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Using mark-recapture models to estimate survival from telemetry data

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In this section, we illustrate the flexibility with which mark–recapture models and telemetry systems can be tailored to unique situations in aquatic environments. We describe simple to complex mark–recapture models and the associated telemetry system design needed to support each model. We simulate a tagged population of fish migrating downstream past a dam and apply different models to this simulated data set. Although we present a specific application, the general framework and flexibility of using mark–recapture models with telemetry data applies to many situations. We then discuss the assumptions associated with each model, and give examples showing how assumptions can be violated. Since our goal is to introduce the application of mark–recapture models to telemetry data, we do not present a complete treatment of the statistical theory or software used to implement mark–recapture models. This section will be useful to readers familiar with either telemetry or mark–recapture models to design telemetry studies within a mark–recapture framework.
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453
Using Mark-Recapture Models to Estimate
Survival from Telemetry Data
Russell W. PeRRy, TheodoRe CasTRo-sanTos, ChRisToPheR
M. holbRook, and benjaMin P. sandfoRd
Section 9.2
INTRODUCTION
Analyzing telemetry data within a mark–recapture framework is a powerful approach for
estimating demographic parameters (e.g., survival and movement probabilities) that might
otherwise be difcult to measure. Yet many studies using telemetry techniques focus on sh
behavior and fail to recognize the potential of telemetry data to provide information about sh
survival. The sophistication of both mark–recapture modeling and telemetry has dramatically
improved since the 1980s, largely due to technological advancements in computing power
(for mark–recapture models) and electronic components (for telemetry). Such advances now
allow mark–recapture models to take advantage of the detailed information that telemetry
techniques can provide.
The key feature of mark–recapture models is simultaneous estimation of detection and
survival probabilities. With telemetry, a “capture” event consists of detecting a given tag
code one or more times at a specic location or time. By contrast, in some studies interest
may focus on the probability of detecting a single tag transmission (see Sections 7.2 and
9.1). Compared to conventional mark and recapture methods, telemetry methods often have
greater detection probabilities due to large detection ranges, increased “effort” (i.e., continu-
ous monitoring with autonomous receivers), and ability to simultaneously monitor multiple
locations. Nonetheless, perfect detectability is rare in telemetry studies because both random
(e.g., from electronic noise) and nonrandom processes (e.g., receiver loses power temporar-
ily) can allow a sh to pass a receiver undetected. Failure to account for imperfect detection
can lead to serious bias in survival estimates. When using telemetry to estimate survival, it
is therefore critical to explicitly estimate detection probabilities to ensure unbiased estimates
of survival (see Section 7.2). Fortunately, using telemetry techniques and mark–recapture
models together yields the best of both worlds: Well-designed telemetry systems deliver high
detection probabilities that result in precise estimates from small sample sizes. Mark–recap-
ture models ensure estimates of the demographic parameters are unbiased with respect to the
detection process.
454
Section 9.2
In this Section, we illustrate the exibility with which mark–recapture models and telem-
etry systems can be tailored to unique situations in aquatic environments. We describe simple
to complex mark–recapture models and the associated telemetry system design needed to sup-
port each model. We simulate a tagged population of sh migrating downstream past a dam
and apply different models to this simulated data set. Although we present a specic applica-
tion, the general framework and exibility of using mark–recapture models with telemetry
data applies to many situations. We then discuss the assumptions associated with each model,
and give examples showing how assumptions can be violated. Since our goal is to introduce
the application of mark–recapture models to telemetry data, we do not present a complete
treatment of the statistical theory or software used to implement mark–recapture models. This
Section will be useful to readers familiar with either telemetry or mark–recapture models to
design telemetry studies within a mark–recapture framework.
Telemetry system design (e.g., the spatial arrangement of antennas or hydrophones) dic-
tates the parameters that can be estimated using a mark–recapture model. More complex mark–
recapture models require more detailed information from a telemetry system. Therefore, study
design is crucial to successfully apply mark–recapture models to telemetry data and involves
1) identifying the demographic parameters of interest, 2) designing a mark–recapture model
to estimate these parameters, and 3) implementing a telemetry system that provides detection
data required by the mark–recapture model. In our experience, many telemetry studies fail to
follow these important steps. Therefore, the organization of this Section emphasizes the critical
linkage between telemetry system design and mark–recapture model design.
A SIMULATED FISH PASSAGE STUDY
As described in Section 9.1, we simulated a sh passage study and then applied both time-
to-event and mark–recapture analyses to the same simulated data set. Section 9.1 describes the
simulation from the perspective of time-to-event analysis; here we describe details of the simu-
lation relating to mark–recapture modeling. In summary, simulated tagged sh were released
upstream of a dam, arrived at the dam over time during experimental treatments consisting of
either a low spill or high spill dam operation, and then migrated downstream to the ocean. The
proportion of tagged sh passing through the dam’s turbines and spillway arises as a function
of 1) treatment-specic rates of passage, and 2) arrival timing in the forebay (see Section 9.1).
Survival was simulated as a function of three different mortality processes occurring at
different locations in the study area: 1) mortality upstream of the dam, which was a function
of residence time in the forebay, 2) mortality as a result of passing through either turbines or
spillway, and 3) mortality between the dam and river mouth at ocean entry. Specically, mor-
tality upstream of the dam is determined by time-dependent mortality in the forebay relative
to the amount of time required to pass the dam (see Section 9.1). The probability of surviv-
ing dam passage was set at 0.70 for the turbines and 0.90 for the spillway. Survival from the
dam to the river mouth declined exponentially over time as S = e
rt
where r = 0.005 and t is
the date of dam passage. This pattern was selected to simulate the effects of migratory delay
on physiological and ecological preparedness for the marine environment (McCormick et al.
1998). Fates of individual sh were simulated by using random number generators to draw
Bernoulli outcomes (i.e., we “ipped a coin”) with associated survival probabilities (i.e., a
probability of “heads”).
455
Estimating Survival Using Telemetry
As described in Section 9.1, 300 sh were released upstream of the dam, but we also sim-
ulated a release of 300 tagged sh in the tailrace corresponding to release times upstream of
the dam. Fish released in the tailrace were considered a “control” group because their survival
between the dam and river mouth was inuenced only by in-river mortality processes (i.e.,
day of year). In contrast, for sh passing the dam, survival between the dam and river mouth
was inuenced by both mortality due to dam passage and mortality due to in-river processes.
The telemetry system was simulated to detect sh in the forebay, within each passage
route (turbines and spillway), in the tailrace just downstream of the dam, and at the river
mouth to detect sh that survived to the ocean (see Section 9.1 for details). Detection prob-
abilities often vary among monitoring sites due to variable effort (e.g., number and detection
range of antennas) and site conguration (e.g., channel width). Therefore, we assigned differ-
ent detection probabilities to each monitoring site. The probability of detecting a tagged sh
was set at 0.95 for the turbines, 0.75 for the spillway, 0.80 for the tailrace, and 0.70 at the river
mouth. As with survival, detections of individual sh were simulated as random draws from
a Bernoulli process with associated detection probabilities. Detection probabilities in actual
eld studies may be higher or lower than those simulated here.
By simulating the data, we know exactly the survival, passage, and detection fates of each
individual. Therefore, by comparing estimates from the mark–recapture model to the calcu-
lations based on the known fates, we can verify that the estimated parameters are unbiased.
APPLYING MARK-RECAPTURE MODELS TO THE SIMULATED DATA SET
Given the telemetry system deployed to detect sh in our simulated study, a number of
different mark–recapture models could be t to these data. It is important to recognize that the
complexity of any mark–recapture model is limited by the complexity of the telemetry system
used to monitor the tagged population. Thus, models may be only as complex as the telemetry
system allows, but may be simplied as desired by pooling or ignoring detection data appro-
priately. We illustrate this concept by rst tting simple mark–recapture models to telemetry
data and showing which parameters can be estimated. We then show how adding information
from the telemetry system allows for more complex models with additional parameters.
THE CORMACK–JOLLY–SEBER MODEL
The Cormack–Jolly–Seber (CJS) model has been the foundation of mark–recapture mod-
eling for over forty years (Cormack 1964; Jolly 1965; Seber 1965). Two types of parameters
are estimated by the CJS model: S
i
, the probability of surviving from telemetry station i to i +
1, and p
i
the probability of being detected at station i conditional on surviving to station i. For
our simulated telemetry study, a CJS model can be constructed using detections at two sites:
one at the dam (combining detections within the spillway and turbines) and one at the river
mouth (Figure 1). This is the minimum telemetry system design that allows us to obtain any
estimate of survival, and for the moment, we ignore information from other telemetry stations
in our study. As will be seen, however, the CJS model for this telemetry design reveals little
about our simulated population.
Relating the CJS parameters to our study, S
1
estimates the probability of sh surviving
from the release point to a passage route entrance (spillway or turbines), and S
2
represents
456
Section 9.2
survival of sh from a passage route entrance to the river mouth. For detection, p
2
represents
the average detection probability of both passage routes (spillway and turbines), and p
3
rep-
resents detection probability at the river mouth (Figure 1). By convention, the release site is
considered the rst detection location and p
1
is set to 1.
With the telemetry design for this model, we cannot estimate all of the parameters just
described. Detection probability at telemetry station i is estimated from detections at loca-
tions downstream of station i. Therefore, the detection probability at the last station cannot be
estimated. Since detection probability is needed to estimate survival, survival alone cannot be
estimated through the nal reach. The best we can do is to estimate λ = S
2
p
3
, the joint prob-
ability of surviving and being detected at the river mouth. This confounding of the parameters
limits inferences about survival in the nal reach.
Maximum Likelihood Estimation
Here, we briey present the statistical theory used to estimate the parameters of all mark–
recapture models, using our simple CJS model as an example. Most mark–recapture models
are based on the multinomial distribution, a probability distribution describing the probability
of occurrence of events falling into discrete categories. In our case, the categories are detec-
tion histories of each sh’s migration through the study area. Detection histories are com-
posed of an alpha-numeric string describing whether a sh was detected at each location in
the study area. For the CJS model, a “1” means a sh was detected at a given location and a
“0” means it was not detected. For the simple CJS model above, a “101” history means the
sh was released upstream of the dam (“1”), not detected in either the spillway or turbines
(“0”), but subsequently detected at the river mouth (“1”).
The set of all possible detection histories forms the categories of a multinomial distribu-
tion. In turn, each detection history has a probability of occurrence that depends on the un-
derlying survival and detection probabilities (Table 1). For example, for the detection history
“101” to have occurred, a sh must have survived to a passage route (S
1
), not been detected in
Release
S
1
S
2
p
2
p
3
Dam
River mouth
Figure 1. Schematic of CJS model showing survival (S
1
and S
2
) and detection probabilities (p
2
and p
3
)
that can be estimated from the telemetry arrays installed at the dam and river mouth. In a CJS model, only
the joint probability, λ = S
2
*p
3
can be estimated in the last reach. For this example, p
3
is estimated from
a separate model for replicate arrays (described in detail below). Given an estimate for p
3
, S
2
can be
obtained as λ/p
3
.
457
Estimating Survival Using Telemetry
either passage route (1 – p
2
), survived to the river mouth (S
2
), and detected at the river mouth
(p
2
). Thus the probability of observing detection history “101” is the product of these event
probabilities: S
1
* (1 – p
2
) * S
2
* p
3
= S
1
* (1 – p
2
) * λ.
For any given data set of detection histories, there is a unique set of parameter values that
is most likely to have produced the observed detection history frequencies. To nd the most
likely parameter values, the next step in mark–recapture analysis is to dene the likelihood
function. For multinomial models, the likelihood takes the form
()
|,
i
n
ii
i
LnR
π
θ
where
()
|,
i
n
ii
i
LnR
π
θ
is the likelihood of the parameters (θ) given the data (n
i
, R): π
i
is the prob-
ability of observing each detection history, n
i
is the number of sh with each detection history,
and R is the total number of sh released. Given the probabilities of each detection history and
the frequencies of occurrence in the simulated data set (Table 1), the full likelihood function
for this example is:
()()()
()
()
()
()()( )
( )
11 18
35
2
54
12 12 12 11 2
|, 11111
i
LnRSpSpSpSSp
λλ
λλ
∝− −−+− θ
The likelihood function yields the relative likelihood of one set of parameter values over
another for a given data set of detection histories. By calculating and comparing likelihoods
among different values for S
1
, p
2
, and λ, we can identify the most likely parameter set for the
data. These “maximum likelihood estimates” will be closest to the true values of the param-
eters.
In practice, two approaches are used to nd the maximum likelihood estimates: 1) de-
riving equations that provide the maximum likelihood estimates (“analytical” approach),
and 2) using an optimization routine (e.g., Newton-Raphson method) that will system-
atically calculate the likelihood of different parameter values to nd the set of param-
eter values that maximizes likelihood (“numerical” approach). The two methods are akin
to estimating the mean using Σx
i
/n (the “analytical” approach) or by using least-squares
methods (i.e., the value of the mean that minimizes the total sum of squares—the “numeri-
cal” approach). The numerical approach is by far the most common means of obtaining
maximum likelihood estimates because analytical expressions for maximum likelihood
estimates do not exist for many mark–recapture models. However, both approaches can be
used for the CJS model. We adapted the maximum likelihood estimators of the CJS model
from Seber (1982) and Burnham et al. (1987) and present those equations here to illustrate
both approaches.
Table 1. All possible detection histories, their probability of occurrence (π
i
), and the number of sh with
each capture history in the simulated data set (n
i
) for our simple CJS model.
Capture history Probability of occurrence (π
i
) n
i
111 S
1
*p
2
54
101 S
1
*(1 – p
2
)*λ 11
110 S
1
*p
2
*(1 – λ) 183
100 (1 S
1
) + S
1
*(1 – p
2
)*(1 – λ) 52
458
Section 9.2
The maximum likelihood estimators for detection probability (p
i
), survival probability
(S
i
), and joint survival-detection probability (λ) of the CJS model are:
ˆ
i
i
ii
r
p
rz
=
+
1
1
ˆ
ˆ
ˆ
i
i
i
M
S
M
=
ˆ
ˆ
ii i
i
i
mr z
m
M
+
1
1
ˆ
k
k
r
m
λ
=
Where variables are dened as follows:
i = detection site, i = 1, 2, …, k, numbered from upstream to downstream (i = 1 is the release
site, and k = 3 in our simple CJS model).
m
i
= number of sh detected at site i.
r
i
= number of sh detected downstream of site i (i.e., at sites i + 1…k), of those detected at
site i.
z
i
= number of sh not detected at site i, but detected downstream of site i.
Note that M
1
= R (the number of sh released), p
1
= 1, and r
i
+ z
i
= total number of sh
detected downstream of site i and therefore, known to be alive at site i.
The maximum likelihood estimators for the CJS model make intuitive sense. The detec-
tion probability is the fraction of sh detected at site i of those known to have migrated past
site i. The parameter
ˆ
i
M
estimates the number of sh alive at site i, which is the number sh
detected at site i divided by the detection probability at site i. The survival probability from
site i–1 to site i is the estimated number of sh alive at site i divided by the estimated number
of sh alive at the previous site, i–1. These maximum likelihood estimators show the direct
connection between the survival probabilities and the numbers of sh detected at each loca-
tion. Standard errors for these estimators can be found in Seber (1982) and Burnham et al.
(1987).
Applying the Model to the Simulated Data Set
The frequencies of the capture histories in Table 1 provide all the information needed to
apply the CJS maximum likelihood estimators:
459
Estimating Survival Using Telemetry
m
2
= n
111
+ n
110
= 237,
r
2
= n
111
= 54,
z
2
= n
101
= 11,
ˆ
i
M
2
= 237/0.8307 = 285.3,
M
1
= R = 300
In Table 2, we compare the analytical estimates from the simulated data set to the known
fates of sh surviving to each detection site over the entire study period. We also t the CJS
model with the software program USER (User-Specied Estimation Routine, Lady and Skal-
ski 2009). This program maximizes the likelihood via an optimization routine and returns
maximum likelihood estimates and their standard errors.
The analytical and numerical estimators produced the same parameter estimates (Table 2),
and both are within 1.1 percentage points of the known-fate estimates. Note that λ accurately
estimates S
2
*p
3
from the known fate data. An important insight here is that our interpretation
about S
2
from λ depends completely on assumptions about the value of p
2
. Given that p
2
can-
not be estimated with the current design, if we had assumed perfect dectability (p
3
= 1), then
our estimate of S
2
is 0.230 when the true value is really 0.365. Similarly, our estimate of S
1
would have been negatively biased if we had not used the mark–recapture model but instead
assumed that p
2
= 1. In this case, substantial negative bias is the consequence of assuming per-
fect detectability. This example highlights the importance of explicitly estimating detection
probabilities using a mark–recapture model in order to obtain unbiased estimates of survival.
Extensions to the Simple CJS Example
In the simple example above, we did not use telemetry arrays located in the forebay or in
the immediate tailrace as shown in Figure 1 of Section 9.1. However, adding data from those
receivers allows us to estimate new parameters and improve precision of existing parameters.
For example, by including the forebay receiver detections as an additional occasion, we can
separately estimate survival 1) between release and the forebay, and 2) between the forebay
and the entrances to the spillway or turbines. Spatial partitioning of survival is the most com-
mon reason to add new receiver locations to a telemetry system.
ˆ
Table 2. Known-fate survival of simulated sh, analytical estimates of parameters, and numerical estimates
obtained from maximizing the likelihood via an optimization routine.
Parameter Known-fate estimates Analytical estimates Numerical estimates (SE)
S
1
282 of 300 = 0.940 285.3/300 = 0.951 0.951 (0.048)
p
2
238 of 282 = 0.844 54/(54 + 11) = 0.831 0.831 (0.047)
S
2
103 of 282 = 0.365 NA NA
p
3
65 of 103 = 0.631 NA NA
λ 65 of 282 = 0.230 54/237 = 0.228 0.228 (0.027)
460
Section 9.2
Extending the model with an additional sampling occasion can also improve precision of
survival estimates. For example, antennas in the tailrace would likely detect transmitters in
sh that had died during dam passage, making it impossible to estimate survival over such a
short distance. Nonetheless, including this detection location in the model could substantially
improve the precision of p
2
and S
1
because precision of the parameters is inuenced by the
number of surviving sh detected downstream of the dam. In the current model, only the 65
sh detected at the river mouth were used to estimate p
2
(Table 1). In contrast, in the simulated
data set, 217 sh were detected in the tailrace. It makes sense to use these detections in the
tailrace to improve precision of S
1
.
To use the tailrace array, a detection location can be added to the CJS model as the next
sampling location after the dam (see Figure 1). Then the survival probability for the reach
between the dam and tailrace array is set to 1, assuming that all sh passing the dam will
also pass the tailrace array, whether alive or dead. With this approach, detections on both
the tailrace and river-mouth telemetry stations are used to estimate p
2
and S
1
, and all other
parameters are interpreted as above. When this model is applied to the data set,
ˆ
S
1
= 0.940
with a standard error of 0.018. This estimate is exactly equal to the known-fate estimate for S
1
,
but the standard error is less than half that obtained from our original model (Table 2)—this
represents a substantial improvement in precision. Another alternative approach is to lump the
dam and tailrace detections together into one occasion. However, maintaining separate occa-
sions provides spatially-explicit detection probabilities that may be insightful for evaluating
the performance of the telemetry system. Further, the approach used here can be applied to
more complex models that estimate route-specic survival (see Multistate models). These
extensions to our simple CJS model illustrate the exibility of using telemetry in conjunction
with mark–recapture models.
Assumptions
CJS models are subject to seven assumptions. These assumptions relate to inferences to
the population of interest, error in interpreting transmitter signals, and statistical t of the data
to the model’s structure:
1) Tagged individuals are representative of the population of interest. For example, if the
target population is wild-origin Chinook salmon Oncorhynchus tsawytscha then the sample
of tagged sh should be drawn from that population. If hatchery-origin Chinook salmon
must be tagged then inferences about survival of wild sh from hatchery sh must be
based on subject matter arguments. Other areas that require representation include size
distribution, health proles, and migration timing. In many telemetry studies this may
be difcult to perfectly achieve, requiring caveats or discussion for proper inference. For
example, the size of an acoustic- or radio-tag may preclude tagging the smallest individuals
of a particular population due to tag-burden concerns, or a study may need to be
implemented during a three-week window of a ve-week migration distribution.
As we’ve noted, careful study design and candid discussion of limitations maximizes
the value of inference.
2) Survival of tagged sh are the same as that of untagged sh. For example, the tagging
procedures or detection of sh at telemetry arrays should not inuence survival or
461
Estimating Survival Using Telemetry
detection probabilities. If the transmitter negatively affects survival, then estimates
of survival rates will be negatively biased relative to untagged sh. Also, using our
simulated example, if the tagging process or transmitter presence affect swimming
performance, tagged sh may not pass through spillway or turbine routes in the
same proportions as untagged sh.
3) All sampling events are instantaneous. When the spatial or temporal scale of sampling
occasions approaches the scale of time or distance between occasions, then it becomes
difcult to correctly attribute mortality to a specic sampling period. For example, for
spatial mark–recapture models, if the detection range of a telemetry station is 200 m, but
the distance between telemetry stations is only 100 m, it becomes difcult to determine
where mortality occurs. Thus, when estimating survival through space, sampling should
take place over a short distance (e.g., hundreds of meters) relative to the distance between
telemetry stations (e.g., kilometers). Likewise, when estimating survival over time,
sampling occasions should be short (e.g., one day) relative to the time between sampling
occasions (e.g., weeks). One way to address this assumption is to place telemetry stations
in locations, if possible, where tagged sh move relatively quickly past an array and do
not reside with the detection range for an extended period.
4) The fate of each tagged sh is independent of the fate of other tagged sh. If sh exhibit
some sort of group behavior (i.e., schooling), sample size is effectively reduced and
precision estimates become biased downward.
5) The prior detection history of a tagged sh has no effect on its subsequent survival. For
example, in mark–recapture studies requiring physical recapture of animals, the capture
process may inuence subsequent survival. For telemetry, this assumption is usually
satised by the passive nature of detecting tags.
6) All tagged sh alive at a sampling location have the same detection probability.
7) All tags are correctly identied and the status of tagged sh (i.e., alive or dead) is known
without error. This assumes sh do not lose their tags and that the tag is functioning while
the sh is in the study area. Additionally, this assumption implies that all detections are
of live sh, that dead sh are not detected and interpreted as live, and that spurious
detections (e.g., false positive detections) are not interpreted as live sh. This assumption
is violated in cases where a detection location occurs close enough below a point-source
mortality (e.g., dam passage) such that dead sh are detected at the downstream telemetry
array. The consequence of violating this assumption is that mortality in one reach is
attributed to a downstream reach.
Assumptions 5 and 6 can be formally tested using χ
2
Goodness of Fit tests known as Test
2 and Test 3 (Burnham et al. 1987). Both Test 2 and 3 are implemented as a series of contin-
gency tables. Test 2 is informally known as the “recapture test” because it assesses whether
detection at an upstream array affects detections at subsequent downstream arrays (assump-
tion 6). Test 3 is known as the “survival test” because it assesses assumption 5 that sh alive
at array i have the same probability of surviving to array i + 1. With telemetry data, these tests
462
Section 9.2
are sometimes uninformative because detection probabilities are so high that cell frequencies
are too small for valid contingency table statistics (i.e., many zero frequencies in the “nonde-
tected” category).
Tag failure can be evaluated to formally test Assumption 7 that tags do not fail prior to sh
exiting the study area. A controlled tag life study can be conducted to estimate the probability of
tag failure at any point in time after tags were turned on. The methods of Townsend et al. (2006)
can then be used to estimate the average probability that a tag was alive while sh were in the
study area. If tags fail prior to exiting the study area, then information from the tag life study
can be used to adjust survival estimates for the probability of tag failure (Townsend et al. 2006).
It should be noted that nearly all telemetry studies violate most or all of these assumptions
to some degree. It is logical to assume that the handling, tagging, and release protocols of a
study, though carefully implemented, still have biological and behavioral effects on tagged
individuals. This does not mean that inference cannot be made from the results of the study,
only that researchers must report, and take care in how they present, the “challenges” of their
data and results and put proper caveats on the potential limitations on their inference.
USING REPLICATED TELEMETRY ARRAYS TO ESTIMATE DETECTION
PROBABILITY
In our simple CJS model, we were unable to estimate survival from the dam to the river
mouth (S
2
) without making assumptions about detection probability at the last telemetry sta-
tion (p
3
). To estimate p
3
and S
2
, we need an additional telemetry array downstream of our nal
array. In some cases, another receiver could simply be installed further downstream, and the
model extended accordingly to estimate S
2
, p
3
, and a new λ = S
3
*p
4
. However, since the nal
station is already at the mouth of the river, our best option is to implement two independent
telemetry stations at the same location. Let us assume that this detection station always con-
sisted of two sets of independent telemetry stations, but that we did not explicitly use this
information in our CJS model. The receivers at this location are so close together, that we can
safely assume that no mortality occurs between them. As with the tailrace example above, one
approach is to modify the CJS model by adding an additional reach and setting the survival
probability to one. Another option is to use a separate model based on the Lincoln-Peterson
method (Seber 1982) that estimates detection probability from the two monitoring stations.
The detections at each array yield a two-digit capture history dening whether sh were
detected on both arrays (“ab”), or only one or the other array (“a0” or “0b”). The detection
history probabilities are given in Table 3. Because we have an independent likelihood for the
nal monitoring station, the total likelihood for combined CJS and dual-array model is the
product of the two likelihoods (i.e., L
1
*L
2
). The idea is to use the dual-array model to estimate
p
3
, and then feed p
3
into the likelihood for the CJS model, allowing us to estimate S
2
. The
advantage of combining the models is that the resulting estimates incorporate uncertainty
arising from both models. This technique is analogous to the robust design of Pollock (1982).
Applying the combined CJS with dual array model to the simulated data, we nd that
ˆ
p
3a
=
0.40 (SE = 0.077),
ˆ
p
3b
= 0.39 (SE = 0.076),
ˆ
p
3
= 0.634 (SE = 0.083), and
ˆ
S
= 0.359 (SE =
0.064). Our estimates for p
3
and S
2
agree well with the known fates (Table 2). By adding the
dual array to the telemetry system and extending the model with the second likelihood, we are
now able to estimate survival of tagged sh through the entire system. We know that 34.3% of
463
Estimating Survival Using Telemetry
simulated sh (103 of 300) survived to the ocean, whereas our estimate using a mark–recap-
ture model is
ˆ
S
1
ˆ
S
2
= 0.951*0.359*100 = 34.1%. This example shows how the design of the
telemetry system is directly linked to the structure of the mark–recapture model.
Assumptions
The primary additional assumption of the replicate array model is that the two telemetry
arrays are statistically independent. That is, sh detected by one array must constitute a ran-
dom sample of sh passing through the detection eld of the second array. If the replicate
arrays are not independent, then estimates of detection probability may be biased, which may
also induce bias in estimates of survival. In the example above, this assumption can be violat-
ed if the detection elds of the dual arrays do not encompass the entire channel cross section,
or if they were both using the same power source and service was interrupted simultaneously.
Perry and Skalski (2008) assess the consequences of failing the assumption of independent
arrays and provide guidelines for conguration of telemetry arrays.
PARTITIONING COMPONENTS OF SURVIVAL
Survival between the dam and the ocean arises as a function of multiple mortality pro-
cesses. In our simulation, sh experienced mortality due to passage through the dam and
additional mortality during migration in the river below the dam. The estimate of S
2
above
includes both of these processes, but what if we were interested in separating mortality due to
dam passage from other sources of mortality? As previously mentioned, estimating survival
to the tailrace using telemetry arrays is difcult because all tagged sh, alive and dead, may
be detected at this location. The solution is to add a second release site immediately below
the dam. The idea is that the tailrace release group experiences only mortality unrelated to
dam passage in the river segment between the dam and the river mouth. Survival estimates
from both release sites can then be used to partition S
2
for the upstream release into dam and
in-river components (Figure 2). This is the “paired-release” model rst dened by Burnham et
al. (1987) and tailored to radio telemetry by Skalski et al. (2001). Although we present a spe-
cic example of the paired release model, the model has general application to any two groups
of sh that experience some common source of mortality, with one of the groups experiencing
some additional source of mortality that we wish to estimate.
To implement the paired release model for this study, we re-write S
2
as a function of its
underlying components and add subscripts for clarity (Figure 2):
Table 3. Probability of occurrence for each detection history of the dual-array model, where p
a
and p
b
are
the detection probability on the rst array and second array. The probability of being detected by at least
one of the arrays is p
3
= 1 – ( 1 – p
3a
)(1 – p
3b
). Also given is the number of sh with each capture history
in the simulated data set.
Detection history Probability of occurrence (π
i
) n
i
ab p
3
a
p
3
b
/p
3
16
a0 p
3
a
(1 – p
3
b
)/p
3
25
0b (1 p
3
a
)p
3
b
/p
3
24
464
Section 9.2
S
2,1(dam & river)
= S
2,1(dam)
S
2,1(river)
(1)
where S
2,1(dam & river)
is the survival probability through reach 2 for release site 1 (see Figure 2)
and “dam & river” indicates that this survival probability is comprised of mortality due to
passing the dam and also due to factors unrelated to dam passage in the stretch of river below
the dam. The right hand side of Eqn. 1 separates the mortality sources into survival probabili-
ties due to each source. Survival from the tailrace to the ocean is estimated from the second
release group (S
2,2(river)
). If we assume S
2,1(river)
= S
2,2 (river)
then Eqn. 1 can be rearranged to yield:
2,1(dam & river) 2,1(dam & river)
2,1(dam)
2,1(river) 2,2(river)
SS
S
SS
==
(2)
This model now has four likelihoods and eight unique parameters: a two-reach CJS model
for the upstream release (3 parameters), a one-reach CJS model for the tailrace release (1
parameter), and two dual-array models (2 parameters each) to estimate p
3,i
for each release
location. Note that the parameter S
2,1(dam)
is not part of the likelihood, but rather is derived as a
function of model parameters. Derived parameters are often of main interest, as illustrated in
this example. However, since these parameters are not explicitly estimated, the standard error
of derived parameters is usually estimated using the Delta method (Seber 1982).
The paired release model estimates the known-fate proportions closely (Table 4) and
yields insights about mortality processes affecting our simulated population. The estimates
suggest that mortality in the river below the dam is greater than mortality due to dam passage.
Release 1
S
1,1
S
2,1 (river)
p
2
p
3, j
Release 2
S
2,2 (river)
Dam
Ocean
S
2,1 (dam)
Figure 2. Schematic of the paired release model showing S
i,j
for reach i and release location j. The
dashed line indicates the release location in the tailrace where survival is partitioned into dam-related and
in-river components. Note that S
2,1(dam)
and S
2,1(river)
can be estimated only by assuming that S
2,1(river)
= S
2,2(river)
.
465
Estimating Survival Using Telemetry
Assumptions
The paired release model is subject to the same assumptions of the CJS model plus the
following additional assumption:
1) Survival is equal for upstream and tailrace release groups between the tailrace release
point and the rst downstream telemetry array. Inequality between these groups can result
in either positive or negative bias, depending on the direction of the inequality (see
Equation 2). This assumption implies that any handling and tagging mortality has
been expressed prior to release.
MULTISTATE MODELS
In our example of sh passing through a dam, we have yet to address the fact that sh
passed through different routes and under different treatments of dam operations. What is the
survival of sh passing through each route? What fraction of the population passes through
each route? Do the treatments inuence passage and survival? Multistate models offer a ex-
ible framework in which to answer these questions (Nichols and Kendall 1995; Lebreton and
Pradel 2002). We rst present the general framework of multistate models, and then show
how our example can be cast as a multistate model. Last, we extend the model to include dif-
ferent kinds of state variables and illustrate how the multistate model can be used as a basis
for evaluating hypotheses.
Like CJS models, multistate models estimate survival and detection probabilities. How-
ever, multistate models also allow individuals to be stratied into groups (states) to estimate
survival and detection parameters separately for each group. Individuals are allowed to move
(transition) from one state to another between sampling occasions. Multistate models estimate
the probability that an individual transitions from one state to another, and also the state-spe-
cic survival probabilities. For example, these models were rst designed to examine move-
ment probabilities of animals among different geographic locations and survival probabilities
for each geographic location (Brownie et al. 1993; Schwarz et al. 1993).
To describe the model, we use the notation of Brownie et al. (1993), but this notation is
later adapted for models specic to our simulated study. The fundamental parameters esti-
mated by the multistate model are:
rs
i
φ
= joint probability of surviving from sampling occasion i to i + 1 and moving from state
r at sampling occasion i to state s at occasion i + 1.
s
i
p
= probability of detection in state s at occasion i.
Table 4. Estimates of selected parameters from the simulated data set compared against the known fates
of sh for the paired-release model shown in Figure 2.
Parameter Known-fate estimate Maximum likelihood estimate (SE)
S
2,1(dam)
224 of 282 = 0.794 0.777 (0.179)
S
2,1 (river)
103 of 224 = 0.460 0.463 (0.068)
466
Section 9.2
Given multiple states, it is convenient to express these parameters in matrix form, here
using three states for simplicity:
11 12 13
21 22 23
31 32 33
iii
ii
ii
iii
φφφ
φφφ
φφφ
=





φ
and
1
2
3
00
00
00
.
i
i
i
i
p
p
p
=





p
When all states are sampled at every occasion and animals move among all states, all
parameters are estimable except for the nal interval, similar to the CJS model. In the param-
eterization above,
rs
i
φ
includes the underlying probabilities of both surviving and moving
between states. To estimate the underlying parameters, the model can be reparameterized as
a function of
r
i
S
, the probability of surviving from occasion i to i + 1 conditional on being in
state r at occasion i; and
rs
i
Ψ
, the probability of transitioning from state r at occasion i to state
s at i + 1, conditional on surviving to i + 1. Using the three-state example, the reparameteriza-
tion is
123rr
rr
ii
ii
S
φφφ
=++
and
.
rs
rs
i
i
r
i
S
φ
Ψ=
Note that
rs
i
Ψ
for a particular state s must be specied as one minus the sum of the others
because the transition probabilities are constrained to sum to one.
The Route-Specific Survival Model
The route-specic survival model introduced by Skalski et al. (2002) is a multistate model
designed to estimate the proportion of sh passing through each route at a dam, and then sur-
vival conditional on the route of passage. In our study, we have two passage routes, yielding
the model shown in Figure 3.
The route-specic survival model requires replicate telemetry arrays with each passage
route in order to estimate route-specic detection probabilities. With only a single detection
array in each route, some sh will pass through a route undetected and although detected
downstream, the route of passage of these sh is unknown. Without replicate telemetry arrays,
we can estimate the average detection probability over both routes, but if detection probabili-
ties differ between routes, then estimates of the proportion of sh passing each route will be
biased. By installing two sets of antennas in each passage route, the detection probability of
each route can be estimated, allowing the remaining parameters to be estimated without bias.
Assumptions
Multistate models are subject to the same assumptions as the CJS model plus two addi-
tional assumptions:
1) In order to separate state-specic survival from state transition probabilities, we must
assume that all mortality occurs rst and then sh transition “instantaneously” at
467
Estimating Survival Using Telemetry
the end of the sampling period. For the route-specic model described above, this
assumption is fullled because mortality occurs upstream of the dam prior to sh
passing (transitioning) through either the spillway or turbines. An example where
this assumption may be violated is when a river splits into multiple channels and the
goal is to estimate the probability of entering a given river channel. In this case, if
monitoring stations are located considerably downstream of the river junction, transition
probabilities may be biased if survival probabilities differ in the two channels
downstream of the river junction (see Perry 2010).
2) There is no error in assignment of states to individuals. For example, for the route-specic
model described above, we assume that no sh are wrongly assigned to a passage route.
TAILORING MARK–RECAPTURE MODELS TO STUDY GOALS
Armed with the suite of models discussed thus far, we have the ability to tailor the te-
lemetry system and mark–recapture model to answer specic questions in a study. In our
simulated study, the dam was operated at high and low spill discharge implemented over six
48-h treatment periods. The primary objective is to determine dam operations that maximize
rates of passage through the safest possible route. A priori, it is assumed that survival through
turbines is lower than the spillway. Therefore, dam operations may affect survival at the popu-
lation level by inuencing the proportions of sh passing through each route. We also suspect
a temporal component to survival, with sh passing the dam later in the study having lower
S
1
Ψ
Tu
Ψ
Sp
=1- Ψ
Tu
p
Tu
p
Sp
S
Tu
S
Sp
Release
p
3
Ocean
Figure 3. Schematic of the route-specic survival model (Skalski et al. 2002) showing route passage
probabilities (Ψ) for the turbines (Tu) and spillway (Sp), route-specic detection probabilities (p
Tu
, p
Sp
), and
survival from the dam to the ocean conditional on having passed through the turbines (S
Tu
) or spillway (S
Sp
).
Route-specic detection probabilities are estimated by the dual array model presented earlier.
468
Section 9.2
survival to the ocean. We would like to quantify this relationship to link migration delay
above the dam to survival below the dam.
How should we structure the model to assess these hypotheses? One can envision a num-
ber of ad hoc approaches to address the questions at hand. The rst that comes to mind is
constructing separate models for each of the six release groups above the dam. Most of the
sh pass within a given treatment period, but some do not. This approach would likely capture
major differences among treatments, but the estimates are not unique to a given treatment and
represent a mixture of sh passing during multiple treatments. Another approach is to con-
struct a separate model for each treatment, assigning sh to a treatment based on when they
pass the dam. This approach is undesirable because we must exclude sh that are not detected
in passage routes and estimating S
1
is impossible (by denition, only sh surviving to a pas-
sage route are included).
The answer is to generalize the multistate model to represent different kinds of state vari-
ables. State variables need not represent geographical locations. States can be dened as any
classication variable that can be identied at the time of recapture. For example, multistate
models have been used to estimate transitions among weight classes (Letcher and Horton
2009) and between breeding and nonbreeding status (Nichols et al. 1994). In our case, we
dene an additional state variable representing each 48-h time period. Thus, sh detected
in a passage route were assigned to one of the six 48-h time periods; the low-spill treatment
occurred during time periods 1, 4, and 5 and the high-spill treatment during the other time
periods (see Figure 2 in Section 9.1). Although sh not detected in passage routes cannot be
assigned to one of the six time periods, the probability function for this detection history in-
corporates the possibility that sh could have passed during any of the six time periods.
This design yields 12 unique combinations of the state variables (2 routes × 6 time-peri-
ods), and thus 12 unique transition and survival probabilities. We must also construct 12 dual
array likelihoods to estimate the detection probabilities for each route in each time period.
Next, we want to estimate the fraction of the population that passes through each route during
each time period. The branching process in Figure 4 shows how this model can be structured.
The parameter τ
t
estimates the probability of sh passing the dam during time period t, Ψ
Tu,t
estimates the probability of passing through the turbines given the sh passed the dam during
time period t, and S
Tu,t
is the probability of surviving from the dam to the ocean given passage
through the turbines during time period t.
At this point in designing the model, S
Tu,t
and S
Sp,t
includes both mortality due to dam
passage and in-river mortality below the dam. Therefore, we included the second release in
the tailrace, so that we could estimate S
Tu,t (dam)
and S
Tu,t (river)
using a paired release design. Of
primary concern here is whether mortality due to dam passage differs between routes.
This model now has 68 parameters, many more than previous models: The multistate
model for the upstream release has 24 parameters, the CJS models for sh released below the
dam has 6 parameters, and the replicate array models estimate 38 parameters. Since param-
eters are estimated for many subgroups, precision of parameters under this full model will
be poor. Furthermore, we would like to evaluate hypotheses about the effect of dam opera-
tions on survival and use of passage routes. We will use model selection approaches based
on Akaike’s Information Criteria (AIC) to both improve parameter precision and evaluate
biological hypotheses (Burnham and Anderson 2002). Among candidate models, the model
with the smallest AIC value represents the model with the most favorable tradeoff between
precision and accuracy of the estimates (i.e., over-tting versus under-tting). The difference
469
Estimating Survival Using Telemetry
in AIC values (ΔAIC) between two models represents the degree of support for one model
over another. As a general rule of thumb, ΔAIC < 2 indicates little or no evidence that either
model is more appropriate than the other (Burnham and Anderson 2002).
To compare models using AIC, we constructed a number of reduced models that rep-
resent different hypotheses about the parameters. For example, detection probabilities are
estimated for each route and time period, but if detection probabilities do not vary over time
then estimating a single detection probability for all time periods will improve precision by
pooling information over all time periods. Comparing AIC of the full model against a model
with detection probability constant over time provides evidence of support for one hypothesis
over another.
We used a step-down approach to form and compare models (Lebreton et al. 1992). First,
we evaluated whether detection probabilities varied by time period or release location. We
compared the full model where all detection probabilities were estimated separately for each
time period and release location to a model where detection probabilities for each route were
constant over time and among release locations (denoted model “p(.),” see Table 5). The
best-t model for detection probability was then used as a basis for evaluating remaining hy-
potheses. Next, we compared three models for the tailrace release group to evaluate whether
survival in the reach below the dam was constant over time, unique for each time-period, or
declined exponentially over time (i.e., S = e
rt
). Using the best t model based on detection
probabilities and downriver survival, we then evaluated whether survival for each passage
S
1
Release
τ
1
τ
2
...
...
Ψ
Tu, 1
Ψ
Sp,1
p
Tu, 1
p
Sp,1
S
Tu,1
S
Sp, 1
p
3
Ocean
5
1
1
t=
τ
t
τ
6
=
Dam
Ψ
Tu, 6
Ψ
Sp,6
p
Tu, 6
p
Sp,6
S
Tu,6
S
Sp, 6
p
3
Figure 4. Schematic of multistate model for estimating passage (Ψ) and route-specic survival (S
Tu
and S
Sp
)
probabilities conditional on the time period of dam passage.
470
Section 9.2
route was 1) constant over time, 2) constant within each treatment but different between treat-
ments, and 3) unique for each time period. Finally, using the best-t model at this stage, we
evaluated the same three hypotheses for passage probabilities (i.e., constant for each route,
treatment-specic, or time-specic). We then present the estimates for the best t overall
model.
The AIC-best model (Table 5) included constant detection probabilities at each location,
survival below the dam declining exponentially with time (Figure 5), no effect of treatments
on route-specic survival, but differences between treatments in the proportion of sh pass-
ing through the turbines. The probability of passing the turbines
Tu
) was much higher dur-
ing the low-spill treatment compared to the high-spill treatment, and the estimates match the
known-fates well (Table 6). Survival was lower through the turbines than through the spill-
way. Although some of the parameter estimates differ considerably from the known fates, the
standard errors are large and the estimates fall well within the 95% condence intervals (i.e.,
the estimate ± 1.96 standard errors). The exponential rate of decline in downstream survival
also t the known-fate data quite well (Figure 5). Last, the best-t model selected by AIC fol-
lowed the structure of the true model from which the data were generated.
Dam-passage survival can be estimated as a function of the route-specic survival and
passage probabilities as
S
2,1 (dam)
= Ψ
Tu
S
Tu,(dam)
+ (1 – Ψ
Tu
)S
Sp,(dam)
.
In our simulated population, the estimate of S
2,1
(dam) was 0.671 (SE = 0.105) during
the low-spill treatment, but 0.906 (SE = 0.123) during the high spill treatment. Since route-
specic survival did not differ between treatments, the lower overall survival during the low-
spill treatment can be attributed to a higher fraction of tagged sh passing through the turbines
during this treatment.
OTHER COMPONENTS OF MARK-RECAPTURE MODELING
In this Section, we have illustrated the exibility of using mark–recapture models to
analyze telemetry data. The examples provided here provide a launching pad for developing
Table 5. Model selection statistics for evaluating different hypotheses about detection (p), survival (S), and
routing probabilities (Ψ). Statistics are as follows: NLL = negative log likelihood, AIC = Akaike’s Information
Criterion, and ΔAIC = difference in AIC between model i and the lowest-AIC model.
Model Number of parameters NLL AIC ΔAIC
Full 63 107.4 340.9 52.7
p(.) 35 120.1 310.1 21.9
p(.), S
2,2 (river)
(.) 30 137.4 334.9 46.7
p(.), S
2,2 (river)
(exp(t)), 30 120.6 301.2 13.0
p(.), S
2,2 (river)
(exp(t)), S
route, (dam)
(treat) 23 124.4 294.8 6.6
p(.), S
2,2 (river)
(exp(t)), S
route, (dam)
(.) 21 124.4 290.8 2.6
p(.), S
2,2 (river)
(exp(t)), S
route,(dam)
(.), Ψ
Tu
(treat) 17 127.1 288.2 0.0
p(.), S
2,2 (river)
(exp(t)), S
route,(dam)
(.), Ψ
Tu
(.) 16 165.0 362.0 73.8
471
Estimating Survival Using Telemetry
mark–recapture models to apply to your own telemetry studies. However, given the introduc-
tory nature of this Section, we have yet to discuss a number of topics that every practitioner
of mark–recapture modeling must address. In closing, we briey identify some of these topics
so integral to mark–recapture modeling.
Evaluating Goodness of Fit
All models are capable of producing estimates, but not all estimates are good estimates
in the sense of optimizing the balance between precision and bias. In our model selection
example, we used AIC to identify the “best” model, but we still need to determine if the best
model is a good one. Evaluating goodness of t is necessary to assess how well the model ts
the data. Lack of t may be induced by overdispersion or by failure of model assumptions.
Overdispersion is dened as more variability in the data than is expected given the model
0.00.2 0.40.6 0.8
04896 144 192 240 288
Time (h) since first release
S
2,2 (river)
Figure 5. Known-fate proportions surviving to the ocean for each tailrace release group (circles) compared
to the tted survival function from the best-t model (line).
Table 6. Maximum likelihood estimates (MLE) of selected parameters from the lowest-AIC model from the
simulated data set compared against the known fates of sh.
Parameter Known-fate estimate MLE (SE)
Ψ
Tu, low spill
90 of 117 = 0.769 0.773 (0.044)
Ψ
Tu, high spill
36 of 165 = 0.218 0.217 (0.034)
S
Tu,(dam)
84 of 126 = 0.667 0.575 (0.121)
S
Sp,(dam)
140 of 156 = 0.897 0.998 (0.146)
r 0.005 0.006 (0.0007)
472
Section 9.2
structure and is common when the parameter values are dependent upon some state or trait
that is not incorporated in the model. Typically, when data are overdisperse, estimates from
a mark–recapture model will be unbiased, but variance estimates will tend to be negatively
biased. Therefore, it is important to test for overdispersion and adjust (inate) the variance
when overdispersion has been detected. There are a number of methods for estimating over-
dispersion (Lebreton et al. 1992; White et al. 2001), but most involve estimation of
ˆ
C
, the
variance ination factor. When the data t the model’s structure perfectly, then
ˆ
C
= 1.0. Lack
of t is indicated by
ˆ
C
> 1, in which case the common practice is to inate the variance, as
well as model selection statistics, by a factor of
ˆ
C
(Burnham and Anderson 2002).
Mark–Recapture Software
Most biologists lack the mathematical and computer programming experience that are re-
quired to construct, t, and diagnose mark–recapture models of practical scale without the help
of automated software routines. Fortunately, computer software packages are available for con-
structing models and tting them to the data. A good software program should: 1) be exible
(allow custom models to be built), and 2) provide standard errors and prole likelihood con-
dence intervals of parameter estimates (see Lebreton et al. 1992), including derived parameters.
Integrated diagnostics and model selection criteria are also preferred. In our examples we used the
software program USER (Lady and Skalski 2009) to t the models to the data (i.e., maximize the
likelihood function) and obtain estimates for derived parameters (e.g., S
2,1(dam)
). USER is a exible
environment for model development because the model structure and parameters are completely
user-specied. Thus, any custom model can be tailored to a specic telemetry study and then t
to the data. However, complete specication of model likelihoods can be a cumbersome task and
requires a minimum level of quantitative savvy, especially for large, complex models.
Perhaps, the most common mark–recapture software program is MARK (White and Burn-
ham 1999). MARK has many well-developed model structures, including diagnostic tools,
goodness of t tests, and model selection criteria (White et al. 2001). However, dening the
structure of complex or unique models can be cumbersome or impossible with MARK. Recent
development of the R package RMark (Laake and Rexstad 2011) has improved the ability to
build models and perform model selection with MARK, but requires some knowledge of the R
language.
MARK and USER are simply two of several available software packages that have en-
hanced the eld of mark–recapture modeling by simplifying the cumbersome process of model
building and tting. Behind every graphical user interface, however, is a host of complex pro-
cesses that affect the output. Practitioners should be familiar with likelihood estimation (includ-
ing variances and condence intervals), model selection, and linear model terminology. We
discourage use of software without an understanding of the underlying statistical theory, such as
optimization routines, link functions, model selection criteria, and goodness of t tests. When
used appropriately, these tools can free biologists from the time-consuming task of model con-
struction and allow them to focus on the biological hypotheses behind the models.
Sample Size and Precision
Determining sample size required to achieve some desired level of precision is an im-
portant aspect of study design, particularly for telemetry studies where scare resources must
473
Estimating Survival Using Telemetry
be allocated between transmitters and telemetry monitoring stations. So then, how many sh
need to be tagged to obtain a given level of precision? The answer depends on the model
structure, the number of sampling occasions, the true values of the parameters, and biological
variation in the parameters. For example, lower detection probabilities require higher sample
sizes to obtain a given level of precision.
To determine sample sizes, simulation studies can be used to estimate precision of esti-
mated parameters (Devineau et al. 2006). The rst step is to assume a set of detection and
survival probabilities that might be obtained in a given study. These values can be obtained
from a pilot study, literature search, or expert opinion. Next, the expected value of capture
history frequencies is calculated based on a given sample size and the assumed detection and
survival probabilities. Finally, the mark–recapture model is applied to this simulated data set
to yield the expected standard errors of survival and detection probabilities.
Fortunately, a number software programs exist to facilitate sample size and precision
analysis. The software program SampleSize (Lady et al. 2003) is a simple-to-use program for
determining precision and sample size with CJS and paired-release models. Program MARK
also contains a simulation module that can be used for conducting sample size analysis for
many of the models implemented in MARK. These software packages allow users to quickly
run a number of different scenarios that vary parameter values, model structures, and sample
sizes to develop a robust study design likely to achieve desired levels of precision.
Modeling Biological Hypotheses
In our application of the route-specic survival model to the simulated sh passage study,
we introduce the idea of tting mark–recapture models that represent different biological hy-
potheses. For example, model selection results showed strong evidence that riverine survival
was not constant, but declined exponentially over time (Table 4; Figure 5). This conclusion
may lead biologists to further evaluate time-dependent mortality processes, such as physi-
ological condition, water temperature, or changes in predator densities.
Hypothesis-driven model selection begins with a general model (i.e., full model) as a rep-
resentation of the underlying biological system. We then develop a set of reduced models that
each represents a specic set of constraints. Each constraint represents an alternative view
(i.e., hypothesis) of the biological system. Identifying candidate models requires knowledge
of the biological system, and is therefore not a statistical exercise (Lebreton et al. 1992).
Reduced models may estimate parameters as functions of categorical or continuous co-
variates, such as rearing history or size at release. The simplest constraints involve categorical
covariates that are xed throughout the study. This would be used, for example, to determine
if rearing history (e.g., hatchery, wild) has a “signicant” effect on survival. Multistate models
offer the ability to evaluate similar hypotheses when the categories (i.e., states) can change
for a given sh during the study. For example, in our route-specic survival model, the route
state for an individual sh could not be known at the time of release. Further, the route assign-
ment was never known for sh that passed the dam undetected. Still, model selection in the
multi-state framework allowed us to test the hypothesis that survival differed among passage
routes (Table 4). In this way, multi-state models are a powerful tool for examining the effects
of categorical state variables on demographic parameters.
Categorical predictor variables can also represent characteristics of the sampling occa-
sion, rather than of the tagged sh. In a classic example, Lebreton et al. (1992) showed that
474
Section 9.2
survival of European dippers Cinclus cinclus was lower during ood years versus nonood
years. Analogous examples for migrating salmon might be to compare survival through river
reaches that are freshwater versus estuarine, or through industrialized versus natural habitats.
In the dipper example, sampling took place at relatively regular intervals (annually). How-
ever, telemetry receivers are never uniformly spaced throughout a study system. Therefore, to
appropriately compare survival among individual river reaches, reach-specic survival esti-
mates must be standardized (by reach length) within the likelihood. An intuitive approach is
to estimate instantaneous survival instead of reach-specic survival, allowing direct compari-
son and hypothesis testing. Further, any other reach-specic survival rate (e.g., reach-specic
survival or survival rate per river km) can also be derived from the instantaneous rates.
Survival is a key demographic parameter that describes the dynamics of a population.
Combined with other data sources and analyses (e.g., time-to-event analysis in Section 9.1),
telemetry and mark–recapture models can be used to understand the inuence of external
processes on populations (e.g., hydro-system regulation, habitat alternation, articial rear-
ing, water quality improvements). These analytical frameworks provide powerful tools for
analyzing telemetry data, and we hope that our introduction to these tools encourages more
biologists to consider them as an integral component of telemetry studies.
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