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Using complex random effect models in epidemiology and ecology Dr William J. Browne School of Mathematical Sciences University of Nottingham

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Outline Background to my research, random effect and multilevel models and MCMC estimation. Random effect models for complex data structures including artificial insemination and Danish chicken examples. Multivariate random effect models and great tit nesting behaviour. Efficient MCMC algorithms. Conclusions and future work.

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Background – PhD in Statistics, University of Bath. Applying MCMC methods to multilevel models – Postdoctoral research positions at the Centre for Multilevel Modelling at the Institute of Education, London – Lecturer in Statistics at University of Nottingham Associate professor of Statistics at University of Nottingham Professor in Biostatistics, University of Bristol. Research interests: Multilevel modelling, complex random effect modelling, applied statistics, Bayesian statistics and MCMC estimation.

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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 might index pupils and j index schools. Alternatively in another example i might index cows and j index herds. The important thing is that the model and statistical methods used are the same!

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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, 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. One possible computer program to use for multilevel models which incorporates both approaches is MLwiN.

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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 estimation methods (IGLS) for fitting models. MLwiN launched in 1998 also included MCMC estimation. My role in the team was as developer of the MCMC functionality in MLwiN in my time at Bath and during 4.5 years at the IOE. Note: MLwiN core team relocated to Bristol in 2005.

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MCMC Algorithm Consider the 2-level normal response model MCMC algorithms usually 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) One possible MCMC 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 averaged 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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So why use MCMC? Often gives better (in terms of bias) estimates for non- normal responses (see Browne and Draper, 2006). Gives full posterior distribution so interval estimates for derived quantities are easy to produce. Can easily be extended to more complex problems as we will see next. Potential downside 1: Prior distributions required for all unknown parameters. Potential downside 2: MCMC estimation is much slower than the IGLS algorithm. For more information see my book: MCMC Estimation in MLwiN – Browne (2003).

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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 also 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) 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 pupils. 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 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) azoospermia0.22(0.11) semen quality0.19(0.03) womens age> (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) (0.162) (0.187) hatchery (0.255) hatchery (0.252) hatchery (0.388) parent flock variance0.895(0.179) house variance0.208(0.108) farm variance0.927(0.197) Results: Response is cases of salmonella 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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Random effect modelling of great tit nesting behaviour An extension of cross-classified models to multivariate responses. Collaborative research with Richard Pettifor (Institute of Zoology, London), and Robin McCleery and Ben Sheldon (University of Oxford).

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Wytham woods great tit dataset A longitudinal study of great tits nesting in Wytham Woods, Oxfordshire. 6 responses : 3 continuous & 3 binary. Clutch size, lay date and mean nestling mass. Nest success, male and female survival. Data: 4165 nesting attempts over a period of 34 years. There are 4 higher-level classifications of the data: female parent, male parent, nestbox and year.

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Data background SourceNumber of IDs Median #obs Mean #obs Year Nestbox Male parent Female parent Note there is very little information on each individual male and female bird but we can get some estimates of variability via a random effects model. The data structure can be summarised as follows:

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Diagrammatic representation of the dataset.

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Univariate cross-classified random effect modelling For each of the 6 responses we will firstly fit a univariate model, normal responses for the continuous variables and probit regression for the binary variables. For example using notation of Browne et al. (2001) and letting response y i be clutch size:

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Estimation We use MCMC estimation in MLwiN and choose diffuse priors for all parameters. We run 3 MCMC chains from different starting points for 250k iterations each (500k for binary responses) and use the Gelman-Rubin diagnostic to decide burn-in length. We compared results with the equivalent classical model using the Genstat software package and got broadly similar results. We fit all four higher classifications and do not consider model comparison.

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Clutch Size Here we see that the average clutch size is just below 9 eggs with large variability between female birds and some variability between years. Male birds and nest boxes have less impact.

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Lay Date (days after April 1 st ) Here we see that the mean lay date is around the end of April/beginning of May. The biggest driver of lay date is the year which is probably indicating weather differences. There is some variability due to female birds but little impact of nest box and male bird.

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Nestling Mass Here the response is the average mass of the chicks in a brood at 10 days old. Note here lots of the variability is unexplained and both parents are equally important.

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Human example Sarah Victoria Browne Born 20 th July 2004 Birth Weight 6lb 6oz Helena Jayne Browne Born 22 nd May 2006 Birth Weight 8lb 0oz Fathers birth weight 9lb 13oz, Mothers birth weight 6lb 8oz

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Nest Success Here we define nest success as one of the ringed nestlings captured in later years. The value 0.01 for β corresponds to around a 50% success rate. Most of the variability is explained by the Binomial assumption with the bulk of the over-dispersion mainly due to yearly differences.

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Male Survival Here male survival is defined as being observed breeding in later years. The average probability is and there is very little over-dispersion with differences between years being the main factor. Note the actual response is being observed breeding in later years and so the real probability is higher!

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Female survival Here female survival is defined as being observed breeding in later years. The average probability is and again there isnt much over-dispersion with differences between nestboxes and years being the main factors.

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Multivariate modelling of the great tit dataset We now wish to combine the six univariate models into one big model that will also account for the correlations between the responses. We choose a MV Normal model and use latent variables (Chib and Greenburg, 1998) for the 3 binary responses that take positive values if the response is 1 and negative values if the response is 0. We are then left with a 6-vector for each observation consisting of the 3 continuous responses and 3 latent variables. The latent variables are estimated as an additional step in the MCMC algorithm and for identifiability the elements of the level 1 variance matrix that correspond to their variances are constrained to equal 1.

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Multivariate Model Here the vector valued response is decomposed into a mean vector plus random effects for each classification. Inverse Wishart priors are used for each of the classification variance matrices. The values are based on considering overall variability in each response and assuming an equal split for the 5 classifications.

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Use of the multivariate model The multivariate model was fitted using an MCMC algorithm programmed into the MLwiN package which consists of Gibbs sampling steps for all parameters apart from the level 1 variance matrix which requires Metropolis sampling (see Browne 2006). The multivariate model will give variance estimates in line with the 6 univariate models. In addition the covariances/correlations at each level can be assessed to look at how correlations are partitioned.

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Partitioning of covariances

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Correlations from a 1-level model If we ignore the structure of the data and consider it as 4165 independent observations we get the following correlations: CSLDNMNSMS LD -0.30XXXX NM XXX NS XX MS X FS Note correlations in bold are statistically significant i.e. 95% credible interval doesnt contain 0.

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Correlations in full model CSLDNMNSMS LD N, F, O XXXX NM F, O F, O XXX NS Y, F 0.20 N, F, O O 0.16 XX MS Y 0.07 X FS F, O F, O Y, F 0.11 Y, O 0.21 Key: Blue +ve, Red –ve: Y – year, N – nestbox, F – female, O - observation

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Pairs of antagonistic covariances at different classifications There are 3 pairs of antagonistic correlations i.e. correlations with different signs at different classifications: LD & NM : Female 0.20 Observation Interpretation: Females who generally lay late, lay heavier chicks but the later a particular bird lays the lighter its chicks. CS & FS : Female 0.48 Observation Interpretation: Birds that lay larger clutches are more likely to survive but a particular bird has less chance of surviving if it lays more eggs. LD & FS : Female Observation 0.11 Interpretation: Birds that lay early are more likely to survive but for a particular bird the later they lay the better!

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Prior Sensitivity Our choice of variance prior assumes a priori No correlation between the 6 traits. Variance for each trait is split equally between the 5 classifications. We compared this approach with a more Bayesian approach by splitting the data into 2 halves: In the first 17 years ( ) there were 1,116 observations whilst in the second 17 years ( ) there were 3,049 observations We therefore used estimates from the first 17 years of the data to give a prior for the second 17 years and compared this prior with our earlier prior.

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Correlations for 2 priors CSLDNMNSMS LD 1. N, F, O 2. N, F, O (N, F, O) XXXX NM 1. F, O 2. F, O (F, O) 1. O 2. O (F, O) XXX NS 1. Y, F 2. Y, F (Y, F) 1. Y, F, O 2. N, F, O (N, F, O) 1. O 2. O (O) XX MS M 2. M, O Y 2. Y (Y) X FS 1. F, O 2. F, O (F, O) 1. F, O 2. F, O (F, O) Y, F 2. Y, F (Y, F) 1. Y, O 2. Y, O (Y, O) Key: Blue +ve, Red –ve: 1,2 prior numbers with full data results in brackets Y – year, N – nestbox, M – male, F – female, O - observation

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MCMC efficiency for clutch size response The MCMC algorithm used in the univariate analysis of clutch size was a simple 10-step Gibbs sampling algorithm. The same Gibbs sampling algorithm can be used in both the MLwiN and WinBUGS software packages and we ran both for 50,000 iterations. To compare methods for each parameter we can look at the effective sample sizes (ESS) which give an estimate of how many independent samples we have for each parameter as opposed to 50,000 dependent samples. ESS = # of iterations/,

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Effective Sample sizes ParameterMLwiNWinBUGS Fixed Effect Year Nestbox Male3633 Female Observation Time519s2601s The effective sample sizes are similar for both packages. Note that MLwiN is 5 times quicker!! We will now consider methods that will improve the ESS values for particular parameters. We will firstly consider the fixed effect parameter.

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Trace and autocorrelation plots for fixed effect using standard Gibbs sampling algorithm

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Hierarchical Centering This method was devised by Gelfand et al. (1995) for use in nested models. Basically (where feasible) parameters are moved up the hierarchy in a model reformulation. For example : is equivalent to The motivation here is we remove the strong negative correlation between the fixed and random effects by reformulation.

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Hierarchical Centering In our cross-classified model we have 4 possible hierarchies up which we can move parameters. We have chosen to move the fixed effect up the year hierarchy as its variance had biggest ESS although this choice is rather arbitrary. The ESS for the fixed effect increases 50-fold from 602 to 35,063 while for the year level variance we have a smaller improvement from 29,604 to 34,626. Note this formulation also runs faster 1864s vs 2601s (in WinBUGS).

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Trace and autocorrelation plots for fixed effect using hierarchical centering formulation

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Parameter Expansion We next consider the variances and in particular the between-male bird variance. When the posterior distribution of a variance parameter has some mass near zero this can hamper the mixing of the chains for both the variance parameter and the associated random effects. The pictures over the page illustrate such poor mixing. One solution is parameter expansion (Liu et al. 1998). In this method we add an extra parameter to the model to improve mixing.

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Trace plots for between males variance and a sample male effect using standard Gibbs sampling algorithm

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Autocorrelation plot for male variance and a sample male effect using standard Gibbs sampling algorithm

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Parameter Expansion In our example we use parameter expansion for all 4 hierarchies. Note the parameters have an impact on both the random effects and their variance. The original parameters can be found by: Note the models are not identical as we now have different prior distributions for the variances.

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Parameter Expansion For the between males variance we have a 20-fold increase in ESS from 33 to 600. The parameter expanded model has different prior distributions for the variances although these priors are still diffuse. It should be noted that the point and interval estimate of the level 2 variance has changed from (0.002,0.126) to (0.000,0.172). Parameter expansion is computationally slower 3662s vs 2601s for our example.

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Trace plots for between males variance and a sample male effect using parameter expansion.

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Autocorrelation plot for male variance and a sample male effect using parameter expansion.

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Combining the two methods Hierarchical centering and parameter expansion can easily be combined in the same model. Here we perform centering on the year classification and parameter expansion on the other 3 hierarchies.

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Effective Sample sizes ParameterWinBUGS originally WinBUGS combined Fixed Effect Year Nestbox Male33557 Female Observation Time2601s2526s As we can see below the effective sample sizes for all parameters are improved for this formulation while running time remains approximately the same.

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Conclusions In this talk we have considered using complex random effects models in three application areas. For the bird ecology example we have seen how these models can be used to partition both variability and correlation between various classifications to identify interesting relationships. We then investigated hierarchical centering and parameter expansion for a model for one of our responses. These are both useful methods for improving mixing when using MCMC. Both methods are simple to implement in the WinBUGS package and can be easily combined to produce an efficient MCMC algorithm.

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Further Work Incorporating hierarchical centering and parameter expansion in MLwiN. Investigating their use in conjunction with the Metropolis-Hastings algorithm. Investigate block-updating methods e.g. the structured MCMC algorithm. Extending the methods to our original multivariate response problem.

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References Browne, W.J. (2002). 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): Browne, W.J. (2006). MCMC Estimation of constrained variance matrices with applications in multilevel modelling. Computational Statistics and Data Analysis. 50: Browne, W.J. and Draper D. (2006). A Comparison of Bayesian and likelihood methods for fitting multilevel models (with discussion). Bayesian Analysis. 1: Browne, W.J., Goldstein, H. and Rasbash, J. (2001). Multiple membership multiple classification (MMMC) models. Statistical Modelling 1: Browne, W.J., McCleery, R.H., Sheldon, B.C., and Pettifor, R.A. (2006). Using cross-classified multivariate mixed response models with application to the reproductive success of great tits (Parus Major). Statistical Modelling (to appear) Chib, S. and Greenburg, E. (1998). Analysis of multivariate probit models. Biometrika 85, Gelfand A.E., Sahu S.K., and Carlin B.P. (1995). Efficient Parametrizations For Normal Linear Mixed Models. Biometrika 82 (3): Kass, R.E., Carlin, B.P., Gelman, A. and Neal, R. (1998). Markov chain Monte Carlo in practice: a roundtable discussion. American Statistician, 52, Liu, C., Rubin, D.B., and Wu, Y.N. (1998) Parameter expansion to accelerate EM: The PX-EM algorithm. Biometrika 85 (4): Rasbash, J., Browne, W.J., Goldstein, H., Yang, M., Plewis, I., Healy, M., Woodhouse, G.,Draper, D., Langford, I., Lewis, T. (2000). A Users Guide to MLwiN, Version 2.1, London: Institute of Education, University of London.

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