Constrained MLE Formulation¶

This page provides the mathematical specification of the Risk Bridge statistical model, following the framework established by Cao et al. (2024) and extended by Wang & Chen (2026).

1. Problem Formulation¶

Let \(X \in \mathbb{R}^p\) denote discrete baseline predictors, \(Y \in \{0, 1\}\) denote the binary outcome of interest, and \(Z \in (0, 1]\) denote an intermediate continuous risk marker.

We partition \(Z\) into \(K+1\) ordered categories \(Z_{\text{cat}} \in \{0, 1, \dots, K\}\) using cutpoints \(0 = c_0 < c_1 < \dots < c_K < c_{K+1} = 1\):

\[ Z_{\text{cat}} = k \iff c_k \le Z < c_{k+1} \]

Target Risk Model¶

The primary goal is to predict risk in the target population using the logistic model:

\[ P(Y = 1 \mid X, Z_{\text{cat}}; \theta) = \frac{1}{1 + \exp\left(-\left(\alpha + X^\top \beta_X + \beta_Z Z_{\text{cat}}\right)\right)} \]

where \(\theta\) collects the unknown parameters.

2. Joint Likelihood on Source Data¶

The source cohort provides observations \((X_i, Y_i, Z_i)\). We factor the joint distribution as:

\[ f(Y, Z \mid X; \theta) = P(Y \mid X, Z_{\text{cat}}; \alpha, \beta) \cdot f(Z \mid X; \gamma, \sigma) \]

Truncated-Lognormal Distribution for \(Z \mid X\)¶

Because \(Z\) is restricted to \((0, 1]\), we model \(Z \mid X\) using a lognormal distribution truncated to the interval \((0, 1]\).

Let \(\tau(X) = \gamma_0 + X^\top \gamma_X\). The conditional density of \(Z \in (0, 1]\) given \(X\) is:

\[ f(z \mid X; \gamma, \sigma) = \frac{\frac{1}{z \sigma \sqrt{2\pi}} \exp\left(-\frac{(\log z - \tau(X))^2}{2\sigma^2}\right)}{\Phi\left(-\frac{\tau(X)}{\sigma}\right)} \]

where \(\Phi(\cdot)\) denotes the standard normal cumulative distribution function.

Negative Log-Likelihood¶

For an analysis sample of \(n\) subjects, the negative log-likelihood \(\ell(\theta)\) is:

\[ \begin{aligned} \ell(\theta) = &-\sum_{i=1}^n \Big[ Y_i \log p_i(\theta) + (1 - Y_i) \log (1 - p_i(\theta)) \Big] \\ &-\sum_{i=1}^n \left[ -\log z_i - \log \sigma - \frac{1}{2} \log(2\pi) - \frac{(\log z_i - \tau(X_i))^2}{2\sigma^2} - \log \Phi\left(-\frac{\tau(X_i)}{\sigma}\right) \right] \end{aligned} \]

The package implements exact closed-form analytic gradients \(\nabla_\theta \ell(\theta)\) in risk_bridge.likelihood.

3. Reference Calibration Constraints¶

Let \(\mathcal{D}_{\text{reference}}\) denote a representative external cohort where \(X\) and \(Y\) are observed.

We fit a base model \(\phi(X) = P(Y=1 \mid X)\) on the reference data and partition the predicted risk range into \(K\) risk strata \(\mathcal{S}_1, \dots, \mathcal{S}_K\) (typically defined by quantiles of \(\phi(X)\)).

For each stratum \(\mathcal{S}_k\), the reference cohort provides an external event rate benchmark:

\[ p_k^* = P_{\text{ext}}(Y = 1 \mid X \in \mathcal{S}_k) \]

Model-Implied Risk¶

Under the joint model parameterized by \(\theta\), the model-implied marginal risk within stratum \(\mathcal{S}_k\) is obtained by integrating over the discrete support of \(X\):

\[ P(Y = 1 \mid X \in \mathcal{S}_k; \theta) = \frac{\sum_{x \in \mathcal{S}_k} P(Y = 1 \mid X = x; \theta) P(X = x)}{\sum_{x \in \mathcal{S}_k} P(X = x)} \]

where the conditional probability marginalizing over \(Z_{\text{cat}}\) is:

\[ P(Y = 1 \mid X = x; \theta) = \sum_{j=0}^K P(Y = 1 \mid X = x, Z_{\text{cat}} = j; \alpha, \beta) \cdot P(Z_{\text{cat}} = j \mid X = x; \gamma, \sigma) \]

Constraint Functions¶

We define the stratum-specific calibration discrepancy function \(g_k(\theta)\) as:

\[ g_k(\theta) = \sum_{x \in \mathcal{S}_k} P(Y = 1 \mid X = x; \theta) P(X = x) - p_k^* \sum_{x \in \mathcal{S}_k} P(X = x) \]

4. Constrained Optimization Problem¶

The constrained Maximum Likelihood Estimator (cMLE) \(\hat{\theta}_{\text{cMLE}}\) solves:

\[ \begin{aligned} \min_{\theta} \quad & \ell(\theta) \\ \text{subject to} \quad & -\epsilon \le g_k(\theta) \le \epsilon, \quad k = 1, \dots, K \end{aligned} \]

where \(\epsilon > 0\) is the user-specified calibration tolerance (--calibration-tolerance, default 0.02).

The package implements exact analytic constraint Jacobians \(J_{kj} = \frac{\partial g_k}{\partial \theta_j}\) in risk_bridge.constraints, ensuring rapid, numerically stable convergence in non-linear interior-point algorithms.