Recently, I received a slide deck extolling the virtue of an exciting new classification system with a purported accuracy of 62.5%. While the number itself is not very high to begin with, the value of that 62.5% begins to diminish further once we evaluate what accuracy really represents. Accuracy is defined as
(number of items correctly classified) / (total number).
Suppose the classes are not equally represented, and rather they are represented in a ratio of 2 to 1. That is, class 1 is the right classification for 2/3rd of the items, and the class 2 is the correct classification for 1/3rd of the items. Consider a degenerate classification system that simply assigns class 1 to all items. The accuracy of that degenerate system is then 67%. And that system does not even do anything!
This simply observation is the reason that there are so many other objective functions – for example, kappa statistic, matthews correlation coefficient, F1 measure, etc, that are considered so much more appropriate than the “accuracy”. Kappa statistic, for example, compares the accuracy of the system to the accuracy of a random system.
We looked at Kappa Statistic previously, and I have been evaluating some aspects of it again.
To remind ourselves, Kappa statistic is a measure of consistency amongst different raters, taking into account the agreement occurring by chance. The standard formula for kappa statistic is given as:
Firstly, an observation that I omitted to make: the value of kappa statistic can indeed be negative. The total accuracy can be lesser than random accuracy, and as CMAJ letter by Juurlink and Detsky points out, this may indicate genuine disagreement, or it may reflect a problem in the application of a diagnostic test.
Secondly, one thing to love about Kappa is the following. Consider the case that one actual class is much more prevalent than the other. In such case, a classification system that simply outputs the more prevalent class may have a high F1 measure (a high precision and high recall), but will have a very low value of kappa. For example, consider the scenario that we are asked if it will rain in Seattle and consider the following confusion matrix:
This is a null hypothesis model, in the sense that it almost always predicts the class to be “T”. In the case of this confusion matrix, precision is 0.9008 and recall is 0.9989. F1 measure is 0.9473 and can give an impression that this is a useful model. Kappa value is very low, at 0.0157, and gives a clear enough warning about the validity of this model.
In other words, while it may be easier to predict rain in Seattle (or sunshine in Aruba), kappa statistic tries to take away the bias in the actual distribution, while the F1 measure may not.