Neutral Bayesian reference models for incidence rates of (rare) clinical events

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Neutral Bayesian reference models for incidence rates of (rare) clinical events Jouni Kerman Statistical Methodology, Novartis Pharma AG, Basel BAYES2012, May 10, Aachen

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 2 BAYES2012 J Kerman May 10 Neutral reference analyses

3 BAYES2012 J Kerman May 10 Neutral reference analyses Motivation

Reference analyses for comparison We do more and more complex analyses... E.g., meta-analyses Reality check: are the results reasonable? 4 BAYES2012 J Kerman May 10 Neutral reference analyses

Reference analyses for comparison Comparing with point estimates to reveal discrepancies Are the results reasonable? Any excessive shrinkage? 5 BAYES2012 J Kerman May 10 Neutral reference analyses

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! 6 BAYES2012 J Kerman May 10 Neutral reference analyses

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) 7 BAYES2012 J Kerman May 10 Neutral reference analyses

Stratified and pooled reference analyses Looking at the raw data 8 BAYES2012 J Kerman May 10 Neutral reference analyses

Stratified and pooled reference analyses Looking at the differences 9 BAYES2012 J Kerman May 10 Neutral reference analyses

Reference ( default ) analyses - Example: Safety Example: Kidney transplantation; one single study Treatment Deaths at 12 months A 7 / 251 B 9 / 274 C 6 / 384 10 BAYES2012 J Kerman May 10 Neutral reference analyses

Considering selection criteria for the reference models 11 BAYES2012 J Kerman May 10 Neutral reference analyses

Binomial/Poisson models and shrinkage Shrinkage is unavoidable! Consider y=0 Illustration: Binomial-beta conjugate model with prior Beta(a, a) 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...) 12 BAYES2012 J Kerman May 10 Neutral reference analyses

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 Illustration: Binomial-beta conjugate model with prior Beta(a, a) 13 BAYES2012 J Kerman May 10 Neutral reference analyses

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? 14 BAYES2012 J Kerman May 10 Neutral reference analyses

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 MLE=0.2; median = dotted line Pr( θ > MLE y ) = 50.2% 15 BAYES2012 J Kerman May 10 Neutral reference analyses

Neutrality for the differences A neutral default model Pr(θ 1 - θ 2 > 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 A reference model should provide neutral inferences for both rates and differences 16 BAYES2012 J Kerman May 10 Neutral reference analyses

Investigating candidates for reference models 17 BAYES2012 J Kerman May 10 Neutral reference analyses

Candidates for reference models (Binomial) Conjugate models y i ~ Binomial(n i, θ i ), i=1, 2 θ i ~ 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 Model A Model B Model C logit(θ 1 ) = µ 1 µ µ - Δ / 2 logit(θ 2 ) = µ 2 µ + Δ µ + Δ / 2 18 BAYES2012 J Kerman May 10 Neutral reference analyses

Candidates for reference models(poisson) Conjugate models y i ~ Binomial(n i, θ i ), i=1, 2 θ i ~ 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 Model A Model B Model C log (θ 1 ) = µ 1 µ µ - Δ / 2 log (θ 2 ) = µ 2 µ + Δ µ + Δ / 2 19 BAYES2012 J Kerman May 10 Neutral reference analyses

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 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) 20 BAYES2012 J Kerman May 10 Neutral reference analyses

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 Median odds Pr( odds > 3 y ) Model Priors (a) 3.9 58% C (b) 2.95 49% A (c) 2.25 39% B µ ~ N(0,100 2 ) Δ ~ N(0,10 2 ) µ 1 ~ N(0,5 2 ) µ 2 ~ N(0, 5 2 ) µ ~ N(0,5 2 ) Δ ~ N(0,2.5 2 ) 21 BAYES2012 J Kerman May 10 Neutral reference analyses

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 What is your point estimate? Median odds Pr( odds > 3 y ) Logistic Model Priors (a) 0.64 65% B (b) 0.90 47% B (c) 1.00 50% B µ ~ N(0,5 2 ) Δ ~ N(0,5 2 ) µ ~ t(0,10, 5) Δ ~ t(0,5, 5) µ ~ N(0,100 2 ) Δ ~ N(0, 5 2 ) 22 BAYES2012 J Kerman May 10 Neutral reference analyses

A proposal for default models 23 BAYES2012 J Kerman May 10 Neutral reference analyses

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 24 BAYES2012 J Kerman May 10 Neutral reference analyses

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 25 BAYES2012 J Kerman May 10 Neutral reference analyses

Neutral models for differences and ratios Treatment groups are estimated separately, then differences computed E.g., the Binomial-beta model: ( θ 1 y ) ~ Beta(1/3 + y 1, 1/3 + n 1 - y 1 ) ( θ 2 y ) ~ Beta(1/3 + y 2, 1/3 + n 2 y 2 ) Compute δ = θ 2 - θ 1 Compute Δ = logit(θ 2 ) - logit(θ 1 ) 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 26 BAYES2012 J Kerman May 10 Neutral reference analyses

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) Beta(1/3, 1/3) 27 BAYES2012 J Kerman May 10 Neutral reference analyses

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) Gamma(1/3, 0) 28 BAYES2012 J Kerman May 10 Neutral reference analyses

Neutral models for differences and ratios Examples of worst cases (one group has y=1) Data 1 Data 2 Median point estimate θ 1 Median point estimate θ 2 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% 29 BAYES2012 J Kerman May 10 Neutral reference analyses

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 31 BAYES2012 J Kerman May 10 Neutral reference analyses

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 posterior median m m + n p + n m + n sample mean 32 Statistical Methodology Science VC Jouni Kerman Nov 9, 2010 Analyzing Proportions and Rates using Neutral Priors

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 33 Statistical Methodology Science VC Jouni Kerman Nov 9, 2010 Analyzing Proportions and Rates using Neutral Priors

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 34 Statistical Methodology Science VC Jouni Kerman Nov 9, 2010 Analyzing Proportions and Rates using Neutral Priors

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) 35 BAYES2012 J Kerman May 10 Neutral reference analyses

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 36 BAYES2012 J Kerman May 10 Neutral reference analyses

Reference model candidates investigated Binomial & Poisson regression models Normal model For µ For Δ 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 37 BAYES2012 J Kerman May 10 Neutral reference analyses

Possible reference models (Binomial) y i ~ Binomial(n i, θ i ), i=1, 2 Beta Normal Scaled t A θ i ~ Beta(a, a) δ = θ 2 - θ 1 logit(θ i ) ~ N(0, σ 2 ) δ = logit(θ 2 ) - logit(θ 1 ) logit(θ i ) ~ N(0, σ 2 ) δ = logit(θ 2 ) - logit(θ 1 ) B logit(θ 1 ) ~ N(0, σ 12 ) δ ~ N(0, σ 22 ) θ 2 = logit(θ 1 ) + δ C logit(µ) ~ N(0, σ 12 ) δ ~ N(0, σ 22 ) θ 1 = logit(µ) - δ / 2 θ 2 = logit(µ) + δ / 2 logit(θ 1 ) ~ t(0, σ 1, df 1 ) δ ~ t(0, σ 2, df 2 ) θ 2 = logit(θ 1 ) + δ logit(µ) ~ t(0, σ 1, df 1 ) δ ~ t(0, σ 2, df 2 ) θ 1 = logit(µ) - δ / 2 θ 2 = logit(µ) + δ / 2 38 BAYES2012 J Kerman May 10 Neutral reference analyses

Possible reference models (Poisson) y i ~ Poisson(X i θ i ), i=1, 2 Gamma Normal Scaled t A θ i ~ Gamma(a, ε) δ = θ 2 - θ 1 log (θ i ) ~ N(0, σ 2 ) δ = log (θ 2 ) - log (θ 1 ) log (θ i ) ~ N(0, σ 2 ) δ = log (θ 2 ) - log (θ 1 ) B log (θ 1 ) ~ N(0, σ 12 ) δ ~ N(0, σ 22 ) θ 2 = log (θ 1 ) + δ C log (µ) ~ N(0, σ 12 ) δ ~ N(0, σ 22 ) θ 1 = log (µ) - δ / 2 θ 2 = log (µ) + δ / 2 log (θ 1 ) ~ t(0, σ 1, df 1 ) δ ~ t(0, σ 2, df 2 ) θ 2 = log (θ 1 ) + δ log (µ) ~ t(0, σ 1, df 1 ) δ ~ t(0, σ 2, df 2 ) θ 1 = log (µ) - δ / 2 θ 2 = log (µ) + δ / 2 39 BAYES2012 J Kerman May 10 Neutral reference analyses

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