What it does

Multiple linear regression fits y = b₀ + b₁x₁ + … + bkxk by ordinary least squares and tests each coefficient. It quantifies how several formulation or process variables act together on a response and gives a prediction with uncertainty for new settings.

Typical uses

• Tablet hardness or disintegration as a function of compression force, binder level and moisture
• Drug release at a time point versus polymer ratio, plasticiser and coating level (screening designs)
• Particle size or encapsulation efficiency versus stirring speed, surfactant and lipid concentration
• Preliminary analysis before response-surface modelling; add squared or interaction columns yourself if needed

How to read the output

• Coefficients: expected change in y per unit of that x with the others held constant; p < α marks a term the data support.
• Model F test: whether the predictors jointly explain anything. R² grows with every added term; adjusted R² penalises useless ones.
• VIF: above 10 the predictors are strongly correlated and individual coefficients become unstable even when the model predicts well.
• Residual plot: standardized residuals should scatter randomly around zero; a curve suggests a missing quadratic term, a funnel suggests non-constant variance. Durbin-Watson near 2 means no serial correlation in the entered order.
• Prediction: the confidence interval covers the mean response at those settings; the wider prediction interval covers a single future observation. Both are unreliable outside the data range.

Prerequisites

• Clearly more observations than parameters (a rule of thumb is at least 5–10 per predictor)
• Linear relationship, independent observations, roughly constant variance and normal residuals
• No predictor may be constant or an exact combination of the others; the tool refuses such a design

References: Draper N.R., Smith H. (1998) Applied Regression Analysis, 3rd ed., Wiley, doi:10.1002/9781118625590; Montgomery D.C., Peck E.A., Vining G.G. (2012) Introduction to Linear Regression Analysis, 5th ed., Wiley. Results are compared with statsmodels OLS in the unit tests.