Bayesian dynamic borrowing of external information: What can be - PowerPoint PPT Presentation

Bayesian methods in the development and assessment of new therapies Workshop of the IBS- DR working group “Bayes Methods” Göttingen, Dec 6-7 2018 Bayesian dynamic borrowing of external information: What can be gained in terms of frequentist power? Annette Kopp-Schneider, Silvia Calderazzo and Manuel Wiesenfarth Division of Biostatistics, German Cancer Research Center (DKFZ) Heidelberg, Germany

Motivation • Adult trial in subjects with previously treated advanced or recurrent solid tumors harboring DNA repair deficiencies: Endpoint: response to treatment (dichotomous) Two arms: Targeted therapy vs. Physician’s choice • DNA repair deficiencies also occur in children → investigate targeted therapy in a single -arm pediatric trial Question: Should this single pediatric arm be designed as stand-alone arm or can power gain be expected when borrowing information from the adult targeted therapy arm? 7 Dec 2018 2 WG Bayes Methods Annette Kopp-Schneider

Planning the pediatric arm with stand-alone evaluation: Bayesian approach (1) • Number of responders in children, 𝑆 𝑞𝑓𝑒 ~ Bin ( 𝑜 𝑞𝑓𝑒 , 𝑞 ) • Test 𝐼 0 : 𝑞 = 𝑞 0 vs. 𝐼 1 : 𝑞 > 𝑞 0 , 𝑞 0 = 0.2 • Type I error rate 𝛽 = 0.05 • 𝑜 𝑞𝑓𝑒 = 40 • Bayesian approach: Use beta-binomial model 𝑆 𝑞𝑓𝑒 | 𝑞 ~ Bin ( 𝑜 𝑞𝑓𝑒 , 𝑞 ), 𝜌 𝑞 = Beta ( 0.5 , 0.5 ) • Evaluate efficacy based on Bayesian posterior probability: 𝑄 𝑞 > 𝑞 0 |𝑠 ≥ 𝑑 , e.g., 𝑑 = 0.95 . 𝑞𝑓𝑒 7 Dec 2018 3 WG Bayes Methods Annette Kopp-Schneider

Planning the pediatric arm with stand-alone evaluation: Bayesian approach (2) Posterior probability 𝑄 𝑞 > 𝑞 0 |𝑠 𝑞𝑓𝑒 as a function of 𝑠 𝑞𝑓𝑒 For 𝒐 𝒒𝒇𝒆 = 𝟓𝟏 : 𝑄 𝑞 > 𝑞 0 |𝑠 𝑞𝑓𝑒 ≥ 0.95 𝑠 𝑞𝑓𝑒 ≥ 13 7 Dec 2018 4 WG Bayes Methods Annette Kopp-Schneider

Planning the pediatric arm with stand-alone evaluation: Bayesian approach (3) Posterior probability 𝑄 𝑞 > 𝑞 0 |𝑠 𝑞𝑓𝑒 as a function of 𝑠 𝑞𝑓𝑒 For 𝒐 𝒒𝒇𝒆 = 𝟓𝟏 : 𝑄 𝑞 > 𝑞 0 |𝑠 𝑞𝑓𝑒 ≥ 0.95 𝑠 𝑞𝑓𝑒 ≥ 13 In general: For every 𝑑 ∈ 0, 𝑄 𝑞 > 𝑞 0 |𝑠 𝑞𝑓𝑒 = 𝑜 𝑞𝑓𝑒 there exists a unique 𝑐 ∈ 0,1, … , 𝑜 𝑞𝑓𝑒 with 𝑄 𝑞 > 𝑞 0 |𝑠 𝑞𝑓𝑒 ≥ 𝑑 𝑠 𝑞𝑓𝑒 ≥ 𝑐 (Kopp-Schneider et al., 2018) 7 Dec 2018 5 WG Bayes Methods Annette Kopp-Schneider

Planning the pediatric arm with stand-alone evaluation: Frequentist approach • Test 𝐼 0 : 𝑞 = 𝑞 0 vs. 𝐼 1 : 𝑞 > 𝑞 0 • Type I error rate 𝛽 , e.g., 𝛽 = 0.05 • Uniformly most powerful (UMP) level 𝛽 test is given by: reject 𝐼 0 𝑠 𝑞𝑓𝑒 ≥ 𝑐 UMP 𝛽 • Here: 𝑐 UMP 0.05 = 13 Power: 𝑄 𝑆 𝑞𝑓𝑒 ≥ 𝑐|𝑞 𝑢𝑠𝑣𝑓 b=13 7 Dec 2018 6 WG Bayes Methods Annette Kopp-Schneider

Planning the pediatric arm with stand-alone evaluation: Power function (1) Power = 𝑔 𝑞 𝑢𝑠𝑣𝑓 = 𝑄 𝑆 𝑞𝑓𝑒 ≥ 𝑐|𝑞 𝑢𝑠𝑣𝑓 𝑜 = 𝑄 𝑆 𝑞𝑓𝑒 = 𝑠 𝑞𝑓𝑒 |𝑞 𝑢𝑠𝑣𝑓 1 𝑠 𝑞𝑓𝑒 ≥𝑐 𝑠 𝑞𝑓𝑒 =0 7 Dec 2018 7 WG Bayes Methods Annette Kopp-Schneider

Planning the pediatric arm with stand-alone evaluation: Power function (2) 𝑞𝑓𝑒 Power = 𝑔 𝑞 𝑢𝑠𝑣𝑓 𝑄 𝑞 > 𝑞 0 |𝑠 = 𝑄 𝑆 𝑞𝑓𝑒 ≥ 𝑐|𝑞 𝑢𝑠𝑣𝑓 𝑜 = 𝑄 𝑆 𝑞𝑓𝑒 = 𝑠 𝑞𝑓𝑒 |𝑞 𝑢𝑠𝑣𝑓 1 𝑠 𝑞𝑓𝑒 ≥𝑐 𝑠 𝑞𝑓𝑒 =0 𝑠 𝑞𝑓𝑒 𝑜 = 𝑄 𝑆 𝑞𝑓𝑒 = 𝑠 𝑞𝑓𝑒 |𝑞 𝑢𝑠𝑣𝑓 1 𝑄 𝑞>𝑞 0 |𝑠 𝑞𝑓𝑒 ≥𝑑 𝑠 𝑞𝑓𝑒 =0 (c selected appropriately) 7 Dec 2018 8 WG Bayes Methods Annette Kopp-Schneider

Borrowing from adult information for the pediatric arm Use information from adults to inform the prior for the pediatric trial. Hope If treatment is successful in adults, then power is increased for pediatric trial: pediatric with borrowing from adult Pediatric only ? Power 𝑞 𝑢𝑠𝑣𝑓 7 Dec 2018 9 WG Bayes Methods Annette Kopp-Schneider

Adaptive power parameter (1) Power prior approach with power parameter 𝜀 ∈ 0, 1 : 𝑏𝑒𝑣 𝜀 𝜌 𝑞 𝜌 𝑞|𝑠 𝑏𝑒𝑣 , 𝜀 ∝ 𝑀 𝑞; 𝑠 Adapt 𝜀 = 𝜀 𝑠 𝑞𝑓𝑒 , 𝑠 𝑏𝑒𝑣 such that information is only borrowed for similar adult and pediatric data: → 𝜀 𝑠 𝑞𝑓𝑒 , 𝑠 𝑏𝑒𝑣 large when adult and children data are similar → 𝜀 𝑠 𝑞𝑓𝑒 , 𝑠 𝑏𝑒𝑣 small in case of prior-data conflict. 7 Dec 2018 10 WG Bayes Methods Annette Kopp-Schneider

Adaptive power parameter (2) Result from adult trial: e.g., 𝑠 𝑏𝑒𝑣 = 12 among 𝑜 𝑏𝑒𝑣 = 40 ( 𝑞 𝑏𝑒𝑣 = 0.3) Use an Empirical Bayes approach where 𝜀 𝑠 𝑞𝑓𝑒 ; 𝑠 𝑏𝑒𝑣 = 12 maximizes the marginal likelihood of 𝜀 (Gravestock, Held et al. 2017): 𝜀 𝑠 𝑞𝑓𝑒 ; 𝑠 𝑏𝑒𝑣 = 12 : 7 Dec 2018 11 WG Bayes Methods Annette Kopp-Schneider

Adaptive power parameter (3) 𝑏𝑒𝑣 , 𝑄 𝑞 > 𝑞 0 |𝑠 𝑞𝑓𝑒 , 𝑠 𝜀 𝑠 𝑞𝑓𝑒 ; 𝑠 > 𝑑 = 0.95 corresponds to 𝑠 𝑞𝑓𝑒 ≥ 𝑐 = 11 𝑏𝑒𝑣 Without adults 7 Dec 2018 12 WG Bayes Methods Annette Kopp-Schneider

Adaptive power parameter (4) 𝑏𝑒𝑣 , 𝑄 𝑞 > 𝑞 0 |𝑠 𝑞𝑓𝑒 , 𝑠 𝜀 𝑠 𝑞𝑓𝑒 ; 𝑠 > 𝑑 = 0.95 corresponds to 𝑠 𝑞𝑓𝑒 ≥ 𝑐 = 11 𝑏𝑒𝑣 b=11 b=13 → power gain but type I error inflation 7 Dec 2018 13 WG Bayes Methods Annette Kopp-Schneider

Adaptive power parameter (5) 𝑏𝑒𝑣 , • For this situation: 𝑄 𝑞 > 𝑞 0 |𝑠 𝑞𝑓𝑒 , 𝑠 𝜀 𝑠 𝑞𝑓𝑒 , 𝑠 is monotonically 𝑏𝑒𝑣 increasing in 𝑠 𝑞𝑓𝑒 𝑏𝑒𝑣 , • 𝑄 𝑞 > 𝑞 0 |𝑠 𝑞𝑓𝑒 , 𝑠 𝜀 > 𝑑′ = 0.99 corresponds to 𝑦 𝑞𝑓𝑒 ≥ 𝑐 = 13 → type I error controlled but no power gained 7 Dec 2018 14 WG Bayes Methods Annette Kopp-Schneider

Robust mixture prior (1) • Another way of discounting prior information is given by the use of robust mixture prior as convex combination of an uninformative prior and a prior that incorporates external information (e.g., Schmidli et al. (2014)) = 𝑥 Beta ( 0.5+𝑠 𝑏𝑒𝑣 , 0.5+𝑜 𝑏𝑒𝑣 −𝑠 𝑏𝑒𝑣 ) + 1 − 𝑥 Beta ( 0.5 , 0.5 ) 𝜌 𝑞 • Here: 𝑥 = 0.5 • Posterior is convex combination of Beta distributions with weight 𝑥 7 Dec 2018 15 WG Bayes Methods Annette Kopp-Schneider

Robust mixture prior (2) 7 Dec 2018 16 WG Bayes Methods Annette Kopp-Schneider

Robust mixture prior (3) 𝑄 𝑞 > 𝑞 0 |𝑠 𝑞𝑓𝑒 , 𝑠 𝑏𝑒𝑣 , 𝑥 > 𝑑 = 0.95 corresponds to 𝑠 𝑞𝑓𝑒 ≥ 𝑐 = 11 → type I error inflation → select 𝑑′ = 0.98 → 𝑐 = 13 → type I error controlled but no power gained. 7 Dec 2018 17 WG Bayes Methods Annette Kopp-Schneider

“Extreme borrowing” (1 ) • Artificial method for illustration of not monotonically increasing 𝑄 𝑞 > 𝑞 0 |𝑠 𝑞𝑓𝑒 , 𝑠 𝑏𝑒𝑣 : borrow adult information 𝑞 𝑏𝑒𝑣 = 𝑞 𝑞𝑓𝑒 • Assume 𝑜 𝑏𝑒𝑣 = 100 , 𝑠 𝑏𝑒𝑣 = 30 𝑞 𝑏𝑒𝑣 = 0.3 • Here: borrow all adult information if 𝑞 𝑞𝑓𝑒 = 0.3 𝑠 𝑞𝑓𝑒 = 12 7 Dec 2018 18 WG Bayes Methods Annette Kopp-Schneider

“Extreme borrowing” (2) • Artificial method for illustration of not monotonically increasing 𝑄 𝑞 > 𝑞 0 |𝑠 𝑞𝑓𝑒 , 𝑠 𝑏𝑒𝑣 : borrow adult information 𝑞 𝑏𝑒𝑣 = 𝑞 𝑞𝑓𝑒 • Assume 𝑜 𝑏𝑒𝑣 = 100 , 𝑠 𝑏𝑒𝑣 = 30 𝑞 𝑏𝑒𝑣 = 0.3 • Here: borrow all adult information if 𝑞 𝑞𝑓𝑒 = 0.3 𝑠 𝑞𝑓𝑒 = 12 For 𝑑 = 0.95 𝑐 = 12 type I error rate = 0.088 7 Dec 2018 19 WG Bayes Methods Annette Kopp-Schneider

“Extreme borrowing” (3) • Artificial method for illustration of not monotonically increasing 𝑄 𝑞 > 𝑞 0 |𝑠 𝑞𝑓𝑒 , 𝑠 𝑏𝑒𝑣 : borrow adult information 𝑞 𝑏𝑒𝑣 = 𝑞 𝑞𝑓𝑒 • Assume 𝑜 𝑏𝑒𝑣 = 100 , 𝑠 𝑏𝑒𝑣 = 30 𝑞 𝑏𝑒𝑣 = 0.3 • Here: borrow all adult information if 𝑞 𝑞𝑓𝑒 = 0.3 𝑠 𝑞𝑓𝑒 = 12 For 𝑑 = 0.9976 reject H 0 if 𝑐 = 12 or 𝑐 ≥ 16 For 𝑑 = 0.95 𝑐 = 12 type I error rate = 0.047 type I error rate = 0.088 7 Dec 2018 20 WG Bayes Methods Annette Kopp-Schneider

“Extreme borrowing” (4) Reject H 0 if 𝑐 ∈ 12 ∪ 16, 17, … , 40 Compare to: Reject H 0 if 𝑐 ∈ 13, 17, … , 40 → type I error controlled but power decreased 7 Dec 2018 21 WG Bayes Methods Annette Kopp-Schneider

Borrowing from adult information in general (1) • If 𝑄 𝑞 > 𝑞 0 |𝑠 𝑞𝑓𝑒 , 𝑠 𝑏𝑒𝑣 is monotonically increasing in 𝑠 𝑞𝑓𝑒 , then there exists 𝑑′ with 𝑄 𝑞 > 𝑞 0 |𝑠 𝑞𝑓𝑒 , 𝑠 𝑏𝑒𝑣 ≥ 𝑑′ 𝑠 𝑞𝑓𝑒 ≥ 𝑐 UMP 𝛽 ( ∗ ) and 𝑐 UMP 𝛽 is the level 𝛽 UMP test boundary. 7 Dec 2018 22 WG Bayes Methods Annette Kopp-Schneider