Maximum Likelihood Inference Of Meta Regression With Partial Normal Distribution

It is well known that the purpose of normal random effect Meta regression model is to explore the influence of some case characteristics and other covariates on the merger effect.The purpose is to clarify the source of heterogeneity among different studies.However,in asymmetric data modeling,it may violate the hypothesis of normality,and may lead to false-positive results,which can not fully explain all the heterogeneity.Because of its limitations,the relevant scholars have proposed the Meta regression model of partial normal random effect.Its remarkable feature is that bias is introduced into the model,and the random effect items obey the partial normal distribution.Meta regression of partial normal random effect can eliminate the heterogeneity caused by bias in the overall effect,making the overall effect value closer to the actual situation.This is the advantage of Meta regression of random effects with partial normal distribution.As for the parameter estimation of partial normal Meta regression,the relevant scholars have proposed the weighted OLS estimation method,but the estimation method is too simple and does not use more distribution information,and the maximum likelihood estimation method is usually used in the parameter estimation of partial normal distribution.Therefore,for the partial normal Meta regression model,this paper proposes EM iterative algorithm for maximum likelihood estimation,which shows great flexibility in asymmetric data modeling.In addition,we know that under the assumption of Gaussian distribution,maximum likelihood estimation and least squares estimation are equivalent,but here we want to focus on which estimation method is better under the assumption of partial normal distribution.Therefore,for the partial normal Meta-regression model,experiments are conducted to verify the advantages and disadvantages of maximum likelihood estimation and weighted OLS estimation.The implementation process of this study is mainly divided into two parts.The first part is the establishment of the model and parameter estimation.It includes the derivation of partial normal Meta regression model,maximum likelihood estimation of EM algorithm,weighted least squares parameter estimation and maximum likelihood parameter estimation of normal Meta regression.The emphasis is on the derivation of maximum likelihood estimation of partial normal Meta regression and the implementation of EM iterative algorithm.The other part is experiment.Including simulation experiments and empirical research,EM iterative algorithm is mainly used to calculate the maximum likelihood parameter estimation,and according to AIC,BIC criteria and weighted least squares parameter estimation are compared.Simulation experiments are mainly designed from three aspects,including the convergence test of log likelihood function under EM algorithm,the error performance of maximum likelihood estimation under different sample sizes(the average relative error of 100 experiments is used as the standard of estimation accuracy),and the comparison between maximum likelihood estimation and weighted least squares estimation.The simulation results show that for the partial normal Meta regression model,the log likelihood function under EM algorithm is gradually increasing and converging,and with the increase of sample size,the error of parameter estimation is gradually decreasing and finally tends to be stable.According to AIC and BIC criteria,the maximum likelihood estimation is better than the weighted least squares estimation,and the estimation accuracy is higher.The empirical study is based on the BCG clinical experimental data,the results verify the convergence of the log likelihood function,and also verify that the maximum likelihood estimation is more accurate than the weighted least squares estimation.The results of the empirical study are consistent with the results of the simulation experiment.The innovation of this paper lies in the theoretical derivation of maximum likelihood estimation for partial normal Meta regression,and then the EM iterative algorithm is used to solve the maximum likelihood parameter estimation,which solves the problem that the log likelihood function has no explicit analytical solution.At the same time,under the assumption of partial normal distribution,the maximum likelihood estimation method is better than the weighted least squares estimation method.