Glossary

Average treatment effect (ATE)

The difference between the average potential outcome under one treatment and the average potential outcome under another treatment.

Ancestor

A node is an ancestor of another node when a directed path runs from the first node to the second.

Bias

The difference between an estimator’s expected value under a probability model and the quantity it is intended to estimate.

Causal estimand

A probabilistic quantity involving potential outcomes that we want to estimate.

Child

In a graph, a node is a child of another node when it receives an arrow directly from that node.

Confounding by indication

Confounding that arises when the clinical reasons for giving a treatment also influence the outcome.

Confounder

A common cause of the exposure and outcome that can create or distort their observed association.

Conditional exchangeability

The independence of a potential outcome and the original treatment conditional on baseline variables: \(A\mathrel{\perp\!\!\!\perp}Y(a)\mid X\).

Outcome regression

Estimation of the outcome’s conditional mean given treatment and baseline variables. For causal estimation, the fitted mean is evaluated under specified treatment values and averaged over baseline values.

Positivity

For an intervention value \(a\), treatment \(a\) has positive conditional probability at almost every baseline value in the target population: \(P(A=a\mid X)>0\) for discrete treatment.

Descendant

A node is a descendant of another node when a directed path runs from the other node to it.

Directed acyclic graph (DAG)

A directed graph with no directed cycles.

Directed cycle

A sequence of arrows that can be followed in their directions and eventually returns to the node where it began.

Directed edge

A connection between two nodes with a direction, shown by an arrowhead. It is also called an arrow.

Directed graph

A collection of nodes joined only by directed edges.

Directed path

A path whose arrows line up head to tail, so that every arrow can be followed in its direction.

Exclusion restriction

The instrument affects the outcome only through the treatment.

Exposure

The condition, action, or factor whose causal effect is being studied. Depending on the setting, it may also be called the treatment or intervention.

Fundamental problem of causal inference

For each person, we observe the potential outcome under the treatment received but not the potential outcome under the other treatment. We therefore cannot observe both potential outcomes for the same person.

Identified

A causal estimand is identified when it can be written entirely in terms of observed variables and their probability model.

Instrument independence

The instrument and outcome have no common causes other than those accounted for by the measured covariates.

Independence

Learning the value of one random variable does not change the probability distribution of the other.

Estimator

A rule that uses sample data to estimate an unknown quantity in a population or probability model.

Observational study

A study that examines naturally occurring exposures or treatments rather than assigning them.

Node

A labeled circle in a graph that represents a random variable.

Outcome

The event, condition, or response that the exposure might affect.

Parent

Another variable used as an input to a variable’s assignment function. In a causal graph, a parent has an arrow pointing directly to its child.

Path

A sequence of arrows joined end to end. We may travel either with or against the arrows’ directions.

Positivity

Each treatment under study has a positive probability of being assigned.

Random variable

A function that assigns a numerical value to each possible outcome of an uncertain experiment or scenario.

Risk difference

The risk of an outcome under one exposure or treatment minus the risk under another.

Relative risk

The risk of an outcome among exposed people divided by the risk among unexposed people.

Simpson’s paradox

A reversal in which the direction of an association in the combined data is opposite to its direction within groups defined by a third variable.

Single-world intervention graph (SWIG)

The graph of the potential-outcome SCM under one collection of intervention values. An intervened node is split into two nearby circles. The random part retains its incoming arrows, and the fixed part inherits its outgoing arrows.

Structural causal model (SCM)

A model that defines each variable through an assignment function of its parents and a noise variable, with the noise variables independent as an entire collection.

Topological order

An ordering of a list of random variables, with one assignment for each variable, in which every variable used in an assignment appears earlier in the list.

Treatment relevance

An instrument is associated with treatment after conditioning on the measured covariates.

Unbiased estimator

An estimator whose expected value under a probability model equals the quantity it is intended to estimate.

Unmeasured confounding

Confounding due to a common cause of treatment and outcome that is not measured in the study data.

Unmeasured confounder

A confounder that was not measured in the study data.

Well-defined (mathematical sense)

A definition is well-defined when each possible input determines one unambiguous object.