Estimation of a modified logistic-Weibull model with time-dependent covariates via the generalized method of moments

A modified cure model is proposed in which the survival function of a cured patient is not fixed at one but is given a specific function. This modification has no bearing on parameter estimation; instead, allowing it to reach 0. Using the modified cure model arises when time-dependent covariates influence survival times. In such cases, estimators become inconsistent using the generalized estimating equations with an incorrect working correlation matrix. Without increasing excessive estimation and computational time, we propose a novel algorithm using the generalized method of moments with three modified restricted basis matrices to obtain consistent estimators. The simulation involves a two-algorithm comparison. The first algorithm employs the first proposed mean full and original restricted basis matrices for Type I time-dependent covariates, while the second algorithm utilizes the second proposed mean lower triangular and original restricted basis matrices for Type II time-dependent covariates. The simulation results demonstrate that the root of mean square errors of parameter estimators derived from the algorithm with the modified restricted basis matrices are smaller than those obtained via the algorithm with the original restricted basis matrices. Analyzing primary biliary cirrhosis data set illustrates the utility of the proposed methodology.