Why Stratify an Encouragement Study?
An encouragement study assigns an intervention that can change whether people receive a treatment, but does not force treatment uptake. The encouragement’s effect on treatment uptake and its effect on the outcome may differ across baseline covariate groups. A single overall instrumental-variable estimate can therefore hide meaningful differences in first stages and complier effects.
Stratifying on a baseline covariate makes these differences explicit. It also makes the target of any combined estimate easier to state: for example, an average of the stratum-specific complier effects using the covariate distribution of a particular population. Such an average is not automatically the same as the effect among all compliers in that population.
Here, \(\pi_x\) is the population first stage in stratum \(x\), \(\rho_x\) is the population reduced form, and \(\widehat{\tau}_x\) is the sample Wald estimate. With \(D\) binary, the first stage is a difference in treatment proportions. The outcome difference retains the units of \(Y\), so the Wald ratio is measured in outcome units per treatment, conventionally described as the effect of receiving treatment on the outcome.
When the Within-Stratum Wald Ratio Is a Complier Effect
For a causal interpretation, the instrumental-variable assumptions must hold within each stratum used in the analysis. Let \(D(z)\) denote the treatment that a person would receive under encouragement assignment \(z\), and let \(Y(d)\) denote the outcome under treatment status \(d\). The relevant assumptions are as follows.
- Conditional independence: within stratum \(x\), encouragement assignment is independent of the potential treatments and potential outcomes. In a randomized encouragement study, this follows from randomization within strata, provided the stratum is defined using baseline information.
- Exclusion restriction: within the stratum, assignment affects the outcome only through treatment received. Thus the outcome under assignment \(z\) is \(Y(D(z))\).
- Monotonicity: \(D(1)\geq D(0)\) for every person in the stratum. There are no people who would take treatment when not encouraged but refuse it when encouraged.
- Relevance: the first stage is nonzero in the stratum, \(\pi_x\ne 0\). For the usual positive-encouragement interpretation, the first stage should be positive.
- Consistency and no interference: observed treatment and outcome correspond to the potential values under the assignments and treatments received, and one person’s assignment does not change another person’s outcome.
Under these assumptions, define a complier in stratum \(x\) as a person with \(D(1)=1\) and \(D(0)=0\). The stratum-specific complier average causal effect, often called the stratum-specific LATE, is \(E[Y(1)-Y(0)\mid D(1)>D(0),X=x]\). It is an average among people whose treatment uptake is changed by encouragement in that stratum—not necessarily among all people in that stratum.
Why the Ratio Identifies the Stratum-Specific Effect
The identification argument can be seen by comparing encouragement groups within one fixed stratum. Conditional independence ensures that the groups have the same distribution of potential values. Exclusion means any outcome difference operates through treatment. Monotonicity divides people into compliers, always-takers, and never-takers; there are no defiers.
For always-takers, \(D(1)=D(0)=1\); for never-takers, \(D(1)=D(0)=0\). Both groups contribute zero to the differences. Compliers contribute \(Y(1)-Y(0)\) to the outcome difference and one to the treatment difference. Dividing the first displayed expression by the second therefore gives
This is a point-identification result for each stratum with a nonzero first stage. It does not say that the effect is constant across strata or across compliers. If effects vary, each ratio averages effects over the compliers in its own stratum.
Worked Example: Calculating Stratum-Specific Wald Estimates
Suppose a hypothetical randomized encouragement study defines two baseline age strata. Within each stratum, 100 people are assigned encouragement and 100 are not. The outcome is a hypothetical symptom score, where larger values indicate more symptoms. The following are the observed group means and treatment proportions.
| Stratum | Treatment proportion, \(Z=1\) | Treatment proportion, \(Z=0\) | Mean outcome, \(Z=1\) | Mean outcome, \(Z=0\) |
|---|---|---|---|---|
| A | 0.70 | 0.30 | 52 | 40 |
| B | 0.50 | 0.20 | 32 | 26 |
In stratum A, the estimated first stage is \(0.70-0.30=0.40\), or a 40-percentage-point increase in treatment uptake. The estimated reduced form is \(52-40=12\) symptom-score points. Thus
In stratum B, the first stage is \(0.50-0.20=0.30\), and the reduced form is \(32-26=6\) points. The Wald estimate is
If the assumptions hold within both strata, these are estimated average effects of treatment among stratum A’s and stratum B’s compliers, respectively. Because higher scores indicate more symptoms, the positive estimates correspond to higher symptom scores under treatment for those compliers. The group means alone cannot verify the exclusion restriction or monotonicity; those assumptions require design knowledge and substantive justification.
Aggregating Effects for a Stated Target
A combined estimate requires an explicit target. Suppose the target population has stratum proportions \(q_x\), with \(q_x\geq 0\) and \(\sum_x q_x=1\). One useful standardized estimand is the target-population-weighted average of stratum-specific complier effects:
This answers: “What is the average of the stratum-specific complier effects when each stratum receives its share of the target population’s weight?” Estimation replaces each \(\tau_x\) with its estimated Wald ratio and uses the target weights. The weights need not be the proportions in the study sample. If the study deliberately oversampled a stratum, for example, its sample share does not represent the target share.
A different estimand is the treatment effect among all compliers in the target population. The proportion of target-population compliers contributed by stratum \(x\) is proportional to \(q_x\pi_x\), not just \(q_x\). Consequently, when the stratum effects and first stages are identified, the target complier average is
The two targets generally differ because strata may have different encouragement-induced complier shares. The first uses target-population stratum proportions; the second weights strata by their contributions to the target population’s compliers. State which target is being reported rather than calling both simply “the overall effect.”
Worked Example: Standardizing to Two Different Targets
Use the hypothetical estimates from the previous example: \(\widehat{\tau}_A=30\) and \(\widehat{\tau}_B=20\) symptom-score points, with first stages \(\widehat{\pi}_A=0.40\) and \(\widehat{\pi}_B=0.30\). Suppose a hypothetical target population is 25% stratum A and 75% stratum B.
For the target-population-weighted average of stratum effects, use \(q_A=0.25\) and \(q_B=0.75\):
For the target-population average among compliers, first calculate the stratum contributions to the target complier share: \(0.25(0.40)=0.10\) for A and \(0.75(0.30)=0.225\) for B. The total is \(0.325\). The normalized complier weights are \(0.10/0.325\) and \(0.225/0.325\), so
The estimates differ because stratum B has both a larger target-population share and a smaller first stage than stratum A. The first calculation gives each person’s stratum effect the target stratum weight; the second gives more weight to strata contributing more target-population compliers.
Worked Example: A Weak First Stage in One Stratum
Consider a hypothetical three-stratum study. Suppose within-stratum estimates are as follows; the outcome is again a symptom score, and the entries are differences in randomized-group means and treatment proportions.
| Stratum | Estimated first stage \(\widehat{\pi}_x\) | Estimated reduced form \(\widehat{\rho}_x\) | Wald estimate \(\widehat{\rho}_x/\widehat{\pi}_x\) |
|---|---|---|---|
| A | 0.40 | 12 | 30 |
| B | 0.30 | 6 | 20 |
| C | 0.05 | 1.5 | 30 |
Suppose the target proportions are \(q_A=0.20\), \(q_B=0.50\), and \(q_C=0.30\). The target-population-weighted average of the three point estimates is
The arithmetic is correct, but stratum C’s first stage is only 0.05. A Wald estimate divides the reduced form by the first stage, so even modest sampling variation in either difference can produce substantial variation in the ratio when the first stage is small. The point estimate of 30 is not evidence that the effect is precisely estimated. Before presenting the standardized result, inspect each stratum’s first stage and its uncertainty, as well as whether both encouragement assignments are represented adequately in each stratum. If the target includes stratum C but its effect is not credibly identified, the data do not justify silently treating the stratum as irrelevant or replacing its estimate with another stratum’s effect.
Assumption Checks and Practical Interpretation
Stratification does not make an invalid instrument valid. Instead, it clarifies the conditions needed for interpretation within each group. Check that \(X\) is measured before assignment, that the encouragement was randomized within the relevant strata (or that conditional independence has another defensible basis), and that both assignment groups occur in every target stratum. Explain why assignment can affect the outcome only through treatment and why encouragement cannot reduce uptake for any individual. These last two claims are substantive assumptions; differences in observed means cannot establish them.
Relevance should also be assessed stratum by stratum. A substantial pooled first stage does not imply a substantial first stage in every stratum. Conversely, a small stratum may have a noisy estimated first stage even if its population first stage is nonzero. Report the stratum-specific first stages and reduced forms alongside the ratios so readers can see what generates each estimate. A ratio should not be interpreted as an ordinary treatment effect for everyone in its stratum.
For a target-population standardization, specify the target stratum proportions and how they were obtained. If a target stratum has no study observations, or its first stage is zero, its stratum-specific Wald effect is not identified by this calculation. Any estimate for that stratum would need additional assumptions or information; assigning it an estimate by convenience does not solve the identification problem.
Common Mistake
A frequent error is to calculate each stratum’s Wald estimate, average the ratios using the study’s sample proportions, and describe the result as “the overall complier effect.” This can be wrong for two reasons: the study proportions may not match the target population, and averaging stratum effects by target population shares does not generally give the effect among target-population compliers. The correction is to name the target estimand, use its appropriate weights, and report the stratum-specific first stages. For an average among target-population compliers, the weights are proportional to \(q_x\pi_x\); for a target-population-weighted average of stratum effects, they are \(q_x\).
Check Your Understanding
Answer each question using the definitions and assumptions in this tutorial.
- In a stratum, treatment uptake is 0.62 under encouragement and 0.22 without encouragement, while mean outcomes are 18 and 12. Calculate the first stage, reduced form, and Wald estimate, including its outcome units.
- Why does the within-stratum Wald ratio identify an effect among compliers rather than an average treatment effect for everyone in that stratum?
- A target population is 60% stratum A and 40% stratum B. Explain which weights are used for a target-population-weighted average of stratum effects and how they differ from weights for a target complier average.
- Why is randomization within strata not sufficient to verify the exclusion restriction or monotonicity?
- What is a key interpretive concern if one stratum has a very small first stage, even when its calculated Wald ratio is finite?