The First Stage and Reduced-Form Contrast

Two Contrasts Across Instrument Values

In an encouragement study, an instrument such as an offer, reminder, or invitation may change whether people receive a treatment without determining receipt perfectly. A useful first step is to compare the two instrument groups separately on treatment receipt and on the outcome. These are distinct contrasts: one describes how strongly the instrument shifts receipt, and the other describes how outcomes differ across instrument values.

Let \(Z\in\{0,1\}\) denote the instrument, with \(Z=1\) representing the encouragement, and let \(D\in\{0,1\}\) denote treatment receipt. Let \(Y\) be the outcome. For a population, the relevant quantities are conditional means or probabilities. In a sample, they are estimated by the corresponding group proportions and means. Keep the direction fixed throughout: each contrast is the \(Z=1\) group minus the \(Z=0\) group.

The name “reduced form” refers here to the outcome contrast by instrument value, without conditioning on treatment receipt. It is not the treatment contrast between those who did and did not receive treatment. Receipt is affected by the instrument, and people in the two receipt groups may differ for reasons that also relate to outcomes.

When the Contrasts Have a Causal Interpretation

The two differences can always be calculated from observed group data when both instrument groups are represented. A causal interpretation requires more. For an encouragement randomly assigned to participants, assume that the assignment is independent of each participant’s potential treatment receipts and potential outcomes. Also assume consistency (the observed receipt and outcome equal the potential values corresponding to the instrument actually received) and no interference (one participant’s assignment does not change another participant’s receipt or outcome). Both instrument values must occur with positive probability.

Write \(D(z)\) for the treatment receipt a participant would have under instrument value \(z\), and \(Y(z)\) for the outcome under that value. Under these assumptions, the first-stage contrast identifies the average change in receipt caused by assignment to the encouragement, \(E[D(1)-D(0)]\). The reduced-form contrast identifies the average change in outcome caused by that assignment, \(E[Y(1)-Y(0)]\). These are effects of changing the instrument value, not automatically effects of changing treatment receipt.

The first-stage contrast is also a relevance check: a value near zero indicates little observed change in receipt across instrument groups. A nonzero first stage alone does not establish that the outcome contrast is caused only through treatment. That interpretation requires additional instrumental-variable assumptions, including exclusion; independence and monotonicity are also central to the later local-effect interpretation. Here, the task is to define and calculate the contrasts, not to combine them into a treatment-effect estimate.

Calculating the Contrasts from Different Data Summaries

The first stage is a difference in proportions when treatment receipt is binary. The reduced form is a difference in means, whether the outcome is binary or quantitative. For a binary outcome, its group mean is the proportion with outcome \(Y=1\), so the reduced-form contrast is a difference in outcome proportions. Group sizes need not be equal: each group’s proportion or mean must use its own denominator.

The sign carries information, but only after the coding and subtraction order are clear. A positive first stage means receipt is more frequent under \(Z=1\); a negative first stage means it is less frequent. A positive reduced form means the outcome mean is higher under \(Z=1\). If the scientific outcome is undesirable, “higher” may represent a worse outcome. Always report the outcome’s meaning and units rather than treating a positive sign as inherently beneficial.

Why the Group Differences Identify Assignment Effects

The causal justification for the raw differences is short but important. Under random assignment, the instrument value is independent of the potential receipts and outcomes. Consistency connects the observed values to the potential values for the assigned instrument. For the first stage:

The first equality uses consistency, and the second uses random assignment. The final equality follows from linearity of expectation. Replacing \(D\) with \(Y\) gives the corresponding result for the reduced form:

This argument identifies effects of assignment to the instrument. It does not require the exclusion restriction, because assignment may affect outcomes directly and the reduced-form contrast still measures the total effect of assignment. Exclusion matters when seeking to attribute outcome changes solely to treatment receipt. Without it, the reduced form can include effects of the encouragement itself, such as added information or attention.

Diagnostics, Alternatives, and Limitations

Start with a simple audit of the data: verify which value is coded \(Z=1\), check that both groups have observations, confirm the denominators, and ensure outcome summaries use the same measurement and follow-up definition in both groups. Then calculate each group’s receipt rate and outcome mean before subtracting. This makes direction errors and denominator mistakes easier to catch.

If the first stage is near zero, the instrument may be weakly related to receipt in the study population. If a supposedly strong encouragement has an unexpected sign, check the coding, assignment process, and receipt definition before offering a substantive explanation. Imbalance in baseline characteristics after a randomized assignment can occur by chance; extensive imbalance may also signal problems with implementation, missing data, or analysis. In a nonrandomized instrument study, observed group differences may reflect confounding, so raw contrasts do not automatically identify causal effects.

The raw mean difference is a transparent way to report each contrast. With a binary instrument, a regression of \(D\) on an intercept and \(Z\) gives a coefficient on \(Z\) equal to the raw first-stage difference; regressing \(Y\) on an intercept and \(Z\) similarly gives the raw reduced-form difference. Regression can be useful for uncertainty estimates or adjustment for prespecified covariates. In randomized studies, covariate adjustment may improve precision, but the adjusted analysis must be stated clearly and should not conceal the unadjusted group summaries. In observational studies, adjustment only supports a causal interpretation if the required conditional independence and modeling assumptions are credible.

The contrasts have limitations. They summarize average group differences and do not describe every participant’s response. A nonzero first stage does not prove instrument validity, and a reduced-form difference does not isolate the pathway through treatment. Sampling uncertainty matters, especially with small groups or a small first stage. Report standard errors or confidence intervals when making inferential claims, and keep those claims separate from the descriptive arithmetic.