Proximal and Cumulative Treatment Effects in SNMMs

One decision or an entire treatment history?

A structural nested mean model (SNMM) describes treatment effects by asking what changes when treatment at one decision is altered, while the treatment history before that decision and the treatment plan after it are held fixed. This is a proximal effect: it is local to one decision, but it can depend on past history and on the specified future treatment. A cumulative effect instead compares outcomes under two complete treatment regimes. It combines changes at multiple decisions, generally averaging each local effect over a history distribution determined by the regime.

These targets are related, but they are not interchangeable. In particular, a unit-treatment blip is defined for treatment 1 versus treatment 0. If the target treatment at a decision is 0, that decision contributes zero to the contrast against 0, even though the unit-treatment blip itself need not be zero.

Defining a proximal blip

Consider decisions at times $t=0,\ldots,K$, binary treatments $A_t$, and an outcome $Y$. Write $\bar a_{t-1}=(a_0,\ldots,a_{t-1})$ for the treatment history before time $t$, and $\bar a_{t+1:K}$ for the future treatment sequence after time $t$. Let $H_t(\bar a_{t-1})$ be the covariate and treatment history just before $A_t$, under the indicated past treatment history. The potential outcome under a full sequence $\bar a$ is $Y(\bar a)$.

Definition: The unit-treatment mean blip at decision $t$ is $$ \gamma_t(h_t;\bar a_{t+1:K}) = E\!\left[ Y(\bar a_{t-1},1,\bar a_{t+1:K}) - Y(\bar a_{t-1},0,\bar a_{t+1:K}) \mid H_t(\bar a_{t-1})=h_t \right]. $$ It is the conditional mean contrast for setting $A_t$ to 1 rather than 0, with the past history and all future treatments fixed. It is defined on histories with positive probability (or, for continuous histories, in a specified regular conditional version on the relevant support).

The notation suppresses the dependence of the history on the past regime only when that regime is clear. The blip is not a comparison of the two treatment groups as observed; it is a contrast between potential outcomes under interventions. For a specified binary treatment value $a_t$, the corresponding treatment-specific contrast against 0 is

E\!\left[ Y(\bar a_{t-1},a_t,\bar a_{t+1:K}) - Y(\bar a_{t-1},0,\bar a_{t+1:K}) \mid H_t(\bar a_{t-1})=h_t \right] = a_t\gamma_t(h_t;\bar a_{t+1:K}).

The equality holds because $a_t$ is either 0 or 1: when it is 0 the contrast is zero, and when it is 1 it is the unit-treatment blip. It does not assert that a blip is multiplied by the treatment value for arbitrary continuous treatments. A continuous-treatment model requires a treatment-specific contrast function, not this binary scaling rule.

From blips to cumulative regime contrasts

A cumulative contrast compares $E[Y(\bar a)]$ with $E[Y(\bar 0)]$, where $\bar 0$ is the all-zero treatment regime. One exact way to decompose it is to move from $\bar 0$ to $\bar a$ in time order. Define $\bar a^{(t)}=(a_0,\ldots,a_t,0,\ldots,0)$, and let $\bar a^{(-1)}=\bar 0$. Consecutive regimes differ only at decision $t$.

Theorem: Suppose $A_t$ is binary, the potential outcomes and histories in the definition above are well-defined, and the relevant conditional expectations exist. Then $$ E[Y(\bar a)]-E[Y(\bar 0)] = \sum_{t=0}^{K} a_t E\!\left[ \gamma_t\{H_t(\bar a_{t-1});\bar 0_{t+1:K}\} \right]. $$ Each expectation is taken over the distribution of $H_t$ under the past treatment history $\bar a_{t-1}$. The future treatments in each blip are zero because this particular telescoping path leaves future decisions at zero.

This identity needs no statistical model for the blips: it follows from their definition and linearity of expectation. Its interpretation as an observed-data estimand additionally requires identification conditions, discussed below.

Insert intermediate regimes.
Write $E[Y(\bar a)]-E[Y(\bar 0)]$ as the sum of $E[Y(\bar a^{(t)})]-E[Y(\bar a^{(t-1)})]$ over $t$.

Isolate the changed decision.
The two regimes in each difference have the same past and future. At time $t$, the treatment changes from 0 to $a_t$.

Apply the blip definition.
Conditional on the shared pre-treatment history, the difference is $a_t\gamma_t$. Average over that history under the shared past regime, then sum the increments.

For completeness, the intermediate differences sum exactly because every interior term cancels:

\sum_{t=0}^{K} \left\{E[Y(\bar a^{(t)})]-E[Y(\bar a^{(t-1)})]\right\} = E[Y(\bar a^{(K)})]-E[Y(\bar a^{(-1)})] = E[Y(\bar a)]-E[Y(\bar 0)].

For a contrast between arbitrary regimes $\bar b$ and $\bar a$, use intermediate regimes that replace one component of $\bar b$ at a time by the corresponding component of $\bar a$. The increment at $t$ is then a treatment-specific contrast between $a_t$ and $b_t$, evaluated at the past history induced by the intermediate past and with the intermediate future held fixed. It is not generally just $a_t\gamma_t$, which is the special contrast against reference treatment 0.

Worked examples

Worked Example: A Two-Decision Model With Interaction

Suppose the conditional mean potential outcome under two binary decisions is $$ E[Y(a_0,a_1)]=10+2a_0+3a_1+1.5a_0a_1. $$ There are no covariates, so the conditional mean blips are constant for each specified treatment history. Find the proximal blips and the cumulative contrast for regime $(1,1)$ versus $(0,0)$.

At time 1, compare $a_1=1$ with $a_1=0$, holding $a_0$ fixed: $$ \gamma_1(a_0)=\{10+2a_0+3+1.5a_0\}-\{10+2a_0\}=3+1.5a_0. $$ At time 0, the blip depends on the future value $a_1$: $$ \gamma_0(a_1)=\{10+2+3a_1+1.5a_1\}-\{10+3a_1\}=2+1.5a_1. $$

Use the time-ordered decomposition from $(0,0)$ to $(1,1)$. First change $a_0$, leaving $a_1=0$: the increment is $\gamma_0(0)=2$. Then change $a_1$, with $a_0=1$: the increment is $\gamma_1(1)=4.5$. Thus the cumulative contrast is $2+4.5=6.5$. Direct calculation checks it: $E[Y(1,1)]=10+2+3+1.5=16.5$, while $E[Y(0,0)]=10$, giving $16.5-10=6.5$. The interaction means that the time-0 blip is 3.5 if future treatment is 1, not 2; the decomposition uses 2 because its first step fixes future treatment at 0.

Worked Example: A Zero Target Is Not a Unit Blip

In the same model, compare regime $(0,1)$ with $(0,0)$. The target at time 0 is 0, and the target at time 1 is 1. The time-0 increment is $0\cdot\gamma_0(0)=0$. The time-1 blip is $\gamma_1(0)=3$, so the cumulative contrast is 3.

Checking the potential outcome means gives $E[Y(0,1)]=10+3=13$ and $E[Y(0,0)]=10$, hence a contrast of 3. The unit-treatment blip at time 0 is still $\gamma_0(0)=2$; it is not the increment for a target treatment of 0. Multiplying by the target value is essential in this binary-treatment comparison against 0.

Worked Example: Averaging an Effect-Modified Blip

Let $L$ be a binary baseline covariate with $P(L=1)=0.25$, and suppose $$ E[Y(a_0,a_1)\mid L] = 10+a_0(1+4L)+a_1\{2+a_0(1+2L)\}. $$ Find the time-0 proximal blip with future treatment fixed at 0, the time-1 blip after $a_0=1$, and the cumulative mean contrast for $(1,1)$ versus $(0,0)$.

With $a_1=0$, changing $a_0$ from 0 to 1 gives $\gamma_0(L;0)=1+4L$. Its average is $0.75(1)+0.25(5)=2$. At time 1 after $a_0=1$, the blip is $\gamma_1(1,L)=2+1+2L=3+2L$, whose average is $0.75(3)+0.25(5)=3.5$. The cumulative contrast is $2+3.5=5.5$.

A direct check gives a regime contrast of 4 when $L=0$ and 10 when $L=1$, so its population mean is $0.75(4)+0.25(10)=5.5$. The calculation averages conditional effects over the population distribution of $L$; using only the effect for $L=0$, or averaging over a different covariate distribution, would target a different quantity.

Assumptions, diagnostics, and limits

The potential-outcome decomposition is algebraic once the blips and regimes are defined. To identify these effects from observed longitudinal data requires additional assumptions. In the usual setup, consistency links observed outcomes to potential outcomes under the treatment actually received; sequential exchangeability rules out unmeasured confounding of each treatment decision given the measured pre-treatment history; and positivity requires support for the treatment choices being contrasted at histories reached by the target regime. The outcome and conditional means must also be integrable. These assumptions apply to the histories and regimes used in the target, not merely to the overall sample.

A proximal blip is local in the treatment sequence, but it need not be a constant or an immediate biological effect. It may vary with measured history and with future treatment when there are treatment interactions. A cumulative contrast is marginal over histories under its specified regime. The time-ordered decomposition is exact, yet another order of changing regime components can produce different intermediate blips; the total contrast remains the same when all terms are defined consistently.

Useful diagnostics include checking whether fitted blips leave residual treatment associations of the kind implied by the g-estimation moment conditions, examining overlap at histories reached by target regimes, and checking sensitivity to alternative blip specifications. Sparse treatment support can make a blip or regime mean depend heavily on extrapolation. A misspecified blip can also obscure effect modification or interactions, even when a cumulative contrast appears plausible. These diagnostics do not establish exchangeability or rule out unmeasured confounding.

A marginal structural model typically parameterizes regime-specific marginal means directly, whereas an SNMM parameterizes treatment effects through conditional blips and combines them over histories. The approaches can address related regime contrasts, but their model parameters and assumptions are not automatically identical. Neither parameterization removes the need for identification assumptions or adequate support. Also, a cumulative mean difference is scale-specific: it is not automatically a risk ratio, odds ratio, or an effect on a different outcome scale.

Common pitfalls

Using a unit blip as the increment for any target value. For a binary target $a_t$ against reference 0, the increment is $a_t\gamma_t$. A zero target contributes zero.
Ignoring future treatment in an interaction model. The same time-$t$ treatment contrast can differ under different future regimes; state which future sequence is fixed.
Adding conditional blips without specifying the history distribution. Each blip must be averaged over the history distribution induced by the relevant past regime.
Calling the blip an observed treated-versus-untreated comparison. It is a potential-outcome contrast; causal identification from observed data needs assumptions such as sequential exchangeability and positivity.

Key takeaway: A proximal SNMM blip is a conditional unit-treatment contrast with past history and future treatment specified. A cumulative regime contrast is built by changing decisions along a path, multiplying a binary unit blip by the target treatment when contrasting against 0, and averaging over the appropriate regime-specific histories.

Check Your Understanding

Answer each question, then compare with the solutions below.

If $A_t$ is binary and the target value is 0, what is its contribution to a contrast against reference treatment 0?
In the model $E[Y(a_0,a_1)]=5+2a_0+4a_1+3a_0a_1$, calculate $\gamma_0(a_1)$ and $\gamma_1(a_0)$.
Using that model, calculate the contrast for $(1,1)$ versus $(0,0)$ by changing decisions in time order.
Why must the histories used to average a blip be linked to the past regime in the decomposition?

Solutions. 1. The contribution is $0\cdot\gamma_t=0$. 2. Subtracting the means at $a_0=1$ and 0 gives $\gamma_0(a_1)=2+3a_1$; subtracting at $a_1=1$ and 0 gives $\gamma_1(a_0)=4+3a_0$. 3. First change $a_0$ with $a_1=0$, giving 2; then change $a_1$ with $a_0=1$, giving 7. The total is 9, also verified by $E[Y(1,1)]=14$ and $E[Y(0,0)]=5$. 4. The conditional blip is evaluated among histories generated under that past intervention; averaging under another history distribution generally targets a different population contrast.