Overview: causal inference

Collider, confounder, mediator and M-bias

Effects

\(Y^{a=1}, Y^{a=0}\) are the potential outcomes under treatment 1 and 0. They are random variables. Treatment A has causal effect if \(Y^{a=1} \neq Y^{a=0}\).

For individual \(i\), \(Y_i^{a=1}, Y_i^{a=0}\) are deterministic.

In reality we do not observe both potential outcomes for an individual, since we only have ONE outcome. We observe \(Y\) and \(A\). For a population, average treatment effect (ATE) can be estimate

Estimands

Greifer, N., & Stuart, E. A. (2021). Choosing the estimand when matching or weighting in observational studies. arXiv preprint arXiv:2106.10577.

ATE: average treatment effect in the population

\(E[Y(1) - Y(0)]\)

ATT: average treatment effect among the treated

\(E[Y(1) - Y(0) | Z = 1]\)

ATC: average treatment effect among the controls

\(E[Y(1) - Y(0) | Z = 0]\)

ATM: average treatment effect among the matched

Graphical representation

Data generation	Correct causal model	Correct causal effect
Collider	Y ~ X	1
Confounder	Y ~ X; Z	0.5
Mediator	Direct effect: Y ~ X; Z. Total effect: Y ~ X	Direct: 0; total: 1
M-Bias	Y ~ X	1

Selection bias

This bias is the result of selecting a common effect of 2 other variables (collider): a treatment, an outcome.

• non-response, missing data
• self-selection, volunteer bias
• selection affected by treatment before study started

A form of lack of exchangeability between the treated and untreated.

Correct for selection bias: IP weighting