Bayesian MCPMod F. Fleischer, C. Loley, S. Bossert, Q. Deng, J. Knig - - PowerPoint PPT Presentation

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

2 Bayesian MCPMod, Göttingen Dec 2018

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: 𝑒1, … , 𝑒𝐿 plus placebo: 𝑒0 -> K+1 dose groups

𝑍

𝑗𝑘 = 𝜈𝑒𝑗 + 𝜗𝑗𝑘, 𝜗𝑗𝑘 ∼ 𝒪(0, 𝜏2), 𝑗 = 0, … , 𝐿, 𝑘 = 1, … , 𝑜𝑗

– 𝜈𝑒𝑗: mean response at dose 𝑒𝑗 – 𝑜𝑗: number of patients treated with doses d𝑗 – 𝜗𝑗𝑘: error term of patient 𝑘 within dose group 𝑗; assumed to be independent

• Mean response for each dose can be represented as 𝜈𝑒𝑗 = 𝑔(𝑒𝑗, 𝜾) for

some dose-response model 𝑔 ⋅

• 𝜾: parameter vector of the unknown dose-response model 𝑔
• > 𝜈𝑒𝑗 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, 𝜾𝑛)

– 𝜾𝑛: parameter vector of model 𝑛

• 𝑔 𝑒, 𝜾 = 𝜄0 + 𝜄1𝑔0(𝑒, 𝜾∗)

– 𝑔0(𝑒, 𝜾∗): standardised model – 𝜾∗: parameter vector of the standardised model – 𝜄0: location parameter – 𝜄1: 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

– 𝜐𝑛 = 𝑏𝑠𝑕𝑛𝑏𝑦𝑑𝑛 𝑑𝑛

𝑈 𝜈𝑛 2/ σ𝑗=0 𝐿 𝑑𝑛𝑗

2

𝑜𝑗

• 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

• ach:

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 = 𝜈𝑒𝑗 + 𝜗𝑗𝑘, 𝜗𝑗𝑘 ∼ 𝒪(0, 𝜏2), 𝑗 = 0, … , 𝐿, 𝑘 = 1, … , 𝑜𝑗

• Error terms are assumed to be independent
• Set of candidate models as defined in the MCPMod approach

Cha hanged: ged:

• 𝜈𝑒𝑗 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
• Plac

acebo bo group: up: 𝜈𝑒0 ∼ 𝑥1𝒪 𝜄𝑞1,0,

𝜏2 𝑜𝑞1,0 + ⋯ + 𝑥4𝒪 𝜄𝑞4,0, 𝜏2 𝑜𝑞4,0

• 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}

• True curve flat then: 𝑑𝑛

𝑈 𝜈 = 0

• Calculate the probability of 𝑑𝑛

𝑈 𝜈 > 0 for the posterior distribution of 𝜈

• Model 𝑛 is significant if: 𝑄 𝑑𝑛

𝑈 𝜈 > 0 𝒵 > 1 − 𝛽∗

• Posterior probability of 𝑑𝑛

𝑈 𝜈:

– conjugate prior: – mixture prior:

𝜄𝒵

𝑚 : mean vector of component l of the posterior distribution of 𝜈

𝜐𝑚: 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

• n of the Poster

erior

• r probab
• babili

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:

• Test statistic: max

𝑛=1,…,𝑁 𝑄(𝑑𝑛 𝑈 𝜾 > 0|𝒵)

• Test decision: PoC is established if: max

𝑛=1,…,𝑁 𝑄(𝑑𝑛 𝑈 𝜾 > 0|𝒵) > 1 − 𝛽∗ , i.e. at least

• ne 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(𝑢1−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

• r 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

Comparison – BMCPMod and MCPMod

• Comparing operating characteristics for BMCPMod and MCPMod
• Dummy example – not necessarily fully realistic

Ass ssumpt ptions: ions:

• 𝐿 = 4 active dose groups + a placebo group:

𝑒0 = 0, 𝑒1 =

1 30 , 𝑒2 = 3 30 , 𝑒3 = 10 30 , 𝑒4 = 1

• Standard deviation of the data: 𝜏 = 0.9
• Significance level 𝛽 = 0.05 (one-sided)

Bayesian MCPMod, Göttingen Dec 2018 19

Comparison – BMCPMod and MCPMod

Bayesian MCPMod, Göttingen Dec 2018 20

Prior

• r Infor
• rmation:

tion:

• Consider three different conjugate priors
• All priors only differ for the placebo group
• For the active

ive dose se group ups: Prior mean: 𝜄𝑞,𝑗 = 0, ESS: 𝑜𝑞,𝑗 = 1 (𝑗 = 1, … , 𝐿)

• For the plac

acebo ebo group up:

• Allocation used (N=300):

– MCPMod/non-informative: 𝐨 = (81,33,44,48,95) – Informative 1: 𝐨 = 56,37,49,53,105 – Informative 2: 𝐨 = 25,41,55,60,119

Prior Mean an (𝜾𝒒,𝟏) ESS (𝒐𝒒,𝟏) non-inf informativ ive 1 informative ative 1 30 informative ative 2 60

Comparison – BMCPMod and MCPMod

Candida didate models els:

• 𝑁 = 6 candidate models
• Three different Emax model
• One Exponential model
• One Linear model
• One Logistic model

Bayesian MCPMod, Göttingen Dec 2018 21

Comparison – BMCPMod and MCPMod

Result lts:

• Increase in power for informative prior scenarios
• Similar values for the Bayesian PoC test using the non informative prior and the

frequentist PoC test

• Ideal scenario where true mean is identical to median prior

Bayesian MCPMod, Göttingen Dec 2018 22

tr true e model el Approach Emax 1 Emax 2 Emax 3 Exponential Linear Logistic Frequentist 0.8140 0.7633 0.8363 0.8930 0.8813 0.9085 Bayes non- informative 0.8137 0.7621 0.8362 0.8928 0.8812 0.9083 Bayes informative 1 0.8947 0.8876 0.9000 0.9205 0.9142 0.9436 Bayes informative 2 0.9386 0.9472 0.9357 0.9348 0.9322 0.9612

Results for Prior–Data Conflict

• Calculation of OCs for different true curves with five different placebo

means and for the three conjugate priors:

– true placebo mean -0.25 or -0.1 -> too high prior mean – true placebo mean 0 -> correct prior mean – true placebo mean 0.25 or 0.1 -> too low prior mean – Expected treatment benefit kept constant

• How extreme are these different true placebo means under the different

prior distributions?

Bayesian MCPMod, Göttingen Dec 2018 23

Prior distribu bution 𝑸 𝝂 < −𝟏. 𝟑𝟔 = 𝑸(𝝂 > 𝟏. 𝟑𝟔) 𝑸 𝝂 < −𝟏. 𝟐 = 𝐐(𝝂 > 𝟏. 𝟐) 𝑸(𝝂 > 𝟏) non- informat mativ ive 0.3906 0.4558 0.5 informat mative ve 1 0.0641 0.2714 0.5 informat mative ve 2 0.0157 0.1947 0.5

Results for Prior–Data Conflict

Result lts for a true flat dose-respo pons nse curve ve (“type one error”):

• For non-informative prior type-1 error is kept below 5%
• Slight decrease of error for informative priors if true placebo mean is 0
• Inflation of type-1 error if true placebo mean is larger than expected
• Deflation of type-1 error if true placebo mean is smaller than expected

Bayesian MCPMod, Göttingen Dec 2018 24

tr true e plac aceb ebo mea ean Prior

• 0.25
• 0.1

0.1 0.25 non-inf informativ ive 0.0465 0.0478 0.0484 0.0494 0.0501 informative ative 1 0.0135 0.0250 0.0384 0.0591 0.1140 informative ative 2 0.0083 0.0192 0.0355 0.0711 0.2066

Results for Prior–Data Conflict

Result lts for a true Emax curve ve of the candida didate set (“Power”):

• Noticeable increase in power if true placebo mean is correct
• For larger true placebo mean even larger increase in power

– Has to be weighted against inflation of type-1 error

• For smaller true placebo mean smaller increase or even reduction in power

– Has to be weighted against deflation of type-1 error

Bayesian MCPMod, Göttingen Dec 2018 25

tr true e plac acebo ebo mea ean Prior

• 0.25
• 0.1

0.1 0.25 non-inf informativ ive 0.8088 0.8117 0.8137 0.8153 0.8185 informative ative 1 0.7332 0.8391 0.8947 0.9362 0.9743 informative ative 2 0.6758 0.8590 0.9386 0.9794 0.9977

Mixture Prior

• Consider four different mixture priors
• 𝑀 = 3 component mixture prior for placebo
• weak/non-informative conjugate priors for the active

ve dose group ups: Prior mean: 𝜄𝑞,𝑗 = 0, ESS: 𝑜𝑞,𝑗 = 1 (𝑗 = 1, … , 𝐿)

• mixture prior for the plac

acebo ebo group up:

– Each mixture prior has two weak informative components

Bayesian MCPMod, Göttingen Dec 2018 26

Prior Mixtu ture e 1.1 .1 Mixtu ture e 1.2 .2 Mixtu ture e 2.1 .1 Mixtu ture e 2.2 .2 Component 1 2 3 1 2 3 1 2 3 1 2 3 Weight 0.5 0.25 0.25 0.8 0.1 0.1 0.5 0.25 0.25 0.8 0.1 0.1 Mean 0.25

• 0.25

0.25

• 0.25

0.25

• 0.25

0.25

• 0.25

ESS 30 2 2 30 2 2 60 2 2 60 2 2

Mixture Prior

• Example for the density of a mixture prior:
• > distribution gets heavier tails through

the weak informative components

• How extreme are these different true placebo means under the different prior

distributions?

Bayesian MCPMod, Göttingen Dec 2018 27

Prior 𝑸 𝝂 < −𝟏. 𝟑𝟔 = 𝑸(𝝂 > 𝟏. 𝟑𝟔) 𝑸 𝝂 < −𝟏. 𝟐 = 𝑸(𝝂 > 𝟏. 𝟐) 𝑸 𝝂 > 𝟏 mixt xtur ure 1.1 .1 0.2110 0.3568 0.5 mixtu ture e 1.2 .2 0.1229 0.3056 0.5 mixt xtur ure e 2.1 .1 0.1869 0.3184 0.5 mixt xtur ure e 2.2 .2 0.0842 0.2442 0.5

Results – Mixture Prior

Results for a true flat dose-response curve (“type one error”):

• Dampening of inflation/deflation of type-1 error effects

– More weight on robust elements => more dampening

Bayesian MCPMod, Göttingen Dec 2018 28

True placebo bo mean

• 0.25
• 0.1

0.1 0.25

Non Non-inf informativ ive

0.0465 0.0478 0.0484 0.0494 0.0501

Informati ative 1

0.0135 0.0250 0.0384 0.0591 0.1140

Mixtu ture e 1.1 .1

0.0272 0.0279 0.0425 0.0520 0.0824

Mixtu ture e 1.2 .2

0.0190 0.0266 0.0411 0.0594 0.1013

Informati ative 2

0.0083 0.0192 0.0355 0.0711 0.2066

Mixtu ture e 2.1 .1

0.0241 0.0260 0.0387 0.0589 0.1020

Mixtu ture 2.2 .2

0.0164 0.0213 0.0366 0.0659 0.1442

Results – Mixture Prior

Results for the true Emax 1 curve of the candidate set (“Power”):

• Increase in power if true placebo mean is 0 for all informative scenarios

– For small weights of robust component very small / non-existing loss of power

• Similar effects as for one-component prior overall

Bayesian MCPMod, Göttingen Dec 2018 29

True placebo bo mean

• 0.25
• 0.1

0.1 0.25

Non Non-inf informativ ive

0.8088 0.8117 0.8137 0.8153 0.8185

Informati ative 1

0.7332 0.8391 0.8947 0.9362 0.9743

Mixtu ture e 1.1 .1

0.7650 0.8346 0.8636 0.8835 0.8826

Mixtu ture e 1.2 .2

0.7459 0.8393 0.8832 0.9192 0.9224

Informati ative 2

0.6759 0.8590 0.9386 0.9794 0.9977

Mixtu ture e 2.1 .1

0.7309 0.8472 0.8883 0.9043 0.8673

Mixtu ture 2.2 .2

0.6998 0.8495 0.9191 0.9430 0.9144

Summary / Outlook

• Bayesian MCPMod deliver a possibility to include historical data for the placebo

group in the analysis – Reproducing results of MCPMod for non-informative scenarios – Possible to extend to more than one dose group with historical information

• Derivation of prior? => e.g. multivariate MAP

– Alternative to functional approach of historical data integration

• In particular if only one treatment group with historical data (control)
• Benefit of applying BMCPMod depends on

– Amount of historical data / informativeness of prior – Believe in prior-data exchangeability – Time/Runtime available

• Extension to generalized linear models possible (Pinheiro et al. 2014)

– Binary, TTE, recurrent event data

• Application in clinical trials
• Combining MCPMod PoC with dose group specific Go
• Interim analyses?
• Confirmatory settings?

Bayesian MCPMod, Göttingen Dec 2018 30

Questions?

References

Bayesian MCPMod, Göttingen Dec 2018 32

Papers and Books:

• Bornkamp, B., Pinheiro, J., Bretz, F.. MCPMod: An R Package for the Design and Analysis of Dose-Finding Studies. Journal of

Statistical Software 29(7), pp. 1-23. 2009.

• Bretz, F., Pinheiro, J., Branson, M.. Combining Multiple Comparison and Modeling Techniques in Dose-Response Studies.

Biometrics 61(3), pp. 738-748. 2005.

• Genz, A., Bretz, F.. Computation of Multivariate Normal and t Probabilities. Lecture Notes in Statistics, Vol. 195. Springer-

Verlag, Heidelberg. ISBN 978-3-642-01688-2.

• Pinheiro, J., Bornkamp, B., Bretz, F.. Design and Analysis of Dose-Finding Studies combining Multiple Comparison and

Modeling Procedures. Journal of Biopharmaceutical Statistics 16(5), pp. 639-656. 2006.

• Pinheiro, J., Bornkamp, B., Glimm, E., Bretz, F.. Model-based dose finding under model uncertainty using general parametric
• models. Statistics in Medicine 33(10), pp. 1646-1661. 2014.
• Schmidli, H., Gsteiger, S., Roychoudhury, S., O’Hagan, A., Spiegelhalter, D., Neuenschwander, B.. Robust Meta-Analytic-

Predictive Priors in Clinical Trails with Historical Control Information. Biometrics 70(4), pp. 1023-1032. 2014.

• Schorning, K., Bornkamp, B., Bretz, F., and Dette, H. (2016) Model selection versus model averaging in dose finding studies.
• Statist. Med., 35: 4021–4040. doi: 10.1002/sim.6991.

R-packages:

• DoseFinding by Bornkamp, B., Pinheiro, J. and Bretz,F., see https://CRAN.R-project.org/package=DoseFinding
• mvtnorm by A. Genz et al., see https://cran.r-project.org/web/packages/mvtnorm/index.html
• RBesT by Weber, S., see https://CRAN.R-project.org/package=RBesT

MCPMod – Candidate models

• Frequently used parametric models for the candidate set (Bornkamp et al.

2009):

• candidate model set could contain two models of the same type (e.g. two

Emax models) but then with different guesstimates

Bayesian MCPMod, Göttingen Dec 2018 33

MCPMod – Application in R

• MCPMod is implemented in the package DoseFinding
• Useful functions:

Bayesian MCPMod, Göttingen Dec 2018 34

Functi nction

• n

descr cription iption

guesst Calculates guesstimates for standardised model parameters fitMod fits for given data a parametric dose-response model (e.g. Emax) MCPMod Implementation of the complete MCPMod approach MCTtest Performs the multiple comparison contrast test Mods Definition of the candidate set

• ptContr

Calculation of the optimal contrast vectors for given models

• ptDesign

Calculates the optimal design for a given set of models powMCT Calculates the power of the multiple comparison test sampSize Calculates the necessary sample size for a given target function

BMCPMod – Conjugate Prior

• for each dose group a normal prior is assumed:

𝜈𝑒𝑗 ∼ 𝒪(𝜄𝑞,𝑗, 𝜏𝑞,𝑗2 )

• 𝜄𝑞,𝑗: mean of the prior distribution for dose group 𝑒𝑗
• 𝜏𝑞,𝑗

2 : variance of the prior distribution for dose group 𝑒𝑗

• because of the independence between the dose groups:

𝜈 ∼ 𝒪

𝐿+1(𝜾𝑞, 𝝉𝑞 2𝐽)

• Prior distribution could also be expressed through the Effective Sample

Size (ESS): 𝜈𝑒𝑗 ∼ 𝒪(𝜄𝑞,𝑗,

𝜏2 𝑜𝑞,𝑗) ⇒ 𝜈 ∼ 𝒪 𝐿+1(𝜾𝑞, 𝜏2 𝒐𝑞 𝐽)

• 𝒐𝑞: vector which contains the prior ESS for each dose group
• non-informative/weak informative Prior: ESS 1-2
• > ESS small: then the standard deviation of the prior is big and so the information of the

prior small

Bayesian MCPMod, Göttingen Dec 2018 35

MCPMod – MCP step

• Aims :

– establish PoC (Proof of Concept) via multiple comparison techniques – selection of model(s) which are most likely to represent the dose-response shape

• PoC test:

– each model of the candidate test is tested using single contrast test – procedures are used which control the Familywise Error Rate (FWER) – for each contrast test (i.e. for each model): decision procedure to determine weather the given dose-response shape is statistically significant , based on the observed data – finally decision: at least one model significant, PoC is established – no model of the candidate set statistically significant, the procedure stops

• Model selection:

– best model(s) will be selected out of all significant models – two different main strategies:

• select one model out of all significant ones
• use all significant models applying model averaging techniques

– different approaches for model selection and averaging will be explained later

Bayesian MCPMod, Göttingen Dec 2018 36

MCPMod – PoC Test

single e cont ntrast test : for model 𝑛, 𝑛 = 1, … , 𝑁

• 𝐼0

𝑛: 𝑑𝑛 𝑈 𝜈 = 0 vs. 𝐼1 𝑛: 𝑑𝑛 𝑈 𝜈 > 0 (one-sided test)

• 𝜈 = (𝜈𝑒0, … , 𝜈𝑒1) unknown treatment means
• test statistic:
• distribution of the test statistic:

– Under 𝐼0

𝑛: central t distributed with 𝑂 − 𝐿 degrees of freedom

– 𝐼0

𝑛 not true: non-central t distributed with 𝑂 − 𝐿 degrees of freedom and non-centrality

parameter: 𝜐𝑛 = 𝑑𝑛

𝑈 𝜈/ 𝜏2 σ𝑗=0 𝐿

𝑑𝑛𝑗

2 /𝑜𝑗

– model 𝑛 significantly different from a flat dose-response curve if: 𝑈

𝑛 > 𝑟

– Can be approximated by a normal distribution for a high degree of freedom

Bayesian MCPMod, Göttingen Dec 2018 37

MCPMod – PoC Test

final l test for PoC:

• C:
• test statistic: 𝑈

𝑛𝑏𝑦 = max 𝑛 𝑈 𝑛

• distribution of the test statistic:

– can be considered as the joint distribution of 𝑈

1, … , 𝑈𝑁

– the distribution of 𝑈 = 𝑈

1, … , 𝑈𝑁 𝑈 is a multivariate t distribution with 𝑂 − 𝐿 degrees of

freedom and a correlation matrix 𝑺 = 𝜍𝑗𝑘 – 𝜍𝑚𝑘 =

σ𝑗=0

𝐿 𝑑𝑚,𝑗𝑑𝑘,𝑗 𝑜𝑗

σ𝑗=0

𝐿 𝑑𝑚,𝑗 2 𝑜𝑗

σ𝑗=0

𝐿 𝑑𝑘,𝑗 2 𝑜𝑗

– Can be approximated by a multivariate normal distribution for a high degree of freedom

• test decision: PoC is established, if: 𝑈

𝑛𝑏𝑦 > 𝑟, i.e. at least one of the single

contrast tests (i.e. one model of the candidate set) is significant multi tiplicity plicity adjusted ed criti tical al value: ue:

• 𝑟 = 𝑢1−𝛽,𝑂−𝐿

𝑁

; 1 − 𝛽 Quantile of the 𝑁-dimensional t-distribution with 𝑂 − 𝐿 degrees of freedom and correlation matrix 𝑺

Bayesian MCPMod, Göttingen Dec 2018 38

MCPMod – Modeling step

• the best model(s) of the candidate set is/are fitted to the data using

generalised least squares (GLS) estimators for the model parameters

• for non-linear models: iterative optimisation techniques are needed
• after the model(s) has/have been fitted to the data the target dose will be

estimated based on this/these models

• possible target doses:

– Minimum effective dose (𝑁𝐹𝐸): dose that produces a certain clinical relevant difference in the outcome over placebo (𝑁𝐹𝐸 = 𝑏𝑠𝑕𝑛𝑗𝑜𝑒∈ 𝑒0,𝑒𝐿 {𝑔 𝑒, 𝜄 > 𝑔 𝑒0, 𝜄 + Δ}) – Effective dose (𝐹𝐸𝑞): smallest dose resulting in 𝑞% of the maximum effect

• possible estimator for 𝑁𝐹𝐸:

– ෣ 𝑁𝐹𝐸 = 𝑏𝑠𝑕𝑛𝑗𝑜𝑒∈ 𝑒0,𝑒𝐿 {𝑞𝑒 > 𝑞𝑒0 + Δ, 𝑀𝑒 > 𝑞𝑒0} – 𝑀𝑒: lower boundary of the 1 − 2𝛿 confidence interval of the predicted mean at dose 𝑒 based on the estimate model – 𝑞𝑒: predicted mean for dose 𝑒 based on the estimated model

Bayesian MCPMod, Göttingen Dec 2018 39

MCPMod – Optimal Study Design

• Also the optimal study design for MCPMod can be calculated
• optimal selection of dose groups and optimal allocation ratio can be

calculated based on two different criteria:

– D-Optimality: optimisation with regard to the estimation of the model parameters -> includes the variance of the estimations – TD-Optimality: optimisation with regard to the target dose estimation -> – also a combination of both criteria is possible

• necessary sample size 𝑂 can also be calculated based on two different

criteria:

– necessary sample size to get a certain target power to establish PoC – necessary sample size to get a prespecified precision for the target dose estimation – again a combination of both criteria is possible – if the main focus of the trial is to establish PoC, the first criterion should be used – if the main focus is to estimate the target dose, the second criterion should be used

Bayesian MCPMod, Göttingen Dec 2018 40

BMCPMod – Go Probability

• Go Probability: probability of a significant PoC test decision
• therefore we assume that the true mean vector is 𝜈𝑢
• the probability can only be calculated for conjugate priors, for mixture

priors it needs to be simulated Conju njugat ate e Prior:

• r:

– While 𝑎 is a M-dimensional standard normal distributed random variable – Can be implemented using pmvnorm of the package mvtnorm

Mixtur ure e Prior:

• r:
• Simulate 10.000 data sets based on the true mean vector 𝜈𝑢
• count how often the PoC test is significant

Bayesian MCPMod, Göttingen Dec 2018 41

MCPMod – Optimal Contrast Vectors

• For model 𝑛: 𝑑𝑛 is chosen in order to maximise the power of the single

contrast test for model 𝑛 under the assumption that 𝑛 is the correct model and that σ𝑗=0

𝐿

𝑑𝑛𝑗 = 0

• Power of the single contrast test if model 𝑛 is true:

– 𝜈𝑛 = (𝜈𝑛0, … , 𝜈𝑛𝐿) mean of the response variable for all dose groups and model 𝑛 (𝜈𝑛𝑗 = 𝑔

𝑛(𝑒𝑗, 𝜾𝑛))

– 𝑄 𝑈

𝑛 ≥ 𝑟 𝜈 = 𝜈𝑛 = 1 − 𝑄(𝑈 𝑛 ≤ 𝑟|𝜈 = 𝜈𝑛 ) -> can be calculated through the distribution

function of a non-central t-distribution with 𝑂 − 𝐿 degrees of freedom

• instead of maximising the Power, we could maximise the non-centrality

parameter: 𝑏𝑠𝑕𝑛𝑏𝑦𝑑𝑛 𝑑𝑛

𝑈 𝜈𝑛 2/ σ𝑗=0 𝐿 𝑑𝑛𝑗

2

𝑜𝑗

• if a standardised version of the model exists, 𝜈𝑛 could be replaced by 𝜈𝑛
• For uniqueness: assumption: 𝑑𝑛

= 1

Bayesian MCPMod, Göttingen Dec 2018 42

BMCPMod – Posterior distribution

• Conjug

njugat ate e prior:

• r:
• 𝒐: vector with the actual sample sizes for each group, I: identity matrix
• Mixtur

ure e prior:

• r: Posterior also a mixture with L components

Bayesian MCPMod, Göttingen Dec 2018 43