Prognostic analysis for patient survival often employs gene expressions obtained from high-throughput screening for tumor tissues from patients. When dealing with survival data, a dependent censoring phenomenon arises, and thus the traditional Cox model may not correctly identify the effect of each gene. A copula-based gene selection model can effectively adjust for dependent censoring, yielding a multi-gene predictor for survival prognosis. However, methods to assess the impact of various types of dependent censoring on the multi-gene predictor have not been developed. In this article, we propose a sensitivity analysis method using the copula-graphic estimator under dependent censoring, and implement relevant methods in the R package “compound.Cox”. The purpose of the proposed method is to investigate the sensitivity of the multi-gene predictor to a variety of dependent censoring mechanisms. In order to make the proposed sensitivity analysis practical, we develop a web application. We apply the proposed method and the web application to a lung cancer dataset. We provide a template file so that developers can modify the template to establish their own web applications.