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Simple and multiple linear regression

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.

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.

CheckRevised 11/03/2026
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The optimizer searches within the observed range of each factor; outside it the model was not tested.

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—How much the model explains
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Fitted model plot

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.

Tap the chart to see the values.

“What if” simulator: prediction profiler

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.

Response contour and surface

Predicted response for two continuous factors; the others stay at the simulator values.

Tap the chart to see the values.

Standardized effects

Pareto of the effects (|t| for each term) with the critical t line, or normal plot of the effects.

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How much each factor adds

Increase in R² when the factor enters the model last (incremental impact).

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Main effects and interactions

Fitted means for each factor (others at the mean) and interaction curves with the other factor low and high.

Interactions

Residual plots: four in one

Normal probability, versus fits (with LOWESS), histogram and versus order. Red points: |standardized residual| > 2, with the row number.

Residuals versus each predictor

Looks for curvature or a funnel against each factor.

Outliers and influential points

Leverage vs standardized residual; bubble size is Cook’s distance.

Tap the chart to see the values.

Regression equation

Coefficients

VIF computed with centered continuous predictors.

Model summary

Analysis of variance

Adjusted SS for each term (type III). Lack of fit appears when there are repeated runs at the same X.

Model selection

Fits and diagnostics for unusual observations

R: large standardized residual; X: high leverage.

Prediction of new values

Response optimizer

Best factor combination within the data range, with desirability d (Derringer and Suich).

Next step

Take this page’s results to another calculator with the fields already filled in.

Correlation and matrix plot →Simple chart for presentations (scatter plot) →ANOVA (categorical factors) →Calibration curve →Simulate the response with Monte Carlo (fitted equation) →

The calculation report gathers this page's inputs, results, equations, assumptions and references, with fields for project, engineer and revision. The full report also includes the charts and interpretations, in color. In the print dialog, choose Save as PDF.

Method, assumptions and references

Equations used

y = Xb + e; b = (XᵀX)⁻¹Xᵀy via QR decomposition; S = √[SSE ÷ (n − p)]
SE(bⱼ) = S√[(XᵀX)⁻¹]ⱼⱼ; T = bⱼ ÷ SE(bⱼ); VIFⱼ = 1 ÷ (1 − R²ⱼ)
R² = 1 − SSE/SST; R²adj = 1 − MSE ÷ [SST/(n − 1)]; PRESS = Σ[eᵢ ÷ (1 − hᵢ)]²; R²pred = 1 − PRESS/SST
CI for the mean: ŷ ± t·S√(x₀ᵀ(XᵀX)⁻¹x₀); PI: ŷ ± t·S√(1 + x₀ᵀ(XᵀX)⁻¹x₀), t with n − p DF
rᵢ = eᵢ ÷ [S√(1 − hᵢ)]; Dᵢ (Cook) = rᵢ²hᵢ ÷ [p(1 − hᵢ)]; DW = Σ(eᵢ − eᵢ₋₁)² ÷ Σeᵢ²
Term: F = [(SSE without the term − SSE) ÷ DF] ÷ MSE; stepwise: enter if p < α, remove if p > α (hierarchy kept)

Assumptions and limits

  • Residuals independent, zero-mean and with constant variance; normality matters for small samples (fewer than 15 rows).
  • The model is valid only within the observed factor range. A statistical relationship does not prove causation: confirm the effect with a designed trial.
  • Optimization is a numerical search (grid + refinement) and gives a good combination, not necessarily the global optimum. Excluding points by tapping is only a visual test.

Standards and technical documents

  • Montgomery, D. C.; Peck, E. A.; Vining, G. G. Introduction to Linear Regression Analysis, 5th ed. Wiley, 2012
  • Kutner, M. H.; Nachtsheim, C. J.; Neter, J.; Li, W. Applied Linear Statistical Models, 5th ed. McGraw-Hill, 2005
  • Draper, N. R.; Smith, H. Applied Regression Analysis, 3rd ed. Wiley, 1998
  • Cook, R. D. Detection of influential observation in linear regression. Technometrics 19(1), 1977, p. 15–18
  • Durbin, J.; Watson, G. S. Testing for serial correlation in least squares regression II. Biometrika 38, 1951, p. 159–178
  • Derringer, G.; Suich, R. Simultaneous optimization of several response variables. Journal of Quality Technology 12(4), 1980, p. 214–219
  • Minitab. Assistant white papers: Simple Regression; Multiple Regression (critérios do cartão de verificação)
  • Brownlee, K. A. Statistical Theory and Methodology in Science and Engineering, 2nd ed. Wiley, 1965 (dados de stack loss)
  • NIST/SEMATECH e-Handbook of Statistical Methods, seção 4 (Process Modeling)

U.S. regulations

  • ISO 9001:2015, clause 9.1.3: analysis and evaluation
  • ASTM E3080: Standard Practice for Regression Analysis with a Single Predictor Variable
  • ISO/TR 10017:2003: Guidance on statistical techniques for ISO 9001:2000

Frequently asked questions

What is the difference between R² and adjusted R²?

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.

What is the prediction interval and when should I use it?

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.

How much data do I need for multiple regression?

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.

What should I do with unusual observations?

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.

How do I use the what-if simulator and the optimizer?

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.

When should I use quadratic terms and interactions?

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.

Do the results match statistical software?

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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