Effect Modification In Structural Nested Models

Why Effect Modification Belongs in a Blip Function

A treatment need not have the same causal effect for everyone. Its effect may differ by a baseline characteristic, such as disease severity before follow-up, or by a time-varying covariate, such as a biomarker measured before a later treatment decision. A structural nested mean model (SNMM) can represent this variation directly in its blip function: the conditional causal contrast associated with a treatment at a particular time.

The interaction terms in a blip function have a causal interpretation on the model’s outcome scale. For example, a treatment-by-baseline-covariate term can describe how the treatment contrast differs between baseline groups. A treatment-by-time-varying-covariate term can describe how that contrast varies across covariate values in a treatment history. Neither term, by itself, is an overall treatment effect or an association estimated by ordinary regression.

The counterfactual histories in that definition matter. The covariates in \(H_t\) are values available before the time-\(t\) treatment decision under the past treatment history being considered. The blip is therefore a history-specific causal contrast, not a comparison of people selected after treatment according to their observed covariates. Setting later treatments to the reference value defines the nested contrast; the blip is not automatically the effect of changing \(A_t\) while leaving every later treatment unchanged.

For clarity, the examples use an additive mean blip: the outcome is measured on a scale where differences in conditional means are meaningful, such as points on a score or days of symptom-free time. A blip of \(-2\) points means that the specified treatment contrast lowers the conditional mean outcome by 2 points. The sign depends on the outcome and reference-treatment coding.

Writing Baseline and Time-Varying Effect Modification

For binary treatment, a simple additive SNMM with one treatment time and a binary baseline modifier \(V\) is

Here \(A_t=0\) is the reference treatment and \(A_t=1\) is the active treatment. The bracket gives the treatment blip when active treatment is assigned. The parameter \(\psi_{t0}\) is the blip for the reference group \(V=0\); \(\psi_{tV}\) is the difference between the blips for \(V=1\) and \(V=0\). The treatment indicator multiplies the whole expression so that the blip is zero at the reference treatment.

A pre-treatment time-varying covariate can be included in the same way. For example, if \(L_t\) is continuous, a model with both modifiers is

where \(c\) is a chosen reference value for \(L_t\). The coefficient \(\psi_{tL}\) describes the change in the treatment blip for a one-unit increase in \(L_t\), holding the other terms fixed. Centering \(L_t\) at \(c\) makes \(\psi_{t0}\) the blip at \(V=0\) and \(L_t=c\); it does not change the model’s fitted contrasts.

More terms can be included when the scientific question requires them. For example, \(A_tV(L_t-c)\) allows the way the blip varies with \(L_t\) to differ across levels of \(V\). A model with no such term assumes that the \(L_t\)-related change in the blip is the same for both groups. As with any interaction model, the meaning of each coefficient depends on the other terms and on the values chosen as references.

Worked Examples

Why the Interaction Has This Interpretation

The interpretation follows directly from the contrast form of an SNMM blip. In the binary-treatment model with baseline and time-varying modifiers, the reference treatment has no blip because its treatment value is zero. Subtracting that reference blip from the active-treatment blip leaves the bracketed terms.

Now compare the active-treatment blips at \(V=1\) and \(V=0\), holding \(L_t\) and the rest of the history fixed:

Likewise, increasing \(L_t\) by one unit changes the blip by \(\psi_{tL}\), provided the other covariates and history are held fixed. For a continuous treatment \(a\), a linear dose blip of the form \(a\{\psi_0+\psi_VV+\psi_L(L_t-c)\}\) gives a per-unit dose contrast equal to the bracket. A change from dose \(a_0\) to \(a_1\) multiplies that per-unit contrast by \(a_1-a_0\). These results are algebraic interpretations of the specified model; they do not establish that the model or causal assumptions are valid.

In a longitudinal analysis, the blip parameters are estimated using g-estimation rather than by treating the blip as an ordinary regression coefficient. Under sequential exchangeability and the other identification assumptions, the correctly blipped-down outcome removes the modeled causal contribution of treatment. The treatment residuals then supply conditional moment information for the parameters. With effect modification, the moments must identify the treatment-by-covariate terms as well as the main treatment term. This requires adequate treatment variation across the histories and modifier values for which the contrasts are sought.

History-Specific Effects Are Not Overall Effects

A time-varying covariate may itself have been affected by earlier treatment. Including \(L_t\) in a time-\(t\) blip can still describe how the current treatment contrast varies across the pre-treatment histories at time \(t\), when the identification assumptions support that interpretation. But it does not turn \(\psi_{tL}\) into the effect of intervening on \(L_t\), and it does not automatically give the overall effect of a treatment strategy in a population.

This distinction is especially important when comparing groups defined by a post-baseline history. People with different \(L_t\) values may also have different earlier treatment histories and prognoses. The conditional blips describe contrasts indexed by those histories; a population-level regime contrast must account for how histories arise under the regimes being compared. The numerical value of an interaction also depends on the outcome scale: effect modification on an additive mean scale need not match effect modification on a ratio scale.

Common Mistake

A common error is to read \(\psi_{tV}\) as the treatment effect for \(V=1\), or to say that it is the effect of changing \(V\). In the model \(\gamma_t=A_t(\psi_{t0}+\psi_{tV}V)\), the treatment blip for \(V=1\) is \(\psi_{t0}+\psi_{tV}\), while \(\psi_{tV}\) is the difference between that blip and the blip for \(V=0\). It describes effect modification on the modeled scale; it is not generally a causal effect of intervening on \(V\). The correction is to substitute the covariate value into the full blip function before interpreting the result.