Structural Nested Mean Models and Blip Functions

Why Structural Nested Mean Models Use Blips

With treatment that can change over time, an outcome contrast need not describe a single, timeless effect. A treatment given at one decision time may have a different effect from the same treatment given later, and its effect may depend on the treatment and covariate history up to that decision. A blip function expresses one such treatment-time-specific contrast: the change in the mean counterfactual outcome when treatment at a particular time is changed, while the past history and a specified future treatment regime are held fixed.

A structural nested mean model (SNMM) describes these contrasts directly, rather than beginning with a model for the observed conditional mean outcome. This focus makes the model’s parameters causal in their intended interpretation. It does not, by itself, make those parameters identifiable from observed data; assumptions connecting observed data to counterfactual outcomes are still needed.

Definition and Reference Regimes

Consider decision times $t=1,\ldots,T$, with treatment $A_t$, covariates $L_t$ measured before treatment at time $t$, and terminal outcome $Y$. Write $\bar a_t=(a_1,\ldots,a_t)$ for a treatment history and $\bar L_t=(L_1,\ldots,L_t)$ for a covariate history. Let $Y^{\bar a_T}$ denote the counterfactual outcome under the full treatment regime $\bar a_T$. For a regime specified only through time $t$, write $Y^{\bar a_t,\underline 0_{t+1:T}}$ for the outcome if treatment follows $\bar a_t$ through time $t$ and is set to the reference level $0$ at every later time. The reference $0$ is a treatment level, not necessarily “no care” or “no treatment” in a substantive setting.

The history $\bar L_t^{\bar a_{t-1}}$ is the covariate history that would occur under the specified past treatments $\bar a_{t-1}$, before the treatment decision at time $t$. The same prior regime and this same counterfactual history are used on both sides of the contrast below.

Definition: For a treatment level $a_t$, a reference level $0$, and a parameter vector $\psi$, a treatment-time-specific blip function is $$ \gamma_t(\bar a_{t-1},a_t,\bar l_t;\psi) = E\!\left[ Y^{\bar a_{t-1},a_t,\underline 0_{t+1:T}} - Y^{\bar a_{t-1},0,\underline 0_{t+1:T}} \mid \bar L_t^{\bar a_{t-1}}=\bar l_t \right]. $$ An SNMM specifies a model for this conditional causal contrast, typically by setting it equal to a function of treatment history, covariate history, and $\psi$. It is a model for the blips, not automatically a model for the observed outcome mean.

This definition makes several features explicit. The comparison is conditional on covariates immediately before decision $t$, under a common past treatment regime. Treatment after time $t$ is set to the reference level in both potential outcomes. Thus this blip is not, without further assumptions, the effect of changing $A_t$ while following an arbitrary later treatment policy. If the scientific question concerns a particular future policy, the contrast must be defined using that policy in both potential outcomes.

A basic setup assumes that the potential outcomes in the definition are well-defined and that the relevant conditional means exist. For a causal interpretation, the treatment levels and regimes must be sufficiently well specified; consistency and no interference are common assumptions. These definitions do not establish identification from observed data. In particular, sequential exchangeability and positivity are among the assumptions that may be needed for identification, and the model alone does not supply them.

Worked Example: A Binary Treatment Blip

Suppose there is one treatment decision, so $T=1$, and let $L_1=x$ denote the covariate history before treatment. In a population with this history, suppose $E[Y^1\mid L_1=x]=3.2$ and $E[Y^0\mid L_1=x]=2.5$. The time-1 blip for treatment $1$ relative to $0$ is $$ \gamma_1(1,x)=3.2-2.5=0.7. $$

The value $0.7$ is a conditional mean contrast on the outcome scale: for people with the specified pre-treatment history, the mean counterfactual outcome under treatment $1$ is $0.7$ units higher than under treatment $0$. A model $\gamma_1(a,x;\psi)=\psi a$ represents this contrast when $\psi=0.7$, since $\gamma_1(1,x;0.7)=0.7$ and $\gamma_1(0,x;0.7)=0$. The numerical contrast is causal by definition of the potential outcomes; observed data do not reveal both means for the same person or history without additional identification assumptions.

What a Blip Model Implies About Contrasts

The reference treatment is a convenient anchor. Once a model gives the mean contrast for each treatment level against that anchor, a contrast between two non-reference levels follows by subtraction, provided the history and future regime are the same in both comparisons.

Result: Fix $t$, $\bar a_{t-1}$, and $\bar l_t$. If the conditional means exist and the relevant counterfactuals are defined, then for treatment levels $a$ and $a'$, $$ \begin{aligned} & E\!\left[ Y^{\bar a_{t-1},a,\underline 0_{t+1:T}} - Y^{\bar a_{t-1},a',\underline 0_{t+1:T}} \mid \bar L_t^{\bar a_{t-1}}=\bar l_t \right] \\ &\qquad = \gamma_t(\bar a_{t-1},a,\bar l_t;\psi) - \gamma_t(\bar a_{t-1},a',\bar l_t;\psi), \end{aligned} $$ when the specified blip model correctly represents both reference contrasts.

Proof. Denote the conditional mean of $Y^{\bar a_{t-1},a,\underline 0_{t+1:T}}$, given the stated history, by $m(a)$. By the definition of the blip, $\gamma_t(\bar a_{t-1},a,\bar l_t;\psi)=m(a)-m(0)$, and likewise the blip for $a'$ is $m(a')-m(0)$. Subtracting gives $(m(a)-m(0))-(m(a')-m(0))=m(a)-m(a')$, which is the stated contrast. $\square$

This result depends on comparing the same conditional history and the same reference future regime. Subtracting blips evaluated at different histories or under different future regimes does not, in general, produce the causal contrast in the result.

Worked Example: Two Non-Reference Treatment Levels

Suppose the treatment has levels $0,1,2$, and a proposed blip model at a particular history $x$ is $\gamma(a,x;\psi)=a(0.4-0.1x)+0.05a^2$. At $x=2$, the coefficient of the linear term is $0.4-0.1(2)=0.2$. The reference-level blip is $\gamma(0,2;\psi)=0$. For level $1$, $\gamma(1,2;\psi)=1(0.2)+0.05(1^2)=0.25$. For level $2$, $\gamma(2,2;\psi)=2(0.2)+0.05(2^2)=0.4+0.2=0.6$.

Therefore the model-implied contrast of level $2$ versus level $1$, at this same history and with the same future reference treatment, is $0.6-0.25=0.35$ outcome units. The contrast is not the level-2 blip alone: that blip compares level $2$ with level $0$.

From Conditional Blips to an Average Contrast

A blip is conditional on a history, but a scientific question may ask for an average contrast across the histories that would arise under a particular past intervention. The averaging distribution matters: histories under one past regime need not have the same distribution as histories under another.

Result: Fix a past intervention $\bar a_{t-1}$ and a treatment level $a$. If the relevant potential outcomes are integrable and the conditional blip is defined, then $$ E\!\left[ Y^{\bar a_{t-1},a,\underline 0_{t+1:T}} - Y^{\bar a_{t-1},0,\underline 0_{t+1:T}} \right] = E\!\left[ \gamma_t(\bar a_{t-1},a,\bar L_t^{\bar a_{t-1}};\psi) \right]. $$ The expectation on the right is over the covariate-history distribution under the specified past intervention.

Derivation. Let $D$ be the difference between the two potential outcomes in the left-hand side. By the definition of conditional expectation, the blip evaluated at the realized counterfactual history is $E[D\mid\bar L_t^{\bar a_{t-1}}]$. Taking expectations of both sides and applying the law of iterated expectations gives $E[E[D\mid\bar L_t^{\bar a_{t-1}}]]=E[D]$. Integrability ensures these expectations exist. This proves the result. $\square$

Worked Example: Averaging History-Specific Blips

Under a fixed past intervention, suppose a pre-treatment history summary $H$ takes values $0,1,2$ with probabilities $0.2,0.5,0.3$. For a binary current treatment $a$, let the blip model be $\gamma(a,H)=a(0.4+0.1H)$. For treatment $1$, the conditional blips at $H=0,1,2$ are $0.4,0.5,0.6$, respectively; for treatment $0$, each is $0$.

The average contrast for treatment $1$ versus $0$, standardized to this history distribution, is $(0.2)(0.4)+(0.5)(0.5)+(0.3)(0.6)=0.08+0.25+0.18=0.51$. Thus the average contrast is $0.51$ outcome units. It is not the unweighted average of the three values, and it applies to the specified past-intervention distribution, not automatically to another population or regime.

Interpreting and Checking an SNMM

A useful blip model respects its reference contrast: setting $a_t=0$ must give $\gamma_t=0$, because the two potential outcomes in the definition then coincide. Beyond that restriction, the model can allow the effect to vary by time, past treatment, or pre-treatment covariates. For example, an interaction between $a_t$ and a component of $\bar l_t$ represents effect modification on the conditional mean-difference scale. The choice of scale is substantive: an additive blip is a difference in means, not a risk ratio or odds ratio.

The main alternative approach in this course is the parametric g-formula, which models outcome and covariate distributions under interventions and then uses those models to calculate intervention-specific quantities. An SNMM instead models the treatment-time-specific contrasts directly. Neither approach is automatically reliable: each depends on appropriate modeling and identification assumptions. A blip model can be attractive when the target is a treatment effect contrast, but it does not by itself supply every regime-specific outcome mean. Conversely, g-computation can target such means but requires correctly specified models for the relevant longitudinal distributions.

A few practical checks help expose conceptual or modeling problems:

Verify that the reference treatment gives a zero blip and that the treatment contrast matches the intended outcome scale.
Confirm that both potential outcomes in a contrast share the same prior regime, covariate history, and future regime.
Check whether the fitted functional form gives implausible contrasts over treatment and covariate values with support in the data.
Do not interpret a modeled conditional contrast as identified merely because a regression or estimation procedure returns a number. Identification requires assumptions about treatment assignment and support in addition to a blip model.

A common mistake is to call $\gamma_t$ the effect of treatment at time $t$ “regardless of what happens later.” The definition here fixes later treatments to the reference level. If later treatment is allowed to respond differently under the two current treatment choices, the resulting contrast may differ, and it must be defined for that future policy. Another mistake is to treat a conditional blip as a population-average effect without specifying how histories are weighted. The averaging result shows why the history distribution must be named.

The definition also has limits. It describes a conditional mean contrast for specified regimes; it does not guarantee that all treatment levels occur at all histories, that the contrast is empirically estimable, or that the chosen parametric form is adequate. A zero reference level must be scientifically meaningful or replaced by another explicitly chosen reference. Finally, when outcomes or treatment effects involve more complicated scales or future policies, the blip definition and model must be adjusted to match the target contrast rather than assumed to carry over unchanged.

Check Your Understanding

Answer each question, then check the solutions below.

In the blip definition, what is held common between the two potential outcomes?
If a model gives $\gamma_t(1,\bar l_t)=0.3$ and $\gamma_t(2,\bar l_t)=0.8$, what is the model-implied contrast of level $2$ versus level $1$ at that same history?
Why is the average blip not generally the unweighted average of its values at a few selected histories?
A proposed blip model has $\gamma_t(a,\bar l_t)=0.2+0.5a$. What reference-level check does it fail?

Solutions. (1) The past treatment regime, the covariate history under that past regime, and the future reference regime are common; only treatment at time $t$ changes. (2) The contrast is $0.8-0.3=0.5$ outcome units, since both blips use the same history and reference regime. (3) The population average weights history-specific blips by the history distribution under the specified past intervention; equal weights are correct only if that distribution assigns equal probability to the selected histories. (4) At $a=0$, the model gives $0.2$, but the two potential outcomes defining the reference blip coincide, so the blip must be zero. The proposed model needs a form that satisfies this constraint.