Matthews correlation coefficient

The Matthews correlation coefficient is used in machine learning as a measure of the quality of binary (two-class) classifications. 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. 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 measure was introduced in 1975 by Matthews. The original formula equal to above was:

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

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

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

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

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 (deltap) and informedness (deltap').

Confusion Matrix

Main article: Confusion matrix

Terminology and derivations
from a confusion matrix

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 or true positive rate (TPR) eqv. with hit rate, recall $\mathit{TPR} = \mathit{TP} / P = \mathit{TP} / (\mathit{TP}+\mathit{FN})$ specificity (SPC) or True Negative Rate $\mathit{SPC} = \mathit{TN} / N = \mathit{TN} / (\mathit{FP} + \mathit{TN})$ precision or positive predictive value (PPV) $\mathit{PPV} = \mathit{TP} / (\mathit{TP} + \mathit{FP})$ negative predictive value (NPV) $\mathit{NPV} = \mathit{TN} / (\mathit{TN} + \mathit{FN})$ fall-out or false positive rate (FPR) $\mathit{FPR} = \mathit{FP} / N = \mathit{FP} / (\mathit{FP} + \mathit{TN})$ false discovery rate (FDR) $\mathit{FDR} = \mathit{FP} / (\mathit{FP} + \mathit{TP}) = 1 - \mathit{PPV}$ Miss Rate or False Negative Rate (FNR) $\mathit{FNR} = \mathit{FN} / (\mathit{FN} + \mathit{TP})$ accuracy (ACC) $\mathit{ACC} = (\mathit{TP} + \mathit{TN}) / (P + N)$ F1 score is the harmonic mean of precision and sensitivity $\mathit{F1} = 2 \mathit{TP} / (2 \mathit{TP} + \mathit{FP} + \mathit{FN})$ Matthews correlation coefficient (MCC) $\frac{ TP \times TN - FP \times FN } {\sqrt{ (TP+FP) ( TP + FN ) ( TN + FP ) ( TN + FN ) } }$ Informedness $TPR + SPC - 1$ Markedness $PPV + NPV - 1$ Source: Fawcett (2006).

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:

		Condition (as determined by "Gold standard")
	Total population	Condition positive	Condition negative	Prevalence = Σ Condition positive Σ Total population
Test outcome	Test outcome positive	True positive	False positive (Type I error)	Positive predictive value (PPV, Precision) = Σ True positive Σ Test outcome positive	False discovery rate (FDR) = Σ False positive Σ Test outcome positive
	Test outcome negative	False negative (Type II error)	True negative	False omission rate (FOR) = Σ False negative Σ Test outcome negative	Negative predictive value (NPV) = Σ True negative Σ Test outcome negative
	Positive likelihood ratio (LR+) = TPR/FPR	True positive rate (TPR, Sensitivity, Recall) = Σ True positive Σ Condition positive	False positive rate (FPR, Fall-out) = Σ False positive Σ Condition negative	Accuracy (ACC) = Σ True positive + Σ True negative Σ Total population
	Negative likelihood ratio (LR−) = FNR/TNR	False negative rate (FNR) = Σ False negative Σ Condition positive	True negative rate (TNR, Specificity, SPC) = Σ True negative Σ Condition negative
	Diagnostic odds ratio (DOR) = LR+/LR−

See also

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

References

General References

• Baldi, P.; Brunak, S.; Chauvin, Y.; Andersen, C. A. F.; Nielsen, H. Assessing the accuracy of prediction algorithms for classification: an overview" Bioinformatics 2000, 16, 412–424. [1]
• Matthews, B.W., Comparison of the predicted and observed secondary structure of T4 phage lysozyme" Biochim. Biophys. Acta 1975, 405, 442–451
• Carugo, O., Detailed estimation of bioinformatics prediction reliability through the Fragmented Prediction Performance Plots. BMC Bioinformatics 2007. [2]