Introduction to WinBUGS Olivier Gimenez

A brief history  1989: project began with a Unix version called BUGS  1998: first Windows version, WinBUGS was born  Initially developed by the MRC Biostatistics Unit in Cambridge and now joint work with Imperial College School of Medicine at St Mary's, London.  Windows Bayesian inference Using Gibbs Sampling  Software for the Bayesian analysis of complex statistical models using Markov chain Monte Carlo (MCMC) methods

Nicky Best Imperial College Faculty of Medicine, London (UK) Thomas Andrew University of Helsinki, Helsinki (Finland) David Spiegelhalter MRC Biostatistics Unit Institute of Public Health Cambridge (UK) Wally Gilks MRC Biostatistics Unit Institute of Public Health Cambridge (UK) Who?

How to obtain and install WinBUGS 1.downloadable from: (... see section Obtaining the File to download WinBUGS14.exe ) 2.to install WinBUGS: Exit all other programs currently running (particularly if using Windows XP); Copy WinBUGS14.exe to your computer; Double click on WinBUGS14.exe and follow the instructions in the dialog box; You should have a new directory called WinBUGS14 within Program Files; Inside the WinBUGS14 directory is a program called WinBUGS14.exe; Right-click on the pretty WinBUGS icon, select `create shortcut', then drag this shortcut to the desktop; Double click on WinBUGS14.exe to run WinBUGS14.

How to obtain and install WinBUGS  To obtain the key for unrestricted use: Fill in the registration form ~1h later, you will receive an from Bugs with subject WinBUGS registration - AUTOMATIC RESPONSE! Follow the instructions...  Congratulations, you're ready to use WinBUGS

Principle  You specify the prior and build up the likelihood  WinBUGS computes the posterior by running a Gibbs sampling algorithm, based on: (|D) / L(D|) ()  WinBUGS computes some convergence diagnostics that you have to check

Demographic components (fecundity, breeding success, survival, etc…) A biological example throughout White stork (Ciconia ciconia) in Alsace Climate (Temperature, rainfall, etc…)

WinBUGS & Linear Regression Y Number of chicks per pairs T Temp. May (°C) R Rainf. May (mm)

1. Do temperature and rainfall affect the number of chicks? Y i =  +  r R i +  t T i +  i, i=1,...,23  i i.i.d. ~ N(0, 2 ) 2. Regression model: 3. Estimation of parameters: ,  r,  t,  4. Frequentist inference uses t-tests Y i i.i.d. ~ N( i, 2 ), i=1,...,23  i =  +  r R i +  t T i WinBUGS & Linear Regression

Y Number of chicks per pairs T Temp. May (°C) R Rainf. May (mm) EstimateStd. Error t value Pr(>|t|) temperature rainfall * Linear Regression using Frequentist approach Y = T R

Y Number of chicks per pairs T Temp. May (°C) R Rainf. May (mm) Linear Regression using Frequentist approach  Influence of Rainfall only EstimateStd. Error t value Pr(>|t|) temperature rainfall * Y = T R

Running WinBUGS What do you need? 1 - a model giving the likelihood and the priors 2 - some data of course3 - initial values to start the MCMC algorithm

Running WinBUGS The model use the WinBUGS command 'model' don't forget to embrace the model with {...} Define the likelihood... Y i ~ N( +  r R i +  t T i, 2 ) Note:  2 = 1/ Specify the priors We use noninformative or vague or flat priors here Monitor any other parameter you'd like to... e.g.  2 = 1/

Running WinBUGS Data and initial values Use 'list' structures (R/Splus syntax)......and 'vector' structures (R/Splus syntax)

Running WinBUGS Overall 1 - a model giving the likelihood and the priors 2 - data3 - initial values

1- check model 2- load data 3- compile model 4- load initial values 5- generate burn-in values 6- parameters to be monitored 7- perform the sampling to generate posteriors 8- check convergence and display results Running WinBUGS At last!!

Running WinBUGS 1. Check model

Running WinBUGS 1. Check model: highlight 'model'

Running WinBUGS 1. Check model: open the Model Specification Tool

Running WinBUGS 1. Check model: Now click 'check model'

Running WinBUGS 1. Check model: Watch out for the confirmation at the foot of the screen

Running WinBUGS 2. Load data: Now highlight the 'list' in the data window

Running WinBUGS 2. Load data: then click 'load data'

Running WinBUGS 2. Load data: watch out for the confirmation at the foot of the screen

Running WinBUGS 3. Compile model: Next, click 'compile'

Running WinBUGS 3. Compile model: watch out for the confirmation at the foot of the screen

Running WinBUGS 4. Load initial values: highlight the 'list' in the data window

Running WinBUGS 4. Load initial values: click 'load inits'

Running WinBUGS 4. Load initial values: watch out for the confirmation at the foot of the screen

Running WinBUGS 5. Generate Burn-in values: Open the Model Update Tool

Running WinBUGS 5. Generate Burn-in values: Give the number of burn-in iterations (1000)

Running WinBUGS 5. Generate Burn-in values: click 'update' to do the sampling

Running WinBUGS 6. Monitor parameters: open the Inference Samples Tool

Running WinBUGS 6. Monitor parameters: Enter 'intercept' in the node box and click 'set'

Running WinBUGS 6. Monitor parameters: Enter 'slope_temperature' in the node box and click 'set'

Running WinBUGS 6. Monitor parameters: Enter 'slope_rainfall' in the node box and click 'set'

Running WinBUGS 7. Generate posterior values: enter the number of samples you want to take (10000)

Running WinBUGS 7. Generate posterior values: click 'update' to do the sampling

Running WinBUGS 8. Summarize posteriors: Enter '*' in the node box and click 'stats'

Running WinBUGS 8. Summarize posteriors: mean, median and credible intervals

Running WinBUGS 8. Summarize posteriors: 95% Credible intervals tell us the same story EstimateStd. Error t value Pr(>|t|) temperature rainfall *

Running WinBUGS 8. Summarize posteriors: 95% Credible intervals tell us the same story EstimateStd. Error t value Pr(>|t|) temperature rainfall *

Running WinBUGS 8. Summarize posteriors: click 'history'

Running WinBUGS 8. Summarize posteriors: click 'auto cor'  Problem of autocorrelation

Coping with autocorrelation use standardized covariates

Coping with autocorrelation use standardized covariates

Re-running WinBUGS 1,2,...7, and 8. Summarize posteriors: click 'auto cor'  autocorrelation OK

Re-running WinBUGS 1,2,...7, and 8. Summarize posteriors: click 'density'

Re-running WinBUGS 1,2,...7, and 8. Summarize posteriors: click 'quantiles'

Running WinBUGS 8. Checking for convergence using the Brooks-Gelman-Rubin criterion A way to identify non-convergence is to simulate multiple sequences for over-dispersed starting points Intuitively, the behaviour of all of the chains should be basically the same. In other words, the variance within the chains should be the same as the variance across the chains. In WinBUGS, stipulate the number of chains after 'load data' and before 'compile' (obviously, as many sets of initial values as chains have to be loaded, or generated)

Running WinBUGS 8. Checking for convergence using the Brooks-Gelman-Rubin criterion The normalized width of the central 80% interval of the pooled runs is green The normalized average width of the 80% intervals within the individual runs is blue

Re-running WinBUGS 1,2,...7, and 8. Summarize posteriors: others... Click 'coda' to produce lists of data suitable for external treatment via the Coda R package Click 'trace' to produce dynamic history changing in real time

/ / / / / / / / / / / / / / / / / / / / / / / 23 Another example: logistic regression Y Proportion of nests with success (>0 youngs) T Temp. May (°C) R Rainf. May (mm)

Performing a logistic regression with WinBUGS

model # succ. in year i ~ Bin(p i, total # couples in year i) where p i the probability of success in year i logit(p i )= +  r R i +  t T i, i=1,...,23 Performing a logistic regression with WinBUGS

noninformative priors Performing a logistic regression with WinBUGS

data & initial values Performing a logistic regression with WinBUGS

the results influence of rainfall, but not temperature (see credible intervals) lowerupper

additional parameters as a by-product of the MCMC samples: just add them in the model as parameters to be monitored - geometric mean: geom <- pow(prod(p[]),1/N) - odds-ratio: odds.rainfall <- exp(slope.rainfall) odds.temperature <- exp(slope.temperature) Performing a logistic regression with WinBUGS the results

additional parameters as a by-product of the MCMC samples - geom. mean probability of success around 82% [81%;84%] - odds-ratio: -16% for an increase of rainfall of 1 unit Performing a logistic regression with WinBUGS the results

Climatic conditions affect survival in the wild a) European dippers in Eastern France ( ) Practical session 1 Estimating survival using capture-recapture models CJS model: time-dependent survival and detection rates Case study a): A flood occured in the 1983 breeding season so that survival rates are expected to differ between flood ( f ) and non-flood ( n ) years  use constraints to estimate parameters  f and  n

Climatic conditions affect survival in the wild b) Lapwings in UK ( ) Practical session 2 Estimating survival using ring-recovery models Case study b): Number of days that the mean temperature is below freezing in central England (fdays) is expected to affect Lapwings survival  use fdays as a covariate Strong assumption: variation in survival is totally explained by the covariate alone  incorporate random effects