Worked Wald Estimates with Compliance-Type Tables

From Compliance Types to a Wald Estimate

A Wald estimate is a ratio of two assignment contrasts: the change in treatment receipt associated with changing the instrument, divided by the corresponding change in the outcome. A compliance-type table makes both contrasts concrete. It shows which individuals change treatment when assignment changes, and how their outcomes contribute to the assignment contrast.

The calculation is straightforward; its interpretation is conditional. Under the instrumental-variable assumptions, the ratio identifies an average treatment effect for compliers. Without those assumptions, the same arithmetic still produces a number, but that number need not be a causal effect for compliers. This tutorial works through both situations using hypothetical population tables.

The tables below report type counts or proportions and, where needed, average potential outcomes within types. They describe hypothetical populations, not real studies. Under independence of assignment from the relevant potential outcomes and treatment responses, the type composition is the same across assignment groups in expectation. Under exclusion, assignment affects the outcome only through treatment, so the outcome under treatment \(d\) is \(Y(d)\), regardless of \(Z\).

Monotonicity rules out defiers, and relevance requires a nonzero first-stage contrast. In this setting the first stage is the difference in treatment rates between assignment groups; the reduced form is the difference in their mean outcomes. The Wald ratio divides the latter by the former.

Worked Calculation Under the IV Assumptions

For each table, use the assigned-treatment pattern to work out who receives treatment under each value of \(Z\). Then calculate treatment and outcome means separately for \(Z=1\) and \(Z=0\). Do not confuse the reduced form, which is an outcome contrast, with the Wald ratio, which rescales that contrast by the first stage.

Why the Ratio Identifies a Complier Effect

The table calculation follows a general cancellation argument. Under independence, assignment groups have the same compliance-type proportions and potential outcome distributions in expectation. Under exclusion, an individual’s outcome depends on treatment rather than directly on assignment. Under monotonicity, only compliers change treatment status when \(Z\) changes. Never-takers and always-takers therefore contribute no treatment-rate change.

Let \(C\) denote the event that a person is a complier, and let \(P(C)\) be the complier proportion. The treatment-rate difference is the proportion who move from untreated to treated:

To see what happens to the outcome contrast, compare the two assignments by compliance type. Never-takers receive treatment 0 under either assignment, so their outcomes cancel from the contrast. Always-takers receive treatment 1 under either assignment, so their outcomes also cancel. Compliers have \(Y(1)\) under \(Z=1\) and \(Y(0)\) under \(Z=0\). Their contribution to the population outcome contrast is their proportion times their mean treatment effect:

When \(P(C)>0\), dividing the reduced form by the first stage gives

This short derivation explains why each assumption matters to interpretation: independence makes the assignment groups comparable in their potential responses, exclusion removes a direct assignment effect on outcomes, and monotonicity ensures that the first stage represents compliers rather than a net of people moving in opposite directions. Relevance ensures the denominator is nonzero.

A Smaller First Stage and an Outcome in Days

A smaller first stage means the reduced form is divided by a smaller number. This can produce a larger Wald ratio even when the outcome contrast is modest. In this example, the assumptions still support the complier interpretation, but the ratio must be read in the outcome’s units.

When the Arithmetic Does Not Give a Complier Effect

The same recipe can be used when an assumption fails, but the usual causal interpretation no longer follows automatically. In particular, if assignment can affect the outcome directly, the reduced form includes more than the pathway through treatment. Dividing that combined contrast by the first stage does not generally remove the direct effect.

Practical Checks and Common Mistakes

Before interpreting a computed ratio, check the table’s treatment patterns and the assumptions behind its outcome entries. A useful calculation order is to find treatment rates first, then outcome means, then contrasts, and only then the ratio. This makes it easier to locate errors and keeps the estimand distinct from its components.

A common mistake is to stop at the reduced form and report it as the treatment effect. The reduced form is the outcome contrast between assignment groups; it is not generally the effect of treatment because not everyone changes treatment when assignment changes. Another mistake is to compute the ratio correctly but label it an average treatment effect for the entire population. Under the assumptions used here, its interpretation is specific to compliers. Finally, a nonzero first stage alone does not establish that the other assumptions hold.