Deriving the Complier Average Causal Effect

Why Assignment Can Reveal an Effect for Compliers

In a randomized experiment, assignment to an offer or encouragement can be randomized even when treatment receipt is not. Some people assigned to the offer may not take up treatment, while some assigned to the control condition may obtain it. Comparing outcomes by actual treatment receipt can therefore lose the protection of random assignment. The assignment itself, however, remains randomized.

The key result in this tutorial is that, under specific assumptions, the effect of assignment on the outcome divided by the effect of assignment on treatment receipt identifies the complier average causal effect (CACE). The ratio adjusts the assignment effect for the fact that assignment changes treatment receipt only for some individuals. The derivation also clarifies which causal effect this ratio targets—and which assumptions make that interpretation possible.

Potential Outcomes and the Target Effect

Let \(Z_i\in\{0,1\}\) denote individual \(i\)'s randomized assignment, where \(Z_i=1\) is assignment to the offer or encouragement and \(Z_i=0\) is assignment to control. Let \(D_i(z)\in\{0,1\}\) be the treatment receipt that individual \(i\) would have under assignment \(z\). Let \(Y_i(d)\) be the outcome individual \(i\) would have under treatment receipt \(d\). The observed receipt and outcome are \(D_i=D_i(Z_i)\) and \(Y_i=Y_i(D_i)\).

An individual's compliance type is determined by the pair \((D_i(0),D_i(1))\). A complier has \((0,1)\): they receive treatment if assigned to the offer and do not receive it if assigned to control. An always-taker has \((1,1)\), and a never-taker has \((0,0)\). A defier has \((1,0)\). Monotonicity, stated below, rules out defiers.

The causal effect in this definition is the effect of receiving treatment, not the effect of being assigned to the offer. The distinction matters because assignment and receipt differ for individuals who do not comply with their assignment.

Assumptions for the Derivation

The following conditions are used to connect randomized assignment to the CACE. They concern the design, the relationship between assignment and outcomes, and how assignment changes receipt.

• Random assignment and positivity. Assignment is independent of the joint collection of potential receipts and outcomes, \(\{D_i(0),D_i(1),Y_i(0),Y_i(1)\}\), and each assignment arm has positive probability. Thus, the two assignment groups represent the same distribution of these potential quantities in expectation.
• Consistency and SUTVA. Each individual's observed receipt and outcome equal their potential receipt and outcome under the assignment and treatment actually received. There are no relevant multiple versions of treatment or interference between individuals.
• Exclusion restriction. Assignment affects the outcome only through treatment receipt. In particular, the outcome under assignment \(z\) is \(Y_i(D_i(z))\); there is no additional direct effect of \(Z_i\) on the outcome.
• Monotonicity. \(D_i(1)\geq D_i(0)\) for every individual. Assignment to the offer does not cause anyone who would otherwise receive treatment to stop receiving it. This rules out defiers.
• Relevance. Assignment changes treatment receipt in the population. Under monotonicity, this requires a strictly positive proportion of compliers.

Randomization and consistency let us express differences between assignment groups as differences in potential quantities. Exclusion lets us write outcomes as depending on receipt rather than directly on assignment. Monotonicity then makes the change in receipt attributable only to compliers.

Deriving the Wald Ratio

Define the population assignment effect on the outcome, also called the reduced-form effect, as the difference in mean outcomes by assignment. Define the first-stage effect as the corresponding difference in mean treatment receipt:

By random assignment and consistency, the first contrast is the average difference between outcomes under assignment to the offer and assignment to control. By exclusion, those assignment-specific outcomes are \(Y_i(D_i(1))\) and \(Y_i(D_i(0))\). Therefore,

Under monotonicity, each individual is an always-taker, never-taker, or complier. For an always-taker, receipt is 1 under either assignment, so their contribution to the outcome contrast is \(Y_i(1)-Y_i(1)=0\). For a never-taker, receipt is 0 under either assignment, so their contribution is \(Y_i(0)-Y_i(0)=0\). A complier's receipt changes from 0 to 1, so their contribution is \(Y_i(1)-Y_i(0)\). Thus only compliers contribute to the assignment-outcome contrast. This does not require always-takers or never-takers to have zero treatment effects; exclusion makes their outcomes unchanged by assignment because their receipt does not change.

Writing \(C_i\) for the indicator that individual \(i\) is a complier, the contrast can be written as

The first-stage contrast follows the same type-by-type reasoning. Always-takers receive treatment under both assignments, and never-takers receive it under neither. Compliers alone change receipt from 0 to 1. Consequently,

When the first stage is positive, divide the outcome contrast by the receipt contrast. The complier share cancels:

From Identification to a Sample Estimate

In data, the population contrasts are unknown. A common estimator replaces each population mean with its observed sample mean in the relevant assignment group. If \(\bar{Y}_{Z=1}\) and \(\bar{Y}_{Z=0}\) denote the sample outcome means, and \(\bar{D}_{Z=1}\) and \(\bar{D}_{Z=0}\) the sample receipt means, the sample Wald estimator is

This is an estimate of the identified population CACE, not generally the exact CACE in a finite sample. A nonzero sample denominator is needed to calculate it, and a small denominator can make the estimate unstable. The assumptions justify the population target; randomization alone does not establish exclusion or monotonicity.

Common Mistakes and Interpretation

• Calling the Wald ratio the overall average treatment effect. The derivation divides out the complier share; it does not make the complier effect representative of always-takers, never-takers, or the full population.
• Assuming exclusion means no treatment effect for noncompliers. Exclusion says assignment affects outcomes only through receipt. Always-takers and never-takers have no assignment-outcome contrast because their receipt is unchanged. Their outcomes under treatment and control may still differ.
• Confusing assignment with receipt. The numerator compares outcomes by randomized assignment; the denominator compares receipt by assignment. A direct comparison by observed treatment receipt is not the Wald ratio and is not protected by randomization in the same way.
• Ignoring the first stage. If assignment does not change treatment receipt in the population, the denominator is zero and this ratio does not identify a CACE. In a sample, a very small estimated first stage can also make the ratio imprecise.
• Treating the assumptions as consequences of randomization. Randomization supports the assignment contrasts. Exclusion and monotonicity are additional substantive assumptions, not automatic features of a randomized design.

The Wald result is valuable because it gives a causal interpretation to a ratio of two assignment contrasts when actual treatment receipt is imperfectly controlled by assignment. Its validity depends on the target population and assumptions, not merely on the arithmetic. The numerator and denominator should therefore be reported and interpreted alongside the ratio, with outcome and treatment-receipt units stated clearly.