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In a sales model with a promotion x weekend interaction, what does the promotion coefficient mean?

level: middleimportance: must knowfreq 72%

answer

  1. there is no single promotion effect any more
  2. the slope depends on the other variable
  3. ask: at what value of weekend?
  4. the other group needs a sum of coefficients
  5. the product term is a difference of slopes

basics

~20 s

It is the promotion effect when the weekend indicator equals zero, that is on weekdays only. The interaction coefficient is the extra promotion effect on weekends, so the weekend promotion effect is the sum of the two.

solid answer

~50 s

Write the model as `sales = b0 + b1*promo + b2*weekend + b3*promo*weekend`. Taking the difference in predicted sales with and without a promotion gives `b1 + b3*weekend`, so `b1` is a conditional slope: the promotion effect at `weekend = 0`, meaning weekdays. On weekends the promotion effect is `b1 + b3`, and `b3` alone is the *difference* between the two promotion effects, not the weekend effect itself. That is why calling `b1` "the promotion effect" is wrong once a product term is in the model - there is no single promotion effect any more, only a slope that depends on the other variable. Practically I report both conditional effects with their uncertainty, and I test `b3` to ask whether the promotion works differently on weekends. Note the standard error of `b1 + b3` needs the covariance of the two estimates, not just their separate variances.

go deeper

for a junior

Be able to say that with a product term in the model the promotion effect is no longer one number, and that the plain promotion coefficient applies only when the other indicator is zero.

for a middle

Derive the conditional effect b1 + b3*weekend from the model equation, explain that the interaction is a difference of slopes, and note that the interaction reads symmetrically in both directions.

for a senior

Show how you would test the effect in the other group, including the covariance term or a recode-and-refit, and handle a null interaction as a power question rather than as evidence of equality.

for a principal

Own the reporting and decision framing: which conditional effects the business actually acts on, how many interactions you allow before the model is fitting noise, and how you keep segment-level claims honest.

## The model and the derivative Start from `sales = b0 + b1*promo + b2*weekend + b3*(promo * weekend) + error` where `promo` and `weekend` are 0/1 indicators. The effect of switching the promotion on is the difference between predicted sales at `promo = 1` and at `promo = 0`, everything else held fixed: `effect of promo = b1 + b3 * weekend` This single line answers the whole question. When `weekend = 0` the effect is `b1`; when `weekend = 1` it is `b1 + b3`. So `b1` is not an average or a headline effect - it is the effect *evaluated at a specific value* of the other variable, namely zero. ## Reading each coefficient - `b0` - predicted sales with no promotion on a weekday. - `b1` - promotion effect on weekdays (`weekend = 0`). - `b2` - weekend effect when there is no promotion (`promo = 0`). - `b3` - how much larger (or smaller) the promotion effect is on weekends than on weekdays. Equivalently, how much larger the weekend effect is under promotion. The interaction is symmetric: it is one number that both readings share. A candidate who says "`b3` is the promotion effect on weekends" has dropped `b1` and will report an effect that is often wildly wrong in size and sometimes wrong in sign. ## Why the zero point suddenly matters Without an interaction, a slope is the same everywhere and the value of the other predictors is irrelevant to it. With an interaction, the main effect is anchored at zero on the interacting variable. For a 0/1 indicator that anchor is meaningful - weekdays are a real group. For a continuous partner such as price or age, zero may be impossible or far outside the data, which makes the main effect a statement about a point nobody occupies. That is the motivation for centering continuous variables before multiplying them, so the main effect becomes the effect at the average of the partner rather than at zero. ## Testing the conditional effects Three different questions get three different tests: 1. Is the promotion effective on weekdays? Test `b1`. 2. Is the promotion effective on weekends? Test `b1 + b3`. This is a linear combination, so its variance is `Var(b1) + Var(b3) + 2*Cov(b1, b3)` - you cannot add the two standard errors, and the covariance is usually strongly negative here. The practical shortcut is to recode the indicator (make weekday the 1 category) and refit; the new main effect *is* the weekend promotion effect, with the correct standard error printed for you. 3. Does the promotion work differently on weekends? Test `b3`. ## Power, and the meaning of a null interaction Interactions are estimated from the contrast of contrasts, so they need far more data than main effects to reach the same precision - a rule of thumb often cited is that detecting an interaction of a given size takes several times the sample needed for a main effect of that size. A non-significant `b3` therefore does not establish that the promotion works identically on weekends; it says the data cannot distinguish the two slopes. Report the estimate and interval, not just the verdict. ## Keep the main effects When `b3` is significant, the temptation is to drop a non-significant `b1` for tidiness. Resist it. Removing a main effect while keeping the product constrains the promotion effect on weekdays to be exactly zero, which is a strong claim you did not intend to make, and it distorts the remaining coefficients. The convention - marginality, or the hierarchy principle - is to keep every lower-order term contained in an interaction you retain. The same rule applies to any higher-order term built from a variable already in the model. ## How to present it The coefficient table is the wrong artefact for a stakeholder. Report the two conditional effects side by side with intervals - "promotion lifts sales by X on weekdays and by Y on weekends" - and state the difference `b3` separately as the evidence that they differ. That framing removes almost every misreading in one step. ## What interviewers listen for The phrase "at weekend equals zero", the sum `b1 + b3` for the other group, the symmetry of the interaction, and awareness that the standard error of a sum needs a covariance. Bonus points for the power caveat and for refusing to drop main effects.

  • How do you test whether the promotion has a non-zero effect on weekends specifically?
    The weekend promotion effect is `b1 + b3`, so I test that linear combination rather than either coefficient alone. Its variance is `Var(b1) + Var(b3) + 2*Cov(b1, b3)`, and the covariance is typically strongly negative, so adding standard errors gives the wrong answer. The simplest reliable route is to recode the indicator so weekday becomes the 1 category and refit: the printed main effect is then the weekend promotion effect with a correct standard error.
  • The interaction term is not significant, but you believe the effect differs. What do you check?
    Power first. Interactions are contrasts of contrasts and need much more data than main effects for the same precision, so a null result is often uninformative rather than reassuring. I look at the confidence interval on the interaction: if it spans effect sizes that would change the decision, the honest report is inconclusive. I also check how many observations fall in the smaller cell, since an unbalanced promotion schedule can leave one combination almost empty.
  • Should you drop a non-significant main effect while keeping its interaction?
    No. Dropping the promotion main effect while keeping the product forces the promotion effect on weekdays to be exactly zero, which is a claim the data did not support, and it shifts the remaining coefficients to compensate. The marginality convention is to keep every lower-order term contained in an interaction you retain, so the model stays interpretable and the conditional effects are still recoverable.

It is like a train fare that depends on the day: quoting only the weekday price and calling it the fare is accurate for half the week and misleading for the rest.

saying these in an interview costs you the question

  • Calls the main effect the average promotion effect
  • Reads the interaction coefficient as the weekend promotion effect
  • Drops a main effect while keeping the product term
  • Reports a main effect without naming where the other variable sits
  • Treats a non-significant interaction as proof the effects are equal
  • Adds standard errors instead of using the covariance for a sum

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