Semiparametric Inference Of Complier Causal Effect Using Density Ratio Model With Covariates

Recently,causal inference becomes a popular research field.Causal inference is the process of drawing a conclusion about a causal connection based on the conditions of the occurrence of an effect.The main difference between causal inference and inference of association is that the former analyzes the response of the effect variable when the cause is changed.The mostly used method to do causal inference is random experiment.Because random experiment can remove the confounding effect,it is considered to be a gold method.During an experiment,not every unit takes the treatment due to user experience or ethical reasons.This situation raises problems for statistical inference on experiment data.The main interest of an experiment is whether treatment changes units'metrics significantly.In literature,there are several methods can deal with this problem,for example,Instrumental Variable,Per-Protocol and As-Treated methods.However,these methods need very strong assumptions and performs bad if the assumptions are violated.Cheng,Qin and Zhang?2009?^[6]introduced a density ratio model but their method loses efficiency because of ignoring covariates.Thus,it's an interesting question how to induce covariates information and meanwhile keep the advantages of density ratio model.In this paper,we use covariates to do statistical inference on complier average causal effect under density ratio model assumptions.Using Rubin Causal Model,we further as-sume different sub-populations obey density ratio assumptions.Then,we use empirical likelihood methods to formulate our semi-parametric likelihood.We can estimate mod-el parameters and do hypothesis test on complier average causal effect.We proof the asymptotic properties of our estimator and calculate the limiting distribution of our test-ing statistics.Our simulation shows method in this paper performs better than existing methods when covariates are explainable.At last,we give a real example on JOBS II trials data to show implement of our method and then compare result of different meth-ods.Our analysis shows that treatment has little influence on the population with high base depression,which means researcher show find other intervention to help these people reducing depression.