GMDS-CEN conference Satellite Webinar “Long -run behaviour of Bayesian procedures “ 16 September 2020 The effect of prior information on frequentist properties of Bayes test decisions Annette Kopp-Schneider, Silvia Calderazzo and Manuel Wiesenfarth Division of Biostatistics, German Cancer Research Center (DKFZ) Heidelberg, Germany
Motivation • Trial in adults with solid tumors harboring DNA repair deficiencies treated by targeted therapy, evaluation of response. • DNA repair deficiencies also occur in pediatric tumors → investigate targeted therapy in a pediatric arm Question: Should this pediatric arm be designed as stand-alone arm or can power gain be expected when borrowing information from the adult trial? 16 Sept 2020 2 CEN Satellite Webinar Annette Kopp-Schneider
Planning the pediatric arm with stand-alone evaluation • Number of responders in children, 𝑆 𝑞𝑓𝑒 ~ Bin ( 𝑜 𝑞𝑓𝑒 = 40 , 𝑞 ) • One-sided test 𝐼 0 : 𝑞 ≤ 𝑞 0 vs. 𝐼 1 : 𝑞 > 𝑞 0 , 𝑞 0 = 0.2 • Type I error rate 𝛽 = 0.05 Bayesian approach (1) • 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: Reject 𝐼 0 ֞ 𝑄 𝑞 > 𝑞 0 = 0.2|𝑠 ≥ 𝑑 , e.g., 𝑑 = 0.95 . 𝑞𝑓𝑒 16 Sept 2020 3 CEN Satellite Webinar 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 𝑄 𝑞 > 𝑞 0 |𝑠 𝑞𝑓𝑒 ≥ 0.95 ֞ 𝑠 𝑞𝑓𝑒 ≥ 13 16 Sept 2020 4 CEN Satellite Webinar 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 𝑄 𝑞 > 𝑞 0 |𝑠 𝑞𝑓𝑒 ≥ 0.95 ֞ 𝑠 𝑞𝑓𝑒 ≥ 13 16 Sept 2020 5 CEN Satellite Webinar Annette Kopp-Schneider
Planning the pediatric arm with stand-alone evaluation: Frequentist approach • Uniformly most powerful (UMP) level 𝛽 test is given by: reject 𝐼 0 ֞ 𝑠 𝑞𝑓𝑒 ≥ 𝑐 UMP 𝛽 • Here: 𝑐 UMP 0.05 = 13 • All possible power curves for 𝑜 𝑞𝑓𝑒 = 40 for varying threshold 𝑐 (and type I error probability): Power: 𝑄 𝑆 𝑞𝑓𝑒 ≥ 𝑐|𝑞 𝑢𝑠𝑣𝑓 𝑐 = 13 𝑞 𝑢𝑠𝑣𝑓 16 Sept 2020 6 CEN Satellite Webinar Annette Kopp-Schneider
Borrowing from adult information for the pediatric arm Use results from adult trial to inform the prior for the pediatric arm. Hope If treatment is successful in adults, then power is increased for pediatric arm: Pediatric with borrowing from adult Pediatric only ? Power 𝑞 𝑢𝑠𝑣𝑓 16 Sept 2020 7 CEN Satellite Webinar 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. 16 Sept 2020 8 CEN Satellite Webinar Annette Kopp-Schneider
Adaptive power parameter (2) Result from adult trial: 𝑠 𝑏𝑒𝑣 = 12 among 𝑜 𝑏𝑒𝑣 = 40 ( Ƹ 𝑞 𝑏𝑒𝑣 = 0.3) Use an Empirical Bayes power prior approach where መ 𝜀 𝑠 𝑞𝑓𝑒 ; 𝑠 𝑏𝑒𝑣 = 12 maximizes the marginal likelihood of 𝜀 (Gravestock, Held et al. 2017): መ 𝑏𝑒𝑣 = 12 : 𝜀 𝑠 𝑞𝑓𝑒 ; 𝑠 16 Sept 2020 9 CEN Satellite Webinar Annette Kopp-Schneider
Adaptive power parameter (3) 𝑏𝑒𝑣 , መ 𝑄 𝑞 > 𝑞 0 |𝑠 𝑞𝑓𝑒 , 𝑠 𝜀 𝑠 𝑞𝑓𝑒 ; 𝑠 > 𝑑 = 0.95 corresponds to 𝑠 𝑞𝑓𝑒 ≥ 𝑐 = 11 𝑏𝑒𝑣 Without adults 16 Sept 2020 10 CEN Satellite Webinar Annette Kopp-Schneider
Adaptive power parameter (4) 𝑏𝑒𝑣 , መ 𝑄 𝑞 > 𝑞 0 |𝑠 𝑞𝑓𝑒 , 𝑠 𝜀 𝑠 𝑞𝑓𝑒 ; 𝑠 > 𝑑 = 0.95 corresponds to 𝑠 𝑞𝑓𝑒 ≥ 𝑐 = 11 𝑏𝑒𝑣 𝑐 = 11 𝑐 = 13 → power gain but type I error inflation 16 Sept 2020 11 CEN Satellite Webinar Annette Kopp-Schneider
Adaptive power parameter (5) 𝑏𝑒𝑣 , መ 𝑄 𝑞 > 𝑞 0 |𝑠 𝑞𝑓𝑒 , 𝑠 𝜀 𝑠 𝑞𝑓𝑒 , 𝑠 is monotonically increasing in 𝑠 𝑏𝑒𝑣 𝑞𝑓𝑒 𝑏𝑒𝑣 , መ → 𝑄 𝑞 > 𝑞 0 |𝑠 𝑞𝑓𝑒 , 𝑠 𝜀 > 𝑑′ = 0.99 corresponds to 𝑠 𝑞𝑓𝑒 ≥ 𝑐 = 13 Without adults → type I error controlled but no power gained 16 Sept 2020 12 CEN Satellite Webinar Annette Kopp-Schneider
Ƹ “Extreme borrowing” (1) • Artificial method for illustration of not monotonically increasing 𝑄 𝑞 > 𝑞 0 |𝑠 𝑞𝑓𝑒 , 𝑠 𝑏𝑒𝑣 : ֞ 𝑞 𝑏𝑒𝑣 = Ƹ 𝑞 𝑞𝑓𝑒 borrow adult information • Assume 𝑜 𝑏𝑒𝑣 = 100 , 𝑠 𝑏𝑒𝑣 = 30 ֜ Ƹ 𝑞 𝑏𝑒𝑣 = 0.3 • Here: borrow all adult information if Ƹ 𝑞 𝑞𝑓𝑒 = 0.3 (𝑠 𝑞𝑓𝑒 = 12 for 𝑜 𝑞𝑓𝑒 = 40 ) don‘t borrow for 𝑠 𝑞𝑓𝑒 ≠ 12 16 Sept 2020 13 CEN Satellite Webinar Annette Kopp-Schneider
“Extreme borrowing” (2) Borrow all adult information iff 𝑠 𝑞𝑓𝑒 = 12 For 𝑑 = 0.95 ֜ 𝑐 = 12 ֜ type I error rate = 0.088 Without adults 16 Sept 2020 14 CEN Satellite Webinar Annette Kopp-Schneider
“Extreme borrowing” (3) Borrow all adult information iff 𝑠 𝑞𝑓𝑒 = 12 For 𝑑 = 0.9976 ֜ reject H 0 For 𝑑 = 0.95 ֜ 𝑐 = 12 if 𝑠 𝑞𝑓𝑒 = 12 or 𝑠 𝑞𝑓𝑒 ≥ 16 ֜ type I error rate = 0.088 ֜ type I error rate = 0.047 Without adults 16 Sept 2020 15 CEN Satellite Webinar Annette Kopp-Schneider
“Extreme borrowing” (4) Reject H 0 if 𝑠 𝑞𝑓𝑒 ∈ 12 ∪ 16, 17, … , 40 Compare to: Reject H 0 if 𝑠 𝑞𝑓𝑒 ∈ 13, 14, … , 40 → type I error controlled but power decreased 16 Sept 2020 16 CEN Satellite Webinar Annette Kopp-Schneider
Borrowing from adult information (1) If 𝑄 𝑞 > 𝑞 0 |𝑠 𝑞𝑓𝑒 , 𝑠 𝑏𝑒𝑣 is monotonically increasing in 𝑠 𝑞𝑓𝑒 , then there exists 𝑑′ with 𝑏𝑒𝑣 ≥ 𝑑 ′ ֞ 𝑠 𝑄 𝑞 > 𝑞 0 |𝑠 𝑞𝑓𝑒 , 𝑠 𝑞𝑓𝑒 ≥ 𝑐 UMP 𝛽 and 𝑐 UMP 𝛽 is the level 𝛽 UMP test boundary. 16 Sept 2020 17 CEN Satellite Webinar Annette Kopp-Schneider
Borrowing from adult information (2) If 𝑄 𝑞 > 𝑞 0 |𝑠 𝑞𝑓𝑒 , 𝑠 is not monotonically increasing in 𝑠 𝑞𝑓𝑒 , then either: 𝑏𝑒𝑣 (1) a threshold 𝑑′ can still be identified with 𝑄 𝑞 > 𝑞 0 |𝑠 𝑞𝑓𝑒 , 𝑠 𝑏𝑒𝑣 ≥ 𝑑′֞𝑠 𝑞𝑓𝑒 ≥ 𝑐 UMP 𝛽 ( ∗ ) (2) if no 𝑑′ with ( ∗ ) can be identified, then either the - test does not control type I error or - test controls type I error but is not UMP. 16 Sept 2020 18 CEN Satellite Webinar Annette Kopp-Schneider
Borrowing from adult information: Summary View decision rule in Bayesian approach as test function φ 𝑠 𝑞𝑓𝑒 = 1 𝑄 𝑞>𝑞 0 |𝑠 𝑞𝑓𝑒 ,𝑠 𝑏𝑒𝑣 ≥𝑑 → There is nothing better than the UMP test! • This holds for all situations in which UMP tests exist: exponential family distribution one-sided tests, two-sided tests (equivalence situation) one-sided comparison of two means of normal variables … • This also holds in situations in which UMP unbiased tests exists: two-sided comparisons comparison of two proportions … • True for any (adaptive) borrowing mechanism (power prior, mixture prior, hierarchical model, test-then- pool,…) ( see Viele et al. (2014)) • Proven by Psioda and Ibrahim (2018) for one-sample one-sided normal test with borrowing using a fixed power prior. 16 Sept 2020 19 CEN Satellite Webinar Annette Kopp-Schneider
In general • 𝑒 𝐷 : realizations of current data 𝐸 𝐷 collected to test: ϑ 𝐷 ∈ 𝐼 0 vs. ϑ 𝐷 ∈ 𝐼 0 • Without historical data: Lehmann (1986) notation: the UMP hypothesis test is ( 𝑈 sufficient test statistic) 1 if 𝑈 𝑒 𝐷 ∈ RejectionRegion (reject 𝐼 0 ) 𝜒 𝑒 𝐷 = ቊ 0 if 𝑈 𝑒 𝐷 ∈ AcceptanceRegion (accept 𝐼 0 ) → power function 𝐹 𝜘 𝐷 𝜒 𝐸 𝐷 → type I error control: 𝐹 𝜘 𝐷 𝜒 𝐸 𝐷 ≤ 𝛽 for all 𝜘 𝐷 ∈ 𝐼 0 • With historical data: Borrow from observed historical data 𝑒 𝐼 (from 𝐸 𝐼 ) by: 1 if 𝑈 𝑒 𝐷 ∈ RejectionRegion 𝑒 𝐼 𝜒 𝐶 𝑒 𝐷 ; 𝑒 𝐼 = ቊ 0 if 𝑈 𝑒 𝐷 ∈ AcceptanceRegion 𝑒 𝐼 | → power function 𝐹 𝜘 𝐷 𝜒 𝐶 𝐸 𝐷 ; 𝑒 𝐼 = 𝐹 𝜘 𝐷 ,𝜘 𝐼 𝜒 𝐶 𝐸 𝐷 ; 𝐸 𝐼 𝐸 𝐼 = 𝑒 𝐼 → type I error: max ϑ 𝐷 ∈𝐼 0 𝐹 𝜘 𝐷 𝜒 𝐶 𝐸 𝐷 ; 𝑒 𝐼 (note: 𝜘 𝐷 may be multidimensional) 16 Sept 2020 20 CEN Satellite Webinar Annette Kopp-Schneider
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