Identifying Treatment Effect Modifiers

Why Prognostic Factors and Effect Modifiers Are Different

A covariate can strongly predict a clinical outcome without changing the effect of treatment. For example, baseline disease severity may predict poorer outcomes in both treatment groups, while the treatment risk difference remains the same at every severity level. Conversely, a covariate is a treatment effect modifier when the treatment contrast differs across its values. That distinction matters in extrapolation: a prognostic factor can be important for predicting target-population outcomes, but a change in the distribution of true effect modifiers can change the target treatment effect.

Effect modification is always defined relative to a particular outcome and effect scale. A covariate may modify a risk difference but not a risk ratio, or the reverse. Therefore, a statement such as “age modifies treatment effect” is incomplete unless it says which outcome and which treatment contrast are being compared.

In a randomized trial, observed outcomes can estimate conditional treatment contrasts, subject to adequate data and appropriate analysis. Causal interpretation also relies on consistency (the observed outcome under the received treatment equals the corresponding potential outcome), correct treatment assignment and outcome measurement, and the trial’s randomization or other valid basis for comparing treatment groups. For transport, the earlier material on exchangeability assumptions for extrapolation adds a separate requirement: after conditioning on the relevant covariates, the source conditional treatment contrast must apply in the target. Identifying effect modifiers does not itself establish that assumption.

Interaction Contrasts Make the Distinction Explicit

For a binary covariate \(X\), an additive interaction contrast is the difference between the treatment effect in the \(X=1\) group and the treatment effect in the \(X=0\) group. Write \(\mu_{ax}=E(Y\mid A=a,X=x)\), with causal interpretation when the design and assumptions support it. The additive interaction contrast is:

The first bracket is the treatment-minus-comparator contrast among people with \(X=1\); the second is that contrast among people with \(X=0\). A zero interaction contrast means the two additive effects are equal. It does not mean that \(X\) has no association with outcome: the outcome levels may differ substantially between \(X\) groups in both treatment arms.

A simple model shows why the interaction coefficient represents this difference. Suppose the conditional mean is linear in treatment and binary \(X\):

At \(X=0\), the treatment contrast is \((\beta_0+\beta_A)-\beta_0=\beta_A\). At \(X=1\), it is \((\beta_0+\beta_X+\beta_A+\beta_{AX})-(\beta_0+\beta_X)=\beta_A+\beta_{AX}\). Subtracting the first contrast from the second gives \(\beta_{AX}\). The main effect \(\beta_X\) describes a difference in conditional outcome means at \(A=0\); it is not the treatment interaction.

For a continuous covariate, the same principle applies. Under the model \(E(Y\mid A,X)=\beta_0+\beta_A A+\beta_X X+\beta_{AX}AX\), the treatment contrast at \(X=x\) is \(\beta_A+\beta_{AX}x\). Thus \(\beta_{AX}\) is the change in the additive treatment contrast per one-unit increase in \(X\). If a linear form is not plausible, interaction can instead be represented with prespecified categories, splines, or another suitable function. The form should be chosen to reflect the scientific question and data support, not selected solely because it produces a small p-value.

Effect Scale Determines What Counts as Modification

The interaction calculation must use the scale of the estimand. For binary outcomes, common conditional contrasts include the risk difference, risk ratio, and odds ratio. Equality of treatment effects across covariate values on one scale does not imply equality on another. For example, the previous table has equal risk differences but different risk ratios. A model’s interaction coefficient is therefore meaningful only together with its link function and the contrast being interpreted.

For a risk difference, the conditional effect is \(R_1(x)-R_0(x)\), where \(R_a(x)=P(Y^a=1\mid X=x)\). For a risk ratio it is \(R_1(x)/R_0(x)\), when the denominator is positive. To assess modification on either scale, compare the conditional treatment contrasts across \(x\) on that scale. A product term in a logistic regression, for example, describes interaction on the log-odds scale; it is not automatically an interaction on the risk-difference or risk-ratio scale.

Which Distributions Matter When Transporting an Effect?

Suppose the target estimand is the target-population average risk difference. If the conditional risk difference applies in the target given the relevant covariates \(X\), the target average risk difference is the mean of those conditional risk differences over the target distribution:

This expression follows from linearity of expectation: the target mean risk under treatment minus the target mean risk under comparator equals the target mean of the conditional risk differences. Consequently, for an additive average treatment effect, it is the target distribution of covariates that modify the conditional risk difference that must be represented correctly in the averaging. If several covariates jointly modify the effect, their joint distribution matters; matching separate marginal distributions can still give the wrong average if their association differs.

A covariate that is purely prognostic does not change this average risk difference if the conditional risk difference is constant over it. But prognostic covariates can still matter to other target quantities, such as target risks under each treatment. For relative measures, one must not generally average conditional relative effects and call the result the marginal relative effect. Instead, calculate target-standardized risks first and then form their ratio:

This distinction is important even when the covariate is not a modifier on the risk-ratio scale. Different baseline risks across covariate strata can make a marginal risk ratio differ from a conditional risk ratio. Thus, the distribution required for transport depends on the estimand and scale: for an additive contrast, average conditional differences; for a marginal relative contrast, standardize the outcome risks and then calculate the relative contrast. Other scales require their own clearly defined marginal estimand and calculation.

Practical Identification and Diagnostics

A defensible modifier assessment starts by fixing the outcome, contrast scale, and target estimand. Then use clinical and pharmacological knowledge to identify plausible baseline modifiers, and estimate their conditional treatment contrasts with uncertainty. In a model-based analysis, inspect predicted contrasts across clinically meaningful covariate values rather than interpreting a product-term coefficient in isolation. For continuous covariates, check whether the chosen functional form describes the observed pattern; a single linear interaction can miss a threshold or curved relationship.

Common warning signs include very small subgroup counts, wide intervals for interaction contrasts, unstable estimates when a few observations are removed, and apparent modification that depends on an arbitrary categorization or effect scale. A non-significant interaction test is not proof of homogeneous effects; interaction tests often have limited precision. Conversely, a small p-value from many unplanned subgroup searches may be a chance finding. Report the estimated contrasts, uncertainty, scale, and analysis choices, and distinguish exploratory patterns from well-supported modifiers.

The main alternative to an interaction model is to estimate treatment contrasts separately within prespecified strata. This is transparent for a small number of clinically meaningful groups, but can be imprecise and does not automatically handle continuous covariates or multiple modifiers. A regression model can borrow information and represent continuous patterns, but depends on its functional form. Either approach identifies conditional patterns; neither removes the need to justify source-to-target applicability. The detailed mechanics of checking whether source and target covariate support is adequate are a separate issue.