A Worked Wald Calculation From an Encouragement Trial

From an Encouragement Table to a Wald Estimate

In an encouragement trial, assignment changes how likely people are to receive a treatment, but it does not determine treatment receipt for everyone. The randomized assignment can therefore be used as an instrument for treatment receipt. A basic calculation compares the outcome difference between assignment groups with the difference in treatment receipt between those same groups. The resulting ratio is the Wald estimate.

The calculation has two distinct contrasts. The outcome contrast measures the effect of assignment on the outcome, often called the reduced-form or intention-to-treat contrast. The treatment contrast measures how much assignment changed treatment receipt, called the first stage. Dividing the first by the second puts the outcome contrast on the scale of treatment receipt.

In a sample, use the observed mean treatment receipt and outcome in each assignment group: \(\widehat{\Delta}_D=\bar{D}_{Z=1}-\bar{D}_{Z=0}\) and \(\widehat{\Delta}_Y=\bar{Y}_{Z=1}-\bar{Y}_{Z=0}\). The sample Wald estimate is \(\widehat{\Delta}_Y/\widehat{\Delta}_D\). For binary \(D\), each group’s mean \(D\) is its treatment-receipt proportion. Keep the assignment groups as the basis for both contrasts; comparing people by treatment actually received is a different, generally confounded comparison.

Why the Ratio Can Represent a Complier Effect

The ratio has a causal interpretation only under assumptions. Write \(D(z)\) for the treatment receipt a person would have under assignment \(z\), and \(Y(d)\) for the outcome that person would have under treatment status \(d\). The relevant assumptions are that assignment is independent of these potential outcomes and potential treatment receipts (as supplied by valid randomization); assignment affects the outcome only through treatment receipt (the exclusion restriction); treatment receipt and outcomes are consistent with the defined interventions, with no interference between participants; and there are no defiers, so \(D(1)\ge D(0)\) for every person. There must also be a nonzero first stage: the encouragement changes treatment receipt for some people.

The principal compliance types are defined by \((D(1),D(0))\): compliers are \((1,0)\), always-takers are \((1,1)\), never-takers are \((0,0)\), and defiers are \((0,1)\). Under monotonicity, defiers are absent. Random assignment makes the assignment-group contrasts identify the corresponding expectations over the randomized population. Exclusion means assignment itself does not change \(Y\), apart from any change caused by treatment receipt.

The first equality for \(\Delta_D\) follows because random assignment identifies the mean difference in potential treatment receipt, and because, without defiers, \(D(1)-D(0)\) is one for compliers and zero for everyone else. For the outcome contrast, exclusion and consistency allow the observed outcome under assignment \(z\) to be written as \(Y(D(z))\). Compliers contribute \(Y(1)-Y(0)\) to the difference across assignments. Always-takers receive treatment either way, and never-takers receive no treatment either way, so both types contribute zero. Averaging therefore gives the second line. Dividing the two contrasts yields:

This is the local average treatment effect (LATE) for compliers: their average causal effect of treatment, not automatically the average treatment effect for everyone. The derivation explains why the denominator matters. The outcome contrast is diluted because only some participants change treatment receipt when encouraged; dividing by the size of that change rescales the contrast to the complier group.

Calculating Carefully From Counts and Totals

When the table gives counts and outcome totals rather than means, first calculate a mean within each randomized assignment group. Do not calculate an outcome mean among treatment recipients and compare it with one among nonrecipients: those groups are formed after assignment and can differ in ways that affect the outcome.

Diagnostics, Uncertainty, and What the Table Cannot Show

Before interpreting a Wald ratio, check the calculation and the design. Confirm that the rows are randomized assignment groups, that the receipt rates use each row’s correct denominator, and that the outcome means use the same assignment groups. Check the direction and units of both contrasts. If assignment increases receipt, \(\widehat{\Delta}_D\) should be positive under the chosen coding; the Wald estimate’s sign should then match the outcome contrast. A small denominator deserves particular attention: even a modest outcome difference can produce a very large ratio.

A nonzero first stage is necessary, but a weak first stage makes the ratio unstable. Sampling variation in a denominator near zero can produce a highly variable estimate and unreliable conventional confidence intervals. A large Wald estimate in that setting is not, by itself, evidence of a large treatment effect. Report the first-stage and reduced-form contrasts alongside the ratio, and assess uncertainty using methods appropriate for an instrumental-variables ratio; do not treat the point calculation as a complete inferential analysis.

The assumptions also have different evidential status. Randomization can support instrument independence, and the observed treatment rates show whether assignment changed receipt in the sample. But a table cannot demonstrate that the offer has no direct effect on the outcome, that no one is a defier, or that treatment versions and interference are innocuous. Those are design and substantive questions. If an assumption fails, the ratio may not estimate the complier effect even when every arithmetic step is correct.

How the Wald Estimate Differs From Other Comparisons

The Wald estimate is useful when the scientific target is the effect of receiving treatment for people whose receipt is changed by encouragement. It does not generally recover the effect for always-takers, never-takers, or the whole population. The complier group is defined by potential responses to assignment, so membership cannot usually be identified for each individual from the observed table.

Common mistake: “The offer group’s outcome is lower by 0.6 points, so treatment lowered the outcome by 0.6 points for everyone.” This confuses the randomized assignment contrast with the treatment effect and ignores that encouragement changes receipt for only some participants. The correction is to report the 0.6-point assignment contrast as the reduced form, calculate the first stage, and divide the former by the latter only when the IV assumptions are defensible. Even then, interpret the result as a complier effect, not an all-participant effect.