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The Matthews correlation coefficient (MCC) or phi 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. The MCC is defined identically to Pearson's phi coefficient, introduced by Karl Pearson, also known as the Yule phi coefficient from its introduction by Udny Yule in 1912. Despite these antecedents which predate Matthews's use by several decades, the term MCC is widely used in the field of bioinformatics and machine learning. The coefficient 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. However, if MCC equals neither −1, 0, or +1, it is not a reliable indicator of how similar a predictor is to random guessing because MCC is dependent on the dataset. MCC is closely related to the chi-square statistic for a 2×2 contingency table 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: 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 MCC can be calculated with the formula: using the positive predictive value, the true positive rate, the true negative rate, the negative predictive value, the false discovery rate, the false negative rate, the false positive rate, and the false omission rate. The original formula as given by Matthews was: 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'). Markedness and Informedness correspond to different directions of information flow and generalize Youden's J statistic, the [math]\displaystyle{ \delta }[/math]p statistics and (as their geometric mean) the Matthews Correlation Coefficient to more than two classes. Some scientists claim the Matthews correlation coefficient to be the most informative single score to establish the quality of a binary classifier prediction in a confusion matrix context.