Transcript SSC2001 - Department of Mathematics

Adjusting Radio-Telemetry Detection Data for Premature Tag-Failure
L. Cowen and C.J. Schwarz
Department of Statistics and Actuarial Science, Simon Fraser University, Burnaby, BC
Introduction
Mark-recapture theory studies a cohort of marked individuals that are
recaptured at a later time and/or space. Animals are marked with unique
tags allowing the estimation of both survival and capture rates. A fairly
simple mark-recapture experiment would be to release fish above a dam
and recapture them before a second dam.
Capture History
Recaptured
at dam 2
Survival
to dam 2
11
DAM 1
R1
1 p2 

ti
n11,t
0
L1 , p2   1 p2 n 1  1 p2 n
11
10
(1)
Notation
n11= the number of fish released at dam 1, recaptured at dam 2.
n10= the number of fish released at dam 1, not recaptured at dam 2.
1 = Pr(survival from dam 1 to dam 2).
We can now estimate D via by taking the ratio of the parameters for each
release group (3) and then substituting in the maximum likelihood estimator
(4).
 D R p2
D 
(3)
 R p2
1
2








1 p2 
(4)
ti = the travel time of the fish from dam 1 to dam 2, i=1,2; 0ti.
n11,ti = the number of fish released at dam 1, traveled to dam 2 in ti time and
were recaptured at dam 2, i=1,2.
n10 = the number of fish released at dam 1, not recaptured at dam 2.
1 = Pr(survival from dam 1 to dam 2).
p2 = Pr(recapture at dam 2).
g(ti) = travel time distribution for the fish from dam 1 to dam 2.

t i
n
11,t1

i
n
11,t 2
(8)
NS (t1 ) NS (t2 )
This estimate can intuitively be thought of as inflating each observed
history by the survival probability of the radio-tags.
Case Study
Historically, mark-recapture studies involving fish have used PIT-tags.
These tags have a low recapture rate, thus large sample sizes are needed to
get robust survival estimates. For animals that are listed as endangered,
large sample sizes are not ideal. With the advent of radio-telemetry, this
problem is somewhat alleviated.
In radio-telemetry studies, small radio-transmitters are attached to the
animal. Associated with each radio-tag is a unique radio frequency. Radiotags are relatively new to the study of salmon survival.
The major problem with the use of radio tags is their reliance on battery
power. Each radio-tag requires a battery and failure of the battery before the
end of the study can negatively bias survival estimates. If information is
available on the life of the radio-tags used in the study, a tag-failure curve
can be developed. Given the tag-failure curve, adjustments can be made to
the known detections to account for the proportion of tags that could not be
detected because a proportion of the tags were no longer active.
A simulation study was done using failure data of 19 radio-tags. The data
is show in table 1. Fish travel times were generated from a lognormal with
mean 10 days, standard deviation 1 day. Survival was generated from a
binomial(10000, 0.95) and capture probabilities were generated from a
binomial(10000, 0.8) to determine if adjusted survival estimates were
similar to actual estimates. Thus 1p2=0.76 for this data set.
For this simulation, the radio-tag survival curve was estimated using
Kaplan-Mier estimates.
Table 1. Failure times of 19 radio-tags provided.
Days from Tag
Activation
6
8
17
18
19
20
21
22
23
26
Tag Failure
Pattern
1
1
2
3
2
3
3
1
2
1
Tag Failure
Percent
5.3
10.5
21.1
36.8
47.4
63.2
78.9
84.2
94.7
100.0
Again, we have simplified the problem by having only 2 dams (figure 3).
Since we are interested in the time of failure of the radio-tag, we must also
keep track of the time it took the fish to travel from dam 1 to dam 2. To
simplify the problem even more, we only allow for 2 possible travel times
for the fish, from dam 1 to dam 2 (although this is not the case for the
simulation study).
Results
Figure 4 describes the possible outcomes and capture histories for a fish in
this radio-telemetry study. Note that figure 4 does not take into
consideration the travel time of the fish.
The estimated 1p2 before adjustments was 0.4059 and the adjusted
estimate was 0.4962. With more tag-failure data, the adjusted estimate
should come closer to the real value.
Capture History
= the total number of fish released.
i
We use maximum likelihood estimator’s of 1p2 (2), which are inseparable
due to identifiability problems.
n11
n11
1 p2 

n11  n10
N

Figure 4. Capture histories and outcomes for a fish in a radio-telemetry
study.
S(ti) = survival distribution of the radio-tag; equal to 1- F(ti).
N   n11,t n10
?
i
t i
Objective: to develop a method to adjust survival estimates given radio-tag
failure-time data.
R2 = release group 2.
(6)
1 , p2    n11,t ln1 p2 g (ti ) S (t i )  n10 ln 1  1 p2 ut g (u ) S (u ) (7)
p2 = Pr(recapture at dam 2).
R1 = release group 1.
(5)
i
First, we make the somewhat bold assumption that the support for this
integral consists of the times that we have observed for the fish (i.e. the
ti’s). Thus our integral becomes a summation as we discretise the problem.
The log-likelihood becomes (7), and our estimator of 1p2 becomes (8).
Motivation
So the probability of seeing a ’11’ would simply be 1p2 and the
probability of seeing a ’10’ would be 1(1-p2) + (1-1) = 1-1p2. Thus for
this simple case we can model these data using a binomial distribution,
equation (1).
i
i
 g (u)S (u)du
DAM 2
2
10
2
N  g (u ) S (u )du
How do we estimate
p2
2
Fish Dies
1


n11, R

n

 n10 , R
11, R

D 

n11, R

n
 11, R  n10 , R
10
D
R
1
Not recaptured
at dam 2.
1   p g t S t 
n10
Due to identifiability problems we cannot get an estimate for 1 on its own.
However we can get an estimate for 1p2 as shown in (6).
R1
1
Released
at dam 1.
L(1 p2 )   1 p2 g ti S ti 
n11 ,ti
ti
Capture histories are a series of 1’s and 0’s. A ‘1’ signifies a recapture and
a ‘0’ signifies a non-recapture. For our simple experiment we have 2
possible event histories: 11 and 10. A ’11’ means that fish were released at
dam 1 and recaptured at dam 2. A ’10’ means that fish were released at
dam 1 and not seen again. Figure 1 outlines the possible events that can
occur between dam 1 and dam 2 and the associated capture histories.
Figure 1. Flow chart of possible histories of a fish. Included in the chart is
the associated survival (1) and recapture probabilities (p2)as well as the
final capture history associated with each event.
We model this type of data using a multinomial distribution. The
likelihood is given in equation (5).
Figure 2. Pathway of fish in release group 1, R1 and release group 2, R2
between dam1 and dam2. The components of survival and recapture
probabilities are shown.
Radio
survives
(2)
Fish
survives
to dam 2
Dam Survival
If we are to have 2 release groups, 1 before dam 1 and 1 after dam 1 we
can estimate dam survival (which is how we fund these projects). 1 for
group 1, released above the dam (figure 2) would be made up of 2
components: survival through the dam, D and survival between dam 1 and
dam2, R. In the second release group, released after the dam, survival is
only that between dam 1 and dam 2, R.
Radio
fails
Released
at dam 1
Fish dies
Recaptured
at dam 2
11
Not
recaptured
at dam 2.
10
Not
recaptured
at dam 2.
10
References
Lebreton, J-D., Burnham, K.P., Clobert, J. and Anderson, D.R. (1992).
Modeling Survival and testing biological hypotheses using marked
animals: a unified approach with case studies. Ecological Monographs,
62(1): 67-118.
English, K.K., Skalski, J.R., Lady, J., Koski, W.R., Nass, B.L., and
Sliwinski, C. An assessment of project, pool and dam survival for run-ofriver steelhead smolts at Wanapum and Priest Rapids projects using radiotelemetry techniques, 2000. Public Utility District No. 2 of Grant County,
Washington, draft report.
Acknowledgements
10
We would like to thank Karl English of LGL Limited for providing the
data for the radio-tag failure curve.