Why the Treatment Question Must Include Its Timing
In a diabetes cohort, a comparison of patients currently receiving a glucose-lowering drug with those not receiving it can refer to different causal questions. Does changing treatment at the present visit improve glycemic outcome, while leaving prior care as it occurred? Or does starting now and continuing treatment at every visit improve outcome compared with never following that treatment strategy? The first question changes current treatment only; the second intervenes on a treatment history.
These questions need not have the same answer. Prior treatment can alter later health, and those health changes can affect both current treatment and outcome. The weighted sustained-treatment contrasts in the earlier material on marginal glycemic outcomes address treatment paths. Here the focus is how that target differs from a current-treatment contrast standardized over the history that patients actually reached.
The current-treatment contrast is standardized over the pre-\(A_2\) history distribution in the target cohort. It does not compare a selected group who happened to receive current treatment with a selected group who did not. The sustained contrast instead changes earlier treatment too, and therefore generally changes the distribution of \(L_1\) that contributes to the outcome.
Identification and Weighting
Write \(Y^a\) for the potential outcome under setting current treatment to \(a\), with earlier treatment left natural. Write \(Y^{a_1,a_2}\) for the potential outcome under the specified two-visit treatment path. The following formulas require consistency, sequential exchangeability, and positivity. Consistency means the observed outcome equals the potential outcome for the treatment history actually received. Sequential exchangeability means there is no unmeasured confounding of treatment assignment at each decision time, conditional on the measured history then available. Positivity means every treatment value required by the intervention has positive probability at histories occurring in the target population.
For the current-treatment intervention, the relevant exchangeability condition is \(Y^a \perp A_2 \mid H_2\), and positivity requires \(P(A_2=a\mid H_2)>0\) almost surely in the target population. Iterated expectation, exchangeability, and consistency give:
The third line uses conditional exchangeability: within a given pre-current history, the potential outcome under \(a\) has the same conditional mean among those assigned \(a\) as in the full history stratum. The last line uses consistency for those who received \(a\). This formula averages the conditional outcome mean over the natural distribution of \(H_2\), not the history distribution among current-treatment recipients.
An equivalent inverse-probability expression makes the weighting explicit. Let \(p_2(a\mid H_2)=P(A_2=a\mid H_2)\). Then:
In the middle equality, conditional on \(H_2\), the indicator has probability \(p_2(a\mid H_2)\), which cancels the denominator. Thus weighting corrects the current-treatment groups to represent the same natural history distribution. In a finite sample, a ratio of weighted sums is often used; the expression above establishes the population target.
For sustained treatment, let \(p_1(a_1\mid X)=P(A_1=a_1\mid X)\) and \(p_2(a_2\mid X,A_1,L_1)=P(A_2=a_2\mid X,A_1,L_1)\). Under sequential exchangeability at both visits, consistency, and positivity at both decision times, the longitudinal g-formula is:
Unlike the current-treatment formula, this expression averages \(L_1\) over its distribution under the specified earlier treatment \(a_1\). The corresponding inverse-probability identity is:
To see the identity, condition first on \(X,A_1,L_1\). The current-treatment indicator and its probability cancel, leaving \(I(A_1=a_1)E[Y\mid X,A_1=a_1,L_1,A_2=a_2]/p_1(a_1\mid X)\). Conditional on \(X\), averaging over \(A_1\) cancels \(p_1\) for the stratum \(a_1\); averaging over \(L_1\) then uses its distribution given \(X,A_1=a_1\). The result is exactly the g-formula above. This derivation also shows why sustained-regime weights include treatment probabilities at both visits, whereas current-treatment weights include only the current decision probability.
Worked Example: Weighting a Current-Treatment Comparison
Suppose the pre-current history is summarized by \(H\in\{0,1\}\), with \(P(H=0)=P(H=1)=0.5\). Current treatment is more common for \(H=0\): \(P(A_2=1\mid H=0)=0.8\) and \(P(A_2=1\mid H=1)=0.2\). Assume conditional exchangeability and the following potential-outcome means:
| History \(H\) | \(E[Y^0\mid H]\) | \(E[Y^1\mid H]\) |
|---|---|---|
| 0 | 10 | 8 |
| 1 | 9 | 6 |
The current-treatment means standardize both treatment options to the natural history proportions. Thus \(\mu_{\mathrm{cur}}(1)=0.5(8)+0.5(6)=7\), and \(\mu_{\mathrm{cur}}(0)=0.5(10)+0.5(9)=9.5\). The current-treatment contrast is \(7-9.5=-2.5\).
The unweighted treated group has \(P(H=0\mid A_2=1)=0.8\) and \(P(H=1\mid A_2=1)=0.2\): the joint probabilities are \(0.5(0.8)=0.4\) and \(0.5(0.2)=0.1\), whose sum is \(0.5\). Its mean is \(0.8(8)+0.2(6)=7.6\). Among untreated patients, the joint probabilities for \(H=0\) and \(H=1\) are \(0.1\) and \(0.4\), so the history proportions are \(0.2\) and \(0.8\). The untreated mean is \(0.2(10)+0.8(9)=9.2\). The raw contrast is therefore \(7.6-9.2=-1.6\), not the standardized contrast \(-2.5\).
The current-treatment weights are \(1/0.8=1.25\) for treated patients with \(H=0\), \(1/0.2=5\) for treated patients with \(H=1\), \(1/0.2=5\) for untreated patients with \(H=0\), and \(1/0.8=1.25\) for untreated patients with \(H=1\). For treatment \(1\), the weighted history masses are \(0.4(1.25)=0.5\) and \(0.1(5)=0.5\), reproducing the target proportions. The weighted mean is \(\{0.4(1.25)(8)+0.1(5)(6)\}/(0.5+0.5)=7\). For treatment \(0\), the weighted history masses are \(0.1(5)=0.5\) and \(0.4(1.25)=0.5\); the weighted mean is \(\{0.1(5)(10)+0.4(1.25)(9)\}/1=9.5\). Their difference is \(-2.5\), as required.
The Same Cohort Can Have Different Causal Contrasts
The next calculation makes the distinction especially clear. For a current-treatment contrast, the outcome means must be specified within the naturally occurring prior-treatment strata. Marginal means under fixed treatment paths alone do not supply those conditional means: they answer a different question and cannot automatically be substituted.
Worked Example: Current Treatment Versus a Sustained Strategy
Suppose the natural prior treatment \(A_1\) is equally likely to be 0 or 1. The following are the conditional means under an intervention on current treatment, retaining the natural prior treatment:
The current-treatment mean under \(A_2=0\) is \(0.5(12)+0.5(8)=10\). Under \(A_2=1\), it is \(0.5(10)+0.5(7)=8.5\). The current-treatment contrast is \(8.5-10=-1.5\).
Now suppose the means under the two sustained strategies are \(E[Y^{(0,0)}]=12\) and \(E[Y^{(1,1)}]=7\). The sustained-treatment contrast is \(7-12=-5\). These are not alternative calculations of the same effect: the current contrast changes only \(A_2\) and averages across natural prior treatment, while the sustained contrast changes both \(A_1\) and \(A_2\). In this example the effect of current treatment is more modest than the contrast between the sustained strategies. The conditional current means and the sustained path means have each been stated explicitly, so neither estimand is being inferred from insufficient marginal information.
History, Positivity, and Weight Diagnostics
A current-treatment analysis asks whether both current treatment choices occur across the histories that define its target population. A sustained-treatment analysis asks whether the complete treatment path occurs across the histories needed for that regime. As the number of visits grows, a sustained path may become rare even when each individual treatment decision seems reasonably common. The product of treatment probabilities can then produce large weights.
Worked Example: A Rare Sustained Path
Suppose that under the strategy \(A_1=1\), half the target population has low \(L_1\) and half has high \(L_1\). Treatment at visit 1 has probability \(p_1(1\mid X)=0.5\). Among those with \(A_1=1\), suppose \(p_2(1\mid X,A_1,L_1)\) is \(0.8\) for low \(L_1\) and \(0.2\) for high \(L_1\). The probabilities of observing the full path \((1,1)\), conditional on the respective \(L_1\) strata, are \(0.5(0.8)=0.4\) and \(0.5(0.2)=0.1\). The corresponding unstabilized weights are \(1/0.4=2.5\) and \(1/0.1=10\).
Suppose the regime-compatible observations have stratum outcomes 4 for low \(L_1\) and 10 for high \(L_1\). Their unconditional population masses are \(0.5(0.4)=0.20\) and \(0.5(0.1)=0.05\). Weighting gives masses \(0.20(2.5)=0.5\) and \(0.05(10)=0.5\), so the weighted mean is \(\{0.5(4)+0.5(10)\}/(0.5+0.5)=7\). The high-history observations each receive a much larger weight because the sustained path is less likely there. This calculation illustrates both the correction and its instability: if very few such observations are available, the estimate can be sensitive to one or two outcomes.
Useful diagnostics include the treatment-probability distributions by history, the largest weights, weight summaries by treatment path, and the effective sample size after weighting. A lack of observed treatment overlap is not repaired by a finite-looking average weight; where positivity fails, the desired mean is not identified from the observed data without additional restrictions or a narrower target population. Weight truncation can reduce variance but changes the estimator and may introduce bias; it does not make a structural absence of treatment choices disappear.
A regression that adjusts for measured current history can also estimate a current-treatment contrast if its model is correctly specified and its predictions are standardized to the target history distribution. The weighting and regression approaches differ in how they model the observed data, but they must target the same estimand to be comparable. A raw comparison generally standardizes to neither the common natural history distribution nor the history distribution under a sustained regime. The choice among these approaches does not remove the need for exchangeability, positivity, consistency, and a clearly stated target population.
Check Your Understanding
Answer each question, then compare with the solutions.
- If \(P(A_1=0)=0.4\), \(E[Y^{A_2=0}\mid A_1=0]=10\), and \(E[Y^{A_2=0}\mid A_1=1]=7\), what is the current-treatment mean for \(A_2=0\)?
- Why does a sustained-treatment weight include a probability for treatment at an earlier visit?
- Suppose \(P(A_2=1\mid H)=0\) for a history that occurs in the target population. Can the current-treatment mean under \(A_2=1\) be identified by inverse-probability weighting over that full target population?
- In one sentence, distinguish the history distribution used for a current-treatment mean from the history distribution used for a sustained-treatment mean.
Solutions. (1) The remaining probability is \(P(A_1=1)=0.6\), so the mean is \(0.4(10)+0.6(7)=4+4.2=8.2\). (2) The sustained intervention requires receiving the specified earlier treatment as well as the current one, so the weight must account for the probability of following the full path. (3) No. The required treatment option is never observed for that history, violating positivity; weighting cannot recover its outcome mean from the observed data without additional assumptions. (4) A current-treatment mean averages over the natural pre-current history distribution, whereas a sustained-treatment mean uses the history distribution that would arise under its specified earlier treatment strategy.