How does standardisation, the g-formula, turn a fitted outcome model into an average treatment effect?

## The idea in one sentence A model gives you predictions conditional on covariates; a causal effect is a contrast between two whole worlds. Standardisation — also called the g-formula, or g-computation — is the bridge: predict everyone under each treatment value, then average. ## The recipe 1. Fit one model for the outcome as a function of treatment `A` and covariates `X`. Include interactions between `A` and `X` if the effect plausibly varies; the recipe accommodates them. 2. Take a copy of the whole dataset and set `A = 1` for every row, leaving covariates untouched. Predict. Call the mean of those predictions `m1`. 3. Take another copy with `A = 0` for every row. Predict. Call the mean `m0`. 4. The estimated average treatment effect is `m1 - m0`. The two averages estimate `E[Y^1]` and `E[Y^0]` — mean outcome if everyone were treated and if nobody were. The formula behind it is `E[Y^a] = average over X of E[Y | A=a, X]`, where the average uses the covariate distribution of the population you care about. ## Why not just read the treatment coefficient On a linear model with no treatment-covariate interaction and a correctly specified form, the coefficient and the standardised estimate agree, and the extra work buys nothing. Everywhere else they part company: - **Effect heterogeneity.** If the model contains `A*X` interactions, no single coefficient is the population effect. The `A` coefficient is the effect at `X = 0`, which may not even be a covariate value that exists. - **Non-linear links.** A logistic or log-link model's coefficient is a conditional contrast on the model's scale — a conditional odds ratio, say. Averaging predicted probabilities instead gives you a marginal risk difference or risk ratio, which is what most decisions actually need. - **Choice of estimand.** Averaging over the whole sample gives the ATE. Averaging the same predicted difference over the treated rows only gives the effect of treatment on the treated. Same fitted model, different question, and the recipe makes the choice explicit rather than implicit. ## What has to hold Standardisation is an estimation strategy, not an identification argument. It gives a causal quantity only if: - **No unmeasured confounding given `X`** — treatment is as good as randomised within covariate strata. - **Positivity** — every covariate pattern has some chance of both treatment values. This one deserves attention: step 2 asks the model to predict outcomes under treatment for units that were never treated. Where such units genuinely do not exist in the data, the prediction is extrapolation dressed up as an estimate, and nothing in the output flags it. Checking the overlap of covariate distributions between arms before running the recipe is part of the job. - **Consistency** — a well-defined intervention that matches the treatment as observed. - **Correct outcome model** — the whole estimate rides on the model being right, since every counterfactual mean comes from it. There is no second line of defence in this estimator. ## Inference The naive standard error from the fitted model does not apply, because the estimate is a nonlinear function of the fitted parameters averaged over the sample's own covariates. The standard approach is a bootstrap of the entire procedure: resample rows, refit the model, redo the two predictions and the averaging, and take the spread of the resulting estimates as the sampling distribution. Bootstrapping only the prediction step while holding the fit fixed understates the uncertainty, because it ignores the fact that the model itself was estimated. ## How to talk about it A crisp interview answer is: I fit an outcome model, then predict the whole sample twice with treatment forced on and forced off, and take the mean difference. That gives me a marginal effect on the scale I care about, it survives interactions and nonlinear links, and I bootstrap the whole thing for the interval. The assumption I lean on hardest is that the outcome model is right, and I check overlap first so the counterfactual predictions are interpolation rather than invention.