Why must a linear model be handed an explicit product column to capture a discount-by-loyalty-tier interaction?

## Additivity is an assumption, not a technicality Write the model out: `demand = b0 + b1*discount_depth + b2*is_gold + error` Whatever numbers the fit chooses, the structure says something strong about the world: the contribution of discount depth is added on, and the contribution of tier is added on, and neither term knows anything about the other. The model can say gold customers buy more overall, and it can say deeper discounts sell more units, but it cannot say that deep discounts barely move gold customers who would have bought anyway while transforming bronze-tier behaviour. That statement requires a term whose value depends on both inputs simultaneously. That is exactly what an **interaction** is, and in a linear model it is created as a **product column**: multiply the two feature columns together, row by row, and add the result to the design matrix. Because the multiplication happens before fitting, the model remains linear in its parameters and the ordinary least-squares machinery is unchanged — this is basis expansion again, with the basis function being a product of two features rather than a power of one. ## Categorical partners multiply the columns If tier has four levels, it is already represented by three indicator columns (one level absorbed into the intercept). Interacting a continuous discount depth with that tier means three product columns, one per indicator. The result is a model that can trace a different discount response inside each tier while sharing everything else. Interacting two continuous features gives a single product column and a fitted surface that twists rather than staying flat in the way a plane does. ## The cost side: columns and cells Two costs decide how far you take this. **Column count grows quadratically.** With 30 features there are 30 x 29 / 2 = 435 distinct pairwise products, before considering three-way terms or products with the polynomial expansions of the same features. Adding them all is rarely defensible: most encode no real mechanism, and each is a parameter estimated from the same finite data. **Every product column needs data in its cells.** A discount-by-tier interaction is estimated from customers who actually appear at various discount depths *within* each tier. If platinum customers were never given a deep discount, the corresponding product column is being fitted from a handful of rows, and the estimate will be unstable no matter how sensible the term looks on paper. Before adding an interaction, cross-tabulate and check the cells are populated. There is also a modelling convention worth knowing: the **hierarchy principle**, which says that if a product term is in the model you keep both of its parent terms too. Dropping a main effect forces the fitted surface through a constrained shape that the data rarely supports, and makes the fit depend on arbitrary details of how the inputs are scaled. ## Which interactions to build The interactions worth spending parameters on come from the domain, not from a search over all pairs. Promotions are the classic source: discount depth by customer tier, discount depth by channel, promotion by day-of-week. Ask the business owner where they believe one factor changes another's effect, build those, and leave the rest. ## Why other model families do not need this This is a limitation of the additive form specifically. A decision tree represents an interaction without being told: it splits on loyalty tier near the top, and then splits on discount depth separately inside each branch, so the discount response can differ by branch by construction. An ensemble of trees does this routinely across many feature pairs, which is a large part of why boosted-tree models often beat a hand-built linear model on tabular data with many interactions. The counter-argument for staying with the linear model and its hand-built terms is that the resulting model is small, stable, auditable and easy to reason about term by term — which matters when the model must be explained to a regulator, a pricing committee or an auditor. That is a deliberate trade: you pay in feature-engineering effort and in the interactions you failed to think of, and you are paid in transparency and stability. ## The one-line answer Additive means additive. A linear model can only express what its columns encode, so an effect that depends on two features at once exists only if you multiply those features into a column yourself.