When Does a Pathway Score Preserve a Mediated Effect? Aggregation Geometry, Minimax Adequacy, and Gap Inference for High-Dimensional Mediation

When collapsing high-dimensional mediators into low-dimensional scores for mediation analysis, maintaining the full mediated effect is crucial. The study investigates when a linear score preserves this effect. Exact preservation occurs when the score includes the covariance-adjusted exposure-predictive mediator direction, which serves as the classical sufficient reduction for exposure under a Gaussian inverse-regression model.

The researchers introduce a distribution-free minimax adequacy loss, calculated as the fraction of exposure-relevant mediator information lost by the score. This loss is directly related to the overlap recovered through aggregation. To ensure unbiased estimators of retained and full exposure information, a split-sample inverse-Wishart correction is proposed, providing accurate results in moderate dimensions.

Under spherically symmetric outcome-effect benchmarks, the relative aggregation error follows a Cauchy location-scale profile. For paired influence-function inference, the realized gap between retained and comprehensive exposure information is targeted. Simulations show that paired interval coverage ranges from 93.3% to 97.5%. In an outcome-free analysis of 1,314 smoking-responsive genes from a lung adenocarcinoma audit, retaining promoter and non-promoter means separately reduces pooled loss compared to a dimension-only random-subspace benchmark.

Independent half-sample screening proves unstable, but top-100 and top-250 cross-screen pooled gene-mean losses range from 0.827 to 0.932. The framework distinguishes structural adequacy, benchmark fragility, and outcome-specific discrepancy.