1. What is the Phi Correlation Coefficient?

Definition: The Phi coefficient (φ) is a measure of association between two binary variables (2×2 contingency table).

Purpose: It's used in statistics to determine the degree of association between two binary variables, similar to Pearson's correlation for binary data.

2. How Does the Calculator Work?

The calculator uses the formula:

Where:

• \( \phi \) — Phi coefficient (ranges from -1 to +1)
• \( TP \) — True positives (correct positive predictions)
• \( TN \) — True negatives (correct negative predictions)
• \( FP \) — False positives (incorrect positive predictions)
• \( FN \) — False negatives (incorrect negative predictions)

Explanation: The numerator measures the difference between concordant and discordant pairs, while the denominator normalizes this difference.

3. Interpretation of Phi Coefficient

Details:

• φ = 1: Perfect positive association
• φ = -1: Perfect negative association
• φ = 0: No association
• Values between 0 and ±1 indicate varying degrees of association

4. Using the Calculator

Tips: Enter all four values from your 2×2 contingency table (TP, TN, FP, FN). All values must be ≥ 0.

5. Frequently Asked Questions (FAQ)

Q1: What's the difference between Phi and Pearson correlation?
A: Phi is specifically for binary variables, while Pearson's is for continuous variables. Phi is mathematically equivalent to Pearson's for binary data.

Q2: When is Phi coefficient undefined?
A: When the denominator is zero, which happens if any marginal total is zero (e.g., TP+FP=0).

Q3: What's a "good" Phi value?
A: Interpretation depends on context, but generally:

• 0-0.3: Weak association
• 0.3-0.7: Moderate association
• 0.7-1: Strong association

Q4: Can Phi be negative?
A: Yes, negative values indicate an inverse relationship between variables.

Q5: What fields use Phi coefficient?
A: Psychology, medicine, machine learning (especially for binary classification evaluation), and social sciences.