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Regression Analysis in DMAIC: Simple and Multiple Regression

Simple and multiple regression in DMAIC Analyze — model building, R-squared, residual analysis, and using the equation to predict Y from X. It sits in Section VI (Analyze) of the 2022 CSSBB Body of Knowledge. For the underlying math, see the CQE page on linear regression →

What regression answers in Analyze

After the fishbone and multi-vari work suggest candidate Xs, regression quantifies the relationship: how much Y changes per unit change in X, how much of Y's variation the model explains, and whether the relationship is statistically significant. It shows association; causation still needs process knowledge or a designed experiment.

Simple linear regression: Ŷ = b₀ + b₁X b₁ = Σ(x − x̄)(y − ȳ) / Σ(x − x̄)² b₀ = ȳ − b₁x̄ R² = 1 − SSE/SST R²(adj) = 1 − [SSE/(n − p − 1)] / [SST/(n − 1)] (p = number of predictors) t for slope = b₁ / SE(b₁), with n − 2 df (simple regression)

Worked example

X = coating pan inlet temperature (°C), Y = coating defect rate (%), n = 8 (illustrative data) X: 40 42 44 46 48 50 52 54 Y: 3.9 3.6 3.5 3.0 2.9 2.5 2.4 2.0 x̄ = 47.0, ȳ = 2.975 Ŷ = 9.24 − 0.133X R² = 98.4%, R²(adj) = 98.1%, s = 0.090 Slope t = −19.3 (6 df), p < 0.001 → temperature is a significant predictor Prediction at X = 51 °C: Ŷ = 9.24 − 0.133(51) ≈ 2.44% Do NOT predict at 70 °C: that is outside the 40–54 °C range studied (extrapolation).

Multiple regression essentials

Adjusted R² penalizes extra predictors; plain R² never decreases when you add an X, so use R²(adj) (or predicted R²) to compare models with different numbers of terms.
Multicollinearity: correlated Xs inflate coefficient standard errors and make signs unstable. Check the variance inflation factor (VIF); values above about 5–10 are a common warning threshold.
Model building: stepwise or best-subsets selection, then confirm the reduced model makes engineering sense. Keep terms that are significant and meaningful.
Overall F-test tests whether any predictor is useful; individual t-tests test each coefficient given the others are in the model.
Coefficients in multiple regression are the change in Y per unit X holding the other Xs constant.

Residual analysis (check every model)

Residuals vs fits: random scatter with constant spread (no funnel = constant variance; no curve = correct form).
Normal probability plot of residuals: roughly straight line.
Residuals vs order: no trends or cycles (independence).
Unusual points: large standardized residuals (outliers) and high-leverage points; investigate before deleting anything.

Common exam traps

r vs R²: r = −0.99 in the example; R² = r² = 0.984 in simple regression. R² has no sign.
Significant ≠ important: with large n a tiny slope can be significant; judge practical size too.
Correlation is not causation: confirm the X with a controlled study (DOE) before building the Improve solution on it.
Attribute Y: a pass/fail Y calls for logistic regression, not ordinary least squares.

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More: DMAIC overview · Project charter · VOC & CTQ · SIPOC · MSA in DMAIC · Process capability · Data collection planning · Graphical analysis · Root cause analysis · Multi-vari analysis · Hypothesis testing · Regression · DOE · FMEA · Define tollgate · Change management · Financial benefits · Team dynamics · Improve tollgate · Piloting & scaling · Poka-yoke (BB) · Solution selection · Control plan · SPC in DMAIC · Sustaining gains

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