AGE DEPENDENT TAG RECOVERY ANALYSES OF PACIFIC HALIBUT DATA
by
Heidi Chen, Cavell Brownie,
Kenneth H. Pollock, and William L. Kendall
Institution of Statistics Mimeograph Series No. 1994
May, 1991
NORTH CAROLINA STATE UNIVERSITY
Raleigh, North Carolina
MIMEO SERIES 111'194 MAY 1991
AGE DEPENDENT TAG RECOVERY ANALYSI
OF PACIFIC HALIBUT DATA
I
NAl1E
DATE
"';
I
.,~
-. --.
--
Department of Statistics Library
-,
Age Dependent Tag Recovery Analyses
of Pacific Halibut Data
Heidi Chen, Cavell Brownie,
Kenneth H. Pollock, and William 1. Kendall
Department of Statistics
North Carolina State University
Box 8203
Raleigh, NC 28695-8203
-,
Abstract
We generalized the Brownie 3-age class model for band-recovery data to one allowing
for 8-age classes, and applied it to the Pacific Halibut tagging data collected by the
International Pacific Halibut Commission from 1979-1986 for areas 2B, 2C and 3A.
Because of the small sample size, the data collected from three areas were combined
together and analyzed as from one population. For the halibut data, age information was
not recorded at banding, but an age-length key was available. The age-length key was used
to form 8 size classes that approximately corresponded to I-year age classes. The structure
of the tag recovery models is generally based on three types of parameters : the annual
survival rates, the direct recovery rates ( for newly tagged fish) and the indirect recovery
rates ( for previously tagged fish). We found that the indirect recovery rate estimates are
much higher than the estimates of direct recovery rates, and that the annual survival rates of
Pacific Halibut are high ( at least greater than 50% in general). According to the results of
the model selection, the survival rates and indirect recovery rates are year dependent; the
indirect recovery rate increases linearly with fish size; and there is no difference between
the survival rates of size classes 5 to 8 which correspond to fish of"ages 10 to 13+ years.
The results of the goodness of fit tests show that all the models developed in the project do
not fit the data well. In general there are two causes of an unrealistic model: ( 1 ) all the
factors that are involved in the system are not included in the model; and ( 2) violation of
the model assumptions. The calculation of maximum likelihood estimates in the more
complex models caused us some computational difficulties. This is not surprising
considering the large number of parameters ( up to 162 parameters) to be estimated using
iterative numerical methods.
2
Table of Contents
Abstract
1. Introduction
II. The Tag Recapture Data
III. Model Structure
IV. Results
Model Estimates and Computer limitations
Model Selection
General Trend on Survival and Recovery Rates of Pacific Halibut
V. Discussion
Model assumptions
Computational Limitations
Future Work
Special Studies
References
Figures
Tables
3
I. Introduction
Capture-recapture methods have been widely used in the estimation of demographic
parameters of fish and wildlife populations. The basic model which allows estimation of
population parameters for an open population is the Jolly-Seber model ( Jolly 1965, Seber
1965). The Jolly-Seber model makes the assumption that all animals in the population
have the same survival and capture probabilities for each sampling period.
This
assumption may be true for some species of animals, but is more questionable for fish
where the survival rates and recovery rates may depend on size, age, sex,etc. Pollock (
1981 ) generalized the Jolly-Seber model to allow for age-dependence of survival and
capture probabilities. See also Pollock et al. ( 1990 ) for a general review of capturerecapture models.
Brownie et al. ( 1985 ) developed a basically equivalent model for tag recovery data
where 3 age-classes are identified at tagging and survival and recovery probabilities are
allowed to depend on all 3 age classes. In order to examine the relationship between
survival and size or age in more detail, we need models that are more general than those in
Brownie et al. ( 1985). In this paper we show how to generalize the 3-age class model to
one allowing for 8-age classes. We also examine a series of reduced-parameter versions of
the most general model which make increasingly restrictive assumptions about the effect of
calendar year and age on survival and recovery rates. Hypotheses about the effects of year
or age on survival and recovery are tested by comparing different models. These models
are applied to the tag-recovery data of Pacific Halibut provided by the International Pacific
Halibut Commission ( IPHC). Pollock and O'Connell ( 1989) used simpler versions of
the Brownie et al. ( 1985 ) models to analyze the Halibut tagging data which did not
adequately allow for dependence of survival or recovery rates or size.
4
II. The Ta~ Recapture Data
Over the past 60 years the IPHC has carried out an extensive tagging program. The
tagging has usually been opportunistic during the course of setline or trawl survey
conducted for other purposes. Tag returns have typically been from commercial fishermen
although recently there have been some from research cruises. Sport caught Halibut tag
returns are probably insignificant.
There have been two primary objectives of the tagging studies. The fIrst is to get a clear
idea of migration patterns of the Halibut while the second is to estimate natural and fIshing
mortality. R. Deriso has carried out some analyses to estimate migration rates but there is
still a lot more that could be done ( Stock assessment documents 11 and 12). Myhre (
1967 ) in IPHC Scientific Report #42 obtained mortality estimates based on a regression
model by Gulland ( 1963 ).
The Pacific Halibut tagging data was collected by the International Pacific Halibut
Commission from 1979 to 1986 in Areas 2B, 2C and 2A. The data collected from those
three areas were treated as from one population because of small sample size. In order to
apply tag-recovery models which I1ermit estimation of age-specific survival to these data, it
is necessary to group fish at the time of release into I-year age classes, except for the fmal
class which contains all older fish. For the halibut data, age information is not available,
but the length was recorded for each tagged fIsh at the time of release. In addition to the
length information, an age-length key was available for Pacific Halibut together with
information on selectivity which is a measure of the fishing pressure for each age-length
class ( table 1). The age-length key was used to form 8 size classes that best seemed to
correspond to I-year age classes in the following manner. The age 8, 9, 10, 11 and 12
from the table 1 on the right are equivalent to size class 3, 4, 5, 6 and 7 respectively shown
on the left, and ages greater than 13 were grouped into size class 8. Boundaries for these
5
size classes were the values midway between mean length for age 8 to 13. As the recovery
data included a substantial number of fish less than 86 cm in length, two classes were
created to represent 6 and 7 year old fish assuming a width of 11 cm for each of the
corresponding size chisses 1 and 2. Data of fish less than 64 cm in length were omitted
from analysis because of the greater uncertainty involved in grouping these into age classes
on the basis of size.
Based on the length at release, each tagged fish greater than 64 cm at release was
assigned to one of these 8 size classes, and the tag-recovery data were summarized by
constructing 8 data arrays, one for each size-class ( table 2). Each of these 8 data arrays is
a recovery matrix ( see example of Brownie et al.. 1985, p118 ) containing numbers
released and recovered for the years 1979 to 1986.
6
III. Model Structure
We develop and refine survival and recovery rate estimates based on models developed
by Brownie et al. ( 1985). Let us consider possible fates of a tagged fish at the start of the
year based on the diagrams in Brownie et al. ( 1985, p14):
survives
year
tag found and
reported to IPHC
killed and retrieved
Tagged fish alive
at start of year ~--------- by fishing boat
tag not found
or not reported
dies from "natural" causes
or not retrieved by fishing
boat
where
s =
the survival rate or the probability of surviving the year.
e =
the exploitation rate or the probability of being harvested by the fishing
fleet.
1-s-e =
the natural mortality rate or the probability of dying from natural causes ( if
we can assume that all fish killed are retrieved by the fishing fleet ).
A. =
the tag reporting rate, the possibility that a tag will be found and reported to
the IPHC given that the fish has been harvested.
Note that the type of data we analyze supplies infonnation directly about only those fish
which are harvested and their tags reported. Therefore, only the product f=A.e is estimable
7
but the component rates A and e are not estimable without additional information on
reporting rate. A modified diagram is as follow:
-,
\
\
survives year
killed, retrieved and tag
Tagged fish alive
at start of year ~-------- reported by fishing boat
dies from "natural" causes
or not retrieved or tag not reported
Where f=Ae is called the tag recovery rate.
There are numerous assumptions involved in making inferences from tagging models
which are based on Brownie et al. (1985, p6)
(1) The sample is representative of the target population.
(2) There is no tag loss.
(3) Survival rates are not influenced by the tagging process itself.
(4) The year of recovery is correctly tabulated (sometimes tags may be kept and returned
later outside the fishing season ).
(5) The fate of each tagged fish is independent of the fate of all other tagged fish.
(6) All tagged fish within an identifiable class ( size, age, sex..etc. ) have the same annual
survival and recovery rate.
In addition to the assumptions above, we also assume that
(7) The halibut of size class i will increase its length to the size class i+ I after one year.
Any violation of the assumptions listed above may cause bias in the estimation of
survival and recovery rates. Pollock and Ravelling ( 1982 ) and Nichols et al. ( 1982 )
8
discussed the bias caused by failures of ( 1 ) to ( 6). Assumption ( 7 ) is new and we
discuss possible biases due to its violation later in this article.
The models used in the analysis of the Halibut data in Table 2 are extensions to an 8
age class situation of the models developed by Brownie et al. ( 1985, chapter 4 ) for
analysis of tag-recovery data for a 3-age class situation. We emphasize that the structure
of the tag recovery models is generally based on two types of parameters: the annual
survival rate and the annual recovery rate. In addition, the recovery rates are allowed to
differ for newly tagged and previously released fish, because the newly tagged fish may
not be so readily available for recapture. These parameters may vary by year or by size
class. Here we investigate a series of year or size specific assumptions for the survival and
recovery rates of halibut. Figure 1 presents the outline of the models discussed in the
article. The most general model ( Model 0 ) is discussed first followed by a series of
more restrictive models with reduced numbers of parameters.
Model 0 assumes the survival rate ( s~ ), indirect recovery rate ( ~ ) and direct recovery
rate ( ~ *) are both year and size specific where
the jth year;
~ denotes
s~ denotes the survival rate for size class i in
the indirect recovery rate ( Le. the recovery rate for previously
tagged fish ) for size class i in the jth year;
~ *den0tes the direct recovery rate ( Le. the
recovery rate for newly tagged fish) for size class i in the jth year. Based on these
assumptions, the model structure is expressed in terms of the multinomial cell probabilities
( see Brownie et al. 1985, p119) shown in Table 3. There are 162 identifiable parameters
in Model 0, and the maximum likelihood estimates can be expressed explicitly.
Modell is a restriction of Model 0 with size dependence but no year dependence for the
survival rates, indirect recovery rates and direct recovery rates. There are 23 parameters,
and their MLE's do not have an explicit algebraic form. Therefore they have to be obtained
by numerical maximization of the likelihood function.
9
Model 2.2 is an extension of Model 1. It allows the indirect recovery rates to be both
year and size dependent, but the survival rates and direct recovery rates are size specific
only. Under the assumptions there are 65 parameters, and MLE's have to be obtained
numerically.
Model 2.1 assumes the relationship between the year-specific indirect recovery rate and
size class is linear where
t. = bOj + blj x L
J
Survival and direct recovery rates are size
specific but not year dependent. Thus Model 2.1 is a reduced version of Model 2.2 and
has 30 parameters only ( compared to 65 parameters for Model 2.2 ). Again MLE's have
to be obtained numerically.
Models 1,2.1,2.2 form a series of increasingly general models which allow different
assumptions about the effect of size on recovery rates while assuming survival rates to be
size dependent but independent of year. The tests between these models focus on the
relationship between recovery rates and size ( see Figure 1 ). We also constructed a second
series of nested models to investigate the relationship between survival rates and the size
classes, while allowing year specificity of survival and indirect recovery rates.
Model 3.1 allows us to model the year dependent effect of survival rates and indirect
recovery rates simultaneously. In the model we assume that the survival rates are year
dependent but are not related to size;' the direct recovery rates depend on size but not on
year; and indirect recovery rates are a linear function of size class in each year. Based on
these assumptions there are 33 identifiable parameters in the model.
Model 3.2 is an extension of Model 3.1. In addition to the assumptions of Model 3.1,
we assume that in each year the survival rate on size class 1 is different from the survival
rate for size classes 2 to 8, Le. we assume survival rates not only depend on the year but
also on two groups of size classes
(size class 1 and size class 2 to 8). In this model
there are 39 identifiable parameters.
10
Model 3.3 is aft extension of Model 3.2. In the model the survival rates are year
specific and
differ~t
for three groups of size classes ( size class 1, size class 2 and size
class 3 to 8). The assumptions of indirect and direct recovery rates are the same as in the
Model 3.1. In the model there are 45 identifiable parameters.
Model 3.4 is an extension of Model 3.3. All the assumptions of Model 3.4 are the same
as in Model 3.1 except that we assume the survival rates depend on 4 groups of size classes
( size class 1, size class 2, size class 3 and size class 4 to 8); In the model there are 51
identifiable parameters.
Model 3.5 is an extension of Model 3.4. There are now 5 groups of size classes ( size
class 1, size class 2, size class 3, size class 4 and size class 5 to 8 ) each with different
survival rates. The assumptions of indirect and direct recovery rates are the same as in the
Model 3.1. There are 57 identifiable parameters in the model.
Model 3.6 is an extension of Model 3.5. The size class 5 is split out from the fifth
group of the Model 3.5. Thus there exist 6 groups of size class ( size class 1, size class 2,
size class 3, size class 4, size class 5 and size class 6 to 8) in the model. The assumptions
of indirect and direct recQvery rates are the same as in the Model 3.1.
The MLE's of the parameters of all the models described above do not have an explicit
form, except those in the Model O. The numerical maximum likelihood estimates ·of
survival rate, inderect recovery rate and direct recovery rate were obtained using SAS
PROC NLIN METIIOD=DUD ( version 5 ) as outlined in Burnham ( 1989 ), and also
using the program SURVIV (White 1983) on an IBM personal computer. The differences
in the models above are due to changes in the assumptions about the survival rates or the
indirect recovery rates. Likelihood Ratio Tests and the Akaike Information Criterion (
Lebereton et al., 1991 ) were used for testing between models.
11
IV,-l~sults
eQmputatiQn Qf MLE's
Table 4 are the explicit MLE's under Model 0, In tables 5-13 we present the MLE's
under each Qf the restricted models. The MLE's calculated by numerical QptimizatiQn
using PROe NLIN and prQgram SURVIV agree clQsely fQr MQdel 1, Model 2.2 and
MQdel 3,1, but this is nQt the case fQr Model 3.5. FQr at least 2 reaSQns, the Qutput frQm
SURVIV seems mQre accurate than that from PROe NLIN with the derivative free
METHOD=DUD. These reaSQns are that SURVIV produces the larger value Qf the IQglikelihood, and even when a range Qf initial estimates was tried, PROe NLIN Qften did nQt
gQ beyQnd Qne iteratiQn. UnfQrtunately, due tQ lack Qf memQry Qn the P,c., SURVIV
CQuid handle at mQst 59 parameters fQr a mQdel and SQ CQuid nQt be used fQr the largest
models, Le. models 2.2 and 3.6, with 65 and 63 parameters respectively. Estimates fQr
these tWQ mQdels were therefQre Qbtained using PROe NLIN and may be less accurate
than fQr the other mQdels.
As a final nQte Qn cQmputatiQn, PROe NLIN with
METHOD=DUD and SAS versiQn 6 Qn a SUN gave less satisfactory results than versiQn 5
Qn the IBM main frame.
Model SelectiQn
In Qrder tQ choose a model which has sufficient parameters tQ prQvide an adequate
descriptiQn Qf the data, but nQt SQ many as tQ make estimatiQn inefficient, we examined
several sequences Qf hierarchical models, cQmparing a simpler mQdel with a mQre general
alternative using a likelihood ratiQ test. If the test statistic is significantly large, the simple
mQdel is rejected in favQr Qf the mQre general Qne. Table 14 cQntains the results Qf the
testing between mQdels. MQdel 1 is rejected in favQr Qf MQdel 2.1 indicating that indirect
12
recovery rates depend on fish size. The likelihood ratio test of Model 2.1 v.s. Model 2.2
suggests we can model the indirect recovery rates as a linear function of fish size, because
the chi-square value of 34.19 with 35 df is not significantly large and therefore we accept
the simpler model, Model 2.1.
Assuming this linear relationship between indirect recovery rates and size classes, a
second series of nested models is used to investigate the relationship between survival rates
and size classes.Results for the tests between the nested models ( Model 3.1 , Model 3.2 ,
Model 3.3 , Model 3.4 , Model 3.5 and Model 3.6 ) suggest that there is no difference
between the survival rates of size class 5 to 8.
General Trends in Survival and RecoveIY Rates of Pacific Halibut
Based on estimates in tables 4-13, we summarize some general results for the survival
rates and recovery rates of Pacific Halibut:
(1) Indirect recovery rates are much higher than the direct recovery rates.
(2) The indirect recovery rates increase with fish size ( Figure 2 ).
(3) Most survival estimates of Pacific Halibut are greater than 50%.
(4) The survival rates are not linearly related to fish size. In general, the smaller and larger
size of Pacific Halibut have lower survival rates ( Figure 3 ).
(5) The survival rate estimated for each year is not constant, but varies among years
(Figure 4).
13
V. DiscussiQn
Model Assumptions
There are numerous assumptiQns behind the tagging models discussed in this article.
We discuss the validity of these assumptiQns when applied tQ Pacific Halibut tagging data.
The discussiQn is based Qn PQllock and Raveling ( 1982).
( 1 ) The Ta&ged Sample is Representative of the Tar&et PO-Pulation
This assumption is obvious but very important especially if heterQgeneity of survival
and recovery rates ( Assumption 6 ) occurs. If, fQr example, tagging tends tQ take place in
areas with heavy fishing pressure then this CQuid give the appearance Qf high recQvery rates
and low survival rates for the whole region under study. This suggests designing tagging
studies so that the tagging is dispersed over a wide area of each region under study.
(2) There is NQ Tag LQSS
Nelson et al (1980) examined this assumption using simulatiQn and found that its
violation causes a negative bias Qn survival estimates that is worse for species with high
survival rates. Unfortunately there is tag IQSS fQr Halibut. The recQvery rates and hence the
explQitation rate estimates will also be negatively biased. There is the need for a new
double tagging study to obtain estimates of tag IQSS so that survival and recovery rates
estimates can be adjusted.
In additiQn tQ the discussiQn abQve, emigratiQn might be
another factor that viQlates the assumptiQn of no tag IQss. If emigratiQn does occur in the
system, the survival estimates tend to be biased also. Myre ( 1966 ) is the most recent
reference tQ this tQpic fQr Halibut.
14
( 3) Survival Rates are not Influenced by Ta~~in~
This assumption is obviously important because if there is substantial mortality due to
the tagging process, the survival estimates would not apply to the untagged fish. Tagging
mortality tends to be higher for smaller fish and lower for fish caught on circle hooks
( Gilbert St-Pierre, personal communication). The use of condition codes for stratification
should help but Gilbert St-Pierre also notes that condition codes vary widely among
experiments. We do not know if it is practical to consider holding experiments to evaluate
short term tagging mortality.
( 4) The Year
(Fishin~ Season)
of Ta~ Recovery is Correctly Tabulated
Sometimes a commercial fisherman may report tags in a later year than when the fish
was actually caught. We do not know how likely this is for Halibut, but to the extent such
incidents occur they operate to produce a positive bias on survival estimates.
( 5) The Fate of Each
Ta~~ed Fish
is Independent of the Fate of Other Ta!i!ied Fish
This assumption is probably violated in almost all practical applications of tag return
models. Fish are not independent entities in terms of survival or other characteristics. This
will not bias any estimators, but will mean that true sampling variances are larger than those
estimated by statistical models. Thus, any calculated confidence intervals will be narrower
than they should be.
A simple ( albeit unrealistic) example for illustration is to consider a population
composed of independent pairs of fish that behave as though they are a single individual.
A sample of n individuals from this population is effectively only one half of n and, hence
,any sampling variances will be twice those for the models that assume the sample is n
independent individuals. The actual situation in real populations is much more complex,
with many partially dependent members, but the effective sample size will still be much less
15
than the actual sample size. To get around this problem, Burnham et aI. ( 1987) suggest
obtaining empirical estimates of variances by subdividing release cohort s into batches.
( 6) All Tagged Fish within an Identifiable Class have the Same Annual Survival and
Recovety Rates
We believe heterogeneity of survival and recovery rates is likely to occur in practice but
we do not know how serious it will be in Halibut tagging studies. Pollock and Raveling
( 1982 ) and Nichols et al. ( 1982 ) examined this assumption using analytical methods and
simulation. They found that if only recovery rates are heterogeneous then there is no bias
in survival estimates and the recovery rate estimates can be viewed as averages for the
population. If survival probabilities are heterogeneous over the population, there is likely
to be a strong positive relationship between the survival probabilities of an individual from
year to year. There is also likely to be a negative relationship between survival and
recovery probabilities for an individual. In this situation, survival rate estimators will
generally have a negative bias. The negative bias will be more serious where the average
survival rate is high and for studies of short duration. It is theoretically possible for the
survival rate estimators to have a positive bias. This could occur if there were segments of
the population with markedly different survival rates but similar recovery rates. This
implies that the difference in survival of the segments would have to be mostly due to
differences in natural mortality. This might occur if drastically different environmental
conditions were encountered by the segments ( ego disease level, food supply, water
temperature, etc.). Some of these factors vary on a local or regional scale.
( 7 ) The halibut of size class i will increase its len&th to the size class i+ I after one year.
If this assumption is not satisfied, it will cause the violation of assumption (6). For
example: IPHC released 482 tagged halibut of size class 1 in 1979, and these fish are
treated as one cohort ( Le. an identifiable class). Under the assumption (7 ), we assume
16
the cohort will grow synchronously to size class 2 next year. If some of the fish are still in
size class 1 in 1980, their recovery rates may be lower than those fish which increase their
length to size class 2 in 1980. Therefore we are saying that non-synchronous growth will
cause heterogeneity of recovery rates in a cohort. This could be investigated further using
simulation.
Computational Limitations
We have extended a 3 age classes model to an 8 age classes model.
Theoretically,
Brownie's model can be extended to any number of age classes, if the data are available.
However there are limitations to developing extensions in practice, because the MLE's do
not have closed form in general, and must be obtained by numerical optimization
techniques using a computer. In the research, we have tried both SAS NLIN and program
SURVIV to compute the MLE's and we found that SURVIV seems more accurate than
NLIN as the number of parameters increases. SURVIV, however, is limited in the
maximum number of allowable parameters, and in the ways we can model survival as a
function of age. Thus we need better software to handle models with large numbers of
parameters, or where survival is modeled as a function of age.
Future Work
There are two types of procedures to statistically test various models and to select the
proper one. The first is the goodness of fit test. Here the null hypothesis is that the model
fits the data, while the alternative hypothesis is that the model does not fit the data. The
second test is the likelihood ratio test which is used for selecting the best model. Due to the
results of the model testing, Model 3.5 explains the heterogeneity of the tagging data more
adequately than other simpler models. However, according to the Pearson chi-square test,
Model 3.5 does not fit the observed data well ( the Pearson chi-square of Model 3.5 is
282.4 with 91 degrees of freedom). The results of the Pearson chi-square test suggest
17
that we need further studies to understand the tagging system more clearly. In general there
are two possible explanations for an unrealistic model: (1) Model 3.5 does not include all
the factors that are involved in the system; ( 2) some of the assumptions do not hold. We
note that many other tag-recovery data sets suffer similar problems of poor fit to a
multinomial model.
Since the survival rates of Pacific Halibut are high, the violation of assumptions (2), (6)
and ( 7 ) can cause negatively biased survival estimators. With halibut the possible
violations of the' homogeneity assumption might be caused by fish growth, sexal
differences or the violation of assumption ( 7 ). Because the sex information is not available
for the Pacific Halibut data, we can not test whether heterogeneity exists between male and
female fish. The best way to avoid the violation of assumption ( 7 ) is to collect the age
information at release of the tagged fish if possible. If the age of each tagged halibut is not
easy to identify in practice, an age-length key can be used to convert the length to age. An
unrealistic age-length key will cause a serious violation of assumption (7 ).
Special Studies
It is clear to us as it was to Pollock and O'Connell ( 1989) that there are several special
problems with the Halibut data which need to be addressed either by changing procedures
or by carrying out special studies to estimate parameters which could be used to adjust and
improve survival and exploitation rate estimates.
1.
Ta~
Loss
Both Deriso ( personal communication) and S1. Pierre ( personal communication)
reacted to our earlier work by suspecting that our survival rate estimates were biased low
18
due to tag loss. A double tagging study, although expensive, would be useful in providing
an estimate of tag loss which could be used to adjust out estimates.
2. Tag Induced Mortality
Another potentially serious problem with the Halibut tagging data is tagging and
handling induced mortality. We do not have a good suggestion here. For some species
holding experiments are used to estimate short term tagging and handling mortality. If
practical this would be very valuable for the Halibut studies.
3. Reporting Rate
The parameter we call A., the reporting rate is actually the probability of a tagged Halibut
having the tag .fmilld and reported. This parameter is extremely important in that if it could
be estimated then tag recovery rate can be converted to exploitation rate. It also allows
natural mortality rate to be estimated by subtraction (Pollock et al. 1991).
The probability of a tag being fmm.d is hard to estimate but could possibly be estimated
if a special study were set up where total catch of some boats were searched for tags.
Given a tag has been fmlnd the probability of it being reported could be estimated using a
variable reward tagging study. Some tags would need to. have known stamped rewards of
high amounts so that it would be safe to assume that all of those tags were reported. This
would enable us to obtain an estimate of the reporting probability ( given the tag found) for
regular tags.
19
References
Brownie,
c., Anderson,D.R., Burnham,K.P. and D.S. Robson ( 1985).
Statistical
inference from band recovery data: A handbook, U.S. Fish and Wildlife Service,
Resource Publication No. 156, Washington, D.C.
Burnham, K.P. ( 1989). Numerical survival rate estimation for capture-recapture models
using SAS PROC NLIN. In Manly, B.FJ., Mcdonald, L.L., and Logan J., eds.
Proceedings of Insect Population Dynamics Workshop, Laramie, Wyoming.
Springer Verlag, Amsterdam.
Burham, K.P., Anderson, D.R., White, G.C., Brownie,
c., and Pollock, K.H.
( 1987 ).
Design and analysis methods for fish survival experiments based on releaserecapture. American Fisheries Society Monographs 5, Bethesda, Maryland.
Gulland, J.A. ( 1963). The estimation of fishing mortality from tagging experiments. Int.
Comm. Northwest Atlant. Fish., Spec. Publ. 4, 218-227, Dartmouth.
Jolly, G.M. ( 1965). Explicit estimates from capture-recapture data with both death and
immigration--stochastic model. Biometrika 52,225-247.
Lebreton, J.D., Burnham, K.P., Clobert, J., and Anderson, D.R. ( 1991). Modeling
survival and testing hypotheses using marked animals: a unified approach with case
studies. Ecological Monographs.
Myhre, R.J. ( 1967). Mortality estimates from tagging experiments on Pacific Halibut.
Rep. Int. Pacific Halibut Comm. No. 42, Seattle, Washington.
Nichols, J.D., Stokes, S.L., Hines, J.E. and Conroy, M.J. ( 1982). Additional
comments on the assumption of homogeneous survival rates in modern bird banding
estimation models. J. Wildl. Manage. 46, 953-962.
Pollock, K.H. ( 1991). Modeling capture, recapture, and removal statistics for estimation
of demographic parameters for fish and wildlife population: past, present, and
future. Journal of the American Statistical Association 86, 225-238.
20
Pollock, K.H., Hoenig, J.M., and Jones, C.M. ( 1991 ). Estimation of fishing and
natural mortality when a tagging study is combined with a creel surveyor port
sampling. Proceedings of the·International Angler Survey Symposium, Houston,
TX, March 27-31, 1990.
Pollock, K.H., and O'Connell, M.A. ( 1989). Estimation of survival, exploitation, and
natural mortality rates for Pacific Halibut from multi-year tagging studies. Institute of
Statistics, North Carolina State University, Technical Report No. 1954.
Pollock, K.H. (1981). Capture-recapture models allowing for age-dependent survival and
capture rates. Biometrics 37, 521-529.
Pollock, K.H. and Raveling, D.G. ( 1982). Assumption of modern band-recovery
models with emphasis on heterogeneous survival rates. J. Wildl. Manage. 46, 8898.
SAS Institute Inc. ( 1985). SAS User's Guide: Statistics, Version 5 Edition. Cary,
North Carolina: SAS Institute, Inc.
Seber, G.A.F. ( 1965). A note on the multiple recapture census. Biometrika 52, 249259.
White,G.C. ( 1983). Numerical estimation of survival rates from band-recovery and
biotelemetry data. J. Wildl. Manage. 47, 716-728.
21
/
\...
Model 0
f* •f. s: year and size specific
162 parameters
/
Mode13.6
f*: age specific but not year specific
f: year specific and linearly related to age
s: year specific and 6 groups of size class:
12345678
5J., 5 ., SJ"
J
/
Model 2.2
f: year and size specific
f* • s: age sJXX:ific but not year specific
65 parameters
\.. -Log (likelihood) = 727.80
= Sj =
S., S., S.
J
J
J
S
j
63 parameters
\.. -Log (likelihood) = 698.94
I
/
Model 3.5
f*: age specific but not year specific
f: year specific and linearly related to age
s: year specific and 5 groups of size class:
12345678
=
S., S., S., S.
SJ"
JJJJ
S.
J
=
S.
J
=
S.
J
57 parameters
\.... -Log (likelihood) =703.11
I
/'
Model3.4
f"': age specific but not year specific
f: year specific and linearly related to age
s: year specific and 4 groups of size class:
1
2
3
4_
5_
6_
7
J
J
J
S., S., S., S. - S. - S. - S.
JJJJ
/
Model 2.1
"'
f: year specific and linearly related to size class
f*. s: age sJXX:ific but not year specific
30 parameters
\.. -Log (likelihood) = 745.16
= Sj8
51 parameters
\.. -Log (likelihood) =712.89
r
/
Model 3.3
f"': age specific but not year specific
f: year specific and linearly related to age
s: year specific and 3 groups of size class:
12345678
SJ"
SJ"
S.
J
= S.J = S.J = S.J = s.J = S.J
45 parameters
\.. -Log (likelihood) = 721.73
I
/'
Model 3.2
f*: age specific but not year specific
f: year specific and linearly related to age
s: year specific and 2 groups of size class:
1234567
S., S.
JJ
= S.J = S.J = S.J = Sj
=
S
j
= S8j
39 parameters
\.. -Log (likelihood) =741.26
I
/
Modell
"'
f*. f. s: size specific but not year specific
23 parameters
t-------------I
'" -Log (likelihood) = 1060.96
Figure 1. The flow chart of the models.
Model 3.1
f"': age specific but not year sJXX:ific
f: year specific and linearly related to age
s: year specific and 1 groups of size class:
1
S.
J
= s.J2 =
3
S.
J
4
= s.J
= S.J5 = s.J6 = s.J7'= S8j
33 parameters
\.. -Log (likelihood) = 753.41
Figure 2. The indirect recovery rate* of Pacific Halibut for size class 2 to 7.
I!l
Model 2.1
10
•
Modell
8
0
'S
c
0
>
6
0
u
~
u
4
-
2
...
.~
'"0
c:
OL...--_.&..-_....L.-_-'--_......L.._--'-_---L._---'_ _L.-_...I....--.J
o
2
4
6
8
Size class
* where the indirect recovery rate is the average through years
10
Figure 3. The survival rate* of Pacific Halibut for size class 1 to 8.
I!I
80
Model 3.5
Model 1
70
-E
cd
>
.~
a
60
50
L..--_.L..-_L..-_"'---_J..-_-'--_...L.._"""--_-1.-_"""---.1
o
2
4
6
Size class
* where the survival rate is the average through years
8
10
Figure 4. The survival rate* of Pacific Halibut during 1979 to 1984.
- a - Model 3.5
40
L..---..a.._-'--_'----J.._"""'"-----l'-----'-_-'------'_---L.._...........---l'-----'----l
1~78
1979
1980
1981
1982
1983
Year
* where survival rate is the average over age classes
1984
1985
Table 1: The size class classification of Pacific Halibut
Size class +
1
Length cm +
width cm
Age
*
Selectivity
Length cm
64-74
11
2
75-85
11
3
86-95
10
8
0.16
91.3
4
96-104
9
9
0.30
100.8
5
105-113
9
0.46
109.8
6
7
8
114-121
122-128
8
7
10
11
0.58
117.9
12
0.75
125.2
13
14
0.80
0.92
131.6
137.1
15
16
17
1.00
1.00
1.00
141.8
145.7
149.0
~129
* from age-length key
+ from size classes used in tag recovery models
*
Table 2.1 Recovery infonnation for Pacific Halibut in areas 2B, 2C and 3A
Size Class 8: 2129 cm
Year # Released
1979
334
572
1980
1817
1981
1982
1000
1416
1983
2066
1984
2116
1985
1295
1986
# Recovered
6
9
11
13
12
2
3
9
63
1
2
7
26
35
21
3
12
34
31
74
4
0
7
11
35
87
120
12
2
2
16
25
61
138
196
1
1
3
0
0
9
5
26
38
6
0
0
4
10
16
42
45
0
Size Class 7: 122-128 cm
Year # Released
1979
108
1980
188
1981
475
1982
258
1983
379
1984
603
1985
539
1986
365
# Recovered
1
4
1
1
3
0
0
6
15
0
0
2
9
4
2
8
10
14
1
Table 2.2 Recovery infonnation for Pacific Halibut in areas 2B, 2C and 3A
Size Class 6: 114-121 cm
Year # Released
117
1979
252
1980
1981
516
1982
351
447
1983
1984
881
752
1985
1986
510
# Recovered
2
4
1
3
7
1
1
6
14
1
5
10
0
3
11
0
2
4
0
1
1
1
8
1
8
22
1
14
21
48
15
29
43
52
0
6
Size Class 5: 105-113 em
Year # Released
1979
142
1980
313
1981
574
1982
380
1983
604
1984
1166
1985
1035
1986
702
# Recovered
1
5
3
. 3
2
13
0
8
6
2
0
4
9
11
4
2
6
8
15
28
0
0
2
4
10
20
56
6
1
5
12
10
35
67
68
0
•
Table 2.3 Recovery information for Pacific Halibut in areas 2B, 2C and 3A
Size Class 4: 96-104 cm
Year # Released
177
1979
1980
390
1981
515
512
1982
581
1983
1426
1984
1985
1157
1986
790
# Recovered
3
3
6
4
10
0
1
9
12
0
1
7
10
10
3
1
8
11
19
20
2
6
7
17
29
0
4
12
17
25
3
50
7
67
60
0
Size Class 3: 86-95 cm
Year # Released
1979
280
1980
583
1981
1982
1983
1984
1985
1986
584
676
674
1814
1423
980
# Recovered
1
8
6
5
11
5
2
1
1
1
9
6
3
11
14
10
2
27
22
4
3
10
20
24
5
2
10
10
70
13
8
17
24
67
67
3
Table 2.4 Recovery infonnation for Pacific Halibut in areas 2B, 2C and 3A
Size Class 2: 75-85 cm
Year # Released
445
1979
1032
1980
823
1981
1982
969
881
1983
2419
1984
2287
1985
1509
1986
# Recovered
5
10
2
8
18
0
5
6
6
3
5
3
17
13
13
11
14
0
39
20
1
1
15
9
29
29
57
11
4
10
18
37
25
78
76
3
•
Size Class 1: 64-74 cm
Year # Released
1979
482
1980
1530
1981
978
1982
820
1983
1228
1984
2405
1985
2217
1986
1459
# Recovered
1
2
1
3
5
0
3
7
5
0
3
12
2
3
3
4
12
13
17
10
0
4
15
16
17
17
24
4
4
11
14
24
24
46
18
0
•
Table 3a. The structure of cell probability under the assumption of Model 0
Pacific Halibut tagged and released as size class 8
Year of recovery
Year
tagged
1
1
2
f*
8~~
1
f*
2
2
5
4
3
7
6
8
888888£1 8888888f
8 8 ~8f 88888£1
8 1 82 83 45 81828384856 818283848586 7 81828384858687 8
8 88£1
8
1 23
8 88 88 8f
1 2 34
8~
8 88 8f
2 34
8 88 88 8f
2 3 45
8 8 8 8f
828384856
8283848586 7
f*
8:~
8 88 8f
3 45
8 888£1
8 8
3 4 56
8 8 8 8 7
3 4 5 6
83848586878
8:~
8 88 8f
4 56
8 88 88 8f
4 5 67
888
848586878
8~
s~s~
8 S 8f
85 86 8 7 S
f*
8:~
S
86 87 S
3
3
f*
4
4
{
5
6
88888f
8888f
6
888888f
8283848586878
88888f
sf
sf
8~~
fl*S
f*
7
7
8
Table 3b. The structure of cell probability under the assumption of Model 0
Pacific Halibut tagged and released as size class 7
Year of recovery
Year
tagged
1
2
3
4
5
6
7
1
2
3
4
5
6
7
8
note: The subscripts denote to year
The superscripts denote to size class
t.J * denotes to direct recovery rate for size class i in jth year
8~ denotes to survival rate for size class i in jth year
J
~
denotes to indirect recovery rate for size class i in jth year
8
Table 3c. The structure of cell probability under the assumption of Model 0
Pacific Halibut tagged and released as size class 6
Year of recovery
Year
4
7
6
8
tagged
1
2
1
f*1
6 7 6 7f
6 7 8f
6788f 67888f 678888'- 6788888'8 /2 8 1 82 g 8 18 28 g 4 81828g845 81828g8485 6 81828g8485867 81828g848586878
6 7
6788f
6 788 ~
6 7f
8 68 78888888~
8 68g78 8f
8 2 8g 4
828g8485867
828g84856
8
ig
2g456 8
45
2
2
67
6
78
8f
6
7
8f
67888£!
8 8 8
8 68 7f
838485867
83848586878
3 4 56
3
4
3 45
3
5
t.*
2
t.*
3
81
t*4
4
8~~
8 678f
8 8
4 5 67
848586878
8~
8 68 7f
5 67
8 68 78 8f
5 6 78
8~~
8 68 7f
6 78
67
(
5
6788f
8 68 7f
4 56
(
6
f*7
7
818
f*
8
8
Table 3d. The structure of cell probability under the assumption of Model 0
Pacific Halibut tagged and released as size class 5
Year of recovery
Year
tagged
1
2
1
f*1
8~(
3
4
5
5 6 7
5 6 7f
8 1 8ig 8 1 82 834
5 6 7
2 4
f*2 8~~ 8 81
f*3
8;~
2
3
f*4
4
5
6
7
5 6 7 8f 56788f
567888f 5678888f
818283845 8182838485 6 818283848586 7 818283848586878
8 58 68 71...
2 3 45
5 6 7 8f
828384856
5 6 7 8 ~
82838485867
567888£!
8283848586878
8 58 6i
3 45
8 58 68 7f
3 4 56
838485867
56781...
56788£!
83848586878
8~~
I!*5
8 58 6i
4 56
8 58 68 71...
4 5 67
5 6 7 sf
848586878
8:~
8 58 6t
5 67
5 6 7£!
85 8 6 8 7 S
6
f*6
7
8
note: The subscripts denote to year
The superscripts denote to size class
~ * denotes to direct recovery rate for size class i in jth year
8~ denotes to survival rate for size class i in jth year
J
~
8
denotes to indirect recovery rate for size class i in jth year
8~~
I!*7
8 58 6r
6 78
56
B
5*
fB
8l
Table 3e. The structure of cell probability under the assumption of Model 0
Pacific Halibut tagged and released as size class 4
Year of recovery
Year
tagged
2
1
4*
1
3
45f
8ft: 8 1 82 g
4*
f2
81s
4*
f3
fl
2
3
4
8
7
6
5
4
456788f.. 4567888f
4 5 6 7
4 5 6 7f.. 45678f
8 8 8 8 8 8 8 8
1 2 4 81828g845 81828g8485 6 81828g8485867 1 2 g 4 5 6 7
456788£1
4567f
456 7 ~
45f
8 456t
8 8
82 8g 4
8283848586878
8283848586 7
828384856
2 3 45
45678£1
4 5 6 7f..
8 48 5 8 6(
8 45f
8
8384858687 8
838485867
3 4 56
3 45
4*
4
456 71.
8 48 58 6r
8 8
f4
848586878
8ts
4 5 67
4 56
4*
8 48 58 6(
8 4 8 5f
f5
5 6 78
5 67
4*
8 48 5f
f6
6 78
4*
f7
8rs
4*
f8
8881
8~
5f
8~
5
8~
6
7
8
Table 3f. The structure of cell probability under the assumption of Model 0
Pacific Halibut tagged and released as size class 3
Year of recovery
Year
tagged
1
2
1
f*1
34
8 /2
3
8:8rs
f*2 813 34
f*3
2
3
4
4
6
5
8 38 48 5f
1 2 34
8:8~
8i~
f*4
8
7
71
3456 7 3 4 5 6
3456781.. 3456788f
f 8182838485 6 8182838485867
818283845
818283848586878
8 38 48 5f
2 3 45
3 4 5 6(
828g84856
3 4 5 6 7f..
82838485867
3456781.
8283848586878
ts
8 38 48 5f
3 4 56
g 4 5 6r
838485867
3 4 5 6 7f
83848586878
8~8~
8~
8 38 4 8 5f
4 5 67
3456(
848586878
8:8~
8 38 48 5f
5 6 78
8~~
8~81s
8i8
8~~
1.*5
5
(
6
1.*7
7
8
note: The subscripts denote to year
The superscripts denote to size class
f.J * denotes to direct recovery rate for size class i in jth year
8~ denotes to survival rate for size class i in jth year
J
f.
1
denotes to indirect recovery rate for size class i in jth year
4
813 8
f*8
Table 3g. The structure of cell probability under the assumption of Model 0
Pacific Halibut tagged and released as size class 2
Year of recovery
Year
tagged
1
2
1
~* 8~
i*2
2
4
3
81238134
8%
i*3
3
8 28 38 ~
1 2 34
23 4
82 4
81
8X
f*4
4
6
5
8
7
sf
4f
s 6t
sf
2 3t:
8 3 84 S
2 3 ~
8 3 84 8S 6
2 3 4
8g848S867
234S6t
8g848S86878
5ts
2 3£
5 4 5S 6
5 25 3 5 ~
4 S 67
2 3 4
545S86878
8~
8:8~
8~
8 28 g8 ~
S 6 78
'2 3 4
2 3 4 S 6 7f.. 234S678f
234
8 8 8 8 S 818283848S 6 818283848S867 818283848S86878
1 2 3 4
2 3 4 S 6
234
234S6i
2 3
8 8 8 8 6
8 2 83 84 S
8283848S86878
8283848S86 7
2 3 4 S
(
5
7f
sf
(
6
sf
8 28 gt:
6 78
C;
7
8ra
f*8
8
Table 3h. The structure of cell probability under the assumption of Model 0
Pacific Halibut tagged and released as size class 1
Year of recovery
Year
tagged
1
2
1
fl*1 8~
t*2
2
3
3
4
8:8%
8ts
t*g
5
1 2 34
8 1 82 4
81
8~8~
8~
1*4
4
6
7
1234f 1234S1 1234S6t
2 3 4 S 6 7(
8 1 8 8 8 S 8 8 8g8 8 6 81828g848S867 8 1828g848S8687
8
2 3 4
1 2
4 S
1
1 2 gt:
12 3 ~
12g4S6t
8 18283848S~
8 2 8g 84 S
8 8g 8 8 6
828g848S86878
2
4 S
2 g 4 S 6
ts
8i 8
8X
5
t*S
123£
8 3 8 4 8S 6
sf
12 3 ~
8 3 84 8S 8 6 7
1 2 g 4
8g848S86878
8 123£
8 8
4 S 67
123 ~
848S86878
8i8~
8~
8 18 28 ~
S 6 78
8~8~
8~
i.*6
6
7
8
note: The subscripts denote to year
The superscripts denote to size class
f.J *denotes to direct recovery rate for size class i in jth year
8~ denotes to survival rate for size class i in jth year
J
~
8
denotes to indirect recovery rate for size class i in jth year
t*7
8~8ra
8~
t*8
Table 4: The Estimations of identifiable Parameters(%) for Model 0
Size class: i ; Recovery year: j
Parameter
1
2
3
4
5
6
7
8
~*
0.07
0.20
0.31
0.00
0.00
0.00
0.00
0.08
~*
0.18
0.48
0.91
0.61
0.58
0.80
1.11
0.57
(
0.00
0.04
0.22
0.21
0.00
0.11
0.17
0.19
~*
0.24
0.00
0.30
0.52
0.66
0.22
0.53
1.48
~*
0.00
0.31
0.44
0.00
0.53
0.29
0.00
0.11
~*
0.00
0.00
0.34
0.00
0.00
0.19
0.00
0.11
~*
0.07
0.19
1.03
1.54
0.96
0.40
0.53
2.27
~*
0.21
1.12
0.36
1. 70
0.70
1. 71
0.93
1.80
1.48
2.75
4.58
5.02
5.72
7.05
8.68
1.35
2.86
4.11
5.13
5.69
8.22
7.48
4.34
1.09
3.06
3.88
5.75
4.52
5.59
1.33
1.56
1.65
3.67
3.98
5.50
7.59
0.63
2.09
3.43
2.32
4.28
4.31
4.01
0.71
2.84
5.20
3.73
5.70
5.53
3.56
6.31
2.84
3.80
4.58
5.99
6.12
7.24
8.70
si
6
67.30
84.60
80.03
72.61
89.13
71.51
75.42
70.78
si
56.73
82.45
83.36
77.95
82.22
74.50
72.04
71.77
si
84.21
100.0#
68.76
72.57
53.53
66.80
56.80
63.17
si
38.44
59.43
56.45
66.21
41.14
45.15
47.16
40.5.5
si
2
51.13
74.83
54.54
100.0#
94.44
66.23
56.91
51.80
si
55.85
68.21
52.59
34.10
56.98
48.07
47.00
72.94
~
~
~
~
~
~
it+1
S78
5
4
3
1
# Estimate is above 100%
Table 5: The parameter estimates(%) and standard errors(%) obtained from SURVIV for Modell
Size class: i ; Recovery year: j
Parameter
t
•J
j-l,..,8
t.J j=2,..,8
s} j=1,..,7
4
5
6
7
8
0.40(00.08)
0.33(00.08)
0.34(00.09)
0.38(00.11)
0.57(00.07)
2.90(00.24)
4.56(00.36)
5.21(00.41)
6.48(00.54)
7.40(00.68)
8.85(00.45)
71.27(04.93)
74.28(05.30)
72.54(05.56)
66.78(05.66)
62.62(04.26)
64.84(01.49)
1
2
3
0.09(00.03)
0.24(00.05)
0.48(00.08)
1.23(00.17)
56.46(04.07) 78.01(05.04)
Table 6: The parameter estimates(%) and standard errors(%) obtained from NUN for Model 2.2
Size class: i ; Recovery year: j
Parameter
1
2
3
(j=1,..,8
0.09(00.03)
0.24(00.05)
0.48(00.08)
1.46(00.35)
~
~
f6
~
~
f3
f2
s~ j=1,..,7
J
55.97{04.14)
4
5
6
7
8
0.40(00.08)
0.33(00.08)
0.34(00.09)
0.38(00.11)
0.57(00.07)
4.33(00.47)
6.41(00.64)
8.05(00.79)
10.60(01.03)
12.71(01.32)
16.57(00.97)
1.78(00.38)
3.06(00.41)
5.88(00.66)
6.30(00.69)
7.58(00.88)
9.11(01.10)
11.01(00.71)
1.46(00.48)
3.57(00.62)
5.87(00.74)
5.19(00.96)
6.72(00.88)
8.53(01.23)
8.56(00.65)
0.69(00.35)
1.39(00.33)
2.57(00.47)
3.56(00.61)
4.24(00.81)
4.23(00.88)
4.71 (00.51)
0.91(00.43)
1.14(00.32)
1.36(00.37)
3.22(00.69)
3.00(00.71)
4.64(01.05)
5.41(00.57)
0.58(00.28)
2.06(00.48)
2.97(00.74)
3.78(00.94)
5.85(01.43)
4.75(01.51)
4.54(00.83)
0.74(00.54)
2.88(00.98)
4.11 (0 1.53)
2.42(01.54)
5.21(02.37)
5.64(02.91)
4.90(01.34)
77.69(05.14)
68.75(04.38)
69.78(05.08)
67.67(05.24)
60.83{05.22)
59.49(04.05)
60.99(01.38)
Table 7: The parameter estimates(%) and standard errors(%) obtained from SURVIV for Model 2.1
Size class: i ; Recovery year: j
Parameter
1
2
3
~*j=1, ..,8
0.09(00.03)
0.24(00.05)
0.48(00.08)
1.55
3.93
6.31
8.69
11.07
13.45
15.83
1.82
3.33
4.83
6.33
7.84
9.34
10.84
2.32
3.39
4.46
5.53
6.61
7.68
8.75
0.87
1.58
2.28
2.99
3.70
4.41
5.11
0.55
1.29
2.04
2.78
3.53
4.27
5.02
0.90
1.72
2.54
3.37
4.19
5.01
5.84
1.49
2.20
2.91
3.61
4.32
5.03
5.74
~
~
~
~
~
~
~
s~ j=1,•.,7
4
0.40(00.08)
53.47(03.67) 79.29(03.83) 74.36(03.37) 64.51(03.00)
5
6
7
8
0.33(00.08)
0.34(00.09)
0.38(00.11)
0.57(00.07)
67.41(03.00) 60.66(02.85)
60.85(02.86) 61.33(01.13)
bOj
0.07(00.96)
-0.75(00.49)
-0.94(00.39)
-0.54(00.46)
0.17(00.64)
-1.18(00.48)
-3.21(00.52)
hI i
0.71(00.24)
0.82(00.13)
0.74(00.09)
0.71(00.10)
1.07(00.13)
1.50(00.11)
2.38(00.14)
where
t.J =W:i+b1j xi
Table 8: The parameter estimates(%) and standard errors(%) obtained from SURVIV for Model 3.1
Size class: i ; Recovery year: j
Parameter
1
2
3
4
5
6
7
8
(j=1,..,8
0.09(00.03)
0.24(00.05)
0.49(00.08)
0.40(00.08)
0.33(00.08)
0.34(00.09)
0.37(00.11)
0.56(00.07)
2.08
3.28
4.48
5.68
6.88
8.09
2.35
3.20
4.04
4.89
5.73
6.58
1.10
1.81
2.53
3.24
3.95
4.67
0.91
1.92
2.92
2.93
4.94
5.94
0.80
1.51
2.23
2.94
3.65
4.37
1. 79
2.54
3.30
4.06
4.81
5.57
0.81(00.19)
3.11(00.29)
4.55(00.37)
5.57(00.45)
7.26(00.54)
7.75(0.61)
~
~
{
~
~.
~
sX
k=1,..,8
s} i=1,..,8
58.03(06.22) 77.17(05.54) 43.85(02.76) 73.22(04 .. 05) 76.96(03.90) 68.14(03.74)
bOj
0.28(01.04)
-0.63(00.40)
-1.10(00.54)
-0.33(00.46)
0.66(00.52)
-0.32(00.40)
b1 j
0.76(00.27)
0.71(00.12)
1.01(00.13)
0.71(00.10)
0.85(00.10)
1.20(00.10)
where
t=bOj+bljxi
J
8.72(00.43)
Table 9: The parameter estimates(%) and standard errors(%) obtained from SURVIV for Model 3.2
Size class: i ; Recovery year: j
Parameter
1
2
3
4
5
6
7
8
~*j=1,..,8
0.09(00.03)
0.24(00.05)
0.49(00.08)
0.40(00.08)
0.33(00.08)
0.34(00.09)
0.37(00.11)
0.56(00.07)
2.72
3.70
4.69
5.67
6.65
7.63
2.94
3.62
4.31
5.00
5.68
6.37
1.04
1.81
2.57
3.34
4.11
4.87
0.87
1.87
2.87
3.87
4.87
5.86
0.95
1.57
2.18
2.79
3.40
4.01
1.83
2.62
3.41
4.20
4.98
5.77
3.59(0.36)
4.78(00.40)
5.34(0.44)
6.98(00.53)
7.64(00.61)
~
~
~
~
ig
~
s~ k=2,..,8
0.81(00.19)
sfa
s~ i=2,..,8
56.12(06.51) 82.31(06.39) 43.82(02.93) 69.97(04.04) 80.19(04.21) 70.53(04.00)
s~
56.57(12.95) 64.61(09.28) 53.43(10.77) 77.26(07.18) 46.64(07.18) 46.05(06.90)
J
bOj
0.26(01.16)
-0.27(00.45)
-1.12(00.54)
-0.49(00.48)
1.56(00.66)
0.76(00.54)
blj
0.79(00.30)
0.61(00.12)
1.00(00.14)
0.77(00.11)
0.69(00.12)
0.98(00.12)
where t.=~+blj xi
J
8.45(00.43)
Table 10: The parameter estimates(%) and standard errors(%) obtained from SURVIV for Model 3.3
Size class: i ; Recovery year: j
Parameter
1
2
3
4
5
6
7
8
~*j=1 •..•8
0.09(00.03)
0.24(00.05)
0.49(00.08)
0.40(00.08)
0.33(00.08)
0.34(00.09)
0.37(00.11)
0.56(00.07)
2.48
3.54
4.60
5.67
6.73
7.79
2.22
3.12
4.02
4.92
5.82
6.72
0.58
1.53
2.49
3.45
4.41
5.36
0.72
1.72
2.71
3.71
4.71
5.71
0.96
1.58
2.21
2.83
3.46
4.08
1.36
2.35
3.35
4.34
5.33
6.32
4.79(00.48)
5.25(00.45)
6.61(00.52)
7.56(00.62)
~
~
~
~
~
~
s~ k=3,u.8
s%
sts
0.35(00.37)
0.81(00.19)
s~ i=3....8
51.26(07.13) 79.82(07.10) 44.53(03.28) 62.73(04.01) 78.75(04.38) 70.52(04.24)
s~
83.53(15.00) 100.0(12.32)
s~
53.35(12.12) 48.54(08.41) 41.75(06.88) 82.13(13.60) 52.27(08.47) 48.35(07.60)
J
J
69.13(09.25) 100.0(10.33)
84.14(10.44) 68.87(07.91)
bOj
-0.62(01.16)
-0.29(00.52)
-1.28(00.54)
-1.34(00.42)
0.42(00.65)
0.35(00.6 3)
bl i
0.99(00.34)
0.62(00.14)
1.00(00.14)
0.96(00.11)
0.90(00.13)
1.06(00.14)
where
t.J =bOj+blj xi
8.69(00.46)
Table 11: The parameterestimates(%) and standard errors(%) obtained from SURVIV for Model 3.4
Size class: i ; Recovery year: j
Parameter
1
2
3
4
5
6
7
8
~*j=l,..,8
0.09(00.03)
0.24(00.05)
0.48(00.08)
0.40(00.08)
0.33(00.08)
0.34(00.09)
0.38(00.11)
0.56(00.07)
1.98
3.21
4.43
5.66
6.89
8.11
1.93
2.91
3.90
4.88
5.87
6.85
0.50
1.48
2.45
3.43
4.41
5.39
0.76
1.74
2.72
3.70
4.68
5.66
1.07
1.67
2.27
2.87
3.47
4.06
1.27
2.35
3.43
4.51
5.59
6.67
4.81(00.47)
6.55(00.53)
7.60(00.64)
~
~
~
~
~
~
s~ k=4,u,8
s~~
4.61(00.47)
3.41(00.36)
sra
si~
0.81(00.19)
s} i=4,u,8
47.18(07.53)
80.11 (07.92) 44.24(03.53) 60.88(04.24) 76.54(04.47)
67.38(04.32)
s~
68.99(15.00)
89.82(13.08)
81.41(08.49)
77.81(15.69)
82.78(13.23) 58.38(09.02) 98.21(12.62)
J
s?J
s?J
52.92(12.62) 47.46(08.24)
59.53(08.45) 80.84(09.27) 89.47(10.17)
80.68(10.59)
72.70(08.54)
39.71(06.50) 83.33(13.86) 54.64(08.94) 53.97(08.90)
bOj
-0.89(01.21)
-0.13(00.57)
-1.20(00.59)
-1.46(00.41)
-0.04(00.66)
-0.47(00.63)
b1 j
1.08(00.36)
0.60(00.15)
0.98(00.15)
0.98(00.11)
0.98(00.14)
1.22(00.15)
where
f.=bOj+blj xi
J
9.09(00.49)
Table 12: The parameter estimates(%) and standard errors(%) obtained from SURVIV for Model 3.5
Size class: i ; Recovery year: j
Parameter
1
2
3
4
5
6
7
8
lj=1....8
0.09(00.03)
0.24(00.05)
0.48(00.08)
0.40(00.08)
0.33(00.08)
0.34(00.09)
0.38(00.11)
0.56(00.07)
2.05
3.26
4.47
5.68
6.89
8.10
1. 79
2.82
3.84
4.87
5.90
6.93
0.41
1.44
2.46
3.49
4.51
5.54
0.81
1.77
2.73
3.70
4.66
5.63
1.10
1.73
2.36
2.99
3.62
4.25
1.38
2.39
3.41
4.42
5.44
6.45
6.68(00.60)
7.68(00.66)
J
~
~
~
~
~
~
s~ i=5•..•8
sra
4.82(00.48)
s~~
4.62(00.48)
srs
sfa
3.42(00.37)
0.81(00.19)
s~ i=5,...8
49.50(08.69) 75.46(08.38) 44.47(03.87) 56.04(04.26) 76.93(04.87) 68.18(04.69)
s~
51.91(15.26) 100.0(15.62)
s~
59.57(13.79) 69.56(11.92) 48.27(07.67) 85.48(11.19) 91.25( 11.07) 80.82(08.47)
s~
77.33(15.62) 81.95(13.06) 57.31(08.96) 99.07(12.72) 81.17(10.71) 72.06(08.56)
s~
52.14(12.46) 47.97(08.43)
bOj
-0.65(01.25)
-0.16(00.68)
-1.12(00.61)
-1.64(00.40)
-0.26(00.65)
-0.42(00.66)
bl i
1.01(00.36)
0.63(00.16)
0.96(00.15)
1.03(00.11)
1.03(00.14)
1.21(00.15)
J
J
J
J
where
64.21(08.94) 78.61(08.86) 70.75(07.37) 64.77(06.55)
39.52(06.47) 84.71(14.14) 55.85(09.19) 53.59(08.93)
i.J =bOj +blj xi
,
I
(
.
,
9.00(00.50)
Table 13: The parameter estimates(%) and standard errors(%) obtained from NLIN for Model 3.6
Size class: i; Recovery year: j
Parameter
~*j=1, ..,8
1
2
3
4
5
6
7
8
0.09(00.03)
0.24(00.05)
0.49(00.08)
0.40(00.08)
0.33(00.08)
0.34(00.09)
0.37(00.11)
0.56(00.07)
1.98
3.22
4.46
5.70
6.94
8.18
1.77
2.82
3.87
4.91
5.96
7.01
0.42
1.48
2.53
3.59
4.64
. 5.70
0.81
1.77
2.73
3.69
4.65
5.61
1.02
1. 70
2.38
3.07
3.75
4.43
1.45
2.40
3.34
4.28
5.23
6.17
~
~
~
~
~
~
s~ k=6,..,8
9.14(00.52)
s~~
6.69(00.60)
s1s
4.82(00.48)
s1s
s¥S
sis
4.62(00.4 7)
3.40(00.37)
0.81(00.19)
s~ i=6,..,8
J
51.09(10.18)
66.06(07.94) 45.44(04.09)
53.70(04.40)
78.55 (05 .34)
66.44(04.87)
s~
66.61(19.32) 100.0#(
--
) 45.11(05.78)
60.49(05.78)
76.51(07.85)
73.62(07.44)
s~
42.57(12.23) 100.0#(
--
) 59.86(09.62)
78.79(10.24)
78.70(09.13)
64.58(06.54)
59.60(12.88)
69.31(11.77) 48.10(07.62)
75.90(10.23)
91.48(11.03)
80.66(08.44)
77.31(15.67)
82.54(13.08) 57.85(08.99)
98.33(12.64)
81.25(10.71)
72.17(08.57)
J
J
s?J
s?J
s~
51.67(12.37) 49.91 (08.40)
39.48(06.47)
84.50(14.09)
56.03(09.25)
54.03(09.07)
bOj
b1j
-0.44(01.26)
0.94(00.36)
-0.34(00.59)
0.68(00.15)
-1.11(00.62)
0.96(00.15)
-1.69(00.40)
1.06(00.12)
-0.32(00.67)
1.05(00.14)
J
where
t= bOj+blj xi; # estimate is above 100%
J
-0.50(00.67)
1.:24(00.15)
7.20(00.69)
Table 14. The results of likelihood ratio test
Parameters
Log-likelihood
of SURVIV
Modell
23
-1060.96
2167.92
Model 2.1
30
Model 2.2
Model 3.1
Model 3.2
Model 3.3
65
33
-745.16
(-727.80)
1550.31
1585.60
-753.41
1572.82
1560.52
No. of
Model
Model 3.4
Model 3.5
Model 3.6
Model 0
39
45
-741.26
-721.73
51
-712.89
-703.11
(-698.94)
57
63
162
Ale
1533.47
1527.78
1520.23
1523.88
Likelihood ratio test:
-2Log-likelihood ratio
D.F.
34.69
35
Model 2.1 V.s. 2.2
Model 3.1 V.s. 3.2
Model 3.2 V.s. 3.3
24.30**
39.05**
6
6
Model 3.3 V.s. 3.4
17.68**
6
Model 3.4 V.s. 3.5
19.56**
6
Model 3.5 V.s. 3.6
Modell v.s.2.1
8.34
6
631.61 **
Model 2.1 V.s. 3.5
84.09**
7
27
The value inside () was estimated from NLIN.
AlC : Akaike Information Criterion
AlC= 2(No. of parameter - Log-likelihood)
•