Matthews Correlation Coefficient Calculator

About the correlation coefficient

The Matthews correlation coefficient or the phi-coefficient is a measure of the strength of a correlation, which is the statistical relation between two variables. It was introduced in 1975 by the biochemist Brian W Matthews. The coefficient accounts for true and false positives and negatives and can be used even where classes are of very different sizes.

The Matthews correlation coefficient formula is based on the so called confusion matrix so the variables are:

■ True positive represents the outcome where the model correctly predicts the positive class (condition is detected when present).

■ True negative is the outcome where the model correctly predicts the negative class (condition is not detected when absent).

■ False positive represents the outcome where the model incorrectly predicts the positive class (condition is detected when absent).

■ False negative is the outcome where the model incorrectly predicts the negative class (condition is not detected when present).

The relation between these classifications is expressed with Matthews correlation coefficient formula:

The MCC describes how changing the value of one variable will affect the value of another and returns a value between -1 and 1:

+1 describes a perfect prediction;

0 unable to return any valid information (no better than random prediction);

-1 describes complete inconsistency between prediction and observation.

If any of the four sums in the denominator is zero, the denominator can be arbitrarily set to the value of 1 and so the MCC becomes 0.

If the FP = FN = 0 (perfect classifier) then the MCC is 1, thus indicating perfect positive correlation. Conversely, if TP = TN = 0 (misclassified) then the MCC is -1, thus indicating perfect negative correlation.

MCC is also perfectly symmetric, so no class is more important than the other; even if the positive and negatives are changed, the same value is obtained.

The Matthews correlation formula is used in research, biological sciences as well as in machine learning (the scientific field that combines statistical models and algorithms). It is deemed to be the best determinant of the quality of a binary classifier prediction in a confusion matrix context.

References

Matthews, B. W. Comparison of the predicted and observed secondary structure of T4 phage lysozyme. Biochimica et Biophysica Acta (BBA) - Protein Structure. 1975; 405 (2): 442–451

Powers, David M W. Evaluation: From Precision, Recall and F-Measure to ROC, Informedness, Markedness & Correlation (PDF). Journal of Machine Learning Technologies. 2011; 2 (1): 37–63.

Chicco D, Jurman G. The advantages of the Matthews correlation coefficient (MCC) over F1 score and accuracy in binary classification evaluation. BMC Genomics. 2020; 21 (6).