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

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.

• The paper introduces an autoencoder and local regression framework that partitions learned coefficient variation into shared case-mix effects, site-specific context departures, and covariation.
• In the COPD application, contextual differences dominated the leading latent coefficient surfaces by 74–96%, while between-site context explained 18.6% of outcome-scale variation compared with 0.24% for case mix.
• The findings suggest that standardizing populations alone may not resolve heterogeneity, so researchers should evaluate site-specific models or context-aware synthesis while recognizing the method’s sensitivity to representation, hyperparameters, and limited site numbers.

Motivation and problem statement

When prognostic regression models are synthesized across multiple data sources—hospitals, registries, trials, or studies—analysts must decide whether a single pooled model is adequate or whether site-specific models are required. Established heterogeneity measures such as coefficient-level $\tau^2$ quantify the magnitude of between-site disagreement but do not identify its source. The paper addresses this gap by distinguishing two channels of coefficient heterogeneity. Case-mix heterogeneity arises when sites sample different regions of the covariate distribution while the conditional outcome relationship is shared; it can often be resolved through standardization or flexible pooled modeling. Contextual heterogeneity arises when comparable patients require different regression relationships across sites, reflecting site-level factors such as treatment protocols or measurement standards; it can render pooled estimates misleading even after adjustment.

Existing methods flag disagreement without attributing its source: one-stage IPD meta-analysis with random intercepts and slopes captures between-study coefficient variation but does not separate the two components (2608.12885); meta-regression requires pre-specified moderators; location-scale models assume known structure. The proposed framework instead uses sites with individual participant data (IPD) as a diagnostic sample to attribute coefficient variation to position-driven (case-mix) versus site-specific (contextual) sources.

Methodological framework

The method proceeds in three steps. First, an autoencoder maps $p$ predictors to a low-dimensional latent space $\mathbf{z}_i \in \mathbb{R}^D$ ($D \ll p$), because local regression is infeasible in high-dimensional predictor space. Second, site-specific local GLMs are fitted at each query latent position using Gaussian-kernel weights with adaptive site-specific bandwidths, yielding smoothed intercept surfaces $\alpha^{(k)}(\mathbf{z})$ and slope surfaces $\boldsymbol{\beta}^{(k)}(\mathbf{z})$ per site. Third, each surface is partitioned into a weighted cross-site reference ($q_k = 1/C$) and a site-specific contextual departure:

$$\boldsymbol{\beta}^{(c_i)}(\mathbf{z}_i) = \bar{\boldsymbol{\beta}}(\mathbf{z}_i) + \boldsymbol{\beta}^{(c_i),\mathrm{ctx}}(\mathbf{z}_i).$$

A coordinate-wise variance identity decomposes the empirical variance of the own-site slopes into reference-surface variation, contextual departure variation, and their covariance. This coefficient-surface partition is the primary analysis and is structurally closest to meta-analytic coefficient heterogeneity, though defined on learned rather than original coordinates.

Two derived summaries project the partition onto the linear-predictor scale: an observation-level case-mix component $$M_i^{\mathrm{mix}} = \bar{\alpha}(\mathbf{z}_i) + \mathbf{z}_i^T\bar{\boldsymbol{\beta}}(\mathbf{z}_i)$$ and a contextual component $M_i^{\mathrm{ctx}}$. Their exact variance identity gives the observation-level partition, and a law-of-total-variance aggregation yields $p$0, a descriptive between-site analogue—not a DerSimonian–Laird estimator—of coefficient heterogeneity.

The autoencoder is trained jointly with three objectives: reconstruction fidelity, a local prognostic loss rewarding neighborhoods where local regression outperforms an intercept-only null, and a native-site loss retaining site-specific relationships (down-weighted in the first encoder layer to limit over-separation by site labels).

Application to COPD data

The illustration uses the PREVENT trial (two centers treated as sites), with 300 observations, 38 lung-function and physiological predictors, and subsequent SGRQ total score as outcome. Separate 38-predictor site models confirm substantial original-variable coefficient divergence, but cannot attribute it to case-mix versus context.

Latent representation validation. In five site-stratified folds, the full proposed autoencoder achieved held-out $p$1, exceeding PCA-based linear regression ($p$2) and a reconstruction-only standard autoencoder ($p$3), though below raw-predictor random forest ($p$4) and ridge regression ($p$5). Removing both local losses reduced held-out $p$6 from $p$7 to $p$8 while improving reconstruction MSE, supporting the intended prognostic–reconstruction trade-off. Notably, removing the native-site loss increased held-out $p$9 to $\mathbf{z}_i \in \mathbb{R}^D$0; the authors explicitly state that predictive results do not support a claim that the native-site loss improves aggregate prognosis—its role is to structure site-specific coefficient surfaces, which this comparison does not test. The standard-versus-proposed autoencoder contrast is also confounded by differing optimizer settings and training duration.

Coefficient-surface partition. For the three leading latent slope coordinates (jointly accounting for 76.5% of summed coordinate slope variance), context accounts for 96%, 74%, and 80% of coordinate variance respectively. The leading latent coefficient surfaces therefore vary predominantly through site-specific departures at matched positions rather than through position-driven reference variation.

Outcome-scale summaries. The derived observation-level partition is case-mix-leading: across 100 bootstrap replicates, $\mathbf{z}_i \in \mathbb{R}^D$1 contributes $\mathbf{z}_i \in \mathbb{R}^D$2, context $\mathbf{z}_i \in \mathbb{R}^D$3, and covariation $\mathbf{z}_i \in \mathbb{R}^D$4. All 10 encoder seeds were case-mix-leading at the point estimate, though gradient scaling was needed for stability—without it, three of ten seeds were context-leading. The within/between aggregation then reverses the reading: the between-site case-mix share is near zero ($\mathbf{z}_i \in \mathbb{R}^D$5 of $\mathbf{z}_i \in \mathbb{R}^D$6), while the between-site context share is $\mathbf{z}_i \in \mathbb{R}^D$7, concentrated almost entirely in the average site shift rather than within-site spread. A permuted-site negative control supports this contrast: under permuted labels, the between-site context share falls to $\mathbf{z}_i \in \mathbb{R}^D$8 and the across-observation context share drops roughly threefold. A reconstruction-only autoencoder that never sees site labels reproduces the asymmetry (between-site context $\mathbf{z}_i \in \mathbb{R}^D$9 vs. case-mix $D \ll p$0), indicating the pattern reflects data structure rather than site-aware training.

Model-swap diagnostic. As an exploratory check, predicting each site's observations with the other site's model increased observation-level MSE by a mean of 99.7% (SD 40.4%) across 1,000 seed–replicate estimates, concordant with site-conditional differences in the latent-to-outcome mapping, though not independent validation.

Predictor trace-back. SHAP attributions on the encoder, weighted by matched-position slope contrasts, link the three leading contextual dimensions to predictors including VAPP, VCP, ERVP, TLCLP, DLCOVAMP, and VALP. Seven of fifteen dimension–feature instances overlap with the independent original-space top-ten ranking—a partial descriptive concordance only, since both analyses use the same observations.

Interpretation for evidence synthesis

The paper's central finding is that the two levels of summary give different readings: the primary coefficient-space result and the between-site aggregation both indicate site-conditional differences, even though the across-observation outcome-scale partition is case-mix-leading. The synthesis implication follows directly: because between-site case-mix sits at the permuted-label floor while between-site context substantially exceeds it, standardization to a common population would not resolve the observed heterogeneity; site-specific or context-aware synthesis should be evaluated. The small covariation share ($D \ll p$1) further implies that reweighting case-mix alone remains interpretable here, whereas large covariation would make the two channels act through the same latent regions. The framework diagnoses which concern is present but does not itself select a final estimator or borrowing rule; in asymmetric (transport) settings, contextual differences indicate that source relationships may not transport without target-specific recalibration.

Limitations and open questions

The authors concede several constraints at the points where they bear on results. The coefficient-space partition is conditional on the scale and orientation of the learned representation; its slopes are not original-variable coefficients and cannot be compared dimension-by-dimension across independently trained encoders without alignment. Neither level provides formal tests of whether one component exceeds another. Results depend on hyperparameters ($D \ll p$2, loss weights, neighborhood fraction, kernel scale, gradient scaling), and sensitivity across diverse datasets remains unassessed. Empirically, the application uses two sites—the minimum for a between-site contrast—and one clinical dataset, so multi-site contextual patterns and external validation remain open. All analyses are conditional on fixed upstream preprocessing (k-NN imputation including the contemporaneous outcome, correlation pruning, outlier filtering performed once on the full data), so the held-out comparison is not fully leakage-free. Validation summaries are descriptive: fold–seed standard deviations share five test folds, the model-swap bootstrap is exploratory, and the SHAP overlap is not independent confirmation. Practical value further depends on assembling enough IPD sites, which systematic reviews show is often not achieved. Specific open questions include whether a linear encoder with the same composite loss achieves comparable results, whether contextual signatures can be summarized in second-stage meta-regression against site-level characteristics, and how the diagnostic scales to many-site meta-analyses.

Conclusion

This work integrates local regression with outcome-guided representation learning to partition learned coefficient surfaces into a shared position-driven component, site-specific contextual departures, and their covariation, with derived observation- and site-level outcome-scale summaries. In the COPD application, contextual differences dominate the leading latent coefficient surfaces (74–96%), the derived across-observation partition is case-mix-leading, yet the between-site aggregation is context-dominated and well above the permuted-label null. The method makes model-based source attribution explicit when pooled versus site-specific synthesis is being considered, without selecting the final strategy, and its extension beyond the IPD subset requires assumptions about representativeness that remain to be evaluated externally.