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MCMC Estimation for Random Effect Modelling – The MLwiN experience Dr William J. Browne School of Mathematical Sciences University of Nottingham

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Contents Random effect modelling, MCMC and MLwiN. Methods comparison – Guatemalan child health example. Extendibility of MCMC algorithms: Cross classified and multiple membership models. Artificial insemination and Danish chicken examples. Further Extensions.

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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 schools. 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.

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MLwiN Software package designed specifically for fitting multilevel models. Developed by a team led by Harvey Goldstein and Jon Rasbash at the Institute of 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 IOE.

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Estimation Methods for Multilevel Models Due to additional random effects no simple matrix formulae exist for finding estimates in multilevel models. Two alternative approaches exist: 1.Iterative algorithms e.g. IGLS, RIGLS, EM in HLM that alternate between estimating fixed and random effects until convergence. Can produce ML and REML estimates. 2.Simulation-based Bayesian methods e.g. MCMC that attempt to draw samples from the posterior distribution of the model.

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MCMC Algorithm Consider the 2-level model MCMC algorithms work in a Bayesian framework and so we need to add prior distributions for the unknown parameters. Here there are 4 sets of unknown parameters: We will add prior distributions

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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 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.

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MCMC Algorithm (3) We next sample from to get, then to get, then finally To get. We have then updated all of the unknowns in the model. The process is then simply repeated many times, each time using the previously generated parameter values to generate the next set

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Burn-in and estimates Burn-in: It is general practice to throw away the first n values to allow the Markov chain to approach its equilibrium distribution namely 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 deviations and other summary measures can also be obtained from the chains.

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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 Hastings sampling and Adaptive Rejection sampling are possible. Numerical Quadrature can give ML estimates but is not without problems.

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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 produce. 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.

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The Guatemalan Child Health dataset. This consists of a subsample of 2,449 respondents from the 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 modern (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 simulated 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!

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The Guatemalan Child Health dataset. Goldstein and Rasbash (1996) consider the same problem but use the PQL method. They show that the results produced by PQL 2 nd order estimation are far better than for MQL but still biased. The model in this situation is In this formulation i,j and k index the level 1, 2 and 3 units respectively. The variables x 1,x 2 and x 3 are composite scales at each level because the original model contained many covariates at each level. Browne and Draper (2004) considered the hybrid Metropolis-Gibbs method in MLwiN and two possible variance priors (Gamma -1 (ε,ε) and Uniform.

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Simulation Results The following gives point estimates (MCSE) for 4 methods and 500 simulated datasets. Parameter (True) MQL1PQL2GammaUniform β 0 (0.65)0.474 (0.01)0.612 (0.01)0.638 (0.01)0.655 (0.01) β 1 (1.00)0.741 (0.01)0.945 (0.01)0.991 (0.01)1.015 (0.01) β 2 (1.00)0.753 (0.01)0.958 (0.01)1.006 (0.01)1.031 (0.01) β 3 (1.00)0.727 (0.01)0.942 (0.01)0.982 (0.01)1.007 (0.01) σ 2 v (1.00)0.550 (0.01)0.888 (0.01)1.023 (0.01)1.108 (0.01) σ 2 u (1.00)0.026 (0.01)0.568 (0.01)0.964 (0.02)1.130 (0.02)

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Simulation Results The following gives interval coverage probabilities (90%/95%) for 4 methods and 500 simulated datasets. Parameter (True) MQL1PQL2GammaUniform β 0 (0.65)67.6/76.886.2/92.086.8/93.288.6/93.6 β 1 (1.00)56.2/68.690.4/96.292.8/96.492.2/96.4 β 2 (1.00)13.2/17.684.6/90.888.4/92.688.6/92.8 β 3 (1.00)59.0/69.685.2/89.886.2/92.288.6/93.6 σ 2 v (1.00)0.6/2.470.2/77.689.4/94.487.8/92.2 σ 2 u (1.00)0.0/0.021.2/26.884.2/88.688.0/93.0

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Summary of simulations The Bayesian approach yields excellent bias and coverage results. For the fixed effects, MQL performs badly but the other 3 methods all do well. For the random effects, MQL and PQL both perform badly but MCMC with both priors is much better. Note that this is an extreme scenario with small levels 1 in level 2 yet high level 2 variance and in other examples MQL/PQL will not be so bad.

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Extension 1: Cross-classified models For example, schools by neighbourhoods. Schools will draw pupils from many different neighbourhoods and the pupils of a neighbourhood will go to several schools. No pure hierarchy can be found and pupils are said to be contained within a cross-classification of schools by neighbourhoods : nbhd 1nbhd 2Nbhd 3 School 1xxx School 2xx School 3xxx School 4xxxx School S1 S2 S3 S4 Pupil P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 Nbhd N1 N2 N3

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Notation With hierarchical models we use a subscript notation that has one subscript per level and nesting is implied reading from the left. For example, subscript pattern ijk denotes the ith level 1 unit within the jth level 2 unit within the kth level 3 unit. If models become cross-classified we use the term classification instead of level. With notation that has one subscript per classification, that captures the relationship between classifications, notation can become very cumbersome. We propose an alternative notation introduced in Browne et al. (2001) that only has a single subscript no matter how many classifications are in the model.

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Single subscript notation School S1 S2 S3 S4 Pupil P1 P2 P3 P4 P5 P6 P7 P8 P9 P10 P11 P12 Nbhd N1 N2 N3 inbhd(i)sch(i) 111 221 311 422 512 622 723 833 934 1024 1134 1234 We write the model as Where classification 2 is neighbourhood and classification 3 is school. Classification 1 always corresponds to the classification at which the response measurements are made, in this case patients. For pupils 1 and 11 equation (1) becomes:

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Classification diagrams School Pupil Neighbourhood School Pupil Neighbourhood Nested structure where schools are contained within neighbourhoods Cross-classified structure where pupils from a school come from many neighbourhoods and pupils from a neighbourhood attend several schools. In the single subscript notation we lose information about the relationship(crossed or nested) between classifications. A useful way of conveying this information is with the classification diagram. Which has one node per classification and nodes linked by arrows have a nested relationship and unlinked nodes have a crossed relationship.

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Example : Artificial insemination by donor 1901 women 279 donors 1328 donations 12100 ovulatory cycles response is whether conception occurs in a given cycle In terms of a unit diagram: Donor Woman Cycle Donation Or a classification diagram:

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Model for artificial insemination data We can write the model as ParameterDescriptionEstimate(se) intercept-4.04(2.30) azoospermia *0.22(0.11) semen quality0.19(0.03) womens age>35-0.30(0.14) sperm count0.20(0.07) sperm motility0.02(0.06) insemination to early-0.72(0.19) insemination to late-0.27(0.10) women variance1.02(0.21) donation variance0.644(0.21) donor variance0.338(0.07) Results: Note cross-classified models can be fitted in IGLS but are far easier to fit using MCMC estimation.

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Extension 2: Multiple membership models When level 1 units are members of more than one higher level unit we describe a model for such data as a multiple membership model. For example, Pupils change schools/classes and each school/class has an effect on pupil outcomes. Patients are seen by more than one nurse during the course of their treatment.

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Notation Note that nurse(i) now indexes the set of nurses that treat patient i and w (2) i,j is a weighting factor relating patient i to nurse j. For example, with four patients and three nurses, we may have the following weights: n1(j=1)n2(j=2)n3(j=3) p1(i=1)0.50 p2(i=2)100 p3(i=3)00.5 p4(i=4)0.5 0 Here patient 1 was seen by nurse 1 and 3 but not nurse 2 and so on. If we substitute the values of w (2) i,j, i and j. from the table into (2) we get the series of equations :

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Classification diagrams for multiple membership relationships Double arrows indicate a multiple membership relationship between classifications. patient nurse We can mix multiple membership, crossed and hierarchical structures in a single model. patient nurse hospital GP practice Here patients are multiple members of nurses, nurses are nested within hospitals and GP practice is crossed with both nurse and hospital.

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Example involving nesting, crossing and multiple membership – Danish chickens Production hierarchy 10,127 child flocks 725 houses 304 farms Breeding hierarchy 10,127 child flocks 200 parent flocks Child flock House Farm Parent flock As a unit diagram: As a classification diagram:

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Model and results ParameterDescriptionEstimate(se) intercept-2.322(0.213) 1996-1.239(0.162) 1997-1.165(0.187) hatchery 2-1.733(0.255) hatchery 3-0.211(0.252) hatchery 4-1.062(0.388) parent flock variance0.895(0.179) house variance0.208(0.108) farm variance0.927(0.197) Results: Note multiple membership models can be fitted in IGLS and this model/dataset represents roughly the most complex model that the method can handle. Such models are far easier to fit using MCMC estimation.

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Further Extensions / Work in progress 1.Multilevel factor models 2.Response variables at different levels 3.Missing data and multiple imputation 4.ESRC grant: Sample size calculations, MCMC efficiency & Model identifiability 5.Wellcome Fellowship grant for Martin Green

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Multilevel factor analysis modelling In sample surveys there are often many responses for each individual. Techniques like factor analysis are often used to identify underlying latent traits amongst these responses. Multilevel factor analysis allows factor analysis modelling to identify factors at various levels/classifications in the dataset so we can identify shared latent traits as well as individual level traits. Due to the nature of MCMC algorithms by adding a step to allow for multilevel factor models in MLwiN, cross- classified models can also be fitted without any additional programming! See Goldstein and Browne (2002,2005) for more detail.

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Responses at different levels In a medical survey some responses may refer to patients in a hospital while others may refer to the hospital itself. Models that combine these responses can be fitted using the IGLS algorithm in MLwiN and shouldnt pose any problems to MCMC estimation. The Centre for Multilevel modelling in Bristol are investigating such models as part of their LEMMA node in the ESRC research methods program. I am a named collaborator for the Lemma project. They are also looking at MCMC algorithms for latent growth models.

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Missing data and multiple imputation Missing data is proliferate in survey research. One popular approach to dealing with missing data is multiple imputation (Rubin 1987) where several imputed datasets are created and then the model of interest is fitted to each dataset and the estimates combined. Using a multivariate normal response multilevel model to generate the imputations using MCMC in MLwiN is described in chapter 17 of Browne (2003) James Carpenter (LSHTM) has begun work on macros in MLwiN that automate the multiple imputation procedure.

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Sample size calculations Another issue in data collection is how big a sample do we need to collect? Such sample size calculations have simple formulae if we can assume that an independent sample can be generated. If however we wish to account for the data structure in the calculation then things are more complex. One possibility is a simulation-based approach similar to that used in the model comparisons described earlier where many datasets are simulated to look at the power for a fixed sample size. Mousa Golalizadeh Lehi will be joining me in February on an ESRC grant looking at such an approach. A 4 th year MMath. student (Lynda Leese) is looking at the approach for nested models.

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Efficient MCMC algorithms In MLwiN we have tended to use the simplest, most generally applicable MCMC algorithms for multilevel models. For particular models there are many approaches that may improve the performance / mixing of the MCMC algorithm. We will also investigate some of these methods in the ESRC grant. Browne (2004) looked at some reparameterisation methods for cross-classified models in a bird nesting dataset. A second 4 th year MMath. student (Francis Bourchier) is looking at MCMC methods based around the IGLS representation of nested models which are interesting.

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Model Identifiability The final part of the ESRC grant is to look at whether a model is identifiable/estimable given a particular set of data. Cross-classified datasets where there are few level 1 units per higher level unit can result in each observation being factored into several random effects with very few observations being used to estimate each random effect. We are interested in establishing whether we can really estimate all parameters in such models. An example where we cant would be a dataset of patients who are attended by doctors in wards. Now if there is only one doctor per ward and likewise one ward per doctor then we cannot tease out doctor and ward effects. Again this work was motivated by a bird nesting dataset.

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Wellcome Fellowship Martin Green has been successful in obtaining 4 years of funding from Wellcome to come and work with me. The project is entitled Use of Bayesian statistical methods to investigate farm management strategies, cow traits and decision-making in the prevention of clinical and sub-clinical mastitis in dairy cows. Martin is a qualified veterinary and a specialist farm animal surgeon. He completed a PhD in 2004 at the University of Warwick veterinary epidemiology and used MCMC to fit binary response multilevel models in both MLwiN and WinBUGS to look at various factors that affect clinical mastitis in dairy cows.

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Wellcome Fellowship In the 4 years we are aiming to analyse a huge dataset that Martin has been collecting in a Milk Development Council grant. In particular we will look at how farm management practices, environmental conditions and cow characteristics influence the risk of mastitis during the dry period. We will hope to get interesting applied results and also some novel statistical methodology from the grant. MCMC will again play a big part. Martin has been appointed as a professor in the new vet school and will be working there 1 day a fortnight during the grant before moving there full time after the four years.

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Conclusions In this talk we have attempted to show the flexibility of MCMC methods in attempting to match the complexity of data structures found in real problems. We have also contrasted the methods with the IGLS algorithm. Although we have used MLwiN, all the models considered here could also be fitted in WinBUGS. WinBUGS offers even more flexibility in model choice for complex data structures however it is typically slower in fitting models that can also be fitted in MLwiN. Genstat and the GLLAMM package in Stata also deserve mention as alternatives in the ML/REML world for the models considered. Slides available at http://www.maths.nottingham.ac.uk/personal/pmzwjb/billt alk.html http://www.maths.nottingham.ac.uk/personal/pmzwjb/billt alk.html

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References Browne, W.J. (2003). MCMC Estimation in MLwiN. London: Institute of Education, University of London. Browne, W.J. (2004). An illustration of the use of reparameterisation methods for improving MCMC efficiency in crossed random effect models. Multilevel Modelling Newsletter 16 (1): 13-25 Browne, W.J. and Draper, D. (2004). A Comparison of Bayesian and Likelihood-based methods for fitting multilevel models. University of Nottingham Research Report 04-01 Browne, W.J., Goldstein, H. and Rasbash, J. (2001). Multiple membership multiple classification (MMMC) models. Statistical Modelling 1: 103-124.

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References Goldstein, H. and Browne, W. J. (2002). Multilevel factor analysis modelling using Markov Chain Monte Carlo (MCMC) estimation. In Marcoulides and Moustaki (Eds.), Latent Variable and Latent Structure Models. p 225-243. Lawrence Erlbaum, New Jersey. Goldstein, H. and Browne, W.J. (2005). Multilevel Factor Analysis Models for Continuous and Discrete Data. In Maydeu-Olivares, A and McArdle, J.J. (Eds.), Contemporary psychometrics: a festschrift for Roderick P. McDonald, p 453-475. Lawrence Erlbaum, New Jersey. Goldstein, H. and Rasbash, J. (1996). Improved approximations for multilevel models with binary responses. Journal of the Royal Statistical Society, A. 159: 505-13. Rodriguez, G. and Goldman, N. (1995). An assessment of estimation procedures for multilevel models with binary responses. Journal of the Royal Statistical Society, Series A, 158, 73-89. Rubin, D.B. (1987). Multiple Imputation for Nonresponse in Surveys. New York: J. Wiley and Sons.

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