Linear Interaction Models for Treatment Effect Heterogeneity

Why Interactions Describe Treatment-Effect Heterogeneity

A treatment can have different effects for people with different characteristics. A linear outcome model represents one particular pattern of this variation: the treatment effect changes linearly with a covariate. The treatment-by-covariate interaction is the coefficient that encodes that change. Interpreting the model therefore requires more than reading off the treatment coefficient; it requires calculating the treatment contrast at the covariate value of interest.

The same model also separates effect modification from differences in outcome levels. For example, one subgroup may have higher average outcomes under either treatment condition, yet the treatment effect may be identical across subgroups. A covariate main effect can represent that baseline difference; it does not, by itself, imply treatment-effect heterogeneity.

The Model and Its Interpretation

Let \(Y\) be an outcome, \(D\) a treatment indicator taking values 0 and 1, and \(X\) a covariate. A linear conditional-mean model with one interaction is

The model makes the conditional mean linear in the included regressors. Its two treatment-specific mean functions are obtained by substituting the treatment value. For control, \(D=0\), so \(E[Y\mid D=0,X=x]=\beta_0+\beta_Xx\). For treatment, \(D=1\), so \(E[Y\mid D=1,X=x]=\beta_0+\beta_D+(\beta_X+\beta_{DX})x\). Subtracting control from treatment gives

Thus \(\beta_D\) is the modeled treatment contrast when \(X=0\), not generally the treatment contrast for every unit. The coefficient \(\beta_{DX}\) is the change in the treatment contrast for a one-unit increase in \(X\). If \(Y\) is measured in points and \(X\) in years, for example, \(\beta_{DX}\) has units of outcome points per year. The covariate coefficient \(\beta_X\) describes the covariate slope in the control group; the slope in the treatment group is \(\beta_X+\beta_{DX}\).

These are conditional mean contrasts. To interpret them as causal effects, additional identification conditions are needed. Let \(Y(1)\) and \(Y(0)\) denote the potential outcomes under treatment and control. Consistency requires that the observed outcome equals the potential outcome corresponding to the treatment received, and no interference requires one unit's treatment not to alter another unit's outcome. Conditional exchangeability, \((Y(1),Y(0))\perp D\mid X\), and positivity, \(0<P(D=1\mid X=x)<1\) for covariate values of interest, allow the observed conditional contrast to identify the conditional average treatment effect. Random assignment can provide exchangeability; in observational data it must be justified by the design and measured covariates. Even with identification, interpreting the linear formula at a value of \(X\) depends on the conditional-mean model being adequate there.

Binary Covariates and Subgroup Effects

When \(X\) is a binary subgroup indicator, the interaction model yields a separate treatment effect for each group. Let \(X=0\) denote one group and \(X=1\) another. The treatment effect for the \(X=0\) group is \(\beta_D\); for the \(X=1\) group it is \(\beta_D+\beta_{DX}\). The interaction coefficient is therefore the difference between those two treatment effects. The coefficient \(\beta_X\), in contrast, is the difference in conditional mean outcomes between groups among controls.

This distinction matters because a group can differ in its untreated outcome level without differing in its response to treatment. The treatment effect is a within-group treatment-versus-control contrast. A comparison of group outcome levels under just one treatment condition is not that contrast.

Centering Changes the Reference Point

The treatment coefficient depends on where zero is placed on the covariate scale. That dependence is a matter of parameterization, not a change in the fitted treatment contrasts. Centering a covariate at a meaningful value often makes the treatment coefficient easier to interpret: it becomes the treatment contrast at that reference value.

To see this, write \(X=X_c+c\), where \(X_c=X-c\) is the covariate centered at \(c\). Substituting into the model and collecting terms gives

The coefficient on \(D\) in the centered model is \(\beta_D+c\beta_{DX}\), which is exactly the original model's treatment contrast at \(X=c\). The interaction coefficient remains the same when this is only a shift of the covariate origin. This algebra assumes the same linear model and regressors are being expressed in a different coordinate system.

What the Interaction Does—and Does Not—Assume

A linear interaction is a structured description of heterogeneity, not a guarantee that treatment effects truly vary linearly. With one continuous covariate, it imposes a treatment-effect function of the form \(\beta_D+\beta_{DX}x\). If the real pattern is curved or has thresholds, this model may misrepresent effects at some covariate values. Predictions outside the observed covariate range are particularly dependent on extrapolation and should not be treated as supported simply because the formula produces a number.

With several covariates, a model might include separate treatment interactions with each covariate:

Under this specification, the treatment contrast at a covariate vector \((x_1,\ldots,x_p)\) is \(\beta_D+\sum_{j=1}^{p}\beta_{Dj}x_j\). Each interaction coefficient describes a change in the treatment contrast as its corresponding covariate increases, holding the other included covariates fixed. This particular model does not include interactions among covariates or nonlinear treatment-effect patterns unless they are added explicitly.

A fitted interaction model also does not by itself establish that treatment effects differ reliably across values or groups. Estimation uncertainty matters, and the uncertainty of a covariate-specific contrast involves the uncertainty and covariance of the relevant fitted coefficients. Interpreting a point estimate as a known effect, or inferring a subgroup difference from separate coefficient sizes alone, ignores that uncertainty.

Common Pitfall: Reading the Treatment Main Effect as Universal

A frequent error is to report \(\beta_D\) as “the treatment effect” in an interaction model, regardless of the covariate value. This is only correct at \(X=0\). Another error is to treat \(\beta_X\) as evidence of effect modification. It describes a difference in outcome levels by \(X\) in the control condition; the treatment-by-covariate coefficient, together with the resulting treatment contrasts, describes modeled effect modification.

The main practical value of the interaction parameterization is that it makes the assumed pattern of effect heterogeneity explicit. It also makes the reference point explicit: centering can align that point with a meaningful value, while the covariate-specific contrasts remain the quantities to report when the question concerns effects for particular covariate profiles.