The Wald Ratio as a Scaled Outcome Contrast

From Two Contrasts to One Ratio

A binary instrument can shift both the probability of receiving a treatment and the average outcome. The Wald ratio combines these two contrasts: it scales the outcome change associated with the instrument by the treatment-receipt change associated with the same instrument. This calculation is simple, but its interpretation depends on keeping the numerator, denominator, and assumptions distinct.

Let \(Z\) be a binary instrument, with \(Z=1\) and \(Z=0\) denoting its two values. Let \(D\) be a binary treatment indicator, where \(D=1\) means the person received treatment, and let \(Y\) be the outcome. In a population, the reduced-form outcome contrast is the difference in mean outcomes between the two instrument groups. The first-stage treatment contrast is the difference in treatment-receipt proportions between those groups.

Because \(D\) is binary, \(E[D\mid Z=z]=P(D=1\mid Z=z)\): the average of the treatment indicator is the proportion receiving treatment. The numerator is measured in units of \(Y\). The denominator is a difference in treatment probabilities, or a difference in proportions. Their ratio therefore has the same units as \(Y\), interpreted as the outcome contrast scaled to a full, one-unit change in treatment probability.

For example, if the instrument raises treatment receipt by 0.25 and is associated with a 2-unit increase in mean outcome, the Wald ratio is \(2/0.25=8\) outcome units. The ratio is not the 2-unit outcome contrast itself: that contrast corresponds to a 0.25 change in treatment probability. Nor is the ratio automatically a causal treatment effect. A causal interpretation requires assumptions about the instrument, not just division of two observed contrasts.

What Must Be in Place Before Interpreting the Ratio

The calculation can always be made when the sample first-stage contrast is nonzero. Interpreting it as causal requires more. The assumptions below describe the framework for a binary instrument and treatment; the earlier material on instrumental-variable assumptions develops their roles.

• Relevance: the instrument changes treatment receipt in the population, so \(\Delta_D\ne 0\). A sample difference alone does not guarantee a meaningful population shift.
• Independence: the instrument is independent of the potential treatment receipts and potential outcomes. Informally, instrument assignment does not systematically select people with different potential responses.
• Exclusion: the instrument affects the outcome only through treatment receipt; it has no separate pathway to \(Y\).
• Monotonicity: changing the instrument from 0 to 1 does not make some people less likely to receive treatment while making others more likely. In potential-treatment notation, \(D(1)\ge D(0)\) for every individual.
• Consistency and a well-defined treatment: observed treatment and outcome correspond to the relevant potential values, and the treatment contrast has a coherent meaning.

Relevance is the assumption directly reflected in the denominator, but it is not the only one that matters. Independence, exclusion, and monotonicity are not established by obtaining a nonzero ratio. They require design knowledge and substantive justification. Without those conditions, the Wald ratio remains a scaled association between instrument groups rather than a justified causal effect.

Worked Examples

Why the Scaling Works—and What It Does Not Show

The arithmetic can be seen directly. Write the first-stage contrast as \(\delta\), where \(\delta\ne 0\), and the reduced-form contrast as \(\gamma\). The ratio \(\gamma/\delta\) is the number that, when multiplied by the treatment contrast, reproduces the outcome contrast:

Thus, the Wald ratio answers a scaling question: how large is the observed outcome contrast relative to the instrument-induced treatment contrast? This identity explains the units and the rescaling, but it does not prove that the ratio equals an effect of treatment. Such an interpretation comes from the assumptions connecting instrument assignment, treatment receipt, and outcomes. For instance, if exclusion fails because the instrument changes outcomes through another pathway, the numerator can include that direct pathway even when the denominator is calculated correctly.

The sample version uses the same calculation with observed group summaries:

where \(\bar{Y}_{Z=z}\) is the sample mean outcome in instrument group \(z\), and \(\hat{p}_{Z=z}\) is the sample proportion treated in that group. The numerator and denominator must use the same groups and the same ordering. Using \(Z=1\) minus \(Z=0\) in the numerator but \(Z=0\) minus \(Z=1\) in the denominator flips the reported sign and does not represent the defined Wald ratio.

Diagnostics, Alternatives, and Limits

Begin with the two contrasts rather than reporting only their ratio. Show the instrument-group sample sizes, treatment proportions, and outcome means, and state which group is subtracted from which. These summaries make it possible to detect coding errors, a reversed treatment direction, or an unexpectedly small first stage. If the first-stage contrast is zero, the Wald ratio is undefined. If it is close to zero, the ratio can be highly sensitive to small changes in the data.

The contrasts also have distinct roles. The reduced form describes the outcome difference by instrument group; the first stage describes the treatment-receipt difference by instrument group. Reporting the reduced form alone avoids dividing by a small treatment shift but does not express an outcome change per treatment shift. Reporting only the first stage establishes the observed treatment shift but says nothing about the outcome contrast. The Wald ratio combines them, and so inherits the assumptions and uncertainty relevant to both.

A ratio calculated from group summaries is a transparent binary-instrument calculation. Other IV analyses may use regression or additional covariates, but those approaches do not make a problematic instrument valid merely by changing the calculation. Ordinary comparisons of treated and untreated people answer a different question and can be confounded when treatment receipt is related to outcome determinants. Conversely, if the instrument is not independent or has a direct effect on the outcome, the Wald ratio does not repair those failures.

A final limitation is that the ratio alone does not identify which people or which population average its causal interpretation concerns. That target depends on the IV framework and its assumptions, not on the numerical division. Keep the descriptive calculation, the causal assumptions, and the substantive interpretation separate when presenting results.

Common Mistakes

• Calling the numerator the treatment effect. The numerator is the outcome contrast by instrument status. It can reflect the instrument's effect on treatment uptake, and it is not automatically the effect of treatment itself.
• Forgetting what the denominator measures. The first stage is a difference in treatment probabilities, not a treatment-group mean outcome or a count of treated individuals.
• Reversing only one contrast. Both contrasts must use the same instrument-group order; otherwise their ratio has an erroneous sign.
• Interpreting a large ratio as automatically convincing. A small first stage can inflate the ratio and make it unstable. Check the component contrasts and their uncertainty.
• Treating a nonzero first stage as proof of validity. Relevance does not establish independence, exclusion, or monotonicity.