Proc.
Nat.
Acad.
Sci.
USA
Vol.
72,
No.
1,
pp.
2022,
January
1975
A
Nonidentifiability
Aspect
of
the
Problem
of
Competing
Risks
(crude
survival
probabilities/net
survival
probabilities)
ANASTASIOS
TSIATIS*
Department
of
Statistics,
University
of
California,
Berkeley,
Calif.
94720
Communicated
by
Jerzy
Neyman,
October
17,
1974
ABSTRACT
For
an
experimental
animal
exposed
to
k
>
1
possible
risks
of
death
R1,
R2,
*,
Rk,
the
term
ith
potential
survival
time
designates
a
random
variable
Yi
supposed
to
represent
the
age
at
death
of
the
animal
in
hypothetical
conditions
in
which
Ri
is
the
only
possible
risk.
The
probability
that
Yi
will
exceed
a
preassigned
t
is
called
the
ith
net
survival
probability.
The
results
of
a
survival
experiment
are
represented
by
kI
"crude"
survival
functions,
empirical
counterparts
of
the
probabilities
Qi(t)
that
an
animal
will
survive
at
least
up
to
the
age
t
and
eventually
die
from
Ri.
The
analysis
of
a
survival
experiment
aims
at
estimating
the
k
net
survival
probabilities
using
the
empirical
data
on
those
termed
crude.
Theorems
1
and
2
establish
the
relationship
between
the
net
and
the
crude
probabilities
of
survival.
In
particular,
Theorem
2
shows
that,
without
the
not
directly
verifiable
assumption
that
in
their
joint
distribution
the
variables
Y1,
Y2,
**,
kare
mutually
independent,
a
given
set
of
crude
survival
prob
abilities
Qi(t)
does
not
identify
the
corresponding
net
probabilities.
An
example
shows
that
the
results
of a
customary
method
of
analysis,
based
on
the
assumption
that
Yi,
Y2,
.,
Ykare
independent,
may
have
no
resem
blance
to
reality.
As
recently
summarized
by
David
(1),
the
customary
treat
ment
Qf
competing
risks
is
based
on
the
model
that
we
shall
term
the
model
of
potential
survival
times.
Consider
an
in
dividual
living
organism
born
at
time
t
0,
and
assume
that
through
its
lifetime
it
is
exposed
to
k
>
1
different
"risks"
or
possible
causes
of
death
RI,
R2,
*
,
Rk.
For
i
=
1,
2,
...
,
k
let
Yi
denote
a
random
variable
described
as
the
"potential
survival
time"
of
the
individual
in
hypothetical
conditions
in
which
Ri
is
the
only
risk
of
death,
and
let
Hi(t)
=
P
{
Yf
>
t
}.
The
function
Hi
is
described
as
the
ith
net
survival
proba
bility
or
the
ith
net
"decrement"
function.
The
potential
survival
times
Y,
are
contrasted
with
the
actual
survival
time,
say
X,
when
the
individual
in
question
is
exposed
to
all
the
k
>
1
competing
risks,
so
that
X
=
min
(Y,,
Y2,

*
*,
Yk).
The
function
Qi(t),
described
as
the
ith
crude
survival
function,
is
defined
as
the
probability
that the
individual
considered
will
survive
up
to
age
t
and
then
die
from
cause
Ri.
Obviously,
for
21,
2,
*
k
and
t
>
0,
Qi(t)
=
P{(Yi
>
t)
n
(Yj
>
Yi)1.
[1]
joi
Ordinarily,
the
studies
of
competing
risks
are
based
on
empirical
counterparts
of
the
crude
survival
functions
Qi(t),
perhaps
derived
from
observations
of
a
cohort
of
experimental
animals.