Research On Proportional Hazards Model With Measurement Error Or Interval Censoring Of Covariates Under Length Deviation Data

In survival analysis,there is a special type of left-truncated data,namely lengthbiased data,whose important feature is the truncation variable follows uniform distribution.When the sampling is length-biased,applying these approaches suitable for left-truncated data will decrease the efficiency of the estimator,thus we need to find the appropriate approach to deal with length-biased data.Cox proportional hazards model has been widely used in survival analysis,which was often used to describe the relationship between covariates and hazards function.Meanwhile,under Cox model,different subjects' hazard rate is proportional.At present,this model also has been widely used in many areas,such as economics,social science and biomedicine.On the other hand,how to correct the biased estimator when covariates are measured with error is a popular topic in statistics,and attracts great attention.In this paper,we will focus on two questions.First,when failure time is lengthbiased,we will work on the estimation for Cox regression model with covariate measurement error.In the first place,we estimate the parameter.On the basis of composite estimation,we correct score function via conditional expectation and get corrected estimating equation.A lot of simulation results have shown that our proposed method is more efficient than regression calibration approach in terms of error correction.In order to illustrate our approach,we apply the proposed method to the ACTG019 data,because of the instrumental measurement error,CD4 count is subject to systemic error which is hard to ignore.And then we prove the corrected estimator is consistent and has an asymptotic normal distribution.In the second place,we propose the estimator for the nonparametric part,namely cumulative hazard function,and the simulation results demonstrate the proposed method is effective in estimation.Second,we take advantage of the corrected method for covariate with measurement error to deal with the problem of parameter estimation for Cox regression model when covariate is interval censored and failure time is length-biased.A lot of studies deal with interval-censored covariate by means of nonparametric method which is using the observed interval to estimate the distribution of covariate.It involves huge and complicated calculating.Taking midpoint to surrogate interval-censored covariate is simple and intuitive,but the estimator is biased since the midpoint is not the true covariate.In this paper,we treat midpoint as covariate with measurement error,then we use the error correction method proposed before to estimate parameter and get the estimator.A lot of simulation results reveal that the estimator got by taking the midpoint of the interval-censored covariate is distinguished biased when the interval length is large,while the corrected estimator has high estimation accuracy.