1. Example

Given a sample of 12 pictures, 8 of cats and 4 of dogs, where cats belong to class 1 and dogs belong to class 0,

actual = [1,1,1,1,1,1,1,1,0,0,0,0],

assume that a classifier that distinguishes between cats and dogs is trained, and we take the 12 pictures and run them through the classifier, and the classifier makes 9 accurate predictions and misses 3: 2 cats wrongly predicted as dogs (first 2 predictions) and 1 dog wrongly predicted as a cat (last prediction).

prediction = [0,0,1,1,1,1,1,1,0,0,0,1]

With these two labelled sets (actual and predictions) we can create a confusion matrix that will summarize the results of testing the classifier:

In this confusion matrix, of the 8 cat pictures, the system judged that 2 were dogs, and of the 4 dog pictures, it predicted that 1 was a cat. All correct predictions are located in the diagonal of the table (highlighted in bold), so it is easy to visually inspect the table for prediction errors, as they will be represented by values outside the diagonal.

In abstract terms, the confusion matrix is as follows:

where P = Positive; N = Negative; TP = True Positive; FP = False Positive; TN = True Negative; FN = False Negative.

Plugging the numbers from the formula:

[math]\displaystyle{ \text{MCC} = \frac{6 \times 3 - 1 \times 2}{\sqrt{(6 + 1) \times (6 + 2) \times (3 + 1) \times (3 + 2)}} = \frac{16}{\sqrt{1120}} \approx 0.478 }[/math]

2. Confusion Matrix

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.

3. Multiclass Case

The Matthews correlation coefficient has been generalized to the multiclass case. This generalization was called the [math]\displaystyle{ R_K }[/math] statistic (for K different classes) by the author, and defined in terms of a [math]\displaystyle{ K\times K }[/math] confusion matrix [math]\displaystyle{ C }[/math]^[1] .^[2]

[math]\displaystyle{ \text{MCC} = \frac{\sum_{k}\sum_{l}\sum_{m} C_{kk}C_{lm} - C_{kl}C_{mk}}{ \sqrt{\sum_{k}\left(\sum_l C_{kl}\right)\left(\sum_{k' | k' \neq k}\sum_{l'} C_{k'l'}\right)} \sqrt{\sum_{k}\left(\sum_l C_{lk}\right)\left(\sum_{k' | k' \neq k}\sum_{l'} C_{l'k'}\right)} } }[/math]

When there are more than two labels the MCC will no longer range between −1 and +1. Instead the minimum value will be between −1 and 0 depending on the true distribution. The maximum value is always +1.

This formula can be more easily understood by defining intermediate variables:^[3]

• [math]\displaystyle{ t_k = \sum_i C_{ik} }[/math] the number of times class k truly occurred,
• [math]\displaystyle{ p_k = \sum_i C_{ki} }[/math] the number of times class k was predicted,
• [math]\displaystyle{ c = \sum_{k} C_{kk} }[/math] the total number of samples correctly predicted,
• [math]\displaystyle{ s = \sum_i \sum_j C_{ij} }[/math] the total number of samples. This allows the formula to be expressed as:

[math]\displaystyle{ \text{MCC} = \frac{cs - \vec{t} \cdot \vec{p}}{ \sqrt{s^2 - \vec{p} \cdot \vec{p}} \sqrt{s^2 - \vec{t} \cdot \vec{t}} } }[/math]

Using above formula to compute MCC measure for the dog and cat example discussed above, where the confusion matrix is treated as a 2 × Multiclass example:

[math]\displaystyle{ \text{MCC}=\frac{(6+3)\times{\color{green}12}\;-\;{\color{blue}5}\times{\color{brown}4}\;-\;{\color{purple}7}\times{\color{maroon}8}}{\sqrt{{\color{green}12}^2-{\color{blue}5}^2-{\color{purple}7}^2}\sqrt{{\color{green}12}^2-{\color{brown}4}^2-{\color{maroon}8}^2}}=\frac{32}{\sqrt{4480}}\approx 0.478 }[/math]

4. Advantages of MCC over Accuracy and F1 Score

As explained by Davide Chicco in his paper "Ten quick tips for machine learning in computational biology" (BioData Mining, 2017) and by Giuseppe Jurman in his paper "The advantages of the Matthews correlation coefficient (MCC) over F1 score and accuracy in binary classification evaluation" (BMC Genomics, 2020), the Matthews correlation coefficient is more informative than F1 score and accuracy in evaluating binary classification problems, because it takes into account the balance ratios of the four confusion matrix categories (true positives, true negatives, false positives, false negatives).^[4][5]

The former article explains, for Tip 8:

In order to have an overall understanding of your prediction, you decide to take advantage of common statistical scores, such as accuracy, and F1 score.

[math]\displaystyle{ \text{accuracy} = \frac{ TP+TN } {TP+TN+FP+FN} }[/math]

(Equation 1, accuracy: worst value = 0; best value = 1)

[math]\displaystyle{ \text{F1 score} = \frac{ 2 TP } {2 TP+FP+FN} }[/math]

(Equation 2, F1 score: worst value = 0; best value = 1)

However, even if accuracy and F1 score are widely employed in statistics, both can be misleading, since they do not fully consider the size of the four classes of the confusion matrix in their final score computation.

Suppose, for example, you have a very imbalanced validation set made of 100 elements, 95 of which are positive elements, and only 5 are negative elements (as explained in Tip 5). And suppose also you made some mistakes in designing and training your machine learning classifier, and now you have an algorithm which always predicts positive. Imagine that you are not aware of this issue.

By applying your only-positive predictor to your imbalanced validation set, therefore, you obtain values for the confusion matrix categories:

TP = 95, FP = 5; TN = 0, FN = 0.

These values lead to the following performance scores: accuracy = 95%, and F1 score = 97.44%. By reading these over-optimistic scores, then you will be very happy and will think that your machine learning algorithm is doing an excellent job. Obviously, you would be on the wrong track.

On the contrary, to avoid these dangerous misleading illusions, there is another performance score that you can exploit: the Matthews correlation coefficient [40] (MCC).

[math]\displaystyle{ \text{MCC} = \frac{ TP \times TN - FP \times FN } {\sqrt{ (TP + FP) ( TP + FN ) ( TN + FP ) ( TN + FN ) } } }[/math]

(Equation 3, MCC: worst value = −1; best value = +1).

By considering the proportion of each class of the confusion matrix in its formula, its score is high only if your classifier is doing well on both the negative and the positive elements.

In the example above, the MCC score would be undefined (since TN and FN would be 0, therefore the denominator of Equation 3 would be 0). By checking this value, instead of accuracy and F1 score, you would then be able to notice that your classifier is going in the wrong direction, and you would become aware that there are issues you ought to solve before proceeding.

Consider this other example. You ran a classification on the same dataset which led to the following values for the confusion matrix categories:

TP = 90, FP = 4; TN = 1, FN = 5.

In this example, the classifier has performed well in classifying positive instances, but was not able to correctly recognize negative data elements. Again, the resulting F1 score and accuracy scores would be extremely high: accuracy = 91%, and F1 score = 95.24%. Similarly to the previous case, if a researcher analyzed only these two score indicators, without considering the MCC, they would wrongly think the algorithm is performing quite well in its task, and would have the illusion of being successful.

On the other hand, checking the Matthews correlation coefficient would be pivotal once again. In this example, the value of the MCC would be 0.14 (Equation 3), indicating that the algorithm is performing similarly to random guessing. Acting as an alarm, the MCC would be able to inform the data mining practitioner that the statistical model is performing poorly.

For these reasons, we strongly encourage to evaluate each test performance through the Matthews correlation coefficient (MCC), instead of the accuracy and the F1 score, for any binary classification problem.

Chicco's passage might be read as endorsing the MCC score in cases with imbalanced data sets. This, however, is contested; in particular, Zhu (2020) offers a strong rebuttal.^[6]

Note that the F1 score depends on which class is defined as the positive class. In the first example above, the F1 score is high because the majority class is defined as the positive class. Inverting the positive and negative classes results in the following confusion matrix:

TP = 0, FP = 0; TN = 5, FN = 95

This gives an F1 score = 0%.

The MCC doesn't depend on which class is the positive one, which has the advantage over the F1 score to avoid incorrectly defining the positive class.