Deriving the HDPS Bias-Reduction Heuristic

Why a Covariate’s Priority Depends on Two Associations

In a high-dimensional feature space, many recorded diagnoses, procedures, or history indicators may be candidates for adjustment. A feature is not automatically important because it predicts treatment, nor because it predicts the outcome. To act as a confounder, it must be related to both treatment assignment and outcome, in a way that can contribute to distortion of the treatment contrast. The HDPS prioritization heuristic combines these two kinds of association to rank candidate covariates for further consideration.

The derivation below isolates one binary, pretreatment covariate at a time. It shows exactly what the score approximates under a simple additive model. It does not establish that every high-ranked feature is a true confounder, or that a low-ranked feature is safe to omit. In particular, a ranking score is a screening heuristic, not a proof of causal identification.

Assumptions and the Bias Decomposition

To interpret the decomposition as causal bias, use a clearly stated model and identification assumptions. Suppose consistency holds, there is no interference between individuals, and conditional exchangeability holds given \(U\): \(Y(a)\perp A\mid U\) for \(a=0,1\). Suppose also that treatment positivity holds, \(0<P(A=a\mid U=u)<1\), for each relevant \(a,u\). Finally, assume a constant additive treatment effect and an additive conditional mean:

Under consistency and conditional exchangeability, the observed conditional mean identifies the potential-outcome mean given \(U\). The model therefore implies \(E[Y(1)-Y(0)\mid U]=\tau\), so the average treatment effect is \(\tau\). The positivity assumption ensures both treatment states are represented within each relevant level of \(U\), rather than requiring extrapolation from an absent group.

Now take the observed mean in each treatment group. Since \(U\) is binary, \(E[U\mid A=a]=P(U=1\mid A=a)=p_a\). Thus:

So the crude mean contrast differs from the causal effect \(\tau\) by \(\gamma\Delta_U\), under the assumptions just given. Its absolute bias contribution is \(S_U=|\gamma\Delta_U|\). The two factors have a direct interpretation: \(|\Delta_U|\) measures how differently common the feature is in the two treatment groups, and \(|\gamma|\) measures how strongly it shifts the mean outcome on the chosen outcome scale.

The decomposition is also a short derivation of why both associations matter. If \(p_1=p_0\), then the covariate contributes zero bias in this model even if \(\gamma\) is large. If \(\gamma=0\), it contributes zero even if the feature is much more prevalent in one treatment group. A feature with moderate values of both factors can nevertheless have a larger score than a feature with only one very large factor.

Using the Score to Rank Candidates

For candidate \(U_j\), the same one-covariate argument gives a score \(S_j=|\Delta_j\gamma_j|\), where \(\Delta_j=P(U_j=1\mid A=1)-P(U_j=1\mid A=0)\), and \(\gamma_j\) is its conditional outcome association on the selected scale. Rank candidates by the magnitude of this product when the aim is to prioritize features with larger approximate contributions under the model. Preserve the signs as well as the magnitudes in analysis: the signed quantity \(\Delta_j\gamma_j\) indicates the direction of the contribution, while the absolute value is used for ranking by size.

The score has the units of the outcome because \(\Delta_j\) is a difference in probabilities and is dimensionless, while \(\gamma_j\) has outcome units. If \(Y\) is binary and the model is an additive risk model, \(\gamma_j\) is a risk difference and \(S_j\) is measured in probability points. The risk model must assign valid probabilities for the covariate and treatment values under consideration. Association measures on other scales, such as odds ratios, are not numerically interchangeable with \(\gamma_j\); a product built from them requires its own approximation and interpretation.

Why This Is a Heuristic, Not a Guarantee

The derivation is exact for the specified one-covariate additive setup, but applying individual scores to a large feature set is a screening approximation. Real outcomes may have interactions, nonlinear relationships, or treatment effects that vary with \(U\). The outcome association estimated from data may also be noisy or distorted by other variables. With multiple candidate features, their contributions need not add as if each were isolated: correlated proxies can represent overlapping information, and conditioning on multiple variables can change conditional associations.

A useful comparison is with rankings based on only one association. Ranking by treatment imbalance alone can elevate a feature that predicts treatment but has little outcome association. Ranking by outcome association alone can elevate a prognostic feature that is nearly equally prevalent in both groups. The product addresses both dimensions, but does not replace substantive causal reasoning. A known confounder may merit adjustment even if its estimated score is modest, particularly when the estimates are imprecise or the feature is measured poorly.

Outcome associations used for prioritization should be interpreted carefully. If estimated from the same data, they are subject to sampling variation and can overstate the importance of chance associations. A proxy may rank highly without being the cause of confounding; conversely, a noisy or rare feature may rank low despite being relevant. The score is also scale-dependent: changing from a risk difference to an odds ratio changes what an outcome association represents. Do not treat values from incompatible scales as comparable.

Diagnostics begin with checking how each factor was estimated and whether its coding and timing make causal sense. Verify that features are measured before treatment, inspect treatment-group prevalence differences and outcome associations separately, and examine uncertainty for rare features. Large absolute differences can indicate limited overlap for a covariate level; a score does not repair positivity problems. Finally, review whether high-ranked features are plausible pretreatment confounders or merely proxies, and whether several features appear to duplicate the same history.

A Practical Reading of a Ranking

This score clarifies why HDPS prioritization combines treatment and outcome information: it approximates how much an individual binary covariate can shift a crude contrast away from the causal effect under a transparent model. Its value is explanatory and organizational. It helps identify candidates deserving attention, but causal validity still depends on the design, measurement, assumptions, and final adjustment strategy.