.
i
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.,
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THE COX REGRESSION MODEL. RANDOM CENSORING
AND LOCALLY OPTIMAL RANK TESTS
by
Pranab Kumar Sen
Department of Biostatistics
University of North Carolina at Chapel Hill
Institute of Statistics Mimeo Series No. 1376
February 1982
THE COX REGRESSION MODEL, RANDOM CENSORING
AND LOCALLY OPTIMAL RANK TESTS
by
Pranab Kumar Sen
University of North Carolina, Chapel Hill
Key Words and Phrases:
Asymptotic efficiency; covariates; hazard function;
LMPR test; local optimality; log-rank test; survival function;
withdrawal.
ABSTRACT
Conditions on the hazard functions under which the usual log-rank
test remains locally optimal for the Cox regression model under random
censoring (withdrawal) are examined.
In the light of these, the asymptotic
efficiency results pertaining to the Cox partial likelihood statistic
and the log-rank statistic are studied.
1.
INTRODUCTION
In the Cox (1972) regression model for survival data, it is assumed
that the ith subject (having survival time
Y and a set of covariates
i
Zi ... (Zil'···'Zi )', for some p > 1) has the hazal'd l'ate (given Z... zi)
-
p
-~
-
....
(1.1)
~i
where hO(t), the hazrad rate for
nonnegative function (for which
f
.. 0, is an unknown, arbitrary,
00
o
hO(t)dt- +00)
and
a-
-
(B , •••
1
parameterizes the regression of survival times on the covariates.
,ap )'
If
the conditional distl'ibution fUnction (d.f.) and, its complement, the
sUl'Vival function (s.f.) of Yi , given ~1 = ~i' are denoted by Fi(yl~i)'
-2and Fi(yl~i)' respectively. then by (1.1). we have
_
_
exp(~'~i)
Fi(yl~i) • [FO(Y)]
• i=l ••••• n.
(1.2)
where
(1.3)
In the particular case of p
=
° or 1).
1 and binary zi (assuming the values
(1.2) reduces to the Lehmann (1953) model. so that the Cox model includes
the Lehmann model as a special case.
Also. for scalar zi and
B.
it
follows from a general theorem in Hajek and Sidak (1967. pp. 70-72)
that for testing
° against non-null B close to 0, a locally most
B=
H :
O
powerful rank (LMPR) test is based on the statistic
(1.4)
where
R
ni
is the rank of Y among Y , •.• ,Y , for i=l •••• ,n. and the
i
1
n
°
°
scores a (l), ••. ,a (n) are defined by
n
n
aO(k)
n
= -1
n,
k)}
-1 + Lk_ (n- j +1) -1 ) , for k=l, ••• ,n,
j 1
(=
where U 1 <••• < U
n,
n.n
- E{log(l-U
(1.5)
are the random variables of a sample of size
n
The scores in (1.5) are known as the
from the uniform (0,1) distribution.
o
log-rank scores and Tn as the log-rank statistic or the (generalized)
Savage statistic.
When p
o
T
~n
~
n
1. we may define
-
0
Z
= L i = l(Zi_ -n )a n (Rn i)'
-
Z
~n
=n
-1 n
Li =1Z'
~~
(1.6)
(1. 7)
and
(1.8)
-3-
Then Lo provides a loaaZly maximin rank test for H :
n
O
e close
e=0
against non-null
to O.
In survival analysis. random censoring due to withdrawals is not
uncommon.
Here. we conceive of a set W ••••• W of independent random
1
n
variables (censoring times) and assume that the Wi and Y are independent.
i
The ovservable r.v.'s are then
i > 1 • (1.9)
If G and G be respectively the d.f. and s.f. for the Wi (assumed to be
Si(xl~i)
identically distributed) and if
the d.£. and s.L of Xi' given
~i
..
~i
and Si(xl~i)
be respectively
(ignoring 0i)' then by (1.2) and
(1.9),
Si(xl~i) .. G(x)Fi(xl~i)
_
_
exp(S'zi)
= G(x) [FO(x)]
- - ,i=1 ••••• n. (1.10)
Note that under HO: ~
G FO)'so that
= 0,
S1, ••• ,Sn are all the same (i.e., equal to
i f one ignores the information contained in
o • (0 , •••• 0 )',
1
-n
n
rank tests based on X1, ••• ,Xn are genuinely distribution-free (un?er H ).
O
o
0
The test based on Ln or !n (replacing the Y by Xi' 1 ~ i ~ n) is therefore
i
a valid test for HO: ~ • ~ under random censoring, though it may not be locally
optimal.
If
g
is the density function corresponding to the d.f. G. then the
hazard rate for the Wi is hG(t)
the Xi (conditional on
~i
=
~i)
= g(t)/G(t).
so that the hazard rate for
are
(1.11)
-4This, in general, vitiates the proportionality assumption in (1.1), and
hence, the log-rank test may not be locally optimal.
Note that the likelihood function (of the Xi and 0i' given
~i)
is
given by
(1.12)
so that the joint distribution of the ranks and
F and
O
G, even under HO:
~ • O.
°-n
depends on the unknown
To eliminate this problem, Cox (1972,
1975) considered a partial likelihood function (which takes into account
the indicator variables 0l""'On) and obtained some (non-rank) test
statistics which depend on the covariates
way.
~i'
i=l, ••• ,n in a more involved
Peto and Peto (1972) considered the two-sample problem [as a special
case of (1.10) with binary zi] and, under a somewhat different setup,
concluded that the log-rank statistic is LMPR even under random censoring.
As we shall see in Section 2 that under the model (1.10)-(1.11), the 10grank statistic is not generally LMPR (or maximin), even for the special
case of the two-sample problem.
For this reasom, we will investigate the
conditions (on the hazard rates hO(t) and hG(t) or equivalently on F and
O
G) under which the log-rank statistic is locally optimal among the rank
based tests on X1 ""'Xn (gnoring
~n)'
The question that naturally
arises about the gain in efficiency of the Cox procedure (due to incor-
°),
~ ) over the log-rank procedure (ignoring
-n
_n
be addressed here to 0 •
poration of
and this will
Section 2 is devoted to the study of locally optimal rank tests (for
testing H :
O
~
= ~),
under the model (1.10).
optimality of the log-rank test.
Section 3 deals with the local
Asymptotic efficiency results on the Cox
-5-
procedure are presented in Section 4. and the concluding section is devoted
to some general remarks.
2.
LOCALLY OPTIMAL RANK PROCEDURES UNDER RANDOM CENSORING
Ignoring 0l' .••• On and based on the ranks of Xl •.•.• X • we like to
n
construct suitable tests for H :
O
~
•
9.
having some local optimality
properties.
Let si(xl~i) be the probability density function (p.d.f.) corresponding
to the d.f. Si(xl~i) in (1.10). for i·l ••••• n.
Then. by (1.10) and (1.11).
we have
log Si(xl~i)
= log
Si(xl~i) + log h~(x)
• log G(x) + log Fi(xl~i) + log h~(x)
= log G(x) + exp(~'~i)log FO(x) +
10g[hG(x) + eXP(~'~i)hO(x)]. 1 < i < n
(2.1)
which leads to the log-likelihood function (of the Xi given
~i
= ~i'
ial ••.•• n). ignoring 0, as
-n
n
-
n
-
• ~i=llog G(X i ) + ri.lexP(~'~i)log FO(X i ) +
~~=llog[hG(Xi) + exp(~'~i)hO(Xi)] •
Note that by (2.2)~
where
(2.2)
-6-
Further note that under HO: ~
m
2,
sl(xl~l) • •.•
= sn(xl~n)
= So(x) ...
G(x)FO(x) and this does not depend on ~l' ... '~n.
Thus, under HO'
are independent and identically distributed random variables
X , .•. ,X
1
n
(i.i.d.r.v.), so that R
= (Rn 1, ••. ,Rnn ), the vector of ranks of X , ••• ,X
~n
1
n
among themselves, has the (discrete) uniform distribution over the set
of n! possible permutations of (l, ••• ,n).
= log FO(x)
Let then
1_
~
* = ~l
+
~2'
where
,0< u < 1, (2.5)
SO(x)=l-u
~2(u) ... 7T(X)
1_
O<u<l,
(2.6)
SO(x)=l-u
define the ordered r.v.'s U l' •.• 'U
n,
n,n
Also, let a * (k)
n
= a n, l(k) +
a
n,
as in after (1.5), and let
2(k), for ka 1, ••• ,n.
Then, by (2.3),
(2.4), (2.5), (2.6) and (2.7), we obtain that
EO{(ll/da)log L* (a) I
IR} = n
L Z.a * (Rn i) ... _n
T* , say,
n
a=O n
i=l-~ n
(2.8)
where EO denotes the expectation under H : a = O.
O
Note that by (2.4) and (2.6),
1
~2 • ! ~2(u)du'"
o
=f
-J
7T(x)dS O(x)
FO(x)G(x)hO(x)dX
= -f
...
!7T(x)dS O(x)'"
FO(x)G(x)d log FO(x)
flog FO(x)d.So(x) ...
-J
[by (1. 3) ]
log FO(x)dSO(x)
1
. . -J0
~ (u)du = - ~
11
(2.9)
Further, by (2.7)
(2.10)
-7-
Hence, from (2.9) and (2.10), we have
Thus, by (2.8) and (2.11), we may rewrite T* as
-n
(2.12)
At this stage, we may note that (i) hO(x)exp(~'~i)/{hO(x)exp(~'~i)+
hG(x)}
a,
is a bounded and continuous function of
o ~ - log FO(x) = - log SO(x) + log G(x)
~ - log
and (ii) for every x,
So (x) , where the right
hand side is square integrable (with respect to SO(x».
Hence, for p - 1,
we may appeal directly to a theorem in Hajek and ~idtk (1967, p.71),
verify their basic conditions (1) and (2) (on page 70) and conclude that
Tn* is a LMPR test statistic for testing H
o: a •
ignoring the 0i).
For p
~
0 (based on the Xi and
1, we let
Ln*
=
*
-
*
(2.13)
(T_n )'V
(T )
_n_n
where V Is defined by (1.7). Then, by an appeal to the Union-Intersection
-n
principle, as in Sen (1982), or the maximin theory as in Hajek and ~id'k
(1967), we claim that
when _n
0 is ignored.
Ln* is a locally maximin rank test for HO:
Let us define
*2
A
1
=J
0
2
{tV (u) + tV2(u)} du
l
*2
An
*
Note that A <
~
~ = ~,
and An*
~
=
(n-1)
A* as n
-1
~~.
n
1 *2
= J tV
(u)du,
(2.14)
•
(2.15)
0
*
Li =l[an (i)]
2
Thus, when the
~i
satisfy the
(generalized) Noether condition:
ma
_ "",n )'V-(Zi-Z
. . . n _ "'un ) ~ 0 (a.s.), as n ~~,
1<i<xn (Zi-Z
(2.16)
-8-
then. by an appeal to the permutational central limit theorem and the
Cochran theorem. we conclude that under HO:
as
Bn
~.
-+
00
(2.17)
•
2
p
where X has the (central) chi square distribution with p degrees of
freedom.
In particualr. for p
=
1. U is a scalar (nonnegative) quantity.
n
and under HO'
*-L-~ *
A 11
T ~
n
n
n
N(O.I)
(2.18)
We may note that by (2.5) and (2.6).
f
1
-f
~I(u)~2(u)du •
{log Fo(x)}n(x)dSo(x)
a
.
• -f
{log FO(x)}hO(x)FO(x)G(x)dx
=
~J FO(x)G(x)d(-log FO(x»2
= -~
fe-log FO(x»
2
dSO(x)
= -~ f
1 2
~l(u)du
o
(2.19)
Hence. by (2.14) and (2.19)
(2.20)
Consider now a sequence {K } of alternative hypotheses. where for
n
each n.
K : (1.10) holds with
n
for some (fixed) A
E
~
(2.21)
b
E • and assume that
n
where
a = n-~A
-1
V
-n
-+
is posotive definite (p.d.).
v (a.s.). as n
-+
00
•
Then. in (2.2). writing
(2.22)
-9-
exp(~'~i)
• 1 +
routine steps
n-~(~'~i)
+
~n-l(~'~i)2
3 2
+ 0(n- / ). we obtain by some
tha~
(2.23)
where. under H : 8
O
= O.
the first term on the right hand side of (2.23)
*2
is asymptotically normal with 0 mean and variance A
(~'~~)
[= 0
2
(say)].
the second term converges in probability to 0, while the third term to I(A'vA)
2
A*2 (-10).
Thus, the left hand side is asymptotically normal with mean
_~02 and variance 0 2 .. This, according to LeCam's first Lemma [viz.,
Hajek and ~id~k (1967, p. 204)],establishes the contiguity of the sequence
of probability measures under {Knl to that under H •
O
As such, using
LeCam's third lemma along with the usual projection of T
...n*, it follows
that under {K } ,
n
n -~~_n*
~
*2 , A*2v)
N(VAA
-_
_
(2.24)
Thus, under {K l, L*fA *2 has asymptotically a noncentral chi-square
n
n n
distribution with p degrees of freedom (D.F.) and noncentrality parameter
*
*2
1::.= A
1 2
(~,~~) ... (Jo1jJ2(U)dU)(~'~~)
Note that, in practice, to use the statistic
•
(2.25)
Ln*, one needs to
know the score function 1jJ *• which [by(2.5)-(2.6)] depends on the unknown
F and G. Thus, in general, L* is not an adaptable test statistic.
n
O
Nevertheless, the above result provides a convenient means for studying
the asymptotic efficiency of other tests, and this will be taken up in
the next section.
-10-
3.
LOCAL OPTIMALITY OF THE LOG-RANK TEST
In this section, we shall study the asymptotic
optimality properties of the log-rank test.
efficie~cy
and
By virtue of the contiguity
results of Section 2 and the usual projection results on TO, parallel
-n
to (2.24), we obtain that under {K } and the regularity conditions of
n
Section 2,
(3.1)
where
1
2
=f
- log(l-u)} du
0
<p
Y
2
= J {-I
A
=1
(3.2)
1
{-I - log(l-u)}{-~I(u) - ~2(u)}du
o
::: f 1~ *(u)log(l-u)du
o
•
(3.3)
o
Thus, under {K }, L has asymptotically a noncentral chi-square
n
n
distribution with p
D.F. and noncentrality parameter
f::"
o = Y2 (A'vA) .
(3.4)
- --
By (2.25) and (3.4), the Pitman-efficiency of the log-rank test
with respect to the locally optimal one is
(3.5)
If we write
*(u)
• 10g(l-u), 0 < u < 1, then by (2.14), (2.20), (3.2)
<p
and (3.3), we have
2
Y
IA
*2
1 *
= (J
<p (u)~
*
o
(!
1
*
(u)du)
_*
2
2
I{(! 1~*2 (u)du)A~}
~
0
*
(<p (u) - <p )~ (u)du)
2
o
==----------------(!1 ~*2 (u)du)(!1 [<p* (u)
o
=
* *
p2 (<P ,~ ) ,
-* 2
- <p ] du)
0
(3.6)
-11where
p2 (~ * ,~*)
2
I, with the strict equality sign
holding only for
for some real a
~
*(u)
~ a~
*(u)
+ b, 0 < u < 1,
(1 0) and b.
(3.7)
By (2.5), (2.6) and (3.7), we conclude that P2 (~ * ,~ *) = 1 only when,
log So(x) = k 1 + k 2 log FO(x) + k 2h O(x)/{h O(x) + hG(x)} ,
(3.8)
for almost all x, and since, log So
log FO + log G, (3.8) may be
=
written equivalently as
log G(x)
= k1
+ (k -1) log FO(X) + k 2h (x)/{hO(x) + hG(x)} •
2
O
(3.9)
for almost all x.
Since hO/{hO+hG} is nonnegative and bounded between
o and 1, and log G ~
1.
~oo
as x
+ +00,
in (3.9), k
(3.9) specifies the interrelationship ofF
2
O
has to be different from
and G for which the
log-rank test is a locally optimal rank test under the model (1.10).
An important class of distributions for which (3.9) holds may be
characterized by the two ha7.ard functions hO(x) and hG(x) as follows.
Suppose that
(3.10)
Then, by integration on both sides, we have
log G(x) - clog FO(x) + c', c' real,
.
while by (3.10), hO/{hO+hG}
that (3.9) holds with k2
~
=
(l+c)
c+1.
-1
•
(3.11)
Hence, it is easy to verify
Thus, if the hazard rates for the d.f.
-12-
F andG ~e proportional to each other, then the log-rank test is a
O
locally optimal rank test for the Cox model under randam censoring.
A second situation where (3.9) holds is the degenerate case where
G(x)
=
1, V x <
00,
so that SO(x) ... FO(X) , V x <
and hence, (3.8) holds with k ""1.
2
00,
hG(x) ... 0, V x <
00,
Thus, if the withdrawal distribution
lies completely to the right of the d.f. F ' then the log-rank test is
O
locally optimal
as it is in the case where there is no (random) censoring.
In passing, we may remark that Peto and Peto (1972) considered the
two-sample problem, where for some n
(- N-n 2 ) , Sl ......
= Sn
... So and
1
and showed that the locally most powerful
1
Sn +1 "" ••• "" SN = [So]l+A
1
rank test (for A ... 0, va. A ~ 0) is the log-rank test.
Their model differs
from (1.10) [in the sense that G does not remain the same under alternatives],
and hance, their conclusions may not hold for the model (1.10) even for
the special case of the two-sample problem.
So far, we have considered the local optimality and efficiency of
the log-rank test relative to L*, where the information contained in 0
n
~
has not been incorporated in the testing procedures.
loss of efficiency due to this.
We like to study the
For this, we may note that the joint
density of the (Xi'Oi) in (1.9) is given by
L
on
n -
n
0i
1-0 i
"" i~l{[fi(Xil:i)G(Xi)]
[g(Xi)Fi(Xil~i)]
}
(3.12)
so that by (1.1), (1.2) and (3.12), we have
0i~'~i + 0i log hO(X ) + (1-0 ) log hG(X )} •
i
i
i
(3.13)
-13Thus. under {K } in (2.21),
n
1
+ --2
Eni leA , z.) 2{ log -FO(X i )} + 0 (n -~)
= -
n
-~
v * (in probability. if the
then by (3.14). under H : ~ = ~. as n ~ 00,
O
Thus, if n-1 Ln =l Z Z'
i
-i-i
(3.14)
p
~
~i
~
are stochastic).
(3.15)
where
n
= [G(x)dFO(x)
= !G(X)FO(x)hO(x)dx
= !(hO(x)/{hO(x) + hG(x)})dSO(x)
(3.16)
= !n(x)dSO(x) •
and 'flex) is defined by (2.4).
Now (3.15) establishes the contiguity of
the probability measure under {K } with respect to that under H •
n
O
Further.
by (3.13),
u = (a/aS)log L (S)I
-n
~
n -
a=O
a
Eni=l~i{log FO(X i ) + oil , (3.17)
-
. and hence, it follows by some standard steps that if In stands for the
likelihood ratio test statistic [for testing H :
O
~
=~
vs H:
~ ~ ~
on the
model (3.12)]. then under H '
O
2
(3.18)
-2 log L - X ,
p
n
while under {K }, -2 log I has asymptotically a non-central chi square
n
n
distribution with pD.F. and noncentrality parameter
-~ = n(A'v *A)
.
(3.19)
Note that by (1.7), (2.21) and the definition of v * • v * - v is positive
- -
-14semi-definite (of rank 1 at most), and hence
-A'V-*-A - -A'VA-- 0. V A •
(3.20)
>
-
the~i
[Actually, if
chosen that Z
-0
=0
are non-stochastic, the model in (1.1) may be so
and this will lead to V
~
-
*
= V.
Even, otherwise, in
-
(1.1), ~i may be replaced by (~i - ~n)' 1 ~ i ~ n, and this will lead to
* = v.
~ = ~ *,
V
Thus, we may assume without any essential loss of generality that
so that the equality sign in (3.20) holds.]
By (2.25), (3.5) and (3.19)-(3.20), we obtain that the asymptotic
relative efficiency of the log-rank test relative to the likelihood ratio
test [for the model (3.12)] is
e(LO,T)
=
~O/X = (~0/6*)(6*/X)
2 * *
~~ (¢ ,~ )
=
=
2
J1~2(u)du]/TI
o
[p 2 (¢ *,~*)]{( JTI 2 (x)dSO(x»/ JTI(x)dSO(x)}
* *
p 2 (¢ ,~ )}{P2}
(3.21)
where p2 (¢ *,~ *) < 1, with the equality sign holding under (3.7)-(3.9) and
(3.22)
with the equality sign on the right hand side holding only when TI(x) • 1
almost everywhere (SO), i.e., hG(x)
=0
almost everywhere.
Note that even
if (3.9) holds [viz., (3.10)], (3.21) will be generally less than 1, due
to P2' so that there is always some inherent loss or efficiency due to
ignoring 0 and using a rank test based on the Xi alone.
-n
4.
If
T
ASYMPTOTIC EFFICIENCY OF THE COX PROCEDURE
= {t 1<•.• <t m}
• {Xi (ordered): 0i=l, i=l, ••• ,n}
be the set
of failure points (for which Wi exceeds Y ), then a partial likelihood
i
-15-
function may be defined as in Cox (1972, 1975) as follows.
there is a risk set R of r
j
At time t.-O,
J
individuals which have neither failed nor
j
dropped out by that time, for j.1 •...• m note that Rmc ••• cRl.
Considering
j
the risk set R and the conditional probability of a failure at time t
j
(1
2
j
2
j
m), we obtain the partial log-likelihood function (from (1.11»
log L**
m
= Emj
*
whereg * • (Q1*••••• Qm)
•
l{S'Z
~ ~Q
*-
(4.1)
log(E i R eXP{S'zi})}
j
E
j
- -
is a sub-vector of the anti-ranks. relating to
the indices of the observations corresponding to the ordered failures
(preceding withdrawals).
For testing HO: ~
= 2 against
~
+2,
Cox (1972)
considered the test statistic
Lnm
= U'
J U
",nm-nm",nm
(4.2)
where
(4.3)
U
_nm
Whenever, 7f, defined by (3.16) is > 0, under H ' L has asymptotically
O mn
chi-square distribution with p D.F. Also, it follows from Sen (1981,
Sec. 4) that under {K } in (2.21), L has asymptotically a noncentral
n
nm
chi-square distribution with p D.F. and noncentrality parameter
f::,
= 7f(A'VA)
(4.5)
By virtue of the remarks made after (3.20), (4.4) is quite comparable to
(3.19). and this reveals the asymptotic optimality of L ,under (2.21).
nm
We may note that the information on 0_n is incorporated in the Cox
procedure through the construction of the risk sets R • 1
j
~
j
~
m, and
this explains the better efficiency when the model in (1.1) holds.
unlike the log-rank test, L
nm
is not a rank statistic.
Finally.
-165.
SOME GENERAL REMARKS
The results in Section 3 reveal the loss in efficiency of the log-
. to ignoring the information in 0 and restricting
rank test (or L*) [due
n
-n
to the ranks of the Xi] when the Cox model in (1.1) holds; the Cox
procedure remains asymptotically locally optimal for the same model and
it incorporates the information in O. However, it may be remarked that
-n
whenever under the null hypothesis, X ""'X are i.i.d.r.v., the 10gn
1
rank test is a genuinely distribution-free test.
This is particularly
true when instead of the Cox model in (1.1), the Fi(xl~i) are given by
the conventional regression model F(x-~'~i)' so that Si(x)
= G(x)F(x-~'~i)'
1 < i ~ n, where under HO: S • 0, Sl""'S are the same. In such a case,
n
the log-rank test remains valid for a general class of F, G. On the
other hand the rationality and/or optimality of L for this conventional
nm
regression model may be open to questions.
Thus, if we have random
censoring, it may be a basic issue whether to stick to the Cox model and
adapt the locally optimal Cox procedure (which may not be robust for
departures from the Cox model) or to use the log-rank test which remains
valid (under more general setups) and reasonably efficient for a broad
class of models.
In any case, if
~,
defined by (3.16) is small,
~he
(random) censoring results in substantial loss of efficiency, and, by
(3.22), a greater loss is incurred for the log-rank procedure.
when
~
Hence,
is close to O,rank procedures may not be recommended.
ACKNOWLEDGEMENT
This work was partially supported by the National Heart, Lung and
Blood Institute, Contract NIH-NHLBI-71-2243-L from the National Institutes
of Health.
"
-17BIBLIOGRAPHY
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Regression models and life tables.
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z.
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on
•
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