## Statistical analysis with missing survival data at random

Grant Data

**Project Title**

Statistical analysis with missing survival data at random

**Principal Investigator**

Dr Wang, Qihua
(Principal investigator)

**Duration**

24

**Start Date**

2006-09-01

**Completion Date**

2008-08-31

**Amount**

250000

**Conference Title**

**Presentation Title**

**Keywords**

statistical, missing survival data

**Discipline**

Applied Mathematics

**Sponsor**

RGC General Research Fund (GRF)

**HKU Project Code**

HKU 7050/06P

**Grant Type**

General Research Fund (GRF)

**Funding Year**

2006/2007

**Status**

Completed

**Objectives**

(1) Missing survival data arises in a number of applied fields such as medical study,biology,public health, epidemiology,clinic trial and so on. Statistical inference tools and approaches for various models and statistical characteristics will be developed under random censorship when some data are missing at random. We consider partial linear models, semiparametric transformation models, proportional hazard models with a partial linear link function when covariables are missing at random. Also, we consider the problem of estimating a survival function, hazard function and parameters in semiparametric transformation models when censoring indicators are missing at random. The project has the following objectives. (2) 1.1) Consider the partial linear model with the covariables missing at random when response variables are censored. A model calibration approach and a weighting approach will be developed to define the estimators of the parametric part and nonparametric part in the partial linear model,respectively. I intend to prove that the estimators for the parametric part are asymptotically normal and the estimators of nonparametric part converge with an optimal convergent rate. It is worth noting that linear or semiparametric partial linear models are useful alternatives to the commonly used Cox model when the proportional hazards assumption is not valid. However, no method is available in the literature for checking the linearity assumption of a linear model with right censored data. The partial linear model is a natural compromise between the linear model and the fully nonparametric model. It allows only some of the predictors to be modeled linearly, with others being modeled nonparametrically. It is a dimension reduction model compared to the fully nonparametric model. In practice, the partially linear model has been applied to many fields such as biometrics, medical study, economics and so on. Hence, the work will be of practical significance.(3)1.2). Semiparametric transformation models will be considered when the covariables are missing at random and the life variables are censored. An inverse probability weighted estimating equation approach will be developed for estimating of regression parameters.The estimation procedure of the survival probability at given covariate level also will be provided. Asymptotic properties for the suggested estimators will be investigated. This includes studying the consistency of the estimators and asymptotic distribution. This class of regression models includes the proportional hazards and proportional odds models as two special examples. The research can be applied to medical study and clinic trails. (4)1.3) Consider the extended proportional hazard models with a partial linear function When some components of covariable are missing at random, I intend to develop some methods for estimating the parameter vector and nonparametric part under random censorship. One of these approaches that I will develop is a local-likelihood score equation method by combining a weighted local partial likelihood and a score approach to define the estimators of the parameter vector and the nonparametric part. I intend to investigate their asymptotic properties including strong consistency and asymptotic normality etc. This model is an important extension of the Cox model and the proportional hazard model considered by Fan and Kong (1997, Ann. Statist.). The proposed procedures for this model can be applied to study of some practical problems such as survival analysis, medical study and biometrics. The work will have a practical impact. (5) 1.4) I will develop methods for estimating a survival function and a hazard function with censoring indicators missing at random. e.g., we will define several asymptotically efficient PL estimators. I intend to prove all the estimators to be strongly uniformly consistent, weakly converge to a Gaussian process and be asymptotically efficient. (6) 1.5) I will develop a calibrated estimation equation approach for estimating the parameters in the semiparametric transformation models with censoring indicators missing at random. I intend to give the asymptotic distribution of the estimators.(7) The project deals with important statistical inference problems in survival analysis when data are missing at random. This work will make original contributions to the foundations, theory development and methodology in survival analysis. If successful, it would have a significant and broad impact on the theory, methodology and application of survival analysis, and would be of great theoretical and practical interest.