Why Dimension Reduction Needs Causal Constraints
A high-dimensional feature set can contain thousands of diagnosis, procedure, or medication indicators. Reducing that set can make modeling more manageable, but a compact representation is not automatically a valid basis for causal adjustment. A feature representation may predict treatment well, explain much of the covariate variation, or be clinically convenient while still discarding a variable that confounds the treatment–outcome relationship.
The key question is therefore not simply whether a reduction preserves information overall. It is whether the reduced variables retain enough of the causally relevant information to make treated and untreated units comparable. The earlier material on Treatment Prediction Versus Confounder Control distinguishes those aims: a good treatment predictor is not necessarily a good confounder adjustment set. Here the same distinction guides how codes and domains are compressed.
This definition uses potential outcomes \(Y(1)\) and \(Y(0)\), the outcomes that would be observed under treatment and no treatment. In addition to conditional exchangeability and positivity, causal interpretation requires consistency (the observed outcome equals the potential outcome for the treatment received) and an appropriate no-interference assumption. Dimension reduction cannot repair violations of these assumptions. In particular, it cannot remove unmeasured confounding merely by transforming measured covariates.
Causal structure constrains which information should be retained. A pre-treatment common cause of treatment and outcome is a confounder, so a reduction must preserve its relevant information. A collider can become associated with its causes when conditioned on, so blindly including or combining variables can induce bias. An instrumental feature may strongly predict treatment without being a confounder; including it is not automatically beneficial and can create practical problems such as poor overlap. The purpose is not to compress every feature indiscriminately, but to retain a defensible adjustment set while avoiding variables whose causal roles make adjustment inappropriate.
Three Ways to Reduce Features
Code grouping replaces detailed codes with broader categories, such as combining several diagnosis codes into one disease-family indicator. This can reduce sparsity and make estimates easier to describe. It preserves adjustment only when the grouped information is sufficient for the relevant causal relationships. If the codes within a group differ in severity, timing, or effects on treatment and outcome, a single group indicator can conceal meaningful variation.
Domain-based aggregation uses substantive knowledge to summarize features into constructs such as disease burden or prior healthcare utilization. A domain can be a useful adjustment variable when its definition reflects the causal question and the summary retains the needed information. For example, a count of related diagnoses might be adequate if treatment assignment and potential outcomes depend on those diagnoses only through the count. That is an assumption about the data-generating process, not a consequence of calling the summary a clinical domain.
Unsupervised dimension reduction, such as principal component analysis (PCA), chooses combinations of features to represent their overall variation without using treatment or outcome labels. PCA prioritizes variance, not confounding. A low-variance feature may be essential for adjustment, and a high-variance direction may mostly reflect irrelevant variation. Components can also mix variables with different causal roles, making it difficult to determine what conditioning on a component means.
| Approach | What determines the reduction? | Main causal concern |
|---|---|---|
| Code grouping | Code definitions or a prespecified hierarchy | Important differences among codes can disappear within a broad group |
| Domain aggregation | A substantively defined summary of related features | The summary may fail to retain code-specific causes of treatment or outcome |
| Unsupervised reduction | Patterns of covariate variation, such as PCA variance | High-variance directions need not be the directions that control confounding |
| Causally constrained reduction | Prespecified causal roles and sufficient adjustment information | Requires explicit assumptions and careful checking of what is retained |
A Condition for Safe Reduction
One useful sufficient condition connects a reduced representation to the propensity score. Define \(e(X)=P(A=1\mid X)\), the probability of treatment given the original covariates. If the reduced covariates retain the propensity exactly, so that \(e(X)=e_Z(Z)\) for some function \(e_Z\), then treatment assignment has the same conditional probability given \(X\) as it does given \(Z\). Combined with exchangeability given \(X\), this implies exchangeability given \(Z\).
The argument is short but important. Suppose \(Y(a)\mathbin{\perp} A\mid X\) and \(P(A=1\mid X)=e_Z(Z)\). For either value of \(a\), conditional exchangeability given \(X\) gives \(P(A=1\mid Y(a),X,Z)=P(A=1\mid X,Z)\). Since \(Z\) is a function of \(X\), the right side equals \(e_Z(Z)\). Averaging over \(X\) conditional on \(Y(a)\) and \(Z\) leaves that same value:
Thus \(A\) and \(Y(a)\) are conditionally independent given \(Z\). This result is a sufficient condition, not a recipe for proving that an arbitrary grouping is adequate. In practice, equality of the full propensity given \(X\) and \(Z\) is demanding and cannot generally be verified from a small set of summaries. The broader requirement remains that the reduced covariates support conditional exchangeability and positivity for the target population.
Worked Example: A Broad Code Group Hides a Confounder
Suppose a hypothetical claims dataset records a pre-treatment severity code \(C\), where \(C=1\) indicates severe disease and \(C=0\) indicates mild disease. Analysts group both codes into a single “respiratory disease” indicator \(G\), which equals 1 for everyone in this example. Assume \(C\) is a confounder and the outcome follows \(Y=2A+4C\). The true treatment effect is therefore 2 outcome units for every person.
Suppose the treated group is 75% severe and the untreated group is 25% severe. These proportions are compatible with positive treatment probabilities in both severity strata; take \(P(A=1\mid C=1)=0.75\) and \(P(A=1\mid C=0)=0.25\), with equal prevalence of the two severity levels. The group indicator \(G\) does not distinguish them.
The expected outcome among treated people is \(2+4(0.75)=5\) units. Among untreated people it is \(4(0.25)=1\) unit. The crude contrast, and the contrast after adjustment for the constant \(G\), is therefore \(5-1=4\) units. Its bias relative to the true treatment effect is \(4-2=2\) units. The lost severity information accounts for that bias: \(4(0.75-0.25)=2\) units.
The original severity code is pre-treatment and, by the setup, controls confounding; the assumed treatment probabilities also give positivity. The grouped indicator does not. Grouping would be harmless in this example only if severity differences within the group did not carry relevant information about treatment and potential outcomes. A convenient code hierarchy is not, by itself, evidence for that assumption.
Worked Example: When a Domain Summary Retains Adjustment Information
Consider a hypothetical study with two pre-treatment diagnosis indicators, \(X_1\) and \(X_2\). Define a disease-burden domain \(S=X_1+X_2\), with low burden \(S=0\) and high burden \(S=1\) in this simplified example. Assume that treatment assignment and both potential outcomes depend on the diagnosis pattern only through \(S\). In other words, within each value of \(S\), the two diagnosis indicators carry no additional confounding information. This is the substantive assumption that makes the domain summary plausible.
Suppose treatment probabilities are 0.25 for low burden and 0.75 for high burden, so both treatment levels occur in both strata. Assume conditional exchangeability given \(S\), and that the conditional mean outcomes, in outcome units, are as follows:
| Burden stratum | \(E[Y\mid A=0,S]\) | \(E[Y\mid A=1,S]\) | Target-population proportion |
|---|---|---|---|
| Low, \(S=0\) | 10 | 13 | 0.70 |
| High, \(S=1\) | 20 | 23 | 0.30 |
Under the stated exchangeability assumption, the conditional contrasts estimate the stratum-specific treatment effects: \(13-10=3\) outcome units at low burden and \(23-20=3\) at high burden. Standardizing to the target population gives
The estimated average treatment effect is 3.00 outcome units. This calculation is justified because the example assumes that \(S\) captures all relevant confounding information and has treatment overlap in both strata. If, for instance, \(X_1\) affected treatment while \(X_2\) affected outcomes differently at the same value of \(S\), the summary might not be adequate. The arithmetic alone cannot establish the causal assumptions.
Why Variance Explained Is Not Confounding Controlled
Worked Example: PCA Drops a Low-Variance Confounder
Consider hypothetical covariates \(V\), a common continuous feature with variance 100, and \(C\), a binary pre-treatment confounder with \(P(C=1)=0.10\). Thus \(\operatorname{Var}(C)=0.10(0.90)=0.09\). Assume \(V\) and \(C\) are uncorrelated. An unscaled PCA on these two features ranks the \(V\) direction first because its variance is much larger. The first component accounts for
of the total variance. Keeping only that component discards \(C\), despite its causal importance. Suppose treatment probability is 0.80 when \(C=1\) and 0.20 when \(C=0\), and the outcome model is \(Y=2A+10C+\varepsilon\), where \(\varepsilon\) has conditional mean zero. The true treatment effect is 2 outcome units.
The treated prevalence of \(C=1\) is \(0.10(0.80)/[0.10(0.80)+0.90(0.20)]=0.08/0.26=0.3077\). Among untreated people it is \(0.10(0.20)/[0.10(0.20)+0.90(0.80)]=0.02/0.74=0.02703\). Therefore, the unadjusted treated–untreated outcome contrast is
The contrast is about 2.807 units above the true treatment effect because the treated group has a higher prevalence of the outcome-increasing confounder. Keeping a component that explains 99.91% of the covariate variance did not preserve adjustment. The result depends on the chosen feature scales and the decision to retain only the first component; standardizing the inputs could change the PCA directions, but would not make variance explained a causal criterion.
Practical Checks and Failure Modes
Before using a reduced representation, specify why its construction is compatible with the causal structure. A reduction that is easy to compute should not be treated as an adjustment set until its role is clear. The following checks help expose common weaknesses:
- Check timing and causal role. Use pre-treatment information for baseline adjustment. Do not combine a confounder with a post-treatment variable or collider and assume the resulting domain is safe to condition on.
- Inspect what is merged or discarded. For code groups, identify which codes disappear as separate indicators. For domain summaries, state which distinctions the summary intentionally ignores and why those distinctions should not matter for treatment or potential outcomes.
- Check for mixed causal roles. A component or aggregate can combine confounders, instruments, and colliders. A numerical score does not make those roles interchangeable; conditioning on the score may create a relationship among its inputs.
- Do not use variance or treatment prediction as a substitute for causal justification. A feature can have low variance but strongly affect both treatment and outcome. Conversely, a feature can predict treatment while adding little control of confounding.
- Keep the estimand in view. A domain summary that is adequate for one target population or effect may not be adequate for another if it loses information needed to adjust within the population being analyzed.
These are structural checks, not a guarantee that a proposed representation is valid. Empirical comparisons can reveal whether compression changed model behavior, but observed data alone cannot prove the absence of unmeasured confounding or establish that an aggregation is causally sufficient. Formal balance assessment across many features is a separate task; it should follow, not replace, the reasoning about what the reduced variables mean.
Choosing Among the Approaches
Code grouping is most defensible when categories are defined before outcome analysis, preserve relevant severity and timing distinctions, and correspond to a plausible adjustment variable. Domain aggregation can be more interpretable when clinical or causal knowledge motivates a summary and when the assumptions that make it sufficient are stated explicitly. Unsupervised methods can be useful for describing or compressing large feature matrices, but their objective does not identify confounders. If used for adjustment, they need causal constraints and a reason to believe the retained representation preserves the necessary information.
No reduction method is uniformly superior. Retaining every measured feature is not automatically safe either: inappropriate adjustment can introduce bias, and an unwieldy feature set can create estimation difficulties. The decision is about the causal role of the information being retained, combined, or excluded. A defensible reduction documents its mapping from original features to summaries, the assumed causal structure, and the conditions under which adjustment using the summaries is intended to work.
Check Your Understanding
Answer each question, then compare your reasoning with the solutions.
- A broad code group combines mild and severe disease. What assumption would make the group adequate for adjustment?
- Why does retaining 95% of total variance with PCA not establish that confounding has been controlled?
- In the domain-summary example, what feature of the treatment probabilities supports positivity?
- A component mixes a confounder and a collider. Why is the component not automatically safe to include in an adjustment model?
Solutions. (1) Within the group, the severity distinction must carry no additional information needed to control the treatment–potential-outcome relationship; equivalently, adjustment using the group must still yield conditional exchangeability for the target analysis. (2) PCA ranks directions by covariate variance, not by their contribution to confounding, so a low-variance confounder can be omitted. (3) The probabilities 0.25 and 0.75 are strictly between 0 and 1, so both treatment levels occur in each burden stratum. (4) Conditioning on a function of both variables can condition indirectly on the collider and induce associations among its causes; the component's causal consequences must be assessed rather than inferred from its numerical form.