Matthews correlation coefficient

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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[citation needed]. 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{ P S  ( 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 (deltap) and informedness (deltap').[2][3]

Confusion Matrix[edit]

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 ) } }
TPR + SPC - 1
PPV + NPV - 1

Sources: Fawcett (2006) and Powers (2011).[4][3]

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:

(as determined by "Gold standard")
Total population Condition positive Condition negative Prevalence =
Σ Condition positive
Σ Total population
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
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+) =
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−) =
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) =

See also[edit]


  1. ^ a b 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. ^ 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. 
  3. ^ a b Powers, David M W (2011). "Evaluation: From Precision, Recall and F-Measure to ROC, Informedness, Markedness & Correlation". Journal of Machine Learning Technologies 2 (1): 37–63. 
  4. ^ Fawcelt, Tom (2006). "An Introduction to ROC Analysis". Pattern Recognition Letters 27 (8): 861 – 874. doi:10.1016/j.patrec.2005.10.010.