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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,

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Introduction The Cure-Death Model RL test Application Simulation Summary Current Work

The Cure-Death Model

An approach to increase efficiency

  • f clinical endpoint evaluations

DAGStat G¨

  • ttingen 2016

Harriet Sommer, Martin Wolkewitz, and Martin Schumacher Institute for Medical Biometry and Statistics, Medical Center – University of Freiburg (Germany)

  • n behalf of the COMBACTE consortium

Harriet Sommer The Cure-Death Model March 15, 2016 (G¨

  • ttingen)

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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¨

  • ttingen)

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Introduction The Cure-Death Model RL test Application Simulation Summary Current Work

The cure-death model

randomisation 30 days

randomisation cure death

λ01(t) λ02(t) λ12(t) ◮ 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¨

  • ttingen)

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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 → Chi2 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¨

  • ttingen)

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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: λj(t | Z) = λ0j(t) exp(βjZ), baseline hazard function λ0j(t), regression coefficient βj, treatment indicator Z. The partial likelihood can be factorised:

L(β) =

K01

  • k=1

exp(β′

01Z(k))

  • r∈R(t01(k)) exp(β′

01Zr) × K02

  • k=1

exp(β′

02Z(k))

  • r∈R(t02(k)) exp(β′

02Zr) × K12

  • k=1

exp(β′

12Z(k))

  • r∈R(t12(k)) exp(β′

12Zr)

⇒ analyse each transition separately by treating the others as censored

Harriet Sommer The Cure-Death Model March 15, 2016 (G¨

  • ttingen)

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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 ⇒ χ2

L = χ2 01 + χ2 02 + χ2 12

= (O01 − E01)2 V01 + (O02 − E02)2 V02 + (O12 − E12)2 V12 ∼ χ2(3) with (O − E)2 V := L

l=1 Oal − L l=1 Eal

2 L

l=1 Vl

.

Harriet Sommer The Cure-Death Model March 15, 2016 (G¨

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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: β01 = −β12 = −β02 ⇒ restricted logrank-type (RL) test with embedded structure: χ2

RL = (ORL − ERL)2

VRL ∼ χ2(1) with ORL = O02 − O01 + O12, ERL = E02 − E01 + E12, and VRL = V02 + V01 + V12

Harriet Sommer The Cure-Death Model March 15, 2016 (G¨

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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¨

  • ttingen)

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Introduction The Cure-Death Model RL test Application Simulation Summary Current Work

Risk differences

Harriet Sommer The Cure-Death Model March 15, 2016 (G¨

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Introduction The Cure-Death Model RL test Application Simulation Summary Current Work

Transition probabilities: HAP excluding VAP (N=571)

10 20 30 40 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

CURE

Time from randomisation Probability to be cured and alive Ceftobiprole Linezolid/Ceftazidime 10 20 30 40 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

DEATH

Time from randomisation Probability to die Ceftobiprole Linezolid/Ceftazidime Harriet Sommer The Cure-Death Model March 15, 2016 (G¨

  • ttingen)

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Introduction The Cure-Death Model RL test Application Simulation Summary Current Work

Transition probabilities: Only VAP (N=210)

10 20 30 40 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

CURE

Time from randomisation Probability to be cured and alive Ceftobiprole Linezolid/Ceftazidime 10 20 30 40 0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7

DEATH

Time from randomisation Probability to die Ceftobiprole Linezolid/Ceftazidime Harriet Sommer The Cure-Death Model March 15, 2016 (G¨

  • ttingen)

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Introduction The Cure-Death Model RL test Application Simulation Summary Current Work

Test results

Subgroup Chi2 test cure Chi2 test cure+alive 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

  • nly VAP

4.07 0.04∗ 4.36 0.04∗ 6.74 0.01∗ Note: ∗ < 0.05

Harriet Sommer The Cure-Death Model March 15, 2016 (G¨

  • ttingen)

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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

Favours b Favours a

Scenario 3 Scenario 2 Scenario 1 −20 20

Risk difference (%)

  • cure

cure+alive at day 30

Harriet Sommer The Cure-Death Model March 15, 2016 (G¨

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Introduction The Cure-Death Model RL test Application Simulation Summary Current Work

Transition probabilities

10 20 30 40 0.0 0.2 0.4 0.6 0.8 1.0

CURE

Time from randomisation Probability to be cured and alive Treatment b Treatment a, Scenario 1 Treatment a, Scenario 2 Treatment a, Scenario 3 10 20 30 40 0.0 0.2 0.4 0.6 0.8 1.0

DEATH

Time from randomisation Probability to die Treatment b Treatment a, Scenario 1 Treatment a, Scenario 2 Treatment a, Scenario 3 Harriet Sommer The Cure-Death Model March 15, 2016 (G¨

  • ttingen)

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Introduction The Cure-Death Model RL test Application Simulation Summary Current Work

Test results: Rejection frequencies (%)

Scenario Chi2 cure Chi2 test cure+alive 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¨

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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

  • rdered 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¨

  • ttingen)

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Introduction The Cure-Death Model RL test Application Simulation Summary Current Work

Current work

◮ advanced test for treatment effect

◮ score test based on pseudo-value regression according to Liu (2008) ◮ test based on confidence bands for the difference of transition

probabilities using wild bootstrap resampling according to Lin (1997)

◮ in many trials, cure is measured only at a single time point per patient, e.g.,

when the clinical study investigator performs the test-of-cure visit ⇒ interval censored ⇒ address this issue using pseudo-value regression

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  • ttingen)

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Introduction The Cure-Death Model RL test Application Simulation Summary Current Work

Some references

PK Andersen and N Keiding. Multi-state models for event history analysis. Statistical Methods in Medical Research, 11:91–115,

  • 2002. doi:10.1191/0962280202SM276ra

SS Awad, AH Rodriguez, YC Chuang, Z Marjanek, AJ Pareigis, G Reis, TWL Scheeren, AS S´ anchez, X Zhou, M Saulay, et al. A phase 3 randomized double-blind comparison of ceftobiprole medocaril versus ceftazidime plus linezolid for the treatment of hospital- acquired pneumonia. Clinical Infectious Diseases, 59:51–61, 2014. doi:10.1093/cid/ ciu219. E Bettiol, WC Rottier, MD del Toro, S Harbarth, MJ Bonten, and J Rodr´ ıguez-Ba˜

  • no. Improved treatment of multidrug-resistant

bacterial infections: utility of clinical studies. Future Microbiology, 9:757–771, 2014. doi:10.2217/FMB.14.35 European Medicines Agency. Addendum to the guideline on the evaluation of medicinal products indicated for treatment of bacterial infections (EMA/CHMP/351889/2013). 2013 Food and Drug Administration. Guidance for industry: Hospital-acquired bacterial pneumonia and ventilator-associated bacterial pneumonia: Developing drugs for treatment. 2014 FY Hsieh, John Crowley, and Douglass C Tormey. Some test statistics for use in multistate survival analysis. Biometrika, 70:111–119, 1983 DY Lin. Non-parametric inference for cumulative incidence functions in competing risks studies. Statistics in Medicine, 16:901–910, 1997. doi:10.1002/(SICI) 1097-0258(19970430)16:8¡901::AID-SIM543¿3.0.CO;2-M L Liu, B Logan, and JP Klein. Inference for current leukemia free survival. Lifetime data analysis, 14:432–446, 2008. doi:10.1007/s10985-008-9093-1 Harriet Sommer The Cure-Death Model March 15, 2016 (G¨

  • ttingen)

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Introduction The Cure-Death Model RL test Application Simulation Summary Current Work

Backup: Logrank-type test based on Hsieh et al. (1983)

χ2

L = χ2 01 + χ2 02 + χ2 12

= (O01 − E01)2 V01 + (O02 − E02)2 V02 + (O12 − E12)2 V12 ∼ χ2(3) ⇒ sensitive against deviations from the null hypothesis of any type Restriction to the regression coefficients in the partial likelihood: β01 = −β02 χ2

RL1 = χ2 RL1 + χ2 12

= (ORL − ERL1)2 VRL1 + (O12 − E12)2 V12 ∼ χ2(2) with ORL1 = O02 − O01, ERL1 = E02 − E01 and VRL1 = V02 + V01 ⇒ sensitive against deviations from the null hypothesis: transition to cure is preferred over transition to death Restriction to the regression coefficients in the partial likelihood: β01 = −β12 = −β02 χ2

RL2 = (ORL − ERL )2

VRL ∼ χ2(1) with ORL2 = O02 − O01 + O12, ERL2 = E02 − E01 + E12, and VRL2 = V02 + V01 + V12 ⇒ sensitive against deviations from the null hypothesis: transition to cure is preferred over transition to death; if cured, a prolonged stay in the cure state Harriet Sommer The Cure-Death Model March 15, 2016 (G¨

  • ttingen)

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Introduction The Cure-Death Model RL test Application Simulation Summary Current Work

Backup: Ceftobiprole (ITT)

10 20 30 40 50 100 200 300 400

Ceftobiprole

time from randomization individuals

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  • during treatment
  • cure

death death after cure censored

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Introduction The Cure-Death Model RL test Application Simulation Summary Current Work

Backup: Mathematical background

1 2 λ01(t) λ02(t) 1 2 λ01(t) λ02(t) λ12(t)

competing risks model (multiple absorbing endpoints) ◮ event-specific hazard rate λ0i(t) = limh→0

P(t<T≤t+h; cause i|T>t) h

◮ P00(0, t) = S(t) = exp

  • −

t 2

i=1 λ0i(u)du

  • ◮ cumulative incidence function

P0i(0, t) = t

0 P00(0, u)λ0i(u)du

depends on all event specific hazards! illness-death-model without recovery illness-death-model without recovery ◮ transition probability P01(0, t) = t

0 P00(0, u)λ01(u)

P11(u, t)

  • =exp
  • −

t

u λ12(v)dv

  • du

P02(0, t) = 1 − (P00(0, t) + P01(0, t))

Harriet Sommer The Cure-Death Model March 15, 2016 (G¨

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Introduction The Cure-Death Model RL test Application Simulation Summary Current Work

Backup: Score Test based on pseudo values

Basic idea: Compare two probabilities to be cured and alive based on a score test constructed using pseudo values

  • 1. calculate pseudo values f¨

ur individual i and day tj (when the estimated transition probabilities change values): ˆ θij = ˆ θi(tj) = nˆ P(tj) − (n − 1)ˆ P(−i)(tj), ˆ P(−i) based on N − 1 by deleting the ith observation → extend R-package prodlim to estimate pseudo values

  • 2. define treatment covariate Zi ∈ {0, 1} if patient was in group 1, 0 otherwise
  • 3. transition probability for patient i at time tj is modelled via g(ˆ

θij) = αj + γZi, g as link function, estimate regression coefficents with a GEE

  • 4. set up generalized score statistic for testing the null hypothesis of no treatment

effect, γ = 0

Harriet Sommer The Cure-Death Model March 15, 2016 (G¨

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Introduction The Cure-Death Model RL test Application Simulation Summary Current Work

Backup: Wild bootstrap

Test the equality of two transition probabilities, the probability to be cured and to stay cured

  • r the probability to die, for treament A (active) and B (control):

H0 : PA

0k(0, t) = PB 0k(0, t)

for all t ≤ τ vs. H1 : PA

0k(0, t) PB 0k(0, t)

for some t ≤ τ, where k ∈ {1 : cure, 0 : death} and τ is the largest time on study. For k = 1, under H0, D(t) :=K(t)[ˆ P

A 01(0, t) − ˆ

P

B 01(0, t)]

=K(t)[ (ˆ P

A 01(0, t) − PA 01(0, t))

  • d

−→ t

0 PA 01(0,u)dU(u)PA 01(u,t)/

√ nA

− (ˆ P

B 01(0, t) − PB 01(0, t))

  • d

−→ t

0 PB 01(0,u)dU(u)PB 01(u,t)/

√ nB

] U is the limit process of the multivariate Nelson-Aalen estimator and K a weight function ⇒ Wild Bootstrap resampling: approximate U, replace P by ˆ P ⇒ approximate boundary value q0.05 for confidence band (ˆ P

A 01(0, t) − ˆ

P

B 01(0, t)) ± q0.05 Harriet Sommer The Cure-Death Model March 15, 2016 (G¨

  • ttingen)

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