Home / Lean Six Sigma / ANOVA, regression & simulation
Find out which process factors move the response, how much each one weighs, which combination gives the best result and with what margin of error, from the data you already have. Paste from Excel one column for the response (the result) and one for each factor, one row per run or lot; the page builds the equation, shows which factors really matter and simulates “what if I change the temperature?”.
Pre-sizing and estimating tool: results do not replace design, calculations or reports signed by a licensed professional, nor verification against the applicable codes and standards.
With 1 predictor: fitted line with confidence (CI) and prediction (PI) bands and the equation. With several: observed vs fitted. Tap a point to exclude it from this chart only and watch the line and R² change.
How to read: with 1 predictor, X on the horizontal axis and Y on the vertical; the line is the model, the narrow band (CI) is where the true mean lies and the wide one (PI) where a next individual measurement will fall. With several predictors, each point is a table row: predicted value on the horizontal, measured on the vertical. Points hugging the diagonal = a good model; points far from it are cases the model does not explain.
Move each factor and see the predicted response with CI and PI. Each curve shows one factor’s effect with the others fixed; the green triangle marks the optimum.
How to read: at the top, the predicted response for the chosen values, with CI (mean) and PI (individual part). In each mini-chart, the horizontal axis is one factor and the curve shows the response when only that factor changes; the red line is the slider’s current value. A steep curve = a factor with a big effect; a flat curve = little effect.
Predicted response for two continuous factors; the others stay at the simulator values.
How to read: it is a map: the two axes are two factors and the color (or height, on the surface) is the predicted response. Equal colors form bands of the same result; the darkest or highest region is where the best setting lies. Tilted bands show the two factors compensate each other; if the contours close into an “eye”, there is an optimum inside the tested region.
Pareto of the effects (|t| for each term) with the critical t line, or normal plot of the effects.
How to read: each bar is a model term (factor, interaction or square), strongest to weakest; its length is the effect size in units of error. Bars that cross the vertical line (critical t) are real effects; those that stop short of it may be noise and are candidates to drop from the model.
Increase in R² when the factor enters the model last (incremental impact).
How to read: each bar says how many points of R² the model would lose if that factor were removed. A long bar = an indispensable factor; a near-zero bar = a factor that adds little, because another already carries the same information. Use it to decide where to invest in process control.
Fitted means for each factor (others at the mean) and interaction curves with the other factor low and high.
Interactions
How to read: in the main effects, each panel shows the mean response as a factor goes from the lowest to the highest tested value: the steeper the line, the more the factor weighs. In the interactions, the two lines are the other factor low and high: parallel = each factor acts on its own; spreading or crossing = the best setting of one depends on the other.
Normal probability, versus fits (with LOWESS), histogram and versus order. Red points: |standardized residual| > 2, with the row number.
How to read: a residual is what the model got wrong on each row. A trustworthy model leaves errors with no pattern: points on the line in the normal probability plot; a cloud of constant width around zero against the fits (funnel = growing variance, curve = a missing term); a bell-shaped histogram; and no trend against the order (a trend = something changed over time).
Looks for curvature or a funnel against each factor.
How to read: one panel per predictor: the horizontal axis is the factor and the vertical one the model error. A shapeless cloud around zero = the factor is well represented. A U or ∩ curve = that factor’s quadratic term is missing; a funnel = the spread changes with the factor (consider the log transformation).
Leverage vs standardized residual; bubble size is Cook’s distance.
How to read: each bubble is a table row. Horizontal axis: leverage (how unusual the X combination is; to the right = an isolated point). Vertical axis: standardized residual (how far the measured Y is from the predicted; beyond ±2 = a surprise). A big bubble = a point that alone changes the equation: check that the measurement is correct before trusting the model.
VIF computed with centered continuous predictors.
How to read: Coef = how much the response changes when the factor rises 1 unit (others fixed); SE = the coefficient’s uncertainty; t = Coef ÷ SE; p-value < 0.05 = proven effect; VIF above 5 to 10 = the factor is tangled with another and the coefficient is unstable.
How to read: S = typical error of a prediction, in the response unit; R² = % of the variation explained; R²(adj) discounts useless terms; R²(pred) estimates accuracy on new data: well below R² = the model memorized the data (too many terms).
Adjusted SS for each term (type III). Lack of fit appears when there are repeated runs at the same X.
R: large standardized residual; X: high leverage.
Best factor combination within the data range, with desirability d (Derringer and Suich).
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R² always rises when a term is added, even a useless one. Adjusted R² accounts for the number of terms and rises only if the term really improves the model. Compare models with the adjusted value and also check predicted R²: if it is much lower, the model memorized the data.
It is the range where a single new measurement should fall. The confidence interval is only for the mean. To know whether a part or batch will meet the specification, use the prediction interval, which is wider.
Minitab’s Assistant recommends about 40 rows for up to 3 terms, 45 for up to 6 and 50 for 7 or 8. With fewer, R² can vary a lot from one sample to another. The page shows the recommended number for your model in the report card.
Investigate the cause: typo, gauge, special event or abnormal condition. Do not delete a point just because it is unusual. Tap it in the chart to see how much the equation changes without it.
Drag each factor’s slider and see the predicted response with CI and PI; the curves show each factor’s effect with the others fixed. The optimizer searches, within the data range, for the combination that maximizes, minimizes or hits the target. Confirm the optimum with a test batch.
Use a quadratic term when the response has a maximum or minimum inside the range (a curve in the residual plot) and an interaction when one factor’s effect depends on another’s level (non-parallel lines). Stepwise selection helps keep only the terms that matter.
Yes. Coefficients, standard errors, p-values, VIF, ANOVA, predicted R², Durbin-Watson and intervals were checked against published examples (Minitab, Brownlee and NIST certified data) and statsmodels.
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