Causal inference

Problem

A hospital records 1200 patients. Of those who got a new treatment, 66.0% recovered; of those who didn't, 74.3%. The treatment looks harmful, by 8.3 points. Yet within the mild cases it raises recovery from 80% to 90%, and within the severe cases from 40% to 50%. Both groups gain ten points, so how can the total lose eight?

Because doctors gave the treatment mostly to severe cases: 75% of severe patients got it, against 25% of mild ones, and severe patients recover less whatever they receive. The treated group is sicker, and the comparison mixes the drug's effect with the difference in who got it. This is Simpson's paradox, and it is the everyday condition of observational data.

Supervised learning answers "given what I see, what will happen?" Decisions need "what would happen if I did this?" A classifier trained on this hospital's records would correctly learn that treated patients recover less often, and a policy built on it would withhold a treatment that helps everyone. Causal inference is the set of assumptions and estimators that get from data to the second question.

Intuition

Imagine each patient has two possible futures: the outcome if treated, Y(1), and the outcome if not, Y(0). The effect of the treatment on that patient is Y(1) − Y(0). We only ever see one of the two, the one matching what actually happened. Every causal question is a missing-data problem: fill in the futures that didn't happen.

Comparing treated with untreated people fills each group's missing future with the other group's observed one. That is only fair if the groups were alike before treatment. A confounder is something that affects both who gets treated and the outcome, here severity, and it makes the groups unlike. There are two ways out. Make the groups alike by assigning treatment at random (an experiment). Or measure the confounders and compare like with like: within each severity level, or by reweighting people so that the treated and untreated groups have the same mix of severity.

Mechanics

Potential outcomes

For unit i with treatment T_i ∈ {0, 1}, the potential outcomes are Y_i(0) and Y_i(1) and we observe Y_i = T_iY_i(1) + (1 − T_i)Y_i(0). The quantities of interest are averages, because individual effects are never observed:

The naive comparison splits exactly into the effect on the treated plus a bias term:

The bracket is selection bias: how different the treated were from the untreated before treatment. It vanishes when treatment is independent of the potential outcomes. This framework, with the assumption that one unit's treatment doesn't change another's outcome (no interference), is usually credited to Neyman and Rubin.

Confounding and causal graphs

A causal graph (a DAG) draws an arrow from each variable to the variables it directly affects. Pearl's backdoor criterion says which variables to adjust for: a set Z is enough if it blocks every path from T to Y that starts with an arrow into T, and contains nothing caused by T. Then, within levels of Z, treatment is as good as random, and

The do means "set T to t by intervention", as opposed to "observe T = t". The graph also says what not to adjust for. Conditioning on a mediator removes part of the effect you want to measure; conditioning on a collider (a common effect of T and Y, like "was admitted to hospital") opens a path that wasn't there and manufactures an association. More variables is not automatically safer.

The potential-outcomes version of the same assumption is unconfoundedness: (Y(0), Y(1)) is independent of T given Z. It comes with overlap (positivity): every kind of unit must have some chance of each treatment, 0 < P(T = 1 | Z = z) < 1. No data can test unconfoundedness; it is a claim that every confounder was measured, defended with domain knowledge.

Randomized experiments

Random assignment makes T independent of everything about the unit, measured or not, so selection bias is zero in expectation and the difference in means estimates the ATE without any model. Any one randomization can still be unlucky; the estimate has variance, which is why experiments report confidence intervals and why larger samples help. An A/B test is the same design with a product change as the treatment. Stratifying the randomization on known strong predictors (block randomization) reduces the unlucky draws.

Propensity scores and inverse weighting

When there are many confounders, comparing within every combination of their values runs out of data. Rosenbaum and Rubin (1983) showed that the propensity score e(z) = P(T = 1 | Z = z) is enough: if treatment is unconfounded given Z, it is unconfounded given e(Z), a single number. In practice e is estimated with a classifier, very often logistic regression.

Inverse propensity weighting (IPW) weights each treated unit by 1/e and each untreated unit by 1/(1 − e). A treated patient from a group where treatment was rare stands in for the many similar patients who weren't treated. The weighted sample is a pseudo-population in which treatment no longer depends on Z:

This is the Horvitz–Thompson form. The Hájek form divides each weighted sum by the sum of its weights instead of N, which is more stable when weights are large. Propensities near 0 or 1 give huge weights and a noisy estimate; that is the overlap assumption failing in practice, and common responses are trimming units with extreme scores or clipping the weights. A propensity model is judged by whether it balances the covariates after weighting, not by its accuracy: a propensity model that predicts treatment perfectly is a sign of no overlap, not a good model.

Difference-in-differences

Sometimes treatment hits a whole group at once: a policy in one city, a feature launched in one country. Comparing the group before and after mixes the effect with whatever else changed over time; comparing it with another group after mixes the effect with the groups' standing difference. Difference-in-differences subtracts both:

The assumption is parallel trends: without treatment, the treated group would have changed by the same amount as the control group. It can't be checked after treatment, but several pre-treatment periods that move in parallel make it believable, and diverging pre-trends are a warning. Card and Krueger's study of a minimum-wage rise in New Jersey, with neighbouring Pennsylvania as the control, is the classic example.

Uplift modeling

Marketing and retention teams ask a per-person question: whose behaviour will the action change? A response model predicts who buys if contacted, which picks people who would buy anyway ("sure things") and even people the contact puts off ("sleeping dogs"). An uplift model predicts the conditional effect P(Y | T=1, x) − P(Y | T=0, x) and targets the "persuadables". It is trained on randomized data: the simplest version (a T-learner) fits one model on the treated group and one on the control group and subtracts their predictions. It is evaluated with uplift or Qini curves: sort people by predicted uplift and plot the incremental outcome as more of them are treated.

Worked example

Implementation

Inverse propensity weighting with scikit-learn, with the checks that matter: an unpenalized propensity model, overlap, and covariate balance after weighting. X holds the measured confounders, t the 0/1 treatment and y the outcome.

import numpy as np
from sklearn.linear_model import LogisticRegression

def ipw_ate(X, t, y, clip=0.01):
    model = LogisticRegression(C=np.inf, max_iter=1000).fit(X, t)
    e = model.predict_proba(X)[:, 1]
    print("propensity range", e[t == 1].min(), e[t == 0].max())   # overlap check
    e = np.clip(e, clip, 1 - clip)
    w = np.where(t == 1, 1 / e, 1 / (1 - e))

    treated = np.sum(w * t * y) / np.sum(w * t)                   # Hajek form
    control = np.sum(w * (1 - t) * y) / np.sum(w * (1 - t))

    def smd(weights):                                             # standardized mean difference
        m1 = np.average(X[t == 1], axis=0, weights=weights[t == 1])
        m0 = np.average(X[t == 0], axis=0, weights=weights[t == 0])
        return (m1 - m0) / np.sqrt((X[t == 1].var(axis=0) + X[t == 0].var(axis=0)) / 2)

    return treated - control, smd(np.ones_like(w)), smd(w)        # balance before / after

A common rule of thumb treats an absolute standardized mean difference under about 0.1 after weighting as balanced. Clipping trades a little bias for much lower variance. For better estimates, combine an outcome model with the propensity model (doubly robust estimation), which stays consistent if either model is right. A T-learner for uplift is two classifiers: fit one on X[t == 1], one on X[t == 0], and subtract their predict_proba outputs.

Tradeoffs

• Experiment vs observational study. Randomization removes confounding, measured or not, but costs time and money and is sometimes unethical or impossible. Observational methods use existing data but rest on an untestable claim that every confounder was measured.
• Adjustment vs weighting. Stratifying is transparent but runs out of data as covariates multiply. Propensity methods compress the covariates into one number but depend on that model, and extreme weights make them noisy.
• What to adjust for. Adjusting for confounders removes bias; adjusting for mediators or colliders adds it. Throwing every available column into the model is not safe.
• DiD. Handles unmeasured confounders that are constant over time, but only under parallel trends, and only with comparable control groups.
• Uplift vs response. Uplift models answer the right question but need randomized data, and the difference of two noisy predictions is noisier than either.

Connections

References

• Pearl. Causal Inference in Statistics: An Overview. Statistics Surveys 3, 96–146, 2009. doi:10.1214/09-SS057
• Rubin. Estimating Causal Effects of Treatments in Randomized and Nonrandomized Studies. Journal of Educational Psychology, 1974. (Potential outcomes.)
• Rosenbaum and Rubin. The Central Role of the Propensity Score in Observational Studies for Causal Effects. Biometrika, 1983.
• Horvitz and Thompson. A Generalization of Sampling Without Replacement from a Finite Universe. Journal of the American Statistical Association, 1952. (The inverse-probability estimator.)
• Card and Krueger. Minimum Wages and Employment: A Case Study of the Fast-Food Industry in New Jersey and Pennsylvania. American Economic Review, 1994.
• Radcliffe and Surry. Real-World Uplift Modelling with Significance-Based Uplift Trees. Stochastic Solutions white paper, 2011. PDF
• scikit-learn 1.9.1: LogisticRegression, fit, predict_proba.