Worked Example: Two-Visit Treatment And Confounding

Why the Order of the Blip-Down Matters

When treatment is assigned at two visits, the variable measured between visits may be affected by the first treatment and may also influence both the second treatment and the outcome. It is then a treatment-affected intermediate confounder for the later treatment. A conventional adjustment that treats it as an ordinary baseline covariate can obscure the causal effect of the first treatment. Sequential g-estimation instead uses it in the later-treatment equation, removes the later-treatment effect, and then estimates the earlier-treatment effect.

The example below uses binary treatments and a continuous outcome. All numbers describe a hypothetical dataset created for illustration; they are not results from a real study. The aim is to show the sequence of calculations, not to claim that the assumptions could be confirmed from a table of observed data alone.

Model, Notation, and Assumptions

Let \(A_1\) and \(A_2\) denote treatment at visits 1 and 2, with each taking values 0 or 1. Let \(L\) be measured after \(A_1\) and before \(A_2\), and let \(Y\) be the final outcome. Let \(L_0\) denote baseline covariates; in the example there are none. Write \(H_1=L_0\) for the first-visit history and \(H_2=(L_0,A_1,L)\) for the second-visit history. The treatment probabilities are \(e_1(H_1)=P(A_1=1\mid H_1)\) and \(e_2(H_2)=P(A_2=1\mid H_2)\).

Equivalently, the sequential conditional moment equations are

for the second visit, and

for the first visit. In practice, these conditional moments are commonly used as estimating equations, summed over individuals. The history in the first equation includes \(L\); the history in the second does not, because \(L\) occurs after \(A_1\).

Identification and estimation require consistency, sequential exchangeability, and positivity. Consistency means each observed outcome agrees with the counterfactual outcome under the treatment actually received. Sequential exchangeability means that, given each visit’s measured pre-treatment history, treatment assignment is independent of the relevant future counterfactual outcomes. Positivity means both treatment options have positive probability at each history with support in the population. The additive blip models and treatment-probability models must also be correctly specified for the estimating procedure used. These assumptions are not established by the arithmetic below; they are stipulated for the hypothetical example.

Hypothetical Data

Suppose \(A_1\) was randomized with probability \(1/2\). The intermediate variable \(L\) is more common among those assigned \(A_1=1\), so it is affected by the first treatment. Within each \((A_1,L)\) history, the chance of \(A_2=1\) varies, and every history includes both second-visit treatment values. The table gives sample sizes and mean outcomes in each treatment cell.

For example, the first history has 120 people, of whom 30 received \(A_2=1\); its empirical treatment probability is \(30/120=0.25\). The four empirical probabilities are \(0.25\), \(0.75\), \(0.20\), and \(0.70\), respectively. All are strictly between 0 and 1, so this table has treatment variation in every listed history. The outcomes also vary with \(L\), which makes it important to account for \(L\) when estimating the effect of \(A_2\).

Blip Down, Then Estimate Visit 1

The second-visit estimate is used to remove the modeled contribution of \(A_2\) from each outcome: \(U_2=Y-3A_2\). This step matters because \(A_2\) is not assigned with the same probability across histories. In particular, the probability of later treatment depends on \(L\), which is related to the outcome and was itself affected by \(A_1\).

Now use \(e_1=0.5\), the randomization probability, and solve the first-visit estimating equation with \(U_2\). Do not condition this equation on \(L\): doing so would condition on a variable affected by \(A_1\), rather than use the baseline history that governed the first treatment.

Sequential Estimation as a Procedure

The order is not merely bookkeeping. The later treatment can be confounded by \(L\), so the later-treatment moment needs to compare treatment assignments within histories that include \(L\). Once that later blip is removed, the earlier-treatment moment uses the earlier assignment mechanism. This lets the procedure address treatment-confounder feedback without treating a post-treatment variable as if it were a baseline confounder.

Common Mistake: Adjusting for the Intermediate Variable at Both Visits

A tempting approach is to include \(L\) in both treatment equations, or to compare the observed \(A_1\) groups within levels of \(L\) when estimating \(\psi_1\). The first-visit assignment occurred before \(L\) was measured. Since \(A_1\) affects \(L\), conditioning on \(L\) changes the comparison and can block part of the earlier treatment’s effect or induce bias. The correction is to use the pre-treatment history for each visit: include \(L\) for \(A_2\), but not for \(A_1\).

A second mistake is to interpret the raw difference in final outcomes by \(A_1\) as the sequentially estimated blip. In this table, the observed mean \(Y\) for \(A_1=0\) is \(11.6+3(0.45)=12.95\), because 90 of 200 received \(A_2=1\). For \(A_1=1\), it is \(14.6+3(0.55)=16.25\), because 110 of 200 received \(A_2=1\). The raw difference is \(16.25-12.95=3.30\), not the blip-down estimate \(3.00\). The extra \(0.30\) reflects the different proportions receiving the later treatment. Sequential estimation separates the two treatment contributions under the specified model.