neutral bayesian reference models for incidence rates of ... · for incidence rates of (rare)...
TRANSCRIPT
Jouni Kerman Statistical Methodology, Novartis Pharma AG, Basel BAYES2012, May 10, Aachen
Neutral Bayesian reference models for incidence rates of (rare) clinical events
Outline
§ Motivation – why reference (default) models?
§ Selection criteria for the reference models
§ Investigating candidates for reference models
§ A proposal for Neutral reference models • Augmenting the proposed reference analysis with historical data
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Motivation
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Reference analyses for comparison
§ We do more and more complex analyses... • E.g., meta-analyses
“Reality check: are the results reasonable?”
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Reference analyses for comparison
§ Comparing with point estimates to reveal discrepancies
• Are the results reasonable?
• Any “excessive” shrinkage?
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Reference analyses for comparison
§ Plotting just the data points is not enough
• Must visualize the uncertainty around the point estimates
• Need simple Bayesian models to produce point estimates and “reference” uncertainty intervals !
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Reference analyses for comparison
§ Stratified analyses • Model the rate within a
single treatment (sub)group • Model a rate difference
(e.g., LoR, RR) for two (sub)groups
§ Pooled analyses • Analyses with pooled
studies/subgroups (i.e., assuming identical rates between studies or groups)
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Stratified and pooled reference analyses “Looking at the raw data”
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Stratified and pooled reference analyses “Looking at the differences”
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Reference (‘default’) analyses - Example: Safety
§ Example: Kidney transplantation; one single study
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Treatment Deaths at 12 months
A 7 / 251
B 9 / 274
C 6 / 384
Considering selection criteria for the reference models
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Binomial/Poisson models and shrinkage
§ Shrinkage is unavoidable ! • Consider y=0
• The point estimate and the length of the posterior intervals (with respect to the scale n) are determined completely by the prior
• (Recall: there are no “uninformative” models...)
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Illustration: Binomial-beta conjugate model
with prior Beta(a, a)
Binomial/Poisson models and shrinkage
§ Shrinkage is unavoidable ! • Consider y=1 • The point estimate and the
posterior intervals are strongly influenced by the prior:
Pr( θ > y/n | y ) > 0.74 or Pr( θ > y/n | y ) > 0.37 ?
• As y increases, influence of the prior is diminished, but N can be arbitrarily large
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Illustration: Binomial-beta conjugate model
with prior Beta(a, a)
Choosing a reference model
§ The choice of shrinkage ... is yours • By choosing a reference
model, we are in fact deciding on the amount of shrinkage
• What is an acceptable “default amount of shrinkage” ?
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Neutrality as a criterion
§ A neutral model for rates and proportions • Pr( θ > MLE | y ) ≈ 50%
consistently for all possible outcomes and sample sizes whenever the MLE is not at the boundary of the parameter space
• “A priori doesn’t favor high or low values relative to the MLE (sample mean)”
• Exact neutrality cannot be achieved – but some priors are “more neutral” than others
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MLE=0.2; median = dotted line Pr( θ > MLE | y ) = 50.2%
Neutrality for the differences
§ A neutral default model • Pr(θ1 - θ2 > d | y ) ≈ 50% • where d is the observed
difference – on some scale, e.g. log or logit or original scale
• Equivalently, ‘d’ should be as close to the posterior median as possible
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A reference model should provide
neutral inferences for both rates and
differences
Investigating candidates for reference models
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Candidates for reference models (Binomial)
§ Conjugate models • yi ~ Binomial(ni, θi), i=1, 2 • θi ~ Beta(a, a); a in (0, 1)
§ Logistic regression with different parameterizations and different vague prior distributions (Normal or scaled Student’s t) – total 116 models
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Model “A” Model “B” Model “C”
logit(θ1) =
µ1 µ
µ - Δ / 2
logit(θ2) = µ2 µ + Δ
µ + Δ / 2
Candidates for reference models(Poisson)
§ Conjugate models • yi ~ Binomial(ni, θi), i=1, 2 • θi ~ Gamma(a, 0); a in (0, 1)
§ Poisson regression (log link) with different parameterizations and different vague prior distributions (Normal or scaled Student’s t) – total 116 models
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Model “A” Model “B” Model “C”
log (θ1) =
µ1 µ
µ - Δ / 2
log (θ2) = µ2 µ + Δ
µ + Δ / 2
An apparent ‘bias’ in rate estimates An example
§ A “noninformative” analysis ? • y=1 event out of n=1000 • Statisticians (a), (b), and (c) use
different “noninformative” models
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Median estimate
Pr( est > 0.001 | y ) Model
(a) 0.7 / 1000 36.8% Beta(0.01, 0.01)
(b) 1.0 / 1000 50.8% Beta(1/3, 1/3)
(c) 1.7 / 1000 73.5% Beta(1, 1)
An apparent ‘bias’ in log-risk ratio estimates An example
§ A “noninformative” analysis ? • Experimental: y=3 events out of n=1000 • Placebo: y=1 events out of n=1000 • Statisticians (a), (b), and (c) use different “noninformative” models
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Median odds
Pr( odds > 3 | y )
Model Priors
(a) 3.9 58% “C” µ ~ N(0,1002) Δ ~ N(0,102)
(b) 2.95 49% “A” µ1 ~ N(0,52) µ2 ~ N(0, 52)
(c) 2.25 39% “B” µ ~ N(0,52) Δ ~ N(0,2.52)
Asymmetric estimates in log-risk ratio estimates An example
§ A “noninformative” analysis ? • Experimental: y=1 events out of n=1000 • Placebo: y=1 events out of n=1000 • Statisticians (a), (b), and (c) use different “noninformative” models
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Median odds
Pr( odds > 3 | y )
Logistic Model
Priors
(a) 0.64 65% “B” µ ~ N(0,52) Δ ~ N(0,52)
(b) 0.90 47% “B” µ ~ t(0,10, 5) Δ ~ t(0,5, 5)
(c) 1.00 50% “B” µ ~ N(0,1002) Δ ~ N(0, 52)
“What is your point estimate?”
A proposal for default models
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Neutral models for proportions and probabilities
§ The Binomial-Beta conjugate model with shape parameter 1/3 • y ~ Binomial(θ, n) • θ ~ Beta(1/3, 1/3)
• Behaves consistently, for all sample sizes n and outcomes y
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Neutral models for rates
§ Poisson-Gamma conjugate model with the shape parameter 1/3 • y ~ Poisson(λX) • X = exposure • λ ~ Gamma(1/3, 0)
• Behaves consistently, for all exposures X and outcomes y
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Neutral models for differences and ratios
§ Treatment groups are estimated separately, then differences computed • E.g., the Binomial-beta model:
• ( θ1 | y ) ~ Beta(1/3 + y1, 1/3 + n1 - y1) • ( θ2 | y ) ~ Beta(1/3 + y2, 1/3 + n2 – y2)
• Compute δ = θ2 - θ1 • Compute Δ’ = logit(θ2) - logit(θ1)
• E.g., by simulation
• Δ and δ are neutral – approximately centered at the point estimate - consistently
• Δ and δ are symmetric when y, n are equal in both groups
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Behavior of the Binomial models
§ The Beta(1/3, 1/3) conjugate model behaves the most consistently
§ Displayed: max. absolute bias (%) for estimated rates or odds in all models
§ (Worst case scenario, y=1 for one of the arms)
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Beta(1/3, 1/3)
Behavior of the Poisson models
§ The Gamma(1/3, 0) conjugate model behaves the most consistently
§ Displayed: max. absolute bias (%) for estimated rate or rate ratio in all models
§ (Worst case scenario, y=1 for one of the arms)
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Gamma(1/3, 0)
Neutral models for differences and ratios
§ Examples of ‘worst cases’ (one group has y=1)
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Data 1 Data 2 Median point
estimateθ1
Median point
estimate θ2
Median odds
estimate
Pr( odds > obs | y )
1/1000 2/1000 0.0010 0.0020 2.0 50%
1/1000 3/1000 0.0010 0.0030 3.0 50%
1/1000 4/1000 0.0010 0.0040 3.9 50%
1/1000 5/1000 0.0010 0.0050 4.9 50%
Example: Meta-analysis
§ Viewing posterior intervals from many multilevel models at once
§ Green: pooled
§ Gray: fully stratified reference intervals
30 | Statistical Methodology Science VC | Jouni Kerman | Nov 9, 2010 | Analyzing Proportions and Rates using Neutral Priors
Augmenting the default analysis with external information
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Augmenting the default reference analysis Binomial model
§ A family of informative Beta priors
Beta(1/3 + mp, 1/3 + m(1-p))
• Fix ‘p’ (a priori observed point estimate)
• Use ‘m’ to adjust prior precision • Beta(1/3, 1/3) is the “prior of all
priors” • Neither shape parameter ever < 1/3
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meansamplenm
npnm
mmedianposterior+
++
≈
Augmenting the default reference analysis Poisson model
§ A family of informative Gamma conjugate priors
Gamma(1/3 + ky, kX)
• Fix ‘y / X’ (a priori observed point estimate)
• Use ‘k’ within (0,1) to adjust prior precision
• Gamma(1/3, 0) is the “prior of all priors”
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Conclusion
§ The classical point estimates (sample means and their differences) remain the reference points that are inevitably compared to model-based inferences
§ Recognizing that shrinkage is unavoidable in these count data models, we propose (approximate) neutrality as a criterion for reference models
§ The proposed conjugate models perform consistently for all outcomes and sample sizes • Symmetry and minimal “bias” • Easily computable without MCMC • Intuitively augmentable by external information
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References
§ Kerman (2011) Neutral noninformative and informative conjugate beta and gamma prior distributions. Electronic Journal of Statistics 5:1450-1470
§ Kerman (2012) Neutral Bayesian reference models for incidence rates of clinical events (Working paper)
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A look at the neutral Beta prior (Log-odds scale)
• Beta(1, 1) – Uniform • Beta(1/2, 1/2) – “Jeffreys”
• Beta(1/3, 1/3) – “Neutral” • Beta(0.001, 0.001) – “Approximate Haldane”
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Reference model candidates investigated Binomial & Poisson regression models
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For µ For Δ
Normal model
σ = 3.3, 5, 10, 100 σ = 2.5, 5, 10
Student-t model
Scale = 3.3, 5, 10, 100 Df = 2, 5, 10
Scale = 2.5, 3.3, 5, 10 Df = same as for µ
Possible reference models (Binomial) yi ~ Binomial(ni, θi), i=1, 2
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Beta Normal
Scaled t
A θi ~ Beta(a, a) δ = θ2 - θ1
logit(θi) ~ N(0, σ2) δ = logit(θ2) - logit(θ1)
logit(θi) ~ N(0, σ2) δ = logit(θ2) - logit(θ1)
B logit(θ1) ~ N(0, σ12)
δ ~ N(0, σ22)
θ2 = logit(θ1) + δ
logit(θ1) ~ t(0, σ1, df1) δ ~ t(0, σ2, df2) θ2 = logit(θ1) + δ
C logit(µ) ~ N(0, σ12)
δ ~ N(0, σ22)
θ1 = logit(µ) - δ / 2 θ2 = logit(µ) + δ / 2
logit(µ) ~ t(0, σ1, df1) δ ~ t(0, σ2, df2) θ1 = logit(µ) - δ / 2 θ2 = logit(µ) + δ / 2
Possible reference models (Poisson) yi ~ Poisson(Xiθi), i=1, 2
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Gamma Normal
Scaled t
A θi ~ Gamma(a, ε) δ = θ2 - θ1
log (θi) ~ N(0, σ2) δ = log (θ2) - log (θ1)
log (θi) ~ N(0, σ2) δ = log (θ2) - log (θ1)
B log (θ1) ~ N(0, σ12)
δ ~ N(0, σ22)
θ2 = log (θ1) + δ
log (θ1) ~ t(0, σ1, df1) δ ~ t(0, σ2, df2) θ2 = log (θ1) + δ
C log (µ) ~ N(0, σ12)
δ ~ N(0, σ22)
θ1 = log (µ) - δ / 2 θ2 = log (µ) + δ / 2
log (µ) ~ t(0, σ1, df1) δ ~ t(0, σ2, df2) θ1 = log (µ) - δ / 2 θ2 = log (µ) + δ / 2