Identification Under Randomized Exposure Assignment

Why Randomized Exposure Assignment Can Identify Causal Means

When one person’s treatment can affect another person’s outcome, a treatment indicator alone may not describe the causal condition of interest. An individual’s outcome could depend both on their own treatment and on the treatments received by peers. A randomized design can still identify mean outcomes under such conditions, provided the exposure definition captures the relevant assignment features and every target individual has a positive chance of receiving the exposure being studied.

The key is to distinguish an individual’s potential outcome under an exposure from the outcome actually observed. Randomization does not reveal every individual’s potential outcomes in one realized assignment. Instead, known assignment probabilities let us relate observed outcomes, across the randomization distribution, to a specified finite-population mean.

Assignments, Exposures, and Potential Outcomes

Let \(Z\) denote the full vector of randomized treatment assignments. For individual \(i\), an exposure mapping \(g_i\) converts that vector into the exposure condition assigned to the individual: \(E_i=g_i(Z)\). For example, an exposure might record both the individual’s own treatment and the number of treated peers. The possible exposure conditions are denoted by \(e\).

Write \(Y_i(e)\) for individual \(i\)’s potential outcome under exposure \(e\). These potential outcomes are fixed for the population under the randomization-based framework; the random assignment determines which exposure, and hence which potential outcome, is observed. For a finite target population of \(N\) individuals, the exposure-specific mean potential outcome is

This is a finite-population average, not an average over a hypothetical superpopulation. The target population and its membership are held fixed while the assignment varies according to the randomization design.

Conditions for Identification

Three conditions connect the randomization mechanism to the target mean. They concern how outcomes relate to the exposure mapping, how observed outcomes relate to potential outcomes, and whether the design can assign each target individual to the exposure of interest.

Exposure-mapping sufficiency is a substantive condition, not a consequence of randomization. If two assignments receive the same exposure label for an individual but can produce different outcomes, then \(Y_i(e)\) does not summarize the individual’s response to those assignments. Consistency cannot repair an exposure definition that omits consequential assignment features.

Randomization supplies the relevant probability statements: the assignment vector is drawn according to the specified design, independently of the fixed potential outcomes. It does not require exposure assignments for different individuals to be independent. In cluster or network designs, their exposures are often dependent.

The Identification Result

Let \(A_i(e)=\mathbb{1}(E_i=e)\), an indicator equal to one when individual \(i\) receives exposure \(e\). Under consistency, the product \(A_i(e)Y_i\) equals \(Y_i(e)\) when \(A_i(e)=1\), and equals zero otherwise. Because the randomization probability of that event is \(\pi_i(e)\), for each individual with \(\pi_i(e)>0\),

Here \(\mathbb{E}_Z\) denotes expectation over the assignment mechanism, holding the finite population and all potential outcomes fixed. Averaging this identity over individuals gives the identification result.

Thus the randomization distribution of the observed assignments and outcomes identifies \(\mu(e)\): the expectation of a function of observed data and known design probabilities equals the target mean. This is an identification statement, not a guarantee that a single realized assignment reveals the exact mean. The displayed random quantity can vary across assignments even though its expectation equals \(\mu(e)\).

The argument does not require equal exposure probabilities across individuals, nor does it require independent exposures. It does require the correct individual-specific probabilities and positivity for the whole target population. The identity is the basis for a design-based estimator; the estimator’s construction and properties are a separate question.

How a Two-Stage Design Supplies Exposure Probabilities

In a two-stage randomized saturation design, clusters may first be assigned to treatment saturation levels, and individuals within clusters may then be randomized. The exposure probability for a particular individual is the probability, under both stages, that the resulting assignment maps to the specified exposure. It must be calculated from the actual design, rather than inferred from the observed proportion exposed.

Why the Ordinary Exposed-Group Mean Is Different

A tempting alternative is to average observed outcomes among those who received exposure \(e\). In a particular assignment, that group mean is a ratio whose denominator is the number of individuals exposed. Its expectation over assignments is generally not the same as the target mean, or even the ratio of the expected outcome total to the expected exposed count. A ratio of expectations and an expectation of a ratio are different quantities.

For example, suppose two hypothetical individuals have potential outcomes under \(e\) equal to 10 and 30. Their exposure indicators are independent, with probabilities \(0.8\) and \(0.2\), respectively. The ratio of the expected exposed-outcome total to the expected exposed count is

This is a probability-weighted average of the two potential outcomes, not the expected realized exposed-group mean. To see the difference, the probabilities of both being exposed, only the first, only the second, and neither being exposed are \(0.16\), \(0.64\), \(0.04\), and \(0.16\). When at least one is exposed, the realized exposed-group mean is respectively 20, 10, or 30. Its expectation conditional on at least one exposure is therefore

If the realized mean is instead defined to be zero when nobody is exposed, its unconditional expectation is \(0.16(20)+0.64(10)+0.04(30)=10.8\). Neither quantity equals 14. The identification identity uses the known probability for each individual; it does not claim that the ordinary exposed-group mean averages to the finite-population mean.

Common Mistakes and Practical Interpretation

• Treating randomization as proof that every exposure is supported. A design can randomize treatment while making some exposure conditions impossible for some individuals. Calculate \(\pi_i(e)\) from the design and check it for each member of the target population.
• Assuming an exposure label is automatically adequate. If two assignments with the same mapped exposure can affect an outcome differently, the exposure-specific potential outcome is not well-defined as stated. Revise or defend the mapping before using it.
• Calling the observed exposed-group mean the identified target mean. The exposed group can overrepresent individuals with larger exposure probabilities, and its realized size is random. Use the identification result with the individual-specific probabilities; do not replace it with an unweighted group average without a separate justification.
• Confusing identification with exact recovery in one experiment. The theorem says an observed-data quantity has the target mean over repeated assignments under the design. It does not say a single assignment reveals all potential outcomes or produces the target mean exactly.

These conditions matter because an exposure-specific causal contrast is meaningful only if its component means refer to well-defined exposures and are identified for the intended population. If one exposure mean is unsupported, a contrast involving it cannot be recovered by randomization alone. If the mapping is inadequate, the contrast may not represent a coherent intervention in the first place.