[2]footnote

# On anticommutative algebras for which $[R_{a},R_{b}]$ is a derivation

Ivan Kaygorodov  Federal University of ABC, Brazil<EMAIL_ADDRESS>and  Pasha Zusmanovich  University of Ostrava, Czech Republic
<EMAIL_ADDRESS>
(Date: First written December 1, 2020; last revised January 17, 2021)

###### Abstract.

We study anticommutative algebras with the property that commutator of any two
multiplications is a derivation.

###### Key words and phrases:

Commutator; derivation; anticommutative algebra; Lie algebra; cohomology;
central extension

###### 2020 Mathematics Subject Classification:

17A30; 17A36; 17B40; 17B55

J. Geom. Phys., to appear

## Introduction

Consider the following property of a (nonassociative) algebra: the commutator
of two (say, right) multiplications is a derivation of an algebra. Commutative
algebras with this property were studied in the literature under the names
“Lie triple algebras” and “almost Jordan algebras”: see [B], [Dz], [HP], [JR],
[Si], and references therein. Jordan algebras are properly contained in this
class.

It appears only natural to consider then the anticommutative analog: that is,
anticommutative algebras in which the commutator of any two multiplications is
a derivation. Such algebras are dubbed, for no better term, as CD algebras. It
turns out that the variety of CD algebras lies between Lie algebras and binary
Lie algebras, both inclusions being strict, so it seems a class of algebras
worth studying.

Earlier, low-dimensional nilpotent CD algebras were classified in [ACK],
[ACF], [KK1], and [KK2], and here we continue the study of CD algebras.

##  1\. Notation, conventions, preliminary facts

We consider (nonassociative) algebras over the ground field $K$, which is
assumed to be arbitrary of characteristic $\neq 2,3$. _All algebras and
varieties of algebras are assumed to be anticommutative, without explicitly
mentioning it_. Thus, left and right multiplications differ only by sign, left
and right ideals coincide, etc.

The multiplication in algebras will be always denoted by juxtaposition, with
the exception of §5, where we deal with Lie algebras with multiplication
denoted customarily by brackets $[\,\cdot\,,\cdot\,]$.

Let $A$ be an algebra. For an element $a\in A$, $R_{a}:A\to A$ denotes the
linear map of right multiplication on $a$: $R_{a}(x)=xa$.
$\operatorname{Der}(A)$ denotes the Lie algebra of derivations of $A$, and
$\operatorname{\mathsf{gl}}(A)$ denotes the Lie algebra of all linear maps
$A\to A$, subject to the usual commutator $[f,g]=g\circ f-f\circ g$ (we always
assume action from the right, so $g\circ f(a)=g(f(a))$). For any algebra $A$,
any $a\in A$, and any $D\in\operatorname{Der}(A)$, the following identity in
$\operatorname{\mathsf{gl}}(A)$ holds:

(1) $[D,R_{a}]=R_{D(a)}.$

$Z(A)$ denotes the center of $A$, i.e., the set of elements $z\in A$ such that
$zA=Az=0$; this is obviously an ideal of $A$.

The _Jacobiator_ $J(x,y,z)$ of elements $x,y,z\in A$ is defined as

$J(x,y,z)=(xy)z+(zx)y+(yz)x.$

Thus, an algebra is a Lie algebra if and only if the Jacobiator on it is
identically zero.

An algebra $A$ is called a _CD algebra_ if it satisfies the property that for
any $a,b\in A$, the commutator $[R_{a},R_{b}]$ is a derivation of $A$. This
condition can be written as a homogeneous identity of degree $4$ comprising
$6$ monomials:

(2) $((xy)a)b-((xy)b)a-((xa)b)y+((xb)a)y+((ya)b)x-((yb)a)x=0.$

##  2\. Identities and non-identities

Let us compare the variety of CD algebras with the other known varieties: Lie,
binary Lie, Malcev, and Sagle (for binary Lie and Malcev algebras, see, for
example, [Sa1], [Sa2], [G], [AS], [ACK] and references therein, and for Sagle
algebras, see [F] and references therein). Let us briefly recall their
definitions. An algebra is called _binary Lie_ if it satisfies the identity

$J(xy,x,y)=0.$

Taking into account anticommutativity, the latter identity is equivalent to

(3) $((xy)x)y=((xy)y)x.$

This is also equivalent to the condition that any $2$-generated subalgebra is
Lie.

An algebra is called _Malcev_ , if it satisfies the identity

$J(x,y,xz)=J(x,y,z)x;$

and an algebra is called _Sagle_ , if it satisfies the identity

(4) $J(x,y,z)w=J(w,z,xy)+J(w,y,zx)+J(w,x,yz).$

There are the following well known strict inclusions between these varieties:

$\begin{array}[]{rcl}&&\text{Malcev}\>\subset\>\text{ Binary Lie}\\\
&\rotatebox{45.0}{$\subset$}&\\\ \text{Lie}&\\\
&\rotatebox{-45.0}{$\subset$}&\\\ &&\text{Sagle}\end{array}$

How CD algebras fit into the picture? We are going to prove that

(5) $\begin{array}[]{rcl}&&\text{Binary Lie}\\\
&\rotatebox{45.0}{$\subset$}\\\
\text{Lie}\>\subset\>\text{Malcev}\cap\text{Sagle}\>\subset\>\text{CD}\\\
&\rotatebox{-45.0}{$\subset$}\\\ &&\text{Almost Lie}\end{array}$

where all inclusions are, again, strict, and, moreover,

(6) $\begin{gathered}\text{\rm Malcev}\>\cap\>\text{\rm CD}=\text{\rm
Sagle}\>\cap\>\text{\rm CD}=\text{\rm Malcev}\>\cap\>\text{\rm Sagle}\\\
\text{\rm Binary Lie}\>\cap\>\text{\rm Almost Lie}=\text{\rm
CD}\end{gathered}$

(the graphically inclined reader may wish to draw Venn diagrams representing
all this).

Here the variety of “Almost Lie” (anticommutative) algebras is defined by the
identity*** There are other meanings of “almost Lie” one can encounter in the
literature, but since neither of them seems to be widespread and accepted, and
since almost Lie algebras in our sense are already mentioned in a number of
related papers – for example [KK1] and [KK2] – we choose to keep this
terminology.

(7) $J(x,y,z)w=0.$

######  Lemma 1.

The variety Malcev $\>\cap\>$ Sagle coincides with the variety of almost Lie
algebras which additionally satisfy the identity

(8) $J(x,y,zw)=0.$

###### Proof.

As proved in [Sa1, Lemma 2.10], any Malcev algebra satisfies the identity

(9) $3J(y,z,wx)=J(x,y,z)w-J(y,z,w)x-2J(z,w,x)y+2J(w,x,y)z.$

Expressing via this identity all the summands on the right-hand side of (4),
which are of the form
$J(\vbox{\hbox{\tiny$\>\bullet\>$}},\vbox{\hbox{\tiny$\>\bullet\>$}},\vbox{\hbox{\tiny$\>\bullet\>$}}\vbox{\hbox{\tiny$\>\bullet\>$}})$,
through terms of the form
$J(\vbox{\hbox{\tiny$\>\bullet\>$}},\vbox{\hbox{\tiny$\>\bullet\>$}},\vbox{\hbox{\tiny$\>\bullet\>$}})\vbox{\hbox{\tiny$\>\bullet\>$}}$,
we get the identity (7), i.e., any algebra which is both Malcev and Sagle, is
almost Lie. Then (9) implies that any algebra which is both Malcev and Sagle,
satisfies also the identity (8).

Conversely, any algebra satisfying both (7) and (8) is, obviously, both Malcev
and Sagle. ∎

######  Lemma 2.

Binary Lie $\>\cap\>$ Almost Lie $\subseteq$ CD.

###### Proof.

As proved in [Sa2, §3], any binary Lie algebra satisfies the identity

$3(J(wx,y,z)+J(yz,w,x))=-J(x,y,z)w+J(y,z,w)x-J(z,w,x)y+J(w,x,y)z.$

If the algebra is simultaneously almost Lie, the right-hand side of this
identity vanishes, and we are left with the identity

(10) $J(wx,y,z)+J(yz,w,x)=0.$

Using the last identity and anticommutativity, we get

(11) $((xz)w)y+((yw)z)x\\\
=J(xz,w,y)+((xz)y)w-(wy)(xz)+J(yw,z,x)+((yw)x)z-(zx)(yw)\\\
=((xz)y)w+((yw)x)z$

for any elements $x,y,z,w$ of an algebra which is simultaneously binary Lie
and almost Lie.

Now transform the left-hand side of (2):

$((xy)a)b-((xy)b)a-((xa)b)y+((xb)a)y+((ya)b)x-((yb)a)x\\\
=((xy)a)b-((xy)b)a-((xa)y)b-((yb)x)a+((xb)y)a+((ya)x)b\\\
=J(x,y,a)b-J(x,y,b)a=0,$

where the first equality is obtained by applying the identity (11) twice, to
the pairs formed by the 3rd and 6th summands, and by the 4th and 5th summands.
∎

######  Proposition 1.

All inclusions in the diagram (5) do indeed take place.

###### Proof.

“Lie $\subset$ Malcev $\cap$ Sagle”: Obvious. To see that the inclusion is
strict (we do not need this in what follows, but doing this for completeness),
consider the free anticommutative algebra freely generated by elements
$x,y,z$, such that any product of any $4$ elements vanishes. This
$9$-dimensional nilpotent algebra satisfies any identity of degree $4$, in
particular, it is both Malcev and Sagle, but, obviously, not Lie, as
$J(x,y,z)\neq 0$.

“Malcev $\cap$ Sagle $\subset$ CD”: By Lemma 1,

$\text{\rm Malcev}\>\cap\>\text{\rm Sagle}\subseteq\text{\rm Binary
Lie}\>\cap\>\text{\rm Almost Lie},$

and then apply Lemma 2.

To show that the inclusion is strict, on can take, for example, the one-
parametric family of nilpotent algebras ${\bf B}_{6,1}^{\alpha}$ from [ACK]
(see Theorems 3 and 10 there). These are $6$-dimensional algebras with the
basis $\\{e_{i}\\}_{i=1,\dots,6}$ and multiplication table

$e_{1}e_{2}=e_{4},\quad e_{1}e_{3}=e_{5},\quad e_{2}e_{3}=\alpha e_{6},\quad
e_{4}e_{5}=e_{6},$

where $\alpha\in K$; these algebras are CD, but not Malcev.

“CD $\subset$ Binary Lie”: As noted in [ACK], substituting $a=x$ and $b=y$ in
(2) yields (3). It is not difficult to find examples of binary Lie algebras
which are not CD; for nilpotent such algebras, see, again, [ACK]. Another nice
example is a $7$-dimensional simple Malcev algebra (what follows from
Proposition 3 below).

“CD $\subset$ Almost Lie”: Let $A$ be a CD algebra. Write the Jacobi identity
for elements of $\operatorname{\mathsf{gl}}(A)$:

(12) $[[R_{x},R_{y}],R_{z}]+[[R_{z},R_{x}],R_{y}]+[[R_{y},R_{z}],R_{x}]=0$

for any $a,b,c\in A$.

Write the identity (1) for the derivation $[R_{x},R_{y}]$:

$[[R_{x},R_{y}],R_{z}]=R_{[R_{x},R_{y}](z)}$

for any $x,y,z\in A$. Taking the last identity into account, the identity (12)
can be rewritten as

$[R_{x},R_{y}](z)+[R_{z},R_{x}](y)+[R_{y},R_{z}](x)\in Z(A).$

Due to anticommutativity of $A$ and the fact that the characteristic of the
ground field is different from $2$, the left-hand side in the last inclusion
is nothing but the Jacobiator $J(x,y,z)$, whence the statement.

Examples of almost Lie algebras which are not CD can be constructed by
considering central extensions of Lie algebras in the suitable variety; this
is deferred to §5††† An alternative method would be to use Albert [A] to
construct explicitly the multiplication table of a suitable finite-dimensional
homomorphic image of a free almost Lie algebra, and then verify in some other
general-purpose computer algebra system like GAP, that this homomorphic image
is not CD. A similar procedure – for another set of identities – is described
more thoroughly in [Z, §4]. . ∎

######  Proposition 2.

All equalities in (6) do indeed take place.

###### Proof.

“Binary Lie $\>\cap\>$ Almost Lie = CD”: The inclusion “CD $\subseteq$ Binary
Lie $\>\cap\>$ Almost Lie” follows from (5), and the inverse inclusion is
proved in Lemma 2.

“Malcev $\>\cap\>$ CD = Malcev $\>\cap\>$ Sagle”: By Lemma 1, the variety
Malcev $\>\cap\>$ Sagle coincides with the variety of almost Lie algebras
which, additionally, satisfy the identity (8).

On the other hand, by just proved,

$\text{\rm Malcev}\>\cap\>\text{\rm CD}=\text{\rm Malcev}\>\cap\>\text{\rm
Binary Lie}\>\cap\>\text{\rm Almost Lie}=\text{\rm Malcev}\>\cap\>\text{\rm
Almost Lie}.$

But by (9), any Malcev algebra which is simultaneously almost Lie, satisfies
(8), what shows that

$\text{\rm Malcev}\>\cap\>\text{\rm Almost Lie}=\text{\rm
Malcev}\>\cap\>\text{\rm Sagle}.$

“Sagle $\>\cap\>$ CD = Malcev $\>\cap\>$ Sagle”: The inclusion Malcev
$\>\cap\>$ Sagle $\subseteq$ Sagle $\>\cap\>$ CD follows from (5), so let us
prove the inverse inclusion.

By the already proved, we have

$\text{\rm Sagle}\>\cap\>\text{\rm CD}=\text{\rm Sagle}\>\cap\>\text{\rm
Binary Lie}\>\cap\>\text{\rm Almost Lie}.$

The algebra which is simultaneously binary Lie and almost Lie, satisfies (10),
and by (4), the algebra which is simultaneously Sagle and almost Lie,
satisfies the identity

$J(w,z,xy)+J(w,y,zx)+J(w,x,yz)=0.$

Permuting in the last identity $z$ and $w$, and using (10), we get

$-J(w,z,xy)+J(w,x,yz)+J(w,y,zx)=0.$

The last two identities yield the identity (8), and thus, by Lemma 1,

$\text{\rm Sagle}\>\cap\>\text{\rm Binary Lie}\>\cap\>\text{\rm Almost
Lie}\subseteq\text{\rm Malcev}\>\cap\>\text{\rm Sagle}.$

∎

Perhaps, the most important among all these multiple relations is the
inclusion

(13) $\text{CD}\subset\text{Almost Lie}$

which shows that, after all, CD algebras are not that far from the Lie ones.

An immediate corollary of this inclusion is

######  Proposition 3.

For any CD algebra $A$, the quotient $A/Z(A)$ is a Lie algebra. In particular,
any centerless (and, in particular, simple) CD algebra is a Lie algebra.

Thus, CD algebras are, essentially, central extensions, in the suitable
variety, of Lie algebras. As central extensions should be described by second
degree cohomology, this suggests that there should be a “CD cohomology”,
extending the usual Chevalley–Eilenberg cohomology, responsible for such
central extensions. And indeed, such cohomology is constructed in §4.

It is natural to ask whether any simple binary Lie algebra is Malcev (and
hence is either Lie, or is a $7$-dimensional simple Malcev algebra). For
finite-dimensional algebras over a field of characteristic zero the question
was answered in affirmative in [G], while the cases of positive
characteristic, and of infinite-dimensional algebras remain open (see, e.g.,
[Dn, Problems 2.33 and 3.87] and [AS, p. 263]). Proposition 3 shows that in
the narrower class of CD algebras the answer is also affirmative.

Note also that Proposition 3 implies that CD non-Lie algebras of dimension
$\leq 5$ listed in [ACF] actually exhaust _all_ CD non-Lie algebras in those
dimensions.

##  3\. Further relations with Lie algebras

Let $A$ be a CD algebra. Consider the subspace $R(A)$, spanned by all maps
$R_{a}$, $a\in A$, and the subalgebra $\operatorname{Der}(A)$ in the Lie
algebra $\operatorname{\mathsf{gl}}(A)$ of all linear maps on $A$ (subject to
the usual commutator of linear maps), and consider their formal direct sum
$R(A)\oplus\operatorname{Der}(A)$ (“formal”, as they may intersect, so the sum
$R(A)+\operatorname{Der}(A)$, considered as the linear subspace of
$\operatorname{\mathsf{gl}}(A)$, is not necessarily direct; more on that
below). As $A$ is a CD algebra, we have
$[R(A),R(A)]\subseteq\operatorname{Der}(A)$. Moreover, because of (1), it
holds $[\operatorname{Der}(A),R(A)]\subseteq R(A)$. Thus
$R(A)\oplus\operatorname{Der}(A)$ is a Lie algebra with respect to the usual
commutator, actually a semidirect sum with $\operatorname{Der}(A)$ acting on
$R(A)$.

This construction is completely analogous to those in the commutative case
(cf., e.g., [B, Definition II.1.4]), and similar to the construction of the
structure algebra of a Jordan algebra used in the Kantor–Koecher–Tits
construction (cf., e.g., [J, Chapter VIII, §4]); the construction of the
holomorph of a Lie algebra (i.e., the semidirect sum
$L\oplus\operatorname{Der}(L)$ for a Lie algebra $L$) is somewhat similar in
spirit, but different, as in the holomorph we have $[L,L]\subseteq L$.

This construction can be modified in several ways. For example, instead of the
linear space $R(A)$ we may consider the Lie multiplication algebra $M(A)$
(again subject to the commutator). As $M(A)$ is generated by $R(A)$, the
commutation relations in the Lie algebra $R(A)\oplus\operatorname{Der}(A)$ can
serve as defining relations in the Lie algebra
$M(A)\oplus\operatorname{Der}(A)$.

Another possibility is to consider not formal, but “real” direct sum, i.e.,
the Lie subalgebra of $\operatorname{\mathsf{gl}}(A)$, spanned by $R(A)$ and
$\operatorname{Der}(A)$. To consider this variant more thoroughly, define the
_Lie center_ of $A$, denoted by $LZ(A)$, as the set of all elements $z\in A$
such that $J(a,b,z)=0$ for any $a,b\in A$. Obviously, the Lie center is always
a vector subspace of $A$, and $LZ(A)=A$ if and only if $A$ is a Lie algebra.
In a sense, it serves as a measure of “non-Lieness” of an algebra.

In what follows, it will be convenient to identify elements of $A$ with the
corresponding multiplications in $R(A)$, up to the center. Namely, the kernel
of the linear map $A\to R(A)$, $a\mapsto R_{a}$ coincides with $Z(A)$; hence
we have the isomorphism of vector spaces

(14) $A/Z(A)\xrightarrow{\sim}R(A).$

Due to Proposition 3, this is also an isomorphism of Lie algebras.

######  Lemma 3.

For any CD algebra $A$, there is isomorphism of Lie algebras

$LZ(A)/Z(A)\simeq R(A)\cap\operatorname{Der}(A).$

###### Proof.

It is obvious that $Z(A)\subseteq LZ(A)$. The condition $z\in LZ(A)$ is
equivalent, taking into account the anticommutativity of $A$, to the condition
$R_{z}\in\operatorname{Der}(A)$. Hence the image of $LZ(A)$ under the
isomorphism (14) coincides with $R(A)\cap\operatorname{Der}(A)$. ∎

######  Lemma 4.

For any CD algebra $A$, $LZ(A)$ is an ideal of $A$.

###### Proof.

Let $z\in LZ(A)$, and $x\in A$. By Lemma 3, we have
$R_{z}\in\operatorname{Der}(A)$, and then by (1) we have
$[R_{z},R_{x}]=R_{R_{z}(x)}=R_{xz}$. Then, since $A$ is a CD algebra,
$R_{xz}\in\operatorname{Der}(A)$, and hence, again by Lemma 3, $xz\in LZ(A)$.
∎

Thus, identifying $R(A)$ with $A/Z(A)$ via isomorphism (14), the intersection
between $R(A)$ and $\operatorname{Der}(A)$ is identified with $LZ(A)/Z(A)$ by
Lemma 4, and (generally, non-direct) sum $R(A)+\operatorname{Der}(A)$ can be
identified with the direct sum $A/LZ(A)\oplus\operatorname{Der}(A)$, the Lie
subalgebra of $\operatorname{\mathsf{gl}}(A)$.

##  4\. “Naive” cohomology

The goal of this section is to construct cohomology theory of CD algebras. The
standard approach to construct cohomology in a variety of algebras is an
operadic one: provided that the corresponding operad $\mathcal{P}$ is
quadratic and Koszul, there is a small explicit cochain complex built out of
the Koszul dual cooperad $\mathcal{P}^{\mbox{!`}}$. However, this works only
for quadratic Koszul operads. The operad defined by the identity (2) is cubic.
The standard trick, employed, for example, in the case of the Jordan operad,
is to pass to triple systems; the corresponding category of triple systems
should be equivalent (say, in representation-theoretic sense) to the initial
category of binary algebras, and the corresponding operad will be (ternary)
quadratic. However, it is not immediately clear which triple systems should
correspond to CD algebras, and whether the corresponding operad will be Koszul
(we believe it will be not).

Thus we rely on the “naive” approach to cohomology. Under this, we mean an
attempt to construct (the beginning of) the corresponding cochain complex by
utilizing the low-degree structural interpretations of cohomology in the given
variety: derivations, central extension, deformations, etc. We take cohomology
of Lie algebras as a model.

For that, we need first to define what is a module over a CD algebra is. We
follow a nowadays standard approach which goes back to Eilenberg (see, for
example, [J, Chapter II, §5]). Namely, for an algebra $A$ in a given variety,
a vector space $M$ with an $A$-action on it, is declared a module over $A$, if
the semidirect sum $A\oplus M$, where multiplication between elements of $A$
and $M$ is determined by the given action, and multiplication on $M$ is zero,
belongs to the same variety. According to this approach, a vector space $M$
with a (left) action of a CD algebra $A$, denoted by $a\bullet m$, is called a
_module over $A$_, if the following equality holds:

$(xy)a\bullet m+a\bullet((xy)\bullet m)-x\bullet((ya)\bullet
m)+y\bullet((xa)\bullet m)-x\bullet(a\bullet(y\bullet
m))+y\bullet(a\bullet(x\bullet m))=0.$

for any $x,y,a\in A$ and $m\in V$.

As in the Lie case, $A$, considered as a (left) module over itself, is called
the _adjoint module_.

As we are interested in central extensions and deformations of CD algebras, we
start with the second cohomology. The second cohomology is interpreted as
equivalence classes of square-zero extensions. Namely, let $A$ be a CD algebra
and $M$ an $A$-module, and consider the CD algebra structure on the direct sum
of vector space $A\oplus M$, where multiplication on $A$ is given by the
formula $x*y=xy+\varphi(x,y)$ for some bilinear map $\varphi:A\times A\to M$,
multiplication between $A$ and $M$ is given by action of $A$ on $M$, and
multiplication on $M$ is zero. Then $\varphi$ is skew-symmetric, and

(15)
$\varphi((xy)a,b)-\varphi((xy)b,a)-\varphi((xa)b,y)+\varphi((xb)a,y)+\varphi((ya)b,x)-\varphi((yb)a,x)\\\
+a\bullet\varphi(xy,b)-b\bullet\varphi(xy,a)-x\bullet\varphi(ya,b)+x\bullet\varphi(yb,a)+y\bullet\varphi(xa,b)-y\bullet\varphi(xb,a)\\\
-a\bullet(b\bullet\varphi(x,y))+b\bullet(a\bullet\varphi(x,y))-x\bullet(a\bullet\varphi(y,b))+x\bullet(b\bullet\varphi(y,a))-y\bullet(b\bullet\varphi(x,a))+y\bullet(a\bullet\varphi(x,b))\\\
=0$

for any $x,y,a,b\in A$.

The usual notion of equivalence of square-zero extensions $0\to M\to\cdot\to
A\to 0$ leads to the notion of trivial, or split, extension, which corresponds
to a cocycle of the form

(16) $\varphi(x,y)=\psi(xy)-x\bullet\psi(y)+y\bullet\psi(x)$

for any $x,y\in A$ and some linear map $\psi:A\to M$. Thus the right-hand side
of this equality suggests the definition of the first order cocycles, what
confirms the standard interpretation of the first cohomology as outer
derivations.

The inner derivations of $A$, according to the general approach devised by
Schafer (see, for example, [Sch, Chapter II, §3]), are defined as derivations
lying in the Lie multiplication algebra $M(A)$. Since
$[R_{a},R_{b}]\in\operatorname{Der}(A)$ and due to (1), $M(A)$ is linearly
spanned by linear maps of the form $R_{a}$ and $[R_{a},R_{b}]$ for $a,b\in A$.
On the other hand, by Lemma 3, $R_{a}$ is a derivation of $A$ if and only if
$a\in LZ(A)$. Thus any inner derivation of $A$ is of the form
$R_{z}+\sum_{i}[R_{a_{i}},R_{b_{i}}]$ for some $z\in LZ(A)$ and
$a_{i},b_{i}\in A$.

Another interpretation of the second cohomology with coefficients in the
adjoint module is equivalence classes of infinitesimal deformations of an
algebra. Thus, following nowadays standard approach by Gerstenhaber, for a CD
algebra $A$ consider a deformed algebra over the ring $K[[t]]$, with
multiplication

$x*y=xy+\varphi_{1}(x,y)t+\varphi_{2}(x,y)t^{2}+\dots$

That the deformed algebra is also CD algebra, is equivalent to an (infinite)
series of equalities, obtained by collecting coefficients by powers of $t$ in
the CD identity (2) for the multiplication $*$. The zeroth of these equalities
(coefficients by $t^{0}$) coincides with the CD identity for the original
multiplication in $A$, and thus gives nothing new. The first of these
equalities (coefficients by $t^{1}$) is obtained by ignoring all terms with
powers of $t$ higher than $1$, and thus is equivalent to the CD identity in
the algebra $A\oplus At$, where $At$ is the adjoint module over $A$ with
trivial multiplication. Hence it is equivalent to the identity (15) with $M=A$
and $\bullet$ being multiplication in the algebra. The further equalities
(coefficients by $t^{2}$ and higher degrees) involve Massey brackets of
$\varphi_{i}$’s which can be interpreted as obstructions of prolongations of
an infinitesimal deformation to a global one, and lie in the third cohomology
of $A$ with coefficients in the adjoint module. But as it leads to cumbersome
formulae, and our primary interest is in the second cohomology, we will not
pursue this further.

The condition of triviality of such deformation, i.e., the equivalence to the
initial algebra with respect to a homomorphism of the form

$\psi(x)=x+\psi_{1}(x)t+\psi_{2}(x)t^{2}+\dots$

leads to an infinite series of equalities, the first of which is

$\varphi_{1}(x,y)=\psi_{1}(xy)-\psi_{1}(x)y-x\psi_{1}(y),$

what is the partial case of (16) in the case of the adjoint module.

Putting all this together, we define the initial terms of the cochain complex
associated to a CD algebra $A$ and an $A$-module $M$:

(17) $0\to LZ(M)\oplus(A\otimes
M)\overset{\operatorname{d}^{0}}{\longrightarrow}\operatorname{C}^{1}(A,M)\overset{\operatorname{d}^{1}}{\longrightarrow}\operatorname{C}^{2}(A,M)\overset{\operatorname{d}^{2}}{\longrightarrow}\operatorname{C}^{4}(A,M).$

Here $LZ(M)$ is a “Lie center” of $M$ defined as

$LZ(M)=\\{m\in M\>|\>xy\bullet m-x\bullet(y\bullet m)+y\bullet(x\bullet
m)=0\text{ for any }x,y\in M\\},$

and $\operatorname{C}^{n}(A,M)$ for $n\geq 1$ is a linear space of skew-
symmetric linear maps $\underbrace{A\times\dots\times A}_{n}\to M$. The
differentials are defined as follows:

$\operatorname{d}^{0}(m)(b)=b\bullet m$

for $b\in A$, $m\in LZ(M)$,

$\operatorname{d}^{0}(a\otimes m)(b)=a\bullet(b\bullet m)-b\bullet(a\bullet
m)$

for $a,b\in A$, $m\in M$. The “$LZ(M)$” component of $\operatorname{d}^{0}$ is
similar to the zeroth differential in the Chevalley–Eilenberg complex
computing the Lie algebra cohomology, and the “$A\otimes M$” component is
similar to the zeroth differential in the complex computing cohomology of
quadratic Jordan algebras defined in [M, §I.3].

Further,

$\operatorname{d}^{1}(\varphi)(x,y)=\varphi(xy)-x\bullet\varphi(y)+y\bullet\varphi(x)$

for $\varphi\in\operatorname{C}^{1}(A,M)$, $x,y\in A$, and

(18) $\operatorname{d}^{2}(\varphi)(x,y,a,b)\\\
=\varphi((xy)a,b)-\varphi((xy)b,a)-\varphi((xa)b,y)+\varphi((xb)a,y)+\varphi((ya)b,x)-\varphi((yb)a,x)\\\
+a\bullet\varphi(xy,b)-b\bullet\varphi(xy,a)-x\bullet\varphi(ya,b)+x\bullet\varphi(yb,a)+y\bullet\varphi(xa,b)-y\bullet\varphi(xb,a)\\\
\hskip-2.0pt-a\bullet(b\bullet\varphi(x,y))+b\bullet(a\bullet\varphi(x,y))-x\bullet(a\bullet\varphi(y,b))+x\bullet(b\bullet\varphi(y,a))-y\bullet(b\bullet\varphi(x,a))+y\bullet(a\bullet\varphi(x,b))$

for $\varphi\in\operatorname{C}^{2}(A,M)$, $x,y,a,b\in A$.

The equalities $\operatorname{d}^{1}\circ\operatorname{d}^{0}=0$ and
$\operatorname{d}^{2}\circ\operatorname{d}^{1}=0$ are verified in a
straightforward, if not a bit cumbersome way, or just follow from the
structural interpretations described above.

This is not the only sensible way to define cohomology of CD algebras. For
example, one may argue that a proper notion of derivation in this context is
the following: a linear map $D:A\to A$ such that the semidirect sum $A\oplus
KD$, where multiplication between $A$ and $D$ is determined by action of $D$
on $A$. In the variety of Lie algebras, this leads to the usual notion of
derivation, but in the variety of CD algebras, this leads to what might be
called a _CD derivation_ of a CD algebra $A$: a linear map $D:A\to A$ such
that

$D((xy)a)-D(xy)a-D(xa)y+D(ya)x+(D(x)a)y-(D(y)a)x=0$

for any $x,y,a\in A$. An inner derivation in this context is, as in the
variety of Lie algebras, just a multiplication $R_{a}$, $a\in A$. Accordingly,
one may define the initial terms of the cochain complex responsible for
cohomology of a CD algebra $A$ with coefficients in an $A$-module $M$, as

$0\to
M\overset{\operatorname{d}^{0}}{\longrightarrow}\operatorname{C}^{1}(A,M)\overset{\operatorname{d}^{1}}{\longrightarrow}\operatorname{C}^{3}(A,M),$

where

$\operatorname{d}^{0}(m)(x)=x\bullet m$

for $m\in M$ and $x\in A$, and

(19)
$\operatorname{d}^{1}(\varphi)(x,y,a)=\varphi((xy)a)-a\bullet\varphi(xy)-y\bullet\varphi(xa)+x\bullet\varphi(ya)+y\bullet(a\bullet\varphi(x))-x\bullet(a\bullet\varphi(y))$

for $\varphi\in\operatorname{C}^{1}(A,M)$ and $x,y,a\in A$.

One cannot say which variant of these cochain complexes is “better”. In that
respect, the situation resembles those with cohomology of so-called mock-Lie
algebras (commutative algebras satisfying the Jacobi identity), where also one
cannot define a low-degree cohomology in a canonical and coherent way, basing
on structural interpretations (cf. [Z, §1]). However, as we are interested in
central extensions and deformations, we adopt the first variant of the cochain
complex, and define the second degree cohomology of a CD algebra $A$ with
coefficients in an $A$-module $M$ as
$\operatorname{H}_{CD}^{2}(A,M)=\operatorname{Ker}\operatorname{d}^{2}/\operatorname{Im}\operatorname{d}^{1}$,
where $\operatorname{d}^{1}$ and $\operatorname{d}^{2}$ are as in (17).

One may try to generalize the formulae for differentials above as follows. Let
$n>0$, and
$\operatorname{d}:\operatorname{C}^{n}(A,M)\to\operatorname{C}^{n+2}(A,M)$ is
given by

$\operatorname{d}(\varphi)(x,y,a_{1},\dots,a_{n})\\\
=\sum_{i=1}^{n}(-1)^{i}\Big{(}\varphi((xy)a_{i},a_{1},\dots,\widehat{a_{i}},\dots,a_{n})+a_{i}\bullet\varphi(xy,a_{1},\dots,\widehat{a_{i}},\dots,a_{n})\\\
\hskip
56.0pt-x\bullet\varphi(ya_{i},a_{1},\dots,\widehat{a_{i}},\dots,a_{n})+y\bullet\varphi(xa_{i},a_{1},\dots,\widehat{a_{i}},\dots,a_{n})\\\
\hskip
102.0pt-x\bullet(a_{i}\bullet\varphi(y,a_{1},\dots,\widehat{a_{i}},\dots,a_{n}))+y\bullet(a_{i}\bullet\varphi(x,a_{1},\dots,\widehat{a_{i}},\dots,a_{n}))\Big{)}\\\
+\sum_{1\leq i<j\leq
n}(-1)^{i+j+n+1}\Big{(}\varphi((xa_{i})a_{j},y,a_{1},\dots,\widehat{a_{i}},\dots,\widehat{a_{j}},\dots,a_{n})-\varphi((xa_{j})a_{i},y,a_{1},\dots,\widehat{a_{i}},\dots,\widehat{a_{j}},\dots,a_{n})\\\
\hskip
104.0pt-\varphi((ya_{i})a_{j},x,a_{1},\dots,\widehat{a_{i}},\dots,\widehat{a_{j}},\dots,a_{n})+\varphi((ya_{j})a_{i},x,a_{1},\dots,\widehat{a_{i}},\dots,\widehat{a_{j}},\dots,a_{n})\\\
\hskip
69.0pt+a_{i}\bullet(a_{j}\bullet\varphi(x,y,a_{1},\dots,\widehat{a_{i}},\dots,\widehat{a_{j}},\dots,a_{n}))-a_{j}\bullet(a_{i}\bullet\varphi(x,y,a_{1},\dots,\widehat{a_{i}},\dots,\widehat{a_{j}},\dots,a_{n}))\Big{)}$

for $\varphi\in\operatorname{C}^{n}(A,M)$, and $x,y,a_{1},\dots,a_{n}\in A$.

One can prove, in the absence of analogs of Cartan formulas in the
Chevalley–Eilenberg complex (are there ones?), by direct verification if not
without some pain, that $\operatorname{d}\circ\operatorname{d}=0$, so we get,
in fact, two complexes, which lead to what may be called “odd” and “even” CD
cohomology respectively:

$\displaystyle\operatorname{C}^{1}(A,M)\overset{\operatorname{d}}{\to}\operatorname{C}^{3}(A,M)\overset{\operatorname{d}}{\to}\operatorname{C}^{5}(A,M)\overset{\operatorname{d}}{\to}\dots$
$\displaystyle\operatorname{C}^{2}(A,M)\overset{\operatorname{d}}{\to}\operatorname{C}^{4}(A,M)\overset{\operatorname{d}}{\to}\operatorname{C}^{6}(A,M)\overset{\operatorname{d}}{\to}\dots$

The differential
$\operatorname{d}:\operatorname{C}^{1}(A,M)\to\operatorname{C}^{3}(A,M)$ here
coincides with differential (19), and the differential
$\operatorname{d}:\operatorname{C}^{2}(A,M)\to\operatorname{C}^{4}(A,M)$
coincides with differential (18).

However, we will not pursue this topic further and in the subsequent section
will work exclusively with $\operatorname{H}_{CD}^{2}(A,M)$ as defined above.

##  5\. Central CD extensions of Lie algebras

In this section we discuss central CD extensions, or, in other words,
$\operatorname{H}_{CD}^{2}(L,K)$, for various Lie algebras $L$ with
coefficients in the trivial module $K$. According to the definition, the
vector space $\operatorname{H}_{CD}^{2}(L,K)$ is a quotient of _CD
$2$-cocycles_ by $2$-coboundaries:

$\operatorname{H}_{CD}^{2}(L,K)=\frac{\operatorname{Z}_{CD}^{2}(L,K)}{\operatorname{B}^{2}(L,K)}.$

The space $\operatorname{Z}_{CD}^{2}(L,K)$ of CD $2$-cocycles consists of
skew-symmetric bilinear maps $\varphi:L\times L\to K$ satisfying the condition

(20)
$\varphi([[x,y],a],b)-\varphi([[x,y],b],a)-\varphi([[x,a],b],y)+\varphi([[x,b],a],y)+\varphi([[y,a],b],x)-\varphi([[y,b],a],x)=0$

for any $x,y,a,b\in L$, and the space of $2$-coboundaries
$\operatorname{B}^{2}(L,K)$ consists, as in the Lie case, of bilinear maps of
the form $\varphi(x,y)=\psi([x,y])$ for some linear map $\psi:L\to K$.

Note that for any Lie algebra $L$, the usual Chevalley–Eilenberg cohomology
$\operatorname{H}^{2}(L,K)$ is a subspace in the CD cohomology
$\operatorname{H}_{CD}^{2}(L,K)$.

Our first goal is to obtain examples of an almost Lie algebra which is not CD,
promised at the end of the proof of Proposition 1, by considering one-
dimensional central extensions of a Lie algebra $L$. Such central extensions
can be written as the vector space direct sum $L\oplus Kz$, where
multiplication in $L$ is twisted by a $2$-cocycle $\varphi:L\times L\to K$,
$\\{x,y\\}=[x,y]+\varphi(x,y)z$, and $z$ is a central element. Such an algebra
is almost Lie for any skew-symmetric $\varphi$, while it is a CD algebra if
and only if $\varphi$ is CD $2$-cocycle. Thus any Lie algebra $L$ whose second
CD cohomology $\operatorname{H}_{CD}^{2}(L,K)$ is strictly larger then its
second Chevalley–Eilenberg cohomology $\operatorname{H}^{2}(L,K)$, will lead,
by extending $L$ by any CD $2$-cocycle which is not a Chevalley–Eilenberg
cocycle, to an example of a CD algebra which is not Lie. Similarly, a Lie
algebra $L$, for which $\operatorname{H}_{CD}^{2}(L,K)$ is strictly smaller
than the space $\operatorname{C}^{2}(L,K)/\operatorname{B}^{2}(L,K)$ – which
can be considered as the “2nd almost Lie cohomology” – will lead to an example
of an almost Lie algebra which is not CD.

Obviously, abelian Lie algebras do not qualify for such examples, so let us
look at nonabelian Lie algebras of low dimension. Elementary calculations show
that the two-dimensional nonabelian Lie algebra, and the $3$-dimensional
nilpotent Lie algebra do not qualify either, as for these algebras

$\operatorname{H}^{2}(L,K)=\operatorname{H}_{CD}^{2}(L,K)=\frac{\operatorname{C}^{2}(L,K)}{\operatorname{B}^{2}(L,K)}$

(all these three spaces vanish in the case where $L$ is two-dimensional
nonabelian, and are of dimension $2$ in the case where $L$ is $3$-dimensional
nilpotent), but the direct sum of the two-dimensional nonabelian and the one-
dimensional algebra does qualify, as for this algebra we have

$\operatorname{H}^{2}(L,K)=\operatorname{H}_{CD}^{2}(L,K)\subsetneq\frac{\operatorname{C}^{2}(L,K)}{\operatorname{B}^{2}(L,K)}$

(the corresponding spaces being of dimensions $1$ and $2$).

Further low-dimensional solvable and nilpotent Lie algebras of dimension $3$
and higher could provide a plethora of such examples (including the cases
where $\operatorname{H}^{2}(L,K)$ and $\operatorname{H}_{CD}^{2}(L,K)$ do not
coincide).

On the other hand, for simple Lie algebras we have

######  Theorem (“Second CD Whitehead lemma”).

For any simple finite-dimensional Lie algebra $L$ over a field of
characteristic zero, and any finite-dimensional $L$-module $M$,
$\operatorname{H}_{CD}^{2}(L,M)=0$.

###### Proof.

This follows at once from [G, Theorem 6]. Indeed, it is proved there that any
solvable (and hence abelian) extension of $L$ in the variety of binary Lie
algebras (and hence in the variety of CD algebras) splits. ∎

In the positive characteristic, we merely have a

######  Conjecture.

For any simple finite-dimensional Lie algebra $L$ over a field of
characteristic $\neq 2,3$,
$\operatorname{H}_{CD}^{2}(L,K)=\operatorname{H}^{2}(L,K)$.

The conjecture is supported by computer calculations for algebras of small
dimension.

##  6\. Further questions

1) How “far” a CD algebra can be from Lie algebras? To start with, describe CD
algebras $A$ with the Lie center “as small as possible”, i.e., satisfying the
condition $LZ(A)=Z(A)$.

2) Which “interesting” Lie algebras can be realized as constructions described
in §3?

3) Study free CD algebras. Are they central extensions of free Lie algebras?

4) For an algebra $A$, its minus-algebra $A^{(-)}$ is an algebra defined on
the same underlying vector space $A$ subject to multiplication given by the
commutator $[x,y]=xy-yx$. If the minus-algebra of an algebra $A$ belongs to a
(necessary anticommutative) variety $\mathcal{V}$, then $A$ is called
$\mathcal{V}$-admissible. Most of the distinguished varieties of algebras have
they “admissible” counterparts: thus, associative algebras are Lie-admissible,
alternative algebras are Malcev-admissible, and binary Lie algebras are
assocyclic-admissible, where the variety of assocyclic algebras is defined by
the identity

$(x,y,z)=(z,x,y),$

$(x,y,z)=(xy)z-x(yz)$ being the associator of elements $x,y,z$.

In such situation arises the question whether each algebra in an
anticommutative variety $\mathcal{V}$ is special, i.e., can be embedded into
an algebra of the form $A^{(-)}$, where $A$ is $\mathcal{V}$-admissible
algebra. Thus, Lie algebras are special due to the celebrated
Poincaré–Birkhoff–Witt theorem; the speciality of Malcev algebras was a long-
standing problem whose negative solution was announced recently by Ivan
Shestakov; and binary Lie algebras are not necessarily special either, see
[AS].

What would be a natural variety of CD admissible algebras? (One possible
general approach in operadic language to such sort of questions is described
in [K]). Would CD algebras be special with respect to that variety?

5) Study representations of CD algebras. Does an analog of the Ado theorem
hold, i.e., whether each finite-dimensional CD algebra admits a faithful
finite-dimensional representation?

6) The classical Lie theory establishes correspondence between Lie groups and
Lie algebras. This correspondence has been generalized to Malcev algebras, Bol
algebras, Lie triple systems, Sabinin algebras, etc. (see, for example, [GP]
and references therein). In these “generalized Lie” correspondences Lie groups
are replaced by various kinds of analytic loops; thus, binary Lie algebras
correspond to diassociative loops (i.e., the subloop generated by any two
elements is a group). Which loops would correspond to CD algebras? One of the
possible approaches to this question would be to find the class of loops
corresponding to almost Lie algebras, and then, according to (6), take the
intersection of that class with the class of diassociative loops.

7) Let us drop the commutativity and anticommutativity conditions altogether,
and consider the variety of algebras defined just by the properties that the
commutator of any two left or right multiplications is a derivation (i.e.,
taking all possible combinations of left/right multiplications, we get $3$
defining identities, see, for example, [KK1, §1]). Is this variety of algebras
amenable to study? Low-dimensional nilpotent algebras in this variety were
classified in [KK1] and [KK2].

## Acknowledgements

Thanks are due to Askar Dzhumadil’daev, Alexander Grishkov, and José María
Pérez-Izquierdo for useful discussions. GAP [GAP] and Albert [A] were used to
check some of the computations performed in this paper. This work was started
when Zusmanovich was visiting Federal University of ABC, supported by FAPESP,
grant number 18/00726-5.

## References

  * [ACF] H. Abdelwahab, A.J. Calderón, and A. Fernández Ouaridi, _Central extensions of 4-dimensional binary Lie algebras_ , Rocky Mountain J. Math. 50 (2020), no.5, 1541–1559.
  * [ACK] by same author, by same author, and I. Kaygorodov, _The algebraic and geometric classification of nilpotent binary Lie algebras_ , Intern. J. Algebra Comput. 29 (2019), no.6, 1113–1129.
  * [AS] M. Arenas and I. Shestakov, _On speciality of binary-Lie algebras_ , J. Algebra Appl. 10 (2011), no.2, 257–268.
  * [B] W. Bertram, _The Geometry of Jordan and Lie Structures_ , Lect. Notes Math. 1754 (2000).
  * [Dn] _Dniester Notebook: Unsolved Problems in the Theory of Rings and Modules_ (ed. V.T. Filippov, V.K. Kharchenko, and I.P. Shestakov), 4th ed., Novosibirsk, 1993 (in Russian); Non-Associative Algebra and Its Applications, Lect. Notes Pure Appl. Math., Chapman & Hall/CRC, Vol. 246, 2006, 461–516 (English translation).
  * [Dz] A. Dzhumadil’daev, _Assosymmetric algebras under Jordan product_ , Comm. Algebra 46 (2018), no.1, 129–142.
  * [F] V.T. Filippov, _Simple Sagle algebras_ , Mat. Zametki 65 (1999), no.4, 607–611 (in Russian); Math. Notes 65 (1999), no.4, 506–509 (English translation).
  * [G] A.N. Grishkov, _Structure and representations of binary-Lie algebras_ , Izv. Akad. Nauk SSSR Ser. Matem. 44 (1980), no.5, 999-1030 (in Russian); Math. USSR Izv. 17 (1981), no.2, 243–269 (English translation).
  * [GP] by same author and J.M. Pérez-Izquierdo, _Lie’s correspondence for commutative automorphic formal loops_ , Lin. Algebra Appl. 544 (2018), 460–501.
  * [HP] I.R. Hentzel and L.A. Peresi, _Almost Jordan rings_ , Proc. Amer. Math. Soc. 104 (1988), no.2, 343–348.
  * [J] N. Jacobson, _Structure and Representations of Jordan Algebras_ , AMS, 1968.
  * [JR] P. Jordan and H. Rühaak, _Über einen Zusammenhang der Lie-Tripel-Algebren mit den Osborn-Algebren_ , Akad. Wiss. Lit. Mainz Abh. Math.-Natur. Kl. 1969, no.3, 49–54.
  * [KK1] I. Kaygorodov and M. Khrypchenko, _The algebraic classification of nilpotent $\mathfrak{CD}$-algebras_, Lin. Multilin. Algebra, to appear; arXiv:2006.00734.
  * [KK2] by same author and by same author, _The geometric classification of nilpotent $\mathfrak{CD}$-algebras_, J. Algebra Appl., to appear; arXiv:2007.01829.
  * [K] P.S. Kolesnikov, _Commutator algebras of pre-commutative algebras_ , Matematicheskii Zhurnal (Almaty) 16 (2016), no.2 (Dzhumadil’daev’s Festshrift), 145–158.
  * [M] K. McCrimmon, _Representations of quadratic Jordan algebras_ , Trans. Amer. Math. Soc. 153 (1971), 279–305.
  * [Sa1] A.A. Sagle, _Malcev algebras_ , Trans. Amer. Math. Soc. 101 (1961), no.3, 426–458.
  * [Sa2] by same author, _On simple extended Lie algebras over fields of characteristic zero_ , Pacific J. Math. 15 (1965), no.2, 621–648.
  * [Sch] R.D. Schafer, _An Introduction to Nonassociative Algebras_ , Academic Press, 1966.
  * [Si] A.V. Sidorov, _Lie triple algebras_ , Algebra Logika 20 (1981), no.1, 101–108 (in Russian); Algebra and Logic 20 (1981), no.1, 72–78 (English translation).
  * [Z] P. Zusmanovich, _Special and exceptional mock-Lie algebras_ , Lin. Algebra Appl. 518 (2017), 79–96; arXiv:1608.05861.

## Software

  * [A] _Albert_ ; https://web.osu.cz/$\sim$Zusmanovich/soft/albert/
  * [GAP] The GAP Group, _GAP – Groups, Algorithms, and Programming_ , Version 4.10.2, 2019;
https://www.gap-system.org/


Bayesian sample size determination in basket trials borrowing information
between subsets

HAIYAN ZHENG1,2,∗, MICHAEL J. GRAYLING2, PAVEL MOZGUNOV1, THOMAS JAKI1,3,
JAMES M. S. WASON2 [4pt] 1MRC Biostatistics Unit, University of Cambridge,
U.K. [-2pt] 2Population Health Sciences Institute, Newcastle University, U.K.
[-2pt] 3Department of Mathematics and Statistics, Lancaster University, U.K.
[2pt] ∗Email<EMAIL_ADDRESS>
###### Abstract

Basket trials are increasingly used for the simultaneous evaluation of a new
treatment in various patient subgroups under one overarching protocol. We
propose a Bayesian approach to sample size determination in basket trials that
permit borrowing of information between commensurate subsets. Specifically, we
consider a randomised basket trial design where patients are randomly assigned
to the new treatment or a control within each trial subset (‘subtrial’ for
short). Closed-form sample size formulae are derived to ensure each subtrial
has a specified chance of correctly deciding whether the new treatment is
superior to or not better than the control by some clinically relevant
difference. Given pre-specified levels of pairwise (in)commensurability, the
subtrial sample sizes are solved simultaneously. The proposed Bayesian
approach resembles the frequentist formulation of the problem in yielding
comparable sample sizes for circumstances of no borrowing. When borrowing is
enabled between commensurate subtrials, a considerably smaller trial sample
size is required compared to the widely implemented approach of no borrowing.
We illustrate the use of our sample size formulae with two examples based on
real basket trials. A comprehensive simulation study further shows that the
proposed methodology can maintain the true positive and false positive rates
at desired levels.

Key words:Borrowing strength; Commensurate priors; Master protocol; Mixture
prior; Phase II.

##  1 Introduction

Clinical research in precision medicine (Mirnezami et al.,, 2012; Schork,,
2015) continues to thrive as a consequence of the rapid technological advances
for identifying possible prognostic and predictive disease factors at the
genetic level (Aronson and Rehm,, 2015; Morganti et al.,, 2019). Because of
this, an increasing number of biomarker-driven therapies have been formulated.
In oncology, for example, much attention has been paid to therapies targeting
one or multiple genomic aberrations (Kim et al.,, 2011; Hyman et al.,, 2018;
Redman et al.,, 2020). In contrast to conventional chemotherapy devised for
treating histology-defined populations, such targeted therapies can
potentially be beneficial to patients of various cancer (sub)types. Immune-
mediated inflammatory diseases (IMIDs) (McInnes and Gravallese,, 2021) are
another area where targeted therapies can be useful (Grayling et al.,, 2021).
IMIDs generally involve a clinically diverse group of conditions that share
common underlying pathogenetic features, calling for the development of
effective immune-targeted therapeutics (Pitzalis et al.,, 2020). This paradigm
shift towards precision medicine has challenged the use of traditional one-
size-fits-all approaches to trial design, which aim to estimate the population
average treatment effect.

Master protocols (Woodcock and LaVange,, 2017) comprise a class of innovative
trial designs that address multiple investigational hypotheses. Newly emerging
types include basket trials that can simultaneously evaluate a new treatment
in stratified patient subgroups displaying a common disease trait (Renfro and
Sargent,, 2017; Tao et al.,, 2018). An implication of the stratification is
that patients may respond very differently to the same treatment due to their
distinct disease subtypes, stages or status. Fully acknowledging the
heterogeneity could lead to the use of stand-alone analyses that regard the
stratified subgroups in isolation. Such an analysis strategy, though adopted
in early basket trials, may not be ideal for realising the promise of basket
trials. This is mainly because it (i) fails to treat the combined (sub)trial
components as a single study, and (ii) often yields low-powered tests of the
treatment effect, due to the small sample sizes.

Sophisticated analysis models, which feature borrowing of information between
subgroups (ideally between those with commensurate treatment effects only),
have been proposed in the statistical literature. One pivotal strategy is to
fit a Bayesian hierarchical random-effects model (Thall et al.,, 2003; Berry
et al.,, 2013), assuming that the subgroup-specific treatment effects are
exchangeable, i.e., as random samples drawn from a common normal distribution
with unknown mean and variance. This methodology has been extended to involve
(i) a finite mixture of exchangeability distributions, see, e.g., Liu et al.,
(2017); Chu and Yuan, (2018); Jin et al., (2020); as well as (ii) non-
exchangeability distributions such that subgroups with an extreme treatment
effect can be inferred to their own devices (Neuenschwander et al.,, 2016).
The former reflects the concern that some subsets of trial may be more
commensurate between themselves than others. A highly relevant proposal is to
further cluster the subgroups, so that the corresponding treatment effects are
assumed to be exchangeable within the same cluster. Chen and Lee, (2020)
present a two-step procedure, by which subgroups are clustered using a
Bayesian nonparametric model, before fitting an adjusted Bayesian hierarchical
random-effects model.

A few authors have further recommended to discuss the exchangeability or
commensurability of any two or more subgroups. These proposals are for better
characterisation of the complex trial data structure, which could involve
mixtures of exchangeable or non-exchangeable patient subgroups. Psioda et al.,
(2021) apply a Bayesian model averaging technique (Hoeting et al.,, 1999) to
accommodate the possibility that any configuration of subgroups may have the
same or disparate response rate. Hobbs and Landin, (2018) construct a matrix
containing elements with value of 0 or 1, indicating that any pair of
subgroups can be exchangeable or non-exchangeable. Alternatively, Zheng and
Wason, (2022) propose measuring the pairwise (in)commensurability by
distributional discrepancy to enable an appropriate degree of borrowing from
each complementary subgroup, which yields a largest weight allocated to the
most commensurate one(s).

Development of methods to choose an appropriate sample size for basket trials,
however, appears to fall behind. A widely implemented approach is to sum up
the sample sizes, calculated as if these trial subsets are to be carried out
as separate studies. Whilst this could impair the efficiency of decision
making, alternative approaches to sample size determination that permit
borrowing of information are lacking. In this paper, we propose formal sample
size planning for the design of basket trials. It strikes a balance between
the sample size saving and the need of enrolling a sufficient number of
patients to assure inferences about the subgroup-specific treatment effects.
As the importance of randomised controlled trials has been increasingly
emphasised in oncology (Ratain and Sargent,, 2009; Grayling et al.,, 2019),
IMIDs (Grayling et al.,, 2021) and rare-disease (Prasad and Oseran,, 2015)
research, this paper will focus on randomised basket trial designs with a
primary objective of simultaneously comparing the new treatment against a
control in various patient subgroups. We will thus develop our sample size
formulae, presuming that the analysis is performed using a model adapted from
Zheng and Wason, (2022).

The remainder of this paper is organised as follows. In Section 2, we
introduce a Bayesian model that estimates the treatment effect specific to
subgroups using entire trial data, as well as the derivation of sample size
formulae appropriate for basket trials. Two data examples are presented in
Section 3 to illustrate the use of our formulae for the design of randomised
basket trials. In Section 4, we describe a simulation study that evaluates the
operating characteristics of randomised basket trials. Finally, we conclude
with a brief discussion and highlight several areas that deserve future
research in Section 5.

##  2 Methods

###  2.1 Leveraging complementary subtrial data into commensurate priors

Let us consider the design of a basket trial where patients can be classified
into $K$ subgroups. These patients nonetheless share a common feature (e.g., a
genetic aberration, clinical symptom or mechanism of drug action), on which a
new targeted therapy may potentially improve patient outcomes. Each component
study in a distinct patient subgroup will hereafter be referred to as a trial
subset (i.e., ‘subtrial’ for short). Within each subtrial $k$, patients are
randomised to receive either the experimental treatment (labelled $E$) with
probability $R_{k}\in(0,1)$, or a control (labelled $C$) with probability
$(1-R_{k})$, for $k=1,\dots,K$. We further assume that the measured responses
are normally distributed with their own subtrial-specific parameters:
$X_{ijk}\sim N(\mu_{jk},\sigma_{k}^{2})$, for $j=E,C;k=1,\dots,K$. Letting
$n_{k}$ denote the subtrial sample size, the difference in mean responses is
$\bar{X}_{Ek}-\bar{X}_{Ck}\sim
N\left(\mu_{Ek}-\mu_{Ck},\frac{\sigma_{k}^{2}}{n_{k}R_{k}(1-R_{k})}\right)$,
per subtrial $k$. For the ease of notation, we let
$\theta_{k}=\mu_{Ek}-\mu_{Ck}$ denote the treatment effect for subtrial $k$.
It is important to clarify at the outset that this design aims to estimate the
subtrial-specific treatment effects, i.e., $\theta_{1},\dots,\theta_{K}$,
instead of an overall treatment effect averaged over all subtrials. If
permitting borrowing of information across subtrials, these treatment effects
are to be estimated using the entire trial data (with $\sum_{k=1}^{K}n_{k}$
patients) rather than in isolation (with $n_{1},\dots,n_{K}$ patients,
respectively).

We follow Zheng and Wason, (2022) in specifying commensurate priors for each
$\theta_{k}$, using information from the $(K-1)$ complementary subtrials
indexed by $q\neq k,\,\forall k=1,\dots,K$. This methodology regards any
$\theta_{q}$ as a biased representation of $\theta_{k}$, yet the direction and
the size of such bias are unknown (Hobbs et al.,, 2011). More specifically,
these commensurate priors are formulated as conditional normal distributions
that are centred at $\theta_{q}$s, respectively; whilst the precisions (i.e.,
reciprocal of variances), denoted by $\nu_{qk}$, accommodate the heterogeneity
between two subtrials $k$ and $q$. Our commensurate prior models for the
continuous location parameter $\theta_{k}$ can thus be given by

$\begin{split}\theta_{k}\mid\theta_{q},\nu_{qk}&\sim
N(\theta_{q},\nu_{qk}^{-1}),\qquad\forall k=1,\dots,K,\\\ \nu_{qk}&\sim
w_{qk}\text{Gamma}(a_{1},b_{1})+(1-w_{qk})\text{Gamma}(a_{2},b_{2}),\end{split}$
(1)

where a two-component Gamma mixture prior (with $a_{1}/b_{1}$ and
$a_{2}/b_{2}$ being the respective means of the component distributions),
instead of a spike-and-slab prior in the original proposal, is placed on each
$\nu_{qk}$ for the convenience of analytic tractability (Zheng et al.,, 2022).
In particular, these two Gamma mixture components correspond to extreme cases
of substantial or limited discounting of information from a complementary
subtrial $q$. For illustration, we suppose that the first Gamma mixture
component has its density massively on small values, and the second component
on large values. The prior mixture weight $w_{qk}\in[0,1]$, which plays a role
of balancing between the extreme cases, can thus reflect one’s preliminary
scepticism about the degree of commensurability between $\theta_{k}$ and
$\theta_{q}$. That is, when subtrials $k$ and $q$ are thought of as
incommensurate (commensurate), $w_{qk}$ can be set close to 1 (0), thus
forcing the conditional prior variance $\nu_{qk}^{-1}$ towards large (small)
values for substantial (limited) discounting.

By integrating out $\nu_{qk}$, the conditional prior for $\theta_{k}$ given
$\theta_{q}$ only follows a shifted and scaled $t$ mixture distribution, with
its two components both centred at $\theta_{q}$. This unimodal $t$ mixture
distribution can further be approximated by matching the first two moments to
give

$\theta_{k}\mid\theta_{q}\,\dot{\sim}\,N\left(\theta_{q},\frac{w_{qk}b_{1}}{a_{1}-1}+\frac{(1-w_{qk})b_{2}}{a_{2}-1}\right),\quad\text{
with }a_{1},a_{2}>1,$ (2)

which incorporates the respective variances of the $t$ component
distributions. As has been shown by Zheng et al., (2022), this normal
approximation provides good properties for the coverage of credible intervals
of interest. Note that the location of each commensurate prior, $\theta_{q}$,
is an unknown parameter. It captures the information from a complementary
subtrial $q$, of which the required sample size $n_{q}$ as well as the
allocation proportion $R_{q}$ is yet to be determined.

Let
$\boldsymbol{x}_{q}=\\{x_{1Eq},\dots,x_{n_{q}Eq};x_{1Cq},\dots,x_{n_{q}Cq}\\}$
denote the data of a complementary subtrial $q$. We consider the difference of
sample means, $\bar{X}_{qE}-\bar{X}_{qC}$, as the random variable to draw the
Bayesian inference. With an ‘uninformative’ operational prior $\theta_{q}\sim
N(m_{0q},s_{0q}^{2})$, we derive the posterior as

$\begin{split}\theta_{q}\mid\boldsymbol{x}_{q}&\sim
N\left(\lambda_{q},\left(\frac{1}{s_{0q}^{2}}+\frac{n_{q}R_{q}(1-R_{q})}{\sigma_{q}^{2}}\right)^{-1}\right),\\\
\text{with
}\qquad\lambda_{q}&=\frac{m_{0q}}{1+\frac{s_{0q}^{2}}{\sigma_{q}^{2}}\cdot
n_{q}R_{q}(1-R_{q})}+\frac{\bar{x}_{Eq}-\bar{x}_{Cq}}{1+\frac{\sigma_{q}^{2}}{s_{0q}^{2}}\cdot\left[n_{q}R_{q}(1-R_{q})\right]^{-1}},\end{split}$
(3)

where $\bar{x}_{jq}$ denotes the average response of samples by treatment
group $j=E,C,$ within subtrial $q$. Combining (2) and (3), we obtain a
commensurate prior for $\theta_{k}$ that leverages data of a complementary
subtrial $q\neq k$:

$\begin{split}\theta_{k}\mid\boldsymbol{x}_{q}&\sim
N(\lambda_{q},\xi_{qk}^{2}),\qquad\forall k=1,\dots,K,\\\ \text{with
}\qquad\xi_{qk}^{2}&=\left(\frac{1}{s_{0q}^{2}}+\frac{n_{q}R_{q}(1-R_{q})}{\sigma_{q}^{2}}\right)^{-1}+\frac{w_{qk}b_{1}}{a_{1}-1}+\frac{(1-w_{qk})b_{2}}{a_{2}-1}.\end{split}$
(4)

Consider now borrowing information from all complementary subtrials, with
$K\geq 3$. Let $\boldsymbol{x}_{(-k)}$ denote the data from all subtrials
excluding $k$, that is, all the $(K-1)$ sets of complementary data for
subtrial $k$. By the convolution operator (Grinstead and Snell,, 1997), we
stipulate a collective, commensurate prior for leveraging all complementary
subtrial data:

$\theta_{k}\mid\boldsymbol{x}_{(-k)}\sim N\left(\sum_{q\neq
k}p_{qk}\lambda_{q},\sum_{q\neq k}p_{qk}^{2}\xi_{qk}^{2}\right),\qquad\forall
k=1,\dots,K,$ (5)

where $p_{qk}$ are the synthesis weights, with $\sum_{q}p_{qk}=1$, assigned to
the respective commensurate priors specified using $\boldsymbol{x}_{q}$. These
synthesis weights can be transformed from the chosen values for $w_{qk}$ in
the commensurate prior models, following an objective-directed approach (Zheng
and Wason,, 2022). More specifically, we expect the largest synthesis weight,
$p_{qk}$, is assigned to the most commensurate prior
$N(\lambda_{q},\xi_{qk}^{2})$, specified based on a subtrial $q\neq k$ that
manifests the smallest discrepancy with subtrial $k$ out of all the $(K-1)$
complementary subtrials. Recall each $w_{qk}$, as one key parameter to
determine $N(\lambda_{q},\xi_{qk}^{2})$, would have been chosen to
appropriately reflect the pairwise discrepancy (i.e., incommensurability). One
may apply a decreasing function of $w_{qk}$ to compute $p_{qk}$. A $K\times K$
matrix can be constructed to contain all $w_{qk}$ in column $k$ and row $q$
as:

$\begin{pmatrix}0&w_{12}&\cdots&w_{1K}\\\ w_{21}&0&\cdots&w_{2K}\\\
\vdots&\vdots&\ddots&\vdots\\\ w_{K1}&w_{K2}&\cdots&0\end{pmatrix}.$

We note that this matrix should be symmetric with $w_{qk}=w_{kq}$, since each
is intended to reflect the level of pairwise incommensurability. That is, the
magnitude of incommensurability between subtrials $k$ and $q$ is the same as
that between $q$ and $k$. If stratifying the matrix by column, the off-
diagonal elements in column $k$ represent the postulated levels of discounting
with respect to the complementary subtrial data. Recall that the latter have
been used to specify the respective commensurate priors in the form of (4),
with $q\neq k$. The decreasing function given by

$p_{qk}=\frac{\exp(-w_{qk}^{2}/c_{0})}{\sum_{q}\exp(-w_{qk}^{2}/c_{0})},\qquad\forall
k=1,\dots,K,$

has been illustrated to have satisfactory properties (Zheng and Wason,, 2022;
Zheng et al.,, 2022). The concentration parameter $c_{0}$, if set equal to a
value close to $0^{+}$, appropriately discerns the $(K-1)$ values of $w_{qk}$;
thus, a $p_{qk}\rightarrow 1$ would be assigned to the corresponding
commensurate prior for $\theta_{k}$ based on $\boldsymbol{x}_{q}$, in which
the smallest $w_{qk}$ has been used. Otherwise, a value of $c_{0}\gg w_{qk}$
yields nearly all $p_{qk}$ to equal $1/(K-1)$ irrespective of the values of
$w_{qk}$. Moreover, this transformation yields equal $p_{qk}$ when all
$w_{qk}$ are equal. We generally recommend setting $c_{0}$ to a value that is
substantially smaller than the magnitude of $w_{qk}$; for a thorough
evaluation of performance by varying $c_{0}$, we refer the reader to Zheng and
Wason, (2022).

By using the Bayes’ Theorem, the collective commensurate prior in the form of
(5) will be updated by the contemporary subtrial data $\boldsymbol{x}_{k}$ to
give the posterior distribution as

$\theta_{k}\mid\boldsymbol{x}_{k},\boldsymbol{x}_{(-k)}\sim
N\left(d_{\theta_{k}},\left(\frac{1}{\sum_{q}p_{qk}^{2}\xi_{qk}^{2}}+\frac{n_{k}R_{k}(1-R_{k})}{\sigma_{k}^{2}}\right)^{-1}\right).$
(6)

The posterior mean is a convex combination of the prior mean $\sum
p_{qk}\lambda_{q}\,(q\neq k)$ and the data likelihood. We will give the exact
expression of $d_{\theta_{k}}$ in Section 4 to carry out the decision making
for simulated trials.

###  2.2 Sample size formulae for basket trials comparing two normal means

The frequentist approach to sample size determination makes use of hypothesis
testing, with $H_{0k}:\theta_{k}\leq 0$ against $H_{1k}:\theta_{k}>0$, if
assuming that greater values of $X_{ijk}$ indicate better effect. In this
traditional framework, a study sample size is computed such that the treatment
effect, $\theta_{k}$, will be found significant at a level $\alpha$ with
probability $1-\beta$, given a certain magnitude of the treatment effect
considered clinically meaningful.

We follow the Bayesian decision making framework, presented by Whitehead et
al., (2008), to compute two interval probabilities from our posterior
distribution as derived in (6), so that the subtrial sample sizes,
$n_{1},\dots,n_{K}$, are sought for providing compelling evidence of $E$ being
either superior to or not better than $C$ by some magnitude $\delta$ in each
subtrial $k=1,\dots,K$. The posterior distribution of $\theta_{k}$ specific to
each subtrial $k=1,\dots,K$ will thus be evaluated to declare that $E$ is

(i) efficacious, if
$\displaystyle\mathbb{P}(\theta_{k}>0\mid\boldsymbol{x}_{k},\boldsymbol{x}_{(-k)})\geq\eta,$
(7) (ii) futile, if
$\displaystyle\mathbb{P}(\theta_{k}\leq\delta\mid\boldsymbol{x}_{k},\boldsymbol{x}_{(-k)})\geq\zeta,$
(8)

where $\eta$ and $\zeta$ are probability thresholds for the success and
futility criteria, respectively. By using this decision rule, the two
posterior tail probabilities of $\theta_{k}$ are controlled. Specifically, we
desire the area under the density from $-\infty$ to the left of 0 to be
limited below $1-\eta$, with the area under the density from $\infty$ to the
right of $\delta$ to be below $1-\zeta$. The sample size therefore needs to be
sufficiently large for a decisive declaration of the treatment effectiveness
or futility per subtrial $k$. That is, $d_{\theta_{k}}/\sigma_{\theta_{k}}\geq
z_{\eta}$ or $(\delta-d_{\theta_{k}})/\sigma_{\theta_{k}}\geq z_{\zeta}$
should be guaranteed. Here, $z_{\eta}$ satisfies $\Phi(z_{\eta})=\eta$ where
$\Phi(\cdot)$ denotes the standard normal distribution function, with
$z_{\zeta}$ defined similarly. We thus require $\delta/\sigma_{\theta_{k}}\geq
z_{\zeta}+z_{\eta}$, which leads to

$\frac{1}{\sigma_{\theta_{k}}^{2}}\geq\left(\frac{z_{\eta}+z_{\zeta}}{\delta}\right)^{2}.$
(9)

The left-hand side of (9) is precisely the posterior precision for
$\theta_{k}$.

When borrowing of information is not permitted,
$\mathbb{P}(\theta_{k}>0\mid\boldsymbol{x}_{k})$ and
$\mathbb{P}(\theta_{k}\leq\delta\mid\boldsymbol{x}_{k})$ are computed instead.
Thus,
$\frac{1}{\sigma_{\theta_{k}}^{2}}=\frac{1}{s_{0k}^{2}}+\frac{n_{k}R_{k}(1-R_{k})}{\sigma_{k}^{2}}$,
which can be rearranged to give

$n_{k}^{0}\geq\frac{\sigma_{k}^{2}}{R_{k}(1-R_{k})}\left[\left(\frac{z_{\eta}+z_{\zeta}}{\delta}\right)^{2}-\frac{1}{s_{0k}^{2}}\right],\quad\forall
k=1,\dots,K.$ (10)

By contrast, based on the proposed Bayesian model for borrowing of
information, $\sigma_{\theta_{k}}^{2}$ comes from the closed-form expression
in (6). This leads to

$n_{k}\geq\frac{\sigma_{k}^{2}}{R_{k}(1-R_{k})}\left[\left(\frac{z_{\eta}+z_{\zeta}}{\delta}\right)^{2}-\frac{1}{\sum_{q}p_{qk}^{2}\xi_{qk}^{2}}\right],\quad\forall
k=1,\dots,K,\;q\neq k,$ (11)

which looks similar to (10), but involves the commensurate prior variances
$\xi_{qk}^{2}$ in the form of (4). The latter leverages the complementary
subtrial information. Thus, a smaller integer for $n_{k}$ could be expected if
the complementary subtrials, labelled $q\neq k$, are to collect rich
information and, further, considerable borrowing of information happens. To
ensure the inference in all $K$ subtrials, we require that $\forall
k=1,\dots,K$,

$\frac{n_{k}R_{k}(1-R_{k})}{\sigma_{k}^{2}}+\left[\sum_{q}p_{qk}^{2}\left(\left(\frac{1}{s_{0q}^{2}}+\frac{n_{q}R_{q}(1-R_{q})}{\sigma_{q}^{2}}\right)^{-1}+\frac{w_{qk}b_{1}}{a_{1}-1}+\frac{(1-w_{qk})b_{2}}{a_{2}-1}\right)\right]^{-1}\geq\left(\frac{z_{\eta}+z_{\zeta}}{\delta}\right)^{2},$

with $p_{qk}$ transformed from $w_{qk}$ following the stipulation in Section
2.1. The $K$ nonlinear equations of $n_{k}$ and its $n_{q}$s are continuously
differentiable. We apply Newton’s method for systems of nonlinear equations
(Dennis and Schnabel,, 1983) to find $n_{1},\dots,n_{K}$ that satisfy the $K$
constraints simultaneously, given known $w_{qk}$ and $\sigma_{k}^{2}$.

The importance of $w_{qk}$, in computing subtrial sample sizes is of
particular interest. Whilst the definition of data (in)commensurability can
vary on a case-by-case basis, these values may better be specified in
collaboration with a subject-matter expert. Those conversations may also help
quantify the magnitude of incommensurability, particularly in the absence of
pilot data or relevant investigation. We caution that the choice of $w_{qk}$
should reflect the level of pairwise (in)commensurability a priori in the
practical implementation, rather than for the desire of obtaining a minimal
sample size.

##  3 Application

###  3.1 A UK-based basket trial for treating patients with chronic diseases

The randomised, placebo-controlled Obeticholic acid for the Amelioration of
Cognitive Symptoms trial (known as the ‘OACS trial’, ISRCTN15223158) aims to
assess the efficacy of Obeticholic acid ($E$), as compared to a placebo ($C$),
for treating cognitive deficits. The OACS trial is split into three subtrials,
each focusing on a distinct patient subpopulation defined by the disease stage
and clinical area. Namely, OACS-1 for patients with stabilised primary biliary
cholangitis (PBC) $>2$ years since diagnosis; OACS-2 for patients with new-
onset PBC $\leq 2$ years; and OACS-3 for patients with Parkinson’s disease.
The primary outcome is a composite cognitive test score obtained from the
CANTAB platform (Goldberg,, 2013), which is an extensively used tool in
clinical practice. The reduction in the composite CANTAB score from the
baseline, after 26 weeks of treatment, will be analysed as a normally
distributed primary endpoint. We assume the magnitude of such reduction can be
adequately depicted by values ranging from -5 to 5, where a high value
suggests improvement of cognitive symptoms in a patient.

The sample sizes have been determined as 40 (20 on $E$ and $C$ each) for
OACS-1, 25 (15 on $E$ and 10 on $C$) for OACS-2, 25 (15 on $E$ and 10 on $C$)
for OACS-3, assuming that these subtrials are to be analysed on their own. As
specified in the trial protocol, these were not computed based on hypothesis
testing considerations originally. However, the resulting sample sizes are
consistent with
$\sigma_{1}^{2}=6.177,\,\sigma_{2}^{2}=5.134,\,\sigma_{3}^{2}=5.134$ to ensure
90% statistical power for OACS-1 and 80% for the the other subtrials to detect
a difference of $\delta=2.3$, whilst controlling the type I error rate below
0.05. In the following, we use these quantities to illustrate the application
of the proposed methodology, if the OACS basket trial would have been designed
using the Bayesian decision making framework outlined above.

We set $s_{0k}^{2}=100$ and $\nu_{qk}\sim
w_{qk}\text{Gamma}(1.1,1.1)+(1-w_{qk})\text{Gamma}(54,3)$ for all
$k=1,\dots,3$. Specifically, the two Gamma mixture components correspond to
the mean and a 95% credible interval for each $\nu_{qk}$ as 1 [0.041, 4.286]
and 18 [13.522, 23.108], respectively. This accommodates two extreme cases for
limited or substantial borrowing when $w_{qk}=1$ or 0.

If no borrowing is permitted, $n_{k}^{0}=39.8,\,24.8,\,24.8$ according to
(10). This is about the same as the actual sample sizes of 40, 25, 25,
respectively. For illustration purpose only, we use the following matrix of
$w_{qk}$

$\begin{pmatrix}w_{11}&w_{12}&w_{13}\\\ w_{21}&w_{22}&w_{23}\\\
w_{31}&w_{32}&w_{33}\\\ \end{pmatrix}=\begin{pmatrix}0&0.239&0.417\\\
0.239&0&0.145\\\ 0.417&0.145&0\end{pmatrix},$

and $c_{0}=0.05$ to compute the synthesis weights $p_{qk}$, following the
objective-directed approach outlined in Section 2.1. The subtrial sample sizes
are found to be $n_{k}=33.3,\,11.8,\,18.2$, fixing $\eta=0.95$ and
$\zeta=0.90$ for subtrial 1 or 0.80 for subtrial 2 maintaining the same
treatment allocation ratios $R_{k}=0.5,\,0.6,\,0.6$. These are considerably
smaller than the subtrial sample sizes assuming no borrowing or the
frequentist counterparts.

###  3.2 Simultaneous evaluation of a new inhibitor in seven cancer subtypes

The ongoing SUMMIT basket trial (NCT01953926) adopted a single-arm design to
evaluate a new pan-HER kinase inhibitor neratinib in 141 patients with
HER2-mutant or HER3-mutant tumours (Hyman et al.,, 2018). A binary outcome
(i.e., responder or no responder, corresponding to a tumour shrinkage $\geq$
30% or below) was used in line with the RECIST criteria (Eisenhauer et al.,,
2009). The SUMMIT trial additionally reported analysis on secondary outcomes,
which include the change in tumour volume on a continuous scale of -100% to
100%. We assume a new randomised basket trial would follow, wherein the change
in tumour volume from -100% to 100% is the primary outcome. A negative sign
indicates the clinical benefit, since it is hoped that the tumour shrinks from
the baseline measurement due to the treatment. With $\delta<0$, the trial
decision criterion expressed by (7) and (8) should be altered as (i)
efficacious, if
$\mathbb{P}(\theta_{k}<0\mid\boldsymbol{x}_{k},\boldsymbol{x}_{(-k)})\geq\eta$,
and (ii) futile, if
$\mathbb{P}(\theta_{k}\geq\delta\mid\boldsymbol{x}_{k},\boldsymbol{x}_{(-k)})\geq\zeta$.

We narrow the focus on seven of the originally investigated 21 cancer subtypes
only, of which the mean responses for patients receiving neratinib ($E$) were
approximately $\mu_{Ek}=-0.489,0.226$, $-0.181,0.293$, $0.329,-0.275,-0.136$.
We further assume that the mean responses on a control treatment embedded in
the newly planned basket trial are $\mu_{Ck}=0$, and that patients within each
subtrial have equal probability to receive $E$ or $C$; that is, $R_{k}=0.5$
for $k=1,\dots,7$.

Based on the published SUMMIT trial results (Hyman et al.,, 2018), we assume
the subtrial-specific variances are $\sigma_{k}^{2}=0.587^{2}$,
$0.345^{2},\,0.380^{2},\,0.347^{2},\,0.344^{2},\,0.392^{2},\,0.392^{2}$. The
pairwise discrepancy between the assumed outcome distributions,
$N(\mu_{Ek},\sigma_{k}^{2})$, can be quantified using, e.g., the Hellinger
distance (Dey and Birmiwal,, 1994),

$w_{qk}=\left[1-\sqrt{\frac{2\sigma_{q}\sigma_{k}}{\sigma_{q}^{2}+\sigma_{k}^{2}}}\exp\left(-\frac{(\mu_{Eq}-\mu_{Ek})^{2}}{4(\sigma_{q}^{2}+\sigma_{k}^{2})}\right)\right]^{1/2}.$

By targeting $\delta=-0.40$ and retaining the specification of other
parameters from Section 3.1,
$n_{k}=52.0,\,17.3,\,20.5,\,17.0,\,17.1,\,22.5,\,22.0$. If no borrowing is
permitted, $n_{k}^{0}=53.3,\,18.4,\,22.3,\,18.6$, $18.3,\,23.8,\,23.8$. Only a
small reduction in the sample sizes has been observed, largely because most of
the values of $w_{qk}$ are above 0.30.

##  4 Simulation study

###  4.1 Basic setting

Motivated by the SUMMIT trial, we consider the sample size planning of basket
trials following the same data structure. That is, the basket trial would
enroll $n_{1},\dots,n_{7}$ patients to the respective subtrials, with
$R_{k}=0.5$ in all $K=7$ subgroups, under six possible scenarios. Figure 1
visualises the six simulation scenarios, where the location and length of
lines suggest the distributions of $X_{ijk},\,j=E,C$, while a larger bubble
corresponds to a larger value of $w_{qk}$. Here, we have followed the
specification of $w_{qk}$ given in Section 3.2 to compute the pairwise
Hellinger distance to characterise (in)commensurablity and obtain $w_{qk}$ for
the levels of borrowing/discounting strength. Scenarios 4 and 6 correspond to
two special cases of the treatment being consistently effective (alternative
hypotheses) and consistently futile (null hypotheses), respectively. Both
scenarios feature perfect commensurability; that is, the outcomes $X_{iEk}$
and $X_{iCk}$ have their respective, same distribution across subtrials, so
all $w_{qk}=0$. Scenario 5 represents a mixed null situation, where
$\theta_{k}=0$ holds for four of the subtrials only. The other scenarios are
constructed to reflect various levels of data incommensurability. Exact
configurations of these simulation scenarios, i.e, values of $\mu_{Ek}$ along
with all $\mu_{Ck}=0$ as well as the subtrial-specific variances
$\sigma_{k}^{2}$, are listed in Table S1 of the Supplementary Materials.

[Figure 1 about here.]

We retain the prior specification and the probability thresholds unchanged
from Section 3. Table 1 thus gives the subtrial sample sizes required to reach
a decisive conclusion of either $E$ is superior to or not better than $C$ by
$\delta=-0.4$ using the respective sample size formulae. Because no $w_{qk}$
has been set to 1 in any scenario, the sample sizes $n_{k}$ computed from the
approach of borrowing are generally smaller than $n_{k}^{0}$ from the approach
of no borrowing. Scenario 1 was constructed from the illustrative data example
in Section 3.2. Since relatively large values have been chosen for $w_{qk}$,
only a slight decrease in sample sizes is observed. Unlike Scenario 1 that
displays divergent effects, Scenario 2 is featured with a higher degree of
commensurability that $E$ has an enhanced benefit over $C$ in all subtrials. A
smaller trial sample size is then required. Scenario 3 has all the variances
$\sigma_{k}^{2}=0.3$. Consequently, $n_{k}^{0}$, based on the approach of no
borrowing, are solved to be equal to 46.4 for all subtrials. Whereas, using
the proposed methodology, the sample sizes for subtrials 1, 3, 4 and 6 are
smaller, as these are recognised to be more commensurate between themselves
than with the other three. A similar explanation can be given to Scenario 5:
subtrials 2, 4 and 5 have greater sample size savings because the
corresponding $w_{qk}$ takes smaller values. Scenarios 4 and 6 represent the
situations of perfect commensurability. With all $w_{qk}=0$, a substantial
reduction in the subtrial sample sizes is observed.

[Table 1 about here.]

In the numerical evaluation below, we simulate the outcomes $X_{ijk},j=E,C,$
from $N(\mu_{Ek},\sigma_{k}^{2})$ and $N(\mu_{Ck},\sigma_{k}^{2})$ for patient
$i=1,\dots,n_{k},$ within subtrial $k=1,\dots,7$. For each scenario, 100,000
replicates of the basket trials are simulated to fit:

  * •

the proposed Bayesian model, which yields the posterior distributions for
$\theta_{k}$ in the form of (6), with

$d_{\theta_{k}}=\frac{\sigma_{k}^{2}/(n_{k}R_{k}(1-R_{k}))\cdot\sum_{q}p_{qk}\lambda_{q}+(\bar{x}_{Ek}-\bar{x}_{Ck})\cdot\sum_{q}p_{qk}^{2}\xi_{qk}^{2}}{\sum_{q}p_{qk}^{2}\xi_{qk}^{2}+\sigma_{k}^{2}/(n_{k}R_{k}(1-R_{k}))},$

  * •

a Bayesian stand-alone analysis model for no borrowing of information.
Operational priors, i.e., $N(m_{0k},s_{0k}^{2})$, are placed on each
$\theta_{k}$. This leads to the posterior distributions for $\theta_{k}$ based
on $\boldsymbol{x}_{k}$ alone, which has the same form as (3), with the
subscript $q$ replaced by $k$. We set all $m_{0k}=0$ in the simulations.

###  4.2 Results

We summarise the frequency of simulated trials concluding that $E$ is either
efficacious or futile, based on the 100,000 replicates per scenario and model.
Figure 2 depicts the percentages of (sub)trials declaring effectiveness of $E$
and those declaring futility. Wherever the lengths of bars sum up to 100%,
this means the study is planned with an adequate sample size for decisive
decision making. As we can observe, collecting data from $n_{1},\dots,n_{7}$
patients to fit the proposed analysis model ensures 100% of the (sub)trials to
conclude that $E$ is either superior to or not better than $C$ by $\delta$.
Whereas, it is not the case (i.e., all below 100%) if implementing the
Bayesian model for no borrowing, since larger sample sizes (i.e., $n_{k}^{0}$
in Table 1) would be required to ensure the same level of posterior
distribution informativeness for the trial decision. In Scenarios 1 and 2
where $n_{k}$ and $n_{k}^{0}$ were comparable, these two Bayesian models
yielded comparable proportions of (sub)trials with a decisive trial decision.
Yet in Scenarios 4 and 6 where substantial sample size saving were made, a
disparity is observed, because the posterior distributions for $\theta_{k}$
based on $\boldsymbol{x}_{k}$ alone are far less informative than those based
on $\boldsymbol{x}_{1},\dots,\boldsymbol{x}_{K}$.

[Figure 2 about here.]

In Scenarios 2 and 3, $E$ is potentially superior than $C$ yet the magnitude
tends to be smaller than desired on average. Only subtrial 1 has a mean
treatment effect greater than $\delta$, so about 91.2% of the simulated
(sub)trials have declared $E$ being effective. By contrast, subtrials 2 and 5
have mean treatment effects closest to 0 and $\delta$, respectively.
Therefore, subtrial 2 has higher chance to declare futility than effectiveness
of $E$, but subtrial 5 is on the contrary. Scenario 4 mimics the borderline
case where the mean treatment effect just has the size of $\delta$. Using the
proposed methodology, about 82.1% of the simulated (sub)trials have favoured
$E$ for effectiveness in all seven subtrials. These subtrialwise true positive
rates are about equal to our chosen threshold $\zeta=0.80$. Scenario 5 assumes
a mixture of subtrial-specific treatment effects with $\theta_{k}=0$ or
$\geq\delta$. Referring to subtrials 2, 4, 5 and 7, less than 5% of the
simulated trials conclude effectiveness erroneously. The two Bayesian models
yield similar operating characteristics in this scenario, as the computed
$n_{k}$ and $n_{k}^{0}$ were close. In Scenario 6, the proportion of incorrect
decision of effectiveness is maintained below 5% for all subtrials using the
proposed methodology. Unsurprisingly, using the approach of no borrowing to
analyse the basket trial from only 62.3 patients has much lower chance of
obtaining a definitive conclusion. The overall false positive rate (i.e.,
probability of incorrectly rejecting at least one subtrial that has true
$\theta_{k}=0$), based on the proposed methodology, is 0.150 for Scenario 5
and 0.054 for Scenario 6. These increase to 0.192 and 0.346, respectively, if
the approach of no borrowing is implemented instead. This is not surprising
because the sample sizes were computed to control the error rate at the
subtrial level. For strong control of the overall error rate, multiplicity
adjustment such as the Bonferroni procedure is required. After the correction,
one can still expect more benefit from borrowing information than not.

Focusing on Scenarios 4 – 6 for the true positive and false positive rates at
the subtrial levels, the proportions are not exactly 80% or 5% because of the
simulation randomness. Additional simulations have been carried out for
homoscedastic scenarios by varying the value of $\sigma_{k}^{2}$. Figure 3
shows (i) the subtrialwise sample sizes, $n_{k}$, determined based on our
sample size formula in (11), and correspondingly, (ii) the subtrialwise true
positive and false positive rates based on the simulated 100,000 replicates of
basket trials. Each set of the additional simulations yielded seven points
(for $K=7$ subtrials), which congregate at the levels around $\zeta=0.80$ or
$1-\eta=0.05$. In summary, the proposed methodology can lead to the control of
error rates.

[Figure 3 about here.]

We have also performed a sensitivity analysis to understand the effect of
misspecified values of $w_{qk}$. Table S2 in the Supplementary Materials
reveals that the proposed methodology is reasonably robust to the
misspecification of $w_{qk}$. Nonetheless, care is needed when the value of
$w_{qk}$ in the analysis deviates too far from that used in the design. When
$w_{qk}$ is set to a larger value in the analysis than in the design (i.e.,
less borrowing is implemented than planned), a smaller percentage of trials
conclude with a decisive decision. Whereas, a smaller value of $w_{qk}$ would
yield a more informative posterior distribution, but this sometimes produces
ambiguous conclusion of effectiveness or futility.

##  5 Discussion

The importance of choosing an appropriate sample size can never be
overemphasised (Senn,, 2007). Whilst basket trials have major infrastructural
and logistical advantages, sophisticated statistical models are needed for the
sample size planning to preserve the added efficiency. The most widely-used
approach to date is based on a Bayesian stand-alone analysis model, which does
not support information sharing across subtrials with commensurate treatment
effects. Consequently, the majority of basket trials recruit a higher sample
size than required. This not only causes a waste of resources, but could
sometimes be unethical for exposing more patients than is necessary to a
treatment that is yet to be fully approved (Altman,, 1980). To realise the
promise of basket trials, this paper establishes a closed-form solution to the
simultaneous determination of subtrial sample sizes. The simulation study
shows that the proposed methodology requires a smaller trial sample size
wherever $0\leq w_{qk}<1$, without undermining the chance of detecting if
there exists a clinically relevant difference between the experimental
treatment and the control.

For deriving our sample size formulae, we adopted the Bayesian decision making
scheme elaborated by Whitehead et al., (2008). Specifically, it involves two
probability thresholds $\eta$ and $\zeta$ for reaching a decisive statement on
the treatment’s effectiveness or futility. In our numerical illustration, we
set $\eta=0.95$ and $\zeta=0.80$ because these probability quantities yield
$n_{k}^{0}$, obtained based on the approach of no borrowing, comparable to the
frequentist solution of sample sizes with $\alpha=0.05$ and $\beta=0.20$.
Other choices may certainly be feasible: there is no conventional level to set
these probability thresholds. In practice, these quantities might be difficult
to justify: fixing $\eta$ to, for example, 0.95 might mean a considerable
increase in sample size compared to 0.90. Since the sample sizes also depend
on other parameters, we recommend the user generates plots for their cases
following the pattern of our Figure S2 in the Supplementary Materials.

Two sets of key parameters to implement the proposed methodology are the
variances $\sigma_{k}^{2}$ and the levels of pairwise discounting $w_{qk}$.
Similar to the widely-used frequentist formulae (Chow et al.,, 2007), an
increase in $\sigma_{k}^{2}$ would mean that larger sample sizes are needed to
maintain the same level of precision in data. We have restricted our focus on
known variances throughout, since this is common in conducting clinical trials
for most circumstances. Appropriate values for $\sigma_{k}^{2}$ to compute the
subtrial sample sizes can be informed by pilot data or information from
relevant investigations. If retaining $\sigma_{k}^{2}$ as unknown parameters,
priors about their magnitude would be required. Subtrial sample sizes would
then be sought by controlling the average property of posterior interval
probabilities of $\theta_{k}$ with respect to 0 and $\delta$, since these
nuisance parameters need to be integrated out from the posterior. Zheng et
al., (2022) derived sample size formulae with unknown $\sigma_{k}^{2}$ for a
relevant context. Although different decision criteria were considered, one
could follow their methodology to obtain the marginal posterior distributions,
for which external information on $\sigma_{k}^{2}$ may further be
incorporated. For wider applications, we have extended the proposed
methodology for basket trials using a binary (in both the randomised
controlled and single-arm settings) or time-to-event outcome; see Sections C –
E of the Supplementary Materials for the corresponding sample size formulae.
In the meanwhile, we note there are limitations; for example, the censoring
assumptions for time-to-event data are greatly simplified. We hope this work
will stimulate further research within this Bayesian decesion framework.

In the present work, data are supposed to be analysed after the completion of
all subtrials. However, in practice, certain subtrials may take much longer to
complete recruitment due to low prevalence. One could (a) adopt a ‘first
(subtrials) complete, first analysed’ principle, or (b) alter the constraint
for simultaneously solving $n_{1},\dots,n_{K}$, e.g., by making them
proportional to the prevalence of respective target subpopulations, whilst
maintaining an overall statistical power or decision accuracy. For strategy
(a), more borrowing would be possible from subtrials that complete faster to
those that complete slower. With strategy (b), all subtrials may finish about
the same time to yield a joint data analysis. We should note that it is not
obvious if either strategy leads to a substantial increase in the total sample
size.

When borrowing of information is permitted, a reduced sample size can be
expected by setting $w_{qk}<1$. The smaller the values of $w_{qk}$, the more
borrowing is possible. The present methodology requires that these values be
specified to reflect the pairwise (in)commensurability of subtrial data. This
is especially feasible when a pilot study has been conducted. More details
about the practical implementation, particularly the specification of
parameters, are available in Section F of the Supplementary Materials.
Throughout, we have elaborated the methodology concerning a pre-specified
magnitude relating to the effect size, $\delta$, to find subtrial sample
sizes. Extending the calculation to consider subtrial-specific effective
sizes, say, $\delta_{k}$, is straightforward. A smaller value of $\delta_{k}$
would indicate that a larger $n_{k}$ is needed, if all other parameters are
held fixed. For practical implementation, the user may substitute the
corresponding argument (currently as a single value) by a vector in the openly
available software.

Our sensitivity analysis in Section G of the Supplementary Materials suggests
the proposed methodology is reasonably robust against misspecification of
$w_{qk}$. Nevertheless, when the values deviate too far from the truth, the
resulting sample sizes would not reflect what is needed to achive the trial’s
objectives. One avenue for future research would therefore be developing
methodology for sample size reassessment in basket trials. Practitioners may
start with rather conservative choices of $w_{qk}$ assuming limited borrowing,
and re-estimate $w_{qk}$ using accumulating data from the ongoing trial at
interims. As the reassessment depends on observed early-stage data, there is a
risk that the error rates across stages cannot be maintained at the intended
levels and subsequent work will investigate avoiding this inflation. The
proposed methodology may also be extended to enable mid-course adaptations.
For instance, the basket trial will begin with a few subsets of interest, and
then restrict enrollment to the ones wherein patients benefit satisfactorily
from the treatment based on an interim analysis. Boundaries for early stopping
of certain subsets must be carefully defined to protect the overall error
rates. With a reduction in the number of subsets, the synthesis weights
$p_{qk}$ should be updated to satisfy the constraint of $\sum_{q}p_{qk}=1$ for
the late stage(s).

## Software

All statistical computing and analyses were performed using the software
environment R version 4.0.3. Programming code for implementing the sample size
formulae and reproducing the numerical results, is available at
https://github.com/haiyanzheng/BasketTrialsSSD.

## Acknowledgment

This work was supported by Cancer Research UK through Dr Zheng’s Population
Research Postdoctoral Fellowship  (RCCPDF\100008 ). Prof Jaki and Dr Mozgunov
received funding from the UK Medical Research Council (MC UU 00002/14). This
report is independent research supported by the National Institute for Health
Research (NIHR) Cambridge Biomedical Research Centre (BRC-1215-20014) and the
NIHR Advanced Fellowship (Dr Mozgunov, NIHR300576). The views expressed in
this publication are those of the authors and not necessarily those of the
NHS, the NIHR or the Department of Health and Social Care (DHSC).

Conflict of Interest: None declared.

## References

  * Altman, (1980) Altman, D. G. (1980).  Statistics and ethics in medical research: III How large a sample?  BMJ, 281(6251):1336–1338.
  * Aronson and Rehm, (2015) Aronson, S. and Rehm, H. (2015).  Building the foundation for genomics in precision medicine.  Nature, 526(7573):336–342.
  * Berry et al., (2013) Berry, S. M., Broglio, K. R., Groshen, S., and Berry, D. A. (2013).  Bayesian hierarchical modeling of patient subpopulations: Efficient designs of phase ii oncology clinical trials.  Clinical Trials, 10(5):720–734.
  * Chen and Lee, (2020) Chen, N. and Lee, J. J. (2020).  Bayesian cluster hierarchical model for subgroup borrowing in the design and analysis of basket trials with binary endpoints.  Statistical Methods in Medical Research, 29(9):2717–2732.
  * Chow et al., (2007) Chow, S., Wang, H., and Shao, J. (2007).  Sample Size Calculations in Clinical Research, Second Edition.  Chapman & Hall/CRC Biostatistics Series. Taylor & Francis.
  * Chu and Yuan, (2018) Chu, Y. and Yuan, Y. (2018).  BLAST: Bayesian latent subgroup design for basket trials accounting for patient heterogeneity.  Journal of the Royal Statistical Society Series C, 67(3):723–740.
  * Dennis and Schnabel, (1983) Dennis, J. E. and Schnabel, R. B. (1983).  Newton’s method for nonlinear equations and unconstrained minimization.  In Numerical Methods for Unconstrained Optimization and Nonlinear Equations, Classics in Applied Mathematics, chapter 5. Prentice-Hall, Englewood Cliffs, NJ.
  * Dey and Birmiwal, (1994) Dey, D. K. and Birmiwal, L. R. (1994).  Robust Bayesian analysis using divergence measures.  Statistics & Probability Letters, 20(4):287–294.
  * Eisenhauer et al., (2009) Eisenhauer, E. A., Therasse, P., and Bogaerts, J. et al.. (2009).  New response evaluation criteria in solid tumours: Revised recist guideline (version 1.1).  European Journal of Cancer, 45:228–247.
  * Goldberg, (2013) Goldberg, M. C. (2013).  Cantab.  In Encyclopedia of Autism Spectrum Disorders, pages 519–522. Springer New York, New York, NY.
  * Grayling et al., (2021) Grayling, M. J., Bigirumurame, T., Cherlin, S., Ouma, L., Zheng, H., and Wason, J. M. S. (2021).  Innovative trial approaches in immune-mediated inflammatory diseases: current use and future potential.  BMC Rheumatology, 5(1):21.
  * Grayling et al., (2019) Grayling, M. J., Dimairo, M., Mander, A. P., and Jaki, T. F. (2019).  A Review of Perspectives on the Use of Randomization in Phase II Oncology Trials.  JNCI: Journal of the National Cancer Institute, 111(12):1255–1262.
  * Grinstead and Snell, (1997) Grinstead, C. M. and Snell, J. L. (1997).  Introduction to Probability.  Open Textbook Library. American Mathematical Society.
  * Hobbs et al., (2011) Hobbs, B. P., Carlin, B. P., Mandrekar, S. J., and Sargent, D. J. (2011).  Hierarchical commensurate and power prior models for adaptive incorporation of historical information in clinical trials.  Biometrics, 67(3):1047–1056.
  * Hobbs and Landin, (2018) Hobbs, B. P. and Landin, R. (2018).  Bayesian basket trial design with exchangeability monitoring.  Statistics in Medicine, 37(25):3557–3572.
  * Hoeting et al., (1999) Hoeting, J. A., Madigan, D., Raftery, A. E., and Volinsky, C. T. (1999).  Bayesian model averaging: a tutorial (with comments by M. Clyde, David Draper and E. I. George, and a rejoinder by the authors.  Statist. Sci., 14(4):382–417.
  * Hyman et al., (2018) Hyman, D. M., Piha-Paul, S. A., Won, H., and others. (2018).  HER kinase inhibition in patients with HER2-and HER3-mutant cancers.  Nature, 554(7691):189–194.
  * Jin et al., (2020) Jin, J., Liu, Q., Zheng, W., Lin, T., Gao, L., and Dong, Y. (2020).  A Bayesian method for the detection of proof of concept in early phase oncology studies with a basket design.  Statistics in Biosciences, 12:167–179.
  * Kim et al., (2011) Kim, E. S., Herbst, R. S., Wistuba, I. I., et al. (2011).  The BATTLE trial: Personalizing therapy for lung cancer.  Cancer Discovery, 1(1):44–53.
  * Liu et al., (2017) Liu, R., Liu, Z., Ghadessi, M., and Vonk, R. (2017).  Increasing the efficiency of oncology basket trials using a Bayesian approach.  Contemporary Clinical Trials, 63:67–72.
  * McInnes and Gravallese, (2021) McInnes, I. B. and Gravallese, E. M. (2021).  Immune-mediated inflammatory disease therapeutics: past, present and future.  Nature Reviews Immunology, 21(10):680–686.
  * Mirnezami et al., (2012) Mirnezami, R., Nicholson, J., and Darzi, A. (2012).  Preparing for precision medicine.  New England Journal of Medicine, 366(6):489–491.
  * Morganti et al., (2019) Morganti, S., Tarantino, P., Ferraro, E., D’Amico, P., Duso, B. A., and Curigliano, G. (2019).  Next generation sequencing (ngs): A revolutionary technology in pharmacogenomics and personalized medicine in cancer.  In Ruiz-Garcia, E. and Astudillo-de la Vega, H., editors, Translational Research and Onco-Omics Applications in the Era of Cancer Personal Genomics, pages 9–30. Springer International Publishing, Cham.
  * Neuenschwander et al., (2016) Neuenschwander, B., Wandel, S., Roychoudhury, S., and Bailey, S. (2016).  Robust exchangeability designs for early phase clinical trials with multiple strata.  Pharmaceutical Statistics, 15(2):123–134.
  * Pitzalis et al., (2020) Pitzalis, C., Choy, E. H. S., and Buch, M. H. (2020).  Transforming clinical trials in rheumatology: towards patient-centric precision medicine.  Nature reviews. Rheumatology, 16(10):590–599.
  * Prasad and Oseran, (2015) Prasad, V. and Oseran, A. (2015).  Do we need randomised trials for rare cancers?  European Journal of Cancer, 51(11):1355–1357.
  * Psioda et al., (2021) Psioda, M. A., Xu, J., Jiang, Q., Ke, C., Yang, Z., and Ibrahim, J. G. (2021).  Bayesian adaptive basket trial design using model averaging.  Biostatistics, 22(1):19–34.
  * Ratain and Sargent, (2009) Ratain, M. J. and Sargent, D. J. (2009).  Optimising the design of phase II oncology trials: The importance of randomisation.  European Journal of Cancer, 45(2):275–280.
  * Redman et al., (2020) Redman, M. W., Papadimitrakopoulou, V. A., Minichiello, K., et al. (2020).  Biomarker-driven therapies for previously treated squamous non-small-cell lung cancer (Lung-MAP SWOG S1400): a biomarker-driven master protocol.  The Lancet Oncology, 21(12):1589–1601.
  * Renfro and Sargent, (2017) Renfro, L. and Sargent, D. (2017).  Statistical controversies in clinical research: basket trials, umbrella trials, and other master protocols: a review and examples.  Annals of Oncology, 28(1):34–43.
  * Schork, (2015) Schork, N. J. (2015).  Personalized medicine: Time for one-person trials.  Nature, 520(7549):609–611.
  * Senn, (2007) Senn, S. (2007).  Determining the sample size.  In Senn, S. and Barnett, V., editors, Statistical Issues in Drug Development, Second Edition, pages 195–212. John Wiley & Sons, Ltd, West Sussex, England.
  * Tao et al., (2018) Tao, J. J., Schram, A. M., and Hyman, D. M. (2018).  Basket studies: Redefining clinical trials in the era of genome-driven oncology.  Annual Review of Medicine, 69(1):319–331.
  * Thall et al., (2003) Thall, P. F., Wathen, J. K., Bekele, B. N., Champlin, R. E., Baker, L. H., and Benjamin, R. S. (2003).  Hierarchical bayesian approaches to phase ii trials in diseases with multiple subtypes.  Statistics in Medicine, 22(5):763–780.
  * Whitehead et al., (2008) Whitehead, J., Valdés-Márquez, E., Johnson, P., and Graham, G. (2008).  Bayesian sample size for exploratory clinical trials incorporating historical data.  Statistics in Medicine, 27(13):2307–2327.
  * Woodcock and LaVange, (2017) Woodcock, J. and LaVange, L. M. (2017).  Master protocols to study multiple therapies, multiple diseases, or both.  New England Journal of Medicine, 377(1):62–70.
  * Zheng et al., (2022) Zheng, H., Jaki, T., and Wason, J. M. S. (2022).  Bayesian sample size determination using commensurate priors to leverage pre-experimental data.  Biometrics, pages 1–29.  Epub ahead of print.
  * Zheng and Wason, (2022) Zheng, H. and Wason, J. M. S. (2022).  Borrowing of information across patient subgroups in a basket trial based on distributional discrepancy.  Biostatistics, 23(1):120–135.

Figure 1: Simulation scenarios depicted by the distributions of
$X_{ijk},j=E,C,\,k=1,\dots,7$, along with the visualisation of $w_{qk}$ for
pairwise discounting of incommensurate complementary subtrial data. The green
triangle and red circle mark the normal means $\mu_{Ek}$ and $\mu_{Ck}$,
respectively; the endpoints of each horizontal bar are computed by
$\mu_{jk}\pm 1.96\sigma_{k}^{2}$. The dashed and dotted vertical lines
indicate the effect size of $\delta$ and naught, respectively.

Table 1: The required subtrial sample sizes, computed using the proposed methodology or the approach of no borrowing, to detect a difference of $\delta=-0.4$, when setting $\eta=0.95,\zeta=0.80$. |  | Required subtrial sample size |
---|---|---|---
|  | $k=1$ | $k=2$ | $k=3$ | $k=4$ | $k=5$ | $k=6$ | $k=7$ | Total
Scenario 1 |  $n_{k}^{0}$ (No borrowing) | 53.2 | 18.4 | 22.3 | 18.6 | 18.3 | 23.7 | 23.7 | 178.2
$n_{k}$ (Proposed) | 52.0 | 17.3 | 20.5 | 17.0 | 17.1 | 22.5 | 22.0 | 168.4
Scenario 2 |  $n_{k}^{0}$ (No borrowing) | 53.2 | 18.4 | 22.3 | 18.6 | 18.3 | 23.7 | 23.7 | 178.2
$n_{k}$ (Proposed) | 50.6 | 15.7 | 17.4 | 15.1 | 15.6 | 18.9 | 19.6 | 152.9
Scenario 3 |  $n_{k}^{0}$ (No borrowing) | 46.4 | 46.4 | 46.4 | 46.4 | 46.4 | 46.4 | 46.4 | 324.8
$n_{k}$ (Proposed) | 23.3 | 32.0 | 22.6 | 24.4 | 32.9 | 23.3 | 30.2 | 188.7
Scenario 4 |  $n_{k}^{0}$ (No borrowing) | 46.4 | 46.4 | 46.4 | 46.4 | 46.4 | 46.4 | 46.4 | 324.8
$n_{k}$ (Proposed) |  8.9 |  8.9 |  8.9 |  8.9 |  8.9 |  8.9 |  8.9 |  62.3
Scenario 5 |  $n_{k}^{0}$ (No borrowing) | 53.2 | 18.4 | 22.3 | 18.6 | 18.3 | 23.7 | 23.7 | 178.2
$n_{k}$ (Proposed) | 50.8 | 14.3 | 20.4 | 14.5 | 14.2 | 22.1 | 20.7 | 157.0
Scenario 6 |  $n_{k}^{0}$ (No borrowing) | 46.4 | 46.4 | 46.4 | 46.4 | 46.4 | 46.4 | 46.4 | 324.8
$n_{k}$ (Proposed) |  8.9 |  8.9 |  8.9 |  8.9 |  8.9 |  8.9 |  8.9 |  62.3

Figure 2: Percentage of (sub)trials that conclude $E$ is efficacious (the left
half of each plot) or not better than $C$ by $\delta=-0.4$, i.e., observing a
shrinkage of 40% in the tumour volume (the right half of each plot). True
subtrial-specific treatment effects, $\theta_{k}=\mu_{Ek}-\mu_{Ck}$, have been
indicated in a second $y$-axis.

Figure 3: The true and false positive rates per subtrials $k=1,\dots,7$,
summarised based on the 100,000 replicates of basket trials, wherein $X_{iEk}$
are generated from $N(-0.4,\sigma_{k}^{2})$ and $N(0,\sigma_{k}^{2})$
distributions for panels (i) and (ii), respectively. The dashed lines indicate
the pre-specified level of $\zeta=0.80$ or $1-\eta=0.05$.

See pages 1-9 of Supplementary_basketSSE.pdf