Randomization and finite-population inference

Every unit has treatment probability n₁ divided by N. Express the treated mean as a sum of assignment indicators times fixed treated potential outcomes, divided by the fixed treated-arm size. Taking expectation over assignments turns this into the average of all treated potential outcomes. The same argument applies to the control mean. Subtracting establishes unbiasedness for the finite enrolled-sample average effect without requiring Gaussian outcomes, a linear model, or independent assignment indicators.

The assignment indicators are in fact dependent: once some units are assigned treatment, fewer treatment slots remain. This dependence is part of the randomization variance. It does not undermine unbiasedness, but it makes independent-Bernoulli variance formulas inappropriate for this design. Positivity is guaranteed for each unit when both arm sizes are positive and less than N. Selection into the experiment is a separate process; the argument says nothing by itself about transporting the result to people or units outside the enrolled sample.

Let S₁ squared and S₀ squared be the finite-population variances of the two potential-outcome columns, and let Sτ squared be the variance of the individual effects, always using denominator N minus one. The exact assignment variance subtracts Sτ squared divided by N from the sum of the two arm-specific variance terms. That last quantity cannot generally be identified from observed outcomes because it requires the joint relation between each unit’s two potential outcomes.

When each arm has at least two units, estimating the two arm-specific variance components with their observed sample variances and dropping the nonnegative subtraction gives the usual Neyman variance estimator. Its expectation over randomization is at least the true assignment variance, with equality under constant individual effects. Conservative in expectation does not mean that every realized estimated variance is larger than the true variance, nor does it by itself guarantee an exact small-sample confidence interval. Asymptotic normal inference needs a suitable finite-population central limit theorem and protection against a few dominating units.

Under the sharp null that every individual treatment effect is zero, observed outcomes impute both potential-outcome columns. One can therefore enumerate the assignment distribution of a chosen statistic under the actual design. The randomization p-value is the probability of statistics at least as extreme as the observed one, including ties according to a prespecified rule. Discreteness often makes an unrandomized test conservative; exact validity means its rejection probability is controlled, not that every nominal level is attained exactly.

A null average effect of zero does not impute every missing outcome, so the same elementary permutation construction is not automatically an exact test of that weaker null. Random sampling from a larger population would introduce another source of uncertainty and another target effect. Keeping assignment and sampling explicit prevents an enrolled-sample randomization statement from being silently promoted into a population claim. Design-based inference is powerful because its random mechanism is known, but the conclusion remains tied to that mechanism and its assumptions.