A
R.AIftOI
HORIZON JDlEL POR SEQUDTIAL CLIIIICAL TRIALS
by
Gordon Simons *
University of North Carolina
ry
SOl
The risk in a trial to cOllpare-two-,Mdi.Ga:l-treatllents is borne by
the patients who receive the"1Rle~!&P--t~eatllent during the
experimental ph~se and qy- ~o~!::!!!i!!~!M~"t~et' 'the experimental
phase who rece 1 ve the.. ~Dferror_.:tr.e~tal!Iltt-.n the resu lts are
misleading.
Under the L.8MuaptJ.;oA that ,the total number of
treated patients is ra:htto.·.-·~~·-.ltntwlffi··described and the
exact stopping rule lit; _dlf"'~i[~~~1t:"!~.n,,"_l1_Qw to stop the
exper illental phase so as ~t:o ..:aax;1i1l1z~:':"'ttUl ..~pected number of
successfully treatedpa£iettta·;r.:..\i':MahY·~.ult-s--:.are--derived
w hich
precisely describe th~~·jl~~"ll!n:~J'=l!:~~!!!!!·_o~_.this optimally
designed sequential cllniClU-.. tJ'i~l[._..:._.._-,:.~_~._
d
I
"--..·----T··----~-
.l
AMS 1980 subject
,.... .~"
classl~lcatlons:
Key words and phrases:
stopping, Markov states.
*
......_.'
~
,~.'
.l.
~..__
.,__
Primary 60G40. 62C10. 62L05.
Clinical trials.
Bayes rules,
optimal
This research was supported by the National Sience Foundation
Grant DMS-8400602.
1. Introduction.
Consider the task of assigning two possible treatments to a random number
of patients who suffer from the same illness.
assign
the
treatment with
the
larger
The nominal objective is to
probability
of success to as many
The assignment procedure begins with a testing phase,
patients as possible.
during which both treatments are to be used, and ends with a utili ty phase,
during which the most promising treatment is to be used for all
patients (if any).
remaining
Throughout the testing phase, the treatments are to be
assigned randomly to patients in
receiving distinct treatments.
pairs,
It
with
the
patients
in
each
pair
is assumed the response time is short,
depending on the patient accrual rate, so that a decision to switch to the
utility phase can be made in a timely manner.
Many variants of this model have been described in the literature, but
typically with a "fixed horizon", i.e., with a fixed nUJBber of patients, known
in advance.
Needless to say, this is an unnatural assumption, and it has met
with much criticism.
The assumption used here, of a randoll number of patients
(a "randoll horizon"), is intended to overcome such criticism.
Let
PI
and
pz
denote the probabilities of success for the the first
has an
and second treatments, respectively, and assume the pair
exchangeable
prior
treatment preferred.
distribution;
Further, let
the
clinical
trial
begins
with
neither
M denote the random number of pairs of
- 1 -
GORDON SIMONS
patients who are to be treated.
It
is assumed here that
M
has a known
distribution, and that it is independent of the responses of the patients to
their treatments, and independent of
might be that the distribution of
(Pl,P2)'
(A more realistic assumption
M depends on the perceived superiority of
the preferred treatment when the utility phase begins, or depends on how this
perception evolves during the utility phase.
(Pl, P2 ) . )
might just depend on the value of
the value of
M
is knolm in advance;
More simply, its distribution
Importantly, i t is not assumed
the decision to continue or stop the
testing phase, at each stage, is to be made without knowing how many pairs of
patients remain to be treated.
The
task
minimizes
Pl-P2'
the
is
to
find
the rule for stopping the testing phase which
"expected successes lost"
due to
ignorance of the
Mathematically, this is the expectation of the product of
and the total number of patiat11' as:s}g!mld,
inferior treatment during
'Iby·!th~
bot:h·,.~.,...t.ent;rpJialses.; Here,
stopping rule,
sign
of
I Pl-P21
to the
"expectation" includes
.G
The main objective in
special case: "two-point
this:JpapeJ'Ci~'to
8~etric,
n
pri'lllr,s'f;!anctoieametrically distributed
The presumption is that this theory
case.
\\'d.Crtbe ".a detailed theory for a
rev~lsjiaportant
M.
aspects of the general
This is more than wishful thinking." ';Berry (1978) has shown, for a
two-armed bandit context, that symmetric priors can be replaced by two-point
symmetric priors with little adverse affect.
And Simons (1986) has provided
evidence that Berry's observations are valid in the present setting when
is a known constant, i.e., for the case of a fixed horizon.
- 2 -
M
e-
RANDOM HORIZON MODEL FOR CLINICAL TRIALS
Section
2
describes
a
general
mathematical
formulation.
This
is
specialized in Section 3 to the case of a two-point symmetric prior, and the
optimal stopping rule is described qualitatively (Theorem 1).
theory,
referred to above, appears in Section 4.
that are valid for all values of
E(M).
The detailed
It includes many results
Many of the theorems and corollaries
contain asymptotic results as well (as
~ ~).
E(M)
These provide additional
information, or inforllation that is easier to interpret.
Moreover,
it
is
anticipated that the asymptotic results are valid for a much wider (but yet to
be determined) context.
2. General . .the. .tical
fo~lation.
Since some of what follows requires
Jl'
~o
have a finite expectation, it
will be assumed hereafter that;E(M) < ~t'i';'
It is convenient to thlnk·.;jn .tQ.,",$ ·,;Qf.
n,
one knows either:
(i)
Ml~~:
n
"~. .
; ,eft", n = 0,1,···.
i:-an:d"c~h'e.,J:teQ)0D8e'6oof ,~·,n
pairs of patients
under the testing phase, or (ii) M < nand; th~!;t:!espomU:t8 of
patients under
the
testing phase.
successes lost (so far)
At tille
n,
E.
n
of
the
nand
M.
+
and
+
their
+
2E {(M-n) ·(Pl-PZ)}
n
or
conditional
(VIM
denotes the
the:,utiHty phase at tille
2(M-n) ;"'- ·i·ipatient~:.will
remaining
treatllent,
If"·l)ne.~.itcltes'to
pairs of
denotes conditional
expectation given the infol'uttoDlavai:ltab.he, at -:-ti:.tR.e::;"n, and
minimum of
M
the conditional expected
is:,>{,;Bl{:~ll""IP'I."'Pl4ql~1where
n
At time
expected
+-
2E {(M-n) ·(Pl-PZ)},
n
be siven
the
successes
more
promising
lost
will
whichever is smaller.
the superscripts "+,-" refer, as usual, to positive and negative parts,
- 3 -
n, all
be
Here,
GORDON SIMONS
respectively.
.
Thus the "posterior Bayes risk" becomes
Conveniently, this can be rewritten as
Since the first term is a martingale in
of optimal stopping.
n, it has no bearing on the question
Consequently, the problem of optimal stopping can be
recast in terms of a reward sequence defined by
(1 )
Let
t
n
= 1 or
0
as
M~ n
or
M < n, respectively.
The value of
t
t
n
= 1
n
= O.
R
n
When
'
where
a n = E(M - nlM
Thus, in general,
,~~
n).
~
<
(2)
t! ';
(3)
R·
n
When computing the third factor on the right side of (3), one is free to
act as i f
t
n
= 1, and, hence, to reinterpret
expectation given the hypothetical responses of
- 4 -
liE "
n
as a condi tional
n pairs of patients, taken
..
RANDOM HORIZON MODEL FOR CLINICAL TRIALS
of patients, all treated under a testing phase;
from an infinite pool
joint distribution of the sequence
Ro,R ,R ,···
1 2
will be unaltered.
interpretation will be used hereafter unless stated otherwise.
the
processes
~
{t :n
n
and
O}
the
This new
In particular,
can be viewed as
independent of each other.
A simple argument shows, for the present level of generality, that the
a, defined by Chow, Robbins and Siegmund (1971, page
functional equation rule
63), is finite and optimal:
R
n
is bounded above by the product
is a uniformly integrable superllartingale in
the value
E(M).
(because
E(M) <
V of the optimal stopping proble. Is bounded above by
In particular,
4.10 holds.
times.
n
V is finite.
Consequently,
a
t ·6 , which
n n
Thus
co).
E( t ·6 )
o
0
=
Moreover, condition (d) of their Theorem
is oP'tilll£fyj· the, extended class of stopping
But it is easily seen that
M.
~
a
Thus
a
is finite and optimal.
The process
(n, r, s), where
pairs
of
rand
patients
s
under
denote the nu.bers of successes, produced by
the
tesUnitlj)tla~~,
for
the
first
and
n
second
treatments, respectively.
Likewise, the process
(n, t) .
The transition
n, and the transition
{tn:n
{Rn:n
~
O}
O}
is Markovian with a state paralleter
(n, 0) ... {n+.f·, tl )::occurs with probability one for all
(n,l) ... (n+1,1)
., n
Thus
~
P(M
~
occurs with probability
n+11M
~
n) .
is Markovian with a state para.eter
- 5 -
(4)
(n,r,s,t).
GORDON SIMONS
Notice,
t· R* (n,r,s),
has the form
R
n
independent of "t".
And it is easy to see that the maximum expected reward,
for optimal stopping from the initial state
*
t·S (n,r,s).
where the second factor is
(n,r,s,t), has the si.ilar form
The relationship between the factors
*
Rand
S
*
is described
by the "dynamic equation":
1
S *(n,r,s)
max{R*(n,r,s)'~n'2
1
~ u .. (n,r,s).s*(n+l,r+i,s+j)}, n,r,s
L
IJ
0, (5)
~
i=O j=O
where the transition
Thus,
(n,r,s)
~
., n
wi th the factor
(n+1,r+i,s+j)
has probability
uij(n,r,s).
included in (5), the component
"t"
dropped, and the state parameter of interest can be taken to be
The factor
., n
can be
(n,r,s).
can be thoultht of as a "discount factor".
But such an
interpretation will not be pursued here.
Notice that the testing phase begins in state
=
1.
form
(n,r,s)
sucC~8se.(J.q.'eu~tler,tht1p.Dt~
..al
So the expected
E(M1PI-P21) - S * (0,0,0).
=
(0,0,0)
with
t
stopping rule takes the
S * (0,0,0)
For a general stopping rule,
must
be replaced by an appropriate analogue.
The point
R* (n,r,s).
(n,r,s)
is an optilMl
stopping point
It is an optimal continuation point if
if
S* (n,r,s)
S * (n,r,s) > R* (n,r,s)
or
if
1
2
u
ij
(n,r,s).s*(n+1,r+l,S+j)
j=O
In the latter case, both appellations will be used.
- 6 -
R* (n,r,s).
..
RANDOM HORIZON MODEL FOR CLINICAL TRIALS
3.Two-point s,..etric priors.
Now suppose the prior
of two symmetric points
G for
(a,b)
(Pl'PZ)
and
assigns probability
(b,a), 0<b<a<1.
Then
1/2
to each
R* (n,r,s)
becomes
(see (3»
R* (n,r,s)
where
a =
21 log{a(1-b)/(1-a)b}.
(a
> 0)
(6)
The Markovian state (n,r,s) can be replaced by the simpler Markovian state
(n,k) = (n,r-s).
Then
(a-b) -1R*(n,r,s)
(7)
becomes
R(n,k) = 15 ·tanh Ikla.
(8)
n
Moreover, S *(n,r,s)
takes
can be written as
S(n,k)
U\e·if6rili·J(daflFS{11~'kH'--Jl:and:':
t he
..
dynamic equation
'1 "
maX{R(n,k)'~n·[~·S(n+1,k-l)
where
+ v-S(n+1,k) + "k'S(n+l,k+1)]},
(9)
,",-:'
It.
cosh (k-1)a
u k -- :.."'--...,.-~~cosh lCi
v
=
JJ·cosh (k+1)a
k -- :..----,r-'-r~~
cosh lCi
w
ab + (1-a)(1-b), p
=
{ab(1-a)(1-b)}~.
- 7 -
]
(10)
GORDON SIMONS
Observe, for future use, that
. fJe
(X
=
a (1-b) ,
fJe
-(X
b(l-a),
=
~
(11 )
2fJ·sinh
(X
= a-b, 2fJ·cosh
= 1-v. )
(X
The notions of optimal stopping and optimal continuation points
are defined as for
(n,r,s)
above, with the appropriate changes.
(n,k)
A simple
qualitative description of the class of optimal stopping points can be stated:
Tbeore. 1. Depending on
~
{len'
O}
n
and the distritmtion of
of nonnegative integers such that
point if and only if
Ie
a, b
I k I lien'
is an optimal stopping
(n, k)
Necessar11 y ,
Ie
n+1 -
M, there is a sequence
Ie n ~
-1, n
~
O.
(Ie
-
n+1
does not appear to have a universal upper bound.)
Proof. I t is enough to show this result for
when
M is bounded
~
0
(due to symmetry), and
above~;n~i~.fthellJ,!Jth~egenm~i":result
applying standard truncation :technIques.
for each fixed
k
e'
n, that therr..
Js
8.
'l.l.H_....r~\.it
"t'Opplngi
can be obtained by
is necessary to show,
~'i.;(n·.k),
k
~
The value
0:
.5
Consequently, R(n,k) > ., ·6 1
n n+
(n,k)
for a sufficiently large
is an optima.l ·;.stopping ., bbinit4.:::'!O'~~~jl~Ull4fs·
earlier, that
t
n
·5
optimal stopping point, then
= ~n·6n+1'
S(n,k)
This is impossible since
- 8 -
~
R(n,k).
k,
from the fact, noted
would be bounded above by
S(n,k)
n >.,·5
n n+ l'
k, and, for this
(n,k)
n
k
were not an
E (t
n
n+1
·5
n+1
)
RANDOM HORIZON MODEL FOR CLINICAL TRIALS
P(M > m)
So suppose
= S(n.k)
Q(n.k)
whenever
=
o.
Then, clearly.
- R(n.k). and observe that
(n.k)
K
Q(n.k)
is an optimal stopping point.
n
~
max(O.~
n
(1-~
oQ (n.k) -
n
~
m.
Let
From (9). one obtains
*
{
n
0, with equality holding
~noQ (n.o)*+ (a-b)o~no~n+l
Q(n.k)
for
= 0
) otanh ka)
for
k
O.
for
k > O.
where
Q* (n,k)
u oQ (n+l.k-l) + v oQ(n+l,k) + w oQ (n+l,k+l).
k
k
It is enough to establish that
~
0)
is
for each
(n,k+l).
n
~
m.
Q(n,k)
is a nonincreasing function in
k
(k
For then, H(n-,.k} 'is an optimal stopping point, so
Moreover. since
"!t':.O",~,for.
Q(n+l;lk)
k
~ K +
n 1
, Q* (n,Kn+1+1)
= O.
and hence, K < K 1 + 1.
n - n+
A
straightforward
function
Q(n, k)
backwlUl.4'B!nQupti.Qn:ir:ar~ent'.
;estabHshes
is nonincre••l~(·:.ln ".k·;Jpfor:,oin
~0
tlJ:l;;Clearly,
are needed for the induction '.step tf \ The ~detaUs aneoMitted
0
40 Two-point
8~tric
M
and
(b,a)
the
Q(m, k)
=0
[]
0
pr!ol'$ i with: ....-tr::1cal1~:dj.'tributed Ko
In addition to the assuaption tbat;3f
(a,b)
that
~
,is a two-,point symaetric prior on
(O<b<a<l), i t will be assumed throughout this section that
has a geometric distribution with parameter
This
substantially simplifies
7, 0<7<1, so that
the theory,
and
it
=
P(M=n)
makes
it
possible to obtain many detailed results.
Two small simplifications are:
=~
The next theorem details a major
and
~
n
= E(M)
=
~/(1-'Y)
for
n
~
O.
- 9 -
~
n
GORDON SIMONS
{K ,
simplification: the sequence
Tbeorea 2.
It is optimal
sinh (I k I +1)a
cosh (lkl+l)9' where
n~O},
n
to stop the testing phase as soon as
sinh Ikla
cosh Ikl9 ~
9 > a is defined by
1 -
cosh 9
Proof.
appearing in Theorem 1, is constant.
.,·v
(12)
2·p·" .
Since
there is a unique
E(M)·~(k)
and
9 > a
S(n,k) =
1-"(
1 - v
1 - "tv
cosh 9 - cosh a
2p"
2p"
=
2P
where
= tanh( Ikla)
~(k)
(13)
One can write
that satisfies (12).
E(M)·~(k),
1
= ~2fJr.E'="'7(':":'M"{'"')'
and
R(n ,k)
~
=
satisfy
e'
the recursive relationship
(14)
~(k)
Notice,
~
has to
In·the "continuation region",
be.~ s~M~tric functio~.
0.:.
"',,':'fi.
;
1l0
'
c
constant
c>O.
Since
region", the constant
> O.
c.
Since
c ... 0
Hereafter, let
~(k)
~(k)
c
as
t
= 'l(k)
• tanh( Ikla)
Must hav~J t~~e,lo~~M
l
be a
...
00,
sinh la cosh l<9
cosh f9·cosh ka
Ikla
is the unique solution to (14).
(= ~(k»
sinh la
81i dC~~ht9
for some integer
l
l
which maximizes
Then it is easily checked that
,.,-,.
for
Ikl < t,
for
Ikl
If the maximizing value
- 10 -
for some
has to hold in the "stopping
there Must be a value of
Maxi.izin~ val~~.
{ tanh
cosh k9
c cosh ka
~(k) =
so
~ l,
t
is unique, then
RANDOM HORIZON MODEL FOR CLINICAL TRIALS
the optimal stopping rule is unique and the optimal stopping points are those
points
Ikl
for which
(n,k)
which maximize
~
There may be two adjacent integers
i.
c, in which case there will be points
optimal continuation and optimal stopping points.
(n, k)
which are both
In any event, the stopping
rule described in the statement of the theorem is of the form
Ikj = i".
as
i
"stop as soon
c
So it is an optimal stopping rule as asserted.
Theorem 2 describes a
"sequential probability ratio test with random
truncation" (imposed by M).
The post tive integer
will be called the
f
optimal level.
It seems obvious the optimal level
e
goes to infinity as
E(M)
It
~~.
does, but at a pleasantly slow rate:
Corollary 2.
a - b
For possibly variable
ex
(def ined in
(6»
and
difference
(ex >0 , 0<a-b<1),
(15)
where,
~-;
e*
(2a) -1 10g {4' sinh
"
..,..
cX-di~=-b)'E'(M)}n'JaMJ z:, -J~''a·t~~h2ex.
Besides
and
the corollary states that
whenever
~
.51
~ 00,
v
is sufficiently large.
whenever
v
is integer-valued.
~
45.
(16)
asse~:ii~g tha:~:(I5;)hoids when the prior G is fixed
Interpretation.
E(M)
(a-b) . E(M) .
i
Nume~ical
is well approximated by
evidence indicates
above.
-
11 -
E(M)
*
Ie - e* I
No bound saaller than .5 could be expected since
The corollary has content even when
i
l
is bounded
GORDON SIMONS
t
Proof of upper bound:
Since
rewritten
t
as
t
~
*
+,5 + a ( 1 )
as
II ....
00,
sinh to' > sinh(t-1)a
h' h
b
e
cos h t9 - cosh(t-l)9' w IC can
sinh(t-1)a
Expanding this with hyperbolic
sinh to'
is an optimal level,
cosh(t-1)9
cosh is
~
identities, one obtains:
1
(cosh 9 - cosh a) (coth -2(9+0.)
-
1)
2e -2la.. sinh a
~ ---~~- +
1-e
Since
9 >a > 0
and
cosh 9 - cosh a
1
2,6E(M)
-2ta
2e -2tfJ sinh 9
-U9
1+e
sinh a.
(a-b)-E(M)' one obtains
sinh a.e- 9
1
sinh 9-(a b)-E(M) ~ (cosh 9 - cosh a)(coth 2'(0.+6) -
1)
4e- 2lasinh 9
~
-2ea
1-e
Thus
It remains to show: (i) 9
=
a.. + 0(0.), (ii) log sinh 9 = log sinh a + 0(0.), and
(iii) e-<X/{4-sinh a- (a-bl-E(M1)i} B£!(i'(a)· '8.ecJ JjlC'..c.(Joo)t,;,
Equation (iii) is
triiviar.t>'''EqiM~ioI'iS'-(i-)~d 1fii)
easily follow from
consequences of the mean value theorem:
9 - a.
S
cosh 9 - cosh a
.1
s lnh ex
.= 7'(a--"'b~)~--;:E:-r(';':MT')
(17)
and
log sinh 9 - log sinh a ~ (cosh 9 - cosh a)_Cosh a
2
sinh a
- 12 -
1
tanha..- (a-b)-E(M)'
[)
RANDOM HORIZON MODEL FOR CLINICAL TRIALS
Proof of lower bound:
l
~
l
*
-.5 - 0(1)
as
~ ~.
v
sinh la > sinh( l+l)a
cosh 19 - cosh(t+l)9
Here one begins with the inequality
and proceeds
in much the same way as with the upper bound to obtain
-2la
e
-219
+ e
eO.
~
2 -e
It remains to show the left side equals
o(a)
as
v
~~.
(18)
2- sinh a- (a-b)- E(M)'
-Ua+o(a)
For this, one needs the upper bound for
.
, I.e.,
t
t9 = ta +
and (17). The
key step is:
ax-1t*(9-<X) ~ log{4-:inha-(a-b)-E(M)}
a -(a-b)-E(M)
-1 -1
2
= v
a tanh a- (log(4JJ) + log cosh a -log(a- tanh a)}
=
-1
a
2
-tanh a {I + log cosh a - log(a-tanh a)}-o(l).
The expression preceding the...--If-Q.(.1.)''..,
;'
,
~·"i::?'
2l
* (9-<x) = o(a)
as
v
is:i_.
. ~!td,function
.:;., .':.L i!. ,
So
c
"f(a) - (a-b) -E(M)", where
function with
a, a>O.
~~.
This result is not "best. PPstlib!f:l" . i './fhe
replaced by
of
f(a) = O(a) ndtS
f
~.e, p~oQf
works if
v
is
is any positive and continuous
2
~.;-+ ~h~~,t)tA;};jf(aJ:r:'":.Q{~
/log(l/a»
as
~
a
0
(big O's).
..
.j •. ,. •,
.,-
This corollary suggests.that one. ca~:'::pz:obab11::.determine the optimal level
(that one needs)
illustrate,
suppose
with no more
a
=
.6
and
than a
b
crude estiaate of
= . 4 , so
6,7,8,9,10,11, the possible integer estimates of
- 13 -
E(M)
ex;'
are:
.405.
E(M) .
For
To
l
GORDON SIMONS '
TABLE 1
t
Estimate of E(M)
6
7
8
9
10
273-602
603-1,337
1,338-2,984
2,985-6,684
6,685-15,002
15,003-33,712
11
Thus, for instance, with an estillate "E(M) = 10,000", one would choose (from
the table)
t
=
10.
If the true expected number of pairs is between 6,685 and
15,002, then the choice
=
t
10
still will be optimal.
The actual situation is even aore favorable than this.
with the same example, suppose the true expectation were
of
what
was
E(M)
estimated.
To illustrate
=
5,000, half
is nine,
not
10.
Nevertheless, the expected su:c:(1"stltJuiost:lCble thisgsuboptiJlal choice (t = 10)
is a Mere .25; one could
5,000 pai rs
-ex~)tt
.25'- pa1iberits·j::' oat of an expected total of
to be advente'i:y 'laffected1
expectation were twice the
'.:I~ :If:, '1 J;Gn
,!.the other hand,
:J 'f t L ,. G
e.,t~.~\~rth~expected,D8.Derof
adversely affected patients
of 30,000 pairs!
And even i f
the expected number of
wou):dlbeoBlYlolle:!pati~mtout
of an expected total
To SUllilerize, nuaerical'·'evideD.ce suggests one can sarel}'
choose the optimal level
t, for an esti1ll8.tfi:ld value of
about how poor the est111NJte is,
However,
;fiRS ~J!btnt>eslt'i.a:te"
true
adversely affected
patients would be .33 out ofd 8,n;,!texpebtedH,tOt:a'W of 20:;'000 pairs.
the true expectation were three
the
E(M), without worrying
as long as the error 1s not
the size of the difference
truly gross.
one is trying to detect does make a
difference: a better estiJlate is needed when
- 14 -
a-b
is quite small. Consider an
RANDOM HORIZON MODEL FOR CLISICAL TRIALS
extreme example:
in fact, E(M)
a = .51 and
b = .49.
If one estimates
20,000, then one will choose
t
optimal level
51.
l
=
42
E(M) = 10,000
and,
instead of the actual
This can be expected to adversely affect 3.85 patients
out of an expected 20,000 pairs of patients.
t
The cost of choosing
incorrectly can be deduced from the following
theorem.
Theore. 3.
Suppose one stops the testing phase with the possibly suboptimal
rule "stop as soon as
(a-b) 'E(M)'{ sinh
.
cosh
Ikl
= t'."
tat9 _cosh
sinh t:a}
i 9 '
in the proof of Theorem .:), a
Then the expected successes lost will be
where
t
is the optimal level
is defined"18 (.6) and
e
(described
is defined in (12).
Proof. The quantity
(n,r,s) =
(0,0,0), the state one begins" iii: :be!for,e.,any..2patJi-.nts,chave been treated; the
third factor
Now
sinh fa
cosh i9
S* (0,0,0)
In state
(0,0,0).
since
~to~:!~.tn
tthej'proof of Theorem 2.
.
sinh t' a
Likewbe, O!7d~·).. E:(Ilf'· c'o'Mf>'I',:' ~d:s the expected reward,
(0,0, O;);~ if one, U881l<ltbeppossdbly suboptiJlal rule based on
This can be checked;
Finally,
a~;
is the maxi.al :expe'cted rewardUorppt.cJ:mali' stopping, beginning
beginning in state
t'.
can bec,ident14\ted;
the
.ona!l'Cmt~al'gtle
Mr,.dn ,:the' :proof. of Theorem 4 below.
origiaal iopt4J1al, :stopping
problem
is
concerned
with
sinh
ia - sinh
i'a
"expected successes lost" ,t:m!"dtffer.ence·('a-b)'E.'(U,,),{
fil
cosh i9
cosh t'
9 }
is precisely the expected successes :lost for acting suboptiJlally.
The testing phase can be stopped in two ways.
NorMally, the stopping
rule stops the testing phase, and the utility phase is begun.
- 15 -
0
But the testing
GORDON SIMONS
In the
phase can be stopped prematurely simply by running out of patients.
first case. the testing phase will be said to be completed.
case.
it
Something akin to truncation is
wi 11 be said to be trunca ted.
well-known in practice;
And in the second
protocols are sometimes abandoned
for
a
lack of
suitable patients.
For the stopping rule described in Theorem 3. the testing phase is
Tbeorea 4.
completed with probability
Proot.
Let
cosh l'a/cosh l'8.
X = {X(t). t=O.I.···}
be a Markov chain on the integers which is
independent of
M and has tr.ansition probabilities from state
k+l
~.
given by
v. wk' respectively.
first passage time to
:i: l ' .
I kl <
cosh /.' a
e ( 0) = cosh I; e .
~{uk·e(k-l)
Notice. e(k)
Ikl ~ l'.
Clearly.
M
e(±l')
The
= 1.
cosh 1<9
k,Jl '3 ;~o ; d~(kilnf" \e Lcos h lCi
cosh t.' a.
cosh t.' a.
e(/" ) cosh l's ~ cosh 1'9-
=
Corollary 4. For the optiJINJl
stopping
rule
probability of truncation goes to zero 1 ike
(when
has the "JIlemoryless property."
is symaetric in
But then. e(O) = c
00
T be the
+ v'e(k) + w ·e(k+l)}.
k
because the geometric random variable
E(M) ...
k-l. k.
l' •
e(k) =
c>O.
to
Further. let
:arfd "sttt~:' -e'(k) = P(M~TIX(O)=k).
assertion is established by showing
For
(See (10).)
k
a and
b are held fixed).
- 16 -
for some constant
o
Theorem 2.
the
log E(M)
+ O(E(N)-I)
2(a-6) 'Q'E(M)
as
described
in
e'
RANDOM HORIZON MODEL FOR CLINICAL TRIALS
Proof. The assertion is concerned with the behavior of
as
E(M) ...
00.
From Corollary 2, one obtains
log E(M)
2(a-b) ·a· E(M)
t
(a-b)·E(M)
+
e-
1 - cosh la/cosh 19
2la -_ O(E(M)-1)
O(E'(M) -1)
E(M) ...
as
and
(19)
00.
Thus
Ie -i. (9 ~)
-
cosh la I
cosh 19
So the task is to show
-2ta
e
~
1 - e -i. (9 ~)
O(E(M)-1)
as
E(M) ...
(a-b).l E(M) + O(E(M)-1)
00.
a sE(M)
...
00
.
But (17) and (19) give
and
as
E(M) ...
E(M) ...
00.
00.
Thus it remains to show
" '" ~
-
t
_ O(E(M)-1)
(a-6)· E(M)
sinh 9 - sinh a (
sinh 9
-
:L-AIS',,\r
(;,~f
'"
1 ~ v
--~2:----'
(a-b) ·E(M)
.'i
9
>
1
-a - ( a -6) • E(M)
_
I-v
-(a-_-b-)~3~.E-(-M-)"%2 .
Proof. The mean value theore~~>fll)ana;>{h3)':y'ield'
t ..
> rf 9 \} ~~
:. ;~--:V0":
.
~~
,e ~,;
:..?
"
cosh <1
sinh 9 - sinh a S (cosh 9 - cosh a)·coth a = (a-b).E(M)'
and hence,
sinh
sinh a
sinh 9
~
9 -
- 17 -
as
o
But this follows from the leaaa below.
.. ':.,JI
Le. . 1.
.(.. __ ) >
1 - v
2
.
(a-b) • E(M)
GORDON SIMONS '
In turn,
9-ex
~
cosh 9 - cosh ex
sinh 9
1
(a-b)'E(M)
1 3
1
~ ,(a-b)' E(M)
sinh 9 - sinh ex
sinh 9' (a-b)' E(M)
v
(a-b) . E(M)
c
2'
The next theorem describes how the available patients are divided between
the testing and utility phases,
Tbeorea 5. For the stopping rule described in Theore. 3, the expected nUJJJber
of pairs of patients
where,
a
treated during
is defined in
(6)
and
the utility phase
9
is
is defined in (12).
E(M). cosh l'a
cosh l'9'
For the optimal
stopping rule described in Theorem 2, the expected number of pairs of patients
treated in the testing phase
a
and
b
grow~u! ike .. ~(;_~~)
as
E(M) -.
00
(with
held fixed).
Proof. For the f irstasse~tfC?~r MPf~P~!!J~; aJi
." .
.
e(k)
+ O( 1)
+
.
redefined as E«M-T) IX(O)=k), Ikl
s
~!}:t~f:'\
t'.
ll:r oof
of Theorem 4 but with
The expected number of pairs
of patients is
cosh t' a
e (0) =,!~~~). -CPJlhi Ite.,;:;:; T~,t~i~Ho~~".:~s.sertion is concerned
the behavior of
E(M)' {l - cosh la/cosh te}
as
E(M) -.
00,
,
WI th
and follows as in
o
the proof of Corollary 4.
It is now possible to exaaine how, under the optimal rule, the expected
successes lost are divided between the
te~ting ~nd
~
'~j
utility phases.
It turns
out that the patients treated during the testing phase are forced to assume
most of the burden:
- 18 -
RANDOM HORIZON MODEL FOR CLINICAL TRIALS
Theorea 6_
during
Under
the
optimal
stopping
the expected successes
lost
and
(a-b)-E(M)-{1- cosh ta}
cosh t9
(ax) -l log E(M)
The former grows like
the testing and utility phases are,
-ta
(a-b)-E(M)-e
Icosh t9, respectively.
+ 0(1)
rule,
E(M) ..
and the latter remains bounded as
(with
00
a
and
held
b
fixed) .
Proof _ The expected successes lost is the expectation of the
IP1-P21
product of
and the total number of patients assigned, by the optiaal stopping
rule, to the inferior treatment during both treat.ent phases. So each pair of
patients treated during the testing phase contributes .an amount
expected successes lost_
_ cosh tal
cosh te
"a-b"
Since the expected nuaber of such pairs is
and grows with
E(M)
log E(M)
2(a-6)a
like
pertaining to the testing phase follow.
+
0(1),
to the
E(M)'{l
the assertions
Now the actual expected successes
lost is
-~... , ;){1C~.r~3i': 5,1"; ,.~ -~:~~!';.:':::- :M :"
Thus the expected successes lost during the':uti1?J.!tl phase is
_sinh ta} _ (a-b)'E(M)'{l _ cosh tal
( a- b) 'E ()
M • {1
cosh Ii
cosh t9
-ta
= (a-b)' E(M)' e· Icosh t9 ,
as asserted.
Finally,
*
" '"
sinc~
.', ' ,t .,'.
-ta
r~-b)~E(M)'e
2(a-b)'E(M)'e- U a = (2'sinh a)-l
Icosh t9 S 2(a-b)'E(M)'e
>
-2ta
and
(see Corollary 2), it follows that the
expected successes lost during the utility phase stays bounded as
- 19 -
E(M) ..
~.
a
GORDON SIMONS
So the expected successes lost that are borne by testing-phase patients
approaches
~
0:
as
100%
E(M)
~~.
Surprisingly, the same thing occurs as
For small values of E(M), the optimal level
t
is fixed at
E(M)
1, and the
,
ratio of expected successes lost, between the testing and utility phases,
. h a
ea 'Sln
becomes
(a-b)' E(M)'
This goes to infinity as
o.
E(M) ...
It appears, based
upon some nu.erical evidence, that the minimal percentage occurs when
just before
E(M)
becomes large enough to make
t
t
=
1
= 2.
It should be emphasized that Theorem 6 describes the expected successes
lost during the utility phase from an a priori vantage point,
patients have been treated.
testing phase, 2E(M)
suggested
by
(a-b)'E(M)'e
-ta
Fro/ll
addi~ional
Theorem
Icosh ta
"5,,,
the
vantage point of a
patients are expected, not
';A841J;1lh~,
are
expected
before any
jast cO/llpleted
2E(M).cosh ta
cosh t9
to
result
as
in
.ore successes lost.
Frequentists are likely to be interested in the operating characteristics
of
the
opti.al rule,
:>:-\L.~,r.&ndQ(,~-h~ ~."ba~~~itM(
rejected for use in the utility phase.
rejected,
para.eter
when
the
a particular treatment is
The first
(second)
triEtll:t.en't:_'~h~_1·i..:'..c.o.pli!·tedi;or
';, ';, 'Y'':'· ~
11;'~?·~;
k < 0 (k > 0), and is selected if
treatllent is
ter.inated,
k > 0 (k < 0).
the excess of successes produced by the first treat.ent.
Recall,
k =0
k
the
is
It is mathematically
convenient to reject each treatllent with probabil i ty one-half when
The occurrence of the value
if
,
k =
o.
is usually very unlikely; it never occurs
when the treatment phase is co.pleted.
A Bayesian result will be given first.
- 20 -
..
RANDOM HORIZON MODEL FOR CLINICAL TRIALS
Tbeorea 7. For the stopping rule described in TheoreJl 3, the better treatment
1s rejected with probability
1
B(a, b) -"2
sinh a
,
(O<b<a<I).
2' sinh 9' tanh t 9,
(20)
For the optimal stopping rule described in Theorem 2,
B(a,b)
S
I-v + ~(I-b) .
2(a-b) • E(M)
(21)
PROOF. Proceed as in the proof of Theorem 4, but with
e(k)
defined as the
probability the better treatment is rejected starting in the state
B(a,b)
=
It is
e(O).
easily seen that.""'-~('±t')i= (1 + e
satisfies the nonhomogeneous secontl
U'
)
-1
Thus
,and
e(k)
ord~'~d'iffe'N!nce,equation
e(k)
for
k.
(22)
Ikl < i'.
It is
shown· ilf' lthe~ ~l'eUa ''below! that: ,the unique solution is
given by
e(k)
(23)
Then (20) follows i"IRediatelf/'·:
!
When
t'
B(a,b) S
t, Lea.a 1, (18) and (11) yield
sinh 9 - sinh a
1 - tanh t9
1 - v
-2t9
2' h 9
+
2
S
2
+ e
'Sln
2(a-b) . E(M)
a
e
-----:2~-- + 4' sinh a' (a -6)· E(M)
1 - v
2(a-b) . E(M)
- 21 -
1-v + a(1-b)
2
.
2(a-b) . E(M)
D
GORDON SIMONS
Inequali ty (21) is "asymptotically tight" in that
limsup 2(a-b)2. E (M).B(a,b)
E(M)
COROLLARY 7. For
the stopping rule described
success probabll i ties
probabll i
I-v + a( I-b).
,
~ 00
and
PI
in
the first
P2'
Theorem
3,
treatment
and for
fixed
is rejected with
ty
(24)
where
0':
and
fI'
> 0
{P1 ·(I- P 2 ) }
1
=
2
p. (l-p )
2
1
log
is defined by
cosh'
,
1.J1·v'
= 2
.P' .., ,
and where
v'
as
for
=
Thus
1
-
1l' (
P2 ' PI)
=
E(M)
~ 00
1 - 0 ( max [ E( M)
-1
(25)
, E( M)
-<x'
Ia.
])
•
as
E(M) ~
00
for
PI
< P .
2
- 22 -
RANDOM HORIZON MODEL FOR CLINICAL TRIALS
PROOF. It is easily seen that n(P1,P2)
B(P2,P1)
become
when
a'
P
1
and
~
P " The values of
2
P
1
~
and
9,
P > P .
1
2
9', respectively, when
(24) is still correct when
When
a
defined in (6) and (12),
And it is easily seen that
P "
2
t., the bound described in (25) can be obtained as for (21):
t.'
+ e
-2l9'
and by (18),
e
-2l9'
+
o(1)}
E(M)-a' fa
as
E(M) ...
c
00.
,--_.
-...
'c" "-::,\
..
...-- ,
LBJIJIA 2.
~:;
Equation (23) is the unique solution to the difference equation shown
in (22) which satisfies the OOurpar"i'COndi.~tJ.ods.lJ~(.*l(' l; = (l + e U ' ) -1.
be given.
With the substitution
f(k)
=
2ocosh kaoe(k), equations (22) and
(23) become
f(k) = ".(jJof(k-l) + vof(k) +pof(k+1)} + (1 _ .,).e- 1k1a ,
1
kl <l' ,
and
f(k)
e -Ikla
The former can be rewritten as
f(k)
sinh a osinh (l' -Ikl )9
sinh 9·cosh t'9
Ikl ~t' .
(23' )
(see (12) and (13»
(2.cosh 9)-1.{f(k-1)+f(k+1)} + (1 _ cosh a).e- 1k1a
cosh 9
'
- 23 -
Ikl <t' .
(22' )
GORDON SIMONS
Clearly (23') satisfies the transformed boundary conditions
f(±l')= e
~'a
And i t is a matter of using elementary hyperbolic identities to show (23')
satisfies (22').
IJ
,
5. Concluding co-ents.
It is hoped that the theory presented in Section 4 becomes a stimulus for
future research.
Clearly the assumptions lIade in this section were designed
for mathematical convenience.
In their defense, it could argued that they are
no stronger than those that have been made for fixed horizon versions of the
model.
And a random horizon does appear more reasonable.
they lead to precisely describable results.
Most importantly,
But i t is not clear yet whether
the conclusions drawn fairly approxillate reality.
There are many illportant theoretical questions that were not addressed.
For instance, one should probably assign a cost to the treatments ad.inistered
during the testing phase.
What effect would this have?
It's inclusion would
probably cause a shift of the "expected successes lost" from treat.ent-phase
patients to utility-phase patients.
shift be?
It
(See Theore. 6.)
But how big would the
should be possible to address such a question theoretically
within a convenient .atheaatical context.
•
- 24 -
RANDOM HORIZON MODEL FOR CLINICAL TRIALS
References
Berry, D. A. (1978). Modified two-arm bandit strategies for
certain clinical trials. J. Amer. Statist. Assoc. 73 339-345.
Chow,
Y.
S.,
Robbins,
H.
and
Siegmund,
D.
(1971).
Great
Expectations: The Theory of Optimal Stopping. Houghton Mifflin,
New york.
Simons, G. (1986). A comparison of seven Allocation rules for a
clinical trial model. Sequential Analysis. To appear.
' . : :t.
:
- 25 -
~:...
~
..
,~J
,
_~