What is regression analysis?
A scatter diagram shows whether two variables move together. Regression analysis goes further: it estimates how much Y changes for each unit of every X while holding the others constant, tests which effects are real, and predicts results for new settings. In quality work it is used to find key process inputs, set operating windows and build transfer functions in Six Sigma projects.
Reading the regression output
- Coefficients: the change in Y per unit of X. The p-value tests whether it differs from zero.
- R² and adjusted R²: how much of the variation is explained.
- S: the typical size of a residual, in the units of Y.
- ANOVA F-test: whether the model as a whole explains anything.
- VIF: multicollinearity between predictors.
- Residual plots: whether the model assumptions hold.
Regression example
The example models the bond strength of 30 bonded assemblies from cure temperature, cure time and resin moisture. The model explains 95.7% of the variation (adjusted R² 95.2%). All three predictors are significant: each additional °C adds about 0.21 MPa, each minute about 0.08 MPa, and each additional 0.1% of resin moisture removes about 1.5 MPa. The VIFs are close to 1, so the effects are not confounded. At 195 °C, 30 minutes and 0.08% moisture the model predicts 17.5 MPa, with a 95% prediction interval of 16.2 to 18.8 MPa for a single assembly.
Common pitfalls
- Extrapolation: predictions outside the range of the data are unreliable.
- Correlation is not causation: happenstance data can mislead; confirm with a designed experiment.
- Overfitting: keep at least 10 observations per predictor.
- Ignoring curvature and interactions: a linear model may not be enough. Check the residuals.
How to use this Regression Analysis tool
- Name the variables. Name the response (Y) and add up to five predictors (X).
- Paste the data. Paste one row per observation with Y and the X columns, straight from Excel.
- Read the model. Check the equation, R², the model p-value and the p-value of each coefficient.
- Check the residuals. Look at the residual plots and the unusual observations before trusting the model.
- Predict. Enter X values to get the predicted Y with confidence and prediction intervals.
- Export. Export coefficients, ANOVA, plots and fitted values to PDF or Excel.
Frequently asked questions
What is regression analysis?
Regression analysis fits an equation that describes how a response (Y) depends on one or more predictors (X). Linear regression by least squares is the most common form. It lets you quantify the effect of each predictor, test whether it is significant and predict Y for new X values.
What is the difference between R² and adjusted R²?
R² is the share of the variation in Y explained by the model. It always increases when you add predictors, even useless ones. Adjusted R² penalizes extra predictors, so use it to compare models with a different number of terms.
How do I know which predictors matter?
Look at the p-value of each coefficient: below 0.05 means the predictor has a significant effect, given the other predictors in the model. Remove non-significant predictors one at a time and refit.
What is a VIF?
The variance inflation factor measures multicollinearity: how much a predictor is explained by the other predictors. A VIF above 5 (some use 10) means the coefficients are unstable and hard to interpret. Remove one of the correlated predictors or combine them.
Why check the residual plots?
Regression assumes the residuals are random, have constant spread and are roughly normal. A curved pattern in residuals versus fitted values means a missing term; a funnel shape means non-constant variance; points far from the line in the normal plot are outliers. The tool flags standardized residuals beyond ±2 and high-leverage points.
What is the difference between a confidence and a prediction interval?
The confidence interval is the uncertainty of the average Y at given X values. The prediction interval is the range for a single new observation, and is always wider because it includes the scatter of individual values.