Subgroup Mixable Inference for Targeted Therapies

Transcription

1 Subgroup Mixable Inference for Targeted Therapies Jason C. Hsu The Ohio State University Duke Industry Statistics Symposium September 2017 In collaboration with Hong Tian, Haiyan Xu, Hui-Min Lin, Ying Ding, Szu-Yu Tang

2 Outline Medicine is involved in personalized medicine Phase 3 study design for targeted therapies may require estimating efficacy in mixtures of Subgroups from Phase 2 Efficacy measures may not respect logic among Subgroups Odds ratio & hazard ratio are not logic-respecting Relative response & ratio of medians are logic-respecting Computer packages have issue with stratified testing if outcome is binary or time-to-event regardless of efficacy measure is logic-respecting or not There is a principled solution 1. Prove an efficacy measure is logic-respecting 2. Apply Subgroup Mixable Estimation (SME)

3 Odds Ratio is not logic-respecting g :g + is 1:1 in population Subgroup g g + Rx.25 : :.25 Control.1 :.9.5 :.5 Odds Ratio 3 3 = Mixed g g + Rx.5 :.5 Control.3 :.7 Odds Ratio / / = 3 / / = 3 truth / / = 2 computer stratified logistic log 3 + log 3 log 3 log 2 computer marginal logistic

4 Relative Response is logic-respecting Mixing coefficients are not proportions of g and g + patients + Subgroup g g + Rx Control.1.5 g :g + is 1:1 in population = Mixed g g + Rx.5 Control.3 truth = computer stratified = log + log log 2 = truth

5 Targeted Therapies sometimes called personalized medicines or precision medicines (Janet Woodcock 2015) biomarker identifies patients o likely to benefit o at higher risk o needing a different dose Compounds that end in mab, or -ib 45% of drugs approved by FDA in 2013 are targeted o Immunotherapies Opdivo & Keytruda are targeted companion diagnostic test CDx identifies o marker-positive g +, marker-negative g patients

6 Personalized/Precision Medicines Zalkori Indication: NSCLC Rx: -tinib Indication ALK-positive CDx: break-apart FISH Regular approval 2013 Accelerated approval 2011 CDx: Vysis Rx: Pfizer Keytruda Indication: NSCLC Rx: (immunotherapy) -zumab Patient selection PD-L1 expressed CDx: IHC 1% Regular approval 10/2016 Accelerated approval 2015 CDx: Dako Rx: Merck

7 OPDIVO (Nivolumab) Postow et al (2015) NEJM In order to preserve an experiment-wide type I error rate of 5%, a hierarchical testing approach was applied to key secondary end points after analysis of the primary end point of the objective response rate in all patients with BRAF wild-type tumors who underwent randomization. Of the key secondary endpoints, the objective response rate among all randomly assigned patients was tested first, followed by testing of progression-free survival among all patients with BRAF wild-type tumors who underwent randomization; progression-free survival among all randomly assigned patients was tested last. FWER = 5% with decision-path (Hsu and R. Berger 1999 JASA) 1. ORR BRAF wild-type If p-value <.05 then go to step 2 2. ORR all If p-value <.05 then go to step 3 3. PFS BRAF wild-type If p-value <.05 then go to step 4 4. PFS all

8 Objective Response Rates (ORR) of OPDIVO treating Melanoma Nivolumab +Ipilimumab Ipilimumab BRAF Wild-Type 44 (n=72) 4 (n=37) BRAF Mutation Positive 12 (n=23) 1 (n=10) Relative Response

9 Objective Response Rates Nivolumab versus Docetaxel by PD-L1 Expression Level Nivolumab (n=231) Docetaxel (n=224) Odds Ratio 1% 5% 10% < 1% > 1% < 5% > 5% < 10% > 10% 10 (108) 15 (101) 38 (123) 15 (123) 14 (136) 19 (138) 34 (95) 11 (86) 16 (145) 20 (145) 32 (86) 10 (79)

10 Immunohistochemistry (IHC) since 1940s PD-L1 Expression in Non Small-Cell Lung Cancers. Garon EB et al. N Engl J Med 2015;372:

11 Logic-respecting efficacy measures In a balanced population (parameter space) Efficacy measure ff is Logic-respecting if ff {g +, g } [ ff {g }, ff {g + }] Rationale: if ff ff {g } = 2 and ff {g + } = 3, then ff {g, g + } = 1 illogical ff {g, g + } = 4 illogical So if ff {g }= ff {g + } = 3 but ff {g, g + } = say then efficacy ff is not logic-respecting consistency has no meaning difficult to define correct decision!

12 Original Proportional Hazard (PH) definition Proportional Hazard Survival Functions Lehmann Family Example of 4 survival functions in a Lehmann family Cox and Oaks (1984) Analysis of Survival Data p.24 Translation family, Scale family, Lehmann family, p.40 Proportional Hazard (PH) family Lehmann family PH is not constant Relative Risk t /2 1/3 1/4 1 1/4 1/9 1/16 RR t /2 1/4 1/8 1 1/4 1/16 1/64 RR

13 Hazard Ratio (HR) is not logic-respecting HR of Rx vs. C within g and g + HR of Rx vs. C in {g, g + } g g + Computer Stratified Un-stratified Rx vs. C 2/3 2/3 2/ HR of each subgroup relative to reference subgroup {C, g } Survival functions for {g, g+} will not have PH because g g + Rx 2/3 2/9 C 1 1/3

14 Logic-respecting efficacy measures Logical relationship among the parameters ff[g + ] <.8, ff[g ] <.8, ff[{g +, g }] >.8 illogical Logic-respecting efficacy measure Relative Response Ratio of medians Not logic-respecting efficacy measures Odds Ratio Hazard Ratio

15 Weibull connects Time with Probability Time = Τ indicator of Treatment or Control Μ indicator of Marker + or status Τ Μ interaction between Τ and Μ ε i.i.d. error with extreme value distribution Log-linear model Accelerated Failure Time model (AFT) Cox Proportional Hazard model (PH) Weibull is the only PH & AFT model

16 Ratio of Medians as efficacy measure Ratio of Medians Medians in g +, g, and {g +, g } well-defined logic respecting Aligns with ASCO (2015) Value Framework for Cancer Treatments Under Weibull = HR = 1/2, Shape = 1.25 = 1.74 Methodology published in Statistics in Medicine There is an app for it

17 Is ff logic-respecting or not? Proving ff is not logic-respecting is to Construct (mathematically) a counter-example so that in balanced population (parameter space) ff {g } = ff {g + } ff {g, g + } Check whether computer is logic-respecting with stratified LSmean / Estimate statement and see if it agrees with marginal Means statement Proving ff is logic-respecting is to prove ff {g +, g } [ ff {g }, ff {g + } ] mathematically Will show proof for Relative Response Proof for ratio of medians in Ding, Lin, Hsu (2017 Stats in Med)

18 Ratio of medians is logic-respecting There is an app for it!