Why Predicting Treatment Is Not the Same as Controlling Confounding
High-dimensional propensity-score procedures can surface many features associated with treatment. That is useful information, but it does not by itself say whether a feature should be adjusted for. A feature may predict the outcome, predict treatment, cause both, or predict treatment for reasons unrelated to the outcome. Those roles have different implications for confounding control, precision, and overlap.
The central practical distinction is this: treatment prediction is a statistical association; confounding control is a causal requirement. A feature that strongly predicts treatment may be important to model, but strength of prediction alone does not establish that it removes bias. Conversely, a feature with a modest treatment association may matter if it helps account for a common cause of treatment and outcome.
These labels are not mutually exclusive. A confounder commonly predicts both treatment and outcome; it is therefore both prognostic and treatment-predictive. A prognostic feature need not predict treatment, and a treatment predictor need not predict the outcome. The same observed feature can also be a proxy for an unmeasured cause rather than a cause itself. Its role cannot be determined from its treatment-model coefficient or ranking alone.
The Causal Conditions Behind Adjustment
Write \(Y(1)\) and \(Y(0)\) for the potential outcomes under treatment and no treatment. For a target population, propensity-score adjustment using pre-treatment covariates \(X\) requires, among other design conditions, conditional exchangeability and positivity:
Conditional exchangeability says that, within levels of \(X\), treatment assignment carries no remaining information about the potential outcomes. Positivity says that each covariate pattern relevant to the target population has a positive probability of receiving either treatment. Under consistency, these conditions allow the observed outcome data to identify causal contrasts by adjustment over \(X\). A good prediction of \(A\) is not a substitute for either condition.
A confounder belongs in a sufficient adjustment set because failing to account for it can leave treatment groups with different distributions of potential outcomes. A purely prognostic feature that is independent of treatment given the other adjusted features is not required to remove confounding, although adjusting for a strong outcome predictor can sometimes improve precision. A treatment predictor that is unrelated to potential outcomes, conditional on the appropriate covariates, does not itself control confounding.
A candidate instrument requires stronger causal interpretation than “predicts treatment.” In the usual formulation, it must be relevant to treatment, independent of potential outcomes (possibly conditional on measured covariates), and satisfy exclusion: it has no effect on the outcome except through treatment. These conditions are not established merely because a feature is strongly associated with \(A\), and they can be difficult to verify from observational data.
Worked Example: Classifying Features Before Fitting a Score
Suppose a hypothetical database records patients’ baseline disease severity, a symptom score, distance to a clinic, and a treatment decision. Clinicians use disease severity when deciding treatment, and severity also predicts the outcome. The symptom score predicts the outcome but does not affect treatment decisions after severity is known. Distance predicts treatment because nearby patients can attend the clinic more easily; assume for this illustration that distance has no direct effect on the outcome and is otherwise independent of the potential outcomes.
Disease severity is both prognostic and treatment-predictive. Because it is a common cause of treatment and outcome, it is a confounder and should be represented adequately in a sufficient adjustment set.
Symptom score is prognostic, but under the stated facts it is not a confounder once severity and other relevant causes are controlled. It may still be useful for precision or as a proxy if it partly captures unrecorded severity; that latter claim would require substantive support, not just an outcome association.
Distance to the clinic is a treatment predictor. Under the assumptions given, it is also a candidate instrument, but the label is conditional on those assumptions: if distance affects outcomes through access to other care, for example, exclusion fails. Its treatment association alone does not make it a confounder.
The classification changes what the analyst asks next. For severity, ask whether it has been represented sufficiently. For the symptom score, consider its prognostic value and whether it proxies omitted common causes. For distance, examine whether its causal assumptions are plausible and whether adding it creates practical overlap problems.
Why Instrument-Like Features Can Be Risky in a Propensity Model
A strong treatment predictor can produce propensity scores close to zero or one. This can happen when an instrument-like feature strongly shifts treatment while carrying little information about potential outcomes. Such a feature does not necessarily create bias in an ideal, correctly specified analysis with adequate positivity. However, it can increase the variability of inverse-probability weights, reduce the number of useful comparisons, and make estimates sensitive to model choices or finite-sample imbalance.
For the average treatment effect, the usual inverse-probability weights are \(1/e(X)\) for treated units and \(1/\{1-e(X)\}\) for untreated units, where \(e(X)=P(A=1\mid X)\). If an added feature makes a treated unit’s estimated propensity very small, its treated weight becomes large; if it makes an untreated unit’s propensity close to one, its untreated weight becomes large. This is a mathematical consequence of the weights, not proof that the feature is an instrument or that it must always be excluded.
Worked Example: A Strong Treatment Predictor and Large Weights
Consider a hypothetical binary feature \(Z\), present in half the population. Suppose the fitted treatment probabilities are \(e(Z=0)=0.05\) and \(e(Z=1)=0.95\). The feature is strongly associated with treatment. For illustration, suppose it has no outcome pathway except through treatment and is independent of the potential outcomes, so it acts as an instrument under these assumptions.
For a treated person with \(Z=0\), the unstabilized ATE weight is \(1/e(Z=0)=1/0.05=20\). For an untreated person with \(Z=1\), it is \(1/\{1-e(Z=1)\}=1/(1-0.95)=20\). The corresponding weights for treated people with \(Z=1\) and untreated people with \(Z=0\) are each \(1/0.95\approx1.053\).
Thus, a feature that predicts treatment but not outcomes can produce weights nearly nineteen times larger in some treatment-feature combinations than in others. This warns of a potential precision and stability problem. It does not, by itself, show that the causal estimate is biased, nor does it demonstrate a positivity violation: the probabilities are still strictly between zero and one. In a finite sample, however, few observations may occupy the combinations that receive large weights, making the estimate more sensitive to individual records.
The appropriate response is to examine the feature’s causal role, the target population, and the observed propensity distribution. Do not retain or discard a feature solely because its treatment coefficient is large. Whether restricting the target population or changing the adjustment strategy is defensible depends on the estimand and design, not on a mechanical rule.
A Binary-Treatment Derivation of Residual Confounding
The following calculation shows why leaving a common cause unadjusted can bias a treatment comparison. It uses a binary treatment and binary confounder throughout. Let \(U\) be a binary pre-treatment cause of treatment and outcome, and suppose the conditional mean outcome is \(E\{Y(a)\mid U=u\}=\tau a+\beta u\). Assume the treatment effect is constant, the outcome errors have mean zero within treatment-confounder groups, and \(U\) is not adjusted for. Then the observed treated-control mean difference is
To justify the expression, the conditional mean model gives \(E(Y\mid A=a,U=u)=\tau a+\beta u\). Averaging this equation among treated people yields \(\tau+\beta P(U=1\mid A=1)\); averaging among controls yields \(\beta P(U=1\mid A=0)\). Subtracting gives the stated result. The second term is the bias relative to \(\tau\). It vanishes if \(\beta=0\) or if the prevalence of \(U\) is the same in the two treatment groups; a measured confounder is intended to address precisely this kind of imbalance, subject to the assumptions for adjustment.
Worked Example: Quantifying Confounding From a Binary Feature
Suppose a hypothetical population has \(P(U=1)=0.5\), and the treatment probabilities are \(P(A=1\mid U=1)=0.7\) and \(P(A=1\mid U=0)=0.3\). Let the true constant treatment effect be \(\tau=2\) outcome points, and let \(U\) increase the outcome by \(\beta=5\) points. Assume the conditional mean model and zero-mean errors described above.
The treated proportion with \(U=1\) is \(P(U=1\mid A=1)=0.5(0.7)/\{0.5(0.7)+0.5(0.3)\}=0.35/0.50=0.70\). The untreated proportion with \(U=1\) is \(P(U=1\mid A=0)=0.5(0.3)/\{0.5(0.3)+0.5(0.7)\}=0.15/0.50=0.30\).
Therefore the unadjusted mean difference is \(2+5(0.70-0.30)=2+2=4\) outcome points. The true effect in this model is 2 points, so the unadjusted comparison is biased upward by 2 points because the treated group contains more people with the outcome-predictive confounder. If \(U\) were measured and the stated model and adjustment conditions held, standardizing within \(U\) would remove this particular compositional term; it would not automatically solve confounding by other unmeasured causes.
Practical Feature Priorities and Diagnostics
In a high-dimensional setting, use a causal-role review alongside predictive ranking. A practical priority is to ensure adequate representation of known pre-treatment common causes and credible proxies for them, while considering prognostic covariates that improve outcome adjustment or precision. Strong treatment-only predictors merit scrutiny rather than automatic priority. Their usefulness depends on the design, estimator, target population, and whether they help or hinder the available comparisons.
Confirm that each feature is measured before treatment and determine what process it represents. A post-treatment variable can be affected by treatment and is not a baseline adjustment feature.
For each feature, ask whether it plausibly causes treatment, the outcome, both, or neither; also ask whether it may proxy a missing common cause. Do not infer causal role from a large coefficient alone.
Assess whether adjustment improves control of measured confounding and whether it creates extreme propensities, sparse treatment-feature combinations, or poor comparability. A feature’s inclusion should be justified in relation to the estimand and assumptions.
In practice, warning signs include treatment predictions close to zero or one, a feature that almost determines treatment, and a propensity model whose apparent discrimination improves while the available treated-control comparisons become less credible. These are reasons to investigate, not stand-alone proof of failure. Later balance diagnostics across many features are needed to assess measured balance; balance alone cannot verify exchangeability with respect to unmeasured causes.
| Feature role | What it tells you | Implication for adjustment |
|---|---|---|
| Confounder | Helps explain differences in treatment and potential outcomes | Represent it in a sufficient adjustment set, subject to valid measurement and model specification |
| Prognostic feature only | Predicts outcome but does not cause treatment after relevant covariates | May improve precision; it is not automatically required for confounding control |
| Treatment predictor only | Predicts treatment but not potential outcomes, conditional on relevant covariates | Prediction alone is not a reason to prioritize it; consider effects on overlap and variance |
| Candidate instrument | Predicts treatment and may satisfy independence and exclusion conditions | Do not assume it is a confounder; assess whether propensity adjustment makes comparisons unstable |
Common Mistake and Limits
Common mistake: “This feature has the largest treatment-model coefficient, so it must be the most important confounder and should always be included.” This reasoning confuses the ability to predict treatment with the causal role needed for confounding control. A feature may be an instrument-like predictor, while a weaker treatment predictor may be a genuine common cause. The correction is to assess causal structure and the conditions for identification first, then evaluate model behavior and diagnostics.
These distinctions are not fully resolvable from a high-dimensional feature matrix. Observed associations cannot establish instrument exclusion, rule out unmeasured confounding, or prove that a proxy captures the relevant common causes. Feature selection also involves trade-offs: including a valid prognostic variable may aid precision; adding an instrument-like predictor can increase weight variability; omitting a confounder can leave bias. No universal ranking based only on treatment association settles those issues.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions below.
- A baseline feature predicts treatment but is unrelated to both potential outcomes after conditioning on measured confounders. Is it automatically a confounder?
- In the binary-confounder derivation, what is the unadjusted bias if \(\beta=3\), \(P(U=1\mid A=1)=0.6\), and \(P(U=1\mid A=0)=0.2\)?
- Why can adding a strong candidate instrument to an ATE propensity model increase weight variability?
- Does a feature with \(e(X)=0.05\) establish a positivity violation? What should the analyst conclude from that value alone?
Solutions. (1) No. Under the stated condition it predicts treatment but does not distinguish potential outcomes, so it is not automatically a confounder; consider its causal role and practical effect on the estimator. (2) The bias is \(\beta(0.6-0.2)=3(0.4)=1.2\) outcome units. (3) A treated unit with propensity 0.05 receives weight \(1/0.05=20\), which can make the estimate more sensitive to a small number of observations. (4) No. It is strictly between zero and one, so this value alone does not prove a positivity violation; it is a warning to assess overlap, the target population, and the stability of the intended analysis.