Estimation Of Regression Model With Interval-censored Failure Time Data

Objective:In survival analysis, the time of failure event could be exact or right censored. But in the clinical practice, it is commonly see the censored data of interval-censored failure time data. We usually use the Ad hoc to solve this situation, because there were not statistical methods and software for interval censoring, but sometimes, the inferences of Ad hoc may be inefficient. Our study simply explores the problems in traditional survival analysis used for interval censoring, and introduces the regression models and parameter estimation methods for case Ⅱ interval censored failure time data. The advantages and disadvantages of various methods are briefly discussed. In this paper, we use the Cox model combined with ICM algorithm, and the imputations of right censored is estimated by pseudo-observations, respectively. We further make the simulations under the conditions of different sample size and compared the results.Methods:Based on the Weibull distribution, we use Cox model and ICM arithmetic to estimate the maximum likelihood estimations of covariates. The R function ic_sp is from the package of icenReg. The imputations of right censored is estimated by pseudo-observations, and the R function pseudosurv is from the package of pseudo.Results:The sample size was 50,200 and 500,and the censored proportion was 22.0%, 36.0% and 34.0%,respectively.In the first method,the estimators of parametrics based on weibull model were （?）₁=0.496,（?）₂=-0.366；（?）₁=0.680,（?）₂=-0.586； （?）₁=0.620,（?）₂=-0.504；the variance of β were SE（β₁）=0.304,SE（β₂）=0.165 and SE（β₁）=0.15,SE（β₂）=0.090；SE（β₁）=0.096,SE（β₂）=0.057.while the est.imators of parametrics based on Cox-PH model were （?）₁=0.652,（?）₂=-0.469; （?）₁=0.683,（?）₂=-0.538;（?）₁=0.629,（?）₂=-0.511;the variance of β were SE（β₁）=0.484,SE（β₂）=0.242；SE（β₁）=0.181,SE（β₂）=0.094；SE（β₁）=0.090,SE（β₂）=0.057 .The estimators based on the cox model was （?）₁=0.203,β=-0.227; （?）₁=0.641,（?）₂=-0.514;（?）₁=0.545,（?）₂=-0.446 and the variance of β were SE（β₁）=0.302,SE（β₂）=0.172;SE（β₁）=0.158,SE（β₂）=0.087;SE（β₁）=0.090,SE（β₂）=0.054. The estimators of parametrics based on pseudo-observations were （?）₁=0.217,（?）₂=-0.275;（?）₁=0.796,（?）₂=-0.601;（?）₁=0.561,（?）₂=-0.468；the variance of β were SE（β₁）=0.338,SE（β₂）=0.189 SE（β₁）=0.181,SE（β₂）=0.101; SE（β₁）=0.100,SE（β₂）=0.059.In the instance, the result was βweibull=-0.129,βcoxPH=-0.146,βPseudo=-0.153,βCOX=-0.192:the variance of β were SE（βweibull）=0.080,SE（βcoxPH）=0.078,SE（βPseudo）=0.085,SE（βCOX）=0.084 After comparing,we found that the est.imator based on Cox-PH model and weobull model are better,and the result is more stable in the former.Conclusions:This paper used two methods to analyze the survival data contained the interval censored failure time data. The first one is based on the survival of the Weibull distribution function fitting of Cox model, using hybrid ICM algorithm to get maximum likelihood estimators. The other is the imputation of right censored data and further use Pseudo observations method to estimate survival model. Through the simulation and example analysis, we found similar model fitting results of the two methods, the result is ideal, but the first method the results under the condition of different sample size is stable, while the second result is not stable, that may be due to the loss of information in the transform of censoring, cause the mechanism of the censoring is different in the right and interval censored. In addition, this paper only considers two covariates, but in the clinical practice, the number of covariates can be multiple, the presence of covariates could make high-dimensional data. We should further explore the validation and method in the multiple covariates, and use methods to get a better model.