Cook’s Distance Calculator

Cook's Distance Calculator

Dependent Variable (Y) Observations separated by commas, spaces, or newlines

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Predictor X₁

Predictor X₂

Significance Level (α) Default threshold is 0.05 (5%)

Significance level α must be strictly between 0 and 1 (e.g. 0.05).

Regression Influence Diagnostics

What Is Cook's Distance?

Cook's Distance ($D_i$) is a standard influence diagnostic used in OLS regression to estimate the overall impact of a single observation ($i$) on the fitted regression model parameters ($\hat{\boldsymbol{\beta}}$).

It combines both the residual magnitude (how far the data point is from the predicted regression hyperplane) and leverage ($h_{ii}$, how extreme the predictor values are relative to the sample mean).

When to Use It

Identifying Influential Outliers: Detecting data points that exert a disproportionate pull on regression slopes.
Model Robustness Audits: Checking whether removing specific observations alters substantive research conclusions.
Data Cleaning & Quality Control: Flagging potential data entry errors or anomalous experimental subjects.
Leverage Assessment: Evaluating observations with extreme predictor values that could distort forecasts.

The Mathematical Formulas

$$D_i = \frac{e_i^2}{(k + 1) MS_{\text{res}}} \left[ \frac{h_{ii}}{(1 - h_{ii})^2} \right], \quad h_{ii} = \mathbf{x}_i^T (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{x}_i$$ $$\text{Threshold Guidelines: } D_i > 1.0 \text{ (Highly Influential)}, \quad D_i > \frac{4}{n} \text{ (Inspection Watch)}$$

Where $e_i$ is the OLS residual, $MS_{\text{res}}$ is the residual mean square error, $k$ is the number of predictors, and $h_{ii}$ represents the leverage of observation $i$ from the hat matrix $\mathbf{H}$.

Worked Example

Scenario: Regressing $Y$ on $X_1$ and $X_2$ ($n=6, k=2$) where observation 6 is an extreme outlier ($Y=55, X_1=30, X_2=15$).

Step 1: Fit OLS Model & Extract Residuals:
Residual $e_6 = 18.42, \quad MS_{\text{res}} = 0.5479$.

Step 2: Compute Leverage ($h_{66}$):
$h_{66} = \mathbf{x}_6^T (\mathbf{X}^T \mathbf{X})^{-1} \mathbf{x}_6 = 0.8124$.

Step 3: Calculate Cook's Distance ($D_6$):
$$D_6 = \frac{(18.42)^2}{(2+1)(0.5479)} \left[ \frac{0.8124}{(1 - 0.8124)^2} \right] = 3.4210$$

Conclusion: Since $D_6 = 3.4210 > 1.0$, observation 6 is flagged as a highly influential outlier.

Influence Audit Complete

Cook's Distance Results

Maximum Cook's Distance (D_max)

D_max = 0.5842

No Excessively Influential Observations Detected (All D_i < 1.0)

Max Cook's D 0.5842

Influential Count (D > 1) 0

Watch Count (D > 4/n) 1

Sample Size (n) 6

Predictors (k) 2

Residual MS (s²) 0.3002

Observation Influence Diagnostics Table

Observation (#)	Observed Y	Predicted Ŷ	Residual (eᵢ)	Leverage (hᵢᵢ)	Cook's Distance (Dᵢ)	Influence Status

Cook’s Distance Calculator