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Introduction to Bayesian Statistical Software:
WinBUGS 1.4.3
Beth Devine, PharmD, MBA, PhDRafael Alfonso, MD, PhC
Evidence Synthesis9/22/2011
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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Comparative Effectiveness of Biologic Therapies in Rheumatoid Arthritis (RA): An Indirect Treatment Comparisons Approach
Beth Devine, PharmD, MBA, PhDRafael Alfonso-Cristancho, MD, MSSean Sullivan, BSPharm, PhDPharmaceutical Outcomes Research & Policy Program University of Washington
Pharmacotherapy 2011;31:39–51
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WinBUGS – Model Syntax
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WinBUGS – Load Data
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WinBUGS – Model Compiled
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WinBUGS – Model Initialized
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WinBUGS – Update (Burn-in)
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WinBUGS – Check Convergence
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WinBUGS – Obtaining Posterior Inference
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WinBUGS – Viewing Summary Statistics
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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
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Now Introducing our Practice Dataset
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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/ orLunn, Thomas, Best, Speigelhalter. Stat Comput 2000;10:325-37
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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
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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
Failure/Dead Success/Alive
New Treatment A B
Control C D
RCT
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Fixed Effects Model
• Statistical homogeneity• Formally assume:
Yi = Normal(d,Vi)
• We estimate the commontrue effect, d
True effect=d
Point estimate=Yi
Random error=Vi
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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
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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)]
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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
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Example: Meta-analysis, RCTs of effect of aspirin preventing death after acute MIsStudy Aspirin Group Placebo Group
Deaths Total Deaths Total
MRC-1 49 615 67 624
CDP 44 758 64 771
MRC-2 102 832 126 850
GASP 32 317 38 309
PARIS 85 810 52 406
AMIS 246 2267 219 2257
ISIS-2 1570 8587 1720 8600
Fleiss. Statistical Methods in Medical Research 1993
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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
Survive Die
Aspirin 566 49
Placebo 557 67
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Example: Aspirin Data to be Combined
Study OR LnOR (Yi) Var(lnOR) (Vi) Weight (1/Vi)
MRC-1 1.39 0.33 0.04 25.71
CDP 1.47 0.39 0.04 24.29
MRC-2 1.25 0.22 0.02 48.77
GASP 1.25 0.22 0.06 15.44
PARIS 1.25 0.23 0.02 28.41
AMIS 0.88 -0.12 0.01 103.92
ISIS-2 1.12 0.11 0.002 664.26
Note: ISIS-2 with small variance and large weight (1/0.002)
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Now It’s Your Turn: Practice using WinBUGS!
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Launch WinBUGS
• Click on WinBUGS14.exe• Click File-Open• Load aspirin FE.odc
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Components of WinBUGS .odc file
Model {< Likelihood><Prior distributions>
}#Data<List or column format>#Starting Values<List or mixture of list and column
format>
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Steps for Running a Model in WinBUGS
1. 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.
2. Bring up Model Specification Tool (menu: Model -> Specification)3. Click “check model”
• Should see “model is syntactically correct” in lower left corner of window.
4. Highlight first row of data containing variable labels (if in rectangular format)
5. Click “load data”• Should see “data is loaded” in lower left corner of window.
6. If using multiple chains, enter number in “num of chains” box. Otherwise, proceed.
7. Click “compile”• Should see “model is compiled” in lower left corner of window.
8. Highlight line containing initial values: list(…)9. Click “load inits”
• If using multiple chains, you will need to repeat steps 8-9 for each chain.
• Should see “model is initialized.”10. Bring up Sample Monitor Tool (menu: Inference -> Samples)
• Enter name of each node you wish to monitor and click “set”11. Bring up Update Tool (menu: Model -> Update)12. Enter a number of samples to take and click “update.”
• Should see “model is updating.”
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Load and Check Model
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Load and Check Data
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Compile ModelCompile
Model
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Load Initials
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Pooled OR: median 1.12 (1.05 to 1.19)
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Random Effects Model• Model
– Within studies• Yi ~Normal(i,Vi)
– Across studies• I ~Normal(d,2)
• d=solid line• =dotted lines• 2 = variability between studies• (heterogeneity)
True Mean Effect=d solid line
Trial-specific effects=dotted lines
Y5
5
Vi
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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 2 as well as for d
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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
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Generic Random Effects Model• Load aspirin RE.odc
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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
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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)
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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
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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!
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Looking to the Future (The Future is Here!)
Data Sources
RCT1 RCT2 Obs 1 Routine Care
Meta-Analysis
General Synthesis
Bayes Theorem Combination
Evidence Synthesis
Clinical Effects
AdverseEffects Utility Costs
Model Inputs (w/ uncertainty)
DecisionModel
Utility
Utility
Utility
Utility
Utility
Utility
Utility Cost
Cost
Cost
Cost
Cost
Cost
Cost
Tx A Fib
Warfarin
NoWarfarin
No Strk
Stroke
Stroke
No Strk
Bleed
NoBld
Bleed
NoBld
NoBld
NoBld
Bleed
Bleed
Utility
Cost
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[email protected]@uw.edu
Attribution for Fleiss example to Keith Abrams, University of Leicester, UK