Package Structure
In predictive modeling, the integrity of the evaluation process depends on the rigorous separation between model development and performance assessment. Data are typically partitioned into a training set and a testing set, an essential split to prevent overfitting and ensure that predictive accuracy serves as a reliable proxy for true generalizability. To maintain this independence, all pre-processing operations are calibrated using the training set and subsequently applied to the testing set, ensuring the latter remains entirely external to the model estimation process. Aligning with these principles, the package organizes the analytical workflow into two distinct functional stages:
the training phase to fit the model,
the testing phase to evaluate performance on unseen samples.
Briefly, the process begins by constructing gene networks from training data to perform variable screening. These selected genes are integrated into network-penalized survival models to derive a prognostic gene signature, a prognostic index, and an optimal cut-off.
The resulting signature is validated on the testing set using survival analysis, statistical significance testing, and performance metrics. To ensure biological relevance, the workflow concludes with pathway and functional enrichment analyses, translating the statistical signature into interpretable biological insights.
The steps are summarized in the following figure.
Regularized Estimator
The package applys on survival model a double penalty. Given a sample of size \(n\), \((y_i,\delta_i,\boldsymbol{x}_i)\) for \(i=1,\dots,n\), the package estimate the unkown parameter vector \(\boldsymbol{\xi}\), where:
- \(\boldsymbol{\xi}=\boldsymbol{\beta}=(\beta_1,\dots,\beta_p)^\top\) for the Cox PH model,
- \(\boldsymbol{\xi} = (\boldsymbol{\beta}^\top, \sigma)^\top\) for the AFT model,
where \(\sigma > 0\) is a scale parameter, using the following regularized definition: \[ \hat{\boldsymbol{\xi}} = \underset{\boldsymbol{\xi}}{\operatorname{argmin}} ~ -\frac{1}{n} ~\ell(\boldsymbol{\xi}) + \lambda \left[\alpha \| \boldsymbol{\xi}\|_1 + (1-\alpha) ~\Omega(\boldsymbol{\xi})\right]. \] Here \(\lambda>0\) is the regularization parameter and \(\alpha \in [0,1]\) is the parameter that balances the contribution of the two penalties, \(\ell(\boldsymbol{\xi})\) the negative (partial) log-likelihood of the selected model. Both parameters are selected optimally through a data-driven procedure (CV-LP).
The first term enforces variable selection through a LASSO penalty, while the second preserves gene–gene correlations by incorporating Laplacian-based penalty \(\Omega(\boldsymbol{\xi})\), ensuring that biologically relevant network structures are maintained:
\[ \Omega(\boldsymbol{\xi})=\boldsymbol{\xi}^\top \mathbf{L} ~\boldsymbol{\xi}=\boldsymbol{\xi}^\top\left(\mathbf{D}-\mathbf{A}\right)\boldsymbol{\xi}=\sum_{1\leq j<k\leq p} |a_{jk}|\left(\xi_j-\mathrm{sign}(a_{jk})~\xi_k\right)^2, \] with the adjacency matrix \(\mathbf{A}\), the degree matrix \(\mathbf{D}\), and \(\mathrm{sign}(a_{jk})\) estimate with Pearson correlation or ridge-penalized approach.
The NetSurvProx algorithms use a proximal gradient
discent approach efficiently, and the proposed survival
network-penalized methods are called COXNet and
AFTNet.