Estimation efficiency in continual reassessmentmethodTian Tian

University of Illinois at ChicagoBackgroundIntroduction

Finding an accurate maximum tolerated dose (MTD), themost efficacious dose whose risk of toxicity is tolerable, is acritical step in development of medicine.There are two widely used approaches for identifying MTD.– The conventional 3 + 3 approach.– The continual reassessment method (CRM), which has been shown to

be more efficient that the 3 + 3 approach through many simulationstudies.

Motivation for our designWhy is CRM more efficient? Can we prove it theoretically?Is there a way to further improve on the performance ofCRM?

Description of a dose-finding studyThe target toxicity rate is set to be p0.We have m available dose levels and n patients in total intotal.The binary response Y at dose level x is modeled as:

Prob(Y = 1|x) = φ(x , θ)Design problem: How to assign dose levels to patients ateach stage in a trial such that we can identify MTD(x∗ : φ(x∗, θ) = p0) accurately?

Remark: Remember that in a standard CRM procedure, ateach stage, we assign each patient the dose level withcorresponding toxicity rate p0.—-So could this assignmentfashion be improved?

Combining (locally) optimal designConsider MTD, the parameter of interest, as a function of θ,i.e., MTD = b(θ).The asymptotic variance of the estimator of MTD could beexpressed as

V (M̂TD) =(∂b(θ)

∂θ

)( n∑j=1

Ij(θ, d))−1(∂b(θ)

∂θ

)T.

Then a locally optimal design ξ∗ based on θ̂ is the one thatminimizes V (M̂TD).

Simple power model problemModel

Dose-response model:p = φ(x , α) = xα, α > 0 and 0 < x < 1. (1)

Under design ξ = {(xi,wi), i = 1, ..., k}, the asymptoticvariance for M̂TD is:

V (M̂TD) =[(

p1/α0

)(ln p0

)(− 1α2

)]2[ k∑

i=1

wixαi (log xi)

2

1− xαi

]−1.

Optimal design for simple power model problemTheorem: Under simple power model (1), regardless of thetarget toxicity rate p0 set in the trial, the optimal designalways choose the next dose level with corresponding toxicityrate p̃, where p̃ is the solution to equation log p − 2p + 2 = 0.Remark: Numerical approximation: p̃ ≈ 0.203.

Interpretation(I). Theoretically optimal!

Recall that most clinical studies set p0 at 0.2. Thus ouroptimality result here further confirms their choices ofdesign to be not only medically reasonable, but alsostatistically optimal. Moreover, as long as p0 is chosenfrom a reasonable range (say, from 0.1 to 0.35), thestandard CRM procedure will generate an optimaldesign, or at least a nearly optimal design withnegligible efficiency loss.

Table: Relative efficiency under different target toxicity rate

p0 0.1 0.15 0.2 0.25 0.3 0.35Relative efficiency 0.910 0.981 1 0.990 0.960 0.916

(II). Counter-intuitive!No matter what p0 is, optimal design always collect dataat dose level with toxicity rate 0.2!

Simulation comparisonTable: Comparison of perforemance of standard CRM and optimal CRM

Panel 1: α = 0.5

p0 0.15 0.25 0.3standard CRM (0.505,0.217) (0.391,0.307) (0.422,0.348)optimal CRM (0.504,0.263) (0.433,0.262) (0.388,0.262)

p0 0.7 0.8standard CRM (0.578,0.743) (0.363,0.833)optimal CRM (0.783,0.262) (0.580,0.272)

Panel 2: α = 2

p0 0.15 0.25 0.3standard CRM (0.567,0.134) (0.461,0.214) (0.464,0.260)optimal CRM (0.593,0.180) (0.532,0.173) (0.408,0.174)

p0 0.7 0.8standard CRM (0.786,0.641) (0.458,0.754)optimal CRM (0.857,0.186) (0.578,0.218)

– First value of each entry represents the percentage ofaccurately identifying the MTD while the second representsthe percentage of toxicity occurrence.

Guidance to design choicesPractical situations (when p0 is around 0.2)– When p0 < 0.2, performance of the standard CRM is nearly as good as

that of the optimal CRM; but with lower toxicity occurrence.– When p0 > 0.2, performance of the optimal CRM is similar to that of

the standard CRM; but with lower toxicity occurrence.High target toxicity cases (e.g., p0 = 0.7, 0.8)– Optimal CRM is significantly better from the perspective of estimation

efficiency as well as toxicity occurrence.

Two-parameter logistic model problemModel

Dose-response model:

p = φ(x ,a) =eα+βx

1 + eα+βx (2)

whereβ > 0, α < log

p0

1− p0and x > 0. (3)

We write MTD as a function of θ, i.e.,

MTD def= η = b(θ) =

1β

(log

p0

1− p0− α

).

Optimal design for logistic model problemSince MTD is a scalar function of θ, the c-optimality, which isdesigned to optimize the estimation of linear combination ofparameters, is an appropriate choice here.We adopt the geometric approach provided by Elfving(1952) for identifying c-optimal design.

Figure: Elfving set of model (2) with parameter (α, β) = (−2,2)

-0.4 -0.2 0.2 0.4h1

-0.6

-0.4

-0.2

0.2

0.4

0.6

h2

AB

C

D

R

P QΘ1

Θ2

Theorem: Under logistic model (2) with the assumptions (3),for any 0 < p0 < 0.5, the optimal design selects the next doselevel with target toxicity rate p0.Conclusion: Under two-parameter logistic model, when p0 isset in a reasonable range (0,0.5), the standard CRMalgorithm is exactly optimal!

Future work– To incorporate model uncertainty problem into optimal

design theory by considering model averaging/selectingtechniques.

– To incorporate delayed-response problem into optimaldesign theory by considering the use of EM algorithm orother missing-data handling methods like Bayesiandata-augmentation.

AcknowledgementsThis project is supported in part by NSF GrantsDMS-13-22797 and DMS-14-07518. The author would like tothank Professor Min Yang for many inspiring discussions andinsightful assistance.

Email: ttian3@uic.edu Midwest Biophar. Stat. Workshop 2015 MSCS, UIC