A Worked Comparison of ITT, LATE, and ATE

Three Effects, Three Questions

In an encouragement design, an instrument \(Z\) changes the probability of receiving a treatment \(D\), but it does not necessarily determine treatment for everyone. The effect of being encouraged is therefore not generally the same as the effect of receiving treatment. Instrumental-variables analysis can identify a treatment effect for a particular group under specific assumptions; it does not automatically identify the average treatment effect for the whole population.

This tutorial works through one hypothetical population to distinguish three quantities: the intention-to-treat effect (ITT) of encouragement on the outcome, the Wald estimate of the local average treatment effect (LATE) for compliers, and the average treatment effect (ATE) across the full population. The central comparison is that the same observed encouragement data can be consistent with different ATEs when treatment effects for some compliance groups are unobserved.

Definition: Let \(Z\in\{0,1\}\) denote the instrument or encouragement, \(D\in\{0,1\}\) the treatment, and \(Y\) the outcome. Write \(D(z)\) for the treatment status that would occur under instrument value \(z\), and \(Y(d)\) for the outcome under treatment status \(d\). The outcome ITT is \(E[Y\mid Z=1]-E[Y\mid Z=0]\); the treatment first stage is \(E[D\mid Z=1]-E[D\mid Z=0]\). The population ATE is \(E[Y(1)-Y(0)]\).

Compliance Groups and Identification Assumptions

The pair \((D(0),D(1))\) defines each person's compliance group. Compliers take treatment only when encouraged: \((0,1)\). Always-takers take treatment either way: \((1,1)\). Never-takers do not take treatment either way: \((0,0)\). A defier would do the opposite of the encouragement, with \((1,0)\). The standard binary-instrument Wald interpretation as a complier effect uses monotonicity, which rules out defiers.

The assumptions for the calculation matter. Instrument independence means \(Z\) is independent of the potential treatment statuses and potential outcomes, as under valid random assignment. The exclusion restriction says the instrument affects \(Y\) only through \(D\), so the relevant outcome potential is \(Y(d)\), not a separate direct effect of \(Z\). Monotonicity requires \(D(1)\geq D(0)\) for every individual. Relevance requires a nonzero treatment first stage. Consistency connects observed treatment and outcome to the potential values corresponding to the instrument and treatment actually received; the usual no-interference condition rules out one person's assignment changing another person's outcome.

These assumptions are substantive, not arithmetic conveniences. If random assignment, exclusion, or monotonicity fails, the Wald ratio need not equal the complier average treatment effect. When the assumptions hold, however, the ratio has a precise interpretation.

Worked Encouragement Example

Suppose a hypothetical population has 1,000 people. The instrument is randomized, and the population consists of 200 always-takers, 500 compliers, and 300 never-takers, with no defiers. The potential outcome means, in outcome points, are as follows. These values are stipulated for the example; they are not presented as results from a real study.

Compliance group	Population share	Mean \(Y(0)\)	Mean \(Y(1)\)	Mean treatment effect
Always-takers	0.20	50	54	4
Compliers	0.50	60	70	10
Never-takers	0.30	55	53	-2

Worked Example: Calculating the Outcome ITT and First Stage

Under \(Z=0\), always-takers receive treatment, while compliers and never-takers do not. Under \(Z=1\), always-takers and compliers receive treatment, while never-takers do not. Independence allows the group means to be compared across instrument assignments, and exclusion means there is no additional outcome effect of \(Z\) within a fixed treatment status.

The expected outcome mean under \(Z=0\) is

E[Y\mid Z=0] =0.20(54)+0.50(60)+0.30(55) =10.8+30+16.5 =57.3\text{ points}.

Under \(Z=1\), compliers now receive treatment, so

E[Y\mid Z=1] =0.20(54)+0.50(70)+0.30(55) =10.8+35+16.5 =62.3\text{ points}.

Thus the outcome ITT is \(62.3-57.3=5.0\) points. It is the average effect of assignment to encouragement on the outcome in this population—not the average effect of treatment receipt.

The treatment rate under \(Z=0\) is the always-taker share, \(0.20\). Under \(Z=1\), the treatment rate is the sum of the always-taker and complier shares, \(0.20+0.50=0.70\). Therefore the first stage is \(0.70-0.20=0.50\), or a 50-percentage-point increase in treatment receipt.

Why the Wald Ratio Is the Complier Effect

For a binary instrument and treatment, the Wald ratio divides the outcome ITT by the treatment first stage. To see its interpretation, partition the population into compliance groups. By instrument independence, group shares and their potential outcome means are comparable across the two instrument conditions. By exclusion, always-takers remain treated under either instrument value, and never-takers remain untreated; neither group's outcome mean changes with \(Z\). Compliers switch from untreated to treated.

Consequently, only compliers contribute to the difference in average outcomes across instrument assignments. If \(p_C\) is the complier share and \(\tau_C=E[Y(1)-Y(0)\mid C]\) is their average treatment effect, then

E[Y\mid Z=1]-E[Y\mid Z=0]=p_C\tau_C.

For treatment receipt, compliers change from \(D(0)=0\) to \(D(1)=1\), while the other groups' treatment statuses do not change. Hence

E[D\mid Z=1]-E[D\mid Z=0]=p_C.

Relevance ensures \(p_C>0\) under these assumptions, so division gives the complier effect. This is the binary-instrument Wald result.

\frac{E[Y\mid Z=1]-E[Y\mid Z=0]} {E[D\mid Z=1]-E[D\mid Z=0]} =\frac{p_C\tau_C}{p_C} =\tau_C.

Worked Example: Computing and Interpreting Wald LATE

From the preceding calculations, the outcome ITT is 5.0 points and the treatment first stage is 0.50. The Wald ratio is therefore

\widehat{\tau}_{\mathrm{Wald}} =\frac{5.0\text{ points}}{0.50} =10.0\text{ points per treatment}.

The result agrees with the stipulated complier effect: \(70-60=10\) points. Under the stated IV assumptions, the Wald ratio identifies the average treatment effect for people whose treatment status is changed by encouragement. It does not, from this calculation alone, identify the effect for always-takers or never-takers.

From LATE to the Population ATE

The ATE averages treatment effects over everyone, not just compliers. Since the three compliance groups partition this hypothetical population, the population ATE is a share-weighted average of their group-specific effects:

ATE =p_A\tau_A+p_C\tau_C+p_N\tau_N, \qquad p_A+p_C+p_N=1.

Here, \(p_A\), \(p_C\), and \(p_N\) are the population shares, and each \(\tau\) is the mean \(Y(1)-Y(0)\) in its group. The Wald estimate supplies \(\tau_C\). But encouragement does not change treatment status for always-takers or never-takers. For those groups, one treatment potential outcome is not observed: \(Y(0)\) for always-takers and \(Y(1)\) for never-takers. Their treatment effects therefore cannot be recovered from the ITT and first stage alone.

Worked Example: Calculating ATE From Specified Group Effects

Using all three stipulated group effects in the table, the population ATE is

ATE =0.20(4)+0.50(10)+0.30(-2) =0.8+5.0-0.6 =5.2\text{ points}.

This ATE differs from the 10-point LATE because the groups have different treatment effects and compliers make up only half the population. The calculation is valid because this example stipulates the effects for all three groups. In an actual encouragement design, the data used to calculate the Wald ratio do not by themselves supply those two noncomplier effects.

The table illustrates the missing counterfactuals. For always-takers, the observed treatment status is \(D=1\) under both instrument values, so their mean \(Y(0)\) is not revealed by encouragement. For never-takers, \(D=0\) under both values, so their mean \(Y(1)\) is not revealed. Seeing the observed outcome means for these groups would not, by itself, reveal those missing potential outcome means.

Alternative Noncomplier Effects, Same IV Estimates

The example's observed ITT and first stage are driven by the compliers' response to the instrument. Changing the treatment effects of always-takers or never-takers does not change those two quantities: their treatment status never changes with \(Z\). Their effects can nevertheless change the population ATE because they contribute to its population-wide weighted average.

Worked Example: Comparing ATE Under Alternative Effects

Keep the population shares and the 10-point complier effect fixed. Consider three hypothetical choices for the unobserved effects among always-takers and never-takers. Each row calculates the population ATE using the same formula.

Scenario	\(\tau_A\)	\(\tau_C\)	\(\tau_N\)	ATE calculation	ATE
Original stipulated effects	4	10	-2	0.20(4) + 0.50(10) + 0.30(-2)	5.2 points
Zero effects for both noncomplier groups	0	10	0	0.20(0) + 0.50(10) + 0.30(0)	5.0 points
Different effects in both groups	8	10	-6	0.20(8) + 0.50(10) + 0.30(-6)	4.8 points

For the second scenario, the arithmetic is \(0+5+0=5.0\) points. For the third, it is \(1.6+5.0-1.8=4.8\) points. In every scenario, the outcome ITT remains 5.0 points and the first stage remains 0.50, so the Wald LATE remains 10.0 points. The ATE changes because the assumed always-taker and never-taker effects change.

These scenarios are sensitivity calculations, not estimates of the missing effects. They show what the ATE would be if those group effects took the specified values. The IV calculation alone does not choose among the scenarios. Additional assumptions or information would be needed to justify particular noncomplier effects.

Common Mistake: Treating LATE as the ATE

A tempting but incorrect argument is: “The Wald estimate is 10 points, so treatment raises the outcome by 10 points for the population on average.” The calculation establishes a 10-point effect for compliers under the IV assumptions. It says nothing by itself about the effects for always-takers and never-takers. As the weighted formula shows, the population ATE also depends on those groups' effects and their population shares.

The reverse mistake is to call the 5-point outcome ITT the treatment effect. ITT is the effect of the instrument assignment on the outcome. In this example, the encouragement raises treatment receipt by 0.50, so its 5-point outcome effect corresponds to a 10-point effect per induced treatment among compliers—not a 5-point treatment effect for every person.

Key takeaway: The outcome ITT measures the effect of encouragement; the Wald ratio identifies LATE for compliers under the binary-IV assumptions; and the ATE averages effects across all compliance groups. ATE generally requires information or assumptions about effects that the instrument does not reveal.

Check Your Understanding

Use the hypothetical encouragement design above to answer these questions.

If the treatment rate under \(Z=0\) is 0.15 and under \(Z=1\) is 0.55, what is the treatment first stage, and how would you interpret it?
If the outcome ITT were 4.0 points and the first stage were 0.40, what would the Wald ratio be, and whose average treatment effect would it represent under the stated assumptions?
Why can a change in the never-takers' treatment effect leave the outcome ITT and treatment first stage unchanged while changing the population ATE?
Using shares \(p_A=0.20\), \(p_C=0.50\), and \(p_N=0.30\), calculate the ATE if \(\tau_A=2\), \(\tau_C=10\), and \(\tau_N=1\) points.
Which potential outcome is unobserved for always-takers under the encouragement design, and which is unobserved for never-takers?