Phi coefficient

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Statistical measure of association for two binary variables

title: "Phi coefficient" type: doc version: 1 created: 2026-02-28 author: "Wikipedia contributors" status: active scope: public tags: ["bioinformatics", "cheminformatics", "computational-chemistry", "information-retrieval-evaluation", "machine-learning", "statistical-classification", "statistical-ratios", "summary-statistics-for-contingency-tables"] description: "Statistical measure of association for two binary variables" topic_path: "science/biology" source: "https://en.wikipedia.org/wiki/Phi_coefficient" license: "CC BY-SA 4.0" wikipedia_page_id: 0 wikipedia_revision_id: 0

::summary Statistical measure of association for two binary variables ::

In statistics, the phi coefficient, or mean square contingency coefficient, denoted by φ or r**φ, is a measure of association for two binary variables.

In machine learning, it is known as the Matthews correlation coefficient (MCC) and used as a measure of the quality of binary (two-class) classifications, introduced by biochemist Brian W. Matthews in 1975.

Introduced by Karl Pearson, and also known as the Yule phi coefficient from its introduction by Udny Yule in 1912 this measure is similar to the Pearson correlation coefficient in its interpretation.

In meteorology, the phi coefficient, or its square (the latter aligning with M. H. Doolittle's original proposition from 1885), is referred to as the Doolittle Skill Score or the Doolittle Measure of Association.

Definition

A Pearson correlation coefficient estimated for two binary variables will return the phi coefficient.

Two binary variables are considered positively associated if most of the data falls along the diagonal cells. In contrast, two binary variables are considered negatively associated if most of the data falls off the diagonal.

If we have a 2×2 table for two random variables x and y ::data[format=table]

totaln_{\bullet1}n_{\bullet0}n
::

where n11, n10, n01, n00, are non-negative counts of numbers of observations that sum to n, the total number of observations. The phi coefficient that describes the association of x and y is

: \varphi = \frac{n_{11}n_{00}-n_{10}n_{01}}{\sqrt{n_{1\bullet}n_{0\bullet}n_{\bullet0}n_{\bullet1}}}.

Phi is related to the point-biserial correlation coefficient and Cohen's d and estimates the extent of the relationship between two variables (2×2).

The phi coefficient can also be expressed using only n, n_{11}, n_{1\bullet}, and n_{\bullet1}, as

: \varphi = \frac{nn_{11}-n_{1\bullet}n_{\bullet1}}{\sqrt{n_{1\bullet}n_{\bullet1}(n-n_{1\bullet})(n-n_{\bullet1})}}.

Maximum values

Although computationally the Pearson correlation coefficient reduces to the phi coefficient in the 2×2 case, they are not in general the same. The Pearson correlation coefficient ranges from −1 to +1, where ±1 indicates perfect agreement or disagreement, and 0 indicates no relationship. The phi coefficient has a maximum value that is determined by the distribution of the two variables if one or both variables can take on more than two values. See Davenport and El-Sanhury (1991) for a thorough discussion.

Machine learning

The MCC is defined identically to 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 accounts for 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

: |\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{ \mathit{TP} \times \mathit{TN} - \mathit{FP} \times \mathit{FN} } {\sqrt{ (\mathit{TP} + \mathit{FP}) ( \mathit{TP} + \mathit{FN} ) ( \mathit{TN} + \mathit{FP} ) ( \mathit{TN} + \mathit{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 exactly one 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. In case two or more sums are zero (e.g. both labels and model predictions are all positive or negative), the limit does not exist.

The MCC can be calculated with the formula: : \text{MCC} = \sqrt{\mathit{PPV} \times \mathit{TPR} \times \mathit{TNR} \times \mathit{NPV}}

  • \sqrt{\mathit{FDR} \times \mathit{FNR} \times \mathit{FPR} \times \mathit{FOR}} 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: : \begin{align} N &= \mathit{TN} + \mathit{TP} + \mathit{FN} + \mathit{FP} \[6pt] S &= \frac{ \mathit{TP} + \mathit{FN} } { N } \[6pt] P &= \frac{ \mathit{TP} + \mathit{FP} } { N } \[6pt] \text{MCC} &= \frac{ \mathit{TP} / N - S \times P } {\sqrt{ P S ( 1 - S) ( 1 - P ) } } \end{align}

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 \delta p statistics, while their geometric mean generalizes 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.{{cite journal | vauthors = Chicco D | title = Ten quick tips for machine learning in computational biology | journal = BioData Mining | volume = 10 | issue = 35 | article-number = 35 | date = December 2017 | pmid = 29234465 | doi = 10.1186/s13040-017-0155-3 | pmc= 5721660 | doi-access = free

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: ::data[format=table]

CatDogCatDog
62
13
::

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: ::data[format=table]

PNPN
TPFN
FPTN
::

where P = positive; N = negative; TP = truepositive; FP = false positive; TN = true negative; FN = false negative.

Plugging the numbers from the formula: :\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

Confusion matrix

Main article: 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, as follows:

Multiclass case

The Matthews correlation coefficient has been generalized to the multiclass case. The generalization called the R_K statistic (for K different classes) was defined in terms of a K\times K confusion matrix C .

:\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)} }

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.

TODO: potentially un-comment later, for now just stick with referenced version--

This formula can be more easily understood by defining intermediate variables:

  • i is the actual value index
  • j is the predicted value index
  • K is the total number of classes
  • t_k = \sum_i C_{ik} the number of times class k truly occurred,
  • p_k = \sum_j C_{kj} the number of times class k was predicted,
  • c = \sum_{k} C_{kk} the total number of samples correctly predicted,
  • s = \sum_i \sum_j C_{ij} the total number of samples. This allows the formula to be expressed as:

:\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}} } ::data[format=table] | Cat | Dog | || Sum | Cat | Dog | Sum | |---|---|---|---|---|---| | 6 | 2 | | | | | | 1 | 3 | | | | | | | | | | | | ::

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:

:\text{MCC}=\frac{(6+3)\times{\color{green}12};-;{\color{blue}5}\times{\color{brown}4};-;{\color{purple}7}\times{\color{maroon}8}}{\sqrt\approx 0.478

An alternative generalization of the Matthews Correlation Coefficient to more than two classes was given by Powers by the definition of Correlation as the geometric mean of Informedness and Markedness.

Several generalizations of the Matthews Correlation Coefficient to more than two classes along with new Multivariate Correlation Metrics for multinary classification have been presented by P Stoica and P Babu.

References

should reference in the main text === General References ===

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.
  2. Cramer, H. (1946). ''Mathematical Methods of Statistics''. Princeton: Princeton University Press, p. 282 (second paragraph). {{ISBN. 0-691-08004-6 https://archive.org/details/in.ernet.dli.2015.223699
  3. Yule, G. Udny. (1912). "On the Methods of Measuring Association Between Two Attributes". Journal of the Royal Statistical Society.
  4. (December 16, 2011). "Forecast Verification: A Practitioner's Guide in Atmospheric Science, Second Edition". John Wiley & Sons.
  5. (2016). "Misunderstood and Unattributed: Revisiting M. H. Doolittle's Measures of Association, With a Note on Bayes' Theorem". The American Statistician.
  6. Guilford, J. (1936). ''Psychometric Methods''. New York: McGraw–Hill Book Company, Inc.
  7. Aaron, B., Kromrey, J. D., & Ferron, J. M. (1998, November). [https://eric.ed.gov/?id=ED433353 Equating r-based and d-based effect-size indices: Problems with a commonly recommended formula.] Paper presented at the annual meeting of the Florida Educational Research Association, Orlando, FL. (ERIC Document Reproduction Service No. ED433353)
  8. (1991). "Phi/Phimax: Review and Synthesis". Educational and Psychological Measurement.
  9. Date unclear, but prior to his death in 1936.
  10. Boughorbel, S.B. (2017). "Optimal classifier for imbalanced data using Matthews Correlation Coefficient metric". PLOS ONE.
  11. (2021). "The Matthews correlation coefficient (MCC) is more reliable than balanced accuracy, bookmaker informedness, and markedness in two-class confusion matrix evaluation". BioData Mining.
  12. (10 October 2020). "Evaluation: from precision, recall and F-measure to ROC, informedness, markedness and correlation".
  13. (2004). "The exploitation of distributional information in syllable processing". J. Neurolinguistics.
  14. (February 2023). "The Matthews correlation coefficient (MCC) should replace the ROC AUC as the standard metric for assessing binary classification". BioData Min.
  15. Gorodkin, Jan. (2004). "Comparing two K-category assignments by a K-category correlation coefficient". Computational Biology and Chemistry.
  16. "The Rk Page".
  17. "Matthew Correlation Coefficient".
  18. Stoica P and Babu P (2024), Pearson–Matthews correlation coefficients for binary and multinary classification, Elsevier Signal Processing, 222, 109511, doi = https://doi.org/10.1016/j.sigpro.2024.109511

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