Bayesian MCPMod F. Fleischer, C. Loley, S. Bossert, Q. Deng, J. König Workshop „ Bayesian methods in the development and assessment of new therapies”, Göttingen, Dec 07th, 2018
Overview • Introduction to MCPMod • Bayesian version of the MCP step • Comparison BMCPMod / MCPMod • Summary and discussion Bayesian MCPMod, Göttingen Dec 2018 2
MCPMod - Introduction • MCPMod Multiple Comparison Procedure and Modeling (under model uncertainty) Combines two principles of dose-finding studies: • – Multiple Comparison Procedure (MCP) – Modeling of dose reponse (DR) • Unified approach by Bretz et al. (2005) Set of candidate models for DR-modelling under model uncertainty – Test PoC using classical contrast tests adjusted for multiplicity of models – – Select a model or average across significant models to model DR shape Extended by Pinheiro et al. (2014) to generalized linear models – Bayesian MCPMod, Göttingen Dec 2018 3
MCPMod - Overview Bayesian MCPMod, Göttingen Dec 2018 4
MCPMod – Model definition • Response Y observed for given set of parallel groups of patients • K active doses: 𝑒 1 , … , 𝑒 𝐿 plus placebo: 𝑒 0 -> K+1 dose groups 𝑗𝑘 = 𝜈 𝑒 𝑗 + 𝜗 𝑗𝑘 , 𝜗 𝑗𝑘 ∼ 𝒪(0, 𝜏 2 ) , 𝑗 = 0, … , 𝐿, 𝑘 = 1, … , 𝑜 𝑗 𝑍 – 𝜈 𝑒 𝑗 : mean response at dose 𝑒 𝑗 – 𝑜 𝑗 : number of patients treated with doses d 𝑗 – 𝜗 𝑗𝑘 : error term of patient 𝑘 within dose group 𝑗 ; assumed to be independent • Mean response for each dose can be represented as 𝜈 𝑒 𝑗 = 𝑔(𝑒 𝑗 , 𝜾) for some dose-response model 𝑔 ⋅ • 𝜾 : parameter vector of the unknown dose-response model 𝑔 -> 𝜈 𝑒 𝑗 non-random but unknown parameter of interest Bayesian MCPMod, Göttingen Dec 2018 5
MCPMod – Candidate models • Set ℳ with M parametric candidate models for the unknown dose-response shape • Model function of model 𝑛 : 𝑔 𝑛 (𝑒, 𝜾 𝑛 ) – 𝜾 𝑛 : parameter vector of model 𝑛 • 𝑔 𝑒, 𝜾 = 𝜄 0 + 𝜄 1 𝑔 0 (𝑒, 𝜾 ∗ ) – 𝑔 0 (𝑒, 𝜾 ∗ ) : standardised model – 𝜾 ∗ : parameter vector of the standardised model – 𝜄 0 : location parameter – 𝜄 1 : scale parameter • Usage of “guesstimates” for planning the trial Bayesian MCPMod, Göttingen Dec 2018 6
MCPMod – PoC Test Single contrast test for testing the m-th model: : • Test statistics are jointly multivariate t-distributed with known correlation structure determined by the model contrasts Multiplicity adjusted critical values (or adjusted p-values) Overall significance established if • max 𝑛 𝑈 𝑛 > q • Contrast for specific model m can be optimized based on non-centrality parameter 2 𝑈 𝜈 𝑛 2 / σ 𝑗=0 𝑑 𝑛𝑗 𝐿 – 𝜐 𝑛 = 𝑏𝑠𝑛𝑏𝑦 𝑑 𝑛 𝑑 𝑛 𝑜 𝑗 • Also optimal allocation ratio derivable – Depend on characteristic of interest for optimization (D, TD-optimality) Bayesian MCPMod, Göttingen Dec 2018 7
MCPMod – Model selection Reference set: all significant models • • No model significant => stop If at least one model is significant: two different possibilities: • – Select a single model of the reference set • Choose the “best” model of reference set based on a criterion • Different criteria: – Largest value of the test statistic / smallest p-value – Goodness-of-fit criterion e.g. (g)AIC or BIC – Use model averaging techniques • For each model of the reference set a model weight needs to be calculated, e.g. based on (g)AIC • Fitted model is a weighted average • Different weighting options (Schorning et al. 2016) Bayesian MCPMod, Göttingen Dec 2018 8
BMCPMod - Introduction Aim: : • Include historical information into MCPMod approach • Most often for one treatment group (placebo) only – No “functional” inclusion (compare BLRM or Bayesian Emax) • Approach should mimic MCPMod results for non-informative prior – Ability to embed into general framework used Appr proac oach: h: • Use a Bayesian test procedure in the MCP step to establish PoC • Adjust the model selection and the modeling step appropriately Focus cus here: : • Bayesian version of the PoC test • Comparison with the frequentist PoC test Bayesian MCPMod, Göttingen Dec 2018 9
BMCPMod - Assumptions Not change ged: • K + 1 dose groups • 𝑍 𝑗𝑘 = 𝜈 𝑒 𝑗 + 𝜗 𝑗𝑘 , 𝜗 𝑗𝑘 ∼ 𝒪(0, 𝜏 2 ) , 𝑗 = 0, … , 𝐿, 𝑘 = 1, … , 𝑜 𝑗 • Error terms are assumed to be independent • Set of candidate models as defined in the MCPMod approach Cha hanged: ged: • 𝜈 𝑒𝑗 random with known variability Bayesian MCPMod, Göttingen Dec 2018 10
BMCPMod – Prior information Ass ssumpt ptions ions / se setting: g: Information often restricted to one dose group (placebo) • • Non/vaguely-informative prior for other dose groups Prior generated e.g. via meta-analytic prior approach • => Fit of mixture normal prior Normal-normal conjugate model • • Independence between the dose groups is assumed Bayesian MCPMod, Göttingen Dec 2018 11
BMCPMod – Mixture Prior • Assumption: mixture prior for placebo with L ( ≤ 4 ) components 𝜏 2 𝜏 2 • Plac acebo bo group: up: 𝜈 𝑒 0 ∼ 𝑥 1 𝒪 𝜄 𝑞 1 ,0 , 𝑜 𝑞1,0 + ⋯ + 𝑥 4 𝒪 𝜄 𝑞 4 ,0 , 𝑜 𝑞4,0 • Act ctive ive dose se group ups: s: conjugate normal prior (with very small ESS) • Complete 𝜈: Bayesian MCPMod, Göttingen Dec 2018 12
BMCPMod – PoC test Bayesian single contrast test for model 𝑛 ∈ {1, … , 𝑁} 𝑈 𝜈 = 0 • True curve flat then: 𝑑 𝑛 𝑈 𝜈 > 0 for the posterior distribution of 𝜈 • Calculate the probability of 𝑑 𝑛 𝑈 𝜈 > 0 𝒵 > 1 − 𝛽 ∗ • Model 𝑛 is significant if: 𝑄 𝑑 𝑛 𝑈 𝜈 : • Posterior probability of 𝑑 𝑛 conjugate prior: – mixture prior: – 𝑚 : mean vector of component l of the posterior distribution of 𝜈 𝜄 𝒵 𝜐 𝑚 : variance-covariance matrix of component l of the posterior distribution of 𝜈 Bayesian MCPMod, Göttingen Dec 2018 13
BMCPMod – PoC test Calc lcul ulati tion on of the Poster erior or probab obabili liti ties: • Conjugate Prior: • Mixture prior: analogous Φ : distribution function of the standard normal distribution Bayesian MCPMod, Göttingen Dec 2018 14
BMCPMod – PoC test Overall te test st for PoC: 𝑈 𝜾 > 0|𝒵) • Test statistic: max 𝑛=1,…,𝑁 𝑄(𝑑 𝑛 𝑈 𝜾 > 0|𝒵) > 1 − 𝛽 ∗ , i.e. at least Test decision: PoC is established if: max • 𝑛=1,…,𝑁 𝑄(𝑑 𝑛 one of the Bayesian single contrast tests is significant • 1 − 𝛽 ∗ : a multiplicity adjusted critical value on the probability scale For the critical value ( 1 − 𝛽 ∗ ): • – use the critical value of the MCPMod approach after transforming it on the probability scale: Φ(𝑢 1−𝛽,𝑂−𝐿 𝑁 ) corresponds to the use of a non-informative prior in the Bayesian Approach – depends on the optimal contrast vectors and on the correlation matrix 𝑺 Bayesian MCPMod, Göttingen Dec 2018 15
BMCPMod – Choice of contrast vector • In MCPMod optimal contrast vectors may be derived – Optimal => maximizing power under alternative – Per model • Natural approach for BMCPMod might be to use the optimal contrast vectors from MCPMod – Not necessarily optimal for BMCPMod • Random component • Might be possible to derive more optimal contrasts – E.g. via simulation – Omitted here as choice of contrast vector usually only has a minor influence on power Bayesian MCPMod, Göttingen Dec 2018 16
BMCPMod – Allocation ratio • MCPMod allows for optimization of allocation ratios across treatment groups – Balanced allocation usually suboptimal – D- vsTD-Optimality vs reaching optimal power – Optimality per model => averaging • Keeping allocation ratio while adding historical information suboptimal – Similar to usual Bayesian borrowing designs • Naive/intuitive adjustment for one component prior – Compute ESS of historical information – Subtract from optimal allocation and adjust to reach original sample size again – MCPMod/non-informative: 𝐨 = (81,33,44,48,95) – Informative 1 (ESS=30): 𝐨 = 56,37,49,53,105 – Informative 2 (ESS=60): 𝐨 = 25,41,55,60,119 More complex for mixture priors • – Use mixture ESS (?) – Averaging Bayesian MCPMod, Göttingen Dec 2018 17
BMCPMod – Modeling aspects • As for MCPMod different options here – Select model with largest posterior probability (for contrast test) – Use e.g. (g)AIC AIC-base sed approach replacing likelihood with posterior (either select or average) – Average across models with significant Bayes test • Via posterior probabilities (for contrast test) • Use Bayesian model averaging / Bayes factors across/within different models – Average via (stratified) bootstrapping / bagging • Bootstrap model selection and average across bootstrap predictions Bayesian MCPMod, Göttingen Dec 2018 18
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