A Bayesian hierarchical surrogate outcome model for multiple sclerosis

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1 A Bayesian hierarchical surrogate outcome model for multiple sclerosis 3 rd Annual ASA New Jersey Chapter / Bayer Statistics Workshop David Ohlssen (Novartis), Luca Pozzi and Heinz Schmidli (Novartis) November 13, 2015

2 Overview of presentation Brief review of meta-analysis methods in clinical trials What methods are attempting to do Quick overview of Bayesian methods Extension to Bayesian hierarchical surrogate outcome model Application related to drug development in MS Clinical background Previous research by Sormani and colleagues Applying the Bayesian surrogate outcome model Discussion and conclusions 2

3 3 Overview of meta-analysis models and methods

4 Introduction to meta-analysis methods and models A technique to adjust for imbalances When combining data from several clinical trials, meta-analysis is a technique that aims at adjusting for differences in background risk of patients in the analysis of the effects of treatments Within a single clinical trial this is accomplished by randomization Therefore, if all trials have the same randomization (e.g. 1:1), estimating the treatment effect from a pool of all observations will not introduce bias (although it might not be the most efficient approach) If trials have different unbalanced randomization, treatments recovered from the pool of all patients from all trials will no longer correspond to a common randomized assignment. This potentially leads to biased treatment effects if it is not accounted for in the analysis 4

5 Simpsons paradox A simple illustration of the need for meta-analysis Trial1 2:1 randomization Trial 2: 1:2 randomization In both cases, a consistent moderate improvement in the treatment effect is seen (Odds ratio of cured 1.39 and 1.40) Analysis of a pooled data results in a treatment effect that appears detrimental (Odds ratio of cured is 0.96) Trial 1 (very sick patients) Cured Died Total New Control Trial 2 (less sick patients) Cured Died Total New Control Pooled Cured Died Total New Control

6 Meta-analysis adjustment Adjustment that can be accomplished with summary data Adjustment could be based on building a model to adjust for difference in potential confounding factors based on individual patient baseline characteristics However, in the randomized control trial setting, most meta-analysis methods only apply some kind of adjustment or stratification for study Because each individual study is randomized, this alone can take care of the main potential source of bias In this sense "study" is a surrogate for all differences between the populations tested in different trials 6

7 Some notation for meta-analysis Generic notation for summary or IP data Let T represent the experimental treatment group and C the control group Suppose y i is the data from N studies (i=1,...,n) y i could be: individual patient outcome data associated with the study Could represent summary data (sufficient statistics) from each treatment group (e.g binary outcome data (y it, n it ) (y ic, n ic ) ) Could represent a treatment effect estimate and corresponding standard error (q i s i ) 7

8 Model parameters for an individual study Let C be the control group parameter (e.g background rate or population mean) and T be the corresponding treatment group parameter T Let q Parameter of interest compares T with C absolute metric: e.g. mean difference T C relative metric: Ratio: T / C (risk ratio, odds-ratio, hazard ratio) log-ratio: log( T / C ) 8

9 Multiple studies Parameters and assumptions Modeling structures for the two treatment case Parameters 1,..., N response rate for C (control parameters) in studies 1,...,N q 1,..., q N (e.g log-odds-ratios (T vs. C, effects) for studies 1,...,N Assumptions for for control parameters ( 1,..., N ) - a common (across studies) parameter c - N unrelated (stratification) parameters u - logit( i ) N(, 2 ): N parameters are similar/related r for effect parameters (q 1,..., q N ) (T vs. C) - a common effect parameter C - N unrelated effect parameters (no synthesis!) U - q i N(, 2 ): N effect parameters are similar/related R 9

10 Models assumptions A picture summarizing possible assumptions 10

11 Summary data meta-analysis Normal likelihood Fixed effects metaanalysis model (u-c) q i q i ~ N(q i, s i2 ) q 1 =...=q N = q (u-c) Random effects metaanalysis model (u-r) q i q i ~ N(q i, s i2 ) q i ~ N(q, 2 ) (u-r) 11 q i is the estimated treatment effect associated with study i s i is the corresponding standard error q is the overall treatment effect parameter is the standard deviation among the treatment effect parameters in the random effects model

12 12 Bayesian methods quick review

13 Introduction Bayesian methods Summary Historical Data Publications Expert Knowledge Bayesian Statistics All uncertainty is expressed probabilistically Critical input: Likelihood (Statistical Model) and Prior Bayes Theorem: Posterior Likelihood Prior Bayes (probability calculus) + Contextual Evidence Observed Data Updated Evidence + = Predictions Decisions + = 13

14 Some comments on Bayesian methods A personal perspective For a given problem, Bayesian statistics provides: A framework to combine relevant sources of information, using a realistically complex probability model In addition, if this model is useful: it should be reasonably well calibrated and lead to predictions that can form the basis for rational decision making However, the big challenge for a Bayesian, is convincing others that their model(s) are useful In other words, the posterior distributions and predictive distributions are approximately correct 14

15 The Bayesian modeling strategy used here Priors are carefully selected that we hope are dominated by the data Models fitted using Markov chain Monte Carlo (MCMC) estimation A variety of modeling structures examined Model support measured using the deviance information criteria (DIC) Model diagnostics with frequentist properties used to help show whether a model has good calibration Examine if similar conclusions are reached from well supported models to check inference robustness This work follows the ideas of Box (1980), who advocated the use of an iterative cycle of model criticism and estimation 15

16 16 Case study background

17 Multiple Sclerosis (MS) is a debilitating progressive disease Chronic, inflammatory, degenerative neurological autoimmune disease No known cure; affecting people in the prime of their lives (diagnosis years) Over 1 million sufferers worldwide Disproportionately affects women (70%), Caucasians and those in temperate climates 80% eventually need cane or wheelchair 17

18 MS disease background Multiple sclerosis (MS) is one area where a range of surrogate outcomes are used in various stages of clinical research The aim of treatments in MS is to prevent long term irreversible disability A clinical trial for evaluating a disability progression would require a large sample of patients with many years of follow-up In early stages, the vast majority of MS cases typically present as relapsing disease (Relapsing Remitting MS, RRMS) 18

19 Phase II and Phase III outcomes Relapsing Remitting MS Multiple Sclerosis In phase II MS trials, primary outcomes are typically based on magnetic resonance imaging (MRI) in the form of lesion counts Accepted surrogate marker for learn phase trials in RRMS: objective measure of the inflammatory activity of MS (detects 5--10x more events compared with clinical relapses) effect seen within weeks or a few months In Phase III primary outcomes are typically clinical relapses (count or time to event) and/or time to confirmed disability progression 19

20 log(rrrel) Clinical translation of MRI response R 2 = log(rr MRI ) Adapted from Sormani et al, Ann Neurol

21 Extract from the Motivating data Based on a systematic review by Sormani and colleagues (2010) 21

22 Clinical translation response (y) MRI-> disability Relapse to disability; MRI to disability Adapted from Sormani et al, Neurolgy

23 Previous work by Sormani and colleagues a systematic review and meta-analysis Two separate weighted least-squares regressions were fitted. The first relating the observed treatment effect on MRI lesions to the observed treatment effect on disability (adjusted R 2 =0:57), The second relating the observed treatment effect on relapses to the observed treatment effect on disability (adjusted R 2 = 0:71). The analyses ignored the measurement error with the relapse and MRI treatment effects Combining information from both MRI lesions and relapses may be a better surrogate for disability than each of them alone 23

24 24 Meta-analysis using a Bayesian hierarchical surrogate outcome model

25 Introduction to statistical techniques for surrogate outcomes The seminal work by Prentice subsequently lead to large literature examining statistical evaluation of surrogate outcomes Within this area statistical methods have been divided into three groups: - Those focusing on data from a single trial such as the Prentice criteria; - Those which use meta-analysis to combine summary information from multiple trials - Techniques that use a combination of trial-level and individual-level data In this talk we will provide an analysis of summary data from multiple clinical trials, with the aim of understanding uncertainty associated with drug development decisions rather than formal validation, we shall solely focus on the Bayesian meta-analytic approach that was developed by Daniels and Hughes (1997) 25

26 Two level Surrogate outcome Bayesian meta-analysis Where q i represents the estimated treatment effect on the main outcome of interest, g i denotes the estimated treatment effect associated with the surrogate outcome, σ 2 θi and σ 2 γi are the variances that reflect the sampling uncertainty and ρi is the correlation between the estimated treatment differences conditional on the true differences θi and γi. The simplest trial-level model to describe the relationship between θi and γi is linear: 26

27 Studies with more than two treatment groups If studies have more than two arms the analysis must account for the correlation between the multiple treatment effects within a study. This can be accomplished by adjusting the sampling model, e.g. to a 4-dimensional multivariate normal model in the case of three treatment arms to: 27

28 Some notes on priors When fitting the model in the Bayesian framework prior distributions must be assumed. In the multiple sclerosis case-study, N(0, 10 6 ) priors will be assumed for all fixed effects (α 1, β 1 and γ i i = 1,, n). In terms of the variance component τ 2 ϵ, it is well known that results can be sensitive to the choice of prior distribution Following the suggestions of Spiegelhalter et al (2003) a standard half normal prior, denoted by Half-Normal(1) will be adopted. The parameters of the covariance matrix σ θi, ρi and σ γi are typically assumed to be known in the meta-analytic literature. 28

29 Extension to a three level model In cases such as the study of relapsing remitting MS we can borrow information from two surrogates simultaneously When formulating the trial level model, a natural extension of the Daniels and Hughes model would involve two levels of linear models: 29

30 Alternative three-level models Various alternative structures could be applied when formulating the three level model. Here, we shall consider two alternative models: Firstly, both surrogates contribute directly to the main outcome of interest Secondly, instead of formulating a conditional structure, the second level of the model could be formulated as a multivariate normal random effects meta-analysis model 30

31 Practical issues when extracting trial level data Estimated variance covariance matrix Sampling variances can usually be drawn directly from summary data included in a publication Direct estimation of the correlations among the treatment effect estimates either requires the specification of a joint model or application of a non-parametric bootstrapping technique If only summary data is available the challenge of dealing with missing sampling covariances or correlations must be addressed: Using a range of plausible values; Sensitivity analysis over the entire correlation range Use of an alternative model 31

32 MS case-study: summary data extraction Variance associated with each treatment effect and covariance MRI Treatment effects can be taken directly from the summary data provided (lesion counts log(risk ratio) and exposure) Variance based on NB sampling model with fixed over-dispersion Relapse Treatment effects taken directly from summary data log(risk ratio) Variance provided by a binomial sampling model Disability Treatment effects taken directly from summary data log(risk ratio) Variance provided by a binomial sampling model Cov Based on limited priority data (bootstrap approach) Sensitivity analysis over a range of plausible correlations 32

33 Comparison of our weighting to Sormani et al (2010) Regressor Weights Intercept (s.e.) Slope (s.e.) adjusted R2 g i Sormani 0.1 (0.055) 0.63 (0.087) 0.7 g i Inverse Variance (0.05) (0.083) y i Sormani 0.2 (0.097) 0.36 (0.085) 0.56 y i Inverse Variance (0.091) (0.085)

34 34 Results

35 Comparisons of 2-level and 3-level models Posterior and predictive distributions 35

36 Sensitivity analysis Assessing the impact of different values of ρ 36 Model Param eter ρ = 0.05 [95%CI] 2 level α [-0.096, 0.244] β [0.476, 1.022] τ ϵ [0.027, 0.29] 3 level α [-0.068, 0.256] β [0.466, 0.95] τ ϵ [0.007, 0.24] α [-0.372, 0.124] β [0.28, 0.784] τ δ [0.057, 0.419] ρ = 0.1 [95%CI] 0.08 [-0.095, 0.241] [0.476, 1.01] [0.024, 0.286] [-0.068, 0.256] [0.466, 0.95] [0.007, 0.24] [-0.369, 0.117] [0.284, 0.781] [0.063, 0.415] ρ = 0 [95%CI] 0.08 [-0.097, 0.246] [0.482, 1.033] [0.026, 0.297] [-0.068, 0.256] [0.466, 0.95] 0.1 [0.007, 0.24] [-0.376, 0.131] [0.27, 0.788] 0.21 [0.054, 0.416]

37 Forest plot of relapse treatment effects Estimated Risk ratios associated with relapse (mean and 95% interval) 3-level model (gray circle); extended 3-level model (gray square) multivariate metaanalysis(gray triangle). For comparison purposes, the estimates and 95% are displayed from fixed effects model (black circle). 37

38 3-level models comparison 38

39 Discussion and further work One key advantage of the Bayesian model is the ability to provide predictions for the results of future clinical trials that fully account for parameter uncertainty. Highly valuable in a phase II setting. 39 A completed phase II study, which would typically have a reasonable amount of MRI data but limited relapse data and no information about disability progression Increased access to individual patient data clinical trial information (EMA data transparency) Therefore, combining individual patient data with summary data in a surrogate outcome setting would provide an important area for future research.

40 References Pozzi L, Schmidli H, Ohlssen D, A Bayesian hierarchical surrogate outcome model for multiple sclerosis (Submitted to Pharmaceutical statistics) Sormani M, Bonzano L, Roccatagliata L, Mancardi G, Uccelli A, Bruzzi P. Surrogate endpoints for EDSS worsening in multiple sclerosis. Neurology 2010; 75(4): Prentice R. Surrogate endpoints in clinical trials: definition and operational criteria. Statistics in Medicine 1989; 8(4): Daniels M, Hughes M. Meta-analysis for the evaluation of potential surrogate markers. Statistics in Medicine 1997; 16(17): Gelman A. Prior distributions for variance parameters in hierarchical models. Bayesian Analysis 2006; 1(33): Spiegelhalter D, Abrams K, Myles J. Bayesian Approaches to Clinical Trials and Health- Care Evaluation. Wiley: Chichester, Riley R. Multivariate meta-analysis: the effect of ignoring within study correlation. Journal of the Friede T, Schmidli H. Blinded sample size reestimation with count data: methods and applications in multiple sclerosis. Statistics in Medicine 2010;Royal Statistical Society, 40 Series A 2009; 172(4):

41 41 Back-up

42 Forest plot of EDSS treatment effects 42

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