What do stabilized inverse probability weights change compared with plain 1/e(X) weights?

## The definition An inverse-probability-of-treatment weight is normally written as one over the probability of the arm a unit actually received: `1/e(X)` for a treated unit and `1/(1 - e(X))` for a control, where `e(X) = P(T = 1 | X)` is the propensity score. A **stabilized** weight replaces the 1 in the numerator with the corresponding *marginal* probability, i.e. the probability of that arm ignoring covariates: - treated: `sw = P(T = 1) / e(X)` - control: `sw = P(T = 0) / (1 - e(X))` If 20% of the sample is treated, a treated unit with `e(X) = 0.02` has an unstabilized weight of 50 and a stabilized weight of `0.20 / 0.02 = 10`. ## What it buys you **The weights become interpretable.** A stabilized weight near 1 means the unit's covariates make it about as likely to receive its arm as the population average; a weight of 10 means it was ten times less likely than average. The mean stabilized weight should be close to 1 in each arm — a mean far from 1 is a useful smoke test that the treatment model is misspecified. **The weighted sample size is right.** Unstabilized weights sum to something much larger than n, so the unnormalized estimator's scale depends on a sum that fluctuates from sample to sample. Stabilized weights sum to roughly n, which keeps that estimator's variance down and stops estimates drifting outside the plausible range. **The estimand does not move.** Stabilization is a rescaling of the weights, not a change of question. The target is still the same average treatment effect, and the identification assumptions are exactly the same as before. ## The caveat interviewers are looking for In a single-time-point setting the numerator is a *constant within each arm*. Every treated weight is multiplied by `P(T = 1)` and every control weight by `P(T = 0)`. That has two consequences worth stating. First, if you use the weight-normalized estimator — a proper weighted average within each arm, which is what a weighted regression of the outcome on the treatment indicator computes — the constant cancels and the point estimate is *numerically identical* whether or not you stabilize. The benefit appears in the unnormalized form, whose scale actually depends on the size of the weights. Second, and more important: because it is a constant rescale, stabilization does **not** compress the relative spread of the weights. If one unit held 10% of its arm's total weight before, it holds 10% afterwards. The largest weight is numerically smaller, which can lull you into thinking the problem is gone, but the concentration of influence is unchanged. Extreme-weight problems are addressed by fixing the treatment model, restricting to the region where both arms occur, or capping weights — not by stabilizing. ## Where stabilization really matters The payoff grows with the complexity of the treatment process. When treatment is measured repeatedly over time, a unit's weight is the product of per-period inverse probabilities, and a product of many terms below 1 explodes: after ten periods an unstabilized weight can reach the thousands. The stabilized version multiplies numerators that shrink at a comparable rate, so the product stays in a manageable range. In that setting stabilization is effectively mandatory rather than optional, which is why it is usually introduced alongside repeated-treatment analyses. ## A refinement: baseline covariates in the numerator The numerator does not have to be the plain marginal probability. It can be the probability of treatment given a subset of *baseline* covariates, `P(T = 1 | V)`. This shrinks the weights further, at the price of a rule: any variable placed in the numerator must also be included in the outcome model you fit in the weighted sample, because the weights no longer balance it. Forget that and you reintroduce confounding by exactly those variables. Interviewers like this detail because it separates candidates who have used stabilized weights from those who have only read the formula. ## Diagnostics to quote After stabilizing, report the mean weight (should be near 1), the maximum, the ratio of maximum to median, and the effective sample size. The mean is the stabilization-specific check; the others tell you whether the underlying weight concentration is still a threat. ## Summary answer Stabilized weights swap the 1 in the numerator for the marginal probability of the arm, giving weights that average 1 and sum to about n. They lower the variance of the unnormalized estimator and make the weights readable, they leave the estimand and the assumptions untouched, and they are close to essential for repeated-treatment weights. They are not a remedy for a covariate region where one arm barely exists.