Why Separate Blips Across Treatment Times?
When treatment is assigned repeatedly, a single coefficient can conceal an important feature of the causal question: the effect of treatment at one decision may differ from its effect at another. An additive structural nested mean model (SNMM) represents these time-specific contributions directly. Its parameters describe the mean change associated with treatment at each decision, after later treatment is set to a reference value.
This tutorial considers \(K\) binary treatment decisions, indexed \(t=0,\ldots,K-1\), and a final outcome \(Y\). Write \(A_t\in\{0,1\}\) for treatment at time \(t\), and let \(H_t\) denote the measured history immediately before that decision, including the covariate history and prior treatments. The reference regime is \(\bar 0=(0,\ldots,0)\). For a fixed treatment regime \(\bar a=(a_0,\ldots,a_{K-1})\), \(Y^{\bar a}\) denotes the outcome that would be observed under that regime.
The superscript on \(H_t^{\bar a_{t-1}}\) matters: the history at time \(t\) can itself be affected by earlier treatment. The model does not require that histories be identical across regimes. Rather, it assumes that the conditional mean blip is the same across the histories that arise. This is a substantive restriction, not a consequence of using additive notation.
From Time-Specific Blips to Regime Contrasts
A central interpretation of the parameters follows by comparing a regime with the reference regime one decision at a time. This is a counterfactual mean result; the assumptions needed to identify the blips from observed data are stated separately below.
Proof. Define \(\bar a^{(t)}=(a_0,\ldots,a_t,0,\ldots,0)\), and let \(\bar a^{(-1)}=\bar 0\). Add and subtract the means for these intermediate regimes:
If \(a_t=0\), the two regimes in the \(t\)th difference are identical, so that difference is zero. If \(a_t=1\), they differ only in treatment at time \(t\), with prior treatments set to \(\bar a_{t-1}\) and later treatments set to zero. Conditional on the history under that prior regime, the mean difference is \(\psi_t\) by the blip definition. Taking expectations over that history therefore gives an unconditional difference of \(\psi_t\). The \(t\)th term is consequently \(a_t\psi_t\). Summing the terms proves the result. \(\square\)
This result gives each parameter a direct reference-regime interpretation. It does not say that a treatment has the same effect at every time: \(\psi_t\) may differ freely across \(t\). Nor does it imply that an effect-modified blip can be replaced by a constant. If the conditional blip varies with history, its contribution generally has to be averaged over the history distribution under the relevant past regime.
Worked Example: Combining Three Time-Specific Contributions
Suppose there are three treatment decisions and the reference-regime mean outcome is 50 units. A fitted constant additive SNMM has \(\psi_0=2\), \(\psi_1=-1\), and \(\psi_2=4\).
Under regime \((1,1,0)\), the contrast with the reference regime is \(1(2)+1(-1)+0(4)=1\). Its mean outcome is therefore \(50+1=51\).
Under regime \((0,1,1)\), the contrast is \(0(2)+1(-1)+1(4)=3\), giving mean \(50+3=53\). Under \((1,1,1)\), the contrast is \(2-1+4=5\), giving mean 55.
The negative value at time 1 reduces the mean relative to the reference regime; it does not erase the distinct contributions from times 0 and 2. These calculations rely on the constant-blip model and concern fixed treatment regimes, not the mean outcome among people who happened to receive those treatments.
Sequential Blip-Down and G-Estimation
To connect the model to observed data, define the outcome with treatment effects from time \(t\) onward removed:
where the empty sum gives \(U_K(\psi)=Y\). At decision \(t\), \(U_{t+1}(\psi)\) has the later treatment contributions removed, while \(A_t\psi_t\) is the contribution to remove at the current decision. Thus \(U_t(\psi)=U_{t+1}(\psi)-A_t\psi_t\). Under the correct model and sequential exchangeability, the conditional mean of this blipped-down outcome does not depend on \(A_t\) once \(H_t\) is held fixed.
For clarity, the usual observed-data conditions are: consistency (the observed outcome agrees with the potential outcome under the received regime); no interference between individuals; sequential exchangeability (at each decision, treatment is independent of the relevant future potential outcomes given the measured history); and positivity (each treatment value has positive probability at histories being analyzed). The blip model must also be correctly specified, and the moments below must exist. Exchangeability and consistency connect counterfactual blips to observed data; positivity supplies treatment variation to identify them.
Derivation. Correct blip-down implies that \(E[U_t(\psi)\mid H_t,A_t]\) is a function of \(H_t\) alone; call it \(q_t(H_t)\). Therefore,
Taking expectations gives the stated moment. Substituting \(U_t=U_{t+1}-A_t\psi_t\) and solving gives the ratio, because \(E[(A_t-p_t(H_t))A_t\mid H_t]=p_t(H_t)\{1-p_t(H_t)\}\). The denominator is positive when there is treatment variation on a set of histories with positive probability. In practice, the equations are commonly solved backward in time: the last blip uses \(U_K=Y\), then earlier equations use outcomes with later blips removed. \(\square\)
Worked Example: Solving Two Sequential Moments
Consider two randomized binary treatment decisions, each with probability \(1/2\), and four observed records:
| Individual | \(A_0\) | \(A_1\) | \(Y\) |
|---|---|---|---|
| 1 | 0 | 0 | 10 |
| 2 | 0 | 1 | 13 |
| 3 | 1 | 0 | 12 |
| 4 | 1 | 1 | 15 |
Use the sample analog of the moment equation, first for time 1. Here \(U_2=Y\), and the numerator is \(\sum_i(A_{1i}-0.5)Y_i=-5-6+6.5+7.5=3\). The denominator is \(\sum_i(A_{1i}-0.5)A_{1i}=0+0+0.5+0.5=1\). Hence \(\hat\psi_1=3\).
For time 0, first remove the estimated time-1 blip: \(U_{1i}=Y_i-3A_{1i}\), giving \(10,10,12,12\). The numerator is \(\sum_i(A_{0i}-0.5)U_{1i}=-5-5+6+6=2\), and the denominator is \(\sum_i(A_{0i}-0.5)A_{0i}=0+0+0.5+0.5=1\). Thus \(\hat\psi_0=2\). Blipping down both contributions gives \(Y_i-2A_{0i}-3A_{1i}=10\) for every record, as expected for this deliberately constructed data set.
This tiny example illustrates the algebra, not the reliability of inference from four observations. In real data, treatment probabilities depend on measured histories, and the empirical moments use those probabilities or estimates of them.
Worked Example: A Contrast Between Two Nonreference Regimes
With three decisions, let \(\psi_0=1.5\), \(\psi_1=2\), and \(\psi_2=-0.5\). Compare regimes \((1,0,1)\) and \((0,1,0)\). Relative to the all-zero regime, their contrasts are \(1.5+0-0.5=1\) and \(0+2+0=2\), respectively. Their mean difference is therefore \(1-2=-1\): the first regime has a mean outcome one unit lower than the second under this model.
Equivalently, the difference is \(1.5-2-0.5=-1\), with each term representing the change in treatment at a particular time. The reference-regime mean cancels in a comparison between two regimes.
Diagnostics, Alternatives, and Limitations
The additive multi-time model is useful when the scientific target is a set of interpretable, decision-specific mean contributions. It differs from the parametric g-formula, which models covariate and outcome distributions to calculate regime means, and from a marginal structural model, which directly models marginal regime means. These are alternative g-method formulations; their assumptions and targets are not interchangeable merely because all can compare treatment regimes.
Several checks are particularly important. Inspect estimated treatment probabilities by time and history: values near zero or one signal weak support for one treatment choice and can make the moment denominator small. Check that the blipped-down outcome has little residual association with current treatment after conditioning on the recorded history, as the model requires. Large or systematic residual associations, or materially different estimates under reasonable model specifications, can indicate misspecification or inadequate adjustment. These checks cannot establish sequential exchangeability, which depends on whether important confounders were measured.
- Common mistake: adding coefficients without checking the regime. The contrast from \(\bar 0\) to \(\bar a\) is \(\sum_t a_t\psi_t\), not necessarily \(\sum_t\psi_t\). The latter is the contrast for the all-treated regime.
- Common mistake: using \(Y\) in every time-specific moment. Earlier blips must be estimated after later treatment contributions are removed. Otherwise later treatment effects can remain in the outcome used to identify an earlier parameter.
- Common mistake: treating a small denominator as a numerical nuisance only. A small \(E[p_t(H_t)\{1-p_t(H_t)\}]\) reflects limited treatment variation, so the data may contain little information about that blip.
Finally, a constant \(\psi_t\) is restrictive: it rules out variation in the time-\(t\) blip across histories. The additive model also focuses on conditional mean contrasts and does not, by itself, describe changes in outcome distributions. Its regime-contrast interpretation depends on the stated model and assumptions; it does not turn an observed treatment-group difference into a causal effect.
Check Your Understanding
Answer each question using the definitions and results above. Full solutions follow the questions.
- For three decisions with blips \((1,-2,3)\), what is the model-implied contrast of regime \((1,0,1)\) with the reference regime?
- Why is the time-\(t\) moment based on \(U_{t+1}\), rather than on the unmodified outcome \(Y\)?
- What does a zero value of \(E[p_t(H_t)\{1-p_t(H_t)\}]\) imply for identifying \(\psi_t\) from this moment?
- Suppose the time-1 blip depends on history rather than being constant. Why might the contrast for a regime with \(a_1=1\) not equal \(\psi_1\) as a single fixed increment?
Solutions. (1) The contrast is \(1(1)+0(-2)+1(3)=4\). (2) \(U_{t+1}\) has later treatment contributions removed, so the current moment isolates the current blip rather than mixing it with effects of later treatment. (3) There is no treatment variation on histories with positive probability, so the moment denominator is zero and this equation cannot identify \(\psi_t\). (4) The conditional blip can vary across histories; the regime contrast then depends on averaging those history-specific effects over the history distribution under the relevant past regime, rather than adding one constant increment.