| Survival data is based on our observational data on the occurrence time of the survival event of interest and covariates.The data is often encountered in many fields such as biology and medicine.The failure time is the occurrence time of the event of interest.The failure time is often not accurately recorded in practice,but it can be known that is within a certain interval.Therefore,we call such data as interval-censored data.In practical applications,we find that the covariates related to the failure time sometimes change over time,and the value of the covariates at different times will have different effects on the survival time of the event at this time.Therefore,it is of practical significance to deal with the missing covariates in the research.This article discusses the Cox model under Case II interval-censored data commonly seen in life,gives the imputation process when the time-dependent covariates are missing,and the estimation method.The main content of this article is as follows:Firstly,aiming at the missing of the time-dependent covariates,this paper proposes a fully conditional definition multiple imputed method based on the complete data likelihood of the model.That is,according to multiple imputation,fully conditional specification is applied to the process of obtaining the imputation model.In this process,this paper selects normal covariates for analysis.Iteratively obtains the final imputation value through three steps of fitting the baseline cumulative hazard function,extracting the initial value from the approximate distribution of the parameters,and calculating the imputed value.The simulation part of this paper gives the corresponding imputation steps under the assumption that the covariates obey the normal distribution,and adopts the other two methods to deal with the missing covariates.Secondly,aiming at the Cox model under Case?interval-censored failure time Data with missing time-dependent covariates,this paper proposes a corresponding parameter estimation method.We use monotonic spline and EM algorithm for calculation.Among them,we use monotonic spline fitting the baseline cumulative hazard function.In step E,we obtain the expectation of each latent variable by setting two levels of Poisson latent variables;in step M,the parameters are solved iteratively by maximizing the expected complete data,.The Q-Q plots is used in the simulation to evaluate the proposed method.Finally,under the assumption of the covariates,the above imputation methods and estimation methods are applied to simulations and examples.The simulation results show that under the same missing at random,when N is 50,100,and 200,the deviation of the imputation method is smaller than that of the complete case method,and the mean square error of the coefficients is also smaller than that of the complete data method.When the missing mechanism is the same and N is 100 and 200,the coverage rate of the 95%confidence interval of the two parameters in the simulation is higher than that of the complete data method.Compared with decision trees imputation,the deviation and mean square error obtained by the method proposed in this paper are lower than the latter,but the confidence interval coverage is higher than the latter.Therefore,it can be considered that the larger the value of N,the better the estimation effect of the proposed imputation method.In addition,when N gets larger and larger,the scattered points of the coefficient estimates are more concentrated around the straight line,therefore,it can be considered that the estimated coefficients approximately obey the normal distribution.Based on this,it can be considered that the estimation method proposed in this paper is effective.In addition,the analysis of the results of the example shows that the 95% confidence interval of the estimated coefficients calculated after the imputation processing is narrower.That is,the imputation method proposed in this paper has achieved certain effects in practical applications. |
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