From Instrument-Specific Rates to a Complier Effect
When an instrument changes treatment uptake, it can also change the probability of a binary outcome. The outcome contrast between instrument groups is not generally the causal effect of treatment for everyone: the instrument may change treatment only for some people. Under the instrumental-variable assumptions, however, comparing the outcome contrast with the treatment-uptake contrast identifies an average treatment effect for the people whose treatment status the instrument changes. These people are called compliers.
For a binary outcome, this complier average causal effect is naturally expressed as a difference in outcome probabilities. If the outcome is recovery, for example, a CACE of \(0.12\) means that treatment increases the probability of recovery by 12 percentage points on average among compliers. It is not a 12% relative increase, and it does not describe every person offered treatment.
The calculation uses two instrument-specific contrasts. The first is the difference in treatment rates, often called the first stage. The second is the difference in outcome rates, often called the reduced-form or intention-to-treat contrast. When the first stage is nonzero, their ratio is the Wald estimand:
Because \(Y\) is binary, \(E[Y\mid Z=z]=P(Y=1\mid Z=z)\); likewise, \(E[D\mid Z=z]=P(D=1\mid Z=z)\). Thus the numerator is an instrument-associated difference in outcome probabilities, and the denominator is an instrument-associated difference in treatment probabilities. The ratio is measured on the outcome-probability scale: it is a probability difference for compliers, not a ratio of risks.
Assumptions and Why the Ratio Identifies the CACE
The formula has a causal interpretation only under substantive assumptions. In the notation above, \(Z\) must be independent of the potential treatments and potential outcomes, \((D(0),D(1),Y(0),Y(1))\), in the target population. Random assignment of the instrument can support this independence; in an observational setting it requires a defensible argument. Consistency requires that observed treatment and outcome agree with the potential values corresponding to the observed assignments. There must also be no interference between units.
The exclusion restriction says that the instrument affects the outcome only through treatment: changing \(Z\) while holding \(D\) fixed does not change \(Y\). Monotonicity requires \(D(1)\ge D(0)\) for every person under this coding of the instrument. It rules out defiers, who would take treatment under \(Z=0\) but not under \(Z=1\). Finally, relevance requires a nonzero difference in treatment rates across instrument groups. These assumptions are not all verifiable from the instrument-specific rates alone.
The derivation follows by dividing the population into principal strata. Under monotonicity, people are either always-takers, with \(D(1)=D(0)=1\); never-takers, with \(D(1)=D(0)=0\); or compliers, with \(D(1)=1\) and \(D(0)=0\). Independence makes the proportions and potential outcomes in these strata comparable across instrument groups. Exclusion means that for always-takers and never-takers, changing \(Z\) does not change the outcome: their treatment status stays fixed. Only compliers have treatment changed by the instrument.
Consequently, the instrument’s treatment-rate contrast equals the proportion of compliers. Its outcome-rate contrast equals the complier proportion multiplied by the average treatment effect among compliers. Dividing the second contrast by the first gives the CACE, provided the complier proportion is positive. In symbols:
This also explains the scale of the answer. Since \(Y(1)\) and \(Y(0)\) are each either 0 or 1, their difference is between \(-1\) and \(1\). Therefore, the population CACE lies in that interval. This bound applies to the population causal effect, not necessarily to a Wald ratio calculated from finite-sample rates.
Worked Example: Treatment Increases Recovery Among Compliers
Suppose a hypothetical randomized encouragement trial has a binary treatment, receiving a program, and a binary outcome, recovery by a specified follow-up. The recovery rate is \(0.42\) in the encouraged group and \(0.33\) in the comparison group. The treatment rates are \(0.60\) and \(0.30\), respectively. Treat these rates as population quantities for this calculation. Assume consistency, no interference, instrument independence, exclusion, monotonicity, and a relevant first stage.
The outcome contrast is \(0.42-0.33=0.09\), a 9-percentage-point increase in recovery probability across instrument groups. The treatment contrast is \(0.60-0.30=0.30\), a 30-percentage-point increase in program receipt. The Wald calculation is:
Under the stated assumptions, the program increases the probability of recovery by \(0.30\), or 30 percentage points on average among compliers. The result is not a 30% increase relative to the comparison recovery rate, nor does it imply that every person’s recovery probability rises by 30 percentage points. The assumptions give this interpretation to the ratio; the rates themselves cannot establish those assumptions.
Further Calculations and Interpretation
Worked Example: A Binary Adverse Outcome
In a second hypothetical setting, \(Y=1\) denotes an adverse event. The event rates are \(0.20\) when \(Z=1\) and \(0.35\) when \(Z=0\). Treatment rates are \(0.50\) and \(0.25\), respectively. Suppose the instrument and outcome coding are set so that higher \(Z\) encourages treatment, and suppose the IV assumptions above are substantively justified.
First calculate both contrasts. The outcome contrast is \(0.20-0.35=-0.15\); the treatment contrast is \(0.50-0.25=0.25\). Therefore:
The estimated population quantity in this hypothetical calculation is a 0.60 reduction in the probability of an adverse event, or a decrease of 60 percentage points among compliers. This is a large effect, but it lies within the possible range for a binary-outcome probability difference. The negative sign is meaningful here because \(Y=1\) denotes harm: treatment reduces the probability of that event among compliers.
A direction check is useful. The instrument raises treatment uptake, so the denominator is positive. It lowers the adverse-outcome rate, so the numerator is negative. A negative ratio is therefore consistent with a protective effect on this outcome scale, subject to the IV assumptions.
Worked Example: A Sample Wald Ratio Outside the Probability Bounds
Now consider hypothetical finite-sample rates from a small study. The observed treatment rates are \(0.42\) for \(Z=1\) and \(0.38\) for \(Z=0\); observed binary-outcome rates are \(0.31\) and \(0.36\). The sample Wald ratio is:
This sample estimate is outside \([-1,1]\), the range for a population CACE with a binary outcome. That fact does not by itself prove a calculation error or refute the population IV model: a finite-sample ratio is noisy, and here a small estimated first stage magnifies the outcome contrast. It does mean that \(-1.25\) cannot be read literally as a plausible population difference in outcome probabilities. The estimate is a warning to check the arithmetic, coding, and assumptions, and to recognize that the data provide an unstable ratio. A confidence procedure suitable for ratio estimands may be needed for inference; a point estimate alone does not show its uncertainty.
Diagnostics, Alternatives, and Limits
Before interpreting a calculated CACE, verify that the instrument direction is coded consistently: \(Z=1\) should correspond to the condition that raises treatment uptake under the monotonicity convention. Compute the two rate differences separately and retain their signs. If the estimated first stage is close to zero, the ratio can be extremely unstable; modest sampling changes in either contrast may substantially change the result. Report the instrument-specific rates and the first-stage contrast alongside the Wald estimate so readers can see its denominator.
A weak or reversed sample first stage is a diagnostic prompt, not a standalone test of all IV assumptions. Randomization or independence, exclusion, and monotonicity are substantive design assumptions. In particular, monotonicity and exclusion generally cannot be confirmed just by comparing these observed rates. A direct instrument effect on the outcome, such as an encouragement that also changes care through a separate pathway, would invalidate the exclusion-based interpretation even if the Wald arithmetic is correct.
The CACE differs from several nearby quantities. The outcome-rate contrast \(E[Y\mid Z=1]-E[Y\mid Z=0]\) is the instrument’s intention-to-treat effect on the outcome; it is not the treatment effect unless the first stage is one or other special conditions apply. The CACE is local to compliers, not automatically the average treatment effect (ATE) for the whole population or the average treatment effect on the treated (ATT). A treatment effect among people induced to take treatment need not equal the effect among people who would take it regardless of the instrument.
For binary outcomes, the Wald CACE is on the risk-difference scale. It should not be described as a risk ratio, odds ratio, or percentage change. If the target question concerns a different population or effect scale, another estimand or analysis may be needed. Further, the simple rate-ratio calculation is not a substitute for an uncertainty analysis: inference must account for the ratio structure, especially when the first stage is weak.
Check Your Understanding
For each question, calculate or interpret the relevant quantity. Solutions follow the questions.
- In a hypothetical study, binary-outcome rates are \(0.48\) for \(Z=1\) and \(0.40\) for \(Z=0\). Treatment rates are \(0.70\) and \(0.50\). Calculate the Wald CACE and state its units.
- Suppose the treatment-rate contrast is \(0.25\) and the CACE is \(-0.20\). What outcome-rate contrast does the IV model imply?
- Why does the outcome-rate contrast alone not generally equal the CACE?
- A finite-sample Wald ratio is \(1.10\) for a binary outcome. What can and cannot be concluded from this value?
Solutions. (1) The outcome contrast is \(0.48-0.40=0.08\), and the treatment contrast is \(0.70-0.50=0.20\). Thus \(\text{CACE}=0.08/0.20=0.40\), a 40-percentage-point increase in outcome probability among compliers under the assumptions. (2) The implied outcome contrast is \(0.25(-0.20)=-0.05\), a 5-percentage-point decrease in outcome probability across instrument groups. (3) The instrument may change treatment for only a fraction of the population; the outcome contrast is the complier proportion multiplied by the CACE under the assumptions. Dividing by the first stage removes that proportion. (4) A population CACE for a binary outcome must be in \([-1,1]\), so \(1.10\) is not a possible population probability difference. But a finite-sample Wald ratio can exceed that range through sampling variability, especially with a small first stage; it warrants checking and careful uncertainty analysis, not an automatic conclusion that the assumptions or arithmetic are wrong.