Why Observational Encouragements Need Adjustment
An encouragement may be a reminder, an offer of assistance, or a recommendation intended to increase uptake of a treatment. When encouragement assignment is randomized, differences in outcomes by assignment have a causal interpretation under the usual consistency conditions. In observational settings, people who receive the encouragement may differ systematically from those who do not. For example, clinicians might offer additional support more often to patients with greater baseline need. A raw comparison can then mix the effect of encouragement with pre-existing differences in need.
Measured pre-encouragement covariates can address this problem if, conditional on those covariates, encouragement assignment is independent of the potential outcomes relevant to the question. This is a substantive assumption, not a consequence of fitting a regression model. Even when it is plausible, causal identification also requires adequate overlap: within the covariate patterns represented in the target population, both encouragement levels must be possible.
The potential outcomes \(Y(z)\) define the encouragement effect: the effect of being assigned or exposed to encouragement, including any consequences that assignment itself has for the outcome. This is distinct from the effect of taking up treatment. Conditional independence can identify encouragement effects without an exclusion restriction. An exclusion restriction is needed for the usual instrumental-variable interpretation that encouragement affects the outcome only through treatment uptake.
Identification by Covariate Standardization
Write \(m_z(x)=E(Y\mid Z=z,X=x)\) for the observed mean outcome among people with covariates \(x\) and encouragement level \(z\). Under consistency, conditional independence, and positivity, \(m_z(x)\) equals the mean potential outcome \(E\{Y(z)\mid X=x\}\). Here, consistency means that a person's observed outcome equals the potential outcome corresponding to their observed assignment. Positivity means that both encouragement levels have positive probability at covariate values in the target population.
The outer expectation averages the conditional contrasts over the covariate distribution of the chosen target population. In a finite set of covariate strata \(x\), with target proportions \(w_x\) that sum to one, the standardized encouragement effect is \(\sum_x w_x[m_1(x)-m_0(x)]\). The same reasoning applies to uptake: replace \(Y\) with \(D\) to identify the average effect of encouragement on treatment uptake, often called the first stage.
A Short Derivation
The key step is to connect an observed conditional mean to a potential-outcome mean. By consistency, among people with \(Z=z\), the observed \(Y\) equals \(Y(z)\). Conditional independence then allows the mean of \(Y(z)\) among those with \(Z=z\) to stand in for its mean among everyone with the same \(X=x\):
Subtract the equality for \(z=0\) from the equality for \(z=1\), then average over the target distribution of \(X\). The result is the standardization formula above. Positivity matters because if one assignment level never occurs at a covariate value, the corresponding observed conditional mean is unavailable there; no observed comparison identifies that stratum's contrast without additional assumptions.
Worked Example: Standardizing an Encouragement Effect
Suppose a hypothetical clinic uses an observational encouragement program. Baseline severity is classified as lower or higher. In a target population, half the patients have lower severity and half higher severity. The observed outcome is a symptom score, where a lower score is better. Suppose the conditional mean scores are as follows:
| Baseline severity | Mean score, \(Z=0\) | Mean score, \(Z=1\) | Conditional contrast, \(Z=1-Z=0\) |
|---|---|---|---|
| Lower | 20 | 18 | -2 |
| Higher | 40 | 34 | -6 |
Assume, for this illustration, consistency, conditional independence given baseline severity, and positivity in both severity groups. The target weights are \(w_{\text{lower}}=0.50\) and \(w_{\text{higher}}=0.50\). The standardized contrast is
Under the stated assumptions, encouragement reduces the mean symptom score by 4 points in this target population. This is an assignment effect, not automatically an effect of treatment uptake. The calculation uses the target severity distribution; using the distribution of the observed encouraged group instead could answer a different question.
The table alone cannot verify conditional independence. In an actual analysis, the assumption would require subject-matter justification that the recorded baseline covariates adequately account for differences associated with encouragement assignment.
From Encouragement Effects to an Instrumental-Variable Contrast
The encouragement effect on the outcome is the reduced form, and its effect on uptake is the first stage. If \(D\) is binary, both can be estimated by standardizing their within-\(X\) contrasts to the same target population. When the first stage is nonzero, their ratio is an adjusted Wald contrast. A causal effect of uptake requires more than conditional independence: in particular, the standard complier interpretation also requires exclusion and monotonicity, as well as consistency and the relevant positivity and relevance conditions.
Under exclusion, assignment has no direct effect on \(Y\) apart from its effect through \(D\). Under monotonicity, no one would take up treatment when not encouraged but fail to take it up when encouraged. With a nonzero first stage, the ratio has a complier-effect interpretation for the population represented by the standardized contrasts. Without exclusion, the numerator can include a direct encouragement effect. Without a credible first stage, the ratio is unstable and does not provide useful treatment-effect information.
Worked Example: Adjusted Reduced Form, First Stage, and Ratio
Consider a hypothetical study with two baseline-risk strata. The target population is 60% lower risk and 40% higher risk. Within each stratum, the reduced-form contrast is the mean outcome difference by encouragement; the first stage is the difference in treatment uptake proportions.
| Risk stratum | Reduced form | First stage |
|---|---|---|
| Lower | 0.03 outcome units | 0.10 |
| Higher | 0.08 outcome units | 0.20 |
The standardized reduced form is \(0.60(0.03)+0.40(0.08)=0.018+0.032=0.050\) outcome units. The standardized first stage is \(0.60(0.10)+0.40(0.20)=0.060+0.080=0.140\), or a 14-percentage-point increase in uptake.
The adjusted Wald contrast is therefore
If consistency, conditional independence, positivity, exclusion, monotonicity, and a nonzero first stage are credible, this is interpretable as a complier-average effect for the target population represented by these weights. If the outcome is a clinical score, the estimate is 0.357 score units per uptake; the favorable or unfavorable direction depends on how that score is defined. Conditional independence alone supports the reduced-form encouragement effect, but does not by itself justify this treatment-effect interpretation.
Overlap: Who Can Be Compared?
Define the encouragement propensity \(e(x)=P(Z=1\mid X=x)\). Positivity requires \(0<e(x)<1\) throughout the covariate support of the target population. In practice, values extremely close to zero or one signal limited overlap: comparisons depend on very few people receiving the less common assignment at those covariate values. A fitted model may still return a number, but that number can rely heavily on extrapolation or a small amount of data.
Overlap is not established merely because the data include at least one encouraged and one unencouraged person overall. It is assessed within covariate patterns relevant to assignment and outcome. Useful checks include distributions of estimated propensity scores by assignment, counts in clinically important strata, and whether the target population contains covariate patterns with little or no representation in one assignment group. Diagnostics can reveal practical support problems, but cannot prove the positivity assumption for every relevant covariate pattern.
Worked Example: Recognizing a Practical Overlap Problem
A hypothetical observational program has three baseline-risk groups. Estimated encouragement probabilities are 0.02 for low risk, 0.50 for medium risk, and 0.98 for high risk. Suppose there are 1,000 people in each group. The expected assignment counts in the low-risk group are 20 encouraged and 980 not encouraged; in the high-risk group they are 980 encouraged and 20 not encouraged.
In an inverse-probability weighted analysis, an encouraged low-risk person receives an assignment weight proportional to \(1/e(x)=1/0.02=50\). A not-encouraged high-risk person receives a weight proportional to \(1/[1-e(x)]=1/0.02=50\). Thus, a small number of observations can carry substantial weight. The medium-risk group, with \(e(x)=0.50\), does not have the same extreme weighting.
These probabilities do not violate strict positivity because none is exactly zero or one. They do indicate weak practical overlap, so estimates may be imprecise and sensitive to a few observations. Restricting analysis to a better-supported population can be reasonable, but changes the target population and therefore the estimand. For example, omitting the low- and high-risk groups changes a population-wide encouragement effect into an effect for the medium-risk group; it does not recover the original population effect.
What Adjustment Can—and Cannot—Resolve
Conditional independence is sometimes called no unmeasured confounding of encouragement assignment, conditional on \(X\). It is not testable from observed data alone: the potential outcomes under the unreceived assignment are not observed for any person. Balance comparisons and propensity-score diagnostics can show whether measured covariates remain imbalanced or whether overlap is poor. They cannot establish that unmeasured causes of encouragement and outcome have been controlled.
The covariates used for adjustment should be determined before encouragement assignment. Adjusting for a variable caused by encouragement can block part of an effect or create bias through selection. Measurement error, coarse covariate categories, omitted interactions, or an incorrectly specified outcome or propensity model can also leave residual differences after adjustment. These issues matter even when a regression model includes many covariates.
Common Mistake
A common error is to say, “We adjusted for age and severity, so encouragement is as good as randomized.” The adjustment only supports that conclusion if age and severity, together with the other measured covariates, suffice to make assignment conditionally independent of the relevant potential outcomes. A regression adjustment cannot remove confounding by unmeasured factors, repair absent overlap, or prove exclusion. State the target population, inspect overlap, and present conditional independence as an assumption that needs substantive support.
Check Your Understanding
Answer each question by identifying the relevant estimand or assumption.
- What does conditional independence of \(Z\) and \(Y(z)\) given \(X\) mean in an observational encouragement study?
- A stratum has no unencouraged people. Which identification condition fails there, and why can standardization not directly recover that stratum's contrast?
- Why can conditional independence identify the reduced-form encouragement effect without an exclusion restriction?
- A standardized reduced form is 0.04 outcome units and the standardized first stage is 0.08. Calculate the adjusted Wald contrast and state one additional assumption needed for a complier interpretation.
- Why does restricting analysis to covariate values with good overlap change the causal question being answered?