简单随机截距模型参数估计与响应预测的最优设计

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简单随机截距模型参数估计与响应预测的最优设计

作者：岳荣先 周晓东

来源：《上海师范大学学报·自然科学版》2013年第05期

Abstract：The paper is concerned with the optimal design problem of estimating linear combinations of the fixed and random effects，and predicting future observations of individual responses in a random intercept model.The variance components in the model are assumed to be known.The design criteria for the predictions are obtained from the mean squared error （MSE） of the estimator and the mean squared prediction error （MSPE） of the predictor.The exact npoint optimal designs and approximate optimal designs are discussed.

Key words：random intercept model； optimal designs； mean squared error 1 Introduction

Mixed models are widely used in various disciplines including agriculture，biology，medicine，physical sciences，education，and social and behavioral sciences.Mixed models

incorporate both fixed effects and random effects.The fixed effects represent the mean values of the parameters in the population of individuals，and the random effects represent the individual deviations in the population.While the statistical analysis for mixed models has been welldeveloped，the optimal design problems of such models have become attractive in the recent two decades. Optimal designs are considered in the presence of random block effects in[1-4].In[5]and[6]optimal designs are considered for linear and quadratic growth mixedeffect models with interclass correlation structure and autocorrelated structure.In[7]and[8]，the maximin Doptimal

designs are discussed for a random intercept and random slope longitudinal mixedeffects models.The Doptimal designs are considered in[9]for linear regression models with a random intercept and first order autoregressive serial correlations.Schmelter[10-11] discussed the optimality of designs for

singlegroup designs for certain mixed models，and then extended his results to groupwise designs for linear mixed models.Debusho and Haines[12] considered the V and Doptimal population designs for the simple linear regression model with a random intercept term.More recently，Cheng et al.[13] considered the optimal designs based on D，G，A，I and Dβoptimality criteria for random coefficient regression models with heteroscedastic errors.

Most research papers on optimal designs in mixed models are for estimating fixed effects and predicting the population mean response.However，the estimation of linear combinations of fixed and random effects and the prediction of individual responses can sometimes be of great interest.Such estimation and prediction problems often arise in many practical applications，such as the estimation of quality index，longitudinal studies，the selection index in quantitative genetics，plant varietal