Introduction The Cure-Death Model RL test Application Simulation Summary Current Work The Cure-Death Model An approach to increase efficiency of clinical endpoint evaluations DAGStat G¨ ottingen 2016 Harriet Sommer, Martin Wolkewitz, and Martin Schumacher Institute for Medical Biometry and Statistics, Medical Center – University of Freiburg (Germany) on behalf of the COMBACTE consortium Harriet Sommer The Cure-Death Model March 15, 2016 (G¨ ottingen) 1
Introduction The Cure-Death Model RL test Application Simulation Summary Current Work Introduction ◮ there is a variety of primary endpoints used in treatment trials dealing with severe infectious diseases ◮ clinical trials with patients that suffer from severe diseases and an additional resistant infection ⇒ in this population, a mortality rate of about 10% up to 30% can be assumed within 30 days ◮ mortality has a considerable influence on the cure process ◮ recommendations given by the existing guidelines are sometimes not consistent, nor is their practical application ◮ EMA proposes clinical cure – clinical outcome documented at a test-of-cure visit ◮ FDA proposes all-cause mortality ◮ we propose to study two primary endpoints, “cure” and “death” , in a comprehensive multistate model Harriet Sommer The Cure-Death Model March 15, 2016 (G¨ ottingen) 2
Introduction The Cure-Death Model RL test Application Simulation Summary Current Work The cure-death model randomisation 30 days cure λ 01 ( t ) λ 12 ( t ) randomisation λ 02 ( t ) death ◮ acknowledges for the time-dependent outcome structure ◮ accounts for the fact that: ◮ patients might die during the time to cure (handles competing risks ) ◮ once cured, patients might still die shortly afterwards ◮ estimates the probability of being cured and remaining cured → highly meaningful for clinicians as well as for patients Harriet Sommer The Cure-Death Model March 15, 2016 (G¨ ottingen) 3
Introduction The Cure-Death Model RL test Application Simulation Summary Current Work Possibilities for a treatment comparison 1. Risk differences with proportions of patients cured ( and alive ) at a pre-specified time point → Chi 2 test 2. Exploratory analysis of transition probabilities via the Aalen-Johansen estimator 3. Confirmatory logrank-type test Harriet Sommer The Cure-Death Model March 15, 2016 (G¨ ottingen) 4
Introduction The Cure-Death Model RL test Application Simulation Summary Current Work Logrank-type test based on Hsieh et al. (1983) For every transition j ∈ { 01 , 02 , 12 } , the Cox model has got the following form: λ j ( t | Z ) = λ 0 j ( t ) exp ( β j Z ) , baseline hazard function λ 0 j ( t ) , regression coefficient β j , treatment indicator Z . The partial likelihood can be factorised: K 01 K 02 K 12 exp ( β ′ exp ( β ′ exp ( β ′ 01 Z ( k ) ) 02 Z ( k ) ) 12 Z ( k ) ) � � � L ( β ) = 01 Z r ) × 02 Z r ) × � � � r ∈ R ( t 01 ( k ) ) exp ( β ′ r ∈ R ( t 02 ( k ) ) exp ( β ′ r ∈ R ( t 12 ( k ) ) exp ( β ′ 12 Z r ) k = 1 k = 1 k = 1 ⇒ analyse each transition separately by treating the others as censored Harriet Sommer The Cure-Death Model March 15, 2016 (G¨ ottingen) 5
Introduction The Cure-Death Model RL test Application Simulation Summary Current Work Logrank-type test based on Hsieh et al. (1983) Comparison of two groups ⇒ score test statistic for L ( β ) ≈ logrank test statistic ⇒ χ 2 L = χ 2 01 + χ 2 02 + χ 2 12 = ( O 01 − E 01 ) 2 + ( O 02 − E 02 ) 2 + ( O 12 − E 12 ) 2 ∼ χ 2 ( 3 ) V 01 V 02 V 12 with �� L � 2 l = 1 O al − � L l = 1 E al ( O − E ) 2 := . � L V l = 1 V l Harriet Sommer The Cure-Death Model March 15, 2016 (G¨ ottingen) 6
Introduction The Cure-Death Model RL test Application Simulation Summary Current Work Restricted logrank-type test based on Hsieh et al. (1983) Therapy goal: ◮ transition to cure is preferred over a transition to death ◮ if cured, a patient should remain in cure-state as long as possible We want a test that achieves high power if a treatment performes as desired ⇒ restriction to the regression coefficients in the partial likelihood: β 01 = − β 12 = − β 02 ⇒ restricted logrank-type (RL) test with embedded structure: RL = ( O RL − E RL ) 2 χ 2 ∼ χ 2 ( 1 ) V RL with O RL = O 02 − O 01 + O 12 , E RL = E 02 − E 01 + E 12 , and V RL = V 02 + V 01 + V 12 Harriet Sommer The Cure-Death Model March 15, 2016 (G¨ ottingen) 7
Introduction The Cure-Death Model RL test Application Simulation Summary Current Work Application: Clin Infect Dis, 59:51–61 (2014) ◮ HAP (hospital-acquired pneumonia) VAP (ventilator-associated pneumonia) ◮ focus on subgroup of patients with ◮ HAP excluding VAP (N=571) ◮ only VAP (N=210) Harriet Sommer The Cure-Death Model March 15, 2016 (G¨ ottingen) 8
Introduction The Cure-Death Model RL test Application Simulation Summary Current Work Risk differences Harriet Sommer The Cure-Death Model March 15, 2016 (G¨ ottingen) 9
Introduction The Cure-Death Model RL test Application Simulation Summary Current Work Transition probabilities: HAP excluding VAP (N=571) CURE DEATH 0.7 0.7 Ceftobiprole Ceftobiprole Linezolid/Ceftazidime Linezolid/Ceftazidime 0.6 0.6 Probability to be cured and alive 0.5 0.5 Probability to die 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.0 0.0 0 10 20 30 40 0 10 20 30 40 Time from randomisation Time from randomisation Harriet Sommer The Cure-Death Model March 15, 2016 (G¨ ottingen) 10
Introduction The Cure-Death Model RL test Application Simulation Summary Current Work Transition probabilities: Only VAP (N=210) CURE DEATH 0.7 0.7 Ceftobiprole Ceftobiprole Linezolid/Ceftazidime Linezolid/Ceftazidime 0.6 0.6 Probability to be cured and alive 0.5 0.5 Probability to die 0.4 0.4 0.3 0.3 0.2 0.2 0.1 0.1 0.0 0.0 0 10 20 30 40 0 10 20 30 40 Time from randomisation Time from randomisation Harriet Sommer The Cure-Death Model March 15, 2016 (G¨ ottingen) 11
Introduction The Cure-Death Model RL test Application Simulation Summary Current Work Test results Chi 2 test cure Chi 2 test cure+alive Subgroup RL test statistic p-value statistic p-value statistic p-value HAP excluding VAP 0.01 0.92 0.08 0.78 0 . 01 0 . 91 0 . 04 ∗ 0 . 04 ∗ 0 . 01 ∗ only VAP 4.07 4.36 6 . 74 Note: ∗ < 0 . 05 Harriet Sommer The Cure-Death Model March 15, 2016 (G¨ ottingen) 12
Introduction The Cure-Death Model RL test Application Simulation Summary Current Work Simulation 1000 independent data sets with 300 individuals in each treatment group a and b: 1. Treatment a is ✓ superior in the cure rate 2. Treatment a is ✓ superior in the transition from cure to death 3. Treatment a is ✓ superior in the cure rate but ✗ worse in mortality rates Non−inferiority margin ● Scenario 1 ● cure ● ● Scenario 2 ● cure+alive at day 30 ● ● Scenario 3 ● Favours b Favours a −20 0 20 Risk difference (%) Harriet Sommer The Cure-Death Model March 15, 2016 (G¨ ottingen) 13
Introduction The Cure-Death Model RL test Application Simulation Summary Current Work Transition probabilities CURE DEATH 1.0 1.0 Treatment b Treatment b Treatment a, Scenario 1 Treatment a, Scenario 1 Treatment a, Scenario 2 Treatment a, Scenario 2 Treatment a, Scenario 3 Treatment a, Scenario 3 0.8 0.8 Probability to be cured and alive Probability to die 0.6 0.6 0.4 0.4 0.2 0.2 0.0 0.0 0 10 20 30 40 0 10 20 30 40 Time from randomisation Time from randomisation Harriet Sommer The Cure-Death Model March 15, 2016 (G¨ ottingen) 14
Introduction The Cure-Death Model RL test Application Simulation Summary Current Work Test results: Rejection frequencies (%) Chi 2 cure Chi 2 test cure+alive Scenario RL test 1: ✓ cure rate 48 7 66 2: ✓ from cure to death 3 55 86 3: ✓ cure , ✗ mortality rates 34 50 13 Harriet Sommer The Cure-Death Model March 15, 2016 (G¨ ottingen) 15
Introduction The Cure-Death Model RL test Application Simulation Summary Current Work Summary ◮ recommendations given by the existing guidelines (see e.g. FDA or EMA) regarding the selection of primary endpoints in anti-infectives studies are sometimes not consistent nor is their practical application ◮ nonmortal endpoints (here: cure) as well as mortality are important for studies including critically ill patients ◮ cure-death model provides suitable conditions, handles competing risks ◮ includes both endpoints ‘cure’ and ‘death’ into one model ◮ provides insight on how a treatment influences the cure process ◮ estimates the time-dependent probability of being cured and alive ◮ restricted logrank-type test introduced by Hsieh et al. (1983) manages the ordered nature of cure and death and adjusts for a desired prolonged stay in the cure state → straightforward and easily understandable Harriet Sommer The Cure-Death Model March 15, 2016 (G¨ ottingen) 16
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