Assignment Effects, First Stages, and Reduced Forms

Why Separate Assignment From Treatment Receipt?

In an encouragement study, people may be randomly assigned to receive an offer, reminder, or other encouragement, but assignment does not guarantee that they take up the treatment. The randomized groups therefore differ directly in assignment; they may also differ in how often people receive the treatment and in their outcomes. Keeping these distinctions clear is essential: a randomized comparison identifies effects of assignment, not automatically effects of treatment receipt.

This tutorial focuses on two contrasts. The first stage is the effect of assignment on treatment uptake. The reduced form is the effect of assignment on the outcome. Both compare people according to the group to which they were assigned, whether or not they followed the assignment.

Potential Outcomes and the Two Assignment Effects

Let \(Z=1\) indicate assignment to the encouragement group and \(Z=0\) indicate assignment to the comparison group. Let \(D=1\) indicate that a person received the treatment and \(D=0\) indicate that they did not. Let \(Y\) be the outcome of interest. For each person, \(D(z)\) is the treatment uptake that would occur under assignment \(z\), and \(Y(z)\) is the outcome that would occur under assignment \(z\). The outcome \(Y(z)\) can reflect everything assignment changes, including any change in treatment uptake.

Because \(D\) is binary, its group mean is the proportion receiving treatment. Thus the first stage is the difference in treatment-uptake proportions. The reduced form is the difference in mean outcomes when \(Y\) is quantitative, or the difference in outcome proportions when \(Y\) is binary. State which group is subtracted from which: throughout, the contrast is assignment to encouragement minus assignment to comparison.

These are effects of the assignment policy. For example, the reduced form answers, “What is the average outcome difference caused by being assigned to the encouragement rather than the comparison group?” It does not, by itself, answer, “What is the effect of receiving the treatment?”

When Does a Randomized Contrast Have a Causal Interpretation?

The key design condition is that assignment is randomized, so assignment is independent of the potential outcomes and potential uptake. The usual interpretation also requires consistency: each person’s observed uptake and outcome equal the values corresponding to the assignment they actually received. It requires no interference between people, so one person’s assignment does not change another person’s potential uptake or outcome. Finally, the assigned groups must have usable outcome data; differential loss to follow-up can undermine a comparison if the observed participants no longer represent the randomized groups.

Under these conditions, the difference in observed group means identifies the corresponding average assignment effect. For example, randomization makes the people assigned \(Z=1\) comparable, on average, to those assigned \(Z=0\) in their potential outcomes. Consistency then links each observed group mean to the potential outcome under that group’s assignment.

The first equality uses consistency, and the second uses random assignment. Replacing \(D\) with \(Y\) gives the same justification for the reduced form. In a sample, the corresponding estimators are the observed difference in group means. In a randomized experiment, these estimators target the average assignment effects; sampling variation means a particular sample difference need not equal the population effect exactly.

Here \(\hat{p}_{Z=1}\) and \(\hat{p}_{Z=0}\) are the observed treatment-uptake proportions in the two assignment groups. For a binary outcome, the second contrast is likewise a difference in proportions. Neither contrast requires that everyone comply with their assignment. Nor does identifying these assignment effects require an assumption that assignment affects the outcome only through treatment receipt. That distinction matters: the reduced form is about assignment itself.

Worked Calculations

How to Report Each Contrast

A clear report names the randomized groups, gives the contrast direction, states the quantity and units, and identifies the effect as an effect of assignment. For example: “Assignment to the encouragement increased treatment uptake by 20.0 percentage points and was associated with an estimated 0.5-unit increase in the outcome mean, compared with assignment to the control.” When randomization and measurement conditions support causal interpretation, “effect of assignment” is appropriate. Avoid wording that turns the reduced form into an effect of treatment receipt.

The first stage also helps describe how much the encouragement changed behavior. A small first stage means the assigned groups had similar treatment-uptake rates; a larger one means uptake differed more. But the first stage is not an outcome benefit, and a large first stage by itself says nothing about whether the outcome improved. The reduced form answers a separate question about the outcome difference between assignment groups.

Common Mistake: Comparing Recipients as if Receipt Were Randomized

A common error is to compare outcomes among people who received treatment with outcomes among those who did not, then attribute that difference to treatment. The reasoning is: “The treatment group had better outcomes, so treatment caused the difference.” The problem is that receipt happens after assignment. People who take up treatment can differ systematically from people who do not, so this comparison does not preserve the balance created by randomization.

The correction is to compare groups by their randomized assignment when estimating assignment effects. Calculate the first stage from uptake rates by assignment and the reduced form from outcome means or proportions by assignment. Report each in its own terms. These contrasts may be useful even when not everyone takes up treatment, because they describe what assignment to the encouragement actually changes.