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1、MCMC Estimation for Random Effect Modelling The MLwiN experienceDr William J. Browne School of Mathematical Sciences University of NottinghamContents Random effect modelling, MCMC and MLwiN. Methods comparison Guatemalan child health example. Extendibility of MCMC algorithms: Cross classified and mu
2、ltiple membership models. Artificial insemination and Danish chicken examples. Further Extensions.Random effect models Models that account for the underlying structure in the dataset. Originally developed for nested structures (multilevel models), for example in education, pupils nested within schoo
3、ls. An extension of linear modelling with the inclusion of random effects. A typical 2-level model is Here i indexes pupils and j indexes schools.MLwiN Software package designed specifically for fitting multilevel models. Developed by a team led by Harvey Goldstein and Jon Rasbash at the Institute o
4、f Education in London over past 15 years or so. Earlier incarnations ML2, ML3, MLN. Originally contained classical IGLS estimation methods for fitting models. MLwiN launched in 1998 also included MCMC estimation. My role in team was as developer of MCMC functionality in MLwiN during 4.5 years at the
5、 IOE.Estimation Methods for Multilevel ModelsDue to additional random effects no simple matrix formulae exist for finding estimates in multilevel models. Two alternative approaches exist: Iterative algorithms e.g. IGLS, RIGLS, EM in HLM that alternate between estimating fixed and random effects unti
6、l convergence. Can produce ML and REML estimates. Simulation-based Bayesian methods e.g. MCMC that attempt to draw samples from the posterior distribution of the model.MCMC Algorithm Consider the 2-level model MCMC algorithms work in a Bayesian framework and so we need to add prior distributions for
7、 the unknown parameters. Here there are 4 sets of unknown parameters: We will add prior distributions MCMC Algorithm (2)The algorithm for this model then involves simulating in turn from the 4 sets of conditional distributions. Such an algorithm is known as Gibbs Sampling. MLwiN uses Gibbs sampling
8、for all normal response models. Firstly we set starting values for each group of unknown parameters, Then sample from the following conditional distributions, firstly To get .MCMC Algorithm (3)We next sample fromto get , thento get , then finallyTo get . We have then updated all of the unknowns in t
9、he model. The process is then simply repeated many times, each time using the previously generated parameter values to generate the next setBurn-in and estimatesBurn-in: It is general practice to throw away the first n values to allow the Markov chain to approach its equilibrium distribution namely
10、the joint posterior distribution of interest. These iterations are known as the burn-in. Finding Estimates: We continue generating values at the end of the burn-in for another m iterations. These m values are then average to give point estimates of the parameter of interest. Posterior standard devia
11、tions and other summary measures can also be obtained from the chains.Methods for non-normal responses When the response variable is Binomial or Poisson then different algorithms are required. IGLS/RIGLS methods give quasilikelihood estimates e.g. MQL, PQL. MCMC algorithms including Metropolis Hasti
12、ngs sampling and Adaptive Rejection sampling are possible. Numerical Quadrature can give ML estimates but is not without problems. So why use MCMC? Often gives better estimates for non-normal responses. Gives full posterior distribution so interval estimates for derived quantities are easy to produc
13、e. Can easily be extended to more complex problems. Potential downside 1: Prior distributions required for all unknown parameters. Potential downside 2: MCMC estimation is much slower than the IGLS algorithm.The Guatemalan Child Health dataset.This consists of a subsample of 2,449 respondents from t
14、he 1987 National Survey of Maternal and Child Helath, with a 3-level structure of births within mothers within communities. The subsample consists of all women from the chosen communities who had some form of prenatal care during pregnancy. The response variable is whether this prenatal care was mod
15、ern (physician or trained nurse) or not. Rodriguez and Goldman (1995) use the structure of this dataset to consider how well quasi-likelihood methods compare with considering the dataset without the multilevel structure and fitting a standard logistic regression. They perform this by constructing si
16、mulated datasets based on the original structure but with known true values for the fixed effects and variance parameters. They consider the MQL method and show that the estimates of the fixed effects produced by MQL are worse than the estimates produced by standard logistic regression disregarding the multilevel structure!The Guatemalan Child He
