Causal Estimands

Summary

Causal estimands are the target quantities in causal inference — specific comparisons of potential outcomes that answer “what is the effect of the treatment?” The main estimands differ by what population they average over and whether they condition on covariates. The choice of estimand is a scientific question, not a statistical one.

Overview

Causal effects are defined as contrasts of potential outcomes under different treatment conditions for the same unit. Since only one potential outcome is ever observed per unit, causal effects are fundamentally about counterfactuals. The choice of estimand should be driven by the scientific question at hand.

Individual Treatment Effect (ITE)

Definition: Individual Treatment Effect (ITE)

For unit $i$, the individual treatment effect is:

$${\tau }_i\equiv Y_i(1)-Y_i(0)$$

the difference in potential outcomes under treatment vs. control for the same unit.

The ITE is never directly observable (fundamental problem of causal inference). Population-level estimands average over ITEs in various ways.

Sample Average Treatment Effect (SATE)

Definition: Sample Average Treatment Effect (SATE)

The average ITE over the observed sample of $N$ units:

$${\tau }^S\equiv N^{-1}\sum _{i=1}^N{\tau }_i=N^{-1}\sum _{i=1}^N[Y_i(1)-Y_i(0)]$$

The SATE is a function of potential outcomes of the specific sample. It is non-random given the sample, though it involves missing potential outcomes.

Conditional Average Treatment Effect (CATE)

Definition: Conditional Average Treatment Effect (CATE)

The average treatment effect for all units with covariate value $X_i=x$:

$$\tau (x)\equiv \mathbb{E}[Y_i(1)-Y_i(0)\mid X_i=x]={\mu }_1(x)-{\mu }_0(x)$$

where ${\mu }_z(x)\equiv \mathbb{E}[Y_i(z)\mid X_i=x]$ for $z=0,1$.

The CATE captures treatment effect heterogeneity — how the average effect varies across covariate subgroups. Estimating $\tau (x)$ as a function of $x$ is a central goal in modern causal inference.

Population Average Treatment Effect (PATE)

Definition: Population Average Treatment Effect (PATE)

Averaging the CATE (or ITE) over a target population $F(x;{\theta }_X)$:

$${\tau }^P\equiv \mathbb{E}[Y_i(1)-Y_i(0)]=\mathbb{E}[\tau (X_i)]$$

• The PATE is a function of the distribution of potential outcomes in a population.
• In observational studies where the target population is the population from which the sample is drawn, PATE is typically the estimand of interest.
• In randomized experiments, SATE is often the primary estimand.

SATE vs. PATE distinction

Both ITE and CATE are important for characterizing treatment effect heterogeneity, but they are obviously different. They are sometimes conflated in the literature.

• SATE = average of ITEs over the specific sample
• PATE = average of ITEs over the population distribution

Mixed Average Treatment Effect (MATE)

Definition: Mixed Average Treatment Effect (MATE)

Replace the population distribution $F(x;{\theta }_X)$ in the PATE with the empirical distribution ${\hat{F}}_X$ of covariates in the sample:

$${\tau }^M\equiv ({\beta }_1-{\beta }_0{)}^{\prime }\bar{X}=N^{-1}\sum _{i=1}^N\tau (X_i;{\theta }_Y)$$

where $\tau (x;{\theta }_Y)=\tau (x)$ evaluated at parameter ${\theta }_Y$.

• The MATE is a convenient approximation to the PATE: it conditions on the observed $X$ values rather than integrating over the population distribution.
• Most Bayesian causal inference in practice focuses on the MATE (rather than PATE or SATE).
• The distinction: PATE has the largest uncertainty; SATE has the smallest; MATE is in between.

Summary Table

Estimand	Formula	Population	Key feature
ITE	$Y_i(1)-Y_i(0)$	Unit $i$	Never observed; target of imputation
SATE	$N^{-1}{\sum }_i{\tau }_i$	Sample	Non-random given sample
CATE	${\mu }_1(x)-{\mu }_0(x)$	Subgroup $X=x$	Captures heterogeneity
PATE	$\mathbb{E}[\tau (X_i)]$	Population	Requires population distribution
MATE	$N^{-1}{\sum }_i\tau (X_i;{\theta }_Y)$	Sample (empirical $X$)	Most used in Bayesian CI

Principal Causal Effects

In complex assignment mechanisms (e.g., instrumental variables), one may define stratum-specific effects:

Definition: Principal Causal Effects

For compliance stratum $U_i\in \{\text{co, at, nt, df}\}$ (compliers, always-takers, never-takers, defiers), the stratum-specific effect is:

$${\tau }_u\equiv \mathbb{E}[Y_i(1)-Y_i(0)\mid U_i=u]$$

These are called principal causal effects.

See Instrumental Variables and Principal Stratification for the full IV/compliance framework.

Connections

• Potential Outcomes Framework — the setup that defines these estimands
• General Structure of Bayesian CI — Bayesian inference for these estimands via posterior imputation
• Frequentist Causal Estimation — frequentist estimators targeting PATE/SATE/CATE
• Bayesian Outcome Models — outcome model ${\mu }_z(x)$ used to estimate CATE

See Also

• Instrumental Variables and Principal Stratification — principal causal effects for IV settings
• Time-Varying Treatments and G-computation — marginal structural model estimands for sequential treatments