Nov., 1987
Revised Aug., 1988
A Unified Theory for Release-Resampling Studies of Animal Populations
Kenneth P. Burnham
Department of Statistics, North Carolina State University
and USDA-Agricultural Research Service
Box 8203, NCSU, Raleigh, North Carolina 27695, U.S.A.
Institute of Statistics
Technical Report Series #1698
Summary
Both capture-recapture and band-recovery studies can be conveniently modelled by one
approach. The key is to view these methods as studies of cohort survival processes. Thus, in
capture-recapture, as in band-recovery, attention is focused on survival rate, not population
size. Probability models for the data are conditional on the known releases initiating each
cohort. The first resampling of an individual after a release (e.g., live-capture, dead-recovery,
live-resighting) effectively removes that individual from its release cohort. That individual may
then be re-released as part of a new cohort. The re-sampling data are modelled as multinomial
counts with a standardized form for the cell probabilities. The different resampling methods
(e.g., live-recapture vs. band-recoveries) correspond to different model parameterizations. For
capture-recapture, the complete model also requires the probability distribution of the
unmarked animals caught on each occasion. That distribution, which is given here under the
time-specific assumptions of the Jolly-Seber model, incorporates the recruitment parameters.
The full distribution of the minimal sufficient statistic for the Jolly-Seber model can be given
as a product of conditionally independent binomials. This representation of the minimal
sufficient statistic greatly facilitates deriving variances and covariances. Goodness of fit tests
for such cohort-survival data have two basic components. One component is identical for
capture-recapture data and band-recovery data. The second component exists only for capturerecapture data; it uses information in subcohorts as defined by capture histories within each
released cohort. The basic notation required in this modelling approach is much less than is
typically used to develop capture-recapture models and the derivation of theory is more
straightforward. Several special cases of the Jolly-Seber model are easily dealt with using this
approach. Also considered here is the generalization of the time-specific model to allow
different parameters to apply during the time interval i to i+l for individuals released at time
i as compared to individuals released before time i and not captured (resampled) at time i.
Key words: Animal marking studies; Bird banding; Capture-recapture; Jolly-Seber model;
Population size estimation; Survival rates.
PREFACE
The purpose of this report is to present some essential elements, as the author perceives
them, of a unified approach to the statistical modelling and analysis of data arising from
animal marking studies. This subject is often referred to as capture-recapture for open
populations. The subject is, however, much broader than just capture-recapture taken literally.
The essence of the methods is that animals are being sampled, somehow, from populations
open to the dynamics of entry and loss. Sampled animals are "marked" in any way that allows
them to be recognized as being marked at a later sampling time and allows complete capture
history information to be known for each animal. Sampling often involves physically capturing
the animals, but this is not necessary; it is only required that their marks be identified at
subsequent sampling times. Typically, each individual has a unique tag.
There is a large literature, both statistical and biological on the estimation of population
parameters from such animal marking studies. One branch of the literature includes birdbanding and fish-tagging studies that yield data from a single terminal harvest-related
recovery, as reviewed by Brownie et al. (1985). Another branch deals with modelling multiple
recaptures of marked animals often referred to as "Jolly-Seber" models (Jolly 1965, Seber
1965); Seber (1982, 1986) reviews this literature. Although I present here some elements of a
unified approach to a mathematical theory for such studies, the emphasis is on theory for
time-specific capture-recapture models. The concentration is on mathematical statistical
models and methods. It is assumed that the reader is familiar with some of the important
capture literature, such as Bailey (1951), Leslie and Chitty (1951), Leslie (1952), Darroch
(1959), Cormack (1964, 1968, 1979, 1985), Jolly (1965, 1982), Seber (1965, 1982), Robson
(1969), Pollock (1975, 1981a), Buckland (1980), Sandland and Kirkwood (1981), Brownie et
al. (1985), Crosbie and Manly (1985) and Burnham et al. (1987).
ACKNOWLEDGMENTS
The thoughts and results in this report have evolved and been refined over several years.
During this time period, quite a few colleagues have contributed to my thinking through a
variety of direct interactions (discussions, correspondence and reviews of ms. drafts). I will not
even try to indicate here the type and extent of influence, or assistance, each of the following
persons have rendered; I do thank these colleagues for their contributions: D. R. Anderson, C.
Brownie, S. T. Buckland, J. Clobert, R. M. Cormack, J. D. Lebreton, P. M. North, K. H.
Pollock, and G. C. White.
Kenneth P. Burnham
(Temporary note: Due to my moving to Colorado, some sections intended to be here are not
yet added. The results are all known to me and will get put in after I settle in Fort Collins at
CSU - USFWS Colorado Cooperative Fish and Wildlife Research Unit.)
-2-
CONTENTS
1.
2.
3.
4.
5.
6.
7.
8.
Introduction
Unifying Concepts and Notation
2.1
Concepts and Philosophy
2.2
Data Representations
2.3
Basic Notation
2.4
Assumptions
Modelling Release-Resampling Data
3.1
A General Approach
3.2
Cohorts and Subcohorts
3.3
Discussion
The Time-Specific Model for a Release-Resampling Data Set
The Minimal Sufficient Statistic and Residual Distributions
4.1
4.1.1 Factorization of Pr{RRDS}
4.1.2 Estimation
4.2
Band Recovery vs. Capture-Recapture
4.3
Capture-Recapture
4.4
Goodness of fit Testing
The Jolly-Seber Model
5.1
A Minimal Sufficient Statistic
5.2
Parameter Estimation
5.2.1 Survival and Capture Rates
5.2.2 Recruitment and Abundance Numbers
5.3
Special Cases of the Jolly-Seber Model
5.3.1 No recruitment
5.3.2 No mortality
5.3.3 No recruitment or mortality
5.4
Discussion
On Deriving Theoretical Variances and Covariances
6.1
A General Methodology for use with Release-Resampling Data
6.2
The Jolly-Seber Model
6.2.1 Deriving Dispersion Formulae for ~i and Pi
6.2.2 Deriving Dispersion Formulae for the Abundance Estimators
6.2.3 Special Cases of the Jolly-Seber Model
The Time-Specific Band Recovery Model
6.3
A Generalized Time-Specific model
Introduction
7.1
7.2
Capture-Recapture
7.3
Band Recovery
Discussion
REFERENCES
-3-
1. Introduction
The origins of comprehensive analysis and inference procedures for band and tag
recovery data are found in the early 1960's and 70's (Seber 1962, 1970, Robson and Youngs
1971; see also Youngs and Robson 1975). In contrast, analyses for live recapture (capturerecapture) sampling were sophisticated even by the early 1950's (see Jolly 1963:113 and
Cormack 1968 for discussions of historical developments). The theories for band recovery
analysis and capture-recapture (CR) analysis were developed separately. However, these two
theories are substantially the same from the standpoint of a basic statistical model describing
the data. This fact has been overlooked because (1), the two resampling methods are very
different (live recaptures by the investigators vs. band recoveries from a virtual army of
hunters); (2), capture-recapture has often focused more on population size N, than on survival
rate, <p; and (3), the two literatures have been separate and use different notation, different
indexing of the data and different statistical modelling methods (see comments by Jolly 1965,
also compare, e.g., Brownie et al. 1985 and Cormack 1985).
In capture-recapture, as realized by Jolly (1965), data are potentially available from
multiple "encounters" of each animal through repeated captures or sightings. An example is
an animal originally captured in year 1 and recaptured in years 3, 5, 6 and 8. Viewed in this
way, i.e., as "multiple" recaptures after the initial release, the data are not represented as
mutually exclusive events, hence they cannot be modelled using simple multinomial
distributions such as are used for band cohort recovery data. There are better ways to
conceptualize CR data to facilitate modelling them.
A unifying way to think of these types of studies is that they all represent primarily
cohort survival processes. At times i = 1, ..., k-1, a cohort of size R i , of marked animals is
released. A sampling process exists to re-sample the extant cohorts. In the case of CR, when
an animal is recaptured (re-sampled) it is usually re-released, aDd thus it has been, in effect,
removed from its original release cohort and has become part of a new cohort. Another way to
view the processes is that there cannot be two recaptures following only one release; there had
to be an intervening "release" after the first recapture. Fundamental to a unifying approach
for release-resampling is to model resampling counts conditional on release and to realize that
upon re-sampling, the animal is conceptually, if not physically, removed from the cohort of its
previous release. If an animal is resampled at occasion i and then re-released, it becomes part
of the new released cohort at occasion i.
Consider the above case of an animal first released on occasion 1 and recaptured on
occasions 3, 5, 6 and 8. We tend to overlook that it was also re-released on occasions, 3, 5, 6
and 8, at which time it was known to be alive. Thus releases and recaptures are paired as
(1, 3), (3, 5), (5, 6), and (6, 8) with no recapture after the final release at time 8. Ignoring the
intervening releases, one is led to portray the data as below (0 denotes not captured, 1 for
captured):
Recapture history at occasion j Release time
1
2
3 4 5 6 7 8 9 10
010110100
Conceived of this way, statistical modelling of the data can be difficult. An effective way to
visualize and tabulate the data is in the manner of band recovery data. The release-recapture
(not just recapture) history of this one animal is shown below:
-4-
Release time i
1
3
5
6
8
First recapture after release time i
2 3 4 5 6 7 8 9 10
o
1
o
1
1
o
1
(not seen again)
o o
Model construction is greatly facilitated by thinking in terms of paired releaserecaptures, rather than an initial release followed by a series of recaptures. Summarizing all the
data like this, it is easy to build up the components of the model required as a basis for
estimation of survival and recapture parameters. This approach has been used by Brownie and
Robson (HJ83) and Brownie and Pollock (1985). Under the time-specific assumptions of the
Jolly-Seber model (Jolly 1965, Seber 1965) the basic array of data is structured as below:
i
-1-
First recaptured at time j
given release at time i
Known releases
at time i
j = 2
m12
2
3
3
m13
m23
k
4
m14
m24
m34
k-l
These data are structured (but not indexed) exactly like band recovery data modelled in
Brownie et al. (1985); note also, this is the Leslie Method B way of tabling the data (Leslie
1952). The situation is different from banding studies because here, any animal recaptured at
time j (j = 2, H" k-l) can be re-released into the cohort of R. animals released at time j,
some of which may have been previously unmarked. The first~recaptures count data (the
mi,i+l' ..., mik given the Rj ) from each cohort can reasonable be modelled as multinomial
random variables, with the cohorts (i
1, 'H' k-l) being independent, given the releases R 1 ,
.H' R k_1. This conditioning on release, or resighting, is not just a mathematical device, rather
it is mandated because one knows the animals are alive at that time.
For this type of capture and re-release sampling, under the Jolly-Seber model, we have
=
,j = i+l,
,j> i+l.
For convenience I will sometimes use E(m· ·), rather than E(m..
1J
1J
model are
p.
1
= the
I R.).
1
The parameters in the
probability of capture on the ith occasion given that the animal
is still in the population at risk of capture on the ith occasion; qi
I-Pi;
=
-5-
¢. = the probability of surviving in the population at risk of capture from
1
occasion i to occasion i+1 given that the animal is alive in the population at
risk at the start of survival period i (¢. = the product of the physical survival
rate times the probability of not emigrating).
=
=
Making the formal re-parameterization of fi
¢i Pi+1 and Si
¢i q i+1' the E(mij) are exactly
in the form of Model M 1 for a band recovery study with k-1 banding and recovery years. In
fact, one can use program ESTIMATE (Brownie et al. 1985) to analyze such an m-array
summary of recapture data and the resulting maximum likelihood estimates (MLEs) Si and f i
can be used to produce the Jolly-Seber (JS) MLEs as
~.1
= f. + S.
1
- 0-·
1
•
=
1
f.1
+ S.)
1
,i
= 1, ..., k-2,
,i
= 1, ..., k-2;
A
f k-1
¢k-1 Pk' cannot be decomposed, i.e., ¢k-1 and Pk are not separably estimable. For
more discussion of this relationship see Brownie and Pollock (1985) and Burnham et al.
(1987:201-204).
These results show that for the purposes of survival rate estimation, capture-recapture
and band recovery data can be modelled in a common framework. In fact, the only difference
is the parameterization; this fact was known years ago, see e.g., Jolly (1965:243-245), Seber
(1973:213, 239-242). The difference in the parameterization is because the two processes for
sampling released animals are so different: live resampling (recapture or resighting) vs.
reported bands from hunter-killed birds. Consequently, the f i represent a real source of
mortality, whereas the Pi represent removal from the release cohort, not actual mortality.
A general statistical model structure for the expectations of such first "resampling" data
from known releases is
E(m..)
IJ
~ =
1
{Q"1
p....
p.J- 2(l·J- 1
1
,j
= i+1,
,j > i+1, i
= 1, ... , k-1.
Here, (l. 1 = the probability that an animal is removed from its most recent release-cohort
J(which will be some i < j) during resampling occasion j, given that it is at risk at release time
j -1. After such a removal, it is either "lost on capture," or re-released into cohort j. The (lparameters are thus a type of cohort loss rate (i.e., a removal rate from the cohort of their last
release). Conversely, Pj is the probability that the animal survives in the cohort of its most
recent release, from release time j to j+1, given that it is alive in the population at risk just
after release time j.
There is yet another class of studies to which these models apply: release of reared
animals (e.g., hatchery raised fish) as part of experiments to study survival processes
(Burnham et al. 1987). This brings into clear focus the fact that the unifying feature here is
that there are known releases in separate, independent cohorts. These animals are then subject
to extant forces of mortality, i.e., they experience a survival process, about which we wish to
make inferences. In addition, the released cohorts are subject to a sampling process, which may
be likened to another source of cohort "mortality;" the resultant counts of live or dead animals
-6-
during each resampling period, cohort by cohort, provide the basis for inferences about the
survival process. The process of sampling released cohorts can be live trapping, kill trapping,
hunter reporting of bands, anglers reporting tags, finding dead animals, or resighting of live
animals without physically recapturing them (see e.g., Cormack 1964, Jolly 1965, Brownie and
Robson 1983).
-7-
2. Unifying Concepts and Notation
2.1
Concepts and Philosophy
Below I summarize my perception of key concepts for this approach to modelling and
analysis of release-recapture data:
1)
Separate the modelling of first captures (i.e., of unmarked animals) from
modelling the recaptures. Just before capture occasion i, there are U j unmarked animals
in the population (and M j marked); of these, Uj are caught. Although N j
U j + M j may
be of interest, this component of the problem is not always present, and it does not
appear possible to have both the tPj and the population size, N j , explicitly appearing as
parameters in the full probability model of open population capture-recapture data; this is
because these two parameters are partially, and sometimes totally, redundant.
Condition on release; this is the key to simplified CR modelling. We know how
many animals are released in any cohort or subcohort. A cohort is all animals released at
one time; a subcohort is just some identifiable (at release time) subset of the cohort,
which we think has the same survival and resampling probabilities as the entire cohort. In
the event that we have individual covariates and we want to model either or both of the
parameter-types tP and p as functions of these covariates, then this conditioning on each
release allows easy construction of the likelihood.
Recognize that releases and resampling are paired: the first encounter (e.g.,
recapture, resighting, tag return, band recovery) after a given release removes the animal
from that cohort. If the animal is re-released, it becomes part of a new cohort. Thus it is
critical in this approach to define recapture to mean the first recapture after any given
release. Viewed alternatively, there cannot be two recaptures following one release
because after the first recapture, the animal must be re-released before it can be
recaptured again.
Concentrate on estimating survival and resampling rates. I maintain that the
fundamental parameters for release-resampling data are survival rate in, and removal rate
from, the cohort of release. Use multinomial models of removal (resampling) counts from
cohorts: an animal can only be resampled once from a given release cohort because if it is
re-released, it has then entered a new cohort.
Assumptions need to be stated explicitly, and thoroughly tested, insofar as
possible. For example, the goodness of fit (g-o-f) of the JS model can now be thoroughly
tested (Pollock et al. 1985). Overall g-o-f tests can, and should, be partitioned into
informative sub-components. This is analogous to single degree of freedom contrasts in
the analysis of variance. It is very important to do a series of such tests focused on
specific alternatives because the omnibus (not partitioned) g-o-f test has very low power.
Think of the analysis of release-resampling data as model fitting in the sense
of seeking a "good" model for the data. With the general ecological theory and all specific
information relating to the study as a starting point, use a combination of g-o-f testing
and testing between specific alternative models to search for the most parsimonious
(fewest parameters) model that statistically fits the data and makes good ecological sense
in the context of the particular study. I recommend that the models so examined should
all be subsets of one general model specified a-priori, and that in the model selection
process, consideration should be given to the multiple significance testing nature of the
process.
=
2)
3)
4)
5)
6)
Often, we are primarily interested in survival rates even when the probability models for
the data are constructed in terms of population size parameters (N and M) as the explicit
-8-
parameters in the models. This situation results in unnecessarily complex theory and analysis.
I maintain that in open population release-resampling we are fundamentally studying a
survival process. Estimation of abundance (N) requires the additional assumption of equal
capture probabilities for the unmarked animals and for some identifiable subset of the marked
animals. I also believe that CR has generally used unnecessarily complicated notation and this
has obscured the simple underlying concepts (Cormack 1973 provides an exception to this
"rule").
In order to deal comprehensively with release-resampling data, a general, flexible
notation is needed. Basically, this notation must recognize different levels of data organization.
We need to recognize 5 levels of organization for release-resampling data; these are
substantially definable by the investigator:
Level
Highest to lowest in terms of data aggregation
family of data sets
,a collection of related data sets,
data set
,a related collection of cohorts,
cohort
,a collection of subcohorts,
subcohort
,a homogeneous set of released individuals,
post stratification
,a partition of resampling counts.
1
2
3
4
5
Levels 1, 2, 3 and 4 are prerelease stratification factors. Level 5 covers any stratification that
can only be done after the resampling occurs, e.g., location of a resampling encounter, or a
type of resampling encounter, such as live recapture vs. a hunter killed recovery (see, e.g.,
Schwarz 1988, Schwarz et al. 1988).
The cohort is the basic organizational level. It is the set of releases, R, from which come
the subsequent first resampling data. Fundamental notation for the ith cohort is
Resample counts
Released
R·1
Here, I am envisioning k capture-release occasions and e ~ k resampling occasions, or intervals,
with no further releases after occasion k. In CR, typically e = k, whereas in bird banding,
e > k can occur. Define
m..
IJ
= the
number of animals resampled between release time j-l and j which
belong to cohort i at the time of resampling, i.e., they were last seen at occasion i; j
i+l, ..., e, i
1, ..., min{e-l, k}.
=
=
Clearly, the usual collection of cohorts is a data set as we are accustomed to thinking of in CR
or bird banding. However, there are categories which we can use to organize entire data sets
into families, e.g., male-female, age classes at time of release, treatment levels, areas of release
and capture history features. Then we can test, within a family of data sets, for parameters
which may be common across the distinct data sets (this was conceptualized in Jolly 1965).
Finally, what is a subcohort? It is just a partition of a cohort into known subsets of
animals which are hypothesized to be homogeneous in their subsequent (i.e., after the release
time that establishes the cohort) survival and capture parameters. Such a partition of a cohort
can be on, e.g., capture history at time of release, or, for band recovery data, on banding
subarea (often, a banding data set is obtained by pooling over several close-by banding sites).
In CR, for example, consider the releases at time 3, R 3 , and their subsequent (first) recaptures
m34' m3S' ..., m 3k; note that R 3 = u3 + m13 + m23 - losses-on-capture. One logical
partition of this set of R 3 animals is into the 4 subcohorts with capture histories, at the time of
third release, as below
-9-
Subcohort
size
Capture history at
release time 3
h
{Ill}
h
{101}
h
{Oll}
h = {OOl}.
=
=
=
A general notation is R ih and mijh ' 1 ~ i < j ~ k, with h ranging over the set of observed
capture histories at release time i. The subcohorts identified at release time i can be used to
test the assumption that the parameters (t/J, p) which apply after release time i do not depend
on capture history over occasions 1 to i-l.
We could consider age or sex as a basis for subcohorts. This is valid, but the resulting
tests for an age or sex effect are not as powerful as it is possible to achieve. It is better to
organize that data into a family of data sets based on factors we believe may affect the
parameters. Thus, a subcohort partition really should be something wherein we believe (or
hope) that the partition is not especially relevant, but we are going to do g-o-f testing based
on this subcohort partition. The overall g-o-f to a model can be partitioned many ways. We
choose a particular partition by, in part, how we choose to structure the data into a family of
data sets and into subcohorts within cohorts.
2.2
Data Representations
For years I have thought of representing capture data by using a listing of the observed
capture histories at the end of the study, either as a "full X-matrix" (one row for each
observed individual animal, giving that individuals capture history and any auxiliary
information relevant to its capture and survival probabilities) or as a "reduced X-matrix" (one
row for each observed capture history, giving both that capture history and the number of
individuals having that capture history). This provides a fully informative representation of
the data in terms of statistical information; it is possible to construct the likelihood, under any
model, from information in the X-matrix. When we have individual, continuous covariates
such as a measure of size, then the full X-matrix must be used as the starting point, no simple
summary statistic may be possible. Such covariates have not generally been available. In this
report I am only considering the case of starting with a reduced X-matrix. For the statistical
modelling of such data I believe we should seek simple, informative sufficient and minimal
sufficient statistics so we can partition the full probability models into a series of components.
Also, we should seek a theory that provides a unified approach (unified over all releaseresampling). I now believe that for open models we need to display (i.e., summarize) the data
conditional on cohort and subcohort of release. Such a representation leads to easy modelling,
testing and estimation. It also unifies the modelling of banding and recapture (or resighting)
data.
The example data below are from S. Crosbie's thesis (pages 217+, his Ph.D. thesis,
1979). These data are from a 7 year ornithological capture-recapture study done by L.M.
Cook. I assumed there were no losses on capture (none were indicated). The data, represented
as a reduced X-matrix, are shown in Table 1; there are 29 distinct capture histories. Table 2
shows how these data can be more profitably displayed as 29 subcohorts in a full m-array.
Under the null hypothesis of subcohort homogeneity, the total recaptures from release
cohort i, mij
E mijh are a sufficient statistic. Examination of the subcohorts can be
conveniently donehconditional on this sufficient statistic (Le., conditional on the cohorts). The
minimal sufficient statistic (MSS) depends on the specific model parameterization of the
E(m i/
=
- 10-
Table
1.
Capture history representation
1979) as a reduced X-matrix.
j
=
1
1
1
1
1
1
1
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
Capture history, h
2 3 4 5 6
0 0 0 0 0
1 0 0 0 0
0 1 0 0 0
1 1 0 0 0
0 0 1 0 0
0 1 0 0 1
1 0 0 0 0
1 1 0 0 0
1 0 1 0 0
1 0 0 1 0
1 1 0 1 0
1 0 0 1 1
1 1 0 1 1
0 1 0 0 0
0 1 1 0 0
0 1 0 1 0
0 1 1 1 0
0 1 0 1 1
0 1 0 1 1
0 0 1 0 0
0 0 1 1 0
0 0 1 0 1
0 0 1 0 1
0 0 0 1 0
0 0 0 1 1
0 0 0 1 0
0 0 0 0 1
0 0 0 0 1
0 0 0 0 0
7
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
0
1
0
0
1
0
1
1
of
L.M.
Cook's
data (from
Crosbie,
Number with
history h, X h
70
2
1
1
1
1
84
9
4
2
1
1
1
69
5
4
2
1
1
54
4
2
1
81
11
4
155
16
143
The cohort data for this example are in Table 3 as an (reduced) m-array (Leslie Method B,
Leslie 1952). It is more informative to show ri' not R i - ri' however displayed this latter way,
the mij data for cohort i have a multinomial distribution given the R i (by assumption).
Additional discussion of these data representations is in Burnham et al. (1987:28-36).
-11-
Table
=1
i
{I}
Table
76
{ll}
{01}
3.
Release-recapture occasion
2
3
4
5
3
3
102
{101}
{1l1}
{Oll}
{001}
2
1
11
2
1
11
82
{100l}
{0101}
{001l}
{0001}
6
7
1
0
0
0
0
0
4
3
0
1
0
0
0
0
0
2
0
0
7
6
0
1
0
0
4
0
0
2
7
0
61
4
3
1
{01001}
3
{01l01}
2
1
{00101}
6
2
{001l1}
2
0
{0001l}
4
0
{00001}
96
11
{101001}
1
{OOO101}
3
1
{0100ll}
{01l01l}
1
{OO10ll}
2
{OOOOll}
11
{000001}
171
{OOOOlOl}
{OOOlOll}
{0010lll}
{OOOOOll}
{OOOOOOl}
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
4
0
1
0
0
1
0
16
4
1
1
16
143
recaptures,
The
cohort
release
and
(essentially, the m-array representation).
mij'
First resampling (i.e., removal from cohort, i)
3
4
5
6
R.
j - 2
76
105
96
73
113
190
3
1
1
2
3
4
5
6
2. The full
m-array
representation
data.
This
of L.M.
Cook's
representation is a 1-to-1 mapping from the X-array in Table 1. Here all
subcohorts and their release-time capture histories are shown.
2
12
1
4
7
0
3
8
6
- 12-
0
0
1
3
15
rih
u·1
(Rih - rih)
70
76
6
2
1
84
102
18
1
1
1
0
2
9
92
13
69
1
0
4
0
2
5
54
61
7
2
1
1
1
4
2
2
0
4
0
81
15
96
1
0
1
2
1
0
1
0
1
1
11
0
155
171
16
(It is not necessary
to tabulate these
releases
-)
143
L.M.
for
7
0
0
0
0
4
18
Cook's
Not seen again
R.r·1
1
70
86
80
64
94
172
data
2.3 Basic Notation
The intention here is to provide a unified approach, to the extent possible, for modelling
and analysis of band recovery and capture-recapture data. A fundamental frustration in this
attempt is that the indexing in the two notations is different. What is m.. in CR would be
IJ
mi,j_l in band recovery data even if the same symbols were used. I choose to use capturerecapture as the basis for the notation here. This is not an arbitrary decision; banding notation
does not extend as easily to cover capture-recapture as vice-versa.
I believe that the key to a simple notation for release-resampling is to keep to a
minimum the basic symbols used and to adopt a logical scheme of subscripting to cope with
the complexities of these types of data. Levels of subscripting that will cover most situations
are given below:
Level
Range set
,.. = {I, ..., V}
1
2
3
j
{I,
, k}
ii,
, e}
Comment
Denotes
classes of release
types,
which
define
separate data sets, e.g. males and females, or young and
adult, or a control-treatment structure; hence this factor
denotes important prerelease "stratification" factors.
Denotes release occasion and indexes released cohorts
within classes of data sets.
Denotes resampling occasion or interval. The range of j
depends on i. This range also depends on context. In
capture-recapture we typically index on j = i+l, ..., k,
whereas in band recovery we would, in the notation being
used here, index on j
i+l, ..., e, for e ~ k. I would
prefer to standardize the indexing on j = i, i+l, ... and
think of j as indexing the interval before the next release,
during which resampling can occur. However, I believe the
literature is too committed to the existing notation to
make such a change now.
Indexes observed subcohorts within cohorts, i.e., any basis
for partitioning cohorts at the time of release. Typically,
capture history at release time is the basis for defining
subcohorts, although there is considerable flexibility for
choosing what determines subcohorts. The set % (and its
size, H) is very dependent on release occasion i when
capture histories define subcohorts. "h" is a prerelease
stratification factor. I recommend avoiding the notation
=
h
4
5
w
%
'W'
= {I, ..., H}
{I, ..., W}
h vi and %vi' Hvi ' even though this is the technically
correct notation.
Denotes post-resampling stratification; e.g., if resampling
can be by several methods, or in several areas (spatial
stratification), then the resampling data can be poststratified. When double tagging is used, resampled
individuals may have either 2 or 1 tags (1 tag was lost);
post-stratification theory is needed to model such data.
The sets % and 'f' can have complex structure (see, e.g., Robson 1969, Pollock 1975). In
particular, there will often be a sequence of sets %vj, i = 2, ..., k-l with complex
interrelationships when subcohorts are determined by capture histories. It is not necessary to
- 13-
worry about this aspect of the problem, or to generate notation for it. In the case of
subcohorts being based on capture histories at release time i, h is a sequence of i values, each
one either a zero or a one, say. h = {61' ... , 6.}, with 6. = 1 (captured) or 0 (not captured) for
I
J
j < 1 and 6j = 1. Thus H S 2,-1.
The only data level that is especially difficult to deal with is w (level 5), Le., poststratification of resample counts. There has not been a lot work done on this level of data
organization (see, however, Schwarz, et al. 1988) and I do not further consider poststratification in this report.
Basic symbols (without subscripts) that suffice as a starting point to develop probability
models for release-resampling data, in particular for the Jolly-Seber model, are given below:
Symbol
u
R
m
Meaning
Denotes captures of previously unmarked animals, hence used for first
captures in capture-recapture. Will be indexed by ij may be indexed by v and in
special cases also by h or w, or both (these individuals were not previously
captured, so the distinction of prerelease vs. post release stratification does not
really exist for "u."
Denotes releases of marked animals. One conditions on R in model
buildingj will be indexed by i, may be indexed by v and h. "Release" does not
have to involve physical recapture. It really means the animal was resampled at
occasion j and was not removed from the study population, hence the animal
changes release cohort (from cohort i to j, i < j).
Denotes resampled animals that are part of a previously released
cohort. These animals are all marked. Will be indexed by i and jj may also be
indexed by v, hand w.
There are also lots of symbols for derived statistics. For example, for CR
r .
VI
k
= j=i+1
E m .. ,
VIJ
which represents all individuals ever recaptured from release cohort Rvij
j-1
mv.j == m vj = i~1 mvij'
which represents all marked animals in class v that were caught on occasion j. Having "m"
serve double-duty this way is unfortunate, but seems to be ingrained in the capture-recapture
literature. Better notation would be C . = m . for a "column" total, as introduced by
VJ
V'J
Robson and Youngs (1971) and used in Brownie et al. (1985).
Additional notation examples for CR:
n.=u.+m.,
VJ
VJ
VJ
which denotes all captures on occasion j for class Vj
k
(j-1
z·=
E
E m .) ,
VJ
t=j+1 i=1 Vlt
which denotes all recaptures of animals in class v which were in cohorts released prior to or at
release occasion j-1 and were subsequently recaptured after capture occasion j, but not
recaptured on occasion j. Finally let
- 14-
T vj
= mvj + Zvj ,
which denotes all recaptures of animals in class v which were in cohorts released prior to or at
release occasion j-l and were subsequently recaptured at or after capture occasion j. An
individual animal may be counted in more than one of the different Z and T because it can be
re-released into a new cohort after each recapture. For one data set (no v subscript), the
quantities up R p ri ' mij' ni' mj' and Zj are defined here the same as in Jolly (1965).
With reference to the complete data set given by the X-matrix, there is also X h = the
number of animals with final capture history h. We can conveniently tabulate the data using
the capture history matrix as "h, X h " as in Table 1; this is an excellent way to code the data
for input and storage in a computer (see e.g., Otis, et al. 1978, White et al. 1982, Burnham, et
al. 1987).
2.4
Assumptions
There are numerous assumptions involved in making inferences from capture-recapture
data. These assumptions vary in their importance and in terms of what the investigator can do
to satisfy them. The following are the important quantitative and statistical assumptions
presented in general terms:
The numbers of animals released are known exactly.
Marking at each occasion occurs in a short time relative to survival rate; marking is
accurate, there are no mark losses, and no misrecorded marks.
The fate of each animal is statistically independent of the fate of any other animal.
All animals in an identifiable subcohort have the same survival and capture
probabilities; this is an assumption of parameter homogeneity. Usually, the stronger
assumption of cohort homogeneity is made.
=
More specific assumptions about how parameters vary with time (i
1, ..., k), data set
(v
1, ..., V) and resampling occasion (j
2, ..., e) define the particular model structures to
be considered. Given the correct model structure, it is also typically assumed that releaserecapture event-pairs are, conditional on the releases, statistically independent events. These
assumptions of "within-animal" (over time) and between-animal independence lead to binomial
and multinomial probability distributions for the data under this modelling approach.
It is not my intention to detail, or discuss, the requisite assumptions here. There is a
substantial literature on this matter, see e.g., DeLury (1954), Cormack (1968), Carothers
(1971, 1973), Youngs and Robson (1975), Nichols et al. (1981, 1982), Pollock (1981b), Pollock
and Raveling (1982), Seber (1982), Burnham et al. (1987).
=
=
- 15-
3. Modelling Release-Resampling Data
3.1 A General Approach
A convenient probability model for one data set can be easily represented provided we
do not get bogged down in the (quite unnecessary) details of the index sets, %i' for subcohorts
within cohorts. The subcohorts that exist in CR at a release occasion are dependent on the
capture histories that have been realized at that occasion. There can be complex relationships
between the sets %1"'" %.~ 1 and %"1 The nature of the set %.1 and all such relationships
between these sets can be almost totally ignored. Also, for the purposes of this report, a data
set means the information in the (reduced) X-matrix as illustrated in Table 1.
Conceptually, it turns out we can get the form
Pr{data set} = Pr{first captures} Pr{ releases
I captures} Pr{ recaptures I releases}.
(1)
First captures are the up ..., Uk' In a sequential development of this expression for Pr{data
set} we end up with Pr{first captures}
=
This conditional form of the probability distribution of the first captures is not useful; an
explicit alternative representation is provided for the JS model in Section 5.1. I expect that the
derivations below used to get the second and third components of (1) have been considered
before in this binomial-multinomial modelling approach, however I am unaware of any
published work giving exactly these results. These results, below, have a very close parallel in
the work, and notation, of Robson (1969) and Pollock (1975, 1981a) who used a
hypergeometric-multiple hypergeometric approach which focuses on numbers, Nand M as the
basic parameters. It is the derivation of an explicit form for Pr{first captures} that is the key
to making the approach I give here really useful.
Losses on capture are just the difference between numbers of animals captured and
numbers released, hence conceptually,
Pr{losses on capture
I captures} ==
Pr{releases
I captures}.
This component of the data is typically ignored because it is usually assumed that losses on
capture contain no information about f/J, p or recruitment. That is probably true, especially if
losses are a relatively very small part of captures. However, this component of the data need
not be ignored; we can test for factors affecting loss numbers, if losses are stochastic. Losses
may be deliberate, in which case the particular individuals "lost on capture" can be considered
to be a random selection from those captured. The question of how to model what Jolly (1965)
called losses on capture has frequently been bothersome, but it is not a fundamental problem.
The complete probability model symbolized in (1) above can be built up by a forward,
sequential conditioning processes. The steps in the derivation are given below for the case of
typical CR data, i.e., using £ = k and with the set %1 being just the one element {1}. For
captures, releases, and recaptures from cohort 1 we have the product of the probability
distributions
- 16-
At release time 2, we potentially release most of the m12 animals recaptured at time 2 along
=
with releases from the u2 new captures. These releases correspond to capture histories h
{ll} and h
{Ol}, respectively, at release time 2. Thus, we get a next set of components for
the probability (i.e., likelihood) model as
=
Pr{u2
= {Ol}} Pr{m23 h' ..., m2kh I ~h' h = {Ol}}x
= {ll} }Pr{m23 h' ..., m2kh I R 2h , h = {ll} }.
I ul}Pr{~h I u2'
Pr{R2h
I m12'
h
h
To show the pattern of the general case we can rewrite this product as
At time 3, one has potential releases from the animals represented by the counts u3'
m13 and m23h for h 2 = {ll} and h 2 = {Ol}. From these four potential sources of releases
2
at time 3, the corresponding release-time 3 capture histories, h 3 , are {OOl}, {lOll, {lll} and
{Oll}, respectively. As stated before, to follow the evolving nature of the %i is more confusing
than helpful, except we do have to treat the capture history corresponding to ui as a special
case; this h has i-I leading zeros and its final element is a 1.
Symbolically at time 3 we have the additional components of the model as
Pr{u3
I ul'
u2}Pr{R:!h 3
I u3'
h3 = {DOl}
II Pr{m 34 h' ..., m3kh
hf%3
}G!l( h~,lG,
Pr{R:!h 3
I mi3h,})}
I R 3h }·
The capture histories of the animals at release time 3, h 3 in R
, are different from the
3h3
release-time i capture histories of the recaptures mi3h. that produce these new releases at
I
occasion 3.
=
Proceeding by induction, we sequentially include the terms in the model in the order i
1, ..., k (releases at time k do not generate any recaptures). At time i, given what has
happened at the previous times, and using the results from all subcohorts released at the
previous times, we can construct the terms to be in the probability model to represent what
happens as a result of time i captures, releases, and (first) recaptures from subcohorts released
at time i. A complete symbolic representation of Pr{data set} is thus derived in terms of the
product of three components:
- 17-
(2.a)
.~ Pr{R'lh lUI·' h = {0· .. 01} })(.~
.jil
(1=1
J=2 1=1
II
hi (%j
Pr{R· h
J
j
I mUh
IJ i
})x
(2.b)
(2.c)
Term (2.b) represents all the losses, or equivalently, releases on capture. Note that there are
two components to (2.b): releases from first captures and releases from recaptures. Term (2.c)
gives the probability distribution of recaptures given releases.
The distribution represented by (2.a, b, c) is general enough to cover a number of
different types of animal capture and marking studies provided we realize that some of the
components of this distribution vanish depending on the type of study. Let the first and
second parts of (2.b) be referred to here as (2.b-u) and (2.b-m), respectively; so symbolically,
(2.b-u)x(2.b-m) = (2.b). Some types of studies and the components in their probability model
are given below (note that "recapture" is used here as a generic term for whatever is the
resampling method):
Study
CR
RR
Model components
(2.a), (2.b), (2.c)
Comments
Capture-Recapture; an example is the traditional
Jolly-Seber model. Animals are captured from an extant
"wild" population; releases and recaptures are used to
estimate c/J and p; in addition first captures are needed to
estimate abundance.
(2.b-m), (2.c)
Release-Recapture;
an
example
is
releases
of
hatchery-raised fish. In these studies there are no first
captures of unmarked animals.
BR
(2.c)
Band Recovery; an example is bird banding studies
on hunted waterfowl. Resampling is by hunting kills, (or
exploitation, in general) so there are no re-releases; that is,
term (2.b-m) is degenerate since no "resampled" birds are
re-released. Although (first) captures are from a wild
population,
the
resampling
process
provides
no
information about the first-captures process, nor is there
usually any other information to allow one to construct a
model for component (2.a). Consequently, all the models
are conditional on releases.
CPUE
(2.a)
Catch Per Unit Effort; examples are commercially
exploited fisheries. Here, sampling is from an extant wild
population and 100% of first captures are removed. This is
a very difficult type of data to analyze well. Term (2.b-u)
exists here, but it is degenerate.
- 18-
One reason I present the above is so I can point out that component (2.c) is common
across all of the data types CR, RR and BR. We will now concentrate on this component of
the full "data set." I therefore give it a name: (2.c) represents the model corresponding to a
release-resampling data set (RRDS). A RRDS may exit essentially by itself (e.g., BR and RR)
or be a major component of a CR data set.
3.2 Cohorts and Subcohorts
No statistical assumptions have been invoked to arrive at the distribution for Pr{data
set} given in (2.a, b, c). In specific results to follow I do make the assumption of independent
fates of individuals, as per Section 2.4. This independence assumption means the distributions
will be binomials and multinomials, In particular, let ?ruh represent the multinomial cell
probabilities for the subcohort distribution in (2.c); hence IJ
E( mijh I R ih )
R
= ?rijh'
ih
Also, I define
k
A'h =
1
E'+1
•
J=l
?ruh;
IJ
note that
Now we have the more explicit result relative to (2.c),
Uh
), II
k
R ih
(?roo) m IJ ) (l-A.) R ih - r ih.
( mi,i+1,h' ..., mikh' R ih - rih '=i+1
IJh
Ih
(3)
Substitution of (3) into (2.c) gives the probability model structure for recaptures given
releases, i.e., for a RRDS.
At this point, formulae (2.c) and (3) are applicable to any type of RRDS. The
probability distribution so represented is fully informative as regards resampling counts given
the releases. The different models, CR, RR, BR, vary in how we eventually parameterize the
?rijh' This RRDS model, as it stands, is "saturated;" we need to reduce the number of
parameters. I initially consider only models independent of capture histories. If some aspect of
capture history is believed to affect survival and resampling probabilities, then we need to
partition the data on that feature (if possible), thereby creating multiple data sets. The topic
of multiple data sets is not considered in this report. For some results on multiple, related CR
and RR data sets, and partitioning a single such data set based on capture histories (thereby
creating multiple related data sets) see, e.g., Pollock (1975, 1981a), Stokes (1984), Brownie et
al. (1985), Burnham et al. (1987).
Assume ?ruh = ?roo, i.e., parameters do not depend on capture histories, hence subcohorts
IJ
IJ
based on capture histories are homogeneous within cohorts. Then a MSS for this hypothesis is
- 19-
R.1 -
, i = 1, ..., k-1,
and
,i
= 1, ..., k-I, j = i+l, ..., k.
The sums ri (also needed) can be computed from the mij or as
r·1
Using known properties of multinomial distributions we can now partition (2.c), getting
Pr{recaptures I releases} = product of terms (4) and (5) below:
k-1
R.1
)( II
k
II (
(11'.. )mij) (I-A.) R.1 -r·1,
i=1 mi,i+I' ..., mik' R i -ri j=i+1 IJ
1
(4)
(5)
This partitions the Pr{RRDS}, given by (3), into a component dealing with cohorts (the m.. )
and a component for subcohorts given cohorts under the hypothesis of subcohort homogeneit~.
We next consider further specification of component (4), as regards model structure.
Because of the nature of the release-resampling process a (still) saturated
parameterization is
,j = i+1
,j > i+l,
for
probability of being resampled (Le., removed from the ith
cohort between time j and j+1 given that the animal was released at time i
and is still at risk of capture at time j,
a .. = the
IJ
{3.. = the probability of survival in the cohort (i.e., both physically survlvlDg and
IJ
not being resampled) from time j to j+1 given that the last release was at
time i and the animal is still at risk of capture at time j.
Various restrictions on these aij and {3ij give models of interest that are not saturated.
The starting point for most developments and analyses is the assumption that the
- 20-
survival and sampling rate parameters are time-specific only, i.e., we assume
=
Q.
, all i, j,
(3.. = (3.
lJ
J
, all i, j.
Q ..
lJ
J
Thus, survival and resampling probabilities do not depend on the time (cohort) of release, only
the time, or time interval, to which the parameters apply. In Section 4.1 we further pursue the
case of a time-specific Pr{RRDS}.
3.3 Discussion
- 21 -
4. The Time-Specific Model for a Release-Resampling Data Set
4.1 The Minimal Sufficient Statistic and Residual Distributions
At this point we will focus on the time-specific model for a RRDS. The general formula
for Pr{RRDS} under independence of animals is given by (3) in Section 3.2. Under the further
assumption of subcohort homogeneity, Pr{RRDS} is the product of terms (4) and (5) in
Section 3.2; it is convenient, and appropriate, to consider term (4) as being Pr{cohorts I
releases} and term (5) as being Pr{subcohorts I cohorts}. "Releases" means the statistics R 1,
..., R k _1. Under the time-specific model for a RRDS we have
Pr{cohorts
I releases}
=
k-1
i!J1 Pr{mi,i+l' ..., mik
I Ri}
=
k-1 (
R.1
)~ ITk (11"..) mij) (1-,\.) (R.1 -r.)
1
i=1 mi,i+1' ..., mik' R i -ri '=i+l IJ
1
n
(6)
with
11"..
IJ
a.
= { p.
... p.
1
1
,j = i+l,
,j > i+l.
2a • 1
J- J-
(7)
Also,
,\. = (j=i+1
~ 11"") = (a. + P''\'+l)
1
IJ
1
1 1
,i
= 1, ..., k-l,
There are 2k-3 parameters (all estimable) in this model: aI' ..., ak_1 and PI' ..., Pk- 2 . The
interpretation of these parameters depends on the specific nature and context of the study.
We can use the parametric form given by (7) for the 1I"ij in the likelihood given by (6)
and factor that likelihood to find the MSS under this time-specific model. The result has been
known since at least Robson and Youngs (1971); see also Brownie and Robson (1983). The
method of deriving the distribution of this MSS is an important result in animal capture
theory; I will give this derivation below. First, however, I give the results (including needed
notation) for what is the MSS and the partition of (6) as
Pr{cohorts
I releases}
= Pr{MSS}Pr{cohorts
I MSS}.
In my opinion, the best form to use for the MSS is the set of 2k -3 statistics r l' ..., rk_l'
m2' ..., mk_l' where
j-1
m· = E m..
J
i=l IJ
is a column sum of recoveries in terms of the m-array display of the cohort data. In order to
easily write a probability distribution for this MSS we need to define the block totals T j , j
2, ..., k-1; these are computable from the MSS as
- 22-
T2
T.J
= r1
= T.J- 1 -
,j
m·J- 1 + r·J- 1
=
,j
= 2,
= 3, ..., k-l.
=
=
0 and note that T k
mk' Also, Zj
T j - mj (so zk == 0);
another relationship is T.
z· 1 + r· l'
J
JJThe distribution of the MSS extracted from of a RRDS for the time-specific model can
be given as a product of conditionally independent binomials:
For convenience, define rk
=
I R.1 ,.."
r·1
, i = 1, ..., k-1,
bin(R.1, A.)
1
m·1 I T.1 ,.." bin(T.,
1 T.)
1
,i
(8)
= 2, ..., k-1,
for
Ai
, i = 1, ..., k-1,
= (ai + ,8i Ai+1)
Ak = 0,
and
_ ai_1 _
ai_1
-1 - A.1- 1 - a·1- 1 +,8.1- 1A.1
, i = 2, ..., k-l.
T·
This MSS has 2k-3 components when there are k-1 release occasions followed by k-1
resampling opportunities.
The results in (8) show the elements of Pr{MSS} in the factorization Pr{RRDS}
Pr{subcchorts I cohorts}Pr{cohorts I MSS}Pr{MSS}
under
the
time-specific
model
assumptions. The second component, Pr{cohorts I MSS}, is free of the parameters under the
assumed model. To write down this actual "residual" probability distribution requires more
notation.
Define mij as a partial column sum of recaptures:
=
m~.
1J
= mI'J +
=
defined for i
1, ..., k-1, j
m~ 1 .
m·. Using this notation,
J- ,J
=
, i < j,
m 2J· + ... + m..
IJ
= i+1, ..., k.
In particular, m1j
= m 1j' j = 2, ..., k,
and
J
Pr{cohorts
I MSS}
z.1
)(
r·1
)
C
C
m · · , ... , m·
k-2 ( mi-1,i+1' ..., mi-1,k
1,1+1
lk
II
i=2
Ti+1
)
(
mi,i+1' ..., mik
(9)
This is a product of k-3 independent hypergeometric distributions. This residual distribution
is used in goodness of fit testing for the time-specific model.
- 23-
4.1.1 Factorization of Pr{RRDS}
To derive the above results we start with the distribution of a RRDS under the timespecific model; this distribution comes from (6) with the structure of the cell probabilities given
by (7):
Pr{RRDS} =
n
k-l (
R.
), k
mij)
(R. -r.)
1
. ~ (11".. )
(1- A.) 1 1
i=1 mi,i+l' ... , mik' R i -ri =1+1 IJ
1
for
11"..
IJ
=
{
O.
1
,j = i+l,
,j > i+l,
p.1 ... p.J- 2 0 J• 1
and Ai = 1I"i,i+l + ... + 1I"ik = 0i + Pi Ai+l for i = 1, ..., k-l, with Ak = O. This probability
distribution is the product of k-l independent multinomials. We can factor this Pr{RRDS}
into the product of the distributions in (8) and (9) (see Robson and Youngs 1971, for the
original proof).
Step one: "peel" off the marginal distributions of the ri I R i . That is, factor each
multinomial into the marginal binomial distribution of ri and the remaining, if it exists,
conditional multinomial distribution of mi,i+l"'" mik given their sum, rio This gives
Pr{RRDS} as the product of the two bracketed terms below:
(I am assuming the reader has some familiarity with the properties of multinomial
distributions at least the level given in Johnson and Kotz 1969:280-282; it will also help to
write down the multinomials in more detail than I am doing here). The first term above, in
brackets, includes the binomials that are part of the MSS reduction of a RRDS under the
time-specific assumptions. The next steps deal with the remaining multinomials in the second
bracketed term above.
Now we sequentially do a series of "peeling and pooling" steps starting with cohorts I
and 2, as regards their conditional distributions given rl and r2' First, "peel" off the
conditional binomial distribution of m 12 (= m2) given rl (=T 2) to get the factorization
Pr{mI2' ..., m1k
I r 1}
I rl} Pr{m I3 ,
, m 1k
I r1 -
m12}
= Pr{m21 T 2 }Pr{mI3'
, m 1k
I T2-
m 2}'
= Pr{ml2
where
for
11"12
T2 = T
1
0 1
= -A-'
1
- 24-
Notice that the conditional distribution of m13' ..., mlk
I T 2- m2
has k-2 cells, the same as
the conditional distribution of m23"'" m2k I r2' Now consider the form of the cell
probabilities in these two multinomials. For Pr{m23' ..., m 2 k I r2} (Le., cohort 2 given r2) the
cell probabilities are
For cohort 1 the cell probabilities in Pr{m 13 , ..., mlk
,j
= 3,
,j
= 4, , .., k.
I T2-m2}
are
,j = 3, ..., k.
Thus, under the time-specific model structure, these two conditional multinomials have the
same cell probabilities; we can therefore "pool" the corresponding cell counts. That is, we can
factor the product of these two conditional multinomials as follows:
=
=
(recall that T'+
T.-m.+r.,
and also that T.-m.
Zl')'
1 l
1
1
1
1
1
So far we have re-expressed Pr{RRDS}, under the time-specific assumption, as the
product of the terms below:
- 25-
r·
n
k-l (R.)
r.1 (A.) 1 (1- A. ) (R.1
[ i=l 1
1
1
-r')J
1
x
[(;;)<T2)T2(l_ T2)(T2 2}
-rn
(ml~~~'~~lk)( m23' ~~, m2k)
(m23 ,
~.~,
x
m2k)
2 I T 3}
Another round of peeling and pooling is now done on Pr{m23' ..., m k
Pr{m34' ..., m3k I r3}' First, we do the partition (i.e., the peeling step)
2 I T 3} =
Pr{m23' ..., m k
Pr{m3
I T 3 }Pr{m24'
and
2 I T 3 - m 3}'
.... m k
=
In writing this we also used the fact that m23
m3' The cell probabilities in the new
conditional multinomial distribution Pr{ m24' ..., m2k I T 3 - m3} are
,j
= 4, ..., k.
Hence, this distribution has the same number of cells, and the same cell probabilities as
Pr{m34' ..., m 3 k I r3}' The explicit partition in this peeling step is
for
- 26-
The "pooling" step takes the convolution of Pr{m24' ..., m2k
I T3-
m 3} and Pr{m34'
..., m3k I r3}, thereby getting an expression for the product of these two distributions as the
product of two different probability distributions:
The next cycle brings in the cohort 4 data and deals with the product
Pr{mS4 ' ..., mSk
I T 4 }Pr{m45' ..., m 4 k I r4}
=
[(;~)<rJ4(1- r) T4 -m4 >Jx
(expressed after the peeling step but before the pooling step). After carrying out the pooling
step of this cycle we can write the re-expressed Pr{RRDS} to this point as below:
n
k-1 (R.)
r.1 (A.) r·1(1- A. ) (R.-r.)j
1 1 x
[ i=1 1
1
1
4 (T.)
T.
(T.-m.)]
n
m~ (T.) J(1-T.) J
J x
[ j=2
J
J
J
4
IT
i=2
T.-m.
)(
r·1
)
1
1
C
C
m . . , ..., m·
( mi-1,i+1'
..., mi-1,k
1,1+1
Ik
Ti+1
)
(
mi,i+1' ... , mik
- 27-
x
(assuming k ~ 7)0
A formal derivation (proof) is by induction on this peeling-pooling cycleo The result is
that under the time-specific hypothesis, Pr{cohortslreleases}
Pr{MSS}Pr{cohortsIMSS}
=
=
o
T.
(To-m )]
k-1 (Ro)
r! (A.) r.l(l_A.) (Ro-r.)][k-1 (To)
m~ (To) J(1-To) J
J x
[ i=l 1
1
I
j=2
J
J
J
n
lin
T.-m.
)(
ro1
1
1
C
C
m.
(
k-2
mi-1,i+1' ..., mi-1,k
1,1+1
0
IT
i=2
,
... ,
m·
lk
(10 a)
0
)
(100b)
Ti+1
)
( mi,i+1' ..., mik
with
,j
= 2, ..., k-1.
401.2 Estimation
Because the MSS for a RRDS under the time-specific model is in the exponential family
and its dimensionality equals that of the parameter space, the MLEs can be obtained by the
method of expectations (see Brownie et al. 1985: Appendix B, and Burnham et al. 1987:14-16).
Hence, to find the MLEs solve the equations
E( r·I
and
I R.)I =
E(m Io IT.)
I
for
the
parameters
R.I A.I
= T.T.
I
I
AI' ..., Ak_l'
, i = 1, ... , k-l
,i
= 2, ..., k-l
T2' ... , Tk_l
and
then
replace
expectations
by
the
corresponding data items (ri and mi) to get the ~ and To The aI' ..., ak_l' (;1' ..., (;k-2 are a
one-to-one transformation of the ~1' ..., ~k-l' 1'2' ..., 1'k_l0 Alternatively, it should be clear
=
=
ri/Ri and 1' i
m/T i .
The solutions for the a and (; are easy to get by using a connected triplet of equations
and using the first of these equations in two different forms:
from (lOoa) that the MLEs of these latter parameters are \
- 28-
r·1
- ft.'
1
and
Solving these equations, the MLEs are found to be
,i
= 1, ..., k-1,
(11)
, i = 1, ..., k-2.
Given the distribution of the MSS as a set of independent binomials, variances and
covariances of the &., p. are relatively easy to derive. I defer that subject to Section 6. It is
also easy to construJt m~dified MLEs that are adjusted for statistical bias which occurs even
when the assumed model is true, see, e.g., Brownie et al. (1978:16,211), Burnham et al.
(1987:207-210).
4.2
Band Recovery vs. Capture-Recapture
For banding studies, as discussed by Brownie et al. (1985), we interpret Pi as equal to
Si = the physical survival of the bird between release time i and i + 1. Also, (ki = fi is the band
recovery rate during the i th year (i.e., between releases i and i+1). The only way a bird is resampled is by a hunting death and hunters are ubiquitous, hence the bird cannot leave the
population at risk of "resampling." So in this case, survival in the cohort is equivalent to
physical survival. Conversely, death is a hidden component of the removal rate, f i , from the
cohort. The structure of both Pr{MSS} and Pr{cohorts I MSS} does not change from what is
given above. In Brownie et al. (1985), the notational equivalents to what is used here (which is
CR-based) are
m..
IJ
r·
R.
1
m·
T~
J
Banding
R·~ J-·1
R.'
~
N·~
ej-1
Tj-1'
Brownie et al. (1985) do not italicize their notation; I do that here to clarify when banding
notation is being used.
Making these substitutions of banding notation into the MLEs given by (11), and
changing k to k+1 (so that in this CR-based notation we then have k releases and k
resampling intervals) we get the MLEs for the time-specific model (Model M 1 ) in Brownie et
al. (1985:16):
- 29-
r.1 --
R·
C·
T·I
.,}-_I_
iY:
I
, i = 1, ..., k,
, i
= 1, ..., k-l.
4.3 Capture-Recapture
Now consider the capture-recapture interpretation of these general theoretical results in
terms of the RRDS component of the Jolly-Seber model. The parameters are capture
probabilities, Pi at occasion i
2, ..., k (PI is not, in general, identifiable), and survival in the
population at risk of capture, l/J., i
1, ..., k-1, between occasion i and i+l. Usually, there
are serious questions of demogr~phic and geographic closure (see White et al. 1982), so that
under the JS assumptions l/J. actually equals the product S.(l-E.), where Si
physical
survival and E i = probability 10f permanent emigration, Le., le~ving tbe population at risk of
capture.
For capture-recapture with this added assumption that any emigration is permanent, we
have the parameterization
=
=
=
Q.
1
/3.1
= l/Ji Pi+1
= l/Ji q i+1
,i
= 1, ..., k-1,
' i
=
(12)
1, ..., k-2.
An animal survives in its release cohort from time i to i + 1 if it survives in the population at
risk of capture from time i to i+1 and is not captured at time i+l. Thus, for JS we have
E(m..
IJ )
_
--a,1
{(l/J'P'+l)
1 1
(l/J·q·+l)·"
(l/J.J- 2 qJ· 1)(l/J·J- 1P')
1 1
J
,j = i+1,
,j>i+l.
Consider now the distribution of the components of the JS MSS (from just the RRDS
part of a full CR data set). Using the parameterizations of (12) we get
Ai
= l/Ji(Pi+1
Ak
= 0,
+ qi+1 Ai+ 1 )
,i
= 1, ..., k-1,
,i = 2, ..., k-1
and, as above, the distribution of the RRDS MSS is 2k-3 independent binomials:
I R.1
- bin(R.,
A.)
1
1
, i = 1, ..., k-1,
m·1 IT.1 - bin(T.,
T.)
1
1
, i = 2, ..., k-l.
r·1
The estimable survival and capture parameters under JS are
- 30-
still 2k-3 of them. The product 4>k-1 P k is more analogous to a survival rate than to a capture
rate, therefore formulae are simpler to develop if one just defines 4>k-1 as equivalent to
(4)k-1 P k)'
It is instructive to derive the (unrestricted) MLEs of these parameters. A quick
derivation is to use the results in (11) along with the formulae below:
~.1
= a. + 13
1
, ,
1
a.
1
a.1 + 13.1
j>.
1 -
If we are working only with the JS parameterization to begin with, the MLEs can be found by
writing three equations derived from equating expectations to statistics:
r·
It
= Ai
1
= ~i(j>i+1 +
qi+1 Ai+1)'
Substitute Ai + 1 into the second of the 3 equations above and solve for
, i+1 = 2, ..., k-l.
This
formula
can
be
simplified,
but
there
are
theoretical
advantages
to
this
representation as regards deriving variances, covariances and bias. Next, find j>i+1 + qi+1 Ai+ 1
and divide this quantity out of the first equation to get
,i = 1, 00"k-l.
The above estimators of p and 4> depend on the assumption that any emigration is
permanent (see e.g., Balser 1984). However, this assumption is not unique in the context of the
time-specific model. A mathematically valid, alternative time-specific parameterization is
- 31 -
ai = Si[(1-Ei )Pi+1]
fJi
=
= Si Pi'+1'
= Si[1- (1- E i )Pi+1] = Si q i'+1
= Si E i + Si (1- E i )qi+1'
=
Here, Pi'+1
(1- E i )Pi+1 and qi+1
1- pi+1' This parameterization corresponds to
assuming emigration is a temporary, random movement out of the area where capturerecapture is occurring. On any resampling occasion each living individual has probability 1-Ei
of being in the area where resampling is occurring. The permanent emigration assumption
renders Pi estimable but confounds Si and (1-E i ). Random, temporary "emigration"
(movements) renders Si estimable and confounds (1-Ei ) and Pi+1' The estimator of Ni
remains valid under either parameterization, but
Ni = n/I\ vs. Ni
= n/pi
must be
interpreted differently. Note, however, that Pi and pi are algebraically identical, it is only the
interpretations of the estimated parameters that change.
On another point of interest, notice that we have made no use of the intermediate
nuisance parameters M i often found in capture-recapture models (see e.g., Jolly (1963),
Cormack 1979, Pollock 1981b). The estimator of Mi is
•
M.1
r·
1
•
= m·I+Z·R
Ii
The concept of the M i does not arise in this approach to capture-recapture theory. Although
the Mi have heuristic value (see, e.g., Cormack 1972, 1973) they complicate theoretic
derivations, especially of variances and covariances, when the focus is on survival rates. If the
focus is on estimating population size, N , then the approach of Robson (1969) and Pollock
i
(1981b) may be better because it directly involves the Ni and Mi as the parameters in the
model (under either approach the
4.4
Ni are identical).
Goodness of fit Testing
To summarize results so far, for a CR data set under the time-specific model we can
write symbolically
I captures}x
I cohorts} Pr{cohorts I MSS} Pr{MSS}.
Pr{data set} = Pr{u1' ..., uk}Pr{losses on capture
Pr{subcohorts
The last three of the five right hand side terms of this partition give Pr{RRDS}. Under the
assumption that the a and fJ are time specific only, we know explicit expressions for the last
three components (we ignore the second component and defer the first component to Section
5). The third and fourth components are free of the survival and re-sampling parameters
- 32-
regardless of the context (BR or JS). Thus the corresponding g-o-f tests are the same for band
recovery model M 1 (Brownie et al. 1985) and the Jolly-Seber model. Note that in band
recovery, we generally have no subcohorts so that component drops out.
I repeat here the formulae for these "residual" distributions which provide the basis for
the g-o-f tests. From (5), Pr{subcohorts I cohorts} =
k-l h!l; (ffii.i+l,h'
IT
i=l
(
···.~~kh. !l.;h -'ih)
~ik' R i -ri
R
mi,i+1' ...,
)
This product of multiple hypergeometric distributions leads to a series of contingency table
tests of homogeneity of subcohort parameters. This null hypothesis of subcohort homogeneity
is implied by the null hypothesis of a time-specific model.
Another set of multiple hypergeometric distributions is used for Pr{cohorts I MSS} -
z·1
)( m . . ,r·1... , m· )
C
C
(
k-2
mi-1,i+1' ..., mi-1,k
1,1+1
Ik
IT
i=2
Ti+1
)
( mi,i+1' ..., mik
(see Robson and Youngs 1971, Seber 1982, Balser 1984, Brownie et al. 1985). The k-3
contingency table tests of homogeneity which arise from this distribution are testing that the
cohort data fit a time-specific model given that there is subcohort homogeneity.
Pollock et al. (1985) is the first paper giving the fully efficient g-o-f test to the JS model;
they use a different representation of that test statistic than is used here. The overall g-o-f test
must be developed as a set of conditionally independent components; there is no unique way to
partition this information. For detailed discussion of g-o-f testing for the JS model see Pollock
et al. (1985) and Burnham et al. (1987:64-77,192-198).
- 33-
5. The Jolly-Seber Model
5.1
A Minimal Sufficient Statistic
I have derived a distribution for the MSS under the Jolly-Seber model for CR data
which incorporates the recruitment into the likelihood. There are two conceptual components:
Pr{u 1 , ..., uk} and Pr{m ij I Ri'i
1, ..., k-1,j
i+1, ..., k}; this latter component is part
of a RRDS. From Section 4.3 we know that a MSS for a RRDS under the time-specific model
reduces to the distributions of ri IR i and mi IT i where
=
I R.1
r.1
and
=
, i = 1, ..., k-1,
"'" bin( R.1 , A.)
1
m.1 IT.1 "'" bin(T.,
T.)
1
1
,i
= 2, ..., k-1,
,i
= 1, ..., k-1,
i
= 1, ..., k-1
for
with Ak
= 0 as a boundary condition, and
p.
T
-
i - p.
1
+ 1q.1 A.1
,
(however, T1 is not estimable in the unconstrained JS model).
People have generally dealt with the new (unmarked) captures, ui' at time i, by using
the model
p.);
u·1 I u.1 ,.., bin(U.,
1
1
then the recruitment does not show up in a general probability distribution model for the data.
Instead, abundance parameters are estimated from the equations below (presented without
"hats", " or expectation operators):
N.1 = n./p.
,
1
1
and
hence
M.1
= m./p.
,
1
1
= </J.(U.
1
1
M'+
1 1 = </J.(M.
1
1 N'+
= </J.(N.
1 l
1
1
U'+
1 1
U.1 = u./p.
,
1
1
u.)
1
+
, i = 1, ..., k-1 ,
B.1
m·1 + R.)
1
n·1
+ R.)1 +
,i
B.1
= 1, ..., k-1 ,
,i=l, ...,k-l.
Existing approaches generally use this last equation to derive an estimator of B i (see however
Crosbie and Manly 1985 who do put recruitment parameters into their likelihood).
The various existing distributions and representations for the JS model are useful, but it
would also be very useful to have an actual distribution for Pr{u1' ..., uk}' incorporating
abundance and recruitment, to use in conjunction with the MSS from the RRDS component of
the data. The derivation of such an Pr{ u1' ..., uk} is given here.
Define recruitment B.1- 1 as the number of unmarked animals which enter the population
at risk of capture between occasions i-I and i and are still alive just before the ith capture
occasion (exactly as in Jolly 1965). We conceptualize these Bi_1 unmarked animals as a cohort
- 34-
from which animals are removed, by capture and marking, at occasions i, i+l, ..., k. Define b ij
as the captures at time j from this unmarked cohort of size B.1- l' We model the capture
process for recruits, i.e., the bij I B i , in the same manner as we model the mij I R i . The
conceptual quantities are given in Table 4 in a way that indicates their relationships.
In Table 4, the row totals are
k
b' l
1- "
= j=i
E b· 1 ·
1-,J
,i
= 1, ..., k.
=
These row totals are not observable, neither are any b i_1,j' except b OI
ul' The column
total Uj is the number of unmarked animals caught on occasion jj these Uj are observable. Note
=
that B O
N1 is the number of animals at risk of capture just before the first occasion. We
would like to consider the N 1 , B 1 , ..., B k _1 and the 4>1' ..., 4>k-l' PI' ..., Pk as the
fundamental parameters, however, they not all estimable in the general JS model.
Next, define some "tail" sums for both the u· and b. 1 :
1
1- "
and so forth. In general,
, i = 1, ..., k-1.
=
We can define uk
uk' but this notation will not be needed. These ui are observable.
Finally, define
b:t'1
= b.l'
+ b'+
1,
1 . + ... + b k - 1 ,.
,i
= 0, ..., k-1.
These bi are sums of lower triangular blocks of the b ij . For example,
b k_2 = (b k_2 ,k-l + b k _2 ,k) + (b k _1,k) = b k _2 ,. + b k_1 ,..
Note that
b O· -- ul* - b*l'
and in general we will be interested in ui - bi which is a rectangular block of the b ij exactly
analogous to T.1 = m.1 + z.1 in the JS model, or the T·J in the time-specific bird banding model.
Moreover, it is easily shown that
u·1
I (u:t'1 -
b:")
"" bin« u:"1 - b:"),
T.)
1
1
1
which is analogous to
m·1 I T.1 "" bin(T.1 , T.).
1
- 35-
, i = 1, ..., k-l,
Table
4. Captures,
b ij ,
of
animals
unmarked
on
occasion
j,
given
the
number
recruited, B.1- l' at occasion i.
B.I- 1
1
BO
2
B1
3
B2
k-l
k
B _
k2
B _
k1
Totals:
=1
2
3
k-l
k
b OI
b 02
b 03
b O,k-l
b Ok
bOo
b 12
b 13
b 1 ,k-l
b 1k
b1.
b 23
b 2 ,k-l
b 2k
b2 .
b k-2,k-l
b k-2 ,k
b k-2,·
b k- 1 ,k
b k-l,·
j
ul
u2
uk_l
u3
b.
I'
uk
To derive the above result, we first need a model for the probability distribution of the
b ij given the cohort sizes B i . Because the Bi animals are by definition alive just before occasion
i+l, there is no survival rate ¢i showing-up in their distribution. Rather, we get expectations
as
E(b i ,i+l I Bi )
Bi Pi+l '
=
E(b i ,i+2
I Bi ) =
E(b i ,i+3
I Bi )
B i qi+1¢i+l Pi+2 '
= Bi Qi+l¢i+l q i+2¢i+2 Pi+3 '
and so forth. Except for the missing ¢i' these E(b ij
E(mij I R i )· In fact, the following holds:
I Bi )
are of the same form as the
E(m.. I R.)
E(b.. 1 B.)
IJ
I=¢.
IJ
1
R.
1
B.
1
1
This is the key to easily deriving the results below.
The b.1- 1 ,1., ..., b.1-,
1 k I B.1- 1 have a multinomial distribution under the time-specific
assumptions and by assumption, cohorts i = 1, ..., k are independent. Therefore, using the
same ideas as in the peeling and pooling method, we can derive the following results:
**
Pr{ul' ..., Uk} = Pr{ul- b 1} (k-l
i~1 Pr{ui
where
(ui- bi>
* *)
I ui
- bi }
- bin(Nl' (PI +Ql Al»
and
, i = 1, ..., k-l.
- 36-
The derivation of these results will be outlined here.
Let 1I'ij = E(mij
I Ri)/Ri
be the cell probabilities in the multinomial distributions for the
cohort data:
(4)i Pi+l)
{
1I'ij =
(4)·q·+I)''· (4). 2 q. 1)(4)· I P ')
I I
J- JJ- J
,j = i+l,
,j > i+l,
for i = 1, ..., k-l, j = i+l, ..., k. The definitions of these cell probabilities can be extended to
allow i = 0 by with the added definitions
4>0 = 1,
AO = PI + ql AI'
This serves to make 1I'0j' j = 1, ... , k, well defined. The unobservable recruitment numbers
b i_1,j' j = i, ..., k, given Bi_l'i = 1, ..., k, are multinomial random variables with Pr{b i_1 ,i'
..., b i - 1,k I B i _1)
=
)( II
k (11'.
1 .) 1-(A. /4>. ) B.1- 1- b.1-,.
1
B.1-1
./4>. )b.I-,J
( b - ,i' ..., b - ,k' B _ - b _ ,. j=i
l-l,J
I-I
(
I-I
1-1)
i 1
i 1
i 1
i 1
Conditional on the total recruits captured, b. 1 ,from the B. 1 recruits entering at occasion i,
1-
"
1-
the b'l - 1 ,I. , ..., b l·- 1 , k have the multinomial distribution Pr{b.1- 1 ,I" ..., b.1-,
1k
b.1- 1"
)( II
k (11"
./A. )b.1- 1,J.)
( b - ,i' ..., b _ ,k j=i
l-l,J
I-I
i 1
i 1
I b.1-,'
1 )
=
, i = 1, ..., k-l.
Using these distributions we can see that for i = 1, the marginal distribution
I B O}
is binomial with cell probability AO' However, B O = N1 and bOo = ui - bi, thus
we get the result
Pr{b O.
The conditional distribution of ul
Pl/ AI' Hence, we have the result
To get the distributions of the ui
I bOo
I (ui -
is seen to be binomial with cell probability
T
1
bi>, i = 2, ..., k-l, and to show that they are
independent, requires using the peeling and pooling algorithm to be used. Also required for the
derivation is notation for partial column sums of the b.1- 1 ,J" say b~1- 1 ,J. for the sum
b .+ b 1 .+ ... + b. 1 '. The key intermediate result, derivable by peeling and pooling, is the
O,J
,J
1- ,J
distribution Pr{b~
1 . , ..., b~1-,
1 k I u:t'I - b:t'}
=
1- ,I
I
- 37-
u:" - b:"
)( k
.)
1
~C
.!!.. (lI'i_l / Ai_I) b~l
1 ,J
( bC
i-l,i' ..., i-l,k J-l
'
,i
= 2, ..., k-1.
From the above, the marginal distribution of btl,i given ui - bi is seen to be binomial with
=
cell probability Ti' Finally, that btl,i
ui'
As a consequence of the above results and those of previous sections, we now know that
a MSS for the unconstrained JS model of a CR data set is ul' ..., uk' rl' ..., rk_l' and m2' ...,
mk_l with its distribution representable as
(13.a)
(ui - bi) ,.." bin(N l' (PI + ql AI)) ,
(13.b)
ull (ui-bi)"'" bin((ui-bi), Tl)'
ri
u·1
,.." bin(Ri , Ai)
,i
= 1, ..., k-l,
(13.c)
b:")
,.." bin(u:"1 - b:",
T.)
1
1
1
,i
= 2, ..., k-l,
(13.d)
,i
= 2, ..., k-1.
(13.e)
I Ri
I (u:"1 -
m·1 IT.1 ,.." bin(T.,
T.)
1
1
=
This is in terms of 3k-3
k+(2k-3) independent binomials. There are 3k-3 estimable
parameters here. These parameters may be taken as
(k-2 parameters; they lead to
and
8 2 , ..., 8 k_2 ),
(2 parameters).
Here, E(ul)
= NlPl
and
E(ui)
= E(bO) = (bi)+Nl(Pl+qlAl)'
In the unconstrained JS
model, none of N1' PI' B l , B k_l , tPk-l and Pk are separately estimable.
This theory now allows us to write down a general likelihood for the JS model based on
a MSS, and that likelihood incorporates the estimable parameters, most of them appearing
explicitly. All special cases of the JS model can be implemented from this starting point, for
example, the case of no recruitment becomes the constraints bi = 0, i = 1, ..., k-l, and the
case of no mortality corresponds to setting all of the parameters
5.2
5.2.1
tP1' ..., tPk-l
equal to 1.
Parameter Estimation
Survival and Capture Rates
In the general JS model no structure or restrictions are placed on the recruitment
parameters B ' B l , ..., B _ . As a result, the estimable parameters from likelihood
O
k l
components (13.a), (13.b), and (13.d) are the product NlPl and b b
b l , or functions
O' 2, ..., k_
of these k "parameters," such as the estimable Bi . Without restrictions on the recruitment
- 38-
process there is no information in likelihood components (13.a), (13.b) and (13.d) about the
survival and capture rate parameters. Thus, in this parameter-saturated JS model the 4>i and
~i derive entirely from the likelihood components (13.c) and (13.e); these components are the
MSS under the time-specific model for a RRDS. The MLEs for a RRDS, and hence for the
saturated JS model, are given in Section 4.3. For completeness, I repeat those estimators here:
, i = 2, ..., k-l.
, i = 1, ... , k-2.
The above formula extends to give the MLE of the product tPk-l Pk by applying it for the case
of i
k-l and then interpreting 4>k-l to really mean
=
(this works because mk = T k ).
As long as the recruitment process is left unspecified, all special cases of JS, in terms of
restrictions on the time-specific parameters tPi and Pi' involve only the likelihood components
(13.c) and (I3.e). The MLEs of these tPi and Pi under any such special cases are not closed
form; these MLEs can be found numerically (see, e.g., Burnham 1989).
5.2.2 Recruitment and Abundance Numbers
In the saturated JS model, moment estimators of (N I P l)' b O' b 2, ..., b k_1 are relatively
easily
found.
From
(I3.a)
and
(I3.b),
and
using
E(bi)+ E(b O.) = E(b O)' we get
and
From (13.d),
hence,
u·1
mi )
= (u.*- b..*) (-T
1
1
.
1
,i
= 2, ..., k-l,
,i
= 2, ..., k-l,
- 39-
E(ui) = E(bi)+ N1(PI + Ql"'l) =
provided T i > O. If mi > 0 then
.*
* uiTi
b.
u 1' ----m:1
=
, i = 2, ..., k-1 ,
1
(14)
Recruitment (B i ) enters the likelihood via the bi'- These are related to the Bi as follows:
E(biJ = B i (Pi+1 +Qi+1'\+1)
,i
= 0, ..., k-1,
,i
= 1, ..., k-2.
= B i Pi+1/ T j+1 '
hence
k-1
=
E(b~)
1
It follows that E(b
.E. B. (P'+1/T'+1)'
J=l J
k) =
J
J
B k - 1Pk and
E(bi - bi+1) = B i Pi+1/ T i+1
We cannot estimate bi, however, b
2,..., bk_1 are estimable, hence
,i
= 2, ..., k -
2.
Using the estimators in (14) we get
,i
Usually
Bi
= 2, ..., k-2 .
(15)
is written as
B.1
=N·+
-¢.(N.+R.-n.)
1 1
1
1
1
1
,i=2, ...,k-2,
(16)
(see Jolly 1965) where
·. =
N
1
n.i
Pi
,1.
= 2 , ..., k - 1.
With some algebra we can show that (15) and (16) are identical. Starting from (15)
write
=
•
( Zi )( ¢i )
Ui+1- u i mi
~i .
This follows because
mi+1.
Pi+1
T i + 1 = Ti+1 = Pi+1 + qi+1 ~i+1 -
- 40-
Continuing to simplify Bi , we add and subtract Mi+1 to get
•
•
• ( Mi+1
•
B. = N·+ 1 - ¢J.
I
I
I
¢J.
I
u·z· )
+-!..."L.
m. .\.
I I
Now note that
z· _ _
q.~.
1_1
mi p.
~
I
and
~.
-
M+
M.I (m.I R.)
I
i 1
--:--....;;....:..~-
I
I leave it to the reader to verify the above equality, just remember that z· + r·
I
I
·
h·IpS,
t h ese re Iatlons
•
B.
I
= T I·+ 1 . Using
•
• (.
uiqi )
= N·+
-¢J· M.-m.+R.+-.I 1
I
I
I
I
Pi
= N·+1-~·(M.-m.+R.+iJ.-u.)
I
I
I
I
I
I
I
= N·+1-~·(M.+R.-n.).
I
I
I
I
I
These same sort of algebraic manipulations can be used to get the alternate form
(17)
which is convenient for deriving var(B i ) and considering its meaning (Section 6.2.2). From
(17), we can see that the stochastic variations of the b ij (given the Bi ) will manifest
themselves entirely in the ui. It is only the variances and covariances of these ui that create
difficulties as regards deriving var(B i ).
In principle we can look at special models wherein we constrain the bi
bi modelling
relationships among the Hi' such as is done by Crosbie and Manly (1985) using a different
approach to getting the likelihood. Similarly, we could model relationships among the ¢J i or Pi'
and/or constrain the estimators of ¢J i and Pi to be ~ 1 as is done by Buckland (1980).
Imposing the constraints B i
bij ~ bi ~ b ~ ~ b 1.
~
0, i = 1, ..., k-1, is accomplished by enforcing the constraints
2 ... k_
Interpretation of B i (i.e., of recruitment) is difficult. If we are really interested in the
population dynamics of recruitment, then I suggest it is necessary to model the recruitment in
the context of the overall population dynamics processes, thereby reducing the number of free
Bi parameters and making them biologically interpretable. In most studies, I do not think we
- 41-
will have enough information to do such modelling.
5.3
Special Cases of the Jolly-Seber Model
There are two special cases of JS that have been particularly investigated in the
= ... =
=
literature: no recruitment (so B 1
B k_
0) as one case, or no mortality as the other
1
case (so tPl
tPk-1
1). These cases lead to closed-form estimators of the remaining
parameters provided no other restrictions are placed on those remaining parameters. A third
case is no recruitment and no mortality, which produces Darroch's model for closed population
capture-recapture (Darroch 1958).
= ... =
5.3.1
=
No Recruitment
The no-recruitment case is especially easy to deal with given the theory developed
above.
The
constraints
B1
= B 2 = ... = B k _1 = 0
imply
bi
= b 2 = ... = bk_1 = O.
Consequently, the distributions in (13.a)-(13.e) provide a sufficient statistic for this reduced
parameter case by simple setting bi
= b2= ... = bk-1 = 0 to get
ui"" bin(Nl' (P1+ q 1A1»'
(18.a)
u1
I ui
,.., bin(ui, T1) ,
r·I
I R.I
,.., bin(R.,
A.)
I
I
,i
= 1, ..., k-1,
(lS.c)
u·I
I u:"I
,.., bin(u:",
T.)
I
I
,i
(18.d)
m·I IT.I ,.., bin(T.,
T.)
I
I
,i
= 2, ..., k-1,
= 2, ..., k-l.
( 18.b)
(18.e)
Starting from these distributions, we see that the MSS for this no-recruitment case is
derived by pooling the data from' (18.d) and (18.e) pairwise on each i. That is, in terms of
probability distributions, we do the factorization
k-1
+
I1 ~(T.In.
i=2
1
u:") (T.) n·
I
l(l-T.)
1
1
k-1 (
(T. + u:" -n')J
1
T.)(u:")
m~
u~
II (T.+u.
*) .
n·
. 2
1=
1
1
I
- 42-
1
1
x
(19)
In writing the above I used n·I = u.I + m.,
which is the total captures on occasion i. The
I
distributions in (19) are hypergeometric.
The MSS for the no-recruitment case of JS is thus
ui "" bin(N l' (PI
n·I
+ qi "I»
(20.a)
,
ul
I ui
"" bin(ui, TI) ,
r·I
I R.I
"" bin(R.I , ".)
I
,i
"" bin((T.I + u:"),
T.)
I
I
, i = 2, ..., k-1.
I (T.I + u:")
I
(20.b)
= 1, ..., k-I ,
(20.c)
(20.d)
The dimensionality of this MSS is 2k -1 and there are 2k -1 estimable parameters:
Algebraic forms are simplified by defining Ti = ui and Ti = T i + ui, i = 2, ..., k-1.
By definition, ul = n i . Hence (20.a)-(20.d) can written as
ui "" bin(N l' (PI
+ qi "I»
(21.a)
,
= 1, ..., k-I ,
ri
I Ri
"" bin(Ri , "i)
,i
n·I
I T:"I
"" bin(T:",
T.)
I
I
, i = 1, ..., k-1.
(21.b)
(21.c)
Estimates are easy to derive here. Just notice that (20.c) and (20.d) are identical in
form and range of index (i) to the distributions in (I3.c) and (I3.e) from which the estimators
of tPI' ..., tPk-2' tPk-IPk' P2' ..., Pk-I are derived for the saturated JS model. Because we also
have here the distribution given by (20.b), PI is now estimable. MLEs of these parameters are
n·I
T:"
Pi -
ni
ni )/r
-T:"
+ (I
1 - - -i
T:" R •
I
Using PI and ~I we get
I
,i=I, ...,k-1.
(22)
,i = I, ...,k-1.
(23)
I
•
P1H1~1 = ( :i )+_1 :i )/(~~)
- 43-
hence, a method of moments estimator of N 1 is (from 21.a)
(24)
The variances and covariances of the estimators in (22)-(24) can be inferred from the results in
Section 6.2.1, using the MSS in (21.a)-(21.c)
Pollock et al. (1974) derived a test for recruitment in the saturated JS model. That test
is, as we might at this point expect, based on the residual distribution, relative to the
unconstrained JS model, given by (19). These hypergeometric distributions provide the basis
of the test of Ho : bi
0, i
2, , k-l, which is a null hypothesis implied by the one of
=
=
Ho : Bi = 0, i = 2,
=
interest to us,
, k (we cannot directly test B 1
0). The test takes the
form of a series of 2 by 2 contingency tables; the chi squares computable from these tables are
independent, hence additive.
By examining the E(bi'>, for i = 2, ..., k-l, as functions of the Pi and <Pi we can
determine some of the sensitivities and robustness of this test for recruitment. For example,
this recruitment test is valid even if there is a release-effect on survival for one period following
release (see Arnason and Mills 1987 and Burnham et al. 1987 regarding release-effects). Also, a
tagging effect (Manly 1971) on survival for one period following tagging and release does not
affect validity of the test for recruitment. This test for recruitment will be affected (hence,
possible biased) if some capture probabilities differ by capture histcry, or if there are longerterm tagging or release effects.
5.3.2
No Mortality
5.3.3
No Recruitment or Mortality
5.4
Discussion
- 44-
6. On Deriving Theoretical Variances and Covariances
6.1
A General Methodology for Use with Release-Resampling Data
We now consider a simple methodology for obtaining estimators of variances and
covariances. This approach is applicable whenever the MSS is distributed as a set of d
independent binomials and correspondingly there are d estimable parameters. The abstract
situation is as follows. Let
,i
= 1, ..., d,
independently for i = 1, ..., d. The parameters of natural interest are ~' = (01' ..., 0d) and the
r1' ..., rd are a 1-to-1 transform of the 1 , ..., 0d' The MLEs of the ri are
°
•
ri
.'
Yi
= ri(~)
= Yo
' i = 1, ..., d.
1
These d equations can be solved uniquely for ~. Also, the delta method applied here to get
variances and covariances gives the same estimated variances as the MLE information matrix
approach.
Often we will be able to find explicit estimators, 0i' as functions of the ratios
y.
a· =
J
~.J '
say
,i = 1, ..., d.
Note that E(aj)
= rj'
Asymptotically, theoretical variances and covariances are
_
d (agi)2(rj(1-rj»)
var«(J.) =.~
y
,
1
J-1
a·
.
J
J
-a-
In the above, partials are evaluated at E(aj)
common, then cov(Oi' On)
6.2
= O.
= rj'
If 0i and On have no "a{ terms in
In fact, in such a case 0i and On are statistically independent.
The JoUy-Seber Model
6.2.1 Deriving Dispersion Formulae for ¢i and Pi
I illustrate and extend this approach with the JS model. In that model the gi functions
take only two forms. It is thus convenient to determine the factors
- 45-
G
{)g.1
-
ij -
()a.
/2
('Y' (1-'Y.»)1
J
1
J
Y.
J
and get
•
var(O.)
2
= J=l
. Ed (G..)
,
IJ
1
••
=
cov(O., On)
1
d
E (G.. G .).
j=l
IJ nJ
For JS, the two forms of these g-functions are below:
~i
Pi
where
= Ai [B i+1 + (1- Bi+1)/Ai+1]
= B/[Bi + (l-Bi )/Ai ]
r·
= 1,
, i = 2,
,i
, k-1,
, k-1,
_I
A 1. nn.,
, j
= 1, .. _, k-1 ,
m·
_1
B1' -T
•
,i
= 2, ..., k-1 ,
1
1
=
=
and we define A k = 1. Note that mk
T k , so B k
1, also. Do not confuse these B j with the
recruitment parameters in Section 5; this notation of Ai and B j is used here for consistency
with Burnham et al. (1987). (Note, also, that there is so much notation needed in CR that it
is not practical to have unique notation for every variable that arises).
Given the binomial model for the ri
I Ri
and mi
I M i , we
know that
The partials of the g's are easily found (Le., the partial derivatives of ~i and Pi wrt the Aj and
Bj ). However, it is more useful to determine and tabulate the scaled partials, the G-functions.
For the JS model these G's are given in Table 5.
To get var(Pi)' for example, one has
var(p.)
1
= (G (A.
1
I p.1 »2 +
= (Piqi)2(~ -
(G(B.1
I p.»2
1
Ji + iii + ~)
,i =
2, ..., k-l.
To get theoretical variances and covariances, substitute parameters for their estimators and
unconditional expected values for statistics (this works here because these variance and
covariance estimators can be derived by first getting the theoretical formulae and then
substituting estimators for parameters).
- 46-
Table 5. The G ij factors for generating variance and covariance formulae for the ~i and 1\ of
the Jolly-Seber model.
For ~i
variable
A.I
variable
.r.CI
1
1\)
A.I
4>i~ii-Ri '
i
G(variable
= 1, ..., k-l.
i = 2, ..., k-l.
B.I
i = 1, ..., k-2
i
i
= 1, ..., k-l.
= 1, ..., k-2.
All other G(·I ~i) = O.
All other G(·
11\)= o.
The general formula for an estimated variance or covariance is
d
. E G(variable.
1=1
1
1
parameter1)G(variable.
1
I
parameter2)'
Terms in the summation will be zero except for variables in common to both of the estimators
parameterl and parameter2' In the case of covariances, if there are no parameters in common,
the covariance is zero. As an example of a covariance, consider cov(~., p.). Of the A 1 , ... ,
A k _1 , and B 2 , ..., B k _ , only Ai is in common in both ~i and Pi; hence 1 I
1
. (1
.)
cov
,/,', p.
1
I
··
=1
'/'.p.q.
111
(1 1)
r; - R
1
i
,i
= 1, ...,k-1.
A few more formulae generated by use of the methodology (i.e., Table 5) presented here:
for i
= 1, ..., k-1,
- 47-
A
A
cov(t/J.,
1
11)
q ·+1 (
t/J'+1)
= - t/J·t/J·+1
r.----R
1
1 1
1
1+1
i+1
A
AA
,i
A
= 1, ..., k-2,
,alli:;6j.
See Burnham et a1. (1987) for a complete list of these variances and covariances given in the
form and notation used here and see Jolly (1965) for the original derivations and results.
6.2.2
Deriving Dispersion Formulae for the Abundance Estimators
Once we have the variances and covariances of the ~i and
quantities like var(N
I Ni ). The derivation
N _
-
ni _
Pi -
1\,
we can then derive
clarifies that this variance is conditional on Ni . Here
mi + ui
Pi
From standard asymptotics:
A
var(N·IN.)=(N.)
1
1
1
2(
(cv(n.»
2
1
•
+(cv(p.»
1
2
-2
cov( ni ' Pi) )
E()
.
n i Pi
Now ni = mi + ui and cov(ui' Pi) = OJ this can be evaluated based on the distribution of the
full MSS for JS CR data. Evaluating cov(mi ' Pi) = cov(ni' Pi) we get
m·1
cov(m 1·, p.) = E T [T. cov«-T
, p.)
1
i i i
1
I T.)]
1
p.q.
= ET i [T.~]
= p.q..
.Li
1
The m~rginal distribution of ni
(assummg 0 < Pi < 1)
I Ni
1 1
is bin(Ni' Pi)' Consequently, the end result can be written
I am grateful to Dr. Cavell Brownie for doing the algebra to show that the above is identical
to Jolly's formula for var(N i I N i ) (Jolly 1965) when we replace R i by E(Ri ).
Whatever the derivation, some conditioning is used. In the above, we conditioned on
a-1
because it appears in var(Pi)' which is conditional on the releases R i and R i + 1 . There is no
place in the derivation of this var(N i I Ni ) that we conditioned on M i or Vi' rather we only
need to condition on Ni .
M
I do not give here all possible variances and covariances involving the various N,
and B. I have derived the results below, this is quite easy to do using the above methods
- 48-
and knowing the distribution of the MSS in Section 5.1. These results below are all conditional
on Ni (or Ni and Nj ):
..
cov(N.,
1
41·)
1
(1 1)
=-
N.4J.q.
- ()-R.
1 l i E r.
1
,i
= 2, ... , k-l,
,i
= 2, ..., k-2,
,i
= 2, ..., k-l.
1
cov(N., p.) = 0 for i =1= j,
1
J
=
Jolly (1965) does show cov(N i , Nj )
0, i =1= j. This follows immediately here because
cov(p., p.) = 0 and cov(n., n.1 N., N.) = O.
1
J
1
J
1
J
6.2.3 Special cases of the Jolly-Seber Model
6.3 The Time-Specific Band Recovery Model
For completeness I apply this approach to variance and covariance estimation for the
time-specific band recovery model. These results apply to Model M 1 in Brownie et al. (1985)
and it is logical to think the results below will be compared to the formulae given in Brownie
et al. (1985). Consequently, I give these derivations and results in the notation of band
recovery data, and I do so for the general case of f ;:: k.
The MSS is representable as
R·a I N·a ....., bin(N.,
a A.)
a
, i = 1, ..., k,
C·a I T·a ....., bin(T.,
a T.)
a
, a
T
f·
a
i -T
= 1, ..., f-l,
, i = 1, ..., f-1
a
(for f > k, the parameters Ak+1' ..., Af_1 are not estimable). There are k+f-1 estimable
parameters, which I take here to be f1' ..., f k , SI' ..., Sk_l and Tk' ... , Tf_l for f > k,.
- 49-
Maximum likelihood estimators are
=
f.
s. i
and if
R·
C·1_
N·I
T·I
_1
1
C·I ) / R.+
• 1
R·I (
-!Vi 1 -T;
+
Ni 1
, I
= 1, ..., k,
, i
= 1, ..., k-l,
e > k we also have
f.
I
=
C·
T.
= k, ..., e-l.
, i
_I_
I
For the case e > k, we could use alternative representations for the additional
parameters, such as the products
rather than Tk' ...,
T
e- k estimable
e-l .
Let us represent these MLEs in a slightly simpler algebraic form as
f.I
= A·B·
, i
= 1, ..., k,
f.I
= B·
, i
= k, ..., e-1 if e > k.
I
I
and
I
Here,
R·
_I
A I' -N
.
, i = 1, ..., k,
I
C·
1
- -T
B I' .
, i
= 1, ..., e-l,
I
and these are mutually independent statistics with theoretical binomial variation. Also, by
definition, Be
l.
The next step is to get the various partials of the MLEs (the g-functions) and then get
the G-functions. For example,
=
{Jf.
I
-{J--
A.
I
B.
and
I
{Jf.
I
-{J--
B.
I
A.
I
i
,i
= 1, ..., k.
Also, all the partials of i wrt Aj or Bj' j =F i are O. Therefore,
- 50-
-B ~A.(I-A.)
I
I
- i
N.
I
and
and G(any variable other than Ai or Bi
I f i)
= O.
The partials of Si are
as.
aA:
I-B·
=
as·
1
Ai
+;
=
s.
AI. '
A·1_ =
oBi -
A i +1
as·
I
A i (1-B i )
aAi+1 -
(Ai+1)2
I
s·
I
I-B.I '
S.I
All partials wrt other Aj and Bj are O.
Given these partials, write the G-functions as, for example
A i+1(I- A i+1)
Ni+1
Ai+1(1-A i+1)
+
Ni 1
- 51-
Table 6 shows the full information on the G(variable I parameter) functions needed to
quickly construct all variances and covariances of the MLEs for the time-specific BR model
(Modell). For example, f i = AiB i , hence
. . = (G(A i I f')2
i)
+ (G(B i I f')2
i)
var(f i)
= (f .)2(.1._.1.+.1._.1.)
R· N· C· T·
I
I
I
I
, i = 1, ..., k.
I
The terms in common to f i and Si are Ai and B i , hence the covariance of these two
estimators is computed as
cov(f.,
S.) = G(A.III
f.)G(A.I
II
IS.)
+ G(B.III
f.)G(B.I IS.)
I
I
.'(1 1 1)
= f.S.
- - & 7 " - mI I R·
lV·
1·
I
I
I
, I
= 1, ..., k.
The interested reader can construct the remammg variances and covariances and compare
them to Modell results in Brownie et al. (1985).
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Table 6. The G ij factors for generating variance and covariance formulae for the f i'
T i of the time-specific band recovery model.
For f i
For
variable
variable
A·I
A·I
B·I
B·I
Si
G( variable
. ~
-So
1Si)
1
IT·-C·
I
I
-1
T·'
I
i = 1, ..., k.
All other G(·I f i)
= o.
I
= 1, ..., k-I.
For Ti
All other G(· 1 Si)=
variable
G( variable
B·I
T·
I T i)
--rrr
.~1
I C·
1·'
I
i = k, ...,
I
e-l if e > k.
All other G(·I f i)
= o.
- 53-
o.
Si
and
8. Discussion
We have concentrated here on a basis for a unified methodology for estimation of
survival rates from capture-type sampling of "open" wildlife populations. "Open" just means
that the population under study may experience unknown mortality and/or recruitment during
the study period. There is also a large literature on capture-recapture for closed populations.
However, that literature is simpler and more complete (in my judgement), partly because the
only biological parameter of interest is population size, N. Good references for an entry into
the capture-recapture sampling literature for closed populations are Otis et al. (1978),
Cormack (1979), Pollock (1982), White et al. (1982), Anderson et al. (1983), Pollock and Otto
(1983), Pollock et al. (1984), Sandland and Cormack (1984) and Seber (1982, 1986).
With open populations, the parameters of interest are survival, recruitment and
population size; these are possibly different at each sampling time or for each interval. When
resampling is by harvest as in bird-banding, survival rate is estimable but not population size,
then the number of models is much more tractable and a good comprehensive reference exists:
Brownie et al. (1985). However, Brownie et al. (1985) does not deal with the classical agespecific life table model which has been much used, especially in Europe (see e.g., Burnham
and Anderson 1979, Lebreton 1977; North and Cormack 1981; Lakhani and Newton 1983;
Anderson et al. 1985; and Clobert et al. 1985).
Classical capture-recapture on open populations has a very large literature. There has
not yet been an accepted unification of this literature (let alone all open population capture
literature). The cutting edge in capture-recapture research now is to produce such a
unification. There is not even any good technical monograph-length exposition on just openmodels capture-recapture (there is in the revision stage: Pollock et aI., in prep.). There are lots
of reviews and extended expositions in the literature (see e.g., Seber 1982, 1986; Cormack
1979; Begon 1979; and Blower et al. 1981). Also there have been attempts to produce a unified
theory, that is, to produce a tractable theoretical approach general enough to incorporate
numerous models as special cases.
Robson (1969) pioneered a general hypergeometric modelling approach, which has been
utilized by Pollock (1975, 1981a) and Pollock et al. (1985). This framework concentrates on
population size, N, in such a way that it appears impossible to extend it to bird-banding or to
incorporate constant survival rate models as special cases.
By using a Poisson-counts model, Jolly (1979, 1982) extended his earlier work (Jolly
1965) to the case of constant survival (I/» and capture rate (p) parameters. However, the
Poisson approximation is not always suitable (see e.g., Brownie et al. 1986); moreover,
multinomial models are just as easy to use and do not require the additional assumption of a
superpopulation.
More promising than the above two approaches (Le., hypergeometric and Poisson) is the
log-linear modelling of Cormack. Cormack's efforts to extend the work of Fienberg (1972), on
closed-populations to open-populations, has been more successful than it originally seemed
possible (see e.g., Cormack 1979, 1981, 1985). However, log-linear modelling implemented in
GLIM does not easily lead to the closed-form sequence of tests, estimators, or variances that
are derivable with the multinomial approach. The initial computational advantage of the loglinear models would also have to be abandoned for iterative numerical solutions for all the lessthan-full rank models (such as constant survival rates). This is of course true for the
multinomial approach. In fact, ease of computation should be abandoned as a criterion for
favoring an approach to capture-recapture models. Rather, I recommend the key criterion
should be ease of model structure representation.
Given the more complex models now in use, good computer programs implementing
these models is the key to their use by ecologists and statisticians. Some available, current,
programs and algorithms for analysis of capture data are provided by Amason and Baniuk
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(1980), Brownie et al. (1985, 1986), Buckland (1980), Burnham (1989), Burnham et al. (1987),
Clobert et al. (1985), Conroy and Williams (1984), Cormack (1985), Lebreton (1977), Pollock
et al. (1985), Schwarz (1988) and White (1983). Over all, there is substantial computing
capability represented by the program these authors discuss for analyses of open-models
capture data.
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