Distinguishing case-mix from context heterogeneity in prognostic regression model synthesis settings

Max Behrens; Janis M. Nolde; Eleni Papakonstantinou; Gabriele Bellerino; Theodoros Evrenoglou; Angelika Rohde; Daiana Stolz; Moritz Hess; Harald Binder

arXiv · 2608.12885

Distinguishing case-mix from context heterogeneity in prognostic regression model synthesis settings

Max Behrens·Janis M. Nolde·Eleni Papakonstantinou·Gabriele Bellerino·Theodoros Evrenoglou·Angelika Rohde·Daiana Stolz·Moritz Hess·Harald Binder

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Abstract

Prognostic regression models often synthesize data from multiple sites, whether within a multi-site study, across federated settings, or in individual participant data meta-analysis. Here, a site is any data source, such as a hospital, registry, trial, or study, and need not be a physical center. Analysts must then decide whether one regression model represents all sites or whether site-specific models are needed. Established measures such as coefficient-level tau^2 quantify heterogeneity but do not distinguish its source. We focus on diagnosing whether coefficient heterogeneity reflects case-mix or site-specific context effects. Case-mix heterogeneity can arise when linear regression terms approximate multivariable non-linear relationships in populations with different covariate distributions. Contextual heterogeneity arises when comparable patients require different regression relationships across sites. We do this by fitting site-specific local regressions in a dimension-reduced space and partitioning the smoothed coefficient surfaces into a cross-site reference and site-specific deviations. An autoencoder and custom loss structure the latent space around local prognostic relationships. We then project this partition onto the outcome scale to derive observation- and site-level summaries. We demonstrate the approach on a COPD trial with two sites. In the three leading latent slope coordinates, coefficient-surface variation was predominantly contextual. The derived observation-level outcome-scale variance partition was case-mix-leading, whereas its between-site aggregation was concentrated in contextual differences rather than case-mix shifts. A permuted-site negative control assesses whether the contextual summary can arise when site labels carry no signal. This diagnostic distinction can inform whether joint or site-specific regression models should be evaluated.

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Max Behrens, Janis M. Nolde, Eleni Papakonstantinou, Gabriele Bellerino, Theodoros Evrenoglou, Angelika Rohde, Daiana Stolz, Moritz Hess, Harald Binder. 2026-08-13. Distinguishing case-mix from context heterogeneity in prognostic regression model synthesis settings. https://arxiv.org/abs/2608.12885

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