Matthews correlation coefficient

The Matthews correlation coefficient is used in machine learning as a measure of the quality of binary (two-class) classifications, introduced by biochemist Brian W. Matthews in 1975.^[1] It takes into account true and false positives and negatives and is generally regarded as a balanced measure which can be used even if the classes are of very different sizes. The MCC is in essence a correlation coefficient between the observed and predicted binary classifications; it returns a value between −1 and +1. A coefficient of +1 represents a perfect prediction, 0 no better than random prediction and −1 indicates total disagreement between prediction and observation. The statistic is also known as the phi coefficient. MCC is related to the chi-square statistic for a 2×2 contingency table

$|{\text{MCC}}|={\sqrt {{\frac {\chi ^{2}}{n}}}}$

where n is the total number of observations.

While there is no perfect way of describing the confusion matrix of true and false positives and negatives by a single number, the Matthews correlation coefficient is generally regarded as being one of the best such measures.^[2] Other measures, such as the proportion of correct predictions (also termed accuracy), are not useful when the two classes are of very different sizes. For example, assigning every object to the larger set achieves a high proportion of correct predictions, but is not generally a useful classification.

The MCC can be calculated directly from the confusion matrix using the formula:

${\text{MCC}}={\frac {TP\times TN-FP\times FN}{{\sqrt {(TP+FP)(TP+FN)(TN+FP)(TN+FN)}}}}$

In this equation, TP is the number of true positives, TN the number of true negatives, FP the number of false positives and FN the number of false negatives. If any of the four sums in the denominator is zero, the denominator can be arbitrarily set to one; this results in a Matthews correlation coefficient of zero, which can be shown to be the correct limiting value.

The original formula as given by Matthews was:^[1]

${\text{N}}=TN+TP+FN+FP$

${\text{S}}={\frac {TP+FN}{N}}$

${\text{P}}={\frac {TP+FP}{N}}$

${\text{MCC}}={\frac {TP/N-S\times P}{{\sqrt {PS(1-S)(1-P)}}}}$

This is equal to the formula given above. As a correlation coefficient, the Matthews correlation coefficient is the geometric mean of the regression coefficients of the problem and its dual. The component regression coefficients of the Matthews correlation coefficient are Markedness (Δp) and Youden's J statistic (Informedness or Δp').^[2][3] Markedness and Informedness correspond to different directions of information flow and generalize Youden's J statistic, the deltap statistics and (as their geometric mean) the Matthews Correlation Coefficient to more than two classes.^[2]

Confusion Matrix

Terminology and derivations
from a confusion matrix

condition positive (P) the number of real positive cases in the data condition negatives (N) the number of real negative cases in the data true positive (TP) eqv. with hit true negative (TN) eqv. with correct rejection false positive (FP) eqv. with false alarm, Type I error false negative (FN) eqv. with miss, Type II error sensitivity, recall, hit rate, or true positive rate (TPR) ${\displaystyle \mathrm {TPR} ={\frac {\mathrm {TP} }{P}}={\frac {\mathrm {TP} }{\mathrm {TP} +\mathrm {FN} }}}$ specificity or true negative rate (TNR) ${\displaystyle \mathrm {TNR} ={\frac {\mathrm {TN} }{N}}={\frac {\mathrm {TN} }{\mathrm {FP} +\mathrm {TN} }}}$ precision or positive predictive value (PPV) ${\displaystyle \mathrm {PPV} ={\frac {\mathrm {TP} }{\mathrm {TP} +\mathrm {FP} }}}$ negative predictive value (NPV) ${\displaystyle \mathrm {NPV} ={\frac {\mathrm {TN} }{\mathrm {TN} +\mathrm {FN} }}}$ fall-out or false positive rate (FPR) ${\displaystyle \mathrm {FPR} ={\frac {\mathrm {FP} }{N}}={\frac {\mathrm {FP} }{\mathrm {FP} +\mathrm {TN} }}=1-\mathrm {TNR} }$ false discovery rate (FDR) ${\displaystyle \mathrm {FDR} ={\frac {\mathrm {FP} }{\mathrm {FP} +\mathrm {TP} }}=1-\mathrm {PPV} }$ miss rate or false negative rate (FNR) ${\displaystyle \mathrm {FNR} ={\frac {\mathrm {FN} }{P}}={\frac {\mathrm {FN} }{\mathrm {FN} +\mathrm {TP} }}=1-\mathrm {TPR} }$ accuracy (ACC) ${\displaystyle \mathrm {ACC} ={\frac {\mathrm {TP} +\mathrm {TN} }{P+N}}}$ F1 score is the harmonic mean of precision and sensitivity ${\displaystyle F_{1}={\frac {2\mathrm {TP} }{2\mathrm {TP} +\mathrm {FP} +\mathrm {FN} }}}$ Matthews correlation coefficient (MCC) ${\displaystyle \mathrm {MCC} ={\frac {\mathrm {TP} \times \mathrm {TN} -\mathrm {FP} \times \mathrm {FN} }{\sqrt {(\mathrm {TP} +\mathrm {FP} )(\mathrm {TP} +\mathrm {FN} )(\mathrm {TN} +\mathrm {FP} )(\mathrm {TN} +\mathrm {FN} )}}}}$ Informedness or Bookmaker Informedness (BM) ${\displaystyle \mathrm {BM} =\mathrm {TPR} +\mathrm {TNR} -1}$ Markedness (MK) ${\displaystyle \mathrm {MK} =\mathrm {PPV} +\mathrm {NPV} -1}$ Sources: Fawcett (2006), Powers (2011), and Ting (2011) [4] [2] [5]

Let us define an experiment from P positive instances and N negative instances for some condition. The four outcomes can be formulated in a 2×2 contingency table or confusion matrix, as follows:

		Predicted condition
	Total population	Predicted Condition positive	Predicted Condition negative	Prevalence = Σ Condition positive/Σ Total population
True condition	condition positive	True positive	False Negative (Type II error)	True positive rate (TPR), Sensitivity, Recall, probability of detection = Σ True positive/Σ Condition positive	False negative rate (FNR), Miss rate = Σ False negative/Σ Condition positive
	condition negative	False Positive (Type I error)	True negative	False positive rate (FPR), Fall-out, probability of false alarm = Σ False positive/Σ Condition negative	True negative rate (TNR), Specificity (SPC) = Σ True negative/Σ Condition negative
	Accuracy (ACC) = Σ True positive + Σ True negative/Σ Total population	Positive predictive value (PPV), Precision = Σ True positive/Σ Test outcome positive	False omission rate (FOR) = Σ False negative/Σ Test outcome negative	Positive likelihood ratio (LR+) = TPR/FPR	Diagnostic odds ratio (DOR) = LR+/LR−
		False discovery rate (FDR) = Σ False positive/Σ Test outcome positive	Negative predictive value (NPV) = Σ True negative/Σ Test outcome negative	Negative likelihood ratio (LR−) = FNR/TNR

See also

• Phi coefficient
• F1 score
• Cramér's V, a similar measure of association between nominal variables.
• Cohen's kappa

References

1. Matthews, B. W. (1975). "Comparison of the predicted and observed secondary structure of T4 phage lysozyme". Biochimica et Biophysica Acta (BBA) - Protein Structure. 405 (2): 442–451. doi:10.1016/0005-2795(75)90109-9.
2. Powers, David M W (2011). "Evaluation: From Precision, Recall and F-Measure to ROC, Informedness, Markedness & Correlation" (PDF). Journal of Machine Learning Technologies. 2 (1): 37–63.
3. Perruchet, P.; Peereman, R. (2004). "The exploitation of distributional information in syllable processing". J. Neurolinguistics. 17: 97–119. doi:10.1016/s0911-6044(03)00059-9.
4. Fawcett, Tom (2006). "An Introduction to ROC Analysis" (PDF). Pattern Recognition Letters. 27 (8): 861 – 874. doi:10.1016/j.patrec.2005.10.010.
5. Ting, Kai Ming (2011). Encyclopedia of machine learning. Springer. ISBN 978-0-387-30164-8.