ICODOE (Tian)

Estimation Eﬃciency in Continual Reassessment
Method
— Optimal Design Theory in Dose-Finding
Problems
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 study
Notations and definitions
CRM procedure
Optimal design
Models
Background
Target toxicity rate (p0) and the maximum tolerated dose
(MTD)
Two widely used approaches for identifying MTD
3+3 approach
Continual reassessment method (CRM)(O’Quigley et al., 1990)
CRM is more efficient (simulation studies)

CRM procedure
Optimal design
Models
Motivation
Why is CRM more eﬃcient?
Can we improve on CRM?

CRM procedure
Optimal design
Models
Description of the problem
Binary response Y at dose level x is modeled as:
Y ∼ Ber(φ(x, θ))
where
Y =
1 tox
0 no-tox
Design question: How to assign dose levels to patients in trial
such that we can identify MTD (x∗ : φ(x∗, θ) = p0) accurately?
Problem: Both φ and θ are unknown!

CRM procedure
Optimal design
Models
The CRM algorithm
(1) Assume model φ(x, θ) and a prior for θ based on preclinical
information;
(2) Calculate the mean of θ, and assign estimated MTD to the
ﬁrst 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 corresponding
toxicity rate p0.

CRM procedure
Optimal design
Models
Optimal design in a clinical study
MTD = b(θ) = φ−1(θ, p0)
V (MTD) = ∂b(θ)
∂θ
n
j=1 Ij(θ, d)
−1 ∂b(θ)
∂θ
T
Choose a design ξ∗ that minimizes V (MTD)

CRM procedure
Optimal design
Models
Locally optimal design
Challenge: The variance-covariance matrix for a nonlinear model
depends on the unknown parameter!
Solution: ”Locally” optimal design - based on the best guess of
the unknown parameter.
Moreover, it ﬁts the sequential design framework.

CRM procedure
Optimal design
Models
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

Under two-parameter logistic model
Model setup
Optimal design
Interpret the result
Simulations
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}, the
asymptotic variance for MTD is:
V = p
1/β
0 ln p0 −
1
β2
]2
k
i=1
ωi
xβ
i (log xi)2
1 − xβ
i
−1
.
Maximize f(x) = xβ(log x)2
1−xβ
x=p1/β
⇐===⇒ g(p) = p(log p)2
1−p

Model setup
Optimal design
Simulations
Optimal design for simple power model
Theorem
Under simple power model (1), for any parameter α ∈ R, regardless
of the target toxicity rate p0 set in the trial, the optimal design
always choose the next dose level with corresponding toxicity rate
˜p, where ˜p is the solution to equation log p − 2p + 2 = 0.
Numerical approximation: ˜p ≈ 0.203.

Model setup
Optimal design
Simulations
Surprising facts:
(I). Theoretically optimal!
Most clinical studies set p0 at 0.2 – not only medically
reasonable, but also (nearly) optimal from statistical point
of view.
(II). Counter-intuitive!
No matter what p0 is, optimal design always collect data at
dose level with toxicity rate 0.203.

Model setup
Optimal design
Simulations
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 0.35
Relative 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 a
reasonable range (say, from 0.1 to 0.35), the standard CRM
procedure will generate an optimal design, or at least a nearly
optimal design with negligible efficiency loss.

Model setup
Optimal design
Simulations
Simulation results
Table: Comparison of perforemance of standard CRM and optimal CRM
when β = exp(α) = 1
p0 0.15 0.25 0.3
standard 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

Model setup
Optimal design
Simulations
Simulation results (Cont.)
when β = exp(α) = 2
p0 0.15 0.25 0.3
standard 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)

Model setup
Optimal design
Simulations
Simulation results (Cont.)
when β = exp(α) = 0.5
p0 0.15 0.25 0.3
standard 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)

Delayed response introduction
Model setup
Optimal design
Model
Dose-response model:
p = φ(x, a) =
eα+βx
1 + eα+βx
(2)
where
β > 0, α < log
p0
1 − p0
and x > 0. (3)

Model setup
Optimal design
Information matrix
Under design ξ = {(xi, ωi), i = 1, ..., k}, the information matrix for
θ = (α, β)T is:
IY (θ) =
k
i=1
ωi(xi, θ)hT
(xi, θ)
where
h(x, θ) =
e
α+βx
2
1 + eα+βx
,
xe
α+βx
2
1 + eα+βx
T
.

Model setup
Optimal design
MTD and c-optimality
We write MTD as a function of θ, i.e.,
MTD
def
= η = b(θ) =
1
β
log
p0
1 − p0
− α .
Optimality criterion: c-optimality.
Tool: Elfving’s geometric approach (1952).

Model setup
Optimal design
Elfving set
0.4 0.2 0.2 0.4
h1
0.6
0.4
0.2
0.2
0.4
0.6
h2
A
B
C
D
R
P QΘ1
Θ2
Figure: Elfving set of model (2) with parameter (α, β) = (−2, 2)

Model setup
Optimal design
Optimal design for two-parameter logistic model
Theorem
Under logistic model (2) with the assumptions (3), for any
0 < p0 < 0.5, the optimal design selects the next dose level with
target toxicity rate p0.
Under two-parameter logistic model, when p0 is set in a reasonable
range (0,0.5), the standard CRM algorithm is exactly optimal!

Likelihood at each study point
Background and motivation
Model the pseudo outcomes
A new problem
Before — Incorporate optimal design theory into dose-ﬁnding
problems under a standard setup.
Now — Problems encountered in real-life clinical trials.
e.g., Monitoring late-onset toxicities.

Real-life clinical studies
In radio-therapy trials, dose-limiting toxicities often occur long
after the treatment is ﬁnished (Coia et al., 1995; Cooper et
al., 1995)
In a trial treating patients with pancreatic cancer, the full
evaluation period was 9 weeks while the accrual rate was 1
per week (Muler et al., 2004).
In the area of molecularly targeted agents, among a total of
445 patients in 36 trials, 57% of the grade 3 and 4 toxicities
were late-onset (Postel-Vinay et al., 2011).
——— It’s taking too long...

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?

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 patient
j by time uk.

Weight function
A weight function wk to depict the relationship between true
response 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)

Outcome by time uk
True responses
after entire evaluation time uK = T
Outcomes after actual follow-up time uk
Row total
No-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)

Probabilities for diﬀerent events
Marginal qj,k = Pr(Zj,k = 0) = 1 − wkφ(xj, θ)
Conditional
˜pj,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)
Joint
pj,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)
Relation
˜pj,k · qj,k−1 = pj,k
˜qj,k · qj,k−1 = qj,k
Table: Probabilities for marginal/conditional/joint events

Preliminary results
Categorize patients
Information matrices
Patient categories
At a certain interim study time point, there are three kinds of
patients involved in the experiment:
Already been fully evaluated X1;
Not been fully evaluated yet X2;
Newly enrolled with dose assignments to be decided X3.

Preliminary results
Categorize patients
Fully evaluated patients
K + 1 groups: X1
1 ,...,XK
1 , and XK+1
1 .
(1). For xj ∈ Xk
1 , 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).

Preliminary results
Categorize patients
Fully evaluated patients (Cont.)
(2). For xj ∈ XK+1
1
Observed event:
{Zj,l = 0, l ≤ K} = {Zj,K = 0};
qj,K = Pr(Zj,K = 0) = 1 − wKφ(xj, θ)
wK =1
===== 1 − φ(xj, θ).

Preliminary results
Categorize patients
Patients with delayed responses
K − 1 groups: X1
2 ,...,XK−1
2 .
For xk ∈ Xk
2 , the random outcome by the next interim study point
is 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).

Preliminary results
Categorize patients
Patients with delayed responses (Cont.)
qj,k · ˜p
˜Zj,k+1
j,k+1 ˜q
1− ˜Zj,k+1
j,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
========⇒ ˜p
˜Zj,k+1
j,k+1 ˜q
1− ˜Zj,k+1
j,k+1 .

Preliminary results
Categorize patients
Newly enrolled patients
D groups: X1
2 ,...,XD
2 .
For xj ∈ Xd
3 , d = 1, ..., D, the random outcome by the next
interim study point Zj,1 ∼ Ber(pd,1); and its contribution to the
likelihood is given by
p
Zj,1
d,1 q
1−Zj,1
d,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
.

Preliminary results
Categorize patients
Information matrices for simple power model
Ip
1 = K+1
k=1 x∈Xk
1
(log x)2
xexp(α)
(1 − xexp(α)
)2 exp2(α).
Ip
2 =
K
k=2 x∈Xk−1
2
wk(log x)2
xexp(α)
(1 − wk−1xexp(α)
)(1 − wkxexp(α)
)
exp2(α).
Ip
3 = D
d=1 ωd ×
w1(log xd)2
x
exp(α)
d
1 − w1x
exp(α)
d
exp2(α).
under design ξ = {(xd, ωd), d = 1, , , D}.

Preliminary results
Categorize patients
Information matrices for logistic model
Let c = exp(α + βx),
Il
1 = K+1
k=1 x∈Xk
1
c
(1 + c2
)
1 x
x x2
Il
2 =
K
k=2 x∈Xk
2
[c(1 + c)(wk − wk−1c) + wk−1wkc2
]
(1 + (1 − wk−1)c)(1 + (1 − wk)c)(1 + c)2
1 x
x x2
Il
3 = D
d=1 ωd × w1cd
(1 + cd)2
(1 + (1 − w1)cd)
1 xd
xd x2
d
under design ξ = {(xd, ωd), d = 1, , , D}.

Preliminary results
Ongoing work
Weight function estimation
On simple power model
On two-parameter logistic model
Two methods:
Linear and time-weighted estimation (Cheung and Chappell,
2000).
˜wk =
uk
T
.
Bayesian estimation (Ji and Bekele, 2009).

Preliminary results
Ongoing work
Bayesian estimation for wk
Conjugacy ⇒ wk ∼ Beta(α0
k + k
l=1 ntoxl , β0
k + ntox − k
l=1 ntoxl )
Posterior mean:
ˆwk =
α0
k +
k
l=1
ntoxl
α0
k + β0
k + ntox .

Preliminary results
Ongoing work
Optimal result on simple power model
Theorem
Under simple power model (1), for any parameter α ∈ R, the
optimal design chooses the next dose level with corresponding
toxicity rate ˜p, where ˜p is the solution to equation
log p − 2w1p + 2 = 0.
Notice that the optimal dose level depends on the estimate of the
ﬁrst-stage weight function w1.

Preliminary results
Ongoing work
Analytical D-optimality result on logistic model
Theorem
At the ﬁrst stage, under logistic model (2), for any given dose
range c = α + βx ∈ [A, B] ⊆ [−10, − log 2], the D-optimal dose
assignment is ξl
opt = {(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 the
solution to equation ˜fl(c) = 0.
Otherwise, c∗ = A.
Notice that the optimal dose level here also depends on the
estimate of the ﬁrst-stage weight function w1.

Preliminary results
Ongoing work
Analytical D-optimality result on logistic model (Cont.)
Speciﬁcally,
˜fl
(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]

Preliminary results
Ongoing work
Thank you
Current algorithm structure
Future work
As for the situations where analytical results are challenging, we
get help from algorithms.
In particular, here we adopt the optimal weight exchanging
algorithm (OWEA; Yang, Biedermann, and Tang, 2013).
Diﬀerent optimality criteria
Multi-stage design
Parameter estimation (MLE/posterior mean)
Diﬀerent targets, i.e., b(θ).

Preliminary results
Ongoing work
Thank you
Future work
Existing problem and future work
Existing problem:
Restrained to situations where the assumed model is close to the
true model.
Future research:
Incorporate the idea of optimal design into dose-ﬁnding studies
where there exist model selection/averaging problem.

Ongoing work
Thank You

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