Why center age and income before adding their interaction to a regression?

## The interpretability problem In `y = b0 + b1*age + b2*income + b3*(age * income) + error` the effect of age is `b1 + b3*income`. The printed `b1` is that effect evaluated at `income = 0`. For an indicator partner, zero is a real group. For income it is not: nobody in the sample has zero income, so `b1` describes a hypothetical person outside the data, and its sign can even look absurd relative to the effect anywhere in the observed range. Centering fixes this by redefining the variables. Let `age_c = age - mean(age)` and `inc_c = income - mean(income)`, then fit `y = a0 + a1*age_c + a2*inc_c + a3*(age_c * inc_c) + error` Now `a1` is the effect of age at *average* income, and `a2` is the effect of income at *average* age - both statements about points the data actually covers. ## What changes and what does not This is the part interviewers probe, because it separates people who have done it from people who have read about it. Unchanged: - The interaction coefficient: `a3 = b3` exactly, along with its standard error and t-statistic. - The fitted values, residuals, R-squared and the residual standard error. The two models span the same column space, so they are the same model in different coordinates. Changed: - The intercept, which now describes an average-age, average-income case rather than a zero-zero case. - The two main effects and their standard errors, because they are now anchored at a different point. A quick sanity check for anyone doubtful: expand `(age - m)(inc - n)` and you get `age*inc` minus linear terms plus a constant, so the centered product differs from the raw product only by a linear combination of variables already in the model. That is why only the coefficients on those lower-order terms move. ## The collinearity story, told accurately A raw product term is mechanically correlated with its own components - if age runs from 30 to 60, `age * income` moves almost in lockstep with income. This shows up as large variance inflation factors and wide standard errors on the main effects, and it can make the fit numerically fragile. Centering largely removes that correlation, because a mean-centered product is roughly uncorrelated with its centered components for symmetric distributions. But this is *nonessential* collinearity: an artefact of where the origin sits, not information missing from the data. The proof is that the interaction's own standard error does not budge. So centering does not make the model better able to separate age from income; if those two are genuinely correlated in the population, they remain just as entangled after centering. Presenting a VIF drop as evidence that a collinearity problem was solved is a misreading interviewers listen for. ## When not to center, or to center elsewhere - Do not center a 0/1 indicator: zero already means something concrete, and centering replaces a clean group contrast with a fractional value nobody has. - Center at a decision-relevant value rather than the mean when one exists - a policy threshold, a target price, the start of a program. The mean is a convenient default, not a rule. - If you report predictions, nothing changes; if you report coefficients, say explicitly what the anchor is. "Effect of age at mean income" is the sentence that makes the table readable. - Standardising (dividing by the standard deviation as well as subtracting the mean) is a further step with its own tradeoffs and is not required to get the interpretability benefit. ## A common failure in practice A subtle bug: centering using the full-sample mean and then evaluating on a later cohort whose mean differs. The coefficients still refer to the *original* centering constant, so "effect at the mean" silently means the old mean. Store the constants with the model and apply the same ones everywhere, exactly as you would with any other fitted transformation. ## What interviewers listen for Two benefits (interpretable main effects, reduced artificial collinearity), one firm invariance claim (the interaction coefficient, its standard error and the fit are untouched), and one limit (it does not cure real collinearity). A candidate who claims centering improves fit or removes genuine multicollinearity has the concept backwards.