Why does a marginal structural model use IPW weights when a confounder is affected by prior treatment?

## The setting A treatment is not applied once but repeatedly: a drug given or withheld each month, a discount applied each week, a policy toggled each quarter. Write the treatment sequence as `A_1, ..., A_T`, the covariates measured before each decision as `L_1, ..., L_T`, and the final outcome as `Y`. The interesting complication is **treatment-confounder feedback**: some `L_t` both influences the next treatment decision and is itself a consequence of earlier treatment. A clinician's dosing depends on a lab value that the earlier dose changed; a targeting rule depends on engagement that last week's discount raised. ## Why one regression cannot work Suppose you fit a single outcome regression on the treatment history and adjust for every `L_t`. **Adjusting for `L_t` blocks part of the effect you want.** Because `L_t` sits on the causal path from `A_{t-1}` to `Y`, conditioning on it removes exactly the portion of the earlier treatment's effect that travelled through it. The coefficient on earlier treatment now measures only the leftover direct path, which is not the total effect anyone asked about. This is over-adjustment, and it happens even with perfectly measured data. **Adjusting can also create bias.** `L_t` is a common effect of earlier treatment and of whatever else drives it. If some unmeasured factor influences both `L_t` and `Y`, conditioning on `L_t` opens a non-causal association between earlier treatment and the outcome — a collider path that did not exist before you adjusted. **Not adjusting is no better.** Leave `L_t` out and the later treatment decisions remain confounded by it, since it drives both `A_t` and `Y`. There is no way to include and exclude the same variable at once, so no single regression on the observed data returns the total effect of the treatment sequence. This is the classic argument for a different machinery. ## What weighting does instead Weighting never conditions on `L_t`; it uses it only to build a weight. For each unit compute, at each period, the probability of the treatment it actually received given its treatment history and covariate history, and take the product over periods. The weight is one over that product, and in practice the stabilized version is used: the numerator is the product of the probabilities of the same treatment history given only past treatment (and possibly fixed baseline variables). In the pseudo-population these weights create, the dependence of `A_t` on the time-varying covariates has been severed: at every period, treatment looks as if it were assigned without regard to the history. Because `L_t` was never conditioned on, the effect flowing through it is preserved. You then fit the **marginal structural model** itself — a deliberately simple model of the outcome as a function of the treatment history alone, for instance on cumulative exposure or on always-treat versus never-treat — in the weighted population. Its coefficients carry a causal interpretation about the counterfactual outcomes under those treatment regimes, which is precisely what the naive regression could not deliver. ## The judgment call a lead has to make This machinery is powerful and expensive, and deciding whether to spend it is the real principal-level question. **Costs.** Weights multiply across periods, so their spread grows quickly with the number of periods; by ten periods a few units can dominate entirely and the effective sample size collapses. Every period needs its own credible treatment model, and every period needs enough units of both kinds at each history for the weights to be meaningful. You also need the time-varying covariates to be measured well and at the right times; a covariate recorded after the decision it supposedly informed silently invalidates the whole construction. **Alternatives worth considering first.** Coarsen the treatment into a small number of regimes that the business actually cares about — started within the first month versus never started — so the weights involve far fewer periods. Shorten the horizon. Restrict to a subpopulation with real variation in treatment at every period. Or, when the intervention is genuinely under your control, run an experiment on the assignment rule rather than reconstructing it from observational history. **Communication.** A weighted time-varying analysis is hard for stakeholders to audit. If you commit to one, commit also to publishing the weight diagnostics per period, the effective sample size over time, and a sensitivity analysis over the truncation rule — otherwise the result is a number nobody can challenge, which is worse than a cruder number everyone can. ## The one-sentence answer When a confounder is also a consequence of earlier treatment, conditioning on it is both necessary and forbidden; weighting resolves the contradiction by using the variable to construct weights rather than as a regressor, and the marginal structural model is then fitted in the resulting pseudo-population.