Phi coefficient

{{short description|Statistical measure of association for two binary variables}} In statistics, the '''phi coefficient''', also known as the '''mean square contingency coefficient''' or '''Yule coefficient of correlation''' and commonly denoted by ''φ'' or ''r''''φ'', is a measure of association between two binary variables. In machine learning and bioinformatics, it is known as the '''Matthews correlation coefficient (MCC)'''. In meteorology and elsewhere, it is referred to as the '''Doolittle Measure of Association''' or '''Doolittle Skill Score'''. Described by Udny Yule in 1912<ref name="Yule 1912 579–652">{{Cite journal|title=On the Methods of Measuring Association Between Two Attributes|journal=Journal of the Royal Statistical Society|last=Yule|first=G. Udny|volume=75|pages=579–652|issue=6|year=1912|doi=10.2307/2340126|jstor=2340126|url=https://zenodo.org/record/1449482}}</ref> and given the name ''phi'' by Karl Pearson in the 1930s,<ref name="Cramer, H. 1946 p. 282">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</ref> it is a special case of the Pearson correlation coefficient.

== Definition == A Pearson correlation coefficient estimated for two binary variables will return the phi coefficient.<ref>Guilford, J. (1936). ''Psychometric Methods''. New York: McGraw–Hill Book Company, Inc.</ref>

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'' {| class="wikitable" style="margin:1em auto; text-align: center; background: inherit; color: inherit;" | || ''y'' = 1 || ''y'' = 0 || total |----- | ''x'' = 1 || <math>n_{11}</math> || <math>n_{10}</math> || <math>n_{1\bullet}</math> |----- | ''x'' = 0 || <math>n_{01}</math> || <math>n_{00}</math> || <math>n_{0\bullet}</math> |----- | total || <math>n_{\bullet1}</math> || <math>n_{\bullet0}</math> || <math>n</math> |} where ''n''11, ''n''10, ''n''01, ''n''00, 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

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

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).<ref name="Ref_">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)</ref>

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

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

== Maximum values == In general, the Pearson correlation coefficient ranges from −1 to +1, where ±1 indicates perfect agreement or disagreement, and 0 indicates no relationship. The range of the phi coefficient—a special case of the Pearson correlation coefficient—is more tightly bound when either of the binary variables are class-imbalanced.<ref>{{cite journal |last1=Davenport |first1=E. |last2=El-Sanhury |first2=N. |title=Phi/Phimax: Review and Synthesis |journal=Educational and Psychological Measurement |volume=51 |issue=4 |pages=821–8 |date=1991 |doi=10.1177/001316449105100403 }}</ref>

== Machine learning == The Matthews correlation coefficient (MCC) is widely used in the fields of bioinformatics and machine learning to evaluate the quality of binary (two-class) classifications. It is named for biochemist Brian W. Matthews, who described the measure in a foundational 1975 paper.<ref name="Matthews1975">{{cite journal|last=Matthews|first=B. W.|title=Comparison of the predicted and observed secondary structure of T4 phage lysozyme|journal=Biochimica et Biophysica Acta (BBA) - Protein Structure|date=1975|volume=405|issue=2|pages=442–451|doi=10.1016/0005-2795(75)90109-9|pmid=1180967}}</ref> An equivalent quantity, the Doolittle Measure of Association or Doolittle Skill Score, was used by M. H. Doolittle in the 1880s to rate the accuracy of meteorologist John Park Finley's tornado predictions and other weather forecasts.<ref>{{cite journal |last1=Armistead |first1=Timothy W. |journal = The American Statistician |date=2016|title = Misunderstood and Unattributed: Revisiting M. H. Doolittle's Measures of Association, With a Note on Bayes' Theorem |volume= 70 |issue= 1 |pages= 63–73 |doi=10.1080/00031305.2015.1086686 |jstor=45118274 }}</ref>

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.<ref name="Boughorbel2017">{{cite journal|last=Boughorbel|first=S.B |title=Optimal classifier for imbalanced data using Matthews Correlation Coefficient metric|journal=PLOS ONE|volume=12 |issue=6 |article-number=e0177678 |date=2017|doi=10.1371/journal.pone.0177678 |pmid=28574989 |pmc=5456046 |bibcode=2017PLoSO..1277678B |doi-access=free }}</ref> 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.<ref name="Chicco2021">{{cite journal|last1=Chicco|first1=D.|last2=Tötsch|first2=N.|last3=Jurman|first3=G.|title=The Matthews correlation coefficient (MCC) is more reliable than balanced accuracy, bookmaker informedness, and markedness in two-class confusion matrix evaluation|journal=BioData Mining|volume=14 |date=2021|issue=1|page=13|doi=10.1186/s13040-021-00244-z|pmid=33541410|pmc=7863449|doi-access=free}}</ref> MCC is closely related to the chi-square statistic for a 2×2 contingency table

: <math>|\text{MCC}| = \sqrt{\frac{\chi^2}{n}}</math>

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.<ref name="Powers2011">{{cite arXiv |last1=Powers |first1=David M. W. |title=Evaluation: from precision, recall and F-measure to ROC, informedness, markedness and correlation |date=10 October 2020 |class=cs.LG |eprint=2010.16061}}</ref> 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:

: <math> \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} ) } } </math>

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: : <math>\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}} </math> 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:<ref name=Matthews1975 /> : <math>\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}</math>

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 (Δ''p'') and Youden's J statistic (informedness or Δ''p''′).<ref name="Powers2011" /><ref name="Perruchet2004">{{cite journal |first1=P. |last1=Perruchet |first2=R. |last2=Peereman |year=2004 |title=The exploitation of distributional information in syllable processing |journal=J. Neurolinguistics |volume=17 |issue=2–3 |pages=97–119 |doi=10.1016/s0911-6044(03)00059-9|s2cid=17104364 }}</ref> Markedness and informedness correspond to different directions of information flow and generalize Youden's J statistic, the <math> \delta p</math> statistics, while their geometric mean generalizes the Matthews correlation coefficient to more than two classes.<ref name="Powers2011" />

Some scientists consider 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.<ref name="Chicco2017">{{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 }}</ref><ref>{{cite journal |vauthors=Chicco D, Jurman G |title=The Matthews correlation coefficient (MCC) should replace the ROC AUC as the standard metric for assessing binary classification |journal=BioData Min |volume=16 |issue=1 |article-number=4 |date=February 2023 |pmid=36800973 |pmc=9938573 |doi=10.1186/s13040-023-00322-4 |doi-access=free }}</ref>

== 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: {| class="wikitable" style="text-align:center;" ! {{diagonal split header|Actual class|Predicted class}} ! Cat ! Dog |- ! Cat | '''6''' | 2 |- ! Dog | 1 | '''3''' |}

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: {| class="wikitable" style="text-align:center;" ! {{diagonal split header|Actual class|Predicted class}} !P !N |- !P |'''TP''' |FN |- !N |FP |'''TN''' |} where P = positive; N = negative; TP = truepositive; FP = false positive; TN = true negative; FN = false negative.

Plugging the numbers from the formula: :<math>\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>

== Confusion matrix == {{Main|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 <math>R_K</math> statistic (for K different classes) was defined in terms of a <math>K\times K</math> confusion matrix <math>C</math><ref name="gorodkin2004comparing">{{cite journal|last=Gorodkin|first=Jan|title=Comparing two K-category assignments by a K-category correlation coefficient|journal=Computational Biology and Chemistry|date=2004|volume=28|number=5|pages=367–374|doi=10.1016/j.compbiolchem.2004.09.006|pmid=15556477}}</ref> .<ref name="GorodkinRk2006">{{cite web|last1=Gorodkin|first1=Jan|title=The Rk Page|url=http://rk.kvl.dk/introduction/index.html|website=The Rk Page|access-date=28 December 2016}}</ref>

:<math>\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 classes 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:<ref>{{cite web|url=https://scikit-learn.org/stable/modules/model_evaluation.html#matthews-corrcoef|title=Matthew Correlation Coefficient|website=scikit-learn.org}}</ref> * <math>i </math> is the actual value index * <math>j </math> is the predicted value index * <math>K </math> is the total number of classes * <math>t_k = \sum_i C_{ik}</math> the number of times class k truly occurred, * <math>p_k = \sum_j C_{kj}</math> the number of times class k was predicted, * <math>c = \sum_{k} C_{kk}</math> the total number of samples correctly predicted, * <math>s = \sum_i \sum_j C_{ij}</math> the total number of samples. This allows the formula to be expressed as:

:<math>\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> {| class="wikitable" style="float:right; text-align:center;" ! {{diagonal split header|Actual class|Predicted class}} ! Cat !! Dog !! rowspan="4" style="padding:0;"| || Sum |- ! Cat | '''6''' || 2 || {{color|maroon|'''8'''}} |- ! Dog | 1 || '''3''' || {{color|brown|'''4'''}} |- style="border-top:2px solid #aaa;" ! Sum | {{color|purple|'''7'''}} || {{color|blue|'''5'''}} || {{color|green|'''12'''}} |}

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>\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>

An alternative generalization of the Matthews Correlation Coefficient to more than two classes was given by Powers <ref name="Powers2011" /> 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.<ref>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</ref>

== See also == * Cohen's kappa * Contingency table * Cramér's V, a similar measure of association between nominal variables. * F1 score * Fowlkes–Mallows index * Polychoric correlation (subtype: Tetrachoric correlation), when variables are seen as dichotomized versions of (latent) continuous variables

== References == {{Reflist}}

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