Review of WINBUGS. 1. Introduction. by Harvey Goldstein Institute of Education University of London

Transcription

1 Review of WINBUGS by Harvey Goldstein Institte of Edcation University of London Introdction This review is based pon the beta release of WINBUGS.4 which has several enhancements over version.3. WINBUGS, a Bayesian MCMC package, is distribted freely and is the reslt of many years of development by a team of statisticians and programmers at the Medical research Concil Biostatistics Research Unit in Cambridge ( Models are represented by a flexible langage, and there is also a graphical featre, DOODLEBUGS, that allows sers to specify their model in terms of a directed graph. For complex models the latter can be extremely sefl, bt we will not se it in this review since all the models dealt with are relatively straightforwardly expressed sing the command langage. The topic of Bayesian estimation sing Markov Chain Monte Carlo (MCMC) methods has an extensive literatre and sers are well advised to introdce themselves to this before sing these methods. The WINBUGS manal provides a good starting place. In this review we shall not be exploring the fll range of possibilities provided by WINBUGS, althogh, where appropriate these will be referred to. We shall give examples of WINBUGS code where this is illminative, bt since this review is not a gide to sing WINBUGS, we will not be exhastive. Users of simlation based methods sch as MCMC shold remember that, nlike maximm likelihood procedres, there is a stochastic element to the estimates obtained and that procedres for jdging convergence and accracy are more complex and a certain amont of jdgement and experience is involved. The WINBUGS manal is a sefl sorce here. The advantage of MCMC estimation, apart from the obvios one for those who wish to adopt a Bayesian approach, is that, like bootstrapping, it provides accrate estimates for parameter qantiles. One disadvantage is the time taken to obtain reslts and often the se of likelihood based methods for initial model exploration prior to sing MCMC will be efficient. Also, there are still many isses whose soltion is less than clearct. For mltilevel models, one of these is the isse of choice of priors for variance and covariance parameters; we have adopted the WINBUGS gamma and Wishart defalts, bt see Browne (998) for a frther discssion. Another isse is that of model comparison and we have sed the Bayesian Deviance Information Criterion for this (Spiegelhalter et al. ).

2 . Data interface Data is inpt to WINBUGS as three components that can be in the same of different files. The first is a series of model specification instrctions, the second contains the observed data and the third contains a set of starting vales. In fact WINBUGS will itself obtain starting vales if reqired, bt generally it is better for the ser to specify these. We shall illstrate the se of these components in the first example below. WINBUGS rns nder varios versions of windows. Older versions will also rn nder other operating systems sch as LINUX see the WINBUGS website for more information. While WINBUGS.4 has many sefl graphical tools, there is also a site of rotines, CODA, that can provide additional plotting featres for WINBUGS otpt in SPLUS or R. The principal featres will be illstrated in or first example.. Model types The range of model types that can be fitted in WINBUGS is very large. The langage allows a wide variety of linear and nonlinear model forms, inclding the standard set of generalised linear models. For the random effects a variety of distribtional forms can be specified, inclding the mltivariate Normal and T-distribtions. Spatial models and latent variable models can be fitted. In addition a range of prior distribtions is available with standard defalts. All of the models in this set of reviews can be fitted in WINBUGS.. Examples. Two and three level Normal models The first model we fit is a variance components -level model sing the EXAM data set and we shall se it to illstrate the se of the WINBUGS modelling langage. The model is y j = ( Xβ ) + + e ~ N(, j e ~ N(, e where i,j respectively index level and level and ( X β ) represents the fixed part contribtion for the i,j - th record. The fixed part here incldes the intercept ( β ) and for covariates, β, β, β3, β 4, being LRT score, gender, school gender composition ( dmmy variables; boys school and girls school contrasted with base category, mixed school). We discss priors below. The WINBUGS model specification code for fitting this model is as follows: ()

3 model { # Level definition for(i in :N) { normexam[i] ~ dnorm(m[i],ta) m[i]<- beta[] * cons[i] + beta[] * standlrt[i] + beta[3] * gender[i] + beta[4] * boysch[i] + beta[5] * girlsch[i] + [school[i]] * cons[i] # Higher level definitions for (j in :n) { [j] ~ dnorm(,ta.) # Priors for fixed effects for (k in :5) { beta[k] ~ dflat() # Priors for random terms ta ~ dgamma(.,.) sigma <- /ta ta. ~ dgamma(.,.) sigma. <- /ta. Notes: Comments (starting with #) are inserted for ease of reading Line 5 specify that the response is Normal, and has a mean m(i) and precision ta. The precision is simply the inverse of the variance and this relationship as specified in the penltimate line by the logical or deterministic (i.e. nonstochastic) relationship sigma. <- /ta.. Lines 6 specify m(i) as a linear additive fnction of the intercept and for covariates and line adds the set of random effects where school(i) is the school ID (.65) for record i and is mltiplied by which is a random variable (see below). This prodct is also mltiplied by cons(i) which here is strictly neccessary bt is retained for consistency with general random coefficient models where the explanatory variable associated with a random coefficient is reqired. Note that constants N, n sed here have vales that are inpt with the data (see later), bt we cold have specified the vales, 459, 65, instead. Under priors for the fixed effects we se a flat i.e. niform prior across the whole real line, for each regression (fixed effect) coefficient. An alternative, proper prior which is fnctionally eqivalent is a Normal with a very large 6 variance, e.g. N(,* ) which is sed in WINBUGS ttorials. Under priors for random terms we give the precisions Gamma (.,.) priors the WINBUGS defalt. Other choices are available see Browne (998) for more discssion. Under higher level definitions the distribtion of (j) (j= n) is specified as Normal mean and precision ta.. The final line relates the level variance to this precision parameter. We can also specify frther logically derived parameters. For example the variance partition coefficient (VPC Goldstein et al, ) which is here

4 eqivalent to the intra-clster correlation can be specified sing the logical relationship: vpc <- sigma./(sigma.+sigma). We can then monitor this (see below) and obtain any reqired estimates. WINBUGS will calclate starting vales bt it is better to inpt sensible vales and we se those from an ML fit sing another package. In fact, MlwiN will also provide the WINBUGS code for many of the mltilevel models that it can fit and this can provide MlwiN sers with a sefl shortct to setting p the specification as well as providing starting vales. The starting vales for the parameters (inclding the random effects residal estimates) can be placed in the same or a different file form the model specification. The following are the REML parameter estimates together with the estimated residals. list(beta= c(-.6473,.5594,.673,.68344,.5358), = c(.46899,.364,.58748,.653,.34638,.467,.3488,-.79,-.4996,-.4358,.79955,.356,-.8845,-.8899,-.95,-.5533, ,-.973,.79768,.947,.86,-.3585, ,.69,-.3736,.64497,.98,-.5769,.96585,.584, ,.84659,.7995,-.4588,-.3997,-.8345,-.937, ,-.6536,-.354,.795,.8898,-.857,-.4797,-.966, ,.96,-.844,-.68,-.7963,.4859,.3845,.5898, , ,.69899,.3463,.75, ,.8485,.4367,.396,.5939,.87875, ), ta=.77647, ta.=.5564) The data can also be placed in a separate file or the same file and the following is the start of this file list(n= 459, n = 65, school = c(,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,3,3,3,3,3,3,3,3,3,3,3,3, Under the specification men the ser first checks the syntax of the model. Assming it is syntactivcally correct (message appears) the data are loaded and then the initial vales. The ser chooses which parameters to monitor from the inference men and chooses the nmber of iterations to convergence to a stationary chain (the brn in). The nmber of following iterations pon which the estimates are based is then chosen and the ser can also observe the chains. In the present case we therefore begin to sample from iteration 5 (set beg to 5 (leave end at defalt) and choose to inclde all the fixed and random parameters as nodes to sample. We cold also sample the residals if we wished. In the model window we select 55 pdates. The otpt choices are flexible in terms of the parameter estimates and statistics derived from the chain sch as qantiles. A fit statistic is also available (DIC) which allows model comparisons. To se this we need to first carry ot the brn in, then set DIC nder the inference men and then sample the 5 pdates. When these are complete we can click on DIC to obtain the vale. After estimation has finished we can obtain or estimates, traces and kernel density

5 plots for the parameter posterior distribtions. Table compares two WINBUGS rns with a 5 brn in and 5 and following iterations, together with the REML estimates. Table. Two level Normal response model for EXAM data Parameter REML estimator (s.e.) WINBUGS (5) (95% interval) β WINBUGS () (95% interval) β.56 (.3).56 (.535,.584).56(.535,.584) β.67 (.34).67 (.99,.34).67 (.98,.33) β.78 (.4).76 (-.36,.383).8 (-.33,.394) 3 β.59 (.89).6 (-.6,.39).6 (-.,.338) 4.86 (.7).88 (.57,.3).88 (.58,.3).563 (.3).563 (.539,.588).563 (.539,.588) e DIC Time.4 secs (SAS) 7 mins. secs* mins. 45 secs* *incldes 5 iterations to convergence (Brn in). MCMC estimates are the chain means. Converted to eqivalent time on 433Mhz Pentim II nder Windows. As can be seen there is very little change between 5 and iterations. The point estimates are similar to the REML estimates and 95% confidence intervals sing a Normal approximation are also close, except for the level variance parameter. If the modal or median rather than mean are sed from MCMC we wold expect closer agreement with REML; in the present case the median for the level variance is.86. The following are the trace and kernel density plot for the level variance for the analysis sing iterations. The kernel density is nsmoothed; this may be changed in later versions. Trace for level variance sigma iteration Kernel density estimate for level variance

6 3.... sg a sa pe In the remaining examples we shall give reslts from rnning iterations with a 5 brn in. We now fit a model where the level variance is a fnction of gender and this is followed by a model where the log(level variance) is a fnction of STANDLRT. The model specifications are given in the following boxes and the reslts in Table. Finally we show below the model specification for a model where STANDLRT has a coefficient random at the school level. model { # Level definition for(i in :N) { normexam[i] ~ dnorm(m[i],ta) m[i]<- beta[] * cons[i] + beta[] * standlrt[i] + beta[3] * gender[i] + beta[4] * boysch[i] + beta[5] * girlsch[i] + [school[i],] * cons[i] + [school[i],] * standlrt[i] # Higher level definitions for (j in :n) { [j,:] ~ dmnorm(zero[:],ta.[:,:]) # Priors for fixed effects for (k in :5) { beta[k] ~ dflat() # Priors for random terms ta ~ dgamma(.,.) sigma <- /ta for (i in :) {zero[i] <- ta.[:,:] ~ dwish(r[:, :],) sigma.[:,:] <- inverse(ta.[,]) Note that we have sed a minimally informative Wishart distribtion as prior for the inverse covariance matrix by giving it degrees of freedom eqal to the order of the matrix and we can choose the Wishart matrix parameters as eqal to the (inverse) covariance matrix based pon ML. These vales, R, will be declared with the data. Other priors are possible inclding a niform.. Complex level variation We fit a separate variance for boys and girls as follows:

7 y e x j = ( Xβ ) = e ( x + ~ N(, = if girl j + e ) + e e e x e ~ N e () Note that WINBUGS can fit other complex level variance fnctions. A sefl one is the logarithm of the level variance as a linear fnction of the STANDLRT score which garantees a positive variance. This wold be written y j = ( Xβ ) e e + ~ N(, j + e e log ( = α + α x ~ N(, e Note that in WINBUGS we need to express the precision as a linear fnction, in which case the coefficients simply reverse their sign. For the first model the WINBUGS specification wold be model { # Level definition for(i in :N) { normexam[i] ~ dnorm(m[i],ta[i]) m[i]<- beta[] * cons[i] + beta[] * standlrt[i] + beta[3] * boysch[i] + beta[4] * girlsch[i] + beta[5] * girl[i] + [school[i]] * cons[i] # Higher level definitions for (j in :n) { [j] ~ dnorm(,ta.) # Priors for fixed effects for (k in :5) { beta[k] ~ dflat() # Priors for random terms for(i in :N) { ta[i] <- /sigma[i] sigma[i] <- va[] * girl[i] * girl[i] + va[] * boy[i] * boy[i] for(i in :) { va[i] ~ dflat() ta. ~ dgamma(.,.) sigma. <- /ta. (3) Note that we se girl in the fixed part and both the dmmy variables girl and boy for level. This reqires that the data for boy is also entered. Note also that we have sed flat priors for the level variance parameters; when we have a general fnction of explanatory variables - we cannot se gamma priors for variances. In fact in the gender case we cold have specified gamma priors since we are really fitting two separate variances.

8 Table. Two level Normal response model for EXAM data with complex level variance Parameter β -.96 Separate gender variances Estimate (95% interval) β.548 (.493,.63) β.69 (.,.3) β.93 (-.3,.43) 3 β.89 (.,.34) 4 β.8 (-.5,.64) 5.88 (.58,.8). (.,.38).5 (.9,.7).589 (.55,.63) e.54 (.499,.556) e α α DIC 987. Time mins 3 secs..3 Three level Normal response model For this example we se the chem97 data set consisting of 3 stdents nested within 79 schools within 3 Edcation athorities. y v k = ( Xβ ) + v + + e ~ N(,, v k j jk k ~ N(, e ~ N(, e (4) where we have the intercept and GCSE average score as predictors. Table 3 presents the reslts.

9 Table 3. Three level Normal response model for CHEM97 data Parameter β WINBUGS () (95% interval) β.473 (.44,.55).9 (.,.45) v.7 (.66,.8) 5.55 (5.69, 5.4) e DIC Time 8 mins. The level 3 variance is very poorly estimated, and in fact the mode of the distribtion is at zero as is shown by the following kernel density estimate A binary response model The dataset sed is the fertility srvey data BANG. In the fixed part the intercept and for covariates are fitted; rban, age, and for dmmy variables for nmber of children in family (,, 3+) with no children as base category. The logit link fnction model is logit( π ) = ( Xβ ) + y j ~ Bin(, π ) ~ N(, j For the probit link fnction we can write the model as: probit( π ) = ( Xβ ) + y j ~ Bin(, π ) ~ N(, j bt we can also estimate the parameters of the nderlying Normal. Consider the -level model y = ( XB) + + e j (5) (6) e ~ N(,) j ~ N(, (7)

10 where y is the probit that determines the probability of a correct response, namely y φ ( tdt ), φ( t) is pdf of N(,) (8) By inserting another step into the MCMC algorithm we can generate a chain from the nderlying Normal distribtion at level. Threshold models sch as (7) & (8) have a nmber of ses where the binary responses can be thoght of as generated from trncation of an nderlying continos distribtion. Frthermore, estimation sing this procedre is often faster since Gibbs sampling rather than, say, Metroplois Hastings or adaptive rejection methods can be sed. A similar device can be sed with the logistic link where we define an nderlying logistic distribtion. See Browne (3) for more details. In the present case we se the standard WINBUGS probit fnction and the model specification is as follows: model { # Level definition for(i in :N) { se[i] ~ dbin(p[i],denom[i]) probit(p[i]) <- beta[] * cons[i] + beta[] * rban[i] + beta[3] * agecentered[i] + beta[4] * one[i] + beta[5] * two[i] + beta[6] * three_[i] + [district[i]] * cons[i] # Higher level definitions for (j in :n) { [j] ~ dnorm(,ta.) # Priors for fixed effects for (k in :6) { beta[k] ~ dflat() # Priors for random terms ta. ~ dgamma(.,.) sigma. <- /ta.

11 Table 4. Two level binary response model for BANG data with logistic and probit link fnctions. Logit Probit Parameter Estimate (95% interval) Estimate (95% interval) β β.735 (.5,.963).45 (.38,.588) β -.8 (-.43, -.) -.7 (-.6, -.7) β.3 (.834,.448).676 (.486,.855) 3 β.43 (.63,.75).84 (.63,.48) 4 β.38 (.,.78).85 (.6,.53) 5.4 (.7,.437).86 (.4,.56) DIC Time 4mins. secs. 6 mins. Note that the DIC vales are almost identical sggesting that the models fit eqally well..5 A mltivariate model with missing responses The data set is GCSEMV, with two responses from ppils to a written and corsework assessment. The model sed is as follows: y y e e j j = β + β x = β + β x ~ N, e ~ N, e + j + j + e + e e The model specification is as follows, sing a similar formlation for the Wishart priors as with the random coefficient model. We have called the response vector MV_RESP. The covariate is gender (girls=). (9)

12 model { # Level definition for(i in :N) { MV_RESP[i,:] ~ dmnorm(m[i,:],ta.[:,:]) m[i,] <- beta[] * CONS[i] + beta[3] * FEMALE[i] + 3[MV_SCHOO[i],] * CONS[i] m[i,] <- beta[] * CONS[i] + beta[4] * FEMALE[i] + 3[MV_SCHOO[i],] * CONS[i] # Higher level definitions for (j in :n3) { 3[j,:] ~ dmnorm(zero3[:],ta.3[:,:]) # Priors for fixed effects for (k in :4) { beta[k] ~ dflat() # Priors for random terms ta.[:,:] ~ dwish(r[:,:],) sigma.[:,:] <- inverse(ta.[,]) for (i in :) {zero3[i] <- ta.3[:,:] ~ dwish(r3[:, :],) sigma.3[:,:] <- inverse(ta.3[,]) In WINBUGS missing vales are denoted by NA and are handled sing imptation for a mltivariate Normal response, althogh this is not available for a mltivariate t- distribtion response. Table 5 gives the reslts Table 5. Bivariate Normal response for written and corsework assessment. Written Corsework Parameter Estimate (95% interval) Estimate (95% interval) β 49.6 (47.8, 5.4) 69.7 (67.5, 7.) β -.53 (-3.6, -.38) 6.73 (5.37, 8.4) 49.7 (33., 7.7) 6.4 (., 47.7) 79.6 (54.4, 5.5) 4.9 (6.8, 33.7) e 73. (65., 8.6) e 8.6 (69.9, 93.4) e DIC 99.3 Time mins. 3 secs.

13 .6 Cross classified data The data set is LEV-XC consisting of examination reslts for stdents cross classified by primary and secondary school. The model is Y = β + β x e e i (3),sec( i) (),prim( i) i ~ N(, ~ N(, ~ N(, i e (3),sec( i) (3) ) () ) (), prim( i) i This ses the classification notation introdced by Browne et al. (). The sperscript denotes the classification nmber (not the level) where the ppil level term is at level, and the sperscript is omitted for convenience. The term sec(i)denotes the secondary school to which ppil I belongs and similarly prim(i) the primary school. The covariate is a dmmy variable for gender (girl=). x The WINBUGS model specification is as follows model { # Level definition for(i in :N) { ATTAIN[i] ~ dnorm(m[i],ta) m[i]<- beta[] * CONS[i] + beta[] * SEX[i] + [PID[i]] * CONS[i] + 3[SID[i]] * CONS[i] # Higher level definitions for (j in :n) { [j] ~ dnorm(,ta.) for (j in :n3) { 3[j] ~ dnorm(,ta.3) # Priors for fixed effects for (k in :) { beta[k] ~ dflat() # Priors for random terms ta ~ dgamma(.,.) sigma <- /ta ta. ~ dgamma(.,.) sigma. <- /ta. ta.3 ~ dgamma(.,.) sigma.3 <- /ta.3 Where PID is the ppil identification. The data are sorted on secondary school within primary school so that the first few ppil records for the secondary school data and for the primary school data are respectively () 9,9,9,9,9,9,,,9,9, 9,9,9,9,9,9,,9,9,9, 9,9,9,9,9,9,9,9,9,9, 9,9,9,9,9,9,,8,,9,,9,9,9,9,9,9,,9,, 9,9,9,9,7,7,7,7,7,7, 7,5,5,5,9,9,9,9,9,9, 6,,,,,,,,,,,,,,,,,,,,

14 ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,3,3,3,4,4,4,4,4,4, 4,5,5,5,5,5,5,5,5,5, The reslts are given in Table 6. Table 6. Two level cross classified model sing LEV-XC attainment data. Parameter β 5.53 Estimate (95% interval) β.498 (.34,.69).4 (.38,.94) (3).3 (.77,.6) () 8.6 (7.687, 8.457) e DIC 74.8 Time 8mins. secs. The wide interval estimate for the secondary school variance reflects the fact that there are only 9 sch schools in the data set..7 Mlticategorical response models The dataset sed is the social attitdes data with the response being the nmber of positive answers to seven qestions with seven codes being or,, 3, 4, 5, 6, 7. This scale is ordered bt for illstration we shall fit both ordered and nordered models. Each respondent has a response in each of 4 years. For the prpose of the present analysis we treat the years as independent replications so that there are jst 4 trials for each respondent. For the nordered case we fit the mltivariate logit model for t categories E( y ) = π ( s) ( s) ( s) π ( s) ( s) log () j, s=,..., t- t = β + π ~ MVN(, Ω ) j where we fit only intercept terms. The covariance matrix Ω is of order ( t ) ( t ) in general bt for present prposes we assme a common ()

15 single variance at level. For the ordered case we fit the model for the cmlative probabilities defined as s ( s) ( s) ( h) h= E( y ) = γ = π, s=,..., t () with γ = {+ exp [ β + β x + β x + β x + ] ( s) ( s) 3 3 j ~ N(, j (3) where we have inclded three covariates which are dmmy variables for the religios categories Roman Catholic, Protestant and others and where only the intercept (threshold) term varies across categories. The WINBUGS model specification code for the nordered and ordered models respectively is and The reslts are given in Table 7.

16 Table 7. Two level ordered and nordered mltinomial response model for SOCATT attitde data with logistic link fnction. Category 7 is the base category. Unordered Ordered (, iterations) Parameter Estimate (95% interval) Estimate (95% interval) () β -.93 (-.456, -.4) (-5.73, -.695) () β -.58 (-.697, -.833) -.6 (-3.45, -.95) (3) β.3 (-.7,.687).669 (-.67,.87) (4) β -.36 (-.76,.47).67 (.33,.83) (5) β -.38 (-.78,.4).58 (.9, 3.746) (6) β -.64 (-.653,.33) 3.57 (.84, 4.83) β -.48 (-3.59, -.86) β (-.33,.554) β (-4.396, -.669) (3.96, 8.69) (4.64, 7.) DIC Time 3. Docmentation, ser spport and general conclsions There is an on line manal spplied with the software which is a fairly basic description of the packages featres. There is no HELP system, and this wold be a very welcome ftre addition. There is a sefl ttorial example in the manal which, together with a series of worked examples with real datasets shold enable new sers to get started. There is a discssion list for sers of WINBUGS where isses are discssed. To access this go to This is an active list where sers share ideas and problems. In addition the developers of WINBUGS are happy to receive feedback; go to WINBUGS is a very impressive package, especially given its free distribtion policy. It is perhaps the best known Bayesian modelling package and is flexible enogh to handle a very wide range of models. In this review we have only toched pon a few of its facilities. Becase of its generality it is not necessarily as comptationally efficient as other packages which can fit MCMC models, sch as MlwiN (see the comparative tables) bt its greater model flexibility does allow new models to be developed. One drawback of WINBUGS is that is rather cmbersome to se if one wishes to carry ot extensive model exploration trying different model formlations. For this prpose one of the likelihood based packages cold be sed with final

17 estimates obtained from WINBUGS. Another isse is that there is a great deal of embedded statistical theory within WINBUGS which will make considerable demands on less statistically experienced sers. There are other isses associated with Bayesian modelling, sch as choice of priors, bt it is not the prpose of this review to enter into these. If yo want a very flexible MCMC Bayesian package for mltilevel modelling, WINBUGS is an excellent choice. References Browne, W. (998). Applying MCMC methods to mltilevel models. Statistics. Bath, University of Bath. Browne, W., Goldstein, H. and Rasbash, J. (). Mltiple membership mltiple classification (MMMC) models. Statistical Modelling : 3-4. Spiegelhalter, D., Best, N., Carlin, B. P. and Van der Linde, A. (). Bayesian measres of model complexity and fit (with discssion). Jornal of the Royal Statistical Society, B 64: