SURVIVAL ANALYSIS IN TELEMETRY STUDIES:
THE STAGGERED
ENTRY
DESIGN
KENNETH
H.
POLLOCK,
Department
of
Statistics,
North
Carolina
State
University,
Box
8203,
Raleigh,
NC
27695-8203
SCOTT
R.
WINTERSTEIN,
Department
of
Fisheries
and
Wildlife,
Michigan
State
University,
Bast
Lansing,
MI
48824
CHRISTINE
M.
BUNCK,
Patuxent
Wildlife
Research
Center,
U.
S.
Fish
and
Wildlife
Service,
Laurel,
MD
20708
PAUL
D.
CURTIS,
Department
of
Zoology,
North
Carolina
State
University,
Box 7617,
Raleigh,
NC
27695-7617
Abstract:
The
estimation
of
survival
distributions
for
animals
which
are
radio-tagged
is
an
important
current
problem
for
animal
ecologists.
Allowance
must
be
made
for
censoring
due
to
radio
failure,
radio
loss,
emigration
from
the
study
area
and
animals
surviving
p88l.
:~the
end
of
the
study
period.
First
we
show
that
the
Kaplan-Meier
.procedure
wid~ly
used
in
medical
and
engineering
studies
can
be
applied
to
this
problem.
An
example
using
some
quail
data
is
given
for
illustration.
As
radios
maItunction
-or
are
lost,
new
radio-tagged
animals
have
to
be
added
to
the
study.
We
show
how
this
modification
can
easily
be
incorpor~.ted
inf.<?
the
basic
Kaplan-Meier
procedure.
Another
example
using
quail
data
is
used
to
illustrate
the
extension.
We
also
show
how
the
log
rank
test
commonly
used
to
compare
two
survival
distributions
can
be
generalized
to
allow
for
additions.
Simple
computer
programs
which
can
be
run
on
a PC
are
available
from
the
authors.
-2-
Radio-telemetry
is
becoming
an
increasingly
popular
methodology
for
studying
wild
animal
populations.
An
animal,
captured
by
trap,
dart
gun
or
some
other
method,
is
fitted
with
a
small
radio
transmitter
and
released.
From
release,
the
animal's
unique
radio
signal
can
be
monitored
until
the
animal
dies
or
is
censored.
An
animal
is
censored
if
we
have
lost
track
of
the
animal
due
to
radio
failure,
radio
loss,
or
emigration
from
the
study
area.
The
most
common
application
of
radio-telemetry
technology
has
been
to
the
study
of
animal
movements
in
relation
to
daily
activity
patterns,
seasonal
changes,
habitat
types,
and
interaction
with
other
animals.
Time
series
approaches
will
become
very
important
to
the
thorough
analyses
of
these
data
(see
for
example
Dunn
and
Gipson,
1977;
Pantula
and
Pollock,
1985)'-
Biologists
have
also
begun
to
use
radio-tagged
animals
to
study
survival.
Present
techniques
for
analysing
the
data
from
these
studies
assume
that
each
survival
event
(typically
an
animal
surviving
a
day)
is
independent
and
has
a
constant
probability
over
all
animals
and
all
periods
(see
Trent
and
Rongstad,
1974;
Bart
and
Robson,
1982;
Heisey
and
Fuller,
1985).
These
assumptions
are
often
believed
to
be
unrealistic
and
restrictive.
White (1983)
has
generalized
discrete
approaches
in
the
framework
of
band
return
models
(Brownie
et
ale
(1978».
He
has
developed
a
flexible
computer
program,
SURVIV,
tor
use
with
his
approach.
Typically
an
animal's
exact
survival
time
(at
least
to
within
one
or
two
days)
is
known
unless
that
survival
time
is
right
censored
(that
is
only
known
to
be
greater
than
some
value).
We
suggest
an
approach
based
on
the
continuous
survival
models
allowing
right
censoring
which
are
widely
used
-3-
in
medical
and
engineering
applications
(Kalbfleisch
and
Prentice,
1980; Cox
and
Oakes,
1984).
Pollock
(1984)
and
Pollock
et
ale (1987)
illustrated
the
usefulness
of
this
approach
and
provided
examples
of
the
Kaplan-Meier
procedure.
This
procedure
does
not
require
specification
of
a
particular
parametric
continuous
distribution
such
as
the
exponential
or
Weibull.
In
this
paper
we
first
present
a
simple
description
of
the
Kaplan-Meier
procedure
complete
with
an
example
of
some
quail
survival
data
collected
by
Curtis.
We
next
show
how
to
generalize
the
Kaplan-Meier
procedure
to
allow
gradual
(or
staggered
entry)
into
the
study.
The
calculations
are
again
illustrated
with
an
example
from
the
Curtis
quail
data.
Finally,
we
present
the
log
rank
test
for
comparison
of
survival
distributions
(modified
for
staggered
entry
of
animals)
and
illustrate
it.
We
also
present
a
discussion
of
model
assumptions
and
directions
tor
future
research.
THE
KAPLAN-MEIER
OR
PRODUCT LIMIT PROCEDURE
The
Kaplan-Meier
or
product
limit
estimator
was
developed
by
Kaplan
and
Meier
(1958)
and
is
discussed
in
many
books
on°
survival
analysis.
See
for
example
Cox
and
Oakes
(1984,
p.
48)
or
Kalbfleisch
and
Prentice
(1980,
p.
13).
The
survival
function
S(t)
is
the
probability
of
an
arbitrary
animal
in
our
population
surviving
t
units
of
time
trom
the
beginning
of
the
study.
A
nonparametric
estimator
of
the
survival
function
can
be
obtained
by
just
restricting
ourselves
to
the
discrete
points
where
deaths
occur
8
1
,
8
2
,
•••
, age
We
define
r
l
,
•••
, r
g
to
be
the
numbers
of
animals
at
risk
at
these
points
and d
1
, d
2
,
•••
, d
g
to
be
the
number
of
deaths
at
-4-
the
same
points.
The
probability
of
surviving
from 0
to
a
l
is
then
estimated
by
Seal)
=1 -
dl/r
l
because
dl/rlis
the
estimated
proportion
dying
in
that
interval.
The
probability
of
surviving
from a
l
to
~
is
similarly
given by
1 - d
2
/r
2
and S
(~)
is
then
given by
the
product
S(~)
=(1 -
dl/rl)(l
- d
2
/r
2
)
Therefore
the
estimated
survivor
function
for
any
arbitrary
time t
is
given by
Set)
=
IT
(1
-
d./r.)
J J
(1)
jla/t
which
is
the
mathematical
way
of
stating
we
are
considering
the
product
of
all
j terms
for
which a
j
is
less
than
the
tiae
t.
Let us
talk
further
about r i which
is
the
number
at
risk
at
time a
i
•
In
this
situation
we
would
start
off
with
a
fixed
sample
of
size
n.
The
number
at
risk
at
a
particular
death
tiae
a
i
will
then
be
n minus
the
number
of
deaths
before
a
j
minus
the
number
of
animals
censored
before
time a
..
J
As an example
of
the
use
of
this
model, we
present
results
from a
radio-tagging
study
on
northern
bobwhite
quail
(Colinus
virginianus)
conducted
by one
of
the
authors
(P.Curtis)
at
Fort
Bragg, North
Carolina.
This
was
a two-year
study,
but
in
this
section
we
just
consider
the
data
collected
in
the
spring
of
1985. This
is
a
small
study,
the
pertinent
data
on
each
of
the
eighteen
radio-tagged
birds
is
included
in
Table 1.
-5-
Six
birds
died
and
five
birds
disappeared
(were censored)
during
the
study,
leaving
seven
birds
which
survived
for
the
thirteen
weeks
of
the
study.
Let us
illustrate
how
the
estimation
of
Set)
was
carried
out
to
obtain
the
last
column
in
Table 1 and
Figure
1.
The
computations
involve
only
the
five
weeks
in
which
deaths
were recorded;
therefore
a
l
=S, a
2
=
6,
as
=
8,
a
4
= 9 and a
5
= 10. Also
recall
that
we
have
eighteen
animals
that
began
the
study
at
time
o.
We
estimate
Seal)
as
Seal)
=S(S) =1 -
dllrl
=1 - 2118
=0.8889,
because
there
are
2
deaths
at
time S and
there
are
18
animals
still
at
risk
(r
l
).
The
next
death
time
~
is
at
6 weeks
(~
=6) and
at
that
time
there
is
one
death
(d
2
= 1) and 16 animals
at
risk
(r
2
=
16).
There
are
16
at
risk
because 2 were
lost
to
death
at
tiae
1.
Therefore
S(a
2
)
is
given by
8(a
2
)
=
8(6)
=(1 -
d1/r1)(1
-
~/r2)
=
(1
- 2118)(1 - 1/16)
=
0.8333
Similarly
S(&s)
is
given by
8(a
S
)
=
8(8)
=(1 -
dl/r1)(1
-
~/r2)(1
-
dS/r
S
)
=(1 -
2/18)(1
-
1/16)(1
- 1/15)
=0.7778,
8(a
4
)
=
8(9)
=(1 -
d1/r1)(1
-
~/r2)(1
-
ds/rs)(1
- d4/r4)
=
(1
-
2/18)(1
-
1/16)(1
-
1/15)(1
-
IllS)
=0.7179,
