Optimal Approximate Designs for Comparison with Control in Dose ...

arXiv:1511.06525v1 [math.ST] 20 Nov 2015

Optimal Approximate Designs for Comparison with Control in Dose-Escalation Studies Samuel Rosa and Radoslav Harman November 23, 2015

Abstract Consider an experiment, where a new drug is tested for the first time on human subjects healthy volunteers. Such experiments are often performed as dose-escalation studies, where a set of increasing doses is pre-selected, individuals are grouped into cohorts and a dose is given to subjects in a cohort only if the preceding dose was already given to previous cohorts. If an adverse effect is observed, the experiment stops and thus no subjects are exposed to higher doses. We assume that the response is affected both by the dose or placebo effects as well as by the cohort effects. We provide optimal approximate designs for selected optimality criteria (E-, M V - and LV -optimality) for estimating the effects of drug doses compared with placebo. In particular, we obtain the optimality of Senn designs and extended Senn designs with respect to multiple criteria. Keywords: dose-escalation studies; approximate designs; comparison with control; optimal designs; block designs

1

Introduction

A dose-escalation study for a new drug consists of a series of trials, where the drug is given to cohorts of individuals, one cohort at a time. The experimenters select a finite number of increasing doses of the drug that are to be tested and each subject is given either one of the doses or placebo. In particular, the subjects in the first cohort may be given only the lowest dose or placebo, the subjects in the second cohort only the two lowest doses or placebo, etc. These studies are often performed when a new drug is tried on humans for the first time and 1

thus the safety of the subjects is the main concern, especially if the drug is given to healthy volunteers - hence the slowly escalating doses. The paper [2], motivated by the Te Genero trial (see [16]), considered designs for doseescalation studies with quantitative responses. Some general principles that reduce variance were presented, as well as a wide range of well performing designs for estimating a system of all pairwise comparisons of treatments. In [4], some numerical methods for constructing optimal designs in the setting proposed by [2] were provided. In this paper, we study optimal approximate designs in the model considered by [2]. The approximate, or continuous, designs do not determine the actual number of subjects that are given a particular treatment, rather they determine the proportions of all subjects that are allocated to individual treatments. Note that when a total number of subjects is selected, optimal or efficient exact designs can usually be constructed from the optimal approximate designs, see e.g. Chapter 12 in [12]. We provide optimal approximate designs for comparing all doses with the placebo (comparing treatments with a control), which is a natural system of contrasts for the drug testing experiments. There is a large amount of literature on comparison with a control in blocking experiments, beginning with [1], see e.g. [8] or [7]. However, these standard results do not seem to be applicable in the model considered in this paper, due to the dose-escalation constraints. We considered the E- and M V -optimality criteria, as well as the latest variance (LV ) criterion proposed by Senn in his commentary [17] on [2] and obtained optimal designs for these criteria. In particular, the Senn designs (which are designs that assign in each cohort half of the subjects to the placebo and the other half to the latest dose, see [2], [15]) stand out as they are optimal with respect to all of the aforementioned criteria. It seems that the Senn designs are a reasonable choice when the effects of the doses relative to placebo are of interest. Moreover, we obtained a complete characterization of the class of E-optimal designs, which allows for selecting the best E-optimal design with respect to secondary criteria chosen by the experimenter, e.g. some other optimality criterion.

1.1

Notation

By 1n and 0n , we denote the column vectors of length n of ones and zeroes, respectively. Furthermore, Jn := 1n 1Tn is the n × n matrix of ones, 0m×n := 0m 0Tn is the m × n matrix

of zeroes and ei is the i-th standard unit vector (the i-th column of the identity matrix In , where n is the dimension of ei ). The symbol C(A) denotes the column space of a matrix A.

2

2

The Model

We consider the model formulated by [2]. A new drug is to be tested on N individuals divided into t cohorts. Each individual is given a treatment u, which is either placebo (u = 0) or one of n doses of the drug (u = 1, . . . , n). We assume that the safety or tolerability outcome, or in general, the measured effects of the drug on the subjects, are quantitative. Furthermore, we assume additivity of treatment and cohort effects, i.e., Yi = µ + τu(i) + θk(i) + εi ,

(1)

i = 1, . . . , N,

where N is the total number of subjects, Yi is the measured response for the i-th subject, µ is the overall mean, u(i) ∈ {0, . . . , n} is the treatment given to the i-th subject (0 denotes no

dose, 1, . . . , n denote escalating doses dose1 ≤ . . . ≤ dosen ), k(i) ∈ {1, . . . , t} is the cohort in

which the i-th subject is; τu is the effect of the u-th treatment, θk is the effect of the k-th cohort and εi is the random error. The errors {εi } are assumed to have zero mean, variance σ 2 and they are independent. Note that the model (1) is in fact a blocking experiment, with

cohorts as blocks and doses as treatments. We assume that the aim of the experiment is to compare the doses with the placebo, that is, to estimate τ1 − τ0 , . . . , τn − τ0 . Adapting the notation of [2], we will consider two cases:

standard designs, where t = n and extended designs, where t = n + 1.

An approximate design (from now on, simply design) ξ for estimating a system AT β in a general linear model Yi = f T (xi )β + εi ,

i = 1, . . . , N P where xi ∈ X, is a mapping ξ : X → [0, 1], such that x∈X ξ(x) = 1; X is the set of all

experimental conditions. A design is feasible if AT β is estimable under ξ. The moment matrix P of a design ξ is M (ξ) = x∈X ξ(x)f (x)f T (x) and the information matrix of a feasible design

ξ for estimating AT β is NA (ξ) = (AT M − (ξ)A)−1 , see [12]. A feasible design ξ ∗ ∈ Ξ, where Ξ is the set of all considered designs, is Φ-optimal if it maximizes (alternatively, minimizes)

a given function Φ(ξ) among all designs ξ ∈ Ξ. We will omit the argument ξ in expressions,

where it is not necessary to preserve it.

In model (1), X = {0, . . . , n} × {1, . . . , t}, f (u, k) = eTu+1 , eTk T

τ = (τ0 , . . . , τn )

T


T

, β = τ T , θT


T

, where

and θ = (θ1 , . . . , θt ) . The objective of the experiment is to estimate

τ1 − τ0 , . . . , τn − τ0 , which can be written as a system of contrasts QT τ , where QT = (−1n , In ),
or, equivalently, AT β, where AT = QT , 0n×t .

The value ξ(u, k) determines the proportion of all subjects that are assigned to cohort k P and given the treatment u. Let ru (ξ) := k ξ(u, k) be the total proportion of trials with 3

treatment u, u = 0, . . . , n, sk (ξ) :=

P

u

ξ(u, k) be the total proportion of trials for cohort k,

k = 1, . . . , t and r(ξ) := (r0 (ξ), . . . , rn (ξ))T , s(ξ) := (s1 (ξ), . . . , st (ξ))T . The moment matrix of a design ξ is M (ξ) =

" M11

T M12

M12 M22

#

,



where M11 = diag(r), M12 = r, X , where X = ξ(u, k) u,k , and M22 =

" 1

sT

s

diag(s)

#

.

Note that r = X1t and s = X T 1n+1 . In accordance with [2], we introduce some constraints on the set of considered designs Ξ. Since the safety is the main concern when a drug is tested on humans for the first time, we consider a dose-escalation constraint (i). Furthermore, we assume that each cohort is of the same size (ii). For each k ∈ {1, . . . , n} we have: (i) ξ(u, k) = 0 for u > k. In cohort k, the highest allowed dose is k. (ii)

P

u

ξ(u, k) = 1/t.

When we consider extended designs, the only condition for cohort k = n + 1 is (ii), i.e. the condition (ii) is assumed for all k ∈ {1, . . . , t}. In other words, the cohort n + 1 is an

additional cohort that is subject to no constraints other than the fixed cohort size.

From now on, we have Ξ = {ξ is feasible | ξ satisfies (i), (ii)}. Bailey [2] considered an

additional constraint (iii) ξ(k, k) ≥ 1/N . That is, in each cohort, the highest allowed dose

must be given to at least one subject. Since (iii) will be naturally satisfied by optimal designs, we will disregard this constraint. From condition (ii) it follows that s = 1t /t. Thus the matrix M22 has rank t and # " 0 0Tt − M22 = 0t tIt − T is its generalized inverse. The Schur complement Mτ = M11 − M12 M22 M12 is then

Mτ = diag(r) − tXX T and the information matrix of a feasible design for estimating QT τ is given by NQ (ξ) = (QT Mτ− (ξ)Q)−1 , see e.g. [14]. Note that QT τ is estimable under ξ if and only if C(Q) ⊆ C(Mτ )

(e.g. [14]). From now on, we will consider only Q = (−1n , In )T , i.e., comparison with control. 4

Furthermore, we will denote Mτ =

" α b

bT C

#

and X =

" # zT Z

(2)

where both C and Z are n × n matrices. Then C = diag(r1 , . . . , rn ) − tZZ T .

Note that it is easy to check that the Schur complement for a given design ξ, and thus the

information matrix, would be the same if the constant term µ was not present in the model (1). It follows that our results hold even in a model without the constant term. In the following sections, we will use a simple technical lemma, which is a well-known result in the exact design literature, see e.g. [1]. Lemma 1. Let ξ be a feasible design for estimating QT τ , where Q = (−1n , In )T . Then NA (ξ) = C. Proof. Since ξ is feasible, rank(Mτ ) ≥ rank(Q) = n. Furthermore, Mτ 1n+1 = diag(r) −
tXX T 1n+1 = r − tX1t /t = r − r = 0n+1 and hence rank(Mτ ) = n. Since C(Q) ⊆ C(Mτ ),

C(In ) ⊆ C(b, C) and thus rank(b, C) = n. It follows that the first row of Mτ is a linear

combination of the remaining n rows of Mτ and thus there exists y ∈ Rn such that b = Cy.

Therefore rank(C) = rank(b, C) = n and

G=

"

0

0Tn

0n

C −1

#

is a generalized inverse of Mτ . It follows that NA (ξ) = (QT GQ)−1 = (C −1 )−1 = C.

3

E-optimality

A design is E-optimal if it maximizes the minimum eigenvalue λmin of NA . The E-optimality criterion has received little attention in the literature on comparing treatments with a control, with a few exceptions, e.g. [10], [9]. The paper [9] advocates for the E-criterion for comparing treatments with a control, particularly in block designs. First, we characterize E-optimal treatment weights, i.e., the total proportions of trials for placebo and the particular doses, following [14]. Lemma 2. Let ξ be an E-optimal design for estimating QT τ , Q = (−1n , In )T , in model (1) without design constraints. Then ξ satisfies r0 (ξ) = 1/2 and ru (ξ) = 1/(2n) for u = 1, . . . , n. Proof. Using Theorems 1 and 6 from [14], with p = −∞ and g = 1, it suffices to prove that the

treatment weights given by Lemma 2 are the only E-optimal treatment weights. That is, we 5

need to prove that there is no treatment proportions design w = (w0 , . . . , wn )T > 0n+1 , other than w∗ = (1/2, 1Tn /(2n))T , satisfying that NQ (w) := (QT diag(w)−1 Q)−1 has its smallest eigenvalue 1/(4n), the optimal value of the E-criterion.


−1 Let w > 0n+1 , then NQ (w) = w0−1 Jn + diag(w1−1 , . . . , wn−1 ) and denote the smallest

−1 eigenvalue of NQ (w) as λmin (w). The largest eigenvalue µmax (w) of NQ (w) satisfies

µmax (w) = max w0−1 uT Jn u + uT diag(w1−1 , . . . , wn−1 )u = max w0−1 (1Tn u)2 + kuk=1

kuk=1

n X

wi−1 u2i .

i=1

P √ For a particular choice of u = 1n / n, we obtain that µmax (w) ≥ w0−1 n + i>0 wi−1 /n. P For given w that does not satisfy w1 = . . . = wn , define w ˜0 = w0 and w ˜i = j>0 wj /n

for i > 0. From the proof of Theorem 6 in [14] it follows that λmin (w) ˜ = w0 (1 − w0 )/n

and thus µmax (w) ˜ = n/[w0 (1 − w0 )] = n/w0 + 1/w˜1 , because nw˜1 = 1 − w0 . From the P arithmetic mean-harmonic mean inequality it follows that i>0 wi−1 /n > 1/w˜1 and therefore

µmax (w) > µmax (w), ˜ i.e., all E-optimal treatment proportions satisfy w1 = . . . = wn . From

the proof of Theorem 6 in [14] it follows that the only E-optimal treatment weights satisfying w1 = . . . = wn are w∗ = (1/2, 1Tn /(2n))T . Note that Lemma 2 holds for any model with nuisance effects as defined in [14].

3.1

Standard Designs

A (standard) design that for each k ∈ {1, . . . , n} satisfies ξ(0, k) = 1/(2n), ξ(k, k) = 1/(2n) and otherwise is zero will be called Senn design (see [2], [15]). That is, in each cohort, half of

the subjects are assigned to the control and half are assigned to the highest allowed dose for that cohort. We show that the Senn designs are the only E-optimal standard designs. u\k

1

2

3

4

0

1/8

1/8

1/8

1/8

1

1/8

0

0

0

2

0

1/8

0

0

3

0

0

1/8

0

4

0

0

0

1/8

Table 1: Senn design for n = 4 doses

6

Theorem 1. A standard design ξ is E-optimal for estimating QT τ , where Q = (−1n , In )T , if and only if ξ is a Senn design. Proof. The Senn design ξ satisfies r = ment is

" 1 1 Mτ = 2 0n " 1 1 = 2 0n

1 2

1, 1Tn /n)T , X =

#

 1 2 −n 2n In /n " # 0Tn 1 n − 4n 1n In /n 0Tn

1 2n

1n , In


T

and its Schur comple-

# i 1Tn h 1n In In # " # 1Tn 1 −1Tn /n 1 = . 4 −1n /n In /n In

"

Note that ξ is connected (see [3]) and therefore feasible. Employing Lemma 1, the information matrix of ξ is NA = In /(4n), with its smallest eigenvalue 1/(4n). From [14], we know that when there are no design constraints, the optimal value of E-criterion is 1/[4(v − 1)], where v

is the number of treatments. Here, v = n + 1 and thus the E-optimal value is 1/(4n), which the Senn design attains. Now, consider the converse part. For a feasible design ξ to be E-optimal, it needs to have the smallest eigenvalue the same as the Senn design, which coincides with the optimal value of E-optimality criterion without design constraints. Thus such ξ is E-optimal in a model without design constraints and from Lemma 2 it follows that ξ needs to satisfy r0 (ξ) = 1/2 and ru (ξ) = 1/(2n), u = 1, . . . , n,

(3)

hence r = (1/2, 1Tn /(2n))T . Observe that C = In /(2n) − nZZ T , as defined in (2). From Lemma 1, NA = C. Then

the smallest eigenvalue λmin of NA satisfies λmin = 1/(2n) − nµmax, where µmax is the largest eigenvalue of ZZ T . Let us denote the columns of matrix Z = (zij ) as z1 , . . . , zn . The largest eigenvalue µmax of ZZ T satisfies µmax = max uT ZZ T u = max kZ T uk2 = max kuk=1

kuk=1

kuk=1

√ and for the particular choice u = 1n / n we obtain

(zjT u)2

j=1

2 1X 2 1X T 2 1 XX zi,j = (1n zj ) = q , n j=1 n j=1 i=1 n j=1 j n

µmax ≥

n X

n

n

n

Pn where qj := i=1 zij , j = 1, . . . , n. From (i) it follows that qj = n1 − ξ(0, j) and from (3), P j qj = 1 − 1/2 = 1/2. Therefore, µmax is not smaller than the optimal value of the following 7

convex optimization problem

n

1X 2 qj q≥0 n j=1

min

n X

qj =

j=1

1 . 2

It can be seen that the single optimal solution q ∗ satisfies q1∗ = . . . = qn∗ = 1/(2n) and then the optimal value is 1/(4n2 ). It follows that if q does not satisfy q1 = . . . = qn = 1/(2n), then µmax > 1/(4n2 ). Therefore, the smallest eigenvalue λmin of NA (ξ) for such ξ satisfies λmin < 1/(4n) and thus ξ is not E-optimal. Hence, any E-optimal design ξ needs to satisfy q1 = . . . = qn = 1/(2n), i.e., ξ(0, 1) = . . . = ξ(0, n) = 1/(2n). From (3) and (i), ξ needs to satisfy ru = 1/(2n) for u > 0 and sk =

P

u

ξ(u, k) = 1/n

for all k. Using (ii), since ξ(n, k) can be nonzero only for k = n, we obtain ξ(n, n) = 1/(2n) and thus ξ(n, k) = 0 for 0 < k < n. Then ξ(n − 1, k) is nonzero only for k = n − 1 and thus

ξ(n − 1, n − 1) = 1/(2n), ξ(n − 1, k) = 0 for 0 < k < n − 1, etc. It follows that the single E-optimal standard design for given n is the Senn design.

3.2

Extended Designs

Now, consider the extended designs, i.e., t = n + 1. We provide the class of all E-optimal extended designs. Since the extra cohort in extended designs provides additional freedom for the experimenter, the class of E-optimal extended designs is richer, compared to the single E-optimal standard design (for given n). The E-optimal extended designs can be completely characterized by the following conditions: the design assigns placebo to half of the subjects in each cohort and it allocates each dose to the same total number of subjects, N/(2n). Note that the E-optimality of standard designs can be described by the same conditions; however, in the standard design case, the Senn designs are the only ones that satisfy these conditions. Theorem 2. An extended design ξ is E-optimal for estimating (−1n , In )τ if and only if it satisfies ξ ∗ (0, k) = 1/(2(n + 1)) for k = 1, . . . , n + 1 and r1 (ξ) = . . . = rn (ξ) = 1/(2n). Proof. Let ξ be a design satisfying conditions of Theorem 2. First, note that ξ is connected 1 1n+1 , (see [3]) and therefore feasible. The partitions of Mτ defined in (2) satisfy z = 2(n+1)
T T T T X1n+1 = r = 1/2, 1n /(2n) , Z1n+1 = 1n /(2n), X 1n+1 = 1n+1 /(n + 1) and Z 1n =

1n+1 /(2(n + 1)).

Using Lemma 1, the information matrix of ξ is NA = In /(2n) − (n + 1)ZZ T . Moreover,

NA 1 n =

1 4n 1n ,

i.e., 1n is an eigenvector of NA with eigenvalue λ1 = 1/(4n), which is, analo-

gously to the proof of Theorem 1, equal to the optimal smallest eigenvalue in a model without 8

constraints, see [14]. Now, it is sufficient to prove that λ1 is the smallest eigenvalue. Denote the smallest eigenvalue of NA as λmin . Then λmin = 1/(2n) − (n + 1)µmax , where µmax is the

largest eigenvalue of ZZ T .
1 ˆ n+1 = 1n+1 and 1Tn+1 Zˆ = 1Tn+1 , i.e., Let Zˆ = 2nZ T , n+1 1n . The matrix Zˆ satisfies Z1 Zˆ is doubly stochastic. Using the Birkhoff-von Neumann theorem we obtain that Zˆ =

X

απ Pπ ,

π∈Pn+1

where απ ≥ 0 for all π,

P

π

απ = 1, Pn+1 denotes the set of all permutations of n + 1

elements and Pπ is the permutation matrix given by π. Let us partition the matrices Pπ as
Pπ = P˜π , vπ , where P˜π is an (n + 1) × n matrix. Then, X

2nZ T =

απ P˜π ,

thus

Z=

π∈Pn+1

1 2n

X

απ P˜πT .

π∈Pn+1

(i)

Let Pn+1 be the set of all permutations π of n + 1 elements, whose matrices satisfy Pπ have their last column equal to ei , i.e., π(n + 1) = i. Then, because the last column of Zˆ is P 1n+1 /(n + 1), we have π∈P(i) απ = 1/(n + 1). n+1

The largest eigenvalue µmax of ZZ T is the same as the largest eigenvalue of Z T Z and thus

it satisfies µmax = max uT Z T Zu = max kZuk2 = kuk=1

=

≤

1 4n2

kuk=1

2

n+1

X X απ P˜πT u max

kuk=1

2

X 1

˜ T u P α max

π π 4n2 kuk=1 π∈Pn+1

i=1 π∈P(i) n+1

n+1

1

X 1 ˜ T 2 u max max P

. (n+1) 4n2 kuk=1 π1 ∈P(1) n + 1 πi n+1 ,...,πn+1 ∈Pn+1 i=1

Pn+1 (1) (n+1) Consider π1 ∈ Pn+1 , . . . , πn+1 ∈ Pn+1 and u ∈ Rn+1 , kuk = 1, that maximize k i=1 P˜πTi uk2 , and let us denote vi = (vi,1 , . . . , vi,n )T := P˜ T u, i = 1, . . . , n + 1. Then µmax satisfies πi

µmax ≤ =

n+1 n  n+1 n 2 X X X X 1 1 2 vi,j = ≤ v (n + 1) i,j 2 2 2 2 4n (n + 1) j=1 i=1 4n (n + 1) j=1 i=1 n+1 n XX 1 v2 , 4n2 (n + 1) i=1 j=1 i,j

where the second inequality follows from the Cauchy-Schwarz inequality. That is, µmax ≤ 1 4n2 (n+1) S,

where S is the sum of squares of all elements of vectors v1 , . . . , vn+1 . For each i ∈ 9

{1, . . . , n + 1}, the vector vi = P˜πTi u consists of all elements of u except ui , therefore, u2i occurs n-times in S. Hence, S = nu21 + . . . nu2n+1 = nkuk2 = n and it follows that µmax ≤ Since µmax ≤

1 4n(n+1) ,

1 4n(n+1) .

1 the smallest eigenvalue of NA is at least 1/(2n)−(n+1) 4n(n+1) = 1/(4n)

and thus λ1 is indeed the smallest eigenvalue of NA . To prove the converse part, note that, similarly to Theorem 1, for ξ to attain the optimal value of E-criterion, 1/(4n), it needs to satisfy Lemma 2 ru (ξ) =

1 for u = 1, . . . , n 2n

and r0 (ξ) =

1 . 2

(4)

Let ξ be a design that satisfies (4) but does not satisfy ξ(0, k) =

1 2(n + 1)

for k = 1, . . . , n + 1.

Let qj := 1/(n+1)−ξ(0, k), k = 1, . . . , n+1, hence

P

j

(5)

qj = 1−1/2 = 1/2. Then, analogously

to the proof of Theorem 1, the largest eigenvalue µmax of ZZ T is not smaller than the optimal value of the following convex optimization problem n+1 1X 2 qj min q≥0 n j=1 n+1 X

qj =

j=1

1 , 2

∗ which has a single optimal solution q1∗ = . . . = qn+1 = 1 4n(n+1) .

1 2(n+1) ,

It follows that if q does not satisfy q1 = . . . = qn+1 =

and the optimal value is

1 2(n+1) ,

then µmax >

1 4n(n+1) .

Therefore, the smallest eigenvalue λmin of NA for such ξ satisfies λmin < 1/(4n) and thus ξ is not E-optimal.

The complete characterization of E-optimal designs allows for choosing the best E-optimal design with respect to secondary criteria, such as other optimality criteria (e.g. the well known criteria of D- and A-optimality, for their definitions see [12]) or a desirable form of the design. The standard Senn designs can be extended to obtain E-optimal extended designs. We say that a uniformly extended Senn design is a design that satisfies ξ(k, k) = 1/(2t) for k = 1, . . . , n, ξ(0, k) = 1/(2t) for k = 1, . . . , n + 1 and ξ(u, n + 1) = 1/(2nt) for u = 1, . . . , n. Such design is given by the standard Senn design in the first n cohorts. In the last cohort, half of the subjects are assigned to the control and the other half are uniformly distributed among all other treatments. The uniformly extended Senn design obviously satisfies the conditions of Theorem 2 and thus is E-optimal. 10

u\k

1

2

3

4

5

0

1/10

1/10

1/10

1/10

1/10

1

1/10

0

0

0

1/40

2

0

1/10

0

0

1/40

3

0

0

1/10

0

1/40

4

0

0

0

1/10

1/40

Table 2: Uniformly extended Senn design for n = 4 doses

An interesting consequence of the linearity of the conditions determining E-optimal extended designs is that we can employ an efficient convex optimization method from [5] to construct A-optimal or D-optimal designs within the class of all E-optimal designs. Example 1. Let n = 4. Then the A-optimal and D-optimal designs in the class of all E-optimal extended designs are given in Tables 3 and 4, respectively. u\k

1

2

3

4

5

0

0.1000

0.1000

0.1000

0.1000

0.1000

1

0.1000

0.0219

0.0031

0.0000

0.0000

2

0

0.0781

0.0287

0.0091

0.0091

3

0

0

0.0682

0.0284

0.0284

4

0

0

0

0.0625

0.0625

Table 3: A-optimal design in the class of all E-optimal extended designs

u\k

1

2

3

4

5

0

0.1000

0.1000

0.1000

0.1000

0.1000

1

0.1000

0.0248

0.0002

0.0000

0.0000

2

0

0.0752

0.0339

0.0079

0.0079

3

0

0

0.0659

0.0296

0.0296

4

0

0

0

0.0625

0.0625

Table 4: D-optimal design in the class of all E-optimal extended designs

11

3.3

Interpretation of E-optimality for Dose-Escalation

Using the relation between E-optimality and c-optimality (see [13], [12]), we may formulate a statistical interpretation of E-optimality in our model. First, we provide Theorem 7.21 from [12] (the General Equivalence Theorem for E-optimality) and Theorem 7.23 from [12]. Theorem 3. A feasible design ξ with its moment matrix M and information matrix NA is E-optimal for estimating AT β if and only if there exist a generalized inverse G of M and a nonnegative definite matrix E, tr(E) = 1, such that T T ˜ ˜ tr(M (ξ)GAN A ENA A G ) ≤ λmin (NA ) for all ξ ∈ Ξ,

(6)

Theorem 4. Let ξ be a feasible design for AT β with its information matrix NA and let h, khk = 1, be an eigenvector of NA corresponding to the smallest eigenvalue of NA . Then ξ is

E-optimal for AT β and E = hhT satisfies (6) if and only if ξ is c-optimal, where c = Ah. If

the smallest eigenvalue of NA has multiplicity 1, ξ is E-optimal for AT β if and only if it is c-optimal, where c = Ah. From Theorem 4, we can characterize E-optimality using c-optimality in the general model with nuisance effects, without design constraints, considered in [14]. Lemma 3. A design ξ in model (1) in [14] is E-optimal for comparing test treatments with a control if and only if it is c-optimal, where c = (−1, 1Tn /n, 0Tn+1 )T . Proof. Consider the set of treatments {0, . . . , n}. From Theorems 2 and 6 in [14] it follows that

there exists an E-optimal (product) design ξ for estimating QT τ , where QT = (−1n , In ) with its information matrix NA = (QT (diag(w))−1 Q)−1 , where w0 = 1/2 and w1 = . . . = wn = 1/(2n). Then NA = (2nIn + 2Jn )−1 =

1 2n In

with multiplicity 1. Observing that NA 1n =

−

1 4n2 Jn

1 4n 1n

with its smallest eigenvalue λmin =

1 4n

and employing Theorem 4 completes the

proof. We denote the vector obtained in Lemma 3 as c˜ := (−1, 1Tn /n, 0Tn+1 )T and the corresponding c-optimality as c˜-optimality. Now, we may formulate the relation between E-optimality and c-optimality in dose-escalation studies. Theorem 5. A standard or extended design ξ is E-optimal in model (1) for estimating QT τ , Q = (−1n , In )T , if and only if it is c˜-optimal, where c˜ = (−1, 1Tn /n, 0Tn+1 )T . Proof. After some calculations, it can be observed that the (standard) Senn design ξS satisfies (6) with NA = In /(4n), G=

"

− −Mτ− M12 M22

Mτ−

− T −M22 M12 Mτ−

− − T M22 M12 Mτ− M12 M22

12

#

,

(7)

where Mτ−

"

0Tn

0

#

− M22

"

0Tn

0

#

, and = 0n nIn 4nIn √ and E = 1n 1Tn /n = hhT , where h = 1n / n. The matrix G is indeed a generalized inverse of =

0n

M , see Theorem 9.6.1 from [6]. From Theorem 4 it follows that any E-optimal standard design √ (i.e., Senn design) is also c-optimal, where c = A1n / n, which is equivalent to A1n /n = c˜. Note that since ξS is E-optimal also in the model without design constraints, it is c˜-optimal in the model without design constraints, using Lemma 3. Let ξ be a c˜-optimal standard design satisfying the conditions (i), (ii). Then it attains the same value of c˜-optimality criterion as ξS and thus ξ is c˜-optimal in the model without design constraints. Employing Lemma 3, ξ is E-optimal even in the model without design constraints. It follows that ξ is E-optimal among all designs satisfying (i), (ii). In the case of extended designs, the information matrix of an Extended Senn design ξ smallest eigenvalue λmin =

1 4n

n+2 4n(n+1) In

1 4n2 (n+1) Jn ,

with its √ with multiplicity 1, and the corresponding eigenvector 1n / n.

(and as such, an E-optimal information matrix) is NA (ξ) =

−

Thus an extended design is E-optimal if and only if it is c˜-optimal.

The c˜-optimality means that the optimal design ξ ∗ minimizes the variance of the least Pn squares estimator of i=1 τi /n − τ0 , i.e., n \  1 X τi − τ0 → min . Var n i=1

(8)

It follows that a standard or extended design is E-optimal for comparing doses with placebo if and only if it minimizes the variance of the average dose effect compared with placebo or the average of the dose effects relative to placebo, cf. Theorem 3.2 in [10]. In [10], the author obtained a class of E-optimal exact block designs, which he showed to be c˜-optimal. Note that this interpretation of E-optimality holds even in the model without design constraints and with general nuisance effects (model (1) in [14]), due to Lemma 3. In contrast, the A-optimality criterion, which is often used in treatment-control experiments, minimizes the average variance of the least squares estimators of the contrasts of interest. That is, it minimizes the average variance of the estimators of the dose effects compared with the placebo effect, which can be expressed as n

1X Var(τ\ i − τ0 ) → min, n i=1 compare with (8) 13

4

LV -optimality

Senn suggested in commentary [17] on [2] that what he calls the latest variance (LV ) criterion may be more appropriate to consider, rather than the traditional optimality criteria. The reasoning is that after each cohort, the experimenter decides whether or not to continue with the experiment, and this decision is based on the results obtained so far (the effect of the latest dose relative to placebo). Therefore, crucial in the experiment is the variance of the comparison of the latest dose with placebo, rather than all variances based on trials which may not even be performed (if the experiment stops before all doses were tried). We denote by Vark (η) the variance of η based on the trials in the first k cohorts. For example, Vark (τ\ 1 − τ0 ) is the variance of the estimator of τ1 − τ0 based on the first k cohorts. Then, an LV -optimal design ξ for comparison with control minimizes Var1 (τ\ 1 − τ0 ), . . . , Varn (τ\ n − τ0 ) among all feasible designs that satisfy the given constraints. Note that by a

feasible design, we mean here a design under which τ1 −τ0 is estimable after one cohort, τ2 −τ0

is estimable after first two cohorts etc. We further remark that the LV -optimality criterion is not defined for a general design problem, it is strongly tied to the particular properties of dose-escalation studies. Note that the criterion of LV -optimality seems to call for simultaneous multicriterial optimization (minimizing n variances) which in general does not need to have a solution. However, we will show that in our model the Senn designs do have this strong property, i.e., no other design has any of the variances in question lower. d T τ ) is proportional to V (ξ) := QT M − (ξ)Q. More precisely, The variance matrix Varξ (Q τ σ2 σ2 d T τ) = Varξ (Q V (ξ) = V (ξ), N tm

where N is the total number of observations and m is the number of subjects in each cohort. The variances calculated from the first k cohorts, Vark , can be obtained from V (ξ), when setting ξ(u, j) = 0 for all u and j > k, i.e., when the trials in cohorts j > k are ignored. This (k)

condition implies that the stage-k moment matrix M (k) satisfies M11 = diag(r1 , . . . , rk , 0Tn−k ), 

(k)

M12

   r1    .   ..  =   rk    0n−k

 ξ(0, 1) ξ(0, 2) . . .  ξ(1, 1) ξ(1, 2) . . .   0 ξ(2, 2) . . .   . ..  . .  . 0

0





 ξ(1, k)   ξ(2, k)   ..   .  . . . ξ(k, k)

0n−k×k

14

ξ(0, k)

0k×t−k

0n−k×t−k

          

and



1Tk /n

1

 (k) M22 =  1k /n Ik /n 0t−k 0t−k×k

0Tt−k



 0k×t−k  . 0t−k×t−k

T T T (k) Then the latest variance Vark (τ\ k − τ0 ) is proportional to dk (ξ) := (ek+1 −e1 , 0t+1 ) M

eT1 , 0Tt+1 )T , where e1 and ek+1 are elementary unit vectors of length n + 1.


−

(eTk+1 −

Theorem 6. Let ξ be a Senn design. Then ξ is LV -optimal for comparison with control among all standard designs. Proof. Let ξ be a design and let k ∈ {1, . . . , n}. For calculating dk (ξ), we disregard cohorts

k + 1, . . . , n and we are not allowed to use doses k + 1, . . . , n in the first k cohorts. Such model coincides with model (1), where n = k and a design ξ ′ given by ξ for doses 0, . . . , k

and cohorts 1, . . . , k. Then the latest variance under ξ is proportional to the inverse of the
−1 T value of the c-optimality criterion for ξ ′ , Φc (ξ ′ ) = cT M − (ξ ′ )c , c = (eTn+1 − eT1 , 0Tn+1 ).

Therefore, without loss of generality, we may assume that k = n and prove that the Senn design has the highest value of c-optimality criterion, where cT = (eTn+1 − eT1 , 0Tn+1 ).

The General Equivalence Theorem in the case of c-optimality becomes (Corollary 5.1 from

[11]): The moment matrix M ∈ M is c-optimal in M if and only if there exists a generalized inverse G of M , such that cT GT BGc ≤ cT M − c for all B ∈ M.

Now, let ξ be a Senn design, let c = (eTn+1 − eT1 , 0Tn+1 )T and let us denote M := M (ξ).
T 1 Then r = 12 1, 1Tn /n)T , X = 2n 1n , In , " n 1 Mτ = 4n −1n

−1Tn In

#

, and let

Mτ−

=

"

0

0Tn

0n

4nIn

#

,

which is indeed a generalized inverse of Mτ . Thus, cT M − c = (en+1 −e1 )T Mτ− (en+1 −e1 ) = 4n. Let G be given by (7), where # " " 0 0Tn 0 − − T − M22 = and − M22 M12 Mτ = 0n nIn 0n It follows that



0

 0n Gc =  0  0n

0Tn 4nIn 0Tn −2nIn



−1





0Tn −2nIn 0n



G12       0n−1   4n  =    1   0    n   G22 0n+1 −2n

and therefore, for any feasible design ξ ′ satisfying conditions (i), (ii), 15

#

.

T

T

′

c G M (ξ )Gc =

h

0Tn

4n



i  −2n 

0Tn

diag(r ′ )

r′ X ′

r′ T X′T

1 1T n /n 1n /n In /n





0n



  4n        0n  −2n

1 + 2 × 4n(−2n)ξ ′ (n, n) = 4n, n because ξ ′ (n, n) = rn′ . Therefore, any design satisfying conditions (i), (ii) satisfies the desired = (4n)2 rn′ + (−2n)2

inequality 4n ≤ cT M − c = 4n. Hence, for any k, the Senn design has the minimum possible

latest variance.

In the case of extended designs, the results are similar. It turns out that the LV-optimal design is the extended Senn design ξ constructed as follows: ξ is given by the Senn design in the first n cohorts and its (n + 1)-st cohort is a replication of the n-th cohort (in the sense of treatment proportions). Formally, ξ(0, k) = ξ(k, k) = 1/(2(n + 1)) for k = 1, . . . , n, ξ(0, n + 1) = ξ(n, n + 1) = 1/(2(n + 1)) and ξ(u, k) = 0 otherwise. We will call such design the highest-dose extended Senn design. u\k

1

2

3

4

5

0

1/10

1/10

1/10

1/10

1/10

1

1/10

0

0

0

0

2

0

1/10

0

0

0

3

0

0

1/10

0

0

4

0

0

0

1/10

1/10

Table 5: Highest-dose extended Senn design for n = 4 doses

Note that the extended designs seem to be less meaningful when considering LV -optimality, as the additional cohort may be used only to improve the estimate of the effect of the highest dose compared with placebo. Theorem 7. Let ξ be a highest-dose extended Senn design. Then ξ is LV -optimal for comparison with control among all extended designs. Proof. For the first n − 1 latest variances, the argument is the same as in the standard design

case, see the proof of Theorem 6. The n-th latest variance depends on the last two cohorts. 16

Optimality of ξ can be proved by using the General Equivalence Theorem for c-optimality, c = en+1 − e1 .

Design ξ satisfies

X=

Then

and



1Tn−1

 1 I 2(n + 1)  n−1 0Tn−1 



1T2

 0(n−1)×2   1T2

1/4

 1 Mτ =  − 4(n+1) 1n−1 1 − 2(n+1)





1 n+1

1 − 2(n+1)

0

0Tn−1 2In−1 0Tn−1



 0n−1  

1 4(n+1) In−1 0Tn−1

 Mτ− = 2(n + 1)  0n−1



  1  and r =   2(n+1) 1n−1  .

1 1Tn−1 − 4(n+1)

0

1 2

1 2(n+1)

0



 0n−1   1

is a generalized inverse of Mτ . Hence, cT M − c = 2(n + 1). Furthermore, # " 0 0Tn+1 − M22 = 0n+1 (n + 1)In+1 is a generalized inverse of M22 . By choosing the generalized inverse G of M given by (7), after some calculations similar to the proof of Theorem 6, the normality inequality becomes cT GT M (ξ ′ )Gc = 2(n + 1) ≤ cT M − c = 2(n + 1) for any design ξ ′ satisfying (i), (ii).

5

MV -optimality

A widely used optimality criterion in designing experiments for estimating a system of contrasts is M V -optimality. A design is M V -optimal if it minimizes the maximum variance of the estimators of the treatment contrasts. That is,   \ T β) , Ψ(ξ) = max Varξ (A i i

and an M V -optimal design minimizes Ψ(ξ) among all ξ ∈ Ξ.

Consider the standard designs for comparison with control, i.e., t = n. Then, same as in

the LV -optimal case, the Senn design is optimal.

17

Theorem 8. Let ξ be a Senn design. Then ξ is M V -optimal among all standard designs for estimating QT τ , where Q = (−1n , In )T . Proof. Let ξ be a standard design and let us denote Vari (ξ) := Varξ (τ\ i − τ0 ), i = 1, . . . , n.

Then Ψ(ξ) = maxi Vari (ξ). Theorem 6 states that the Senn design ξS is LV -optimal, i.e., dk (ξ) ≥ dk (ξS ) for all k = 1, . . . , n. In the case k = n, the latest variance and the final variance (the variance when the entire design is carried out) coincide. It follows that Varn (ξ) ≥ Varn (ξS ).

In the proof of Theorem 1, we may observe that QT Mτ− (ξS )Q = 4nIn . It follows that Vari (ξS ) ≈ 4n for i = 1, . . . , n. To be exact, Vari (ξS ) = 4σ 2 /m for all i, where m is the number

of subjects in each cohort, and thus Ψ(ξS ) = 4σ 2 /m. Since Varn (ξ) ≥ Varn (ξS ) = 4σ 2 /m, we

obtain Ψ(ξ) ≥ 4σ 2 /m = Ψ(ξS ).

It can be numerically demonstrated that in the extended design case, the M V -optimal designs do not enjoy the regular structure, as observed for the standard designs. The M V optimal extended designs even tend to allocate less patients to placebo and more patients to the actual drug, compared with the standard designs. The extra cohort provides enough freedom, so that the M V -optimal designs behave similarly to the A-optimal designs (as is expected, see e.g. Theorem 7 in [14]), i.e., having total treatment proportions similar to A-optimal designs and having a more complex structure, which we are unable to describe analytically. We have proved the optimality of Senn designs with respect to multiple optimality criteria. Another desirable property of these designs is their extremely simple and regular structure, which makes for simple (i) construction of exact designs from approximate Senn designs, and (ii) characterization of these designs for practical use. We obtained an infinite class of E-optimal extended designs, characterized by simple conditions, from which the experimenters can choose the most suitable one, according to some secondary criteria. Furthermore, we provided a statistical interpretation of the E-optimality criterion in the dose-escalation studies.

References [1] Bechhofer, R. E., Tamhane, A. C. (1981): Incomplete Block Designs for Comparing Treatments With a Control: General Theory, Technometrics 23(1), pp. 45-57. [2] Bailey, R. A. (2009): Designs for dose–escalation trials with quantitative responses, Statistics in Medicine 28, pp. 3721–3738 18

[3] Eccleston, J. A., Hedayat, A. (1974): On the theory of connected designs: characterization and optimality, The Annals of Statistics 2(6), pp. 1238-1255. [4] Haines, L. M., Clark, A. E. (2014): The construction of optimal designs for doseescalation studies, Statistics and Computing 24(1), pp. 101-109. [5] Harman, R., Sagnol, G. (2015): Computing D-Optimal Experimental Designs for Estimating Treatment Contrasts Under the Presence of a Nuisance Time Trend, Stochastic Models, Statistics and Their Applications, Springer Proceedings in Mathematics & Statistics 122, pp. 83-91 [6] Harville, D. A. (1997): Matrix Algebra From A Statiscian’s Perspective, Springer-Verlag, New York [7] Kunert, J., Martin R. J., Eccleston J. (2010): Optimal block designs comparing treatments with a control when the errors are correlated, Journal of Statistical Planning and Inference 140, pp. 2719-2738 [8] Majumdar, D.: Optimal and efficient treatment-control designs, Handbook of statistics 13: Design and Analysis of Experiments (1996), pp. 1007-1053. [9] Morgan, J. P., Wang, X. (2011): E-optimality in treatment versus control experiments, Journal of Statistical Theory and Practice 5(1), pp. 99-107 [10] Notz, W. I. (1985): Optimal designs for treatment—control comparisons in the presence of two-way heterogeneity, Journal of Statistical Planning and Inference 12, pp. 61-73. [11] Pukelsheim, F. (1980): On linear regression designs which maximize information, Journal of Statistical Planning and Inference 4(4), pp. 339-364. [12] Pukelsheim, F. (2006): Optimal design of experiments, Classics in Applied Mathematics, SIAM [13] Pukelsheim, F., Studden, W. J. (1993): E-optimal designs for polynomial regression, The Annals of Statistics 21(1), pp. 402-415. [14] Rosa, S., Harman R. (2015): Optimal approximate designs for estimating treatment contrasts resistant to nuisance effects, available on: arXiv:1504.06079 [math.ST] [15] Senn S. J. (1997): Statistical Issues in Drug Development, Wiley, Chichester

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[16] Senn, S., Amin, D., Bailey, R. A., Bird, S. M., Bogacka, B., Colman, P., Garrett, A., Grieve, A., Lachmann, P. (2007): Statistical issues in first-in-man studies, Journal of the Royal Statistical Society: Series A 170(3), pp. 517-579 [17] Senn, S. (2009): Commentary on ‘Designs for dose–escalation trials with quantitative responses’, Statistics in Medicine 28, pp. 3754–3758

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