# penAFT-package: Fit and tune the a semiparameteric accelerated failure time... In penAFT: Fit the Regularized Gehan Estimator with Elastic Net and Sparse Group Lasso Penalties

## Description

This package contains numerous functions related to the penalized Gehan estimator. In particular, the main functions are for solution path computation, cross-validation, prediction, and coefficient extraction.

## Details

The primary functions are penAFT and penAFT.cv, the latter of which performs cross-validation. In general, both functions fit the penalized Gehan estimator. Given (\log(y_1), x_1, δ_1),…,(\log(y_n), x_n, δ_n) where y_i is the minimum of the survival time and censoring time, x_i is a p-dimensional predictor, and δ_i is the indicator of censoring, penAFT fits the solution path for the argument minimizing

\frac{1}{n^2}∑_{i=1}^n ∑_{j=1}^n δ_i \{ \log(y_i) - \log(y_j) - (x_i - x_j)'β \}^{-} + λ g(β)

where \{a \}^{-} := \max(-a, 0) , λ > 0, and g is either the weighted elastic net penalty or weighted sparse group lasso penalty. The weighted elastic net penalty is defined as

α \| w \circ β\|_1 + \frac{(1-α)}{2}\|β\|_2^2

where w is a set of non-negative weights (which can be specified in the weight.set argument). The weighted sparse group-lasso penalty we consider is

α \| w \circ β\|_1 + (1-α)∑_{l=1}^G v_l\|β_{\mathcal{G}_l}\|_2

where again, w is a set of non-negative weights and v_l are weights applied to each of the G (user-specified) groups.

For a comprehensive description of the algorithm, and more details about rank-based estimation in general, please refer to the referenced manuscript.