Estimating Dynamic Regime Contrasts From Blips

From Treatment-Time Effects to Regime Means

A blip function describes the contribution of a treatment at a particular time to an outcome, relative to setting that treatment to zero while holding the relevant treatment history fixed. A policy or treatment regime, however, specifies actions across multiple times. To compare two such regimes, we need to combine their blips over time and average them over the histories that each regime would produce.

This tutorial focuses on the mean outcome under a specified regime and on contrasts between regime-specific means. The calculations use an additive structural nested mean model (SNMM): blips are on the mean-outcome scale, and contributions from the treatment times add. A regime contrast is therefore not generally just the sum of a few coefficients. The blips may depend on covariates or treatment history, and the histories on which they are evaluated can differ between regimes.

Write \(\gamma_t(h_t,a_t)\) for the additive blip at time \(t\): the mean-outcome contribution of setting treatment at that time to \(a_t\), rather than zero, for history \(h_t\), with later treatments set to zero in the nested comparison. In particular, \(\gamma_t(h_t,0)=0\). The exact covariates and treatment-history terms in a blip depend on the chosen SNMM. For an additive binary-treatment model, a common form is \(\gamma_t(h_t,a_t)=a_t\beta_t(h_t)\).

A blip is not automatically a population-average treatment effect. If the blip depends on \(H_t\), its value varies across histories. To obtain a regime mean, evaluate each blip at the history and action prescribed by the regime, then average over the history distribution that would arise under that regime.

The Additive Regime-Mean Identity

Under an additive SNMM, the mean under a regime can be expressed as an all-untreated reference mean plus the mean blip contribution at each time:

Here \(H_t^d\) is the history that would be present just before time \(t\) if regime \(d\) had been followed. The expectation for time \(t\) is thus taken under the history distribution induced by \(d\), not automatically under the observed distribution of \(H_t\). This distinction matters whenever a regime changes which histories arise.

For two regimes \(d\) and \(d^*\), the shared all-untreated reference term cancels:

These identities require an additive blip representation on the outcome-mean scale. For a causal interpretation from observed data, the usual identification conditions also matter: consistency (observed outcomes agree with the counterfactual outcomes for the treatment actually received), sequential exchangeability (at each time, treatment is conditionally independent of relevant future potential outcomes given the measured past), and positivity (the treatments required by the regimes have support within the relevant histories). The blip model and the procedure used to estimate or average its terms must also be appropriate for the target regime means. These conditions do not mean that a contrast is causal by algebra alone; they support interpreting the identified regime means causally.

Why the Identity Holds

The key idea is to move from the all-untreated treatment sequence to the regime one time at a time. At each time, the nested contrast between the regime’s action and zero is represented by that time’s blip. Adding the successive increments gives the total change from the all-untreated outcome to the regime outcome.

In compact form, the telescoping argument is:

This argument uses the additive mean blips as the increments in the nested comparison. It does not assert that every individual’s outcome changes by the fitted conditional mean blip, nor that the blip values are constant across people. The expectations are what turn history-specific contributions into regime-specific mean outcomes.

Effect Modification Changes the Average Blip

When a blip depends on a covariate, the regime’s mean contribution is an average of the blip over that covariate’s distribution. For a regime that treats everyone, an effect-modified blip need not equal the value for a typical person or the value at the covariate mean. It is the mean of the specified blip values in the target population.

Dynamic Regimes Require Regime-Specific Histories

A dynamic regime chooses treatment in response to information available at each time. Its blip contribution depends both on the action it prescribes and on the histories arising under that policy. If a covariate distribution at a later time differs across regimes, a history-dependent blip must be averaged separately for each regime. Simply plugging actions into a blip formula while using one common, observed-history distribution can give the wrong regime contrast.

Turning Estimated Blips Into a Contrast

In an analysis, blip functions and the reference mean are estimated rather than known. To obtain a point estimate of a regime contrast, the calculation follows the same structure: evaluate each estimated blip under the actions and histories specified by each regime, average those contributions in the appropriate target population, sum across times, and subtract the two regime totals. The averaging step may require estimates of the regime-specific history distributions; it is not supplied by the blip coefficients alone.

When an estimated blip is linear in treatment but varies with history, the action under a regime determines whether the contribution is zero, while the history determines its size. For example, with \(\widehat{\gamma}_t(h_t,a_t)=a_t\widehat{\beta}_t(h_t)\), the time-\(t\) term is the average of \(d_t(H_t^d)\widehat{\beta}_t(H_t^d)\) under regime \(d\). A regime that treats only selected histories may have a smaller average contribution than treat-all, even when its blip is large among those selected.

The all-untreated mean is useful as a shared reference, but it is not necessary to interpret a contrast once the same reference population and model are used for both regimes: it cancels. By contrast, the regime-specific averages of blips generally do not cancel. Their differences are the quantities that distinguish the policies.

Common Mistake: Averaging Over the Wrong Histories

A tempting but often incorrect calculation is to average a time-varying blip over the observed histories, then use that same average for every regime. This silently assumes the distribution of those histories is the same under every regime. In a dynamic setting, the actions prescribed earlier can change which later histories occur, so that assumption need not hold.

The correction is to identify the distribution relevant to each term: for the mean under \(d\), evaluate the blip using \(H_t^d\); for the mean under \(d^*\), use \(H_t^{d^*}\). When a blip is constant or the relevant history distribution is genuinely common, the averages may coincide. Otherwise, regime-specific averaging is part of the estimand, not an optional refinement.