With known cause of death, competing risk survival methods are applicable in estimating disease-specific survival. Relative survival analysis may be used to estimate disease-specific survival when cause of death is either unknown or subject to misspecification and not reliable for practical usage. This method is popular for population-based cancer survival studies using registry data and does not require cause of death information. The standard estimator is the ratio of all-cause survival in the cancer cohort group to the known expected survival from a healthy reference population. Disease-specific death competes with other causes of mortality, potentially creating dependence among the causes of death. The standard ratio estimate is only valid when death from disease and death from other causes are independent. To relax the independence assumption, we formulate dependence using a copula-based model. Likelihood based method is used to fit a parametric model to the distribution of disease-specific death without cause of death information, where the copula is assumed known and the distribution of other cause of mortality is derived from the reference population. Since the dependence structure for disease related and other-cause mortality is nonidentifiable and unverifiable from the observed data, we propose a sensitivity analysis, where the analysis is conducted across a range of assumed dependence structures. We demonstrate the practical utility of our method through simulation studies and an application to French breast cancer data.