Compute residuals, standardized residuals, leverage, Cook's distance, Durbin-Watson, skewness, kurtosis, and outlier detection for regression diagnostics.
Fitting a regression line is only half the job — checking whether that line is trustworthy is the other half. Residual analysis reveals problems invisible in summary statistics: non-linearity, heteroscedasticity, autocorrelation, outliers, and influential points. This calculator provides a comprehensive residual diagnostic suite.
For each data point, we compute raw residuals, standardized (internally studentized) residuals, leverage values h_ii, and Cook's distance. Global diagnostics include the Durbin-Watson statistic for autocorrelation, a runs test for randomness, and skewness/kurtosis of the residual distribution.
Load the preset datasets to see healthy vs. pathological residual patterns. The "Heteroscedastic" preset shows increasing residual spread — violating a key OLS assumption. The "Nonlinear Pattern" preset shows residuals with a systematic curve — the linear model is fundamentally wrong. Check the example with realistic values before reporting. Use the steps shown to verify rounding and units. Cross-check this output using a known reference case. Use the example pattern when troubleshooting unexpected results.
Many analysts run a regression, report R², and stop. But a high R² with violated assumptions produces misleading confidence intervals, incorrect p-values, and unreliable predictions. Residual analysis is the safety check.
This calculator puts standard after-regression diagnostics in one place: outlier detection via standardized residuals, influence via Cook's distance, autocorrelation via Durbin-Watson, and normality via skewness/kurtosis. The visual residual bars immediately show patterns that numbers alone might hide.
Residual: eᵢ = yᵢ − ŷᵢ. Standardized: eᵢ* = eᵢ / (s√(1−hᵢᵢ)). Leverage: hᵢᵢ = 1/n + (xᵢ−x̄)²/Sxx. Cook's D: Dᵢ = eᵢ*²·hᵢᵢ / (p(1−hᵢᵢ)). Durbin-Watson: d = Σ(eᵢ−eᵢ₋₁)²/Σeᵢ².
Result: R² = 0.9997, RMSE = 0.117, Durbin-Watson = 2.14, all |std. residuals| < 2.0, max Cook's D = 0.32
Residuals show no pattern, Durbin-Watson near 2.0 (no autocorrelation), no outliers or influential points. This is a healthy regression with all assumptions met.
OLS regression assumes: (1) Linearity — the true relationship is linear. (2) Independence — residuals are uncorrelated. (3) Homoscedasticity — residual variance is constant. (4) Normality — residuals are normally distributed. Each assumption maps to specific diagnostic tests.
Linearity: Plot residuals vs. predicted values. Random scatter = good. Curves = consider polynomial terms. Independence: Durbin-Watson tests first-order serial correlation. Homoscedasticity: Look for fan shapes in residual plots. Normality: Check skewness and kurtosis.
An outlier has a large residual — the model predicts poorly for that point. A high-leverage point has an extreme X value. An influential point changes the regression substantially when removed. A point can be high-leverage without being influential (if it falls on the trend), or an outlier without being influential (if leverage is low). Cook's distance captures the combined effect.
Non-linearity: Add polynomial terms or transform variables. Heteroscedasticity: Use weighted least squares or robust standard errors. Autocorrelation: Use generalized least squares or add lag terms. Non-normality: Transform Y (log, sqrt) or use robust regression. Outliers: Investigate data quality, use robust methods (LAD, Huber), or report with and without.
Raw residuals (eᵢ = yᵢ − ŷᵢ) retain Y-units. Standardized residuals divide by estimated standard deviation accounting for leverage, converting to a unit-free scale where values beyond ±2 indicate potential outliers.
DW tests for first-order autocorrelation in residuals. DW ≈ 2 means no autocorrelation. DW << 2 suggests positive autocorrelation (consecutive residuals similar). DW >> 2 suggests negative autocorrelation (consecutive residuals alternate sign).
The traditional rule: Cook's D > 1 is influential. A stricter rule uses D > 4/n. Remove or investigate high-Cook's-D points — they may be data errors, outliers, or genuinely different observations that shouldn't be modeled together.
Leverage measures how far xᵢ is from x̄. Extreme X values have high leverage: they have outsized potential to pull the regression line. High leverage isn't always bad — compare Cook's D to see if the point actually affects the regression.
Non-normal residuals don't affect coefficient estimates but do affect confidence intervals and p-values. Check skewness (should be near 0) and kurtosis (should be near 0 for excess kurtosis). With n > 30, the Central Limit Theorem provides some protection.
Look for a fan or funnel shape in the residual visual — residuals getting larger (or smaller) as X increases. Our visual bars show this pattern clearly. Formal tests include Breusch-Pagan and White's test.