Calculating Conditional Treatment Effects

Why Calculate a Contrast at a Particular Baseline Value?

When the fitted relationship between a baseline outcome and a follow-up outcome differs by assigned treatment, a single treatment coefficient does not describe the fitted treatment contrast at every baseline value. The product term in an interaction model makes the contrast depend on the baseline value. To answer a question such as “What contrast does the model fit for participants whose baseline score is 20 points?”, we must combine the treatment coefficient with the product-term coefficient.

This calculation describes a comparison between the fitted outcomes under treatment and control for the same baseline value. It is a model-based conditional contrast; it is not an individual participant’s treatment effect. The assignment of treatment in a randomized study gives the comparison its study context, but the calculation itself is an algebraic consequence of the specified model.

The Interaction Model and Its Conditional Contrast

Let \(D\) denote randomized treatment assignment, coded \(D=1\) for treatment and \(D=0\) for control. Let \(X\) be a baseline variable measured before assignment, and let \(Y\) be the outcome. Consider the fitted linear interaction model

Here, \(\widehat{Y}\) is the fitted outcome, \(\hat{\beta}_0\) is the fitted intercept, \(\hat{\beta}_2\) is the baseline slope in the control group, and \(\hat{\beta}_3\) is the change in that slope for the treatment group relative to control. The treatment coefficient \(\hat{\beta}_1\) is the fitted treatment-versus-control contrast specifically when \(X=0\).

To calculate the contrast, substitute \(D=1\) and \(D=0\) into the model separately, using the same baseline value \(x\). Subtract the control fitted outcome from the treatment fitted outcome:

The intercept and the baseline main-effect term cancel because both fitted outcomes are evaluated at the same \(x\). The treatment main effect does not cancel: it is present for \(D=1\) and absent for \(D=0\). The product term also contributes for treatment, since \(D X=x\) when \(D=1\), but is zero under control.

What the Calculation Requires

Before using the formula, check that the model and the comparison match the question. The fitted contrast above applies when treatment is coded 1 and control 0, the baseline value is \(x\), and the model includes both the treatment main effect and the treatment-by-baseline product term. Changing the coding or omitting a term changes the calculation.

For the result to be a useful description of fitted conditional means, the model should represent the mean outcome reasonably well over the baseline values being considered. In particular, this model specifies a linear change in the treatment contrast with \(X\). The baseline variable must be measured before treatment assignment for it to serve as a baseline value in this randomized-study comparison. There should also be observations from both assigned groups across the baseline region of interest; a calculated contrast far beyond the observed baseline values may be an extrapolation rather than a well-supported comparison.

These conditions concern the meaning and support of the fitted contrast. They do not make the calculation an inferential test: the formula gives a point estimate from fitted coefficients. Assessing uncertainty or testing whether the contrast differs from zero requires additional calculations.

Reading the Contrast Across Baseline Values

The contrast formula can be applied separately at any selected baseline values. If \(\hat{\beta}_3\) is positive, the fitted treatment contrast increases as \(X\) increases; if it is negative, the contrast decreases as \(X\) increases. This describes the direction of change in the model’s fitted contrast. The value of the contrast at a particular \(x\) still depends on both \(\hat{\beta}_1\) and \(\hat{\beta}_3\).

The units provide a useful check. If \(Y\) is measured in points and \(X\) is measured in baseline points, then \(\hat{\beta}_1\) is in outcome points and \(\hat{\beta}_3\) is in outcome points per baseline point. Multiplying \(\hat{\beta}_3\) by \(x\) gives outcome points, so it can be added to \(\hat{\beta}_1\). A contrast should be reported in the units of \(Y\), not in the units of \(X\).

Common Mistake: Losing the Treatment Main Effect

A frequent error is to confuse a constant in the written model with the treatment coefficient. For instance, a model written as

does not contain a treatment main-effect term. The \(-3\) is part of the intercept, not a term that switches on when \(D=1\). Since the only treatment-dependent term shown is \(0.05(DX)\), its treatment contrast at \(X=20\) is \(0.05(20)=1\) point, not \(-2\) points.

To represent a treatment coefficient of \(-3\), the model must include \(-3D\), as in \(\widehat{Y}=18-3D+0.60X+0.05(DX)\). Then the contrast at \(X=20\) is \(-3+0.05(20)=-2\) points. A reliable check is to substitute \(D=1\) and \(D=0\) into the entire model and subtract the fitted outcomes. This catches missing terms and sign errors.