Mixed Effect Prediction Of Linear And Generalized Linear Mixed Model And Their Applications

Large-scale sample surveys are usually designed to provide reliable estimates of var-ious numerical characteristics of interest for large geographic areas such as provinces,cities.However,for effective planning of education,public health and other social ser-vices,and for apportioning government funds,there is a growing demand to produce similar estimates for smaller geographic areas included in the large areas.Unfortunate-ly,sampling from all areas can be expensive in resources and time,and this leads to the possibility that the sample size of the small areas could be small and even zero.Here,the term "small area" refers to a small geographical region or a demographic group which has a sample size that is too small for delivering direct estimates of adequate precision.This makes it necessary to "borrow strength" from related areas to find indirect estimators based on models,using auxiliary information such as census data or administrative data.Methods based on models are now widely accepted.Mixed(effects)models are particularly suitable for small area estimation because of its flexibility in effectively combing different sources of information and explaining different sources of errors.In the mixed model,the random effects account for the between-area variation that cannot be explained by including auxiliary variables(that is the fixed effects).This paper aims at the prediction of mixed effect in the mixed model.It can be summarized into five chapters as follows.In Chapter 1,we review firstly the background of small area estimation and the mixed model,including linear mixed model and generalized linear mixed model.Then we introduce the two classical methods of mixed effect prediction,Empirical Best Linear Unbiased Prediction for the linear mixed model and Empirical Best Prediction for the generalized linear mixed model.In the end of this chapter,we present a new method of mixed model prediction for continuous data,Classified Mixed Model Prediction proposed by Jiang et al.in[59].The CMMP method thinks that there is a "match" between the classes of the training data and the potential class of the new observations.By identifying the "class" to which the new observations belong,CMMP can make a better prediction.In addition,regardless of whether the new observations actually have match with the training data,CMMP method tries to find a similar one and make better prediction than regression prediction.In Chapter 2,we develop a classified mixed logistic model prediction(CMLMP)method for clustered binary data by extending the method proposed by Jiang et al.(2017)for continuous outcome data.By identifying a class,or cluster,that the new observations belong to,and "borrow strength" from the same class in the training data,we are able to improve the prediction accuracy of probability associated with a mixed effect of new observation over the traditional method of logistic regression and mixed model prediction without matching the class.Furthermore,we develop a new strategy for identifying the class for the new observations by utilizing covariates information,which improves accuracy of the class identification.This leads to the better performance of prediction.In addition,we develop a method of obtaining second-order unbiased estimators of the mean squared prediction errors(MSPEs)for CMLMP,which are used to provide measures of uncertainty.We prove consistency of CMLMP,and demonstrate finite-sample performance of CMLMP via simulation studies.Our results show that the proposed CMLMP method outperforms the traditional methods in terms of predictive performance.Two applications to real data are also discussed.In Chapter 3,a criterion of optimality in discrete random variable prediction is proposed that requires the predictor to have the same type of values as the predicted random variable.For example,if we want to predict a binary variable,the best prediction is its conditional expectation in the sense of minimum mean squared prediction error,whose value usually lies strictly between 0 and 1.But in fact we wish to take the values 0 or 1.In this chapter,the method based on this criterion is called best look-alike prediction(BLAP).In the case of categorical responses,the BLAP method is similar to the Bayesian classifier with a uniform prior,while the former doesn't need prior information.In addition,the BLAP method can be extended to other cases,such as zero-inflated random variable,as well.Then we apply the BLAP method to small area estimation.We consider the Fay-Herriot model with zero-inflated random effects,after predicting zero-inflated random effects,we can obtain the prediction of mixed effect in each area.The performance of BLAP is also presented by two real-data examples.In Chapter 4,we notice that although the original CMMP outperforms signifi-cantly the classical regression prediction,but it does not utilize covariate information in its matching procedure.As a result,the probability of correct match is low,then the original CMMP method doesn't have obvious advantages over other mixed model prediction methods in prediction performance.Now we propose a new CMMP method that incorporates covariates information,which improves the precision of matching and can borrow strength from the more similar class.Hence the performance of CMMP can be further improved.Furthermore,for assessing the uncertainty of CMMP,we devel-op a method of estimating the mean squared prediction error for CMMP.The method leads to a second-order unbiased estimator of the MSPE of CMMP,denoted by Sumca MSPE.Empirical performance of the new CMMP method as well as that of the Sumca MSPE estimator are carefully studied.The results show that the new CMMP method can improve prediction performance substantially,and the Sumca MSPE is close to the true value.Next,we give the theoretical proof of the property of second unbiased for the Sumca MSPE estimatiorn Finally,the new CMMP method and Sumca MSPE are applied to real data.In Chapter 5,general conclusions and the outlook for future work are presented.