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Estimation Efficiency in Continual ReassessmentMethod
— Optimal Design Theory in Dose-FindingProblems
Tian Tian Min Yang Lei Nie
ICODOE 2016
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 1 / 43
Problem introduction
Under simple power model
Background of dose-finding studyNotations and definitionsCRM procedureOptimal designModels
Background
Target toxicity rate (p0) and the maximum tolerated dose(MTD)
Two widely used approaches for identifying MTD
3+3 approachContinual reassessment method (CRM)(O’Quigley et al., 1990)CRM is more efficient (simulation studies)
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 2 / 43
Problem introduction
Under simple power model
Background of dose-finding studyNotations and definitionsCRM procedureOptimal designModels
Motivation
Why is CRM more efficient?
Can we improve on CRM?
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 3 / 43
Problem introduction
Under simple power model
Background of dose-finding studyNotations and definitionsCRM procedureOptimal designModels
Description of the problem
Binary response Y at dose level x is modeled as:
Y ∼ Ber(φ(x, θ))
whereY =
{1 tox0 no-tox
Design question: How to assign dose levels to patients in trialsuch that we can identify MTD (x∗ : φ(x∗, θ) = p0) accurately?
Problem: Both φ and θ are unknown!
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 4 / 43
Problem introduction
Under simple power model
Background of dose-finding studyNotations and definitionsCRM procedureOptimal designModels
The CRM algorithm
(1) Assume model φ(x, θ) and a prior for θ based on preclinicalinformation;
(2) Calculate the mean of θ, and assign estimated MTD to thefirst entered patient(s);
(3) Collect data and update the posterior distribution of θ;(4) Calculate the posterior mean of θ and assign estimated MTD
to the next patient;(5) Repeat procedure (3) to (5) until some predetermined
stopping rule is met.
Key point: Assign each patient the dose level with correspondingtoxicity rate p0.
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 5 / 43
Problem introduction
Under simple power model
Background of dose-finding studyNotations and definitionsCRM procedureOptimal designModels
Optimal design in a clinical study
MTD = b(θ) = φ−1(θ, p0)
V (MTD) =(∂b(θ)∂θ
)(∑nj=1 Ij(θ, d)
)−1(∂b(θ)∂θ
)T
Choose a design ξ∗ that minimizes V (MTD)
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 6 / 43
Problem introduction
Under simple power model
Background of dose-finding studyNotations and definitionsCRM procedureOptimal designModels
Locally optimal design
Challenge: The variance-covariance matrix for a nonlinear modeldepends on the unknown parameter!
Solution: ”Locally” optimal design - based on the best guess ofthe unknown parameter.
Moreover, it fits the sequential design framework.
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 7 / 43
Problem introduction
Under simple power model
Background of dose-finding studyNotations and definitionsCRM procedureOptimal designModels
Models in CRM
Two models are dominated in CRM-based designs:
Simple power model:
p = φ(x, θ) = xexp(α)
Two-parameter logistic model:
p = φ(x, θ) = eα+βx
1 + eα+βx
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 8 / 43
Problem introduction
Under simple power model
Under two-parameter logistic model
Model setupOptimal designInterpret the resultSimulations
Model and information matrix
Dose-response model:
p = φ(x, α) = xexp(α), α ∈ R and 0 < x < 1. (1)
Let β = exp(α), under design ξ = {(xi, ωi), i = 1, ..., k}, theasymptotic variance for MTD is:
V =[(p
1/β0)(
ln p0)(− 1β2
)]2[ k∑i=1
ωixβi (log xi)2
1− xβi
]−1.
Maximize f(x) = xβ(log x)2
1−xβx=p1/β⇐===⇒ g(p) = p(log p)2
1−p
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 9 / 43
Problem introduction
Under simple power model
Under two-parameter logistic model
Model setupOptimal designInterpret the resultSimulations
Optimal design for simple power model
TheoremUnder simple power model (1), for any parameter α ∈ R, regardlessof the target toxicity rate p0 set in the trial, the optimal designalways choose the next dose level with corresponding toxicity ratep, where p is the solution to equation log p− 2p+ 2 = 0.
Numerical approximation: p ≈ 0.203.
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 10 / 43
Problem introduction
Under simple power model
Under two-parameter logistic model
Model setupOptimal designInterpret the resultSimulations
Interpret the result
Surprising facts:
(I). Theoretically optimal!
Most clinical studies set p0 at 0.2 – not only medicallyreasonable, but also (nearly) optimal from statistical pointof view.
(II). Counter-intuitive!
No matter what p0 is, optimal design always collect data atdose level with toxicity rate 0.203.
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 11 / 43
Problem introduction
Under simple power model
Under two-parameter logistic model
Model setupOptimal designInterpret the resultSimulations
Relative efficiency under different target toxicity rate
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 0.999 0.990 0.960 0.916
Under the simple power model, as long as p0 is chosen from areasonable range (say, from 0.1 to 0.35), the standard CRMprocedure will generate an optimal design, or at least a nearlyoptimal design with negligible efficiency loss.
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 12 / 43
Problem introduction
Under simple power model
Under two-parameter logistic model
Model setupOptimal designInterpret the resultSimulations
Simulation results
Table: Comparison of perforemance of standard CRM and optimal CRMwhen β = exp(α) = 1
p0 0.15 0.25 0.3standard CRM (0.533,0.181) (0.416,0.271) (0.408,0.309)optimal CRM (0.561,0.223) (0.434,0.228) (0.440,0.226)
Similar percentage of correctly identifying MTD
Lower percentage of toxicity occurrence for p0 > 0.2
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 13 / 43
Problem introduction
Under simple power model
Under two-parameter logistic model
Model setupOptimal designInterpret the resultSimulations
Simulation results (Cont.)
Table: Comparison of perforemance of standard CRM and optimal CRMwhen β = exp(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)
Similar percentage of correctly identifying MTD
Lower percentage of toxicity occurrence for p0 > 0.2
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 14 / 43
Problem introduction
Under simple power model
Under two-parameter logistic model
Model setupOptimal designInterpret the resultSimulations
Simulation results (Cont.)
Table: Comparison of perforemance of standard CRM and optimal CRMwhen β = exp(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)
Similar percentage of correctly identifying MTD
Lower percentage of toxicity occurrence for p0 > 0.2
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 15 / 43
Under simple power model
Under two-parameter logistic model
Delayed response introduction
Model setupOptimal design
Model
Dose-response model:
p = φ(x, a) = eα+βx
1 + eα+βx (2)
whereβ > 0, α < log p0
1− p0and x > 0. (3)
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 16 / 43
Under simple power model
Under two-parameter logistic model
Delayed response introduction
Model setupOptimal design
Information matrix
Under design ξ = {(xi, ωi), i = 1, ..., k}, the information matrix forθ = (α, β)T is:
IY (θ) =k∑i=1
ωi(xi, θ)hT (xi, θ)
where
h(x, θ) =( e
α+βx2
1 + eα+βx ,xe
α+βx2
1 + eα+βx
)T.
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 17 / 43
Under simple power model
Under two-parameter logistic model
Delayed response introduction
Model setupOptimal design
MTD and c-optimality
We write MTD as a function of θ, i.e.,
MTDdef= η = b(θ) = 1
β
(log p0
1− p0− α
).
Optimality criterion: c-optimality.
Tool: Elfving’s geometric approach (1952).
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 18 / 43
Under simple power model
Under two-parameter logistic model
Delayed response introduction
Model setupOptimal design
Elfving set
-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
Figure: Elfving set of model (2) with parameter (α, β) = (−2, 2)T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 19 / 43
Under simple power model
Under two-parameter logistic model
Delayed response introduction
Model setupOptimal design
Optimal design for two-parameter logistic model
TheoremUnder logistic model (2) with the assumptions (3), for any0 < p0 < 0.5, the optimal design selects the next dose level withtarget toxicity rate p0.
Under two-parameter logistic model, when p0 is set in a reasonablerange (0,0.5), the standard CRM algorithm is exactly optimal!
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 20 / 43
Under two-parameter logistic model
Delayed response introduction
Likelihood at each study point
Background and motivationNotations and definitionsModel the pseudo outcomes
A new problem
Before — Incorporate optimal design theory into dose-findingproblems under a standard setup.
Now — Problems encountered in real-life clinical trials.
e.g., Monitoring late-onset toxicities.
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 21 / 43
Under two-parameter logistic model
Delayed response introduction
Likelihood at each study point
Background and motivationNotations and definitionsModel the pseudo outcomes
Real-life clinical studies
In radio-therapy trials, dose-limiting toxicities often occur longafter the treatment is finished (Coia et al., 1995; Cooper etal., 1995)In a trial treating patients with pancreatic cancer, the fullevaluation period was 9 weeks while the accrual rate was 1per week (Muler et al., 2004).In the area of molecularly targeted agents, among a total of445 patients in 36 trials, 57% of the grade 3 and 4 toxicitieswere late-onset (Postel-Vinay et al., 2011).
——— It’s taking too long...
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 22 / 43
Under two-parameter logistic model
Delayed response introduction
Likelihood at each study point
Background and motivationNotations and definitionsModel the pseudo outcomes
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 23 / 43
Under two-parameter logistic model
Delayed response introduction
Likelihood at each study point
Background and motivationNotations and definitionsModel the pseudo outcomes
Some current methods
Time-to-event CRM (Cheung and Chappell, 2000)The predicted-risk-of-toxicity method (Bekele et al., 2008)Escalation with overdose control design (Mauguen et al.,2011)Treated as missing data and use a modified EM algorithm(Yuan and Yin, 2011)Treated as missing data and use Bayesian data augmentation(Liu and Ning, 2013)
——– How to bring in some optimal-design ideas here?
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 24 / 43
Under two-parameter logistic model
Delayed response introduction
Likelihood at each study point
Background and motivationNotations and definitionsModel the pseudo outcomes
Description of the problem
Entire evaluation time is T ;
K + 1 interim study time points, u0 = 0, u1, ..., uK−1,uK = T ;
Binary response Zj,k denotes the toxicity outcome for patientj by time uk.
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 25 / 43
Under two-parameter logistic model
Delayed response introduction
Likelihood at each study point
Background and motivationNotations and definitionsModel the pseudo outcomes
Weight function
A weight function wk to depict the relationship between trueresponse by time uK = T and outcome by interim study time uk,k = 1, ...,K (Cheung and Chappell, 2000):
Pr(Tox by time uk) = wkPr(Tox by time uK)⇒wk = Pr(Tox by time uk|Tox by time uK)
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 26 / 43
Under two-parameter logistic model
Delayed response introduction
Likelihood at each study point
Background and motivationNotations and definitionsModel the pseudo outcomes
Outcome by time uk
True responsesafter entire evaluation time uK = T
Outcomes after actual follow-up time uk Row totalNo-tox Tox
No-tox 1− φ(xj , θ) 0 1− φ(xj , θ)Tox (1− wk)φ(xj , θ) wkφ(xj , θ) φ(xj , θ)
Column total 1− wkφ(xj , θ) wkφ(xj , θ) 1
Table: Joint probabilities of outcomes by time uk and uK = T
Outcome for patient j by time uk follows the marginal distribution:
Pr(Zj,k = 1) = wkφ(xj , θ)Pr(Zj,k = 0) = 1− wkφ(xj , θ)
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 27 / 43
Under two-parameter logistic model
Delayed response introduction
Likelihood at each study point
Background and motivationNotations and definitionsModel the pseudo outcomes
Probabilities for different events
Marginal qj,k = Pr(Zj,k = 0) = 1− wkφ(xj , θ)
Conditionalpj,k = Pr(Zj,k = 1|Zj,k−1 = 0) = wkφ(xj ,θ)−wk−1φ(xj ,θ)
1−wk−1φ(xj ,θ)
qj,k = Pr(Zj,k = 0|Zj,k−1 = 0) = 1−wkφ(xj ,θ)1−wk−1φ(xj ,θ)
Jointpj,k = Pr(Zj,k = 1, Zj,k−1 = 0) = wkφ(xj , θ)− wk−1φ(xj , θ)
qj,k = Pr(Zj,k = 0, Zj,k−1 = 0) = Pr(Zj,k = 0) = 1− wkφ(xj , θ)
Relationpj,k · qj,k−1 = pj,k
qj,k · qj,k−1 = qj,k
Table: Probabilities for marginal/conditional/joint events
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 28 / 43
Delayed response introduction
Likelihood at each study point
Preliminary results
Categorize patientsInformation matrices
Patient categories
At a certain interim study time point, there are three kinds ofpatients involved in the experiment:
Already been fully evaluated X1;
Not been fully evaluated yet X2;
Newly enrolled with dose assignments to be decided X3.
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 29 / 43
Delayed response introduction
Likelihood at each study point
Preliminary results
Categorize patientsInformation matrices
Fully evaluated patients
K + 1 groups: X 11 ,...,XK1 , and XK+1
1 .
(1). For xj ∈ X k1 , k = 1, ...,K
Observed event:
{Zj,k = 1, Zj,l = 0, l < k} = {Zj,k = 1, Zj,k−1 = 0};
Contribution to the likelihood:
pj,k = Pr(Zj,k = 1, Zj,k−1 = 0) = wkφ(xj , θ)− wk−1φ(xj , θ).
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 30 / 43
Delayed response introduction
Likelihood at each study point
Preliminary results
Categorize patientsInformation matrices
Fully evaluated patients (Cont.)
(2). For xj ∈ XK+11
Observed event:
{Zj,l = 0, l ≤ K} = {Zj,K = 0};
Contribution to the likelihood:
qj,K = Pr(Zj,K = 0) = 1− wKφ(xj , θ)wK=1===== 1− φ(xj , θ).
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 31 / 43
Delayed response introduction
Likelihood at each study point
Preliminary results
Categorize patientsInformation matrices
Patients with delayed responses
K − 1 groups: X 12 ,...,XK−1
2 .
For xk ∈ X k2 , the random outcome by the next interim study pointis either {Zj,k+1 = 1, Zj,l = 0, l < k + 1} = {Zj,k+1 = 1, Zj,k = 0}or {Zj,l = 0, l ≤ k + 1} = {Zj,k+1 = 0}.
Here we introduce a new ”conditional random variable”,
Zj,k := Zj,k|Zj,k−1 = 0,
to depict the random outcome by the next interim study point,given the observed outcome at the current time.We have Zj,k+1 ∼ Ber(pj,k+1).
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 32 / 43
Delayed response introduction
Likelihood at each study point
Preliminary results
Categorize patientsInformation matrices
Patients with delayed responses (Cont.)
Contribution to the likelihood:
qj,k · pZj,k+1j,k+1 q
1−Zj,k+1j,k+1
which can be interpreted as follows, observed part: Zj,k = 0 contribution========⇒ qj,k
random part: Zj,k+1 ∼ Ber(pj,k+1) contribution========⇒ pZj,k+1j,k+1 q
1−Zj,k+1j,k+1 .
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 33 / 43
Delayed response introduction
Likelihood at each study point
Preliminary results
Categorize patientsInformation matrices
Newly enrolled patients
D groups: X 12 ,...,XD2 .
For xj ∈ X d3 , d = 1, ..., D, the random outcome by the nextinterim study point Zj,1 ∼ Ber(pd,1); and its contribution to thelikelihood is given by
pZj,1d,1 q
1−Zj,1d,1 = (w1φ(d, α)− w0φ(d, α))Zj,1(1− w1φ(d, α))1−Zj,1
w0=0===== (w1φ(d, α))Zj,1(1− w1φ(d, α))1−Zj,1 .
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 34 / 43
Delayed response introduction
Likelihood at each study point
Preliminary results
Categorize patientsInformation matrices
Information matrices for simple power model
Ip1 =∑K+1k=1
∑x∈Xk1
(log x)2xexp(α)
(1− xexp(α))2 exp2(α).
Ip2 =∑Kk=2
∑x∈Xk−1
2
wk(log x)2xexp(α)
(1− wk−1xexp(α))(1− wkxexp(α))
exp2(α).
Ip3 =∑Dd=1 ωd ×
w1(log xd)2xexp(α)d
1− w1xexp(α)d
exp2(α).
under design ξ = {(xd, ωd), d = 1, , , D}.
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 35 / 43
Delayed response introduction
Likelihood at each study point
Preliminary results
Categorize patientsInformation matrices
Information matrices for logistic model
Let c = exp(α+ βx),
Il1 =∑K+1k=1
∑x∈Xk1
c(1 + c2)
(1 xx x2
)
Il2 =∑Kk=2
∑x∈Xk2
[c(1 + c)(wk − wk−1c) + wk−1wkc2]
(1 + (1− wk−1)c)(1 + (1− wk)c)(1 + c)2
(1 xx x2
)
Il3 =∑Dd=1 ωd ×
w1cd(1 + cd)2(1 + (1− w1)cd)
(1 xdxd x2
d
)under design ξ = {(xd, ωd), d = 1, , , D}.
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 36 / 43
Likelihood at each study point
Preliminary results
Ongoing work
Weight function estimationOn simple power modelOn two-parameter logistic model
Weight function estimation
Two methods:
Linear and time-weighted estimation (Cheung and Chappell,2000).
wk = ukT.
Bayesian estimation (Ji and Bekele, 2009).
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 37 / 43
Likelihood at each study point
Preliminary results
Ongoing work
Weight function estimationOn simple power modelOn two-parameter logistic model
Bayesian estimation for wk
Conjugacy ⇒ wk ∼ Beta(α0k +
∑kl=1 n
toxl , β0k + ntox −
∑kl=1 n
toxl)
Posterior mean:
wk =α0k +
k∑l=1
ntoxl
α0k + β0
k + ntox.
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 38 / 43
Likelihood at each study point
Preliminary results
Ongoing work
Weight function estimationOn simple power modelOn two-parameter logistic model
Optimal result on simple power model
TheoremUnder simple power model (1), for any parameter α ∈ R, theoptimal design chooses the next dose level with correspondingtoxicity rate p, where p is the solution to equationlog p− 2w1p+ 2 = 0.
Notice that the optimal dose level depends on the estimate of thefirst-stage weight function w1.
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 39 / 43
Likelihood at each study point
Preliminary results
Ongoing work
Weight function estimationOn simple power modelOn two-parameter logistic model
Analytical D-optimality result on logistic model
TheoremAt the first stage, under logistic model (2), for any given doserange c = α+ βx ∈ [A,B] ⊆ [−10,− log 2], the D-optimal doseassignment is ξlopt = {(c∗, 1/2), (B, 1/2)}; and the choice of c∗would be one of the following two cases:
When 1− d2(A) < w1 and B > A+ d1(A), c∗ ∈ (A,B) is thesolution to equation f l(c) = 0.Otherwise, c∗ = A.
Notice that the optimal dose level here also depends on theestimate of the first-stage weight function w1.
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 40 / 43
Likelihood at each study point
Preliminary results
Ongoing work
Weight function estimationOn simple power modelOn two-parameter logistic model
Analytical D-optimality result on logistic model (Cont.)
Specifically,
f l(c) = 2(1− w1)(c−B − 1) exp(2c)
+ (c−B − 2− 2(1− w1)) exp(c) +B − c− 2
d1(A) = 2(1 + exp(A))(1 + (1− w1) exp(A))(1− exp(A)− 2(1− w1) exp(2A))
d2(A) = (2− log 2−A) exp(A) + 2 + log 2 +A
2 exp(A)[(A+ log 2− 1) exp(A)− 1]
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 41 / 43
Preliminary results
Ongoing work
Thank you
Current algorithm structureFuture work
Current algorithm structure
As for the situations where analytical results are challenging, weget help from algorithms.
In particular, here we adopt the optimal weight exchangingalgorithm (OWEA; Yang, Biedermann, and Tang, 2013).
Different optimality criteriaMulti-stage designParameter estimation (MLE/posterior mean)Weight function estimationDifferent targets, i.e., b(θ).
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 42 / 43
Preliminary results
Ongoing work
Thank you
Current algorithm structureFuture work
Existing problem and future work
Existing problem:
Restrained to situations where the assumed model is close to thetrue model.
Future research:
Incorporate the idea of optimal design into dose-finding studieswhere there exist model selection/averaging problem.
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 43 / 43
Ongoing work
Thank You
T.Tian, M.Yang, L.Nie Optimal design theory in dose-finding problems 44 / 43