Another look at Adjusting Radio-Telemetry Data for Tag-FailureL. Cowen and C.J. Schwarz

Department of Statistics and Actuarial Science, Simon Fraser University, Burnaby, BC

Motivation

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.

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.

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known detections to account for the proportion of tags that could not be detected because a proportion of the tags were no longer active. Thus our objective is to develop a method to adjust survival estimates given radio-tag failure-time data.

To illustrate the method, consider the simple study of having only 2 dams. Fish are released above the first dam (release group 1) and below the first dam (release group 2). They are later recaptured using radio antennas before the second dam. Since we are interested in the time of failure of the radio-tag, we must keep track of the time it takes each 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 dam2 (although this is not the case in the study).

Capture histories are represented by a series of 1’s and 0’s. A ‘1’ signifies a recapture and a ‘0’ signifies a non-recapture. Table 1 describes the possible outcomes and capture histories for a fish in the radio-telemetry study. Note that table 1 does not take into consideration the travel time of the fish.

Table 1. Capture histories and associated expected probabilities of the two release groups.

n11,Ri= the number of fish released at dam 1, recaptured at dam 2 from release group Ri.

n10,Ri= the number of fish released at dam 1, not recaptured at dam 2 from release group Ri.

1 = Pr(survival from dam 1 to dam 2).

p2 = Pr(recapture at dam 2).

R1 = release group 1.

R2 = release group 2.

ti = the travel time of the fish from dam 1 to dam 2, i=1,2,…; 0ti.

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.

S(ti) = survival distribution of the radio-tag.

Nr= the number of initial radio-tags.

nr,ti= the number of radio tags that have failed by time ti.= the total number of fish released.10,11 nnN

i

i

tt

22

2

11

1

,10,11

,11

,10,11

,11

2

2

RR

R

RR

R

D

R

RDD

nn

n

nn

n

p

p

(1)

(2)

ntStgpntStgppL iii

iiit 10,11 )(1)(),( 212121

(3)

For illustration, we only use two time periods for traveling from dam 1 to dam 2: t1 and t2. 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 (4).

duuSugN

nnp tt

)()(21 ,11,11

21 (4)

AcknowledgementsWe would like to thank Karl English of LGL Limited for providing the data for the radio-tag failure curve.

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-of-river steelhead smolts at Wanapum and Priest Rapids projects using radio-telemetry techniques, 2000. Public Utility District No. 2 of Grant County, Washington, draft report.

Cowen and Schwarz. Adjusting survival for premature radio-tag failure. SSC, Burnaby BC, 2001.

Release Group

Capture History

Expected Probability

R1 11

10

R2 01

00

2)( ptSPD

DPDPDPD ptSptS 111)(11)( 22

2)( ptSP

PPP ptSptS 11)(11)( 22

In order to obtain the radio-tag failure curve, tags are placed in a tank. The time of radio failure for each tag is recorded and a radio-failure curve can then be constructed.

If we have 2 release groups, 1 before dam1 and 1 after dam1 we can estimate dam survival. 1 for group1, released above the dam would be made up of 2 components: survival through the dam, D and survival between dam 1 and dam 2, R. In the second release group, released after the dam, survival is only that between dam 1 and dam 2, R.

We can now estimate dam survival by taking the ratio of the parameters for each release group (1) and then substitution tin the maximum likelihood estimator (2).

We model this type of data assuming a multinomial distribution. The likelihood is given in equation (3).

First we make the assumption that the support for this integral consists of the travel times that we have observed for the fish (i.e. the ti’s). Thus our integral becomes a summation as we discretise the problem. Our estimator of 1p2 becomes (5).

This estimate can intuitively be thought of as inflating each observed history by the survival probability of the radio-tags.

Problems arise when the travel time of a fish is large so that the survival probability of the radio is very small and the adjusted counts explode. To account for this we add 1 to the numerator and the denominator of the radio-tag survival estimate. Motivation for this comes from the adjusted Petersen estimate where adding 1 is found to reduce bias. (Note that we are using the Product Limit survival estimate.)

)(ˆ)(ˆ1

2

,11

1

,1121

21

tS

n

tS

n

Np tt

1

1)(ˆ

r

tri N

nNtS i

Case Study

Using data for both fish travel times and radio-tag failure times for the 2001 season we obtained estimates for 1) unadjusted dam survival, 2) adjusted dam survival, and 3) adjusted dam survival adjusting radio survival first.

(5)

(6)

Radios Fish Released (R1/R2)

Before Adjustment

Adjustment 1 Adjustment 2

19 360/324 0.68210 1.33382 1.05763

Notation

The difficulty here is in the estimation of . duuSug )()(

Future WorkWhen making an adjustment for radio-failure, travel time of the fish must be taken into consideration. This is more evident in the 3 dam case (which I am still working on).

As we now have a survival estimate, we have to get a variance estimate. This will be done using bootstrapping.