Partial Correlation Calculator & Analyzer

Exploratory Presets:

🏷️ Step 1: Variable Names

Variable 1 ($X$)

Variable 2 ($Y$)

Variable 3 ($Z$)

📊 Step 2: Zero-Order Correlations ($r$)

[-1.000, 1.000]

r(1, 2) [X ↔ Y]

r(1, 3) [X ↔ Z]

r(2, 3) [Y ↔ Z]

Sample Size (N)

For Student's t-test significance

✅

Valid Correlation Matrix

Determinant |R|

r(1,2 . 3) Ctrl: Z

Zero-order: 0.550

t-stat: --

p-value: --

r(1,3 . 2) Ctrl: Y

Zero-order: 0.600

t-stat: --

p-value: --

r(2,3 . 1) Ctrl: X

Zero-order: 0.700

t-stat: --

p-value: --

Visual Comparison: Zero-Order vs Partial Correlations

Illustrates attenuation, suppression, or spurious correlation effects

Zero-Order Partial

📐 Formula Breakdown & Mathematical Substitutions

Step-by-step evaluation

1. Partial correlation between Var 1 & Var 2 controlling for Var 3:

2. Partial correlation between Var 1 & Var 3 controlling for Var 2:

3. Partial correlation between Var 2 & Var 3 controlling for Var 1:

4. Statistical Significance Test (Two-Tailed Student's t-test):

🧠 What is Partial Correlation?

A first-order partial correlation measures the linear relationship between two continuous variables while mathematically eliminating (holding constant) the linear effect of a third variable.

🛡️ Positive Semi-Definiteness

A $3 \times 3$ correlation matrix must be positive semi-definite ($|R| \ge 0$). In three dimensions, this requires $1 - r_{12}^2 - r_{13}^2 - r_{23}^2 + 2r_{12}r_{13}r_{23} \ge 0$, ensuring geometric consistency.

📈 Degrees of Freedom

For sample size $N$ and first-order partial correlation (1 controlled covariate), the hypothesis test has $df = N - 2 - 1 = N - 3$. As $df \to \infty$, the t-distribution approaches standard normal $Z$.