Instrument Assessment and LATE Calculation Workshop

From Instrument Description to a Defensible LATE

An instrument can strongly change treatment receipt and still fail to identify a causal treatment effect. The assessment requires more than checking whether treatment rates differ: it requires defining what treatment and outcomes each person would have under alternative instrument values, then evaluating the assumptions that connect those potential outcomes to the observed data.

This workshop uses a binary instrument $Z$, a binary treatment $D$, and an outcome $Y$. The instrument may be an encouragement, assignment, or other exposure that can shift treatment receipt. The target, when the assumptions justify it, is the local average treatment effect (LATE) among people whose treatment is changed by the instrument.

Definition: For unit $i$, $D_i(z)$ is the treatment that would be received if the instrument were set to $z$, where $z$ is 0 or 1. $Y_i(d)$ is the outcome that would occur if treatment were set to $d$, where $d$ is 0 or 1. If the instrument could affect the outcome other than through treatment, write the more general potential outcome $Y_i(z,d)$. The observed values satisfy $D_i=D_i(Z_i)$; under exclusion, $Y_i=Y_i(D_i)$.

The Four Assumptions to Audit

For the standard binary-instrument Wald interpretation, the four core conditions are relevance, independence, exclusion, and monotonicity. Consistency and no interference are also needed: observed treatment and outcome agree with the potential outcomes under the conditions actually received, and one unit’s assignment does not change another unit’s outcome or treatment.

Relevance: The instrument changes treatment probability: $E[D_i(1)]\ne E[D_i(0)]$.

Independence: Instrument assignment is independent of the relevant potential outcomes. In the unconditional binary-instrument setting, a standard statement is $Z_i\perp\{Y_i(0),Y_i(1),D_i(0),D_i(1)\}$. Random assignment can support this condition; an observational instrument requires a substantive argument that assignment is not associated with these potential outcomes.

Exclusion: The instrument affects the outcome only by changing treatment. In potential-outcome notation, $Y_i(z,d)=Y_i(d)$ for each $z,d$.

Monotonicity: There are no defiers: $D_i(1)\ge D_i(0)$ for every unit. Thus, nobody would take treatment under $Z=0$ but avoid it under $Z=1$.

Relevance is about a difference in treatment receipt, not whether the instrument is otherwise valid. Independence and exclusion are not established merely because the observed treatment and outcome means differ. They require design knowledge and scientific judgment. Monotonicity is an individual-level condition: a positive average first stage does not rule out some people moving in the opposite direction.

Wald LATE: If the four assumptions hold, the ratio of the instrument’s effect on the outcome to its effect on treatment identifies the average treatment effect among compliers: $$ \text{LATE} = \frac{E[Y_i\mid Z_i=1]-E[Y_i\mid Z_i=0]} {E[D_i\mid Z_i=1]-E[D_i\mid Z_i=0]} = E[Y_i(1)-Y_i(0)\mid D_i(1)>D_i(0)]. $$ Compliers are units who receive treatment when $Z=1$ and do not receive it when $Z=0$.

Why the Ratio Targets Compliers

A short decomposition clarifies both the interpretation and the role of monotonicity. Under independence and consistency, the difference in observed treatment means across instrument groups equals $E[D_i(1)-D_i(0)]$. With binary treatment, units fall into four response types: compliers $(D_i(0),D_i(1))=(0,1)$, never-takers $(0,0)$, always-takers $(1,1)$, and defiers $(1,0)$.

Under exclusion, independence, and consistency, the outcome difference across instrument groups is $E[Y_i(D_i(1))-Y_i(D_i(0))]$. Under monotonicity, the defier type is absent. Never-takers and always-takers have no treatment change and contribute zero to this difference; compliers contribute their treatment effect. Therefore:

E[Y_i\mid Z_i=1]-E[Y_i\mid Z_i=0] = P(\text{complier})E[Y_i(1)-Y_i(0)\mid\text{complier}], $$ $$ E[D_i\mid Z_i=1]-E[D_i\mid Z_i=0]=P(\text{complier}).

When relevance holds, the denominator is nonzero, so dividing the first equality by the second gives the stated LATE. If monotonicity fails, defiers contribute in the opposite direction, and this simple complier interpretation generally does not follow.

Worked Example: Randomized Encouragement With a Defensible LATE

Suppose a hypothetical trial randomly assigns an offer of a support program, $Z=1$, or no offer, $Z=0$. Treatment $D=1$ means attending the program, and $Y=1$ means meeting a prespecified health target. The observed treatment proportions are 0.60 with the offer and 0.35 without it; the observed outcome proportions are 0.74 and 0.68, respectively.

Define the potential outcomes. $D_i(1)$ and $D_i(0)$ indicate whether person $i$ would attend under each offer status. $Y_i(1)$ and $Y_i(0)$ indicate whether that person would meet the target if treated or untreated. Exclusion means that the offer itself, apart from its effect on attendance, does not affect the health target.

Audit the assumptions. Relevance is supported by the observed first-stage difference, $0.60-0.35=0.25$. Random assignment supports independence. Exclusion is plausible only if the offer does not itself provide health advice or services that affect the target independently of attendance. Monotonicity requires that no one is discouraged from attending by receiving the offer; this is a substantive assumption, not something proved by the positive first stage. Suppose the trial’s design and program delivery make exclusion and monotonicity defensible.

Calculate and interpret. The Wald estimate is:

\widehat{\text{LATE}} = \frac{0.74-0.68}{0.60-0.35} = \frac{0.06}{0.25} = 0.24.

Under the assumptions, the estimate is an increase of 0.24 in the probability of meeting the health target due to program attendance among compliers. Since the outcome is binary, this is a 24-percentage-point increase, not a 24% increase for every participant. It is not automatically the effect for all people offered the program or for the full population.

Worked Example: A Strong First Stage With a Threat to Independence

In a hypothetical observational setting, $Z=1$ indicates living near a clinic that provides a treatment, and $D=1$ indicates receiving that treatment. The treatment proportions are 0.70 for people living nearby and 0.50 for those farther away. The outcome proportions are 0.80 and 0.70. Let $D_i(z)$ denote treatment receipt under each location-based instrument value, and $Y_i(d)$ the outcome under each treatment value.

Relevance is present in the observed data: $0.70-0.50=0.20$. The unadjusted Wald ratio can also be calculated:

\frac{0.80-0.70}{0.70-0.50} = \frac{0.10}{0.20} = 0.50.

However, suppose clinic proximity is associated with baseline health, transportation access, or local care quality. Those factors could be related to the potential outcomes, so independence is not credible without a stronger design-based or substantive argument. If proximity also changes access to other care, exclusion is doubtful as well. The value 0.50 is therefore a descriptive ratio of observed differences, not a justified estimate of the treatment LATE. A large first stage does not repair confounding or a direct pathway to the outcome.

Worked Example: An Encouragement That Also Changes Care

Consider a hypothetical randomized reminder system. $Z=1$ means receiving a reminder and $Z=0$ means not receiving one; $D=1$ means attending a treatment session. Define $D_i(z)$ and $Y_i(d)$ as before. Suppose reminders are randomized, treatment proportions are 0.55 and 0.35, and outcome proportions are 0.72 and 0.66.

The observed first stage is $0.55-0.35=0.20$, so relevance holds. Randomization supports independence. But suppose the reminder also connects recipients to a care coordinator, who can improve the outcome even if they do not attend the session. Then $Y_i(z,d)$ can depend on $z$ even when $d$ is held fixed, violating exclusion.

The arithmetic ratio is:

\frac{0.72-0.66}{0.55-0.35} = \frac{0.06}{0.20} = 0.30.

This 0.30 is not identified as a treatment LATE because the outcome difference can include the coordinator pathway. The ratio’s numerator combines the effect of changing treatment with any direct effect of the reminder. Randomization alone does not make exclusion true.

Worked Example: Defiers Change the Ratio’s Meaning

Suppose a hypothetical population has valid independence and exclusion, but some people respond to the instrument in the opposite direction. The proportions of never-takers, always-takers, compliers, and defiers are 0.30, 0.20, 0.40, and 0.10. Let $D_i(0)$ and $D_i(1)$ define each person’s treatment response type, and let $Y_i(0)$ and $Y_i(1)$ define their treatment outcomes. Suppose the average treatment effect is 0.20 among compliers and 0.50 among defiers.

The first-stage difference is the proportion of compliers minus the proportion of defiers:

$$ E[D_i(1)-D_i(0)] = 0.40-0.10 = 0.30. $$

Never-takers and always-takers do not change treatment. Under exclusion, the reduced-form outcome difference is the complier contribution minus the defier contribution:

$$ E[Y_i(D_i(1))-Y_i(D_i(0))] = (0.40)(0.20)-(0.10)(0.50) = 0.08-0.05 = 0.03. $$

The Wald ratio is consequently $0.03/0.30=0.10$. This is not the complier effect of 0.20: the defiers’ larger positive treatment effect offsets part of the compliers’ contribution to the numerator. The ratio is a weighted contrast involving both types, rather than the standard complier LATE. This is why a positive first stage alone does not establish monotonicity.

A Practical Assessment Sequence

Specify the variables and potential outcomes.
State what $Z=1$ and $Z=0$ mean, what treatment $D$ represents, and what $D_i(z)$ and $Y_i(d)$ describe. If a direct instrument effect is possible, consider $Y_i(z,d)$.

Check relevance in the data.
Compare treatment rates across instrument groups. A nonzero difference is necessary for the Wald ratio, but it does not establish the other assumptions.

Audit independence and exclusion through the design.
Ask how instrument assignment was generated and trace plausible pathways from the instrument to the outcome. Randomization supports independence but does not rule out direct effects.

Assess monotonicity and calculate only with a justified interpretation.
Ask whether any units could respond in the opposite direction. If all assumptions are defensible, divide the outcome difference by the treatment difference and describe the result as a complier effect.

Common Mistake: Treating the Ratio as Self-Validating

A common mistake is to calculate a Wald ratio whenever instrument-group means are available, then call the answer a LATE because treatment rates differ. That reasoning checks relevance but silently assumes independence, exclusion, and monotonicity. The correction is to report the assumptions separately, explain what features of the assignment and intervention support them, and withhold a causal LATE interpretation when a key condition is not defensible. Even when a LATE is justified, it refers to instrument-specific compliers; it is not automatically the ATE for everyone.

Key takeaway: A Wald calculation is a causal LATE only when the instrument is relevant, independent of the potential outcomes, excluded from the outcome except through treatment, and monotone in its effect on treatment. The ratio’s target is the people whose treatment status the instrument changes.

Check Your Understanding

For each question, distinguish what the observed data show from what the assumptions allow you to conclude.

In a randomized encouragement study, treatment rates are 0.52 with encouragement and 0.32 without it. Which assumption does this difference support, and which assumptions remain to be assessed?
Write the meanings of $D_i(1)$, $D_i(0)$, $Y_i(1)$, and $Y_i(0)$ for a binary treatment and instrument.
An instrument is randomized, but it also provides participants with advice that can affect the outcome without changing treatment. Which assumption is threatened, and why does randomization not resolve that problem?
Suppose the outcome difference across instrument groups is 0.08 and the treatment difference is 0.20. Calculate the Wald ratio and state the assumptions needed before interpreting it as a LATE.
Why can a positive average first stage coexist with some defiers, and what implication does that have for interpreting the Wald ratio?