Marginal Structural Model (IPTW)

Unstabilized weights are the inverse of the full covariate-conditional probability of the observed treatment; they can become enormous when some treatment histories are rare, inflating variance and letting a few subjects dominate. Stabilized weights add a numerator equal to the marginal (or baseline-covariate) probability of the observed treatment, which keeps the weights centered near one without changing the target estimand. Robins, Hernán, and Brumback show stabilization typically yields far narrower confidence intervals while preserving confounding control, so it is the recommended default. A mean stabilized weight near one is also a useful informal check on the model.

Positivity requires that, at every covariate history, each treatment level has a non-zero probability. When this fails — say, patients with a certain lab value are essentially never treated — the corresponding inverse probability blows up, producing extreme weights. A handful of subjects then carry the whole estimate, variance explodes, and bias can appear because the pseudo-population is built largely from rare cases. Diagnose it by inspecting the weight distribution; remedies include truncating weights, restricting to the region of common support, or switching to the g-formula, though each has its own costs.

Both use a treatment model, but for different problems. Ordinary propensity-score methods address point-treatment (single-time) confounding. An IPTW marginal structural model extends the idea to a time-varying treatment with confounders affected by prior treatment, multiplying inverse-probability weights across visits to build a pseudo-population where the whole treatment sequence is unconfounded. The generic single-time IPW is a special case of this longitudinal construction; the MSM machinery is what makes it valid for treatment-confounder feedback, which static propensity adjustment cannot handle.