Backward G-Estimation Across Multiple Times

Why Estimate Blips Backward?

With treatment at several times, the observed outcome reflects effects from multiple treatment decisions. A later treatment may also be associated with an earlier one because treatment histories influence subsequent care. Estimating an early treatment effect directly from the final outcome can therefore mix the early blip with effects of later treatment.

Backward g-estimation separates these contributions. It first removes the modeled effect of the latest treatment from the outcome, then uses that adjusted outcome to estimate the preceding treatment blip. Repeating the process yields estimates from the final treatment time back to the first.

The Backward Recursion

Suppose there are \(T\) treatment times and a final outcome \(Y\). Define the starting pseudo-outcome by \(R_{T+1}=Y\). For a candidate collection of blip parameters, work backward according to

At time \(t\), the later blips have already been subtracted from \(R_{t+1}\). The current blip is then subtracted to form \(R_t\). If the blip models are correct and their parameters are the true values, \(R_t\) has had the modeled effects of treatment from time \(t\) onward removed. The last quantity, \(R_1\), is an outcome blipped down across all treatment times.

Backward estimation uses this recursion in reverse time order. Estimate \(\psi_T\) from the final outcome, use that estimate to construct \(R_T\), estimate \(\psi_{T-1}\) using \(R_T\), and continue until time 1. At each step, the target is the effect of the current treatment after later treatment effects have been removed.

Identification and the Estimation Step

To estimate a blip, g-estimation searches for parameters that remove the current treatment’s conditional association with the appropriately blipped-down outcome. In the binary-treatment case, this idea can be expressed as equality of conditional means for treated and untreated participants with the same history.

This equality describes the identifying condition at the true parameters; it does not require the treatment groups to have the same distribution of histories. The comparison is within \(H_t\). In practice, the conditional means are estimated from data, and the parameters are chosen so the fitted blipped-down outcome satisfies the corresponding g-estimation condition.

Sequential exchangeability is the causal condition that allows a within-history treatment comparison to identify the corresponding causal blip rather than an association caused by unmeasured confounding. Positivity ensures that the comparison is supported by data: if everyone with a particular history receives treatment, the untreated conditional mean for that history cannot be learned from the observed data. Correct specification matters because subtracting an incorrect later blip leaves some later treatment effect in the pseudo-outcome.

Why the Later Blip Is Removed First

The central justification is that the treatment contrast at time \(t\) should be taken on an outcome from which later modeled effects have already been removed. Under the assumptions above, the correctly blipped-down outcome has no remaining conditional mean effect of the current treatment. For a binary treatment with a zero blip at \(A_t=0\), the conditional mean contrast in \(R_{t+1}\) is therefore the current blip:

The first equality follows because subtracting the true current blip from \(R_{t+1}\) leaves \(R_t\), whose conditional mean does not vary with \(A_t\) under the identifying condition. The second equality uses the zero blip for no treatment. Thus the current blip can be found from the conditional contrast in the outcome after later effects have been removed. Estimating this contrast from the unadjusted final outcome instead can attribute part of a later effect to the earlier treatment.

Uneven Treatment Histories

Treatment probabilities may differ across histories. Backward g-estimation does not require treatment to be equally common in every history, but it does require both treatment levels to occur with positive probability in the histories used for a comparison. The next example shows how removing a later blip can matter even when the marginal treatment groups have very different compositions.

Common Mistakes and Practical Meaning

• Estimating the first blip from the final outcome without adjustment. This can include the effects of later treatment. Estimate later blips first and subtract them before estimating earlier ones.
• Comparing treatment groups without conditioning on history. Sequential exchangeability is stated conditional on \(H_t\); a marginal comparison need not identify the current blip.
• Assuming positivity from an overall treatment count. Both treatment levels must occur within the relevant histories. A large sample cannot repair a history stratum with no treatment variation.
• Treating an estimated pseudo-outcome as an observed counterfactual. It is constructed by subtracting fitted blips from observed outcomes. Its causal interpretation depends on the assumptions and model being adequate.

The backward order is useful because it makes the role of each treatment time explicit: later effects are removed before earlier ones are estimated. It does not make the conclusions automatic. Unmeasured confounding, lack of treatment variation, or misspecified blip functions can undermine identification or estimation, and the recursion cannot correct those problems by itself.