| The problem of analyzing times to events arises frequently in biomedical and epidemiologic studies.Due to the financial limitations or technical difficulties,it is often that some primary exposure covariates can not be observed completely.However,in many cases,there also exist some other variables can be easily obtained for the almost whole study cohort,these variables are informative about the primary covariates,but provide no additional information for the estimating of the regression parameters when the true primary covariates are available.They are referred to as the auxiliary covariates.How to take advantage of the auxiliary information to improve the efficiency of inference is important to the success of such studies.Correlated multivariate failure time data are usually encountered in real studies,when the ages of disease occurrences are recorded for members of families or the times of disease recurrences of patients which are monitored.The main purpose of this article is to incorporate both the auxiliary information and the intra-cluster correlation into the estimation procedure for the multivariate failure time data with auxiliary covariates.In this article,under the framework of marginal hazard model with different baseline hazard function,we estimate respectively the induced relative risk function nonparametrically when the auxiliary covariates are discrete or continuous,derive the estimated generalized estimating equations which utilizes the auxiliary information and incorporates the intra-cluster correlation,and propose the estimated quadratic inference functions(EQIF).We estimate the unknown parameters in the marginal model by minimizing the EQIF.The EQIF method considers both the auxiliary information and the intra-cluster correlation,and can be easily implemented when the cluster size is large.The proposed estimators are shown to be consistent and asymptotically normal with covariance matrices that can be consistently estimated.Furthermore,the EQIF,which is an objective function,can be used to measure how well the model fits the data.Thus we develop a chi-squared test method for the hypothesis testing for the regression parameters.Some simulation studies show that,when the censoring rate is low or moderate,the proposed estimation methods gain efficiency;when the censoring rate is too high,EQIF methods have almost the same efficiency as the methods which just utilize the auxiliary information but ignore the intra-cluster correlation(independent methods).In addition,the proposed chi-squared test methods are more powerful than the Z-test for the independent methods.Finally,we present an application of the two EQIF methods to the dataset from the Left Ventricular Dysfunction(SOLVD)study,respectively.The results of the analysis show that the proposed EQIF methods are more efficient than the independent methods.However,the EQIF method dealing with the problem of discrete auxiliary covariates would lose some efficiency when the auxiliary covariates are continuous but discretizing them first. |
