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Prior Elicitation in Bayesian Clinical Trial Design Peter F. Thall Biostatistics Department M.D. Anderson Cancer Center SAMSI intensive summer research program on Semiparametric Bayesian Inference: Applications in Pharmacokinetics and Pharmacodynamics Research Triangle Park, North Carolina July 13, 2010

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Disclaimer To my knowledge, this talk has nothing to do with semiparametric Bayesian inference, pharmacokinetics, or pharmacodynamics. I am presenting this at Peter Muellers behest. Blame Him!

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Outline ( As time permits ) 1. Clinical trials: Everything you need to know 2. Eliciting Dirichlet parameters for a leukemia trial 3. Prior effective sample size 4. Eliciting logistic regression model parameters for Pr(Toxicity | dose) 5. Eliciting values for a 6-parameter model of Pr(Toxicity | dose 1, dose 2 ) 6. Penalized least squares for {Pr(Efficacy),Pr(Toxicity)} 7. Eliciting a hyperprior for a sarcoma trial 8. Eliciting two priors for a brain tumor trial 9. Partially informative priors for patient-specific dose finding

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Clinical Trials Definition: A clinical trial is a scientific experiment with human subjects. 1. Its first purpose is to treat the patients in the trial. 2. Its second purpose is to collect information that may be useful to evaluate existing treatments or develop new, better treatments to benefit future patients. Other, related purposes of clinical trials: 3. Generate data for research papers 4. Obtain $$ financial support $$ from pharmaceutical companies or governmental agencies 5. Provide an empirical basis for drug or device approval from regulatory agencies such as the US FDA

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Medical Treatments Most medical treatments, especially drugs or drug combinations, have multiple effects. Desirable effects are called efficacy Shrinkage of a solid tumor by > 50% Complete remission of leukemia Dissolving a cerebral blood clot that caused an ischemic stroke Engraftment of an allogeneic (matched donor) stem cell transplant Undesirable effects are called toxicity Permanent damage to internal organs (liver, kidneys, heart, brain) Immunosuppression (low white blood cell count or platelet count) Cerebral bleeding or edema (accumulation of fluid) Graft-versus-host disease (the engrafted donor cells attack the patients organs) Regimen-related death due to any of the above

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Scientific Method Advice from Ronald Fisher Dont waste information Advice From Peter Thall Dont waste prior information when designing a clinical trial Standard Statistical Practice Ignore Fishers advice and just run your favorite statistical software package. And be sure to record lots and lots of p-values.

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A Chemotherapy Trial in Acute Leukemia Complete Remission (CR) YesNo Yes 1 2 No 3 4 T O X I C I T Y Model 1 2, 3, 4 ) ~ Dirichlet(a 1, a 2, a 3, a 4 ) Dir(a) p( a) 1 a1-1 2 a2-1 3 a3-1 4 a4-1, a + = a 1 +a 2 +a 3 +a 4 = ESS TOX = 1 2 ~Be(a 1 +a 2, a 3 +a 4 ) CR = 1 3 ~ Be(a 1 +a 3, a 2 +a 4 ) E( TOX ) = (a 1 +a 2 )/ a + E( CR ) = (a 1 +a 3 )/ a + 4 = 1 – 1 – 2 – 3

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If possible, use Historical Data to establish a prior: CR and Toxicity counts from 264 AML Patients Treated With an Anthracycline + ara-C CRNo CR Toxicity73 (27.7%) 63 (23.9%)136 (51.5%) No Toxicity 101 (38.3%) 27 (10.2%) 128 (48.5%) 174 (65.9%) 90 (34.1%) 264 P(CR | Tox) = 73/136 =.54 P(CR | No Tox) = 101/128 =.79 CR and Tox are Not Independent

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S = Standard treatment E = Experimental treatment S ~ Dir (73,63,101,27) a S,+ = ESS = 264 (Informative) Set E = S with a E,+ = 4 E ~ Dirichlet (1.11,.955, 1.53,.409) (Non-Informative) Dirichlet Priors and Stopping Rules Stop the trial if 1) Pr( S,CR +.15 < E,CR | data) <.025 (futility), or 2) Pr( S,TOX (safety)

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But what if you dont have historical data?!! An Easy Solution: To obtain the prior on S 1) Elicit the prior marginal outcome probability means E( TOX ) = (a 1 +a 2 )/a + and E( CR ) = (a 1 +a 3 )/a + 2) Assume independence and solve algebraically for ( 1, 2, 3, 4 ) = (a 1, a 2, a 3, a 4 )/ a + 3) Elicit the effective sample size ESS = a + that the elicited values E( TOX ) and E( CR ) were based on 4) Solve for (a 1, a 2, a 3, a 4 )

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Sensitivity Analysis of Association in the desirable case where Pr(CR) 0.15 from.659 to.809 and Pr(TOX) =.516 i.e. there is no increase in toxicity. p 11 p 00 p 10 p 01 True E Probability of Stopping the Trial Early Sample Size (25%,50%,75%).007(.027,.489,.782,.102) > (.227,.289,.582,.102) (.427,.089,.382,.102) (.510,.006,.299,.185) Oops!!

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If you dont have historical data... A slightly smarter way to obtain prior( S 1) Elicit the prior means E( TOX ) = (a 1 +a 2 )/a + and E( CR ) = (a 1 +a 3 )/a + 2) Elicit the prior mean of a conditional probability, like Pr(CR | Tox) = 1 /( ), which has mean a 1 /(a 1 + a 2 ), and solve for ( 1, 2, 3, 4 ) = (a 1, a 2, a 3, a 4 )/ a +. That is, do not assume independence. 3) Elicit the effective sample size ESS = a + that the values E( TOX ) and E( CR ) were based on 4) Solve for (a 1, a 2, a 3, a 4 ) Rocket Science!!

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Example Elicited prior values E( TOX ) = (a 1 +a 2 )/a + =.30 E( CR ) = (a 1 +a 3 )/a + =.50 E{ Pr(CR | Tox) } = E{ 1 /( )} = a 1 /(a 1 + a 2 ) =.40 ESS = a + = 120 (a 1, a 2, a 3, a 4 ) = (14.4, 21.6, 45.6, 38.4) ( 1, 2, 3, 4 ) = (a 1, a 2, a 3, a 4 )/ a + = (.12,.18,.38,.32)

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A Fundamental question in Bayesian analysis: How much information is contained in the prior? Prior p( θ ) (((((( (((((( (((((( (((((( Determining the Effective Sample Size of a Parametric Prior (Morita, Thall and Mueller, 2008)

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The answer is straightforward for many commonly used models E.g. for beta distributions Be (1.5,2.5) Be (16,19) Be (3,8) ESS = = 35 ESS = 3+8 = 11 ESS = = 5

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But for many commonly used parametric Bayesian models it is not obvious how to determine the ESS of the prior. E.g. usual normal linear regression model

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Intuitive Motivation Saying Be(a, b) has ESS = a+b implicitly refers to the wel known fact that θ ~ Be(a, b) and Y | θ ~ binom(n, θ) θ | Y,n Be(a +Y, b +n-Y) which has ESS = a + b + n So, saying Be(a,b) has ESS = a + b implictly refers to an earlier Be(c,d) prior with very small c+d = and solving for m = a+b – (c+d) = a+b – for a very small > 0

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General Approach 1) Construct an -information prior q 0 ( θ ), with same means and corrs. as p(θ) but inflated variances 2) For each possible ESS m = 1, 2,... consider a sample Y m of size m 3) Compute posterior q m ( θ |Y m ) starting with prior q 0 ( θ ) 4) Compute the distance between q m (θ|Y m ) and p(θ) 5) The interpolated value of m minimizing the distance is the ESS.

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A Phase I Trial to Find a Safe Dose for Advanced Renal Cell Cancer (RCC) Patients with renal cell cancer, progressive after treatment with Interferon Treatment = Fixed dose of 5-FU + one of 6 doses of Gemcitabine: {100, 200, 300, 400, 500, 600} mg/m 2 Toxicity = Grade 3,4 diarrhea, mucositis, or hematologic (blood) toxicity N max = 36 patients, treated in cohorts of 3 Start the with1 st cohort treated at 200 mg/m 2 Adaptively pick a best dose for each cohort

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Continual Reassessment Method (CRM, OQuigley et al. 1990) with a Bayesian Logistic Regression Model 1) Specify a model for ( x j, = Pr(Toxicity|, dose x j ) and prior on 2) Physician specifies p TOX * = a target Pr(Toxicity) 3) Treat each successive cohort of 3 pats. at the best dose for which E[ ( x j, | data] is closest to p TOX * 4) The best dose at the end of the trial is selected exp( + x j ) ( x j, = 1 + exp( + x j ) using x j = log(d j ) - { j=1,…k log(d j )}/k, j=1,…,6. Prior: ~ N(, 2 ), ~ N(, 2 )

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Elicit the mean toxicity probabilities at two doses. In the RCC trial, the elicited prior values were E { ( 200, )} =.25 and E { ( 500, )} =.75 1)Solve algebraically for = -.13 and = ) = = 2 ~ N(-.13, 4), ~N(2.40, 4) which gives prior ESS = 2.3 Alternatively, one may specify the prior ESS and solve for = = CRM with Bayesian Logistic Regression Model

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Plot of ESS as a function of ESS{ p(, )| } ESS For cohorts of size 1 to 3, =1 is still too small since it gives prior ESS = 9.3 These ESS values are OK, so = 2 to 5 is OK. ? These values give a prior with far more information than the data in a typical phase I trial.

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Prior of = Prob(tox | d = 200, ) p( ) =0.1 =0.5 =2.0 =10.0 ESS=928 ESS=37.1 ESS=2.3 ESS=0.09

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= = a very large number, so ESS = a tiny number, and have a very non-informative prior ? Why not just set = = a very large number, so ESS = a tiny number, and have a very non-informative prior ? Example: A non-informative prior is ~ N(-.13,100) and ~ N(2.40,100), i.e. =10.0 ESS = But this prior has some very undesirable properties : Prior Probabilities of Extreme Values Dose of Gemcitabine (mg/m 2 ) Pr{ ( x, )<.01} Pr{ ( x, )>.99}

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This says you believe, a priori, that 1)Pr { ( x, <.01} = Prob(toxicity is virtually impossible) =.31 to.45 2) Pr { ( x, >.99} = Prob(toxicity is virtually certain) =.30 to.40 Making = = too large (a so-called non informative prior) gives a pathological prior. Making = = too large (a so-called non informative prior) gives a pathological prior. What is too large numerically is not obvious without computing the corresponding ESS. What is too large numerically is not obvious without computing the corresponding ESS.

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Dose-Finding With Two Agents (Thall, Millikan, Mueller, Lee, 2003) Study two agents used together in a phase I clinical trial, with dose-finding based on (x, ) = probability of toxicity for a patient given the dose pair x = (x 1, x 2 ) Find one or more dose pairs (x 1, x 2 ) of the two agents used together for future clinical use and/or study in a randomized phase II trial Elicit prior information on (x, ) with each agent used alone Single Agent Toxicity Probabilities : 1 x 1 1 = x 1 = Prob(Toxicity | x 1, x 2 =0, 1 ) 2 x 2 2 = x 2 = Prob(Toxicity | x 1 =0, x 2, 2 )

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Hypothetical Dose-Toxicity Surface

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Probability Model x 2 =0 1 x 1 1 = 1 x 1 / ( x 1 ) = exp( 1 )/{1+exp( 1 )} x 1 =0 2 x 2 2 = 2 x 2 / (1 + 2 x 2 ) = exp( 2 ) / {1+ exp( 2 )} where j = log( j )+ j log(x j ) for j=1,2 1 2 ), where and have elicited informative priors and the interaction parameters 2 3, 3 have non-informative priors.

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Single-Agent Prior Elicitation Questions 1.What is the highest dose having negligible (<5%) toxicity? 2.What dose has the targeted toxicity * ? 3.What dose above the target has unacceptably high (60%) toxicity? 4.At what dose above the target are you nearly certain (99% sure) that toxicity is above the target (30%) ?

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Resulting Equations for the Hyperparameters Denote g( ) = / (1+ ) so (x, )} = g( x ). Denote the doses given as answers to the questions by { x (1), x (2), x (3) = x *, x (4) }, and z j = x (j) / x *. Assuming ~ Ga(a 1, a 2 ) and ~ Ga(b 1, b 2 ), solve the following equations for (a 1, a 2, b 1, b 2 ) : 1. Pr{ g( z 1 ) <.05 } = E( z*) ) = a 1 a 2 E(1 ) = * / (1 - * ) 3. E( z 3 ) = a 1 a 2 E(z 3 ) = 0.60 / 0.40 = Pr{ g( z 4 ) > * } = 0.99

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The answers to the 4 questions for each single agent Randy Millikan, MD

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An Interpretation of this Prior The ESS of p (θ) = p(θ 1, θ 2, θ 3 ) is 1.5 Since informative priors on θ 1 and θ 2 and a vague prior on θ 3 were elicited, it is useful to determine the prior ESS of each subvector : ESS of marginal prior p(θ 1 ) is for (x 1,0 | 1 1 )} ESS of marginal prior p(θ 2 ) is for (0,x 2 | 2 2 )} ESS of marginal prior p(θ 3 ) is 0.01 for the interaction parameters θ 3 = ( 3 3 )

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This illustrates 4 key features of prior ESS 1.ESS is a readily interpretable index of a priors informativeness. 2.It may be very useful to compute ESSs for both the entire parameter vector and for particular subvectors 3.ESS values may be used as feedback in the elicitation process 4.Even when standard distributions are used for priors, it may NOT be obvious how to define a priors ESS.

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For indices a=0,1 and b=0,1, and x = standardized dose, a,b ( x, ) = Pr(Y E = a, Y T = b | x, ) = E a (1- E ) 1-a T b (1- T ) 1-b + (-1) a+b E (1- E ) T (1- T ) (e -1)/(e +1) with marginals logit T ( x, ) = T x T logit E ( x, ) = E x E,1 x 2 E,2 The model parameter vector is ( T T E E,1 E,2, Probability Model for Dose-Finding Based on Bivariate Binary Efficacy (Response) and Toxicity Indicators Y E and Y T (Thall and Cook, 2004)

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Establishing Priors 1)Elicit mean & sd of T ( x, ) & E ( x, ) for several values of x. 2)Use least squares to solve for initial values of the hyperparameters in prior( | ) 3) Each component of is assumed normally distributed, r ~ N( r, r ), so = ( 1, 1,…, p, p ) 4) m E,j = prior mean and s E,j = prior sd of E (x j, m T,j = prior mean and s T,j = prior sd of T (x j, 5) # elicited values > dim( ) find the vector that minimizes the objective function Penalty term to keep the s on the same numerical domain, c =.15

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A trial of allogeneic stem cell transplant patients: Up to 12 cohorts of 3 each (N max = 36) were treated to determine a best dose among {.25,.50,.75, 1.00 } mg/m 2 of Pentostatin ® as prophyaxis for graft-versus-host disease. E = drop from baseline of at least 1 grade in GVHD at week 2 T = unresolved infection or death within 2 weeks. Example: Elicited Prior for the illustrative application in Thall and Cook (2004) ESS( ) = 8.9 (equivalent to 3 cohorts of patients!!) ESS( E ) = 13.7, ESS( T ) = 5.3, ESS( ) = 9.0

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A Slightly Smarter Way to Think About Priors

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Fix the means and use ESS contour plots to choose Example: A Strategy for Determining Priors in the Regression Model To obtain desired overall ESS = 2.0 and ESS E = ESS T = ESS = 2.0, one may inspect the ESS plots to choose the variances of the hyperprior. One combination that gives this is

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Eliciting the Hyperprior for a Hierarchical Bayesian Model in a Phase II Trial (Thall, et at. 2003) A single arm trial of Imatinib (Gleevec, STI571) in sarcoma, accounting for multiple disease subtypes. i = Pr( Tumor response in subtype i ) Prior: logit( i ) | ~ i.i.d Normal( ), i=1,…,k Hyperprior: ~ N( -2.8, 1), ~ Ga( 0.99, 0.41 ) Stopping Rule: Terminate accrual within the i th subtype if Pr( i > 0.30 | Data ) < Data refers to the data from all 10 subtypes. But where did these numbers come from?

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Eliciting the Hyperprior Denote X i = # responders out of m i patients in subtype i. 1) I fixed the mean of at logit(.20) = , to correspond to mean prior response rate midway between the target.30 and the uninteresting value.10. 2) I elicited the following 3 prior probabilities : Pr( 1 > 0.30 ) = 0.45 Pr( 1 > 0.30 | X 1 / m 1 = 2/6) = Pr( 1 > 0.30 | X 2 / m 2 = 2/6) = 0.47

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Prior Correlation Between Two Sarcoma Subtype Response Probabilities 1 and 2

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Two Priors for a Phase II-III Pediatric Brain Tumor Trial A two-stage trial of 4 chemotherapy combinations : S = carboplatin + cyclophosphamide + etoposide + vincristine E 1 = doxorubicin + cisplatinum + actinomycin + etoposide E 2 = high dose methotrexate E 3 = temozolomide + CPT-11 Outcome (T,Y) is 2-dimensional : T = disease-free survival time Y = binary indicator of severe but non-fatal toxicity Both p(T | Y,Z, ) and p(Y | Z, ) account for patient covariates: Age, I(Metastatic disease), I(Complete resection) I(Histology=Choriod plexus carcinoma)

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Probability Model 1)T| Z,Y, j ~ lognormal with variance T 2 and mean T,j (Z,Y, ) = T,j + T (Z,Y) T,j = effect of trt j on T, after adjusting for Z and Y For j=0 (standard trt), T = ( T,0, T ) 2) logit{Pr(Y=1 | Z, j)} = Y,j + Y Z Y,j = effect of trt j on Y, after adjusting for Z For j=0 (standard trt), Y = ( Y,0, Y )

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Toxicity Probability as a Function of Age Elicited from Three Pediatric Oncologists

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Probability Model for Toxicity logit{Pr(Y=1 | Z,, j=0)} = Y,0 + Y,1 Age 1/2 + Y,2 log(Age) was determined by fitting 72 different fractional polynomial functions and picking the one giving the smallest BIC. Estimated linear term with posterior mean subscripted by the posterior sd is This determined the prior of Y

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64 Elicited EFS Probabilities How do you use these 64 probabilities to solve for 10 hyperparameters?!! Johannes Wolff, MD

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T = ( T,j, T, T ) has prior Regard each prior mean EFS prob as a func of Use nonlinear least squares to solve for by minimizing E( T ) = (0.44, -0.41, 0.56, -0.53) with common variance and log( T ) ~ N(-0.08, ) Prior for T

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Y E = indicator of Efficacy Y T = indicator of Toxicity d = assigned dose Z = vector of baseline patient covariates Model the marginals E ( d, Z) = Prob(E if d is given to a patient with covs Z) T ( d, Z) = Prob(T if d is given to a patient with covs Z) Use a copula to define the joint distribution : a,b = Pr(Y E =a, Y T =b) is a function of E ( d, Z) and T ( d, Z) A Phase I/II Dose-Finding Method Based on E and T that Accounts for Covariates

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E = link{ E ( d,Z) } & T = link{ T ( d,Z) } where E ( d,Z) & T ( d,Z) are functions of [ dose effects ] + [ covariate effects ] + [ dose-covariate interactions ] a,b = Pr(Y E =a, Y T =b) = func( E, T, for a, b = 0 or 1 Model for E ( d,Z) and T ( d,Z)

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For the trial: E ( x, Z) = f( x, E ) + E Z + x E Z For the historical treatment j : E ( j, Z) = E, j + E,H Z + E, j Z Linear Terms of the Model for E (,Z) Dose effectCovariate effects Dose-Covariate Interactions Historical trt effect Historical trt- covariate interactions

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For the trial: T ( x, Z) = f( x, T ) + T Z + x T Z For the historical treatment j : T ( j, Z) = T, j + T,H Z + T, j Z Linear Terms of the Model for T (,Z) Dose effectCovariate effects Dose-Covariate Interactions Historical trt effects Historical trt- covariate interactions

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In planning the trial, historical data are used to estimate patient covariate main effects : Prior( T ) = Posterior( T,H | Historical data) Prior( E ) = Posterior( E,H | Historical data) The estimated covariate effects are incorporated into the model for E ( d,Z) and T ( d,Z) used to plan and conduct the trial Using Historical Data

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For a reference patient Z*, elicit prior means of T ( x j, Z*) and E ( x j, Z*) at each dose x j to establish prior means of the dose effect parameters Assume non-informative priors on dose effects and dose-covariate interactions Use prior variances to tune prior effective sample size (ESS) in terms of E and T Establishing Priors

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Control the prior ESS to make sure that the data drives the decisions, rather than the prior on the dose-outcome parameters

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Application chemo- protective agent (CPA) A dose-finding trial of a new targeted chemo- protective agent (CPA) given with idarubicin + cytosine arabinoside (IDA) for untreated acute myelogenous leukemia (AML)patients age < 60 Historical data from 693 AML patients Z = (Age, Cytogenetics) where Cytogenetics = (Poor, Intermediate, Good) Inv-16 or t(8:21)-5 or -7

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Application Efficacy = Alive and in Complete Remission at day 40 from the start of treatment Toxicity = Severe (Grade 3 or worse) mucositis, diarrhea, pneumonia or sepsis within 40 days from the start of treatment

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Doses and Rationale CPA The CPA is hypothesized to decrease the risk of IDA-induced mucositis and diarrhea and thus allow higher doses of IDA CPA Fixed CPA dose = 2.4 mg/kg and ara-C dose = 1.5 mg/m 2 daily on days 1, 2, 3, 4 IDA dose = 12 (standard), 15, 18, 21 or 24 mg/m 2 daily on days 1, 2, 3 (five possible IDA doses)

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Interactive E ( j, Z) = E, j + E Z + E, j Z T ( j, Z) = T, j + T Z + T, j Z Additive E ( j, Z) = E, j + E Z T ( j, Z) = T, j + T Z Reduced E ( j, Z) = E + E Z T ( j, Z) = T + T Z Models for the linear terms used to fit the historical data No treatment- covariate interactions No differences between the historical treatment effects

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Model Selection for Historical Data

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Posteriors of E (, Z) and T (, Z) based on Historical Data from 693 Untreated AML Patients

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1) Choose each patients most desirable dose based on his/her Z No dose acceptable for that Z : 2) No dose acceptable for that Z : Do Not Treat 3) At the end of the trial, use the fitted model to pick ( d | Z ) for future patients Dose-Finding Algorithm

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dynamically The trials entry criteria may change dynamically during the trial : 1)Different patients may receive different doses at the same point in the trial 2)Patients initially eligible may be ineligible (no acceptable dose) after some data have been observed 3)Patients initially ineligible may become eligible after some data have been observed

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Hypothetical Trial Results : Recommended Idarubicin Doses by Z AGECyto PoorCyto IntCyto Good 18 – – – – > 66None1215

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Currently being used to conduct a 36-patient trial to select among 4 dose levels of a new cytotoxic agent for relapsed/refractory Acute Myelogenous Leukemia Y = (CR, Toxicity) at 6 weeks Z = (Age, [1 st CR > 1 year], Number of previous trts) Marina Konopleva, MD, PhD is the PI

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Bibliography [1] Morita S, Thall PF, Mueller P. Determining the effective sample size of a parametric prior. Biometrics. 64: , [2] Morita S, Thall PF, Mueller P. Evaluating the impact of prior assumptions in Bayesian biostatistics. Statistics in Biosciences. In press. [3] Thall PF, Cook JD. Dose-finding based on efficacy-toxicity trade-offs. Biometrics, 60: , [4] Thall PF, Simon R, Estey EH. Bayesian sequential monitoring designs for single-arm clinical trials with multiple outcomes. Statistics in Medicine 14: , [5] Thall PF, Wathen JK, Bekele BN, Champlin RE, Baker LO, Benjamin RS. Hierarchical Bayesian approaches to phase II trials in diseases with multiple subtypes. Statistics in Medicine 22: , [6] Thall PF, Wooten LH, Nguyen HQ, Wang X, Wolff JE. A geometric select-and-test design based on treatment failure time and toxicity: Screening chemotherapies for pediatric brain tumors. Submitted for publication. [7] Thall PF, Nguyen H, Estey EH. Patient-specific dose-finding based on bivariate outcomes and covariates. Biometrics. 64: , 2008.

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