Deriving the Wald Ratio from Assignment Effects

Why Divide One Assignment Contrast by Another?

An instrument changes the opportunity or encouragement to receive a treatment; it does not necessarily determine who receives it. Two differences therefore matter. The outcome contrast compares average outcomes across instrument levels, while the treatment-receipt contrast compares treatment uptake across those same levels. The Wald ratio divides the first contrast by the second.

This division puts the outcome contrast on a scale per unit of instrument-induced change in treatment receipt. The numerator and denominator are each assignment-based contrasts, so it is important to keep track of which quantity is being reported: the outcome contrast alone, the first-stage contrast alone, or their ratio. The ratio is not itself an assignment effect.

These definitions describe contrasts between instrument groups. To interpret them as causal effects of assignment, one needs an appropriate assignment mechanism. In a randomized encouragement study, a standard condition is that assignment is independent of the relevant potential outcomes and treatment-receipt potential outcomes. Consistency links each person’s observed outcome and treatment to the instrument level actually assigned. Both instrument levels must occur with positive probability in the population being studied. Under these conditions, the group-mean contrasts equal the corresponding average effects of assignment on outcome and treatment receipt.

Those conditions justify calling the two contrasts assignment effects; they do not, by themselves, ensure that the ratio has a particular causal interpretation for treatment. Instrument validity also involves conditions such as exclusion, and a nonzero first stage is needed to define the ratio. Here the derivation is about how the Wald ratio is built from assignment contrasts.

From Assignment Effects to the Wald Ratio

For a binary outcome or treatment-receipt variable, a conditional expectation given an instrument level is the mean within that group. Thus the population contrasts can be written as differences in group means. If \(\bar{Y}_z\) and \(\bar{D}_z\) denote the population means among units with \(Z=z\), then

The numerator measures how much average outcome differs between those assigned to instrument level 1 and those assigned to level 0. The denominator measures how much average treatment receipt differs between the same instrument groups. Dividing one by the other expresses the outcome difference relative to the treatment-receipt difference.

For a sample, replace population group means by observed sample means. Write \(\bar{Y}_{z}\) for the mean observed outcome among the \(n_z\) units with \(Z=z\), and \(\bar{D}_{z}\) for their mean treatment receipt. The sample Wald estimate is

The corresponding population ratio is \(\theta_W=\Delta_Y/\Delta_D\), provided \(\Delta_D\ne 0\). This algebra does not require a model for the outcome conditional on treatment, nor does it turn the numerator into a treatment effect. It uses two contrasts indexed by instrument assignment and forms their ratio.

Why the Denominator Matters

The denominator is not a technical detail: it sets the scale of the ratio. If assignment changes treatment receipt only slightly, a given outcome contrast translates into a much larger ratio. If assignment has no effect on average treatment receipt, the ratio is undefined. A ratio with a very small estimated denominator can also vary substantially when the observed data change modestly.

The sign of the denominator also matters for arithmetic. If the instrument reduces treatment receipt, then \(\Delta_D<0\), and the ratio’s sign is the sign of \(\Delta_Y\) divided by a negative number. One should not silently replace a negative first stage with its absolute value: doing so changes the defined ratio.

Sample Contrasts and a Second Calculation

In practice, the group means are computed from observed data. When \(D\) is binary, its mean within an instrument group is exactly the proportion receiving treatment in that group. For \(Y\), the group mean is the average outcome among the units assigned to that level. The same formula therefore applies whether the data are summarized as means or, for binary treatment receipt, as proportions.

A Short Derivation and an Interpretation Check

The ratio follows directly by substituting the definitions of the assignment contrasts. Starting from the outcome assignment contrast and the treatment-receipt assignment contrast, divide the former by the latter:

Under the assignment conditions stated above, the numerator is the average effect of assignment on the outcome, and the denominator is the average effect of assignment on treatment receipt. Thus, the Wald ratio is the outcome assignment effect divided by the treatment-receipt assignment effect. The derivation is complete: it is a substitution of two assignment-based contrasts, not a claim that either contrast is a treatment effect.

Units provide a useful check. If \(Y\) is measured in blood-pressure points and \(D\) is binary, then \(\Delta_Y\) has units of blood-pressure points and \(\Delta_D\) is a probability difference. The ratio is in blood-pressure points per unit change in treatment receipt. A unit change is a change in the probability from 0 to 1; if the first stage is 0.10, the ratio divides by that 0.10 rather than by 10 percentage points as a number.

The ratio is unchanged if both assignment contrasts are reversed together: switching which instrument level is called 1 and which is called 0 multiplies both contrasts by \(-1\), leaving their quotient unchanged. Reversing only one contrast, by contrast, is an error. This is another reason to compute and label the two differences before dividing.

Common Mistakes and Practical Checks

• Reporting only the outcome difference as the Wald estimate. The outcome difference is the numerator, not the ratio. Divide it by the difference in treatment-receipt means.
• Using the wrong denominator. The denominator compares treatment receipt across instrument levels. It is not the treatment rate in just one group, nor the overall treatment rate.
• Forgetting that a binary-treatment mean is a proportion. If 60% versus 40% receive treatment, the first stage is \(0.60-0.40=0.20\), not a difference of 20 unless the percentage-point scale is explicitly used and the ratio’s units are adjusted accordingly.
• Dividing by a zero first stage. If \(\Delta_D=0\), the Wald ratio is undefined. A ratio should not be reported by treating zero as a negligible denominator.
• Calling the ratio an assignment effect. The numerator and denominator are assignment contrasts; the ratio is their quotient. Its further causal interpretation requires assumptions beyond the arithmetic of the ratio.