Comparing φ and the F-measure as performance metrics for software-related classifications

Context The F-measure has been widely used as a performance metric when selecting binary classifiers for prediction, but it has also been widely criticized, especially given the availability of alternatives such as phi (also known as Matthews Correlation Coefficient). Objectives Our goals are to (1) investigate possible issues related to the F-measure in depth and show how phi can address them, and (2) explore the relationships between the F-measure and phi. Method Based on the definitions of phi and the F-measure, we derive a few mathematical properties of these two performance metrics and of the relationships between them. To demonstrate the practical effects of these mathematical properties, we illustrate the outcomes of an empirical study involving 70 Empirical Software Engineering datasets and 837 classifiers. Results We show that phi can be defined as a function of Precision and Recall, which are the only two performance metrics used to define the F-measure, and the rate of actually positive software modules in a dataset. Also, phi can be expressed as a function of the F-measure and the rates of actual and estimated positive software modules. We derive the minimum and maximum value of phi for any given value of the F-measure, and the conditions under which both the F-measure and phi rank two classifiers in the same order. Conclusions Our results show that phi is a sensible and useful metric for assessing the performance of binary classifiers. We also recommend that the F-measure should not be used by itself to assess the performance of a classifier, but that the rate of positives should always be specified as well, at least to assess if and to what extent a classifier performs better than random classification. The mathematical relationships described here can also be used to re-interpret the conclusions of previously published papers that relied mainly on the F-measure as a performance metric.