Causal Inference

Conditional ignorability requires that within every level of the measured covariates X, which units received treatment is independent of their potential outcomes. Treatment must be as good as randomly assigned inside each X stratum.

Positivity requires every covariate stratum in the target population to contain both treated and untreated units: 0 < P(T=1 | X=x) < 1. A stratum with only one condition supplies no comparison, so no effect is identified there.

ATE averages the effect Y(1) - Y(0) over the whole population, ATT over only the units that actually got treated, and ATU over the untreated ones. ATE is their size-weighted mix, and the three can differ in sign.

Each unit has two potential outcomes, one under treatment and one under control, but only the one matching the arm it actually received is ever observed. The other stays missing, so an individual causal effect is never measured directly.

SUTVA has two parts: one unit's treatment does not affect another unit's outcome, and there is only one version of the treatment, so every unit labelled treated received effectively the same thing.

Simpson's paradox is when a comparison that holds inside every subgroup reverses once the subgroups are pooled. It happens because the two groups sit in the subgroups in very different proportions, so the pooled rate weights those subgroups unequally.

A backdoor path from treatment T to outcome Y is any path starting with an arrow into T; it carries non-causal association. A valid adjustment set contains no descendant of T and blocks every such path.

A collider is a variable that two others both cause. Conditioning on it - filtering to a subgroup, stratifying, or adding it as a control - correlates those two causes in the data even when they are independent.

A mediator sits on the causal path from treatment to outcome, so adjusting for it removes the very effect you set out to measure. A real total effect shrinks toward zero, and new bias can appear.

Tenure and Black-Friday timing each influence who was emailed and who buys, so both sit on backdoor paths and belong in the adjustment set. Clicking is caused by the campaign, so the criterion excludes it.

A propensity score is a unit's probability of receiving the treatment given its observed covariates. Conditioning on that single number balances the covariates that went into it, so matching happens in one dimension instead of many.

Adding a measured confounder turns the treatment coefficient from a raw comparison into a within-strata one: it now compares treated and untreated units that share the same covariate value, removing the part of the gap that covariate explained.

Each unit is weighted by the inverse of its probability of receiving the arm it actually got: treated units by 1/e(X), controls by 1/(1-e(X)). The reweighted sample mimics a population where treatment was assigned independently of the measured covariates.

Compare standardized mean differences for each covariate before and after matching, with an absolute value under about 0.1 as the usual bar, and read them off a love plot. Check variance ratios and interactions too, not just means.

Multiply two signs: the effect of the omitted variable on the outcome, times its correlation with the included regressor. Ability raises wages and is positively correlated with schooling, so the schooling coefficient is biased upward.

It subtracts the comparison group's before-to-after change from the treated group's before-to-after change. That double difference cancels the fixed level gap between the groups and any shock that moved both of them over the same window.

Parallel trends requires that, absent the treatment, the treated and comparison groups' average outcomes would have changed by the same amount. It is a claim about an unobserved counterfactual, so it can never be verified directly.

Two-stage least squares first predicts the treatment from the instrument, then relates the outcome to that prediction. Only the instrument-driven part of the treatment's variation is used, and that part is assumed free of the confounding.

A valid instrument needs relevance, meaning it actually shifts the treatment, and the exclusion restriction, meaning it affects the outcome only through that treatment. It must also be as good as randomly assigned with respect to unmeasured confounders.

Identification rests on continuity: average outcomes with and without treatment must vary smoothly through the cutoff, so nothing but treatment jumps there. The estimate is the average treatment effect for units sitting at the cutoff, not for everyone.

An average pools opposite effects. A drug that raises recovery for patients under 50 and lowers it by a similar amount for patients over 70 posts an overall average near zero while both real effects remain untouched underneath it.

A sensitivity analysis says how strong an unmeasured confounder would have to be to overturn the estimate. It never shows confounding is absent; it prices how much hidden bias the finding can tolerate.

You rerun the analysis on a relationship that must be null: an outcome the treatment cannot cause, or an exposure that cannot cause the outcome, both sharing the study's confounding. Finding an effect where none can exist proves bias remains.

The S-learner fits one outcome model with treatment as a feature and differences its predictions. The T-learner fits separate treated and control models and subtracts them. The X-learner adds a stage that models imputed per-unit effects from each side and blends them.

A churn-risk model ranks who will leave; an uplift model ranks whose behaviour the discount changes. Those are different people: the highest-risk customers often leave regardless, and some contented ones cancel only because the offer reminded them the subscription exists.