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Introduction to Bayesian Statistical Software: WinBUGS 1.4.3

Introduction to Bayesian Statistical Software: WinBUGS 1.4.3. Beth Devine, PharmD, MBA, PhD Rafael Alfonso, MD, PhC Evidence Synthesis 9/22/2011 2:30 pm. Supported by the Institute of Translational Health Sciences, Grant NIH 3 UL1 RR 025014-04S2 and the UW CHASE Alliance.

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Introduction to Bayesian Statistical Software: WinBUGS 1.4.3

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  1. Introduction to Bayesian Statistical Software:WinBUGS 1.4.3 Beth Devine, PharmD, MBA, PhD Rafael Alfonso, MD, PhC Evidence Synthesis 9/22/2011 2:30 pm Supported by the Institute of Translational Health Sciences, Grant NIH 3 UL1 RR 025014-04S2 and the UW CHASE Alliance

  2. Comparative Effectiveness of Biologic Therapies in Rheumatoid Arthritis (RA): An Indirect Treatment Comparisons Approach • Beth Devine, PharmD, MBA, PhD • Rafael Alfonso-Cristancho, MD, MS • Sean Sullivan, BSPharm, PhD • Pharmaceutical Outcomes Research & Policy Program • University of Washington • Pharmacotherapy 2011;31:39–51

  3. WinBUGS – Model Syntax

  4. WinBUGS – Load Data

  5. WinBUGS – Model Compiled

  6. WinBUGS – Model Initialized

  7. WinBUGS – Update (Burn-in)

  8. WinBUGS – Check Convergence

  9. WinBUGS – Obtaining Posterior Inference

  10. WinBUGS – Viewing Summary Statistics

  11. WinBUGS – Interpreting Summary Statistics • Check start and sample columns – 10,000 to 30,000 • Rename your parameters • Assess accuracy of posterior estimates by calculating Monte Carlo error for each parameter: • Rule of thumb: MC error should be < 5% of sample standard deviation • Exponentiate median log odds to odds ratios

  12. Now Introducing our Practice Dataset

  13. Advantage of Bayesian analysis in ITC/ MTC is that it allows calculation of the probability of which treatment is best http://www.mrc-bsu.cam.ac.uk/bugs/ or Lunn, Thomas, Best, Speigelhalter. Stat Comput 2000;10:325-37

  14. Outcome Measures • How is your outcome of interest measured? • Binary (e.g. dead or alive) • Continuous (e.g blood pressure) • Categorical/ordinal (e.g. severity scale) • Binary outcomes most common • We will consider here • Continuous • Similar approach to binary • Ordinal • More complex and more rare

  15. Binary Outcome Measures • Binary outcome data from a comparative study can be expressed in a 2 x 2 table • Three common outcome measures: • Odds ratios, risk ratios, risk differences RCT

  16. Fixed Effects Model Random error=Vi • Statistical homogeneity • Formally assume: Yi = Normal(d,Vi) • We estimate the common true effect, d Point estimate=Yi True effect=d

  17. Generic Fixed Effect • Yi ~ Normal(d,Vi) where i= 1…….N studies • Yi is the observed effect in study i with Variance Vi • All studies assumed to be measuring the same underlying effect size, d • For a Bayesian analysis, a prior distribution must be specified for d

  18. Choice of Prior for d • Often, amount of information in studies is large enough to render any prior of little importance – therefore choice not critical • Often specified as “vague” or “flat” • E.g. If meta-analysis is on ln(OR) scale, could specify d~Normal (0, 105) • This states a priori we would be 95% certain that true value of d is between [0±1.96( 105)]

  19. Fixed Effect with Prior • Yi ~ Normal(d,Vi) where i= 1…….N studies • d ~ Normal(0, 105) • Models are specified in WinBUGS using formulas similar to this algebra • Note: Normal distributions are specified by mean and ‘precision’ • where precision = 1/variance • Estimate model parameter using MCMC, rather than inverse weighting of variance

  20. Example: Meta-analysis, RCTs of effect of aspirin preventing death after acute MIs Fleiss. Statistical Methods in Medical Research 1993

  21. Example: Calculation: Log(OR) & Variance • For MCR-1 • OR=(566*67)/ (557*49) = 1.389 • Log(ln)OR = 0.3289 • VariancelnOR = 1/566 + 1/49 + 1/557 + 1/67 = 0.0389 • Note- this is OR for Survival • If 2x2 table contains any zeros, common to add 0.5 to those cells before calculations

  22. Example: Aspirin Data to be Combined Note: ISIS-2 with small variance and large weight (1/0.002)

  23. Now It’s Your Turn: Practice using WinBUGS!

  24. Launch WinBUGS • Click on WinBUGS14.exe • Click File-Open • Load aspirin FE.odc

  25. Components of WinBUGS .odc file Model { < Likelihood> <Prior distributions> } #Data <List or column format> #Starting Values <List or mixture of list and column format>

  26. Steps for Running a Model in WinBUGS • Make model active. • Doodles: • If in own window, click title bar. • If in compound document, double-click the doodle (should have “hairy” border). • Text: Simply highlight the word “model” at the beginning of your model. • Bring up Model Specification Tool (menu: Model -> Specification) • Click “check model” • Should see “model is syntactically correct” in lower left corner of window. • Highlight first row of data containing variable labels (if in rectangular format) • Click “load data” • Should see “data is loaded” in lower left corner of window. • If using multiple chains, enter number in “num of chains” box. Otherwise, proceed. • Click “compile” • Should see “model is compiled” in lower left corner of window. • Highlight line containing initial values: list(…) • Click “load inits” • If using multiple chains, you will need to repeat steps 8-9 for each chain. • Should see “model is initialized.” • Bring up Sample Monitor Tool (menu: Inference -> Samples) • Enter name of each node you wish to monitor and click “set” • Bring up Update Tool (menu: Model -> Update) • Enter a number of samples to take and click “update.” • Should see “model is updating.”

  27. Load and Check Model

  28. Load and Check Data

  29. Compile Model Compile Model

  30. Load Initials

  31. Pooled OR: median 1.12 (1.05 to 1.19)

  32. Random Effects Model True Mean Effect=d solid line 5 Vi Y5 • Model • Within studies • Yi ~Normal(i,Vi) • Across studies • I~Normal(d,2) • d=solid line • =dotted lines • 2 = variability between studies • (heterogeneity) Trial-specific effects=dotted lines

  33. Generic Random Effect • Yi ~ Normal(,Vi) where i= 1…….N studies • i~ Normal(d,2 ) • As for fixed effect, Yi is observed effect in study i with variance Vi • Now study specific effects, I are allowed to be different from each other and are assumed to be sampled from a Normal distribution with mean d and variance 2 • For a Bayesian analysis, a prior distribution is required for 2as well as for d

  34. Choice of Prior for 2 • This is a little trickier than for d • Variances cannot be negative so Normal distribution is not a good choice • Examples in WinBUGS Manual use Uniform distribution. E.g. ~ Uniform (0,10) •  of 10 is massive, because we are working with ORs; even  of 1 or 2 is large • Specification of vague priors on variance components is complex and is an active area of research

  35. Generic Random Effects Model • Load aspirin RE.odc

  36. Results of Aspirin RE model • Pooled OR: median 1.149 (0.976-1.434) • OR now contains 1 • Bayesian CrI wider than classical CI • 2 is random variable and uncertainty is included in pooled result

  37. Compare our Two Odds Ratios and CrIs • Fixed effects Normal Distribution • OR=1.12 (95% CrI: 1.05, 1.19) • Random Effects Normal Distribution • OR=1.15 (95% CrI: 0.97, 1.44)

  38. MCMC Basics • Now that we’ve run a few models consider sensitivity analyses • Sensitivity to prior distributions • esp. important for distributions of variance/precision parameters • Sensitivity to initial values • Multiple chains using very different starting values & comparing using Brooks Gelman-Rubin Statistics • Length of “burn in”: examine history/trace plots

  39. Interpreting Random Effects • A single parameter cannot adequately summarize heterogeneous effects • Therefore estimation and reporting of 2 is important • This tells us how much variability there is between estimates from the population of studies • In some instances studies contain both beneficial and harmful effects, so important!

  40. Looking to the Future (The Future is Here!) Data Sources Routine Care RCT1 RCT2 Obs 1 Evidence Synthesis Meta-Analysis General Synthesis Bayes Theorem Combination Model Inputs (w/ uncertainty) Clinical Effects Adverse Effects Utility Costs Bleed Utility Cost Stroke NoBld Utility Cost Warfarin Bleed No Strk Utility Cost NoBld Tx A Fib Decision Model Utility Cost Bleed Stroke Utility Cost NoWarfarin NoBld Utility Cost Bleed No Strk Utility Cost NoBld Utility Cost

  41. Questions?bdevine@uw.edu ralfonso@uw.edu Attribution for Fleiss example to Keith Abrams, University of Leicester, UK

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