Multiple Regression

Fitting several drivers at once so each one is read holding the others still.

Also known as: multiple regression, multiple linear regression, multivariable regression

Multiple regression extends linear regression to several predictors at once:

y = a + b1·x1 + b2·x2 + ... + bk·xk

Each coefficient is read holding the other predictors still. With revenue as the outcome and discount, traffic and season as predictors, the discount coefficient answers “among rows with similar traffic and season, what does a one-point discount come with?” That conditional reading is the whole point of the tool, and the reason people fit it.

Why it is worth the trouble: a coefficient from a single-predictor model silently absorbs the effect of everything correlated with that predictor. Discount and traffic probably move together, so a single-predictor discount coefficient is partly a traffic coefficient. Fitting both separates them — at least inside the model.

What the separation does and does not buy:

It controls only for what you measured. If a variable you did not observe drives both traffic and discount, it stays inside both coefficients (confounding variables). Adding predictors does not turn an observational fit into a causal estimate.
It controls only up to the form you fitted. If the true relationship is curved, or the effect differs by segment, a straight additive coefficient is an average over cases that behave differently (segmentation).
Correlated predictors make each individual coefficient unstable without introducing a bias into the fitted values. When two predictors are near-duplicates, their coefficients swing wildly while the predictions stay steady — the classic symptom of collinearity. The check is to refit without the columns a coefficient is entangled with and see whether the story survives.
Holding others still is not the same as changing one thing. The coefficients answer a “comparing like with like” question about the data you have, not “what would happen if I moved this” (causal inference).

Two reporting habits. Quote a coefficient together with its uncertainty rather than as a bare number, and check the residuals before you quote anything, because the interpretation of every coefficient depends on the specification being right.

For the pooled-versus-split question — one model or one per group — the answer is often the same as in Simpson’s paradox: fit both and see whether they disagree.