Maximum Likelihood Estimation of Hierarchical Linear Models from Incomplete Data: Random Coefficients, Statistical Interactions, and Measurement Error
针对连续响应和协变量随机缺失的两层模型，提出一种利用自适应高斯-埃尔米特求积近似似然函数的方法，处理随机系数、交互项和多项式项导致的非正态联合分布，并解决吉布斯采样中的条件分布兼容性问题。
–We consider two-level models where a continuous response R and continuous covariates C are assumed missing at random. Inferences based on maximum likelihood or Bayes are routinely made by estimating their joint normal distribution from observed data Robs and Cobs . However, if the model for R given C includes random coefficients, interactions, or polynomial terms, their joint distribution will be nonstandard. We propose a family of unique factorizations involving selected “provisionally known random effects” u such that h(Robs,Cobs|u) is normally distributed and u is a low-dimensional normal random vector; we approximate h(Robs,Cobs)=∫h(Robs,Cobs|u)g(u)du via adaptive Gauss-Hermite quadrature. For polynomial models, the approximation is exact but, in any case, can be made as accurate as required given sufficient computation time. The model incorporates random effects as explanatory variables, reducing bias due to measurement error. By construction, our factorizations solve problems of compatibility among fully conditional distributions that have arisen in Bayesian imputation based on the Gibbs Sampler. We spell out general rules for selecting u, and show that our factorizations can support fully compatible Bayesian methods of imputation using the Gibbs Sampler.