure,AUC, and so on. In multivariate tests, comparison is done using multiplemeasures simultaneously. For example, error is the sum of false positives andfalse negatives and a univariate test on error cannot make a distinctionbetween these two sources, but a 2-variate test can. Similarly, instead ofcombining precision and recall in $F$ measure, we can have a 2-variate test on(precision, recall). We use Hotelling's multivariate $T^2$ test for comparingtwo algorithms, and when we have three or more algorithms we use themultivariate analysis of variance (MANOVA) followed by pairwise post hoc tests.In our experiments, we see that multivariate tests have higher power thanunivariate tests, that is, they can detect differences that univariate testscannot. We also discuss how multivariate analysis allows us to automaticallyextract performance measures that best distinguish the behavior of multiplealgorithms.

Multivariate Comparison of Classification Algorithms