Coefficient Interpretation

## What a dummy variable is A categorical predictor such as day of week has no numeric meaning — coding Monday as 1 and Tuesday as 2 would tell the model that Tuesday is 'one more' than Monday and that the step from Monday to Tuesday is the same size as the step from Thursday to Friday. Neither claim is intended. Instead each category is represented by a **dummy** (also called an indicator): a column that is 1 when the row belongs to that category and 0 otherwise. With seven days you create **six** dummies, not seven. One level is left out; it is called the **reference level**, the **baseline**, or the omitted category. Here that level is Monday. ## Where Monday went Monday has no column, but it has not left the model. A Monday row has 0 in all six day columns, so the only thing describing it is the intercept. That is what makes the intercept the baseline: with an intercept plus dummies for Tuesday through Sunday, the intercept is the expected outcome for a Monday row whose other predictors are all zero. Every other day's coefficient is then measured *from that anchor*. Write the fitted equation for a row with no other predictors: ``` yhat = b0 + b_Tue*Tue + b_Wed*Wed + ... + b_Sat*Sat + b_Sun*Sun ``` A Monday row gives `yhat = b0`. A Saturday row gives `yhat = b0 + b_Sat`. Subtracting, `b_Sat = (Saturday mean) - (Monday mean)`. That is the whole interpretation. ## Units and direction The coefficient is in the outcome's units. If the outcome is orders per day and `b_Sat = -120`, Saturday is estimated to see 120 fewer orders than Monday. The sign is relative to the baseline only — a negative dummy coefficient does not mean the level is bad or small in absolute terms, it means it sits below the omitted level. The usual ceteris paribus clause applies: the comparison holds every other predictor in the model at the same value. If the model also contains, say, a promotion indicator, `b_Sat` is the Saturday-minus-Monday gap *among days with the same promotion status*, not the raw gap in the data. ## Comparing two non-reference levels Because each coefficient is measured from Monday, the difference between two other days is the difference of their coefficients: ``` Saturday - Sunday = b_Sat - b_Sun ``` The point estimate is easy. The **standard error** is not: `SE(b_Sat - b_Sun)` needs the variances of both estimates *and* their covariance, which is why you cannot eyeball it from the two printed standard errors. The practical routes are to compute the standard error of that linear contrast properly, or simply to refit the model with Sunday as the reference level, which makes the Saturday coefficient *be* the Saturday-minus-Sunday contrast with its correct standard error attached. ## What a significance test on one dummy does and does not say The t-statistic printed beside `b_Sat` tests the null hypothesis that Saturday and Monday have the same expected outcome. Three consequences follow, and interviewers probe all three. First, a non-significant Saturday coefficient does **not** mean Saturday is unremarkable — it means you could not distinguish it *from Monday*. Saturday might sit far from Wednesday. Second, significance depends on which level you omitted. Choose a different baseline and the same data produce a different set of individual t-tests, because they are tests of different comparisons. Third, the question 'does day of week matter at all?' is a joint question about all six coefficients together, not a scan of six individual t-tests. That joint test is unchanged by the choice of baseline. ## Choosing a reference level Nothing in the mathematics prefers one level. The fit is identical whichever you drop. But the *readability* of the output is not identical, so pick a baseline that makes the contrasts you care about the ones printed: a level with plenty of observations (so the anchor is precisely estimated, since every contrast inherits its noise), a level that is a natural status quo or control, and — where there is an order — one end of that order rather than the middle. ## Common mistakes Reading `b_Sat` as Saturday's average outcome is the most common error; that quantity is `b0 + b_Sat`, and only when the other predictors are zero. Reporting the coefficient as a raw fact about Saturday without naming the baseline is the second; a coefficient with no stated reference level is uninterpretable, which is why written model summaries should always say which level was omitted.