Group sequential designs with negative binomial data

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1 Group sequential designs with negative binomial data Ekkehard Glimm 1 Tobias Mütze 2,3 1 Statistical Methodology, Novartis, Basel, Switzerland 2 Department of Medical Statistics, University Medical Center Göttingen 3 DZHK (German Centre for Cardiovascular Research), partner site Göttingen, Göttingen, Germany Thanks to Tim Friede and Heinz Schmidli PSI/BBS One-day meeting Basel September 13, 2016

2 1 Motivating examples 2 Fixed design 3 Group sequential designs 4 Assessing operating characteristics 5 Discussion and outlook 2 / 28

3 Motivating examples Example 1: Clinical trials in heart failure Heart failure (HF) with preserved ejection fraction (HFpEF) Primary endpoint: Number of heart failure hospitalizations (HFH) HFH can be modeled with negative binomial distribution (Rogers et al., 2014) Example: the CHARM-Preserved trial (Yusuf et al., 2003) Table: Heart failure hospitalizations in CHARM-preserved Placebo Candesartan Number of patients Total follow-up years Patients with 1 admission Total admissions Rate ratio for recurrent heart failure hospitalizations according to negative binomial model θ = / 28

4 Motivating examples Example 2: Clinical trials in relapsing-remitting multiple sclerosis Primary endpoint: number of combined unique active lesions (CULAs) CULAs are modeled using the negative binomial distribution Example: Phase II study of Siponimod (Selmaj et al., 2013) Placebo and five doses of Siponimod Equal follow-up times (in general either 3 or 6 months) Table: Monthly number of lesions (at 3 months) Placebo Siponimod 0.25 mg Siponimod 0.5 mg Number of patients Monthly CULAS / 28

5 Statistical model Fixed design Number of counts for patient i = 1,..., n j receiving treatment j = 1, 2 Follow-up per patient: t ij Gamma-mixture for the rates Y ij λ ij Pois(t ij λ ij ) λ ij Γ Marginal distribution of counts Expected value and variance ( ) 1 φ, 1 φµ j Y ij NB (t ij µ j, φ) E [Y ij ] = t ij µ j Var [Y ij ] = t ij µ j (1 + φt ij µ j ) 5 / 28

6 Hypothesis testing I Fixed design Statistical hypothesis H 0 : µ 1 µ 2 1 vs. H 1 : µ 1 µ 2 < 1. Hypothesis is tested using a Wald-type test of the maximum-likelihood estimators ˆβ j of the log-rates β j = log(µ j ) Wald-type test statistic T = ˆβ 1 ˆβ 2 1 Îβ1 + 1 Î β2 H 0 N (0, 1) asymp. 6 / 28

7 Hypothesis testing II Fixed design Fisher information of log-rates β j I βj = n j i=1 t ij exp(β j ) 1 + φt ij exp(β j ) = n j i=1 t ij µ j 1 + φt ij µ j. (Reminder: I 1 β j is the asymptotic variance of the MLE ˆβ j.) Information level I fix describes knowledge about unknown treatment effect 1 I fix = 1 I β1 + 1 = I I β1 β2 I I β1 + I β2 β2 Sample size planning by solving equation I fix! = (q 1 β q 1 α ) 2 (β 1 β 2 ) 2 7 / 28

8 Group sequential designs Group sequential designs: Overview Test the hypothesis H 0 at several interim analyses and stop the trial if H 0 can be rejected (stop for efficacy) The interim analyses are performed with the Wald-type test using all data available up to that point in time Counts of patient i in treatment j at analysis k: Y ijk NB(t ijk µ j, φ) t ijk is the follow-up time until analysis k The final analysis is performed when a prespecified information level I max is attained (maximum information trial) 8 / 28

9 Group sequential designs Type I error Type I error Critical values of the individual tests c k must be chosen such that global type I error α, i.e. α P H0 (T k < c k for at least one k = 1,..., K). Allocate global type I error α = K k=1 π k Type I error rate π k for analysis k P H0 (T 1 c 1,..., T k 1 c k 1, T k < c k ) = π k Choose π k through error spending function f : [0, ) [0, α] with f (0) = 0 and f (t) = α, t 1: π 1 = f (I 1 /I max ), π k = f (I k /I max ) f (I k 1 /I max ) k = 2, 3, / 28

10 Critical values Group sequential designs Type I error First critical value is the normal quantile c 1 = q π1 Joint distribution (T 1,..., T k ) required to calculate critical value c k Asymptotic normality of joint distribution has canonical form [Scharfstein et al., 1997] (T 1,..., T k ) N (0, Σ k ) with (Σ k ) (k1,k 2 ) = (Σ k) (k2,k 1 ) = I k1 I k2, 1 k 1 k 2 k. 10 / 28

11 Group sequential designs Practical considerations Type I error Information level depends on rates µ j, shape parameter φ, follow-up times t ijk, and sample size n j At analysis k, I k not known and is estimated by plugging in the rate and shape maximum-likelihood estimators Critical value c k is not determined prior to the trial but at the time of analysis k Î k is the estimated information level of stage k obtained with the data available at interim k 11 / 28

12 Group sequential designs Type I error Practical considerations continued In practice the following estimators are considered ) ˆπ 1 = f (Î1 /I max ) ) ˆπ k = f (Îk /I max f (Îk 1 /I max (ˆΣk ) (k 1,k 2 ) = Îk1 Î k2 k = 2, 3,... Estimated information might decrease if sample sizes or time between analyses is small, i.e. Î k < Î k 1 then analysis is skipped critical value c k = Locally allocated type I error preserves the global type I error K ˆπ k = α i=1 12 / 28

13 Group sequential designs Planning of group sequential trials Planning of group sequential trials Power for given set of critical values c 1,..., c K Power = 1 P H1 (T 1 c 1,..., T K c K ) For rate ratio θ in alternative, joint distribution (T 1,..., T K ) approximately normal with mean vector log(θ )( I 1,..., I K ) For planning purposes, we write I k = w k I max, k = 1,..., K, w k (0, 1] Calculate maximum information I max required to obtain power of 1 β by solving 1 P θ (T 1 c 1,..., T K c K ) = β Sample size, study duration, etc must be selected such that the maximum information is obtained 13 / 28

14 Assessing operating characteristics Simulation study - preface In the simulation, interim analysis time points are determined by theoretical information levels I k. The actual estimated information levels Î k differ from this. Use of spending functions which imitate critical values of Pocock s test and O Brien & Fleming s test Recruitment times uniform in fixed accrual period 14 / 28

15 Assessing operating characteristics Simulation scenarios - type I error Re-hospitalizations in heart failure Simulation scenarios motivated by the number of hospitalizations from Example 1 Parameter Values Type I error rate α Annual rates µ 1 = µ , 0.1, 0.12 Shape parameter φ 2, 3, 4, 5 Group sample size n 1 = n 2 600, 1000, 1400 Stages K 2, 5 Study duration 3.5 (years) Recruitment period 1.25 (years) Monte Carlo replications per scenario 15 / 28

16 Assessing operating characteristics Re-hospitalizations in heart failure Results - type I error Fix O'Brien & Fleming Pocock Maximal group specific sample size Type I error rate 16 / 28

17 Assessing operating characteristics Simulation scenarios - power Re-hospitalizations in heart failure Parameters for the Monte Carlo simulation study of the power Parameter Values Type I error rate α Annual rate µ Annual rate µ Rate ratio µ 1 /µ Group sample size n 1 = n 2 600, 650,..., 1500 Shape parameter φ 5 Stages K 2, 5 Study duration 3.5 (years) Recruitment period 1.25 (years) Monte Carlo replications per scenario 17 / 28

18 Assessing operating characteristics Re-hospitalizations in heart failure Results - power Maximum number of stages: 2 Maximum number of stages: Maximum group specific sample size n 1 Power Design Fix O'Brien & Fleming Pocock 18 / 28

19 Assessing operating characteristics Results - stopping times Re-hospitalizations in heart failure Rejections by stage (O Brien-Fleming, total power of 80%) Maximum number of stages: 2 Maximum number of stages: Rejection rate Analysis k at which the null hypothesis is rejected 19 / 28

20 Assessing operating characteristics Re-hospitalizations in heart failure Results - gains from stopping early Study times at which in theory 25%, 50%, 75%, and 100% of the maximum information level I max is attained Study duration τ Patient 20 / 28

21 Assessing operating characteristics Simulation scenarios - type I error Lesion counts in multiple sclerosis Simulation scenarios motivated by the CULAs from Example 2 Parameter Values Type I error rate α month rates µ 1 = µ 2 6, 8, 10 Shape parameter φ 2, 3, 4 Group sample size n 1 = n 2 50, 70,..., 150 Stages K 2, 3 Individual follow-up 0.5 (years) Recruitment period 1.5 (years) 18 scenarios per group sample size for group sequential designs and 9 scenarios for the fixed design Monte Carlo replications per scenario 21 / 28

22 Assessing operating characteristics Lesion counts in multiple sclerosis Results - type I error Fix O'Brien & Fleming Pocock Maximum group specific sample size Type I error rate 22 / 28

23 Assessing operating characteristics Simulation scenarios - power Lesion counts in multiple sclerosis Parameters for the Monte Carlo simulation study of the power Parameter Values Type I error rate α month rate µ month rate µ Group sample size n 1 = n 2 70, 75,..., 140 Shape parameter φ 3 Stages K 2 Individual follow-up 0.5 (years) Recruitment period 1.5 (years) Monte Carlo replications per scenario 23 / 28

24 Assessing operating characteristics Lesion counts in multiple sclerosis Results - power Maximum number of stages: 2 Maximum number of stages: Maximum group specific sample size n 1 Power Design Fix O'Brien & Fleming Pocock 24 / 28

25 Assessing operating characteristics Lesion counts in multiple sclerosis Results - analysis specific rejection rate Rate of stopping at a specific analysis at a power of 80% Maximum number of stages: 2 Maximum number of stages: Rejection rate Analysis k at which the null hypothesis is rejected 25 / 28

26 Assessing operating characteristics Results - gains from stopping early Lesion counts in multiple sclerosis Study times at which in theory 25%, 50%, 75%, and 100% of the maximum information level I max is attained 90 Patient Study duration τ 26 / 28

27 Discussion and outlook Discussion and outlook Maximum-likelihood theory for negative binomial data results asymptotically in canonical form of joint distribution of test statistic Information level depends on rates, shape parameter, follow-up times, and sample size Future research on group sequential with negative binomial endpoints Blinded information monitoring Adaptive group sequential designs Optimal designs Extend approach to quasi-poisson models in the future 27 / 28

28 Bibliography Discussion and outlook DO. Scharfstein, AA. Tsiatis, JM. Robins (1997). Semiparametric efficiency and its implication on the design and analysis of group-sequential studies. Journal of the American Statistical Association, 92: S. Yusuf et al. (2003) Effects of candesartan in patients with chronic heart failure and preserved left-ventricular ejection fraction: the CHARM-Preserved Trial. The Lancet, 362: JK. Rogers, SJ. Pocock, JJV. McMurray, CB. Granger, EL. Michelson, J. Östergren, MA. Pfeffer, SD. Solomon, K. Swedberg, S. Yusuf (2014) Analysing recurrent hospitalizations in heart failure: a review of statistical methodology, with application to CHARM-Preserved. European Journal of Heart Failure, 16: K. Selmaj, SKB. Li, HP. Hartung, B. Hemmer, L. Kappos, MS. Freedman, O. Stüve, P. Rieckmann, X. Montalban, T. Ziemssen (2013) Siponimod for patients with relapsing-remitting multiple sclerosis (BOLD): an adaptive, dose-ranging, randomised, phase 2 study. The Lancet Neurology, 12: / 28

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