Diagnosing Non‑Normal Residuals in Minitab Regression: A Step‑by‑Step Guide

Recognizable Condition

After fitting a regression model in Minitab (Stat → Regression → Regression), the residual diagnostics reveal a departure from normality. The Normal Probability Plot of the residuals curves away from the straight line, the histogram shows clear skew, and the Anderson‑Darling test reports a p‑value below 0.05 (often <0.01). These three signals together indicate that the residuals are not normally distributed, violating a core assumption for inference on coefficients and prediction intervals.

Cause / Diagnostic Table

Potential Cause	Diagnostic Signature
Outliers or influential points	Standardized residuals >|2|; high leverage (Hi > 2p/n) or large Cook’s distance.
Incorrect model form (missing curvature or interactions)	Systematic pattern (U‑shape, curve) in Residuals vs Fits plot.
Heteroscedasticity (non‑constant variance)	Funnel or fan shape in Residuals vs Fits; spread of residuals grows with fitted values.
Non‑normal underlying response distribution	Skewed histogram of residuals; S‑shaped Normal Probability Plot even after model corrections.
Measurement error or data‑entry mistakes	Unexpected clusters, gaps, or duplicate values in raw data; isolated extreme residuals.
Autocorrelation (time‑ordered data)	Pattern in Residuals vs Order plot; Durbin‑Watson statistic far from 2.

Ordered Checks (Run in Minitab)

1. Fit the model and store residuals.
   Stat → Regression → Regression → in Response, in Predictors → Storage → Residuals → OK. Requires write permission on the active worksheet.
2. Normal Probability Plot with Anderson‑Darling test.
   Graph → Probability Plot → Single → Residuals (stored column) → OK. Note the AD p‑value displayed in the output pane.
3. Histogram of residuals.
   Graph → Histogram → Simple → Residuals → OK. Look for skew or heavy tails.
4. Residuals vs Fits scatterplot.
   Graph → Scatterplot → Simple → Y: Residuals, X: Fits → OK. Check for curvature, funnel, or random cloud.
5. Residuals vs Order (if data are time‑ordered).
   Graph → Scatterplot → Simple → Y: Residuals, X: Row number → OK. Also run Stat → Time Series → Autocorrelation on residuals.
6. Outlier and influence diagnostics.
   Stat → Regression → Regression → Diagnostics → Standardized residuals, Leverages, Cook’s distance → OK. Flag observations exceeding conventional cut‑offs.
7. Formal heteroscedasticity check (optional).
   Use Stat → Regression → Regression → Options → Test for constant variance (Breusch‑Pagan) if available in your version.

Fixes Tied to Findings

Apply the remedy that matches the dominant diagnostic signature, then repeat the checks from step 2 onward.

• Outliers / influential points – Verify the raw entry (measurement log, sensor record). If the point is a genuine error, delete or correct it; if it is a real but extreme observation, consider robust regression (Stat → Regression → Robust Regression) or a weighted least‑squares fit with weights = 1/variance estimate. Risk: removing points without subject‑matter justification biases coefficient estimates.
• Missing curvature or interaction – Add polynomial terms (e.g., X^2) or interaction terms (X1*X2) via Stat → Regression → Regression → Model → Terms. Use Stat → Regression → Stepwise or Best Subsets to explore candidate terms.
• Heteroscedasticity – Apply a variance‑stabilizing transformation to the response (log, sqrt, or Box‑Cox). In Minitab: Calc → Calculator → log10() or Stat → Regression → Box‑Cox Transformation. Alternatively, fit a weighted least‑squares model (Stat → Regression → Regression → Options → Weights) using the inverse of the fitted variance.
• Non‑normal response distribution – Transform the response (log, sqrt, Box‑Cox) as above. After transformation, re‑fit the model and re‑examine the Normal Probability Plot; a more linear plot and AD p‑value >0.05 indicate improvement.
• Autocorrelation – Add lagged predictors (e.g., Y(t‑1)) or switch to a time‑series method (Stat → Time Series → ARIMA).

Concrete Example

Dataset: 30 observations of turbine efficiency (Eff) vs. inlet temperature (Temp). Initial simple linear regression yields:

• AD p‑value = 0.018
• Histogram of residuals right‑skewed
• Residuals vs Fits shows a widening funnel

Diagnosis: heteroscedasticity + non‑normal response. Fix: apply a natural‑log transformation to Eff (Calc → Calculator → ln(Eff)), store as lnEff, then re‑run regression with lnEff as response. Post‑fix diagnostics:

• AD p‑value = 0.12
• Normal Probability Plot nearly linear
• Residuals vs Fits appears as a random cloud

Result: inference on temperature coefficient is now valid; prediction intervals on the original scale can be obtained by back‑transforming (exponentiating) the fitted values and intervals.

Escalation Criteria

• Anderson‑Darling p‑value remains <0.01 after all reasonable transformations and model expansions.
• Systematic residual patterns persist despite adding curvature, interactions, or weights.
• Sample size <15 observations, limiting the power of diagnostic tests and the reliability of transformations.

When any of the above occur, consult a statistician or consider alternative frameworks such as non‑parametric regression (e.g., Stat → Regression → Nonparametric Regression), quantile regression, or robust methods beyond Minitab’s standard regression module.

Limitations & Practical Verification

• Assumes Minitab 19 or later where the Anderson‑Darling test for residuals and the Probability Plot are available; earlier versions may use the Ryan‑Joiner test or lack the diagnostic menu.
• Automated outlier removal without engineering review can introduce bias; always document any exclusions or transformations.
• Transformation changes the interpretation of coefficients (e.g., log‑response yields multiplicative effects). Verify that the transformed model still answers the original engineering question.

Verification step: After each fix, repeat the Normal Probability Plot (Graph → Probability Plot) and confirm that the AD p‑value rises above 0.05 and the plot aligns with the reference line. If the p‑value improves but the Residuals vs Fits still shows a funnel, iterate with a weighted fit.

Rollback Guidance

If a transformation (e.g., log) degrades model fit or makes interpretation untenable, revert by deleting the transformed column and re‑using the original response column. No persistent state change occurs beyond the worksheet, so a simple Edit → Undo or column deletion restores the prior analysis.