Phi Coefficient¶

The signed Pearson correlation of two varying binary variables, computed from the normalized cross-product difference of their 2×2 table.

Version: v1 · 2026-10-03 · History
Domain-specific #: 13500
Domain group: Formal Sciences
Origin domain: Experimental Design & Statistics
Subdomains: Categorical Association, Binary Statistics → Experimental Design & Statistics

Core Idea¶

The phi coefficient is signed Pearson correlation for two paired, nonconstant binary variables. With counts $a=n_{11}$, $b=n_{10}$, $c=n_{01}$ and $d=n_{00}$, it is $\phi=(ad-bc)/\sqrt{(a+b)(c+d)(a+c)(b+d)}$. All four margins must be positive; otherwise the statistic is undefined. The value is between $-1$ and $1$ when defined, and swapping one variable's labels reverses its sign.[^{ref-391a1f8073b3][}ref-34f5f14e33ec]

Scope of Application¶

NIST's hypothetical two-process by two-outcome table $(2,5;3,2)$ yields $\phi=-11/35$ by applying the formula to its published counts; NIST itself uses the table to teach a separate Fisher test. Scikit-learn's four-case binary prediction example has $TP=2$, $FN=1$, $FP=1$, $TN=0$ and yields binary MCC/phi $=-1/3$, rounded in its guide to $-0.33$. Multiclass MCC is a different extension.[^{ref-7d0c0b608e6c][}ref-34f5f14e33ec]

Clarity¶

State which cells are $11$, \$10$, $01$ and $00$, and report both variables' margins. A zero cell is allowed; a zero margin makes the denominator vanish. Binary MCC is an exact formula match under truth/prediction labeling, but unsigned phi-squared, Youden's $J$, Fowlkes–Mallows and Cramér's $V$ are not signed-phi aliases. “Doolittle Skill Score” remains an unresolved historical naming question and is not adopted as an alias.[^{ref-34f5f14e33ec][}ref-5fe6613b030a]

Manages Complexity¶

Phi compresses a fourfold table into one directional association score, while its algebraic equality to Pearson $r$ checks the calculation. The compression hides marginal constraints: with fixed row-1 total $r_1$ and column-1 total $c_1$, $a$ can range only from $\max(0,r_1+c_1-n)$ to $\min(r_1,c_1)$. Thus a fixed-margin table may not attain both $-1$ and $1$ even though the general mathematical range is $[-1,1]$.[^{ref-7d0c0b608e6c][}ref-34f5f14e33ec]

Abstract Reasoning¶

Pair both binary observations by unit, declare coding, count the four cells, check that every row/column margin is positive, then compute the signed normalized cross-product. Interpret the result as a sample binary correlation. A test, uncertainty statement or causal conclusion requires additional assumptions and methods: zero observed phi does not prove underlying population independence. The exact equality to Pearson correlation justifies the proposed strict live Pearson parent.[^{ref-391a1f8073b3][}ref-7d0c0b608e6c]

Knowledge Transfer¶

Industrial process/outcome comparisons and classifier truth/prediction comparisons have the same paired-binary/four-cell/signed-normalization structure. Only the labels and evaluation purpose change. Do not transfer a classifier-quality verdict to a process comparison, or interpret NIST's separate Fisher-test result as phi's intrinsic significance. The statistic is descriptive in both settings.[^{ref-7d0c0b608e6c][}ref-34f5f14e33ec]

[^ref-391a1f8073b3]: H. P. Edmundson, “A Correlation Coefficient for Attributes or Events,” in Statistical Association Methods for Mechanized Documentation, original National Bureau of Standards proceedings (1964), p.41 onward, Abstract and §3; only official indexed passages were inspectable because the full PDF exceeded viewer size. [^ref-7d0c0b608e6c]: NIST/SEMATECH, Engineering Statistics Handbook, §7.3.3, original fourfold hypothetical process table and separate Fisher test. [^ref-34f5f14e33ec]: Scikit-learn maintainers, “Metrics and scoring,” §3.4.4.13, original binary MCC formula, multiclass extension and worked example. [^ref-5fe6613b030a]: SciPy maintainers, scipy.stats.contingency.association, Notes on related Cramér's $V$ and Tschuprow's $T$.

Relationships to Other Abstractions¶

Current abstraction Phi Coefficient Domain-specific

Parents (1) — more general patterns this builds on

Phi Coefficient is a kind of Pearson correlation coefficient Domain-specific

Pearson correlation specialized to two binary variables.

Hierarchy path (1) — routes to 1 parentless root

Phi Coefficient → Pearson correlation coefficient → Correlation

Neighborhood in Abstraction Space¶

Phi Coefficient sits in a sparse region of the domain-specific corpus (76^th percentile for distinctiveness): few abstractions share its structure, so a faithful description tends to retrieve it precisely.

Family — Clinical Trial & Research Methodology (20 abstractions)

Nearest neighbors

Computed from structural-signature embeddings · 2026-10-08