Worked Example of a Two-Decision Additive SNMM

From Two Treatment Decisions to Two Blips

With treatment decisions at two times, the final outcome can reflect both decisions. A useful way to express the model is to separate the treatment effect at the second decision from the effect at the first. The resulting additive structural nested mean model (SNMM) describes how the mean outcome changes when a treatment is switched on at a particular decision, given the history available then.

Let $A_1,A_2\in\{0,1\}$ be treatments at two successive decisions, $L_0$ baseline covariates, and $L_1$ covariates measured after $A_1$ but before $A_2$. Write $Y^{a_1,a_2}$ for the outcome under treatment regime $(a_1,a_2)$. The histories just before the decisions are $H_1=L_0$ and $H_2=(L_0,A_1,L_1)$. A simple additive model specifies the second-decision blip as $\gamma_2(H_2;\psi_2)$ and the first-decision blip as $\gamma_1(L_0;\psi_1)$.

Definition: For binary treatments, an additive two-decision SNMM specifies the mean blips $$ E[Y^{a_1,1}-Y^{a_1,0}\mid H_2(a_1)] =\gamma_2(H_2(a_1);\psi_2), \qquad E[Y^{1,0}-Y^{0,0}\mid L_0] =\gamma_1(L_0;\psi_1). $$ Here $H_2(a_1)$ denotes the history that would be observed after first-decision treatment $a_1$. The two blips need not be constant: covariates may modify either effect.

The model is additive on the outcome-mean scale. For example, $\gamma_2(H_2;\psi_2)=\psi_{20}+\psi_{21}L_1$ says that the second-decision treatment contrast changes linearly with $L_1$. It does not assert that the outcome itself is a linear function of all covariates, nor does it require the effect of $A_1$ to equal the effect of $A_2$.

Assumptions and the Reverse-Time Calculation

Let $\pi_1(H_1)=P(A_1=1\mid H_1)$ and $\pi_2(H_2)=P(A_2=1\mid H_2)$. These are the treatment probabilities, or propensities, at the corresponding decisions. For the causal interpretation of g-estimation, assume consistency, sequential mean exchangeability and positivity. Consistency means that the observed outcome and covariate history agree with their counterfactual values under the treatments actually received. Sequential mean exchangeability requires that, conditional on each observed history, treatment assignment carries no additional information about the relevant counterfactual mean. Positivity requires $0<\pi_k(H_k)<1$ on histories with positive probability, for $k=1,2$. Also assume the expectations used below exist and that the stated blip models are correctly specified.

Blipping down removes the modeled treatment contribution from the observed outcome. It must be done in reverse time order: first remove the second-decision blip, then remove the first-decision blip. Under the model, the resulting pseudo-outcomes are

U_2(\psi_2)=Y-A_2\gamma_2(H_2;\psi_2), \qquad U_1(\psi_1,\psi_2)=U_2(\psi_2)-A_1\gamma_1(L_0;\psi_1).

At the true parameters, $U_2$ has had the second-decision treatment effect removed. Its conditional mean is therefore not associated with the second treatment once $H_2$ is held fixed. Likewise, after removing the first-decision blip, $U_1$ has no remaining mean association with $A_1$ conditional on $H_1$. This motivates equations that multiply each blipped outcome by a treatment residual.

Definition: A basic pair of g-estimation equations is $$ E[(A_2-\pi_2(H_2))U_2(\psi_2)]=0, \qquad E[(A_1-\pi_1(H_1))U_1(\psi_1,\psi_2)]=0. $$ To estimate effect modifiers, use history-based instruments $Z_2(H_2)$ and $Z_1(H_1)$ and set the corresponding vector moments to zero.

In data, replace expectations by sample averages and the propensities by known probabilities or fitted treatment models. A first-decision model can be logistic in $L_0$; the second-decision model should use the history available at that time, such as $A_1,L_0,L_1$. In particular, a variable affected by $A_1$ may belong in the second model, even though it is not available for the first.

Worked Examples

Worked Example: Solving Both Blips in a Balanced Dataset

Suppose four equally likely records have the treatment combinations and outcomes shown below. There are no covariates, and the treatment probabilities at both decisions are $1/2$. Use constant blips $\gamma_2=\psi_2$ and $\gamma_1=\psi_1$.

$A_1$	$A_2$	$Y$
0	0	10
0	1	13
1	0	12
1	1	15

At decision 2, $U_2=Y-\psi_2A_2$. The sample moment is $$ \frac14\sum_{i=1}^4(A_{2i}-1/2)U_{2i} =\frac18\{(13-\psi_2)+(15-\psi_2)-10-12\} =\frac{6-2\psi_2}{8}. $$ Setting it to zero gives $\hat\psi_2=3$. Thus the estimated second-decision blip is three outcome units.

Now blip down $A_2$. The $U_2$ values, in table order, are $10,10,12,12$. At decision 1, $U_1=U_2-\psi_1A_1$, so $$ \frac14\sum_{i=1}^4(A_{1i}-1/2)U_{1i} =\frac18\{(12-\psi_1)+(12-\psi_1)-10-10\} =\frac{4-2\psi_1}{8}. $$ The second equation gives $\hat\psi_1=2$. The order matters: the first-decision equation uses the outcome already blipped down for $A_2$.

Worked Example: Unequal Treatment Probabilities

Consider a population with $P(L_0=0)=P(L_0=1)=1/2$. Suppose $P(A_1=1\mid L_0=0)=0.25$ and $P(A_1=1\mid L_0=1)=0.75$. At the second decision, let $P(A_2=1\mid A_1=0)=0.4$ and $P(A_2=1\mid A_1=1)=0.6$. These probabilities correspond to the logistic models $$ \operatorname{logit}\pi_1(L_0)=-\log 3+L_0\log 9, \qquad \operatorname{logit}\pi_2(A_1)=\log(2/3)+A_1\log(9/4). $$

Suppose the outcome mean follows $Y=5+2A_1+3A_2$, with no additional outcome noise. At the second decision, conditional on $H_2$, the non-$A_2$ part of $Y$ is fixed. Since a Bernoulli variable $A$ with probability $\pi$ satisfies $E[(A-\pi)A]=\pi(1-\pi)$, the population moment for a candidate $\psi_2$ is $$ E[(A_2-\pi_2)(Y-\psi_2A_2)] =E[\pi_2(1-\pi_2)](3-\psi_2). $$ Here $E[\pi_2(1-\pi_2)]=0.4(0.6)=0.24>0$, so the root is $\psi_2=3$.

After blipping down $A_2$, $U_2=5+2A_1$. At the first decision, the moment is $$ E[(A_1-\pi_1)(U_2-\psi_1A_1)] =E[\pi_1(1-\pi_1)](2-\psi_1). $$ Both baseline strata have $\pi_1(1-\pi_1)=0.25(0.75)=0.1875$; hence the expectation is $0.1875$, which is positive. The root is $\psi_1=2$. This example shows why the residual must be $A_k-\pi_k(H_k)$, not $A_k-1/2$ when assignment probabilities vary by history.

Worked Example: A Second-Decision Effect Modifier

Let $L_1$ be binary, with all combinations of $A_1,A_2,L_1$ equally represented. Suppose $\pi_1=\pi_2=1/2$ and the outcome is $Y=8+2A_1+(1+2L_1)A_2$. Use the second-decision blip $\gamma_2(L_1)=\psi_{20}+\psi_{21}L_1$ and the constant first-decision blip $\gamma_1=\psi_1$.

For a fixed value of $L_1$, the second-decision residual moment is proportional to $$ (1+2L_1)-\psi_{20}-\psi_{21}L_1. $$ Using instruments $Z_2=(1,L_1)^\mathsf{T}$ gives two equations, one for each value of $L_1$. At $L_1=0$, the equation is $1-\psi_{20}=0$. At $L_1=1$, it is $3-\psi_{20}-\psi_{21}=0$. Therefore $\hat\psi_{20}=1$ and $\hat\psi_{21}=2$.

The blipped outcome is then $U_2=8+2A_1$. The first-decision moment is proportional to $2-\psi_1$, so $\hat\psi_1=2$. The modifier estimate says the second-decision blip is one unit when $L_1=0$ and three units when $L_1=1$.

Why the Equations Work and When the Parameters Are Identified

The next result states the moment justification and a useful uniqueness condition. The mean-independence statements in the assumptions are the implications of sequential exchangeability and the correctly specified blip models that g-estimation uses. For the first equation, assume specifically that $E[U_2(\psi_2^*)\mid A_2,H_2]$ does not depend on $A_2$, where $\psi_2^*$ is the true parameter. For the second, assume $E[U_1(\psi_1^*,\psi_2^*)\mid A_1,H_1]$ does not depend on $A_1$.

Theorem: Under these conditional mean-independence assumptions, the true parameters satisfy the two residual-moment equations. For a linear blip $\gamma_k(H_k;\psi_k)=Z_k(H_k)^\mathsf{T}\psi_k$, the parameter at decision $k$ is uniquely determined by its population moment if $$ M_k=E[\pi_k(H_k)\{1-\pi_k(H_k)\}Z_k(H_k)Z_k(H_k)^\mathsf{T}] $$ is positive definite. At decision 1, use the outcome already blipped down at decision 2.

Proof. At decision 2, condition on $H_2$. By the stated conditional mean-independence assumption, $E[U_2(\psi_2^*)\mid A_2,H_2]=m_2(H_2)$ for some function $m_2$. Therefore $$ E[(A_2-\pi_2(H_2))U_2(\psi_2^*)\mid H_2] =m_2(H_2)E[A_2-\pi_2(H_2)\mid H_2]=0. $$ Taking expectations proves the second-decision moment. The same argument, conditioning on $H_1$ and using $U_1$, proves the first-decision moment.

For uniqueness in the linear model, the part of the moment depending on $\psi_k$ has coefficient matrix $$ E[(A_k-\pi_k)A_kZ_kZ_k^\mathsf{T}] =E[\pi_k(1-\pi_k)Z_kZ_k^\mathsf{T}] =M_k, $$ where the equality follows by conditioning on $H_k$. If $M_k$ is positive definite, it is invertible, so the population linear moment equation has at most one solution for $\psi_k$. Positivity alone does not guarantee this condition: the instrument components must also vary enough across histories to distinguish the parameters. $\square$

In a finite sample, equations can be noisy. With as many independent moments as parameters, one may solve the sample equations; with more moments, one may use a specified weighting criterion. Estimated propensities are ordinarily substituted into the equations, and reliable estimation requires adequate overlap and appropriately specified treatment models.

Diagnostics, Comparisons and Limitations

A zero sample moment does not by itself validate the causal model. Check fitted propensities for values near zero or one, because such histories contribute little information and can make estimates unstable. Check whether the second-decision model includes the observed history available at that decision. Compare blip specifications with and without plausible effect modifiers, and inspect residual-moment patterns across covariate strata. Large remaining associations between treatment residuals and blipped outcomes can indicate a misspecified blip, a misspecified treatment model, or inadequate control of treatment-confounder history.

The additive SNMM is useful when the target is a decision-specific mean contrast and treatment-confounder feedback is present. It differs from a conventional outcome regression that conditions on all observed covariates: conditioning on $L_1$ in an ordinary regression may not recover the desired marginal treatment contrast, since $L_1$ can itself be affected by $A_1$. G-estimation instead uses treatment residuals given each decision's history and removes modeled effects sequentially. Other g-methods target related causal questions using different parameterizations or estimation strategies; their estimands and assumptions must be compared explicitly rather than treated as interchangeable.

The method remains limited by the assumptions and models that support its moments. Unmeasured confounding can invalidate the conditional mean-independence conditions; lack of positivity undermines identification; and an incorrect functional form for either blip can distort the answer even when treatment probabilities are well modeled. Conversely, a correct blip does not automatically protect against a misspecified propensity model in the usual implementation. These are model-based causal estimates, not an automatic consequence of solving equations.

Common pitfall: Using the first-decision equation with $Y$ instead of the second-decision-blipped outcome $U_2$ leaves the $A_2$ contribution in the outcome. If later treatment depends on earlier treatment or history, that remaining contribution can distort the first-decision moment. First form $U_2=Y-A_2\gamma_2(H_2;\psi_2)$; then use $U_2$ to form $U_1$ and the first-decision equation.

Check Your Understanding

Answer each question, then compare with the solutions below.

For binary $A_2$, why does the second-decision equation use $A_2-\pi_2(H_2)$ rather than $A_2-1/2$?
Suppose $\gamma_2(L_1)=\psi_{20}+\psi_{21}L_1$, and the true blips are 2 at $L_1=0$ and 5 at $L_1=1$. What are the two parameters?
In a constant-blip model, suppose the second-decision population moment simplifies to $0.18(4-\psi_2)$. Find its root and explain why it is unique.
Name two distinct reasons that a fitted g-estimate may be unreliable, even if its sample moments are close to zero.

Solutions. (1) The residual must have conditional mean zero given the treatment history: $E[A_2-\pi_2(H_2)\mid H_2]=0$. A fixed $1/2$ does not generally have this property when treatment probabilities vary by history. (2) At $L_1=0$, $\psi_{20}=2$. At $L_1=1$, $\psi_{20}+\psi_{21}=5$, so $\psi_{21}=3$. (3) Setting $0.18(4-\psi_2)=0$ and dividing by $0.18>0$ gives the unique root $\psi_2=4$. (4) Examples include near-violations of positivity, unmeasured confounding, a misspecified blip model, and a misspecified treatment model. A near-zero empirical moment cannot rule these out.

\(A_1\)	\(A_2\)	\(Y\)
0	0	10
0	1	13
1	0	12
1	1	15