  The Pearson-Matthews correlation coefficient (usually abbreviated MCC) is
considered to be one of the most useful metrics for the performance of a binary
classification or hypothesis testing method (for the sake of conciseness we
will use the classification terminology throughout, but the concepts and
methods discussed in the paper apply verbatim to hypothesis testing as well).
For multinary classification tasks (with more than two classes) the existing
extension of MCC, commonly called the $\text{R}_{\text{K}}$ metric, has also
been successfully used in many applications. The present paper begins with an
introductory discussion on certain aspects of MCC. Then we go on to discuss the
topic of multinary classification that is the main focus of this paper and
which, despite its practical and theoretical importance, appears to be less
developed than the topic of binary classification. Our discussion of the
$\text{R}_{\text{K}}$ is followed by the introduction of two other metrics for
multinary classification derived from the multivariate Pearson correlation
(MPC) coefficients. We show that both $\text{R}_{\text{K}}$ and the MPC metrics
suffer from the problem of not decisively indicating poor classification
results when they should, and introduce three new enhanced metrics that do not
suffer from this problem. We also present an additional new metric for
multinary classification which can be viewed as a direct extension of MCC.
