Treatment Benefit Across Baseline Fitness Levels

Why Treatment Effects May Vary With Baseline Fitness

An exercise program may not yield the same improvement for everyone. Participants with different starting fitness levels might benefit by different amounts. A regression interaction lets us estimate how the treatment contrast changes with baseline fitness, rather than assuming one common contrast for all participants.

This tutorial uses a hypothetical randomized trial. Participants are assigned to an exercise program or control, and their endurance-test distance is measured after 12 weeks. The outcome is distance in metres; larger values indicate better performance. Baseline fitness is measured before randomization in millilitres of oxygen per kilogram per minute. Random assignment supports comparisons of treatment groups, while the interaction model describes how those comparisons vary with baseline fitness.

The Interaction Model and Its Assumptions

Let \(T=1\) indicate assignment to the exercise program and \(T=0\) indicate assignment to control. Let \(x\) be baseline fitness. We center and scale baseline fitness as \(z=(x-30)/5\), so that \(z=0\) corresponds to a baseline score of 30, and a one-unit increase in \(z\) corresponds to an increase of 5 baseline-score units.

The control-group regression line is \(\beta_0+\beta_2z\). The exercise-group line is \((\beta_0+\beta_1)+(\beta_2+\beta_3)z\). Subtracting the control line from the exercise line at the same \(z\) gives \(\beta_1+\beta_3z\). Thus, \(\beta_3\) describes a difference between the groups’ slopes; it is not the treatment effect at every fitness level.

Interpretation depends on assumptions. The conditional mean of the outcome must be adequately represented by a linear function of baseline fitness within each treatment group. Errors should have mean zero conditional on treatment and baseline fitness, be independent across participants, and have a common variance for the standard errors used here. Normal errors support the exact small-sample \(t\) calculations; with a sufficiently large sample, approximate inference may be reasonable without exact normality. A useful causal interpretation also requires that assignment was randomized, baseline fitness was measured before treatment, and the treatment contrast corresponds to the assigned treatments being compared. These assumptions do not justify extrapolating beyond the baseline-fitness values represented in the trial.

The interaction is scale-specific: it concerns differences in mean endurance distance, measured in metres. A changing difference in metres need not imply the same pattern on another outcome scale.

Worked Trial Results

Suppose 120 participants were randomized, 60 to each group. The observed baseline-fitness scores ranged from 22 to 38. A hypothetical linear-model fit gives the following estimates. The standard errors use the model’s residual degrees of freedom, \(df=120-4=116\), because four coefficients were estimated.

For calculating treatment-effect uncertainty, suppose the fitted model also gives \(\operatorname{Cov}(\hat{\beta}_1,\hat{\beta}_3)=-20\) square metres. This covariance matters because both estimates contribute to treatment contrasts away from baseline fitness 30.

Conditional Treatment Effects and Their Uncertainty

To estimate the treatment effect at a chosen baseline value, substitute its \(z\) value into \(\hat{\beta}_1+\hat{\beta}_3z\). Its standard error must account for the uncertainty in both coefficient estimates and their covariance. For any fixed \(z\),

This follows by expanding the variance of a sum: for random quantities \(A\) and \(B\), \(\operatorname{Var}(A+zB)=\operatorname{Var}(A)+z^2\operatorname{Var}(B)+2z\operatorname{Cov}(A,B)\). A \(t\)-based 95% confidence interval for the conditional treatment contrast is the estimate plus or minus \(t^*SE\), using the model’s residual degrees of freedom.

What the Trial Supports—and What It Does Not

The examples separate two questions. First, is there evidence that the treatment contrast changes with baseline fitness? The interaction estimate and its interval address that question. Second, what is the estimated contrast at a particular baseline value? The conditional estimate and its interval address that question. A clear positive contrast at one value does not, by itself, prove that contrasts differ across values; the interaction comparison is the direct evidence about change.

The model predicts control means of 485 m, 510 m, and 535 m at baseline values 25, 30, and 35, respectively. Its corresponding fitted treatment contrasts are 24 m, 42 m, and 60 m. These are estimated group mean comparisons, not guaranteed outcomes for individuals. The observed baseline range of 22 to 38 includes all three values, so the examples do not require extrapolation beyond the trial’s range. Even within that range, the linear interaction assumption should be considered when interpreting the pattern.

Because treatment was randomized, conditional comparisons can have a causal interpretation as effects of assignment at a given pre-treatment fitness value, provided the trial’s conduct and model assumptions support that interpretation. The results do not show that changing a person’s baseline fitness would cause their treatment benefit to change. Baseline fitness was measured, not randomly assigned.

Common Mistake: Comparing Separate Significance Results

A common mistake is to say, “The effect is significant at fitness 35 but not at fitness 25, so the effects must differ.” That reasoning is not valid: one interval excluding zero and another including zero does not itself test the difference between the effects. The relevant comparison is the interaction, or a direct contrast such as the estimated 36 m difference between fitness 35 and 25 with its own standard error and interval. In this hypothetical trial, that direct interval is positive; in another trial it might not be.