BugsXLA: Bayes for the Common Man
Contents
1. 3l iss 32 x semen Coeffs 35 36 s Ea 38 C Help Output Option s Clear Farm Exit OK 33 WinBUGS MCMC amp HGLM Output Options MEME Options Scam s000 sw i0000 E Import Samples For Fixed Terms Import Stats For Iv Fixed Terms Iv Cnvariate Coeffs Covariate Coeffs E Intercept Exch Goeffs v Intercept Exch Coeffs Thin Over Relax Eee i Auto Qut TE m Use Presek alues Iv variance Comps Var sp Random Terms Iv variance Compts C Var sp Random Terms E Deviance E DIC E Deviance Predicted Averages Predicted Averages scaled Residuals Scaled Residuals C PS Model Complex Model Create BugFolio Restore Exit m Defaults xit no save ave Changes Help Figure 3 MCMC and output options form showing default settings 12 BugsXLA Bayes for the Common Man Prior Distributions Ix Ind Govariates Exch Govariates Ind Factors Errors Exch Factors Prior for Random Factors Exchangeable effects of levels Effect NEO tau tau Dist ni Parm Dist Parm 4 Alrer Prior Distribution Flate HalF M Elicit prior beliefs For tau E directly via level effects Specify maximum credible difference between effects of levels Helpl Exit OK Figure 4 Form used to specify prior distributions Lab2. B C is shorthand for a model with the terms A B C A C B C while A B C is a model with main effects of both A and B plus a factor C nested within the combination of these factors This would be expanded to A B A B C Note that although A B C is written exactly the same here when defining a nested term as it would be written when defining a three way interaction term it may need to be interpreted differently when constructing the appropriate code to fit the model As well as being able to include both independent fixed in classical terminology and exchangeable random effect terms in the model it is possible to specify covariates When specifying the covariate part of the model the symbol is used to denote the product of two covariates Polynomials can be easily specified by using the and operators e g X1 X2 X3 X4 02 X1 X2 X3 X4 2 would fit all terms up to the second order Clearly there is nothing new in developing software that is able to interpret the algebra of a design statement see Heiberger 1989 for a thorough explanation of how this can be done What is novel is to derive code that fits the specified model using a Bayesian analysis This has been achieved by generating the appropriate code that can be run by the WinBUGS Version 1 4 software package By utilising the script facility added to this version of the software it is then possible to call the program from within Excel using V
3. Journal of Statistical Software May 2005 Volume 14 Issue 5 http wunw jstatsoft org BugsXLA Bayes for the Common Man Phil Woodward Pfizer Global Research and Development Abstract The absence of user friendly software has long been a major obstacle to the rou tine application of Bayesian methods in business and industry It will only be through widespread application of the Bayesian approach to real problems that issues such as the use of prior distributions can be practically resolved in the same way that the choice of significance levels has been in the classical approach although most Bayesians would hope for a much more satisfactory resolution It is only relatively recently that any general purpose Bayesian software has been available by far the most widely used such package is WinBUGS Although this software has been designed to enable an extremely wide va riety of models to be coded relatively easily it is unlikely that many will bother to learn the language and its nuances unless they are already highly motivated to try Bayesian methods This paper describes a graphical user interface programmed by the author which facilitates the specification of a wide class of generalised linear mixed models for analysis using WinBUGS The program BugsXLA v2 1 is an Excel Add In that not only allows the user to specify a model as one would in a package such as SAS or S PLUS but also aids the specification of priors and co
4. gerous It could be argued that BugsXLA increases the risks due to the possibility of the WinBUGS output being accepted without checking for convergence or satisfactory mixing These risks can be mitigated by education in how to use such software as well as the software providing appropriate guidance both during the specification of the model and MCMC run as well as during the analysis stage BugsXLA offers some such guidance but future packages aimed at a broad audience need to offer much more Ultimately the most effective way to grow the knowledge needed to feed this advice is to engage a much broader group of people who regularly use GLMMs in business and industry My hope is that BugsXLA can help to encourage even more people to try Bayesian methods and through its constructive criticism play a small part in the ultimate development of the first Bayesian software package truly useable by the average data analyst References Crowder MJ 1978 Beta binomial ANOVA for Proportions Applied Statistics 27 34 37 Gelfand A Smith AFM 1990 Sampling based Approaches to Calculating Marginal Densi ties Journal of the American Statistical Association 85 398 409 Gelman A 2004 Prior Distributions for Variance Parameters in Hierarchical Models Report Columbia University URL http www stat columbia edu gelman research unpublished taub pdf Gilks WR Roberts GO 1995 Strategies for Improving MCMC
5. is initialised to 10 times the precision of the response This equates to an expected R of about 90 e Any independent factor effects are initialised to 0 3 times the minimum of the standard deviation of the response and the prior standard deviation of the effect itself The reason for the 0 3 multiplier will be explained when multiple chains are discussed later e The standard deviation for exchangeable factor effects is initialised to 0 3 times the minimum of the standard deviation of the response and the prior Half Normal sigma parameter for this term e The covariate regression coefficient is initialised to 0 3 times the minimum of the ratio of standard deviations of the response and covariate and the prior standard deviation of this coefficient itself For both the independent factor effects and the covariate regression coefficients the signs of the initial values are alternated between successive terms fitted in the model A maximum of 5 chains can be run using BugsXLA with the following multipliers for the default values defined above used to obtain a range of initial values 1 0 1 10 0 3 and 3 These multipliers were chosen so that initial values differing by one or two orders of magnitude would be obtained without getting values too far into the tails of the prior distribution Currently I have only limited experience on which to comment on the adequacy of this advice I certainly would not unreservedly recommend this adv
6. options available for controlling the MCMC run itself An example is provided in Section 6 to illustrate how the user interacts with the interface The remaining functionality not illustrated by the example is given in Section 7 A few concluding remarks are made in Section 8 Since segments of WinBUGS code are shown throughout the paper Appendix A gives a very brief explanation of the key elements of the language However readers not familiar with WinBUGS may simply wish to skip those parts that discuss the generated code as it is not necessary to understand this in order to use BugsXLA 2 Brief overview of WinBUGS WinBUGS was developed by Spiegelhalter et al 2003b and is an interactive Windows pro gram for Bayesian analysis of complex statistical models using Markov chain Monte Carlo MCMC techniques In the Bayesian approach all quantities in the model are treated as random variables and inferences are made by conditioning on the data to obtain a posterior distribution for the parameters in the model see Lee 2004 for an introduction to Bayesian statistics The MCMC approach is used to marginalise over the posterior distribution in order to obtain inferences on the main quantities of interest WinBUGS contains a programming language that allows an arbitrarily complex statistical model to be defined This code is then compiled into a form suitable for efficient computation and a sampler operates on this structure to generate
7. 1 Default priors consist of a Normal distribution for the independent effects and covariate regression coeffi cients More choice is provided for the exchangeable effects the default is a Half Normal for their standard deviation as suggested by Spiegelhalter Abrams and Myles 2003a but this can be changed to the Uniform Half Cauchy as suggested by Gelman 2004 or Root Inverse Gamma A Half Normal distribution is also offered as a prior for any covariate regression coefficient whose sign is known with certainty For Normal linear models the prior for the lowest level variance parameter is an inverse gamma distribution Initially default priors are set that should under most circumstances allow the likelihood to dominate when forming the posterior distribution These defaults are given in Table 1 where uy is the sample mean of the response and p and ay are the sample variances of the response and covariate respectively When a link function other than the identity is used the response is appropriately transformed before calculating the mean and variance Currently one of the most controversial topics amongst Bayesians fitting these types of mod els is the choice of prior for the exchangeable effects or more specifically for the standard deviation of the Normal distribution used as the prior The problem mainly arises when there is little information in the likelihood with which to estimate this term The default chosen here a
8. appropriate values of the unknown quantities in order to feed the MCMC algorithms Instead of calculating exact or approximate estimates of the necessary numerical integrals MCMC generates a stream of simulated values for each quantity of interest WinBUGS also has tools to monitor the stream for convergence and summarise the accumulated samples Journal of Statistical Software 9 3 Specifying the model using BugsXLA All the major statistical packages used to facilitate classical methods of data analysis allow the user to specify the model to be fitted by means of a relatively concise model statement It is likely that in order to persuade many users of packages such as SAS S PLUS and Genstat to try a Bayesian analysis of their problems amongst other things they will need software that allows models to be specified in a familiar way BugsXLA uses a very similar set of design statement symbols as that used in Genstat addition of a term interaction between two terms or a nested term crossing of two terms i e both main effects and the interaction nesting of second term inside the first grouping used to determine the order of evaluation for complex models subtraction of a term 0 only expand terms to order specified e g A B C02 A B C A B C power term covariates only e g temp 2 is the quadratic term for temp These enable quite complex models to be specified very concisely For example A
9. i in 1 N Z d for j in 1 levs Zlil d Z Eff li j dnorm 0 tau Z il tau Z i lt 1 sigma2 Z i sigma2 Z i lt pow sigma Z il 2 sigma Z i lt abs hyper Z i hyper Z i dnorm 0 tau hyperZlil tau hyperZli lt 1 Var Zzli where levs X and 1evs Z are the number of levels for each independent and exchangeable effects term respectively and Var X and Var Z are the variances and Half Normal parameters of the priors for independent effects and exchangeable effects standard deviations respectively The prior for the covariate coefficients is coded in a very similar way to that for the independent factor effects Currently the priors for the independent factor effects and covariate coefficients are obtained by the user entering values for the parameters that define them Although this can also be done for the exchangeable factor effects a graphical feedback interface GFI is also provided to facilitate this process The GFI is shown in Figure 1 This figure shows the elicitation option for an exchangeable factor s prior in a Normal linear model It is possible to consider the effects standard deviation 7 directly or as shown think in terms of the difference between the effects of two randomly chosen levels of the factor The user is asked to enter the maximum credible size of this difference this being interpreted as the approximate 95 percentile of its prior distribution Simulation work not discussed here h
10. 994 Markov Chains for Exploring Posterior Distributions Annals of Statistics 22 1701 1762 Journal of Statistical Software if A Key elements of the WinBUGS language I explain only enough here to enable anyone familiar with statistical modelling to follow the code segments in the main paper Most of this is directly copied from the BUGS 0 5 Manual written by Spiegelhalter Thomas Best and Gilks 19962 The code provides a declarative description of the probability model which is compiled before running There are two types of relations which means is distributed as which means is to be replaced by These relations are used to define stochastic and deterministic nodes respectively For exam ple alpha dnorm mu tau defines a Normally distributed variable alpha with mean mu and precision 1 variance tau While mu lt beta time defines mu to be equal to the product of the two nodes beta and time these latter nodes could themselves either be variables or predefined constants Vectors and arrays are represented using square brackets e g x i jl The indexing con vention broadly follows those of S PLUS so that for example x 3 indicates all the values of the third column of the two dimensional array x Repetitive structures can be succinctly described using for loops Note that these constitute a declarative description of the model and should not be interpreted in a procedural way
11. BA code and retrieve the results after the analysis is complete All this is possible without the end user having to know how to use the WinBUGS package at all There are some dangers in this approach which I will briefly discuss in the concluding remarks The majority of generalised linear models used in practice are Normal identity Poisson log or Binomial logit These can all be routinely fitted using BugsXLA the generated WinBUGS code being of the form mu i lt BetaO Y i dnorm mu i tau 4 BugsXLA Bayes for the Common Man Or log mu il lt BetaO Ylil dpois qnulil Or logit mu i lt BetaO Yli dbin mulil N il where log and logit are functions and dnorm dpois and dbin are distribu tions intrinsic to WinBUGS corresponding to Normal Poisson and Binomial distributions respectively Note that the BugsXLA user does not need to know how to code in or run WinBUGS at all BugsXLA creates this code automatically after interpreting a much simpler model statement as explained above By only using link functions that allow the parameters to vary over the whole real line it is possible to generalise the code for defining the priors and likelihood contributions of the parameters Software written to use classical methods of statistical analysis converts the list of model terms to be fitted into an appropriate design matrix This matrix provides the substrate on which the n
12. Half Normal with scale parameter equal to 30 is an attempt to keep this prior weak while not giving much support to extremely large values Although the parameters defining these priors are calculated from the data their variances are a sufficiently large multiple of the range of the response that there is little danger that the likelihood is being used twice The priors are specified this way simply to ensure that irrespective of the location and spread of the response the default prior is essentially flat over the range of the likelihood In principle it is not necessary to impose any constraints on the independent factor effects although they will not converge any estimable function of them will and could therefore be successfully monitored during the MCMC run In practice leaving these effects free often severely hampers the performance of the MCMC algorithm Hence BugsXLA imposes corner constraints on these effects similar to those adopted by SAS or Genstat Instead of literally fixing some of the effects to zero in order to make it easier to write a generic code segment these constrained effects are given a prior distribution centred on zero with an extremely small variance The priors for independent and exchangeable factor effects Half Normal prior is coded for i in 1 N X d for j in 1 levs Xlil i1 tau X i j lt 1 Var XL j i Journal of Statistical Software X Eff i j dnorm 0 tau X i jl for
13. In SR W R Gilks D Spiegelhalter eds Markov Chain Monte Carlo in Practice pp 75 88 Chapman amp Hall London 15 16 BugsXLA Bayes for the Common Man Heiberger RM 1989 Computation for the Analysis of Designed Experiments John Wiley amp Sons Inc Lee PM 2004 Bayesian Statistics An Introduction Arnold London 3rd edition Neal R 1998 Suppressing Random Walks in Markov Chain Monte Carlo using Ordered Over Relaxation In MI Jordan ed Learning in Graphical Models pp 205 230 Luwer Academic Publishers Dordrecht Smith AFM Roberts GO 1993 Bayesian Computation via the Gibbs Sampler and Related Markov Chain Monte Carlo Methods with discussion Journal of the Royal Statistical Society D 55 3 24 Spiegelhalter D Abrams K Myles J 2003a Bayesian Approaches to Clinical Trials and Health Care Evaluation John Wiley amp Sons Ltd Spiegelhalter D Best N Carlin B van der Linde A 2002 Bayesian Measures of Model Complexity and Fit with discussion Journal of the Royal Statistical Society B 64 583 640 Spiegelhalter D Thomas A Best N Gilks W 1996a BUGS 0 5 Manual version ii MRC Biostatistcs Unit Cambridge UK Spiegelhalter D Thomas A Best N Gilks W 1996b BUGS Examples Volume 1 version i MRC Biostatistcs Unit Cambridge UK Spiegelhalter D T homas A Best N Lunn D 2003b BUGS User Manual Version 1 4 MRC Biostatistcs Unit Cambridge UK Tierney L 1
14. The syntax follows that of S PLUS For name in expression1 expression2 B Downloading and installing BugsXLA BugsXLA v2 1 is a Microsoft Excel Add In that is freely available via the BUGS website http www mrc bsu cam ac uk bugs or direct from http www pipshome freeserve co uk stats After clicking on Download BugsXLA v2 1 and saving the self extracting zip file to a convenient location on your PC follow these instructions to install BugsXLA as an Excel add in It is necessary to have Windows 98 or higher tested on 98 2000 and 2002 MS Excel tested on 2000 and 2002 will not work on 97 MS Notepad and WinBUGS v1 4 already loaded on your PC It is important that Excel be closed before starting 1 Run the self extracting zip file by default this puts the files in 18 BugsXLA Bayes for the Common Man C Program Files WinBugsXLA 2 Open MS Excel 3 Select Tools Add Ins and Browse in the directory to which the zipped files have been extracted see step 1 ensuring that Files of Type All Files has been selected 4 Select the file BugsXLA sta and make sure that BugsXLA has been checked in the Add Ins dialog box before clicking OK 5 If you are warned that either WinBUGS or Notepad is not in specified location click Yes and alter the directory entries so that these programs can be found 6 A toolbar titled BugsXLA should appear that could be moved to your favoured l
15. as shown that if r Half Normal c Half Normal 27 then o 95 3 where 0 95 is the 95th percentile of the distribution of 6 In BugsXLA the elicited maximum credible difference defined above is interpreted as 3o The GFI is designed such that the graph of the prior distribution as well as some of its percentiles is updated in real time as the user changes the credible difference using the spinner control It is possible to plot the exponential of the X axis this being useful when log or logit link is used When this transformation is applied the axis can then be interpreted as the ratio log or odds ratio logit of two randomly chosen effects 8 BugsXLA Bayes for the Common Man E F G H JS K L L o P a R 1 stribution 3 F 3 EA Prior Distribution for Difference between Rep Effect Levels Prior Elicitation Controller Rep tau HalfN 1 67 2 Link function Identity 6 0 35 Elicit prior beliefs For tau E e directly via level effects 8 0 3 Specify maximum credible difference ERI between effects of levels 10 0 25 z 5 dy z e A s 0 2 Parameters For Elicitation Controller 9 Upper Limit i3 s 0 15 pper Limi 11 EM a Lower Limit 16 0 1 17 18 0 05 Parameters For Distribution Graph 19 Max X value 10 20 0 4 2 Min X value 0 Updat Difference between Rep Ef
16. del statement of the form A Journal of Statistical Software B C into the appropriate X fixed effects or Z random effects design matrix Note that each row of these matrices represents a different observation and each column represents a different term in the model with the entries representing the associated factor level set X Ef L Z Ef L and BetaO are what WinBUGS coders refer to as stochastic nodes representing the statistical parameters of the model and will need to have priors specified see Section 4 It is equally as simple to write generic code for including covariates The only additional issue to consider is the need to centre and scale the covariates which greatly improves the performance of the MCMC algorithm as discussed in Gilks and Roberts 1995 The code required to achieve this in WinBUGS is for i in 1 N V 1 mean V i lt mean V il sd V i lt sd VL il Cov Coeff i lt V Beta i sd V i for j in 1 N obs 1 std V j i lt VLj i mean V il sa V i mean Betal i lt Cov Coeffli mean V il alpha lt BetaO sum mean Betall for j in 1 N obs V rowLj lt inprod std V j V Betall mu j lt BetaO V rowlj where N V is the number of covariates V is a matrix made up of the covariates std V is a matrix of centred and scaled covariates Cov Coeff and V Beta are the vectors of regression parameters for the or
17. el Mean St Dev 2 5 Median 97 5 WinBUGS Name CONSTANT 0 5520 0 2217 1 0120 0 5520 0 1035 Betal Root Bean 0 0000 0 0000 0 0000 0 0000 0 0000 X Ef 1 1 Root Cucumber 1 3590 0 3104 0 7541 1 3500 2 0250 X Ef 1 2 Seed O aeg 75 0 0000 0 0000 0 0000 0 0000 0 0000 X Ef 2 1 Seed O aeg 73 0 0670 0 3478 0 6635 0 0748 0 7320 X Ef 2 2 Root x Seed Bean O aeg 75 0 0000 0 0000 0 0000 0 0000 0 0000 X Ef 3 1 Root x Seed Cucumber O aeg 75 0 0000 0 0000 0 0000 0 0000 0 0000 X Ef 3 2 Root x Seed Bean O aeg 73 0 0000 0 0000 0 0000 0 0000 0 0000 X Eff 3 3 Root x Seed Cucumber O aeg 73 0 8379 0 4787 1 8200 0 8230 0 0934 X Ef 3 4 SD Plate 0 3535 0 1520 0 0928 0 3383 0 6793 sigma Z 1 Model A 1 E 22 Distribution Binomial Link Logit Response R N Fixed Root Seed Random Plate Priors CONSTANT N mu 0 sigma 100 Root N mu D sigma 100 Seed N mu D sigma 100 Root x Seed N mu 0 sigma 100 Plate N D tau 2 tau Half N siqma 5 WinBUGS MCMC Settings Burn In 5000 Samples 10000 Thin 1 Chains 1 Run took 23 seconds BugsXLA 2 1 04 07 13 21 49 BugsXLA is provided without any warranty of any kind either expressed or implied The user is responsible for any consequences arising from use of this program See also the disclaimer provided with the WinBUGS package Figure 5 Summary statistics as imported into Excel Journal of Statistical Software term has been constrained to zero with appropriate constrain
18. fect Levels I Plot exp 25 Stile 0 157 25 ile 0 797 Median 1 69 75 hile 2 88 95 ile 4 9 Delete graph on exit v 26 J Help i 37 ep Exit OK Figure 1 Prior elicitation process via the graphical feedback interface 5 Controlling the MCMC run Probably the most difficult aspect to automate is the settings used to control the MCMC run BugsXLA allows direct control of the parameters in Table 2 the default settings are in parentheses It is not suggested that the default settings for these are adequate even for the majority of GLMM s that are likely to be fit in practice Their use is primarily for the case when it is unsure if WinBUGS will run the model at all providing a quick and dirty run They may also be adequate for models with no exchangeable effects terms fixed effects models For this reason the program offers three alternative pre defined settings with some advice on when to use them Burn In Number of initial MCMC samples to be discarded 1000 Convergence assumed past this point Number of MCMC samples to be generated from Posterior Distribution 1000 Thin If set to K then only every Kth sample is saved Use when high 1 autocorrelation is present to reduce the number of samples required to give good coverage of the posterior distribution Number of separate chains to be generated 1 Determines whether WinBUGS is automatically closed after the MCMC run Yes Table 2 Defa
19. for the histogram using the options on the form 7 Other functionality in BugsXLA e Additional distributions and links Weibull t and log Normal probit and complemen tary log log links for binomial data e Multinomial and ordered categorical data the data can be either grouped into counts or recorded as individual responses e Censored data lower bound upper bound interval or any mixture of these The button labelled Set Variable Types in Figure 2 launches a form that facilitates the classification of variables as one of Factor Variate or Censor e Factor by covariate interactions easy fitting of group regression coefficients e Covariates with exchangeable coefficients i e random coefficients e Allow over relaxed samples to be generated as discussed in Neal 1998 e Calculation and importing of the DIC statistic as defined in Spiegelhalter Best Carlin and van der Linde 2002 e Able to create a Bugfolio using the Create Bugfolio feature shown in Figure 3 By creating a Bugfolio all the files created by BugsXLA are stored in the folder specified This is useful for experienced WinBUGS programmers who wish to use BugsXLA to generate some basic code and then modify the code script or data files to run a more complex model not possible using BugsXLA directly e Help is provided on line by clicking on the Help box found at the bottom left of most of the forms see Figure 2 for exa
20. ice and include it in the program 10 BugsXLA Bayes for the Common Man mainly as a placeholder for better guidance as experience with the program develops I do believe that providing such guidance is an essential feature of any software that is aimed at the average data analyst If nothing else I would hope that this attempt would raise discussion on the general guidance that can be given 6 Worked example To illustrate how the user interacts with BugsXLA the Seeds example originally from Table 3 of Crowder 1978 but also used by Spiegelhalter Thomas Best and Gilks 1996b is reanalysed Figure 2 shows the completed model specification form in this case defining a binomial error model with logistic link The variables R and N contain the number germinated and total number of seeds respectively in a study undertaken to assess the effects of seed and root extract types As well as fitting independent fixed main and interaction effects for the treatment structure over dispersion is accommodated by including an exchangeable random effects term for the 21 plates used in the experiment By clicking on the button labelled MCMC amp Output Options another form is shown that allows the user to alter the default settings as given in Figure 3 Importing Stats or Samples will result in the WinBUGS summary statistics or generated samples from the posterior distribution being imported into Excel re
21. iginal and scaled covariates respectively and alpha and BetaO are the intercept parameters for the original and scaled covariates respectively The terms mean sd and inprod are intrinsic WinBUGS functions with obvious results Note that the semicolon separator allows more than one statement on the same line It should be clear that it is straightforward to combine the different code segments such that linear models of arbitrary complexity can be defined using BugsXLA Of course it does not follow that any model that can be defined will always lead to a successful WinBUGS analysis Also since the files created by BugsXLA are all in text form users knowledgeable in the WinBUGS language can edit these generated code and data files and rerun the script to obtain extensions of the standard models offered by BugsXLA 4 Specifying and eliciting priors The most conspicuous difference between any software that facilitates a Bayesian as opposed to a classical statistical analysis is the need to include prior distributions in the model speci fication Currently BugsXLA has only a limited choice of priors from which to choose These 6 BugsXLA Bayes for the Common Man Constant term Normal uy 1000002 Independent effect contrasts Normal 0 1000007 Exchangeable effect st dev Half Normal 9o 2 Covariate regression coefficient Normal 0 10000 xX Normal residual variance Inv Gamma 0 001 0 001 Table
22. lp Exit Figure 7 Histogram of 10 000 samples generated during the MCMC run This graph should closely approximate the posterior distribution for the between plates standard deviation pa rameter The specified prior distribution has been overlaid solid black line suggesting that the likelihood has dominated in the estimation of this parameter Journal of Statistical Software 8 Concluding remarks The purpose of BugsXLA is to provide a tool that makes it easier for the average data analyst to try Bayesian methods By allowing models to be specified in a manner that is familiar to users of such packages as SAS S PLUS and Genstat it removes one of the major barriers to those people who are curious but uncommitted to the Bayesian approach An initial attempt has been made to provide guidance on appropriate settings for the MCMC run when GLMMs are being fitted It is also shown how a prior elicitation interface could be integrated into a Bayesian analysis program By removing the need to know how to code the models import the data and export the results it allows the user to focus on the more important issues e Is the model appropriate e What priors can I justify e What inferences am I trying to make As well as providing a tool for tackling real problems this could also provide a useful aid for those teaching Bayesian methods As Spiegelhalter et al 2003b state in the BUGS manuals MCMC sampling can be dan
23. lter T homas Best and Lunn 2003b is the most widely used Bayesian data analysis program It provides an easy to learn language that allows a wide variety of models to be specified Its popularity is probably due to the user not needing to know very much about the MCMC methods that underpin it and the fact that it is free However there is inevitably a learning curve barrier associated with any new software that requires sufficient motivation to overcome If it were possible to specify a model in a similar manner to that in SAS S PLUS or Genstat then I believe a much larger number of people used to working with such programs would try a Bayesian approach This paper describes a graphical user interface for WinBUGS that allows a wide class of generalised linear mixed models GLMMs to be specified in a manner familiar to users of more conventional statistical software This interface BugsXLA v2 1 is a Microsoft Excel Add In that is also freely available via the BUGS website http www mrc bsu cam ac uk bugs or direct from http www pipshome freeserve co uk stats see Appendix B for instructions on how to download and install the software Section 2 provides a brief overview of WinBUGS Section 3 describes the types of model that can be fitted and how the model statement is converted into WinBUGS code Section 4 explains how priors are specified including a graphical tool to help in the elicitation process Section 5 briefly discusses the
24. me shows the generic node names used in Win BUGS and is of little use unless an experienced user wished to edit the code directly later Informative names for the parameters are given in the column with the header Label In spection of the summary statistics shows that the first level of each independent main effects Journal of Statistical Software R M FE d EM REL RE M ESL LE Mo M Seed Root Plate 1 E Rm D aeg 5 Bean Hierarchical Generalized Linear Model Bayesian Analysis E x 3 e3 Be LI aeq 75 Bean 2 A S DLaeg 5 Bean 3 Data single column variables all same length 5 eB 5 D aeg 75 Bean 4 Data Ranae B Tr 33 O aeg 75 Bean 5 i ASISESZZ T 5 E D aeg 75 Cucumber Data is 8 53 74 D aeg 75 Cucumber 7 Di v Mames in first raw 7 G ae9 75 Cucumber 7 Ek columns Set Variable Types 10 32 51 D aeg 75 Cucumber 3 Iri rows Blank cells Mys Response only 11 46 73 O aeg 75 Cucumber 10 lz 10 13 OF aeg 75 Cucumber 11 13 8 16 O aeg 73 Bean lz ow m T m DOaeg73 Bean Distribution Binomial Eliciting Priors Only T 15 8 ea D aeg 73 Bean 14 16 23 45 0 aegra Bean 15 Linie Logit Tr 4 D aeq 73 Bean 16 18 d tz Q aeg 73 Cucumber Tr 13 ez 41 D aeg 73 Cucumber 18 Response a0 15 30 DLaeg 3 Cucumber 13 P RIN 2l 32 51 D aeg 73 Cucumber 20 22 3 7 D aeg 73 Cucumber 21 Censored 23 24 Factors zh sd Rnat Seed ZEB 2v T Eadan Plate 30 Covariates
25. mple Features that have help text provided are highlighted and the help is obtained by moving the mouse over the highlighted area 13 14 BugsXLA Bayes for the Common Man Parameter to be plotted Select any cell in data column CODA 41 E Update Parameter Information SD Plate Prior HalF M e sigma 5 For SD Histogram range defined by Default Method C User Specified Minimum number of bins gt p For Histogram of Posterior Update Graph Hide histogram data W Delete graph on exit v E Help Exit Figure 6 Form shown after the Post v Prior icon is selected from BugsXLA toolbar The specified prior is shown in this case a Half Normal constrained to be positive with a scale parameter of 5 Plot Posterior and Prior Distributions SD Plate Parameter to be plotted Prior Half N ve sigma 5 for SD Select any cell in data column CODA 1 A 1 Update Parameter Information SD Plate Prior Half N ve sigma 5 For SD mem Posterior Data Min 0 0075 amp Max 1 17 Prior Histogram range defined by C Default Method User Specified Minimum number of bins 12 For Histogram of Posterior Start o End iz Relative Frequency o o o o o o o o o O oO o o dp for X axis i 5 A o FT D Oo r 00 SH A Update Graph gt o o o O O O O O OQO 0 s s s Hide histogram data IV Value Delete graph on exit v He
26. ntrol of the MCMC run itself Inevitably developing a program such as this forces one to think again about such issues as choice of default priors parameterisation and assessing convergence I have tried to adopt currently perceived good practices but mainly share my approach so that others can apply it and through constructive criticism play a small part in the ultimate development of the first Bayesian software package truly useable by the average data analyst Keywords Bayesian analysis GLMM Microsoft Excel graphical user interface WinBUGS 1 Introduction The use of Bayesian methods in academia appears to have increased at an exponential rate over the last ten to twenty years There can be little doubt that this is due to the availability of Markov chain Monte Carlo MCMC algorithms to implement such methods that simply did not exist previously see Gelfand and Smith 1990 Smith and Roberts 1993 or Tierney 1994 for details of this approach The philosophical debate will never go away but even 2 BugsXLA Bayes for the Common Man amongst academics it is the practicality of being able to solve real problems that has greater influence Although Bayesian methods are used more today in business and industry than they were twenty years ago the increase is nowhere near as marked I suggest that one of the barriers to greater use is the unavailability of Bayesian software with a good user interface The WinBUGS package of Spiegelha
27. ocation on the Excel frame You can find an Excel Workbook with a few example data sets in the unzipped to directory specified in step 1 BugsXLA Egs xls Help is available on some of the BugsXLA forms Click on the Help checkbox and hover the mouse over any areas that are highlighted Affiliation Phil Woodward Biostatistics and Reporting Pfizer Global Research and Development Ramsgate Road Sandwich Kent CT13 9NJ United Kingdom E mail phil woodward pfizer com Journal of Statistical Software Submitted 2004 06 23 May 2005 Volume 14 Issue 5 Accepted 2005 01 31 http www jstatsoft org
28. spectively Selecting Predicted Averages gives the predicted mean response for every possible combination of the factors specified to have independent effects averaged over the levels of any exchangeable effects and with any covariates set to their mean value The scaled residuals are the raw residuals divided by Normal error o Poisson error J Lu Binomial error Nu 1 u After setting these options and confirming that the model is specified correctly the prior distributions for each of the terms can be altered Figure 4 shows the default settings for the exchangeable effects of the plate term The button labelled GFP would bring up the graphical feedback interface as illustrated in Figure 1 In the Seeds example since a logit link has been specified the user would be prompted to think about his or her prior beliefs in terms of the odds ratio of the effects of two randomly chosen plates Once the user is satisfied with the priors by clicking OK the program generates all the files needed by WinBUGS to complete the analysis data files including the constants needed to define the model initial value files the code itself as well as the script file that controls the WinBUGS analysis Provided that a successful exit from WinBUGS is achieved the user is then prompted to import the results which in the case of the summary statistics will be formatted as in Figure 5 The column with the header WinBUGS Na
29. ts on the interaction effects also The output also provides summary of the model fitted plus the prior distributions specified It also records the settings for the MCMC run with the time taken to run Due to the inherent uncertainties currently associated with using MCMC sampling methods to tackle generic problems a final warning message is displayed to ensure the user remains cautious when interpreting the results One of the distinct advantages of using WinBUGS is that it is possible to characterise the whole posterior distribution by viewing and summarising the generated samples for any of the parameters or arbitrarily complex functions of them This is particularly useful for distributions that are distinctly non Normal such as the variance component for the plate effect in this example The imported samples can be subsequently pasted into your favourite statistics package and a histogram or other summaries created Alternatively the Post v Prior icon on the BugsXLA toolbar can be selected which runs a utility program that can do this within Excel Figure 6 shows the form that appears when this option is chosen After identifying the column of data to be graphed and clicking on Update Parameter Infor mation the name and prior specified for this parameter are shown Clicking on update graph will produce a plot of the posterior distribution histogram and prior overlaid Figure 7 shows such a plot after adjusting the bins
30. ult MCMC control settings Journal of Statistical Software 9 Simple model Burn In 5 000 Samples 10 000 Thin 1 Chains 1 Auto Quit Yes Orthogonal design may include one exchangeable effects term with variance estimate having 3 or more effective degrees of freedom Regular model Burn In 25 000 Samples 10 000 Thin 10 Chains 1 Auto Quit Yes Nearly orthogonal design may include covariate s but with each variance estimate having 3 or more effective degrees of freedom Although not part of the default settings I would recommend that Auto Quit be turned off and at least 2 chains run so that MCMC convergence can be checked before using the results Complex model Burn In 100 000 Samples 20 000 Thin 50 Chains 1 Auto Quit No Messy design may include covariate s or some variance estimates having less than 3 effective degrees of freedom Note that Auto Quit is turned off as it is strongly recommended that MCMC convergence be checked before using the results Although not part of the default settings I would recommend that at least 2 chains be run Mixing can often be greatly improved if informative priors are provided for poorly estimated variance components These need not be strong priors just sufficiently precise to effectively rule out incredibly large values Initial values are automatically generated using the following procedure e The lowest level precision parameter Normal models
31. umerical algorithms developed to give the parameter estimates are applied Categorical factors are converted into dummy variables with constraints to ensure estimable functions are defined For example by default SAS constrains the last level of each main effect to be zero while Genstat s default is to constrain the first level A different approach has been taken when generating the data files and code for WinBUGS Each categorical factor term be it main effect or interaction is stored as a single vector consisting of integers indexing the level set for each observation This allows the linear predictor to be specified in WinBUGS code using the generic statements for j in 1 N obs d for i in 1 N X d X row i j lt X Effli Xlj il for i in 1 N Z d Z rowli j lt Z EffLi Z j il mu j lt BetaO sum X rowl jl sum Z rowl jl where N obs is the number of observations N X and N Z are the number of independent and exchangeable effects terms respectively X and Z are matrices made up of the independent and exchangeable level indexing vectors respectively X Eff and Z effL are the parameters defining the independent and exchangeable effects respectively BetaO is the constant parameter and mu L is the linear predictor in the generalised linear mixed model The intrinsic WinBUGS function sum sums over the missing index of its vector argument One of the main tasks underpinning BugsXLA is to convert a mo
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