Dimension Reduction On Partially Nonlinear Index Models With Applications To Medical Costs

Research on medical cost data has focued in health economics field.The difficulty of medical cost data is how to deal with the complex relationship between medical data.At the same time we found the medical cost data is often skewed,heteroscedastic,non-normal,which increases the difficulty of researching on medical cost statistics.In addition,due to the development of science and technology,we are now in the era of big data,medical cost data always has high dimensional structure.Many existed models are only applicable to deal with low dimensional covariates.When the covariates are high-dimensional,owing to the "Curse of dimensionality",many models are no longer applicable.Therefore,this paper proposes a new model which is more flexible to deal with medical cost data containing higher dimensional covariates.The model proposed in this paper is Partially Nonlinear Index Model.This model assumes that covariates are high-dimensional,meanwhile the model can also analysis some covariates with nonlinear effects.Then this paper presents two kinds of estimation methods in order to estimate the Partially Nonlinear Index Model.One method is to consider the use of sufficient dimension reduction combined with the average of surface modified regression method.Firstly,we use partial sufficient dimension reduction to obtain the basis of the partial central subspace and its structural dimension.Then we improve the average regression method(Chen et al.(2011))in order to fit the multi-index model.Finally,using the average surface modified nonparametric estimation to deal with the nonparametric part of Partially Nonlinear Index Model,thus we complete the reduction of Partially Nonlinear Index Model and the estimation of the nonparametric function.The second method is using the average of MAVE(Xia(2002,2008)),which can simultaneous estimtate the part of parametric and nonparametric.Considering the partial sufficient dimension may have stricter requirement for the structure of X,the new method—the expanded MAVE is more adaptive,can effectively avoid the curse of dimensionality problem.In addition,this paper also shows the corresponding numerical simulation for the above two kinds of methods,and prove the asymptotic properties.Through numericalsimulation,we found that the two methods have certain superiority in dealing with high dimensional covariates.And the calculation of the first method is more convenient,the operation speed is faster;but from the operation applicable scope,the second method –the expanded MAVE is better to handle the relevant data.Furthermore,we use the Partially Nonlinear Index Model proposed in this paper to analyze the medical cost data(MPES).Firstly,we found that hospitalization has the most impact on the medical expenses.In addition,death,cardiovascular diseases,respiratory diseases,cancer and other diseases also have a greater impact on medical expenses;Secondly,the age has a stage impact on health cost.Concretely,from the beginning,with the increase of age,the medical costs are declining,and then in 67 to 69 years old,the medical cost has a larger increase;then between 69 to 73,medical expenses will increase with age fluctuates within a certain range,then to reach a small peak near 74;after the decline,in 75 to 80 years old with age growth fluctuation near to increase;and then decreased in 81 to 82 years old near the medical expenses,and then the rapid rise.In conclusion,using the Partially Nonlinear Index Model proposed in this paper and the corresponding estimation method,we can effectively analysis the medical cost data.