Modeling Effect Modification in Blip Functions

Why Let a Blip Depend on Covariates?

An additive SNMM can represent a treatment effect as constant across people with different histories. That assumption may be too restrictive: a treatment might have a different conditional mean effect at different baseline risk levels, or its effect at a later decision might vary with a time-varying covariate. A blip function represents this heterogeneity directly, through treatment-by-covariate terms.

The interpretation is local to the model: an interaction coefficient describes how the blip changes with a covariate, conditional on the history specified in that blip. It is not automatically a marginal treatment effect, nor does it establish that changing the covariate would change treatment response.

For the examples below, treatment is binary and the reference is \(A_t=0\). A simple model allowing both baseline and time-varying effect modification is

where \(V\) is a baseline covariate, \(L_t\) is measured before the treatment decision at time \(t\), and \(H_t\) contains these covariates and the relevant past treatment and covariate history. Only terms appropriate to the application need be included. The model is a model for treatment contrasts, not a model for the covariates’ prognostic associations with the outcome.

Reading Treatment-by-Covariate Parameters

For binary treatment, subtracting the blip at \(A_t=0\) from the blip at \(A_t=1\) gives the conditional mean contrast:

Thus \(\psi_{t0}\) is the contrast when both modifying covariates equal zero; \(\psi_{tV}\) is the change in that contrast per one-unit increase in \(V\), holding the other included history terms fixed; and \(\psi_{tL}\) is the corresponding change per unit of \(L_t\). If a binary covariate is coded 0/1, its interaction coefficient is the difference between the two covariate-specific treatment contrasts.

Proof. At \(A_t=1\), the blip is \(\psi_{t0}+\psi_{tX}x\); at \(A_t=0\), it is zero. Their difference is therefore \(\psi_{t0}+\psi_{tX}x\). Subtracting the contrast at \(x_0\) from that at \(x_1\) gives \((\psi_{t0}+\psi_{tX}x_1)-(\psi_{t0}+\psi_{tX}x_0)=\psi_{tX}(x_1-x_0)\). \(\square\)

A continuous modifier is interpreted per unit of its measurement scale. Centering it can make the main treatment coefficient easier to interpret: with \(X_c=X-c\), the coefficient on treatment is the contrast at \(X=c\). Centering changes the intercept parameterization, not the model’s fitted contrasts.

Time-Varying Effect Modification

A covariate measured before a later treatment decision can modify that decision’s blip. For treatment times \(t=0,\ldots,K\), define \(H_t\) to include the observed history up to, but not including, \(A_t\). A time-varying modifier such as \(L_t\) belongs in the time-\(t\) blip only if it is available before \(A_t\). The conditional interpretation then compares treatment options among histories with the same specified \(L_t\) and other included history values.

When \(L_t\) is affected by earlier treatment, this is not the same as a baseline subgroup effect. The distribution of \(L_t\) can differ under different earlier treatment regimes. Consequently, a conditional contrast at \(L_t=l\) does not by itself give the marginal effect of a treatment strategy. A marginal contrast would additionally require averaging over the covariate histories generated under the relevant strategies.

Why Blipping Down Gives a G-Estimation Moment

The interaction parameters enter the same blipping-down operation as other blip parameters. Under a correctly specified SNMM, sequential exchangeability, consistency, and positivity, removing the treatment effect at time \(t\) makes the resulting conditional mean independent of \(A_t\), given \(H_t\). This gives a moment condition that can be used to estimate the blip parameters.

Proof. Write \(\mu_t(H_t)=E[Q_t(\psi_0)\mid H_t]\). By the stated conditional-mean restriction, \(E[Q_t(\psi_0)\mid H_t,A_t]=\mu_t(H_t)\). Since \(A_t-e_t(H_t)\) is determined by \(H_t,A_t\), iterated expectation gives

because \(e_t(H_t)=E[A_t\mid H_t]\). This proves the moment condition. \(\square\)

In practice, one can solve sample analogues of these moments for the parameters in the blip, including its treatment-by-covariate coefficients. The theorem explains the population condition, not that every finite-sample solution is unique or stable. Identification of the interaction terms requires adequate variation in treatment across the histories and covariate values that distinguish those terms.

Assumptions, Diagnostics, and Alternatives

The causal interpretation of the blip parameters requires consistency, sequential exchangeability (no unmeasured confounding at each decision conditional on measured history), and positivity: each relevant treatment option must have positive probability at histories in the target population. The blip function must also be sufficiently well specified for the conditional contrasts of interest. These requirements are especially consequential for interactions: a model can have enough information to estimate an average contrast while lacking information to estimate how that contrast varies across covariate values.

Signs of trouble include very few treated or untreated observations in parts of the modifier range, extreme fitted treatment probabilities, unstable interaction estimates under modest changes in covariate scaling, and fitted contrasts that depend heavily on extrapolation beyond observed histories. Inspect the treatment distribution across modifier values and relevant histories, and check whether the fitted contrast is plausible across the data-supported range. A coefficient near zero does not establish no effect modification if support is weak or the functional form is inadequate.

The main alternatives answer related but distinct modeling questions. A marginal structural model targets marginal contrasts under treatment strategies, typically using inverse-probability weighting; it does not automatically provide the same history-conditional blip interpretation. The parametric g-formula models covariate and outcome distributions to calculate strategy-specific means. An SNMM instead parameterizes successive treatment blips and uses g-estimation moments. These approaches rely on different modeling and estimation structures; none removes the need for causal identification assumptions or adequate treatment support.

A common mistake is to interpret \(\psi_{tV}\) as the treatment effect for people with \(V=1\). In a binary modifier model \(A_t(\psi_{t0}+\psi_{tV}V)\), the effect at \(V=1\) is \(\psi_{t0}+\psi_{tV}\); the interaction alone is the difference between the \(V=1\) and \(V=0\) contrasts. Another mistake is to put a covariate measured after \(A_t\) into the time-\(t\) modifier. That changes the conditioning structure and can condition on a consequence of treatment. Use only covariates known before the decision in \(H_t\).