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Cohorts & Problem Formulation

Risk Bridge formalizes the challenge of transferring predictive models when populations differ systematically in their covariate distributions and data availability.

The Three-Cohort Architecture

A Risk Bridge analysis requires three distinct cohorts:

                  ┌─────────────────────────────────────┐
                  │          Reference Cohort           │
                  │  - Representative population survey │
                  │  - Baseline covariates X, outcome Y │
                  │  - Yields calibration benchmarks    │
                  └──────────────────┬──────────────────┘
                                     │
                                     ▼ External calibration constraints
┌────────────────────────────────┐       ┌────────────────────────────────┐
│         Source Cohort          │       │         Target Cohort          │
│ - Clinic / EHR repository      │       │ - Destination deployment site  │
│ - Covariates X, outcome Y,     ├──────►│ - Covariates X, outcome Y      │
│   and rich risk markers Z      │       │ - Evaluates transportability   │
│ - Subject to selection drift   │       └────────────────────────────────┘
└────────────────────────────────┘

1. Target Cohort (\(\mathcal{D}_{\text{target}}\))

  • Role: Represents the intended clinical population where risk predictions will ultimately be applied.
  • Data Available: Baseline covariates \(X\) and binary outcome \(Y\). (Marker \(Z\) is not required in the target cohort).
  • Purpose: Evaluates how well models fitted on source data generalize, measuring discrimination (AUC) and calibration (CITL, slope, Brier score).

2. Source Cohort (\(\mathcal{D}_{\text{source}}\))

  • Role: A rich data repository (such as specialized clinic registry or biobank) where comprehensive clinical testing has been performed.
  • Data Available: Baseline covariates \(X\), binary outcome \(Y\), and intermediate risk marker \(Z\) (both continuous zOrigin and categorical zCat).
  • Challenge: The source cohort is often selected (e.g., patients referred for advanced screening), leading to distorted baseline prevalence and biased parameter estimates if standard unconstrained MLE is used.

3. Reference Cohort (\(\mathcal{D}_{\text{reference}}\))

  • Role: A representative external study (e.g., a broad population cohort or health survey) that accurately reflects the true baseline risk in the broader population.
  • Data Available: Baseline covariates \(X\) and outcome \(Y\).
  • Purpose: A base risk model \(\phi(X) = P(Y=1 \mid X)\) is fit on this cohort, defining risk strata and external target prevalence values used as constraints during source model fitting.

Dual Sampling Paths

To assess and mitigate confounding caused by selection into the source cohort, Risk Bridge implements two parallel sampling paths:

Propensity Score Matched Path (PSM)

  1. Fits a propensity score model \(e(X) = P(\text{Cohort} = \text{source} \mid X)\) pooling target and source individuals.
  2. Performs 1:1 nearest-neighbor matching without replacement between target and source subjects.
  3. Fits ordinary ML and constrained cMLE on this matched source subsample.
  4. Benefit: Balances the covariate distribution between source and target cohorts, isolating the effect of calibration constraints.

Random Sampling Path (RS)

  1. Draws a random sample of size \(n\) directly from the source cohort without matching.
  2. Fits ordinary ML and constrained cMLE on the unadjusted source subsample.
  3. Benefit: Serves as an unadjusted baseline, illustrating the degree of miscalibration present in observational clinic data without propensity adjustments.

Estimator Comparison

Within each path, Risk Bridge evaluates two estimators: - ML Baseline: Unconstrained Maximum Likelihood Estimation on \((Y, Z) \mid X\). Maximize the joint log-likelihood ignoring external calibration constraints. - cMLE: Constrained Maximum Likelihood Estimation. Maximizes the joint log-likelihood subject to calibration constraints derived from the reference cohort across risk intervals.