Statistical Inference for Biased Sampling Design under the Proportional Hazards Model with Parameter Constraints

Author:Pan Ying beautiful

Supervisor:Wu Jun


Degree Year:2018





The problem of analysing time to event data arises in a number of applied,such as medicine,biology and so on.A common feature of these practical problems data sets is they contain censored.Censored data arises when an individual’s life length is known to occur only in a certain period of time.Possible censoring schemes are right censored,where all that is known is that the individual is still alive at a given time.Left censoring,when all that is known is that the individual has experienced the event of interest prior to the start of the study.In this paper,we develop inference procedure for regression parameter of the proportional hazards model with constraints under the biased sampling design.The biased sampling design is widely used in large cohort studies to reduce the cost and improve the efficiency.In this paper,we adopt three biased sampling schemes,which are case-cohort design,generalized case-cohort design and outcome-dependent sampling design.Taking prior information of parameters into consideration in modeling process can further raise the efficiency of such studies.To adjust the biased sampling scheme,we develop a weighted estimating equation for the regression parameters for the proposed design,based on the estimating equation,we give a working likelihood function,and then we propose an optimization problem.The Karush-Kuhn-Tucker conditions which is in the convex optimization is applied to derive the asymptotic properties of the constrained estimator.To give a explicit estimator,we use the convexity of the exponential function and the negative logarithm function to construct a surrogate function which is parameter-separate and its Hessian matrix is diagonal,maximizing this surrogate function subject to box constraints is equivalent to separately maximizing several one-dimension concave functions with a lower bound and an upper bound constraint,which has an explicit solution via a median function.Simulation studies show that the estimator which is derived by the biased sampling design is more efficient than that derived by the simple random design design,and also show that the constrained estimator calculated by the proposed minorization-maximization algorithm is more efficient than the unconstrained estimator calculated by the Newton-Raphson algorithm based on the working likelihood function or calculated by the minorizationmaximization algorithm based on the surrogate function,respectively.We caution users to carefully select the constraints before using the proposed method and we provide an example graphic procedures to check the rationality of constraints in the real example.An application to a Wilms tumor study demonstrates the utility of the biased sampling design in practice and also show that such constraints cannot be ignored,otherwise the statistical inference may be misled and an underestimate of the effect may be caused.