Model-based Recursive Partitioning for Subgroup Analyses

Transcription

1 EBPI Epidemiology, Biostatistics and Prevention Institute Model-based Recursive Partitioning for Subgroup Analyses Torsten Hothorn; joint work with Heidi Seibold and Achim Zeileis

2 Subgroup analyses Identifying groups of patients for whom the treatment has a different effect than for others. Effect is: Stronger Lower Contrary than the average treatment effect. Suitable models promise better prediction of treatment effect and thus individualised treatments. University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 2

3 Situations of interest for subgroup analyses EMA (2014, 1): 1. The clinical data presented are overall statistically persuasive with therapeutic efficacy demonstrated globally. It is of interest to verify that the conclusions of therapeutic efficacy (and safety) apply consistently across subgroups of the clinical trial population. 2. The clinical data presented are overall statistically persuasive but with therapeutic efficacy or benefit/risk which is borderline or unconvincing and it is of interest to identify post-hoc a subgroup, where efficacy and risk-benefit is convincing. 3. The clinical data presented fail to establish statistically persuasive evidence but there is interest in identifying a subgroup, where a relevant treatment effect and compelling evidence of a favourable risk-benefit profile can be assessed. University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 3

4 Subgroup analyses Goal: Find predictive factors ˆ= covariate treatment interactions treatment predictive factor prognostic factor primary endpoint primary endpoint University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 4

5 Covariate treatment interactions Subgroups known: incorporate subgroup treatment interaction in linear predictor. (Few) Categorical covariates at few levels: incorporate covariate treatment interaction in linear predictor. Many, potentially numeric covariates: Interactions hard to interpred, difficult to derive subgroups. Common approach: Automated interaction detection. BUT: Ordinary trees don t know treatment effect parameters. We need both parametric models AND trees. Solution: Model-based recursive partitioning (MOB). University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 5

6 MOB Basics MOB: Model-based recursive partitioning (Zeileis, Hothorn, Hornik, JCGS, 2008, 3) Start with model M((Y, X), ϑ) with α ϑ = β γ ν which fits data (Y, X) (Y, X ). intercept(s) treatment effect other parameter(s) of interest nuisance parameter(s), University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 6

7 MOB Basics Estimation ˆϑ = arg min ϑ N Ψ((y, x) i, ϑ) i=1 or equivalently solving the score equation N ψ((y, x) i, ϑ) = 0 i=1 Ψ: objective function ψ: score function; gradient of Ψ University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 7

8 MOB Basics Maybe the treatment effect is not the same for all patients, but depends on their characteristics (covariates). Find partitions {B b } (b = 1,..., B) based on patient characteristics Z = (Z 1,..., Z J ) Z Fit separate models M((Y, X), ϑ(b)) in partitions. M((Y, X), ϑ(1)) M((Y, X), ϑ(2)) ϑ(b) = (α(b), β(b), γ, ν) University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 8

9 MOB Basics If partition {B b } is known, the partitioned model parameters ϑ(b) could be estimated by minimizing the segmented objective function: (ˆϑ(b)) b=1,...,b = arg min ϑ(b) N i=1 b=1 B 1 (z i B b ) Ψ((y, x) i, ϑ(b)) Subgroup-specific intercept and treatment parameters can be written as functions of the partitioning variables α(z) = B 1(z B b ) α(b) and β(z) = b=1 B 1(z B b ) β(b). b=1 University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 9

10 Partitioning How to find the partitions? Test: H α,j 0 : ψ α ((Y, X), ˆϑ) Z j H β,j 0 : ψ β ((Y, X), ˆϑ) Z j, j = 1,..., J ψ α, ψ β partial derivatives of Ψ with respect to α/β. Partition if global permutation p-value smaller than significance level Use as split variable the one with the smallest p-value University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 10

11 Example: Linear model Two-arm trial comparing active (A) to control (C): Y X = x N (α + βx A + γx stratum, σ 2 ). Y X = x, Z = z N (α(z) + β(z)x A + γx stratum, σ 2 ), ψ((y, x), ˆϑ) = Ψ((y,x),θ) α θ=ˆϑ Ψ((y,x),θ) β θ=ˆϑ = 1 σ 2 ( y (ˆα + ˆβx A + ˆγx stratum ) (y (ˆα + ˆβx A + ˆγx stratum )) x A ) University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 11

12 Example: Linear model Prognostic α(1) α(2) ; β(1) = β(2) Predictive α(1) = α(2) ; β(1) β(2) Predictive α(1) α(2) ; β(1) β(2) Both α(1) α(2) ; β(1) β(2) Mean endpoint Subgroup 1 Subgroup 2 C A C A C A C A Treatment University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 12

13 Example: Linear model Linear model: Y X = x N (α + βx B, σ 2 ) Data generating process (Loh, He, Man, 2014, 4): Y X = x, Z = z N ( x A (z 1 < 0) (z 1 > 0) x A, 0.7) 2 2 ψ α 0 2 ψ β 0 2 C A z z 1 University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 13

14 Example: Linear model Linear model: Y X = x N (α + βx B, σ 2 ) Data generating process: Y X = x, Z = z N ( x A (z 1 < 0) (z 1 < 0) x A, 0.7) ψ α ψ β C A z z 1 University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 14

15 Example: Linear model Linear model: Y X = x N (α + βx B, σ 2 ) Data generating process: Y X = x, Z = z N (2 x A + 1(z 1 > 0), 0.7) ψ α 2 0 ψ β 2 0 C A z z 1 University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 15

16 Partitioning effects of Riluzole on ALS patients PRO-ACT database (2) Amyotrophic lateral sclerosis (ALS) patients Data of several clinical trials Treatment of interest: Riluzole Primary endpoints of interest: ALS Functional Rating Scale (ALSFRS) ALSFRS items Survival time University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 16

17 Question Riluzole modestly prolongs life expectancy But: Are there any groups of patients for whom it is better or worse? Subgroup analysis University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 17

18 ALSFRS ALS functional rating scale: Measure of functional status of ALS patients Sum-score of ten items (0 < 1 < 2 < 3 < 4): speech salivation swallowing handwriting cutting food and handling utensils, dressing and hygiene turning in bed and adjusting bed clothes walking climbing stairs breathing University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 18

19 ALSFRS ( ) ALSFRS6 E ALSFRS 0 X = x or equivalently = E(ALSFRS 6 X = x) ALSFRS 0 = exp{α + βx R } E(ALSFRS 6 X = x) = exp{α + βx R } exp{log(alsfrs 0 )} GLM with log-link and offset University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 19

20 ALSFRS Results 1 time_onset_treatment p < > FVC p < phosphorus p < > > 1 3 Value (CI) α ( , ) βr ALSFRS ( , ) n = 625 No Yes Riluzole 4 Value (CI) α ( , ) βr ALSFRS ( , ) n = 378 No Yes Riluzole 6 Value (CI) α ( , ) βr ALSFRS ( , ) n = 292 No Yes Riluzole 7 Value (CI) α ( , ) βr ALSFRS ( , ) n = 1239 No Yes Riluzole University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 20

21 ALSFRS items Unidimensionality of ALSFRS is questionable. Two components to consider: 1. Baseline adjustment Adjust for item score at beginning of treatment compute seperate models 2. Multivariate primary endpoint Look at 10 items simultaneously compute 10 item-models in every node Score matrix of dimension n p 10 University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 21

22 ALSFRS items Each item ordinal with values 0,..., 4 proportional odds model adjusted for baseline: P(Y 6 r Y 0 = k, X = x) = exp(α rk β k x R ) 1 + exp(α rk β k x R ) for k = 0,..., 4 Compute stratified permutation tests treating baseline item values as blocks. University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 22

23 ALSFRS items Results 2 FVC p < > time_onset_treatment p < > lymphocytes p = > n = 348 n = 1005 n = 407 n = 774 University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 23

24 ALSFRS items Results Item No. Start Node 3 Node 4 Node 6 Node 7 Speech Salivation Swallowing Handwriting Cutting University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 23.

25 ALSFRS items Results Item No. Start Node 3 Node 4 Node 6 Node 7 Hygiene Bed Walking Stairs Respiratory University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 23

26 Survival time We present two ways of modeling: Weibull model (parametric survival model) Cox model (semiparametric survival model) University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 24

27 Weibull model ( ) log(y) α1 βx P(Y y X = x) = F R α 2 with F cumulative distribution function of Gompertz distribution ( ) α1 intercept and α = α 2 scale parameter α defines the shape of the baseline hazard use as "intercept" University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 25

28 Weibull model Results 1 age p < Riluzole No Riluzole 55.7 > age p < time_onset_treatment p < > > Value (CI) 4 Value (CI) 6 Value (CI) 7 Value (CI) α ( 6.784, 7.251) α ( , ) α ( , ) α ( 6.565, 6.777) βr ( 0.228, 0.166) βr ( , ) βr ( , ) βr ( 0.090, 0.175) log(α2) ( 1.268, 0.794) log(α2) ( , ) log(α2) ( , ) log(α2) ( 0.615, 0.451) 1.00 n = n = n = n = University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 26

29 Cox model λ(y x) = λ 0 (y) exp(βx R ) Objective function: negative partial likelihood (without baseline hazard) No classical score function Use surrogate score function Martingale residuals (as score with respect to the baseline hazard/"intercept") Score residuals (as score with respect to β) University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 27

30 Cox model Results 1 age p < Riluzole No Riluzole 55.7 > age p < time_onset_treatment p < > > Value (CI) Value (CI) Value (CI) Value (CI) βr 0.11 ( 0.46, 0.68) βr ( 0.537, 0.017) βr ( 0.388, 0.061) βr ( 0.382, 0.065) 1.00 n = n = n = n = University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 28

31 Computational details PRO-ACT data are available at The source code for reading and cleaning the database is provided in the TH.data package. All computations were conducted using partykit (version 0.8-2) in the R system for statistical computing (version 3.1.2). University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 29

32 Simple implementation Weibull model library (" partykit ") ## Function to compute Weibull model and return score matrix mywb <- function ( data, weights, parm ) { mod <- survreg ( Surv ( survival. time, cens ) ~ Riluzole, data = data, subset = weights > 0, dist = " weibull ") ef <- as. matrix ( estfun ( mod )[, parm ]) ret <- matrix (0, nrow = nrow ( data ), ncol = ncol ( ef )) ret [ weights > 0,] <- ef ret } ## Compute tree tree <- ctree ( fm, data = data, ytrafo = my. wb, control = ctree_control ( maxdepth = 2, testtype = " Bonferroni ")) University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 30

33 Summary MOB partitions a large class of models suitable for the treatment effect on the primary endpoint of interest. Score functions capture instabilities and thus help to identify predictive and prognostic variables. It is hard to differentiate between predictive and prognostic variables. Permutation tests suitable for all models (distribution free) with good small sample properties and error control. Multiplicity adjustment for subgroup-specific treatment effects unclear. University of Zurich, EBPI Model-based Recursive Partitioning for Subgroup Analyses Page 31

34 Literature European Medicines Agency EMA Guideline on the investigation of subgroups in confirmatory clinical trials (draft) guideline/2014/02/wc pdf, Massachusetts General Hospital, Neurological Clinical Research Institute Pooled Resource Open-Access ALS Clinical Trials Database A. Zeileis, T. Hothorn, K. Hornik Model-Based Recursive Partitioning Journal of Computational and Graphical Statistics, W. Loh, X. He, M. Man A regression tree approach to identifying subgroups with differential treatment effects