# A New Non-Linear Conjugate Gradient Algorithm for Destructive Cure Rate Model and a Simulation Study: Illustration with Negative Binomial Competing Risks

## TL;DR

This paper introduces a novel projected non-linear conjugate gradient (PNCG) algorithm for estimating survival models with a cure fraction under competing risks, demonstrating its advantages over EM and other methods through simulations and real data application.

## Contribution

The paper develops a new PNCG algorithm for survival models with destruction in competing risks, offering improved performance over EM and existing optimization techniques.

## Key findings

- PNCG outperforms EM in simulation studies.
- PNCG shows computational advantages over other algorithms.
- Application to melanoma data demonstrates practical utility.

## Abstract

In this paper, we propose a new estimation methodology based on a projected non-linear conjugate gradient (PNCG) algorithm with an efficient line search technique. We develop a general PNCG algorithm for a survival model incorporating a proportion cure under a competing risks setup, where the initial number of competing risks are exposed to elimination after an initial treatment (known as destruction). In the literature, expectation maximization (EM) algorithm has been widely used for such a model to estimate the model parameters. Through an extensive Monte Carlo simulation study, we compare the performance of our proposed PNCG with that of the EM algorithm and show the advantages of our proposed method. Through simulation, we also show the advantages of our proposed methodology over other optimization algorithms (including other conjugate gradient type methods) readily available as R software packages. To show these we assume the initial number of competing risks to follow a negative binomial distribution although our general algorithm allows one to work with any competing risks distribution. Finally, we apply our proposed algorithm to analyze a well-known melanoma data.