# Bridging Finite and Super Population Causal Inference

## TL;DR

This paper establishes a theoretical connection between finite and super population causal inference frameworks in randomized experiments, showing they yield identical variance estimators despite conceptual differences.

## Contribution

It introduces a variance decomposition and completeness argument to unify finite and super population causal inference approaches.

## Key findings

- Finite and super population variance estimators are identical.
- A variance decomposition links the two causal inference views.
- The approach provides a template for broader causal inference scenarios.

## Abstract

There are two general views in causal analysis of experimental data: the super population view that the units are an independent sample from some hypothetical infinite populations, and the finite population view that the potential outcomes of the experimental units are fixed and the randomness comes solely from the physical randomization of the treatment assignment. These two views differs conceptually and mathematically, resulting in different sampling variances of the usual difference-in-means estimator of the average causal effect. Practically, however, these two views result in identical variance estimators. By recalling a variance decomposition and exploiting a completeness-type argument, we establish a connection between these two views in completely randomized experiments. This alternative formulation could serve as a template for bridging finite and super population causal inference in other scenarios.