Identification by Random Assignment

Why Random Assignment Identifies Causal Means

A randomized experiment creates a principled link between observed outcomes and potential outcomes. For each unit, only the outcome under the assignment actually received is observed. Random assignment makes the units assigned to treatment representative, in expectation, of the units’ potential outcomes under treatment; the same logic applies to control. This is an identification argument: it explains what the assignment mechanism lets us learn about causal quantities, not how uncertain a particular estimate is.

Let \(Z_i\) indicate whether unit \(i\) is assigned treatment: \(Z_i=1\) for treatment and \(Z_i=0\) for control. Let \(Y_i(1)\) and \(Y_i(0)\) be that unit’s potential outcomes under treatment and control. Under consistency, the observed outcome is \(Y_i=Z_iY_i(1)+(1-Z_i)Y_i(0)\). The average treatment effect for a population is the mean of the individual effects \(Y_i(1)-Y_i(0)\), or, equivalently, the difference between the population means of \(Y_i(1)\) and \(Y_i(0)\).

The independence condition and positivity do different jobs. Independence ensures that assigned groups are not selected according to their potential outcomes. Positivity ensures that each relevant unit can, under the design, appear in either assignment condition. A known assignment probability is useful for describing the design, but knowing it alone does not guarantee that a raw group average identifies the target mean.

Unbiasedness Under Equal-Probability Assignment

First consider a finite population of \(N\) units, with a fixed number \(n_1\) assigned to treatment and \(n_0=N-n_1\) assigned to control. Under complete random assignment, every subset of \(n_1\) units is equally likely to receive treatment. Each unit therefore has treatment probability \(n_1/N\), and control probability \(n_0/N\). Write the finite-population potential-outcome means as \(\bar{Y}(1)=N^{-1}\sum_{i=1}^{N}Y_i(1)\) and \(\bar{Y}(0)=N^{-1}\sum_{i=1}^{N}Y_i(0)\). The observed treatment and control averages are \(\bar{Y}_{1}=n_1^{-1}\sum_{i=1}^{N}Z_iY_i\) and \(\bar{Y}_{0}=n_0^{-1}\sum_{i=1}^{N}(1-Z_i)Y_i\).

For a treated unit, consistency gives \(Y_i=Y_i(1)\); for a control unit it gives \(Y_i=Y_i(0)\). Thus, the observed averages can also be written using the corresponding potential outcomes. Taking expectations over the random assignment gives the result directly:

Here \(E_Z\) means expectation over the assignment mechanism, holding the finite population’s potential outcomes fixed. The matching control calculation replaces \(Z_i\) with \(1-Z_i\), whose expectation is \(n_0/N\), and yields \(E_Z[\bar{Y}_{0}]=\bar{Y}(0)\). These are design-unbiasedness statements: over repeated assignments under the same design, each group average has the corresponding potential-outcome mean as its expectation. They do not say that a particular realized group average must equal that mean.

Identification in a Superpopulation

The same argument has a superpopulation version. Suppose units are drawn from a population, treatment is assigned independently of the pair of potential outcomes, and the probability of treatment is \(p\), where \(0<p<1\). Consistency and independence imply that the observed outcome mean among treated units equals the population mean of the treatment potential outcome; similarly, the observed mean among controls equals the mean of the control potential outcome:

The first equality in each line uses consistency for the assignment group in question. The second uses independence of assignment and potential outcomes. Both conditional expectations are defined only when the corresponding assignment has positive probability: \(P(Z=1)=p>0\) and \(P(Z=0)=1-p>0\). The average treatment effect is therefore identified as the difference between these two observed conditional means. Identification does not require that the treatment and control outcomes for the same person both be observed; it requires a design that lets each assignment group represent the relevant potential outcomes in the population.

Positivity and the Limits of Raw Averages

Positivity is substantive, not merely a technical convenience. If a unit or a relevant subgroup can never receive one assignment, the data contain no outcomes for that condition in that part of the target population. Random assignment cannot reveal a potential-outcome mean where the design never allows the corresponding outcome to be observed. In a design with covariates or strata, positivity must hold within the relevant covariate groups when the target requires comparisons within those groups.

Equal assignment probabilities also matter for the simple raw-average argument. Under complete random assignment, every unit has the same chance of appearing in each group, so each group is an equally representative sample of the finite population in expectation. If assignment probabilities differ across units, the composition of an observed group can differ from the target population’s composition. Random assignment within groups can still identify group-specific means when positivity holds, but the unadjusted overall group average need not identify the overall population mean.

What the Identification Argument Does—and Does Not—Establish

The argument rests on a clear sequence of conditions. Consistency connects observed outcomes to the potential outcome for the assignment received. Random assignment makes assignment independent of potential outcomes, either for the whole population or conditionally within the design’s relevant groups. Positivity gives each group a nonzero chance of appearing in each required assignment condition. Under equal-probability assignment, the raw treatment and control averages are unbiased for the corresponding finite-population means. In a superpopulation formulation, the corresponding observed conditional means equal the potential-outcome expectations.

These conditions establish identification, not precision. A design can identify a mean while producing a noisy estimate in a small sample. Nor does this argument alone establish a standard error, confidence interval, or test. Those require additional inference calculations. Finally, when assignment probabilities vary, identification of the overall target mean may require accounting for the assignment design and target population composition; the raw group average cannot simply be assumed to work.

• Independence without positivity: even if assignment is unrelated to potential outcomes where both assignments occur, a group with no chance of control provides no observed control outcomes.
• Positivity without independence: both assignments may occur, but if assignment favors units with particular potential outcomes, raw group means need not identify the target means.
• Known probabilities without equal representation: knowing probabilities describes how units entered the assignment groups; it does not imply that an unadjusted average has the target population’s composition.