Should a 1-5 satisfaction rating enter a regression as one numeric predictor or as four dummies?

## The two codings A satisfaction rating takes the values 1, 2, 3, 4, 5. There are two natural ways to put it into a regression. **Numeric coding.** Enter the rating as one column of numbers. The model gets one slope, `b`, and the fitted contribution is `b * rating`. **Dummy coding.** Treat the rating as a categorical variable with five levels. With an intercept you add four dummies against a reference level — say rating 1 — and each of the other levels gets its own coefficient, interpreted as that level minus rating 1. ## What the numeric coding assumes The numeric coding is not neutral. It imposes two claims at once. First, **equal spacing**: moving from 1 to 2 is the same distance as moving from 3 to 4. That is a claim about the measurement, and for a rating scale it is a convention rather than a fact — respondents do not calibrate their 2s and 4s to a common ruler. Second, **linearity in the outcome**: each additional rating point shifts the expected outcome by exactly `b`, everywhere on the scale. Real rating scales frequently violate this. Top-box behaviour is the classic case: 1 through 4 barely differ in downstream behaviour while 5 stands well apart, so the true pattern is flat then a jump. A single slope fitted to that pattern splits the difference and understates both the flatness and the jump. The dummy coding assumes neither. It lets the five levels sit at any five heights, in any order. It does *not* even impose monotonicity, which is a mild disadvantage if you have strong prior reason to expect it. ## The degrees-of-freedom tradeoff The numeric coding spends **one** parameter; the dummy coding spends **four**. Three consequences follow. With plenty of data, four parameters are cheap and the flexibility is close to free. With a small sample, or with a scale where some levels are rare — ratings of 2 often are — those four coefficients are estimated from few observations each and come back with wide standard errors. A level with a handful of rows produces a contrast so noisy it carries little information, and if a level has no observations at all the model cannot estimate it. The numeric version also concentrates all the evidence into a single slope, which is more precisely estimated and easier to communicate: 'each extra rating point is worth X'. Precision bought with an assumption is only a bargain if the assumption holds. Finally, the numeric version extrapolates naturally to a rating outside the observed range while the dummy version cannot represent an unseen level at all — a real consideration when scoring new data. ## Deciding between them The two models are **nested**: the numeric specification is the dummy specification with the restriction that the four contrasts lie on an equally spaced straight line. That makes the comparison a standard nested-model test. Fit both, and test the 3 extra degrees of freedom that the dummy model uses beyond the numeric one — 4 parameters versus 1 — with an F-test comparing the two residual sums of squares. A small p-value says the equal-step pattern is not tenable; a large one says the simpler coding is not detectably worse. Don't stop at the test. Plot the four estimated contrasts against the rating level with their confidence intervals. The shape tells you *how* the linear coding fails when it does: a jump at the top, a threshold in the middle, a U-shape. That shape is often the finding the stakeholder actually wants. A middle course is common in practice. If the pattern is flat-then-jump, collapse the scale to a small number of meaningful groups — for example a single 'top box' indicator — which costs one parameter and captures the real structure. Grouping decisions should be made on substantive grounds or on a plot of the coefficients, not by trying every grouping and keeping the one with the best p-value. ## What interviewers listen for The weak answer is a rule ('always use dummies for categories') with no mention of what the numeric coding costs or buys. The strong answer names the assumption by its content — equal spacing and a constant per-point effect — says how many degrees of freedom each coding spends, and gives a concrete way to check the restriction rather than asserting one coding is correct. Mentioning sparse levels and unseen levels at scoring time shows you have actually fitted these models on real data.