Practice: Indicator Variables and Interactions

What ${\beta }_2=2.4$ reports. The group difference at $X=0$, among participants with zero years of experience. The sample runs from 2 to 30 years, so that is an extrapolation to a point no one in the study occupies.

The gap as a function of $X$. Setting $D=1$ and subtracting the $D=0$ line gives $2.4+0.9X$.

• At 5 years: $2.4+4.5=6.9$ points
• At 15 years: $2.4+13.5=15.9$ points
• At 25 years: $2.4+22.5=24.9$ points

The programme's benefit rises steeply with experience, and 2.4 understates it everywhere in the observed range.

What centring changes. Refitting with $X_c=X-12$ gives ${\beta }_2=2.4+0.9(12)=13.2$, the gap at average experience. The fitted values, residuals, $R^2$ and ${\beta }_3$ are all unchanged, only the reference point moved, and with it the question ${\beta }_2$ answers.

The general lesson. ${\beta }_2$ was never the overall effect. It was always the effect at $X=0$, and centring chooses where that zero sits.

Algebra against warrant. With an intercept and a treatment indicator, least squares returns the difference in conditional means exactly, that is an identity, true of any data whatever produced them. What licenses reading it as an effect is that this is a trial: assignment was randomized, so the two arms are comparable in expectation. The regression contributes the arithmetic and, through the interaction, the way the gap varies with experience; it supplies none of the causal warrant.

The first conclusion is unsupported. The coefficient 0.4 is the treatment effect at age zero, which is 45 years below the youngest patient in the study. Its non-significance says nothing about the effect at any age anyone actually had, and reporting it as 'no overall effect' is the error.

The effect across the observed range. The treatment effect at age $a$ is $0.4+0.08a$:

• at 45: $0.4+3.6=4.0$
• at 65: $0.4+5.2=5.6$
• at 80: $0.4+6.4=6.8$

Substantial throughout, and the opposite of what was claimed.

The second conclusion is better supported. The interaction describes how the effect varies with age and does not depend on the reference point, so it is not vulnerable to the same error.

A caveat on it. Whether the age interaction was specified in advance matters. If it emerged from exploring subgroups, it carries the multiplicity of every subgroup analysis that might have been run, and should be treated as a hypothesis for the next study.

What should be reported. The treatment effect at several substantively chosen ages with intervals, or the model refitted with age centred at its mean so the headline coefficient describes the average patient.

One thing the report gets right for the wrong reason. Fitting the model does return the group difference at the reference value exactly, but that is an algebraic property of least squares with an intercept and an indicator. It holds whether or not anyone was assigned to anything, so it cannot be what makes the coefficient an effect. The design has to do that, and a regression run on observational data returns the same identity with no warrant behind it.

The segment coefficient is misread. With the interaction in the model, 180 is the segment difference at a price of zero, outside the £40 to £250 range and not a price anyone is charged. It is not the overall segment gap.

The gap across the actual price range. The segment difference is $180-1.4\times \text{price}$:

• at £40: $180-56=124$ units
• at £145 (mid-range): $180-203=-23$ units
• at £250: $180-350=-170$ units

So enterprise customers buy more only at low prices; above roughly £129 the sign reverses and small-business customers buy more. The headline of 180 describes none of this.

The consequence for the decision. Prioritising enterprise accounts on the strength of a 180-unit advantage is unsound, because that advantage does not exist at most prices in the range. The right question is what price points the business actually operates at, and the segment comparison should be evaluated there.

What I would report. The segment gap at the prices actually charged, with intervals, or the model refitted with price centred at a representative value so the segment coefficient describes a real operating point. The interaction itself is the substantive finding: price sensitivity differs between segments, and that is what should drive segment-specific pricing rather than a blanket prioritisation.

One more check. Whether prices were set by the business in response to expected demand, in which case the coefficients are conditional associations rather than a demand curve, and no pricing decision should be read off them directly.

Why the algebra is not the argument. The enterprise coefficient is exactly the difference in conditional means between segments at the reference price, by construction. Customers were not assigned to segments; they arrived in them. So the number is a description of who buys what, and reading it as what would happen if a customer were moved between segments asks the regression for something only a design could supply.