Bayesian interval dose-finding designs: Methodology and Application

Transcription

1 Bayesian interval dose-finding designs: Methodology and Application May 29, 2018

2 Conflict of Interests The U-Design webtool is developed and hosted by Laiya Consulting, Inc., a statistical consulting firm co-founded by Yuan Ji.

3 Contents 1. Standard and new dose-finding designs 2. Interval-based designs for dose finding 3. Application via the U-Design Webtool

4 Standard and new dose-finding designs

5 Consider trials with fixed doses. Phase I dose-finding (Oncology) Setup Climb up and down a sequence of D ordered doses of a new drug to determine the maximum tolerated dose (MTD). Data At each dose i, n i patients are tested, y i patients experienced toxicity outcome (DLT). Parameters Dose i has a toxicity probability of p i (unknown). Sampling Model Binomial y i p i Bin(n i, p i ) Assumption Toxicity Monotonicity : p i p i+1. Hidden assumption Efficacy Monotonicity : q i q i+1 if not, why escalate when the dose is safe? Goal to find the MTD, defined as the highest dose with toxicity rate lower (or close to) a target rate, p T, e.g., p T = 0.30.

6 A hypothetical dose-finding trial

7 A hypothetical dose-finding trial

8 A hypothetical dose-finding trial

9 A hypothetical dose-finding trial

10 A hypothetical dose-finding trial

11 A hypothetical dose-finding trial

12 A hypothetical dose-finding trial

13 A hypothetical dose-finding trial

14 A hypothetical dose-finding trial

15 A hypothetical dose-finding trial

16 A hypothetical dose-finding trial

17 A hypothetical dose-finding trial

18 A hypothetical dose-finding trial

19 Existing Designs 3+3 Storer (1989). Algorithmic design; simple and transparent Insufficient power to find the MTD even with sufficient resources Contradicting rules : Stay for 1/3; Escalate for 1/6 (MTD not exceeded)

20 Existing Designs 3+3 Storer (1989). Algorithmic design; simple and transparent Insufficient power to find the MTD even with sufficient resources Contradicting rules : Stay for 1/3; Escalate for 1/6 (MTD not exceeded) CRM The first model-based design. First publication in Performs well but could be sensitive to prior models; black-box to physicians mtpi and mtpi-2 Model-based interval designs in an algorithmic presentation (Ji et al., 2010; Guo et al., 2017). Optimal under a formal Bayesian decision framework CCD and BOIN Cumulative cohort design (CCD, Ivanova et al. 2007) a Markov process and a simple rule BOIN (Liu and Ying) is an extension of CCD Their inference is based on point estimates (not accounting for variabilities) of toxicity probabilities; BOIN s asymptotic behavior is strange

21 The Industry-Standard 3+3 Design Yang, Wang, and Ji (2015) An integrated dose-finding tool for phase I trials in oncology.

22 The 3+3 Design: should be a past history? Features Simple; Transparent; FDA review reward for being inferior and naive Problems Target highest dose with no more than 1/6 DLT rate; but STAY when 1 out 3 patients has DLT Cannot target p T values different from 1/6 Cannot have large power (3+3 treats no more than 6 patients per dose) what if one has more resources to spend?

23 The 3+3 Design: it should be a past history Ji and Wang (2013, JCO) showed that with matched sample size, 3+3 is less safe and reliable when compared to the mtpi design, a model-based design. The 2015 FDA/AACR Dose-Finding Symposium concluded that (Nie et al., 2016, Clinical Cancer Research) The MTD/3+3 approach is not optimal and may result in recommended doses that are unacceptably toxic for many patients and in dose reduction/interruptions that might have an impact on effectiveness.

24 The CRM Design a specific model Perhaps the most popular version of the CRM is the power model: The dose-response curve : p i = p exp(α) i0, where p i0 are fixed and prespecified constants, and α is a parameter that describes the dose response curve. The prior for α is N(0, 2). The p i0 s are decided by solving E[p exp(α) i ] = s i, where s i s are a set of prior probabilities that one must determine (called skeletons ). A binomial likelihood: d i=1 pyi i (1 p i) ni yi. Posterior of α is obtained by numerical integration. The next dose is arg min i ˆp i p T, where ˆp i is the posterior mean.

25 The CRM Design trial conduct and decision tables Challenging to implement in practice (logistic and effort support) Coherence (some version is NOT coherence ) and Over-dose control (e.g., no skipping dose when escalation) Team meetings are needed for every patient allocation CRM decisions may be overruled Output CRM Decision Tables for assessment (U-Design Platform (udesign.laiyaconsulting.com))

26 Interval Designs

27 Hallmark of Interval Designs The decision of dose finding is based on toxicity probability intervals. Interval designs : up-and-down decisions based on intervals (mtpi, mtpi-2, CCD, BOIN) Stay Escalate De-escalate p i (p T ɛ 1, p T + ɛ 2 ) p i (0, p T ɛ 1 ) p i (p T + ɛ 2, 1) Non-interval designs : CRM chooses the dose arg min i ˆp i p T, 3+3 uses up-and-down decisions based on y i n i,

28 Interval-Based Decision Setting Given the target toxicity probability p T (e.g., p T = 0.16 or 0.3), and effect size ɛ 1, ɛ 2, the interval-based decision is based on the following framework. Idea Divide (0, 1) into three intervals: (0, p T ɛ 1 ) }{{} Under dosing interval (p T ɛ 1, p T + ɛ 2 ) }{{} Equivalence Interval (p T + ɛ 2, 1) }{{} Under dosing interval [Associate with actions] E, S, D Decision rule Use Bayes rule to decide which action (decision) to take for the next patient. Next: Let us use mtpi and mtpi-2 as examples.

29 Decision Theory 101 Data Y is used to define likelihood function p(y θ). Dose finding : p(y θ) i pyi i (1 p i) ni yi. Parameters are quantities of interests. Dose finding : θ = {p 1,, p D }; Prior p(θ η)p(η) (η are additional parameters) Actions a to be taken (e.g., estimators). Dose finding : a {D, S, E} Optimal Decision Making is based on the following steps: Loss (Utility) function Define loss l(a, θ) as a function of a and θ Optimality criterion define what you want to optimize. For Bayesian, maximize posterior expected loss : l(a, θ)p(θ Y)dθ Optimal decision rule determines the action that achieves the optimality criterion. For Bayesian, we use Bayes Rule a = arg min l(a, θ)p(θ Y)dθ a

30 The mtpi and mtpi-2 Designs Use Bayes Rule Consider mtpi for now. Data Y mtpi : p(y θ) i pyi i (1 p i) ni yi. Parameter θ Tox probs {p i } = {p 1,, p D }; In addition to {p i }, introduce another parameter m i {M D, M S, M E } and Then mtpi : θ = {p i } {m i } Actions a mtpi : a {D, S, E}. Loss function mtpi : l(a = i, m i = M j ) = M D : p i (p T + ɛ 2, 1) M S : p i (p T ɛ 1, p T + ɛ 2 ) M E : p i (0, p T ɛ 1 ) { 1, if i j; 0, if i = j, for i, j {E, S, D}. Optimal decision mtpi uses Bayes rule : For dose i, choose decision a (p T, ɛ 1, ɛ 2 ) = arg max P r(m i = M j Y) j {D,S,E}

31 The mtpi Design Given p T, ɛ 1, ɛ 2, all the decisions a are pre-calculated for all the possible data Y at dose i. An mtpi decision table but in an algorithmic form

32 UPM and Bayes rule If we assume a simple working model (Ji et al., 2010), y i p i Binom(n i, p i ), and p i indep Beta(1, 1) then the working posterior is q(p i y) = Beta(1 + y i, 1 + n i y i ) Turns out the mtpi s rule to max P r(m i = M j Y) is the equivalent tomax the unit probability mass under the working posterior. P r(m i = M j Y) = UPM(M j ) = q(p i M j Y), j {D, S, E} length(m j)

33 Ockham s razor and mtpi In mtpi, when 3 out of 6 patients have DLT and if p T = 0.3, the decision based on UPM is S, to stay at the current dose. Why? Ockham s razor states the principle that an explanation of the facts should be no more complicated than necessary Bayesian model selection requires a prior p(m k ) for the candidate model k and a prior p(θ k M k ). Models are selected based on p(m k data) and automatically applies the Ockham s razor: when two models fit the data equally well, the smaller one wins. The mtpi design considers three intervals that partition the sample space (0, 1) for the probability of toxicity p i at a given dose d: M D : p i (p T + ɛ 2, 1) M S : p i (p T ɛ 1, p T + ɛ 2 ) M E : p i (0, p T ɛ 1 ) (1) Typically, p T ranges from 0.1 to 0.3 in phase I trials, and ɛ s are usually small, say So M S the middle interval is the smallest (shortest).

34 Ockham s razor and interval length Con t mtpi uses Bayes rule : For dose i, choose decision a (p T, ɛ 1, ɛ 2 ) = arg max P r(m i = M j Y) j {D,S,E} Given y i = 3 patients with toxicity events out of n i = 6 patients at dose i, mtpi decision is a = S. This is because the posterior probability P r(m i = M s Y) is the largest. See UPMs for three intervals below: UPM(E) UPM(S) 3/6 UPM(S)

35 Ockham s razor is the SOUND statistical inference Optimality The mtpi design uses Bayes rule for decision, which is optimal in the sense that it minimizes the posterior expected loss (maximize the posterior probability of each interval) Ockham s razor favors the equivalence interval [p T ɛ 1, p T + ɛ 2 ] because it is the most parsimonious (shortest) model Stay when p T = 0.3 and y/n = 3/6 is a result of Ockham s razor, which is statistically sound Ethical consideration However, in practice, one may argue that the decision is aggressive. SOLUTION: To blunt the Ockham s razor! The mtpi-2 design purely for the purpose of practical consideration no consensus!

36 The mtpi-2 design (Guo et al., 2017): Blunt the Ockham s Razor Divid (0, 1) into subintervals with equal length, same as that of (p T ɛ 1, p T + ɛ 2 ). Pick the decision {D, S, E} corresponding to the interval with the largest UPM. Still the Bayes (optimal) rule Density of B(3, 3) Post. Density for x d=3, n d=6 UPM's LI M EI HI HI HI... M 1 M 1 M 2 M

37 The mtpi-2 Design Has Fewer S than mtpi

38 The CCD and BOIN designs

39 Cumulative Cohort Design (CCD) Ivanova et al. (2007) proposed a simple interval-based design: Stay Escalate De-escalate y i n i (0, p T ɛ 1 ) y i n i (p T ɛ 1, p T + ɛ 2 ) y i n i (p T + ɛ 2, 1) The decision is based on the point estimate yi n i (MLE, posterior mode under Unif(0,1) prior) not accounting for variabilities The intervals are used as boundaries for thresholding the point estimate yi n i but not directly related to the estimators mtpi calculates posterior probabilities of the intervals accounting for variabilities The up-and-down decision rules generate a Markov chain of decisions, which is shown to converge asymptotically to a desirable stationery distribution (Ivanova et al., 2007) The CCD uses equal ɛ 1 = ɛ 2 values based on a theoretical derivation.

40 The BOIN Design A hybrid between CCD and mtpi. Use the same type of decision rules as CCD (y i /n i interval) Stay Escalate De-escalate y i y n i (p T ɛ 1, p T + ɛ 2 ) i y n i (0, p T ɛ 1 ) i n i (p T + ɛ 2, 1) The ɛ s are derived based on φ 1 and φ 2 values that are below and above p T, provided by users. BOIN uses the same MTD selection and safety rules as in mtpi. BOIN stands for Bayesian optimal interval design. The parameter space has three points: only allow p i taking three values H 0 : p i = p T, H 1 : p i = p T ɛ 1, and H 2 : p i = p T + ɛ 2 The optimality criterion is ad-hoc and not based on a loss function or a formal decision theoretic framework. The optimization in BOIN is trying to mimic a p-value type of calculation So theoretically the BOIN design is based on an ad-hoc framework, not following the classical optimal decision theory. This leads to some asymptotically strange behavior (two slides later).

41 Two Types of Interval-Based Designs The TPI, mtpi, and mtpi-2 designs anchor decision based on P r(p i interval data) and uses a formal decision theoretic framework to find the optimal decision based on the current trial data. The CCD and BOIN designs anchor decision based on y i n i interval and uses the intervals to threshold yi n i. In small sample phase I trials, the distinction might not make a big difference in the frequentist operating characteristics based on a large number of computer simulations. Theoretically, some concerns about the use of point estimate and the specific setup in BOIN.

42 A pictorial summary of mtpi and BOIN BOIN has a gap that is asymptotically inconsistent When sample size n i, the gap still exists. This creates wrong asymptotic decisions. The mtpi (mtpi-2) Design) The BOIN Design Pr(3 & )'*+,-./) )'*+,-./ % & ' & )'*+,-./ Escalate Stay (retain) De-escalate Escalate Stay (retain) De-escalate $ " p T $ # 0 1 Nominal boundary & Decision boundary 0 $ "! " p T! # $ # 1 Decision boundary gap gap Nominal boundary

43 Practical performance of mtpi-2 and BOIN is great Ying et al. (2016) show BOIN performs more favorably when compared to mtpi in terms of safety, for a small set of scenarios mtpi is wounded by the Ockham s razor Guo et al. (2017) show that mtpi-2 is safer than BOIN, and slightly more reliability in finding the true MTD. Ji and Yang (2017) report a comprehensive review.

44 Operating characteristics of various designs We compared mtpi, mtpi-2, 3+3, CRM, BOIN in terms of reliability (power of finding MTD) and safety. mtpi-2 is the top performer, followed by BION. Ji and Yang (2017; )

45 A New Web-based Integrative Statistical Platform

46 A New Web-based Integrative Statistical Platform Web Based: No need to download any software; works on MAC, PC, ipad, and smart phones just need an internet browser (e.g., Chrome, FireFox) Integrative: Offers up to six designs, 3+3, CRM, mtpi-2, BLRM, etc. Many new features: CRM decision table, etc. User-friendly: One-click download of statistical section writeup for protocols