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IMPROVING STATISTICAL CATCH-AT-AGE STOCK ASSESSMENTS
By

Michael J. Wilberg

A DISSERTATION
Submitted to
Michigan State University
in partial fulﬁllment of the requirements
for the degree of
DOCTOR OF PHILOSOPHY
Department of Fisheries and Wildlife

2005

ABSTRACT
IMPROVING STATISTICAL CATCH-AT-AGE STOCK ASSESSMENTS
By
Michael J. Wilberg

My dissertation addresses three objectives: 1) to estimate ﬁshing mortality rates
and abundance of yellow perch in southwestern Lake Michigan during 1986-2002 to
determine the contribution of ﬁshing to the collapse of yellow perch in southwestern
Lake Michigan, 2) to determine robust methods of dealing with time-varying ﬁshery
catchability within a statistical catch-at-age analysis (SCA) framework, and 3) to
determine whether using Bayesian model selection, speciﬁcally Deviance Information
Criterion (DIC) and an approximation of Bayes factors, results in using accurate models
for prediction of important ﬁsheries management quantities.

In chapter 1, I conducted an age-, size-, and sex-structured stock assessment of
yellow perch to estimate population size and examine historical trends in ﬁshing
mortality in Illinois and Wisconsin waters of southwestern Lake Michigan. Model
estimates indicated that yellow perch abundance in 2002 was less than 10% of 1986
abundance in Wisconsin and about 20% in Illinois. Annual mortality rates for females
age 4 and older averaged 69% during 1986-1996 in Wisconsin and 60% in Illinois during
1986-1997, which are quite high for a species like yellow perch that can live longer than
10 years. Estimated ﬁshing mortality rates on adult females during 1986-1996 exceeded
widely used reference points, suggesting that overﬁshing may have occurred. I believe
unsustainably high ﬁshing mortality rates were a substantial contributing cause of the

rapid decline of mature females in the mid-19903.

The relationship between ﬁshing mortality and ﬁshery effort (catchability) may
change over time through either density dependent or density independent processes. I
used Monte Carlo Simulations in chapter 2 to evaluate how different methods of
estimating ﬁshery catchability within an SCA model performed when models were
confronted with different data generating scenarios. I evaluated performance of the
estimation models by their accuracy and precision in determining quantities of interest
such as biomass in the last year. In many cases, including ﬁshery effort data in the
estimation model and allowing catchability to follow a random walk performed as well or
better than other methods. Exceptions were cases where ﬁshing mortality was low and
catchability trended over time. The estimation model that ignored ﬁshery effort data
performed well in cases with a good survey, but performance degraded as survey
precision decreased. White noise and density dependent estimation models performed
poorly in situations where catchability trended over time. No estimation model was best
for all underlying models of catchability, hence I recommend ﬁtting multiple SCA
models with alternative assumptions.

Structural ﬂaws in SCA models may cause considerable bias in model estimates
of mortality rates, abundance, and recruitment. I used simulations to evaluate whether
using Deviance Information Criterion (DIC) or approximate Bayes factors to select the
best SCA model provided more accurate estimates of quantities important for
management than using a single model in all cases. Using the model selected by DIC or
approximate Bayes factors resulted in estimates with lower mean square errors than using

any single model.

ACKNOWLEDGMENTS

I am indebted to many people for helping me during my doctoral program. I
especially thank my major advisor, Dr. Jim Bence, who was always available and helpful
whenever I had questions or needed guidance. Jim has been a superb mentor and it has
truly been a pleasure to work with him. I thank the rest of my graduate committee
members, Drs. Dan Hayes, Mike Jones, and Rob Tempelman, for providing invaluable
assistance. Emily Szalai, Mike Rutter, Norine Dobiesz, Brain Linton, and Wenjing Dai
provided programming support. Genny Nesslage provided valuable editing help,
encouragement, and general support that helped keep me relatively sane. Members of the
Lake Michigan Yellow Perch Task Group, particularly Brad Eggold, Dave Clapp, and
Dan Makauskas, and three anonymous reviewers provided helpful comments for chapter
1. Data used in chapter 1 were collected and provided by Ball State University, the
Illinois Department of Natural Resources, the Illinois Natural History Survey, the Indiana
Department of Natural Resources, and the Wisconsin Department of Natural Resources.
Dave Glover and John Dettmers provided information on yellow perch movement in
southern Lake Michigan. This research was supported in part by Michigan Sea Grant, the
US. Fish and Wildlife Service Great Lakes Fish and Wildlife Restoration program, the
Great Lakes Fishery Commission, Michigan Sea Grant, the International Association for
Great Lakes Research, and the Michigan State University College of Agriculture and
Natural Resources. The Bence/J ones lab provided encouragement, assistance, and much
needed distractions. Finally, I thank my family for unwavering and unconditional

support.

iv

TABLE OF CONTENTS

LIST OF TABLES ............................................................................................................ vii
LIST OF FIGURES ............................................................................................................ ix
INTRODUCTION ............................................................................................................... 1
Objective 1 ................................................................................................................... 4
Objective 2 ................................................................................................................... 4
Objective 3 ................................................................................................................... 5
References .................................................................................................................... 8
CHAPTER 1
YELLOW PERCH DYNAMICS IN SOUTHWESTERN LAKE MICHIGAN DURING
1986-2002 .......................................................................................................................... 10
Introduction ................................................................................................................ 10
Methods ...................................................................................................................... 12
Model Fitting .................................................................................................. 14
Sensitivity Analyses ....................................................................................... 17
Data ................................................................................................................ 17
Results ........................................................................................................................ 19
Model Fits ...................................................................................................... 19
Model Estimates ............................................................................................. 20
Sensitivity Analyses ....................................................................................... 23
Discussion .................................................................................................................. 24
Management Implications .............................................................................. 30
References .................................................................................................................. 32
CHAPTER 2
PERFORMANCE OF TIME-VARYING CATCHABILITY ESTIMATORS IN
STATISTICAL CATCH-AT-AGE ANALYSIS .............................................................. 49
Introduction ................................................................................................................ 49
Methods ...................................................................................................................... 50
Data Generating Model .................................................................................. 52
Estimation Model ........................................................................................... 54
Model Fitting and Convergence ..................................................................... 56
Evaluation of Estimation Model Performance ............................................... 57
Results ........................................................................................................................ 58
White Noise, First Order Autoregressive, and Density Dependent ............... 59
Linear Increase and Abrupt Change ............................................................... 60
Discussion .................................................................................................................. 62
References .................................................................................................................. 66

CHAPTER 3
PERFORMANCE OF BAYESIAN MODEL SELECTION IN STATISTICAL CATCH-

AT-AGE ANALYSIS ....................................................................................................... 83
Introduction ................................................................................................................ 83
Methods ...................................................................................................................... 85

Data-generating Model .................................................................................. 86
Estimation Model ........................................................................................... 89
Model Fitting and Convergence ..................................................................... 90
DIC Calculations ............................................................................................ 92
Approximate Bayes Factors ........................................................................... 93
Evaluation of Estimation Model Performance ............................................... 94
Results ........................................................................................................................ 95
Discussion .................................................................................................................. 97
References ................................................................................................................ 100

APPENDIX A ................................................................................................................. 113

Description of Yellow Perch Models ....................................................................... 113
Population Submodel ................................................................................... 113
Observation Submodel ................................................................................. 116
Likelihood Equations ................................................................................... 1 17

APPENDIX B ................................................................................................................. I32

vi

LIST OF TABLES

Table 1.1. Model estimates of yellow perch abundance-at—age (in thousands) during
1986-2002 in Illinois and Wisconsin waters of southwestern Lake Michigan. ........ 37

Table 1.2. Model estimates of abundance (N; 10008), biomass (B; 1000 kg), mean rate of
ﬁshing mortality for females age-4 and older (Ii:4+ females), and mean rate of ﬁshing

mortality for males age-4 and older (F44. males) for 2002 under three scenarios of
unreported commercial harvest in Wisconsin waters of southwestern Lake Michigan

during 1989-1992. ..................................................................................................... 39
Table 2.1. Symbols and descriptions of variables for data generating and estimation

models. ...................................................................................................................... 69
Table 2.2. Data generating and estimation model equations ............................................ 72

Table 2.3. Objective function equations for statistical catch-at-age analysis simulation
study. Equations T233 and T2.3.5 were only used in estimation models that
considered survey data. Equations T236 and T2.3.7 were only used in estimation
models that modeled ﬁshery catchability as white noise or a random walk
respectively. ............................................................................................................... 74

Table 2.4. Simulation results for statistical catch-at-age estimation model performance in
cases where data generating models included white noise catchability (WN), ﬁrst
order autoregressive catchability (AR), and density dependent catchability (DD).
Shown are median relative error (MRE) and median of the absolute values of
relative error (MARE) for estimated biomass in the last year (year 15) from four
statistical catch—at-age estimation models: white noise (WN), random walk (RW),
density dependent (power relationship; DD), and freely estimated F at maximum
selectivity (i.e., not ﬁtted to ﬁshery effort data; FF). Data generating models
included three levels of ﬁshing mortality (high [F=2M], medium [F=M], and low
[F =0.5M]), and 3 levels of survey precision (good [CV=25%], poor [CV=100%],
and no survey). Estimation models with the lowest MARE for each treatment are
indicated in bold. ....................................................................................................... 75

Table 2.5. Simulation results for statistical catch-at-age estimation model performance in
cases where data generating models included linearly increasing catchability (LI)
and an abrupt increase in catchability (AC). Shown are median relative error (MRE)
and median of the absolute values of relative error (MARE) for estimated biomass in
the last year (year 15) from four statistical catch-at-age estimation models: white
noise (WN), random walk (RW), density dependent (power relationship; DD), and
freely estimated F at maximum selectivity (i.e., not ﬁtted to ﬁshery effort data; FF).
Data generating models included three levels of ﬁshing mortality (high [F =2M],

vii

medium [F =M], and low [F =0.5M]), and 3 levels of survey precision (good
[CV=25%], poor [CV=100%], and no survey). Estimation models with the lowest
MARE for each treatment are indicated in bold. ...................................................... 77

Table 3.1. Symbols and descriptions of variables for data-generating and estimation
models. .................................................................................................................... 102

Table 3.2. Data-generating and estimation model equations. ........................................ 105

Table 3.3. Objective function equations for statistical catch-at-age analysis simulation
study. ....................................................................................................................... 107

Table 3.4. Mean relative error (MRE) and mean square error (MSE) of models selected
using deviance information criterion (DIC), approximate Bayes factors (ABF), only
white noise catchability estimation model (WN), only random walk catchability
estimation model (RW), and only using the estimation model that estimated ﬁshing

mortality for each year independent of effort (FF) .................................................. 108
Table A. 1. Symbols representing parameters, data, and calculated quantities for

assessment models ................................................................................................... 1 19
Table A2. Equations for population and observation submodels .................................. 123

Table A3. Speciﬁcation of terms for normal and lognorrnal negative log-likelihood
components (see equation A1). ............................................................................... 126

Table A.4. Results of sensitivity analyses of changes of weights of data sources in the
objective function for yellow perch catch-at-age models for Illinois and Wisconsin
waters of southwestern Lake Michigan. Differences from baseline estimates are
displayed as percentages. Baseline model estimates of abundance (N; 10003),

biomass (B; 1000 kg), mean ﬁshing mortality for females age-4 and older ( E4 +

females), and mean ﬁshing mortality for males age-4 and older ( E4 + males) for

2002 are displayed for comparison. In two cases the model’s parameter estimates
failed to converge to values that minimized the objective function and these are
denoted by NC. ........................................................................................................ 127

viii

LIST OF FIGURES

Figure 1.1. Map of Lake Michigan statistical districts with modeled areas shaded. WM
indicates Wisconsin waters, IL indicates Illinois waters, IN indicates Indiana waters,
and MM indicates Michigan waters. ......................................................................... 40

Figure 1.2. Model ﬁts to commercial yield (1000 kg), recreational harvest (10005), and
gill net survey catch—per-effort (CPE; number per 30.5 m) in Illinois and Wisconsin
waters of southwestern Lake Michigan during 1986-2002. Model predictions are
represented by solid lines and observed values are represented by dots. .................. 41

Figure 1.3. Mean age of yellow perch caught in gill net surveys in Illinois and Wisconsin
waters of southwestern Lake Michigan during 1986-2002. Lines represent model
predictions and dots represent observed values. ....................................................... 42

Figure 1.4. Mean length of yellow perch caught in the commercial and recreational
ﬁsheries and gill net surveys in Illinois and Wisconsin waters of southwestern Lake
Michigan during 1986-2002. Lines represent model predictions and dots represent
observed values. ........................................................................................................ 43

Figure 1.5. Model estimates of average instantaneous mortality rates for yellow perch
age—4 and older in Illinois and Wisconsin waters of southwestern Lake Michigan
during 1986-2002. ..................................................................................................... 44

Figure 1.6. Model estimates of yellow perch recruitment (10008) in Illinois and
Wisconsin waters of southwestern Lake Michigan for the 1984-2000 year-classes

and estimates of recruitment plotted against yellow perch spawning stock biomass
(SSB; 1000 kg). ......................................................................................................... 45

Figure 1.7. Estimated abundance (10003), biomass (1000 kg), and spawning stock
biomass (SSB; 1000 kg) of yellow perch age-2 and older in Illinois and Wisconsin
waters of southern Lake Michigan during 1986-2002. Error bars represent 95%
probability intervals (the Bayesian analog of conﬁdence intervals). ........................ 46

Figure 1.8. Model estimates of mean length at age 5 Illinois and Wisconsin waters of
southwestern Lake Michigan during 1986-2002. ...................................................... 47

Figure 1.9. Model estimates of selectivity of the commercial ﬁshery, recreational ﬁshery,
and survey in Illinois and Wisconsin during 1986-2002 ........................................... 48

Figure 2.1. Effort series used for high, medium, and low ﬁshing mortality rate scenarios
in the data generating models. The average ﬁshing mortality rates for fully selected
age classes were approximately 2M for the high scenario, M for the medium
scenario, and 0.5M for the low scenario .................................................................... 79

ix

Figure 2.2. Fishery and survey selectivity patterns used in the data generating model. .. 80

Figure 2.3. Relative performance of the estimation model that ignores ﬁshery effort
versus the random walk estimation model measured by the difference of median of
the absolute value of the relative errors (MARE). Positive values indicate that the
estimation model that ignored ﬁshery effort data had a larger MARE than the
random walk estimation model and vice versa. Data generating models are indicated
by the symbol shape: WN — white noise, AR — autoregressive, DD - density
dependent, LI — linear increase, and AC — abrupt change. Two letters identify each
treatment: the ﬁrst letter for level of ﬁshing mortality (L — low, M - medium, H -
high) and the second letter for level of survey quality (G — good, P - poor). ........... 81

Figure 2.4. Relative performance of white noise versus random walk estimation model
measured by the difference of median of the absolute value of the relative errors
(MARE). Positive values indicate that the white noise estimation model had a larger
MARE than the random walk estimation model and vice versa. Data generating
models are indicated by the symbol shape: WN - white noise, AR — autoregressive,
DD — density dependent, LI - linear increase, and AC — abrupt change. Two letters
identify each treatment: the ﬁrst letter for level of ﬁshing mortality (L — low, M -
medium, H — high) and the second letter for level of survey quality (G — good, P —
poor, N — none) .......................................................................................................... 82

Figure 3.1. Baseline effort series used in data-generating models. ................................ 109
Figure 3.2. Fishery and survey selectivity patterns used in data-generating models. ..... 110

Figure 3.3. Deviance Information Criterion (DIC) differences among models.
Differences from the best model for each data set are shown. Data-generating
models are indicated by WN for white noise catchability, LI for linear increase in
catchability, and UE for the case where observed effort data were uninforrnative.
Estimation model comparisons are indicated by X vs. Y (legend), where Y is the
hypothetical best estimation model for the scenario. Positive DIC differences
indicate that the model Y is better than model X. Points are randomly jittered to
reduce overlap. ........................................................................................................ 1 11

Figure 3.4. Box plots of relative error of estimates of biomass and average ﬁshing
mortality in year 15. The middle line indicates the median, the box indicates the
interquartile range, and the whiskers indicate the 95% quantile range. Estimation
methods are indicated by ABF for approximate Bayes factors, DIC for deviance
information criterion, FF for the estimation model that freely estimated F, RW for
the estimation model with random walk catchability, and WN for the estimation
model with white noise catchability. ....................................................................... 112

Figure A. 1. Estimated catchability coefﬁcients for Wisconsin and Illinois recreational
and commercial ﬁsheries in southwestern Lake Michigan during 1986-2002. ...... 131

INTRODUCTION

Fishery managers need realistic predictions of future population dynamics of
individual ﬁsh stocks and predictions of how these populations will respond to
management actions. Most major ﬁsheries are managed by a process where scientists
estimate population size and other parameters of a ﬁsh population (and uncertainty of
these estimates) and provide this information to ﬁshery managers who then make
decisions regarding which ﬁshery policies to implement (e. g., catch quotas, bag limits,
season or area closures). The process of estimating these quantities is called stock
assessment.

Relatively recent advances in ﬁsheries science have allowed researchers to
estimate total abundance from ﬁshery harvest and age or length composition data, and
other diverse data sources, with a method known as statistical catch-at-age analysis
(SCA; Foumier and Archibald 1982; Deriso et a1. 1985; Megrey 1989; Methot 1990).
This approach is preferable, in many cases, to other stock assessment methods because it
can incorporate many diverse data sources and allows for a rigorous statistical approach
(i.e., promotes explicit modeling of measurement and process error). Hence, SCA can
allow estimation of uncertainty associated with parameter estimates and other model
quantities. SCA methods are being applied worldwide for many ﬁsheries and
predominates applications to major marine ﬁsheries in the northwestern US, New
Zealand, Australia, and South Africa (Radomski et al. in press).

The basic idea behind SCA is that one can infer the effect of ﬁshing on a

population by estimating how absolute removals (e. g., ﬁshery harvest or yield) affect

relative abundance. A model is created that describes the population and the process of
removals, and this model is statistically ﬁt to data from a ﬁshery. Usually, one of the key
assumptions of these models, called the separability assumption, is that ﬁshing mortality
can be described by an overall year effect (how a certain amount of ﬁshing effort affects
a population) and an age effect (the relative vulnerability of different aged ﬁsh to a
ﬁshery). This basic approach has also been extended to species for which there is no
directed ﬁshery (Szalai 2003).

In southern Lake Michigan, yellow perch abundance has declined substantially
since the mid-19803 (Marsden and Robillard 2004). As the abundance of yellow perch
declined during the mid to late 19903, commercial ﬁsheries in Indiana, Illinois, and
southern Wisconsin were restricted to smaller quotas, and were eventually closed during
1996-1997 (Francis et al. 1996). Stricter regulations were also imposed on the
recreational ﬁshery with reductions in daily bag limits implemented in all states during
1996-1998, the incorporation of a slot size limit (i.e., only ﬁsh between 8 and 10in could
be kept) in Illinois during 1997-2000, and seasonal closures of the ﬁshery (Marsden and
Robillard 2004). Reproductive failure has been implicated as the primary cause of the
population collapse, but the role of ﬁshing in the collapse has not previously been
rigorously investigated. My research investigates the role of ﬁshing in the decline of
yellow perch in southwestern Lake Michigan by using SCA models.

A frequent (but somewhat outdated) criticism of SCAs is that they do not allow
for the ﬂexibility to accurately model time-varying ﬁshing mortality at age (NRC 1998).
Speciﬁcally, the relationship between ﬁshery effort and ﬁshing mortality or the age-based

vulnerability to the ﬁshery may change over time (Butterworth et al. 2002; Radomski et

al. in press). Many methods have been developed to account for these changes over time
(e. g., Fournier and Archibald 1982; Fournier 1983; Methot 1990; Hampton and Fournier
2001; Butterworth et al. 2003), but there is not consensus on which methods are best
when faced with different underlying mechanisms for change. My research also aims to
evaluate performance of several SCA methods under many situations and to develop
guidelines to help researchers decide among several SCA model structures.

My dissertation addresses three objectives: 1) to estimate ﬁshing mortality rates
and abundance of yellow perch in southwestern Lake Michigan during 1986-2002 to
determine the contribution of ﬁshing to the collapse of yellow perch in southwestern
Lake Michigan, 2) to determine robust methods of modeling time-varying catchability
within an SCA framework, and 3) to determine whether using Bayesian model selection,
speciﬁcally Deviance Information Criterion (DIC) and an approximation of Bayes
factors, results in choosing models with accurate estimates of ﬁshing mortality rates and
abundance. These objectives arose out of questions that formed during my research
program, and each chapter of my dissertation addresses an objective. Chapter 1
developed from a management need to evaluate the importance of ﬁshing in the
population declines of yellow perch in southern Lake Michigan. In working on chapter 1,
I found that many of the model parameters were likely time-varying and wanted to
determine whether the approaches I used (or alternatives) were robust methods for
modeling these processes. This led to the evaluation of several methods for incorporating
time-varying catchability into SCA models detailed in chapter 2. Based on the results of
chapter 2 (differential performance of SCA models under different data-generatin g

scenarios), the question arose as to whether statistical model selection techniques could

be used to select “good” (i.e., accurate) models when one does not know the true
underlying pattern or mechanism of change.
Objective 1

In chapter 1, I detail my assessment model and describe results from the modeling
efforts. I developed a length-, age-, and sex-based SCA model to estimate ﬁshing
mortality rates and abundance, to determine if ﬁshing mortality rates exceeded the
maximum that could be supported, and to integrate diverse sources of data to get the best
estimates of recruitment and population size. My model allowed ﬁshing mortality rates
at age and sex to change over time 1) in response to changes in ﬁshery effort, 2) by
allowing ﬁshery catchability to change according to random walk models, and 3) by
modeling ﬁshery selectivity as a function of length and allowing growth to change over
time to match observed changes in size at age and sex in southern Lake Michigan.

Model estimates of catchability of the recreational ﬁshery changed approximately
ﬁve-fold during 1986-2002, and commercial ﬁshery catchability changed approximately
four- and eight-fold in Illinois and Wisconsin respectively. However, ﬁshing mortality
rates changed approximately 15-fold for females and eight-fold for males during 1986-
2002, indicating that changes in effort and catchability were both important to changes in
ﬁshing mortality. This leads to questions of whether modeling ﬁshery catchability as a
random walk, as in this application, is the best approach and whether ﬁshery effort data
should be used at all (because of the extreme changes in ﬁshery catchability).

Objective 2
Many SCAs of ﬁsh stocks assume that ﬁshing mortality is directly proportional to

ﬁshing effort (i.e., constant catchability). However, ﬁshery catchability has often

changed in response to changes in population abundance (e. g., Peterman and Steer 1981),
environmental conditions (e. g., Ziegler et al. 2003), or changes in ﬁshing gear or
ﬁsherman experience (e. g., Hilbom and Walters 1992 pp. 126, 130). Likewise,
catchability in yellow perch ﬁsheries in southwestern Lake Michigan has changed
substantially over time, perhaps due to a combination of the factors listed above.
Therefore, my second chapter describes an evaluation of several methods of modeling
time-varying catchability within SCA models. I used Monte Carlo simulations to
compare how four different methods of estimating ﬁshery catchability within an SCA
model performed when models were confronted with ﬁve different data generating
scenarios.

In many cases, including ﬁshery effort data in the estimation model and allowing
catchability to follow a random walk performed better than ignoring ﬁshery effort data.
Exceptions were cases where ﬁshing mortality was low and catchability trended over
time. The estimation model that ignored ﬁshery effort data performed well in cases with
a good survey, but performance degraded as survey precision decreased. White noise and
density dependent estimation models performed poorly in situations where catchability
trended over time. No estimation model was best for all underlying models of
catchability.

Objective 3

Structural ﬂaws in SCA models may cause considerable bias in model estimates
of mortality rates, abundance, and recruitment (McAllister and Kirchner 2002). Often
researchers will make ad hoc decisions about model structure that may cause substantial

biases in their ensuing model estimates (Bumham and Anderson 2002; Gavaris and

Ianelli 2002). Given that a wide variety of models can potentially describe dynamics of a
given stock, methods to decide among several SCA models are needed. Helu et al.
(2000) evaluated performance of Akaike’s Information Criterion (AIC; Akaike 1973) and
Schwartz’s Bayesian Information Criterion (BIC; Schwartz 1978) in SCA models and
found that AIC and BIC both performed well by selecting the candidate model that was
the same as the data-generating model in most of their scenarios. Unfortunately,
although AIC or BIC may perform well in some cases, their implementation is
problematic when models differ in their random effects or hierarchical structures because
the number of parameters in these models is not easy to determine (Bumham and
Anderson 2002). Therefore, to be able to compare structurally complex SCA models
requires alternative model selection approaches that can account for random effects and
priors on parameters. DIC and Bayes factors are generally considered practical methods
to choose the best model from a set of candidate models and do not require the user to
specify the number of model parameters. However, performance of model selection
using DIC and Bayes factors has not been evaluated for SCA models.

My third chapter evaluates whether using DIC or an approximation of Bayes
factors results in choosing accurate models from the set of candidate models.
Speciﬁcally, I was interested in whether Bayesian model selection could determine an
appropriate model structure for time varying catchability because catchability is one of
the most important parameters in SCA models (scales abundance relative to catch),
catchability varied widely over time in yellow perch ﬁsheries in southwestern Lake
Michigan, and the accuracy of different structural forms of SCA models differs

depending on the underlying true changes in catchability, quality of data, and average

ﬁshing mortality rate. To achieve these objectives, I designed a simulation study and
challenged the model selection criteria with three estimation models, which differed in
how catchability was allowed to vary over time, and three scenarios of data accuracy and
time-varying catchability. I evaluated whether using DIC and approximate Bayes factors
to select among SCA model variants provided more accurate estimates of quantities used

for management than an approach of using a single model structure in all cases.

References

Akaike, H. 1973. Information theory as an extension of the maximum likelihood
principle. In B. N. Petrov and F. Csaki eds. Second International Symposium on
Information Theory 267-81. Budapest: Akademiai Kiado.

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environmental, physiological and density-dependent processes. Fisheries
Research 61: 107-123.

CHAPTER 1
YELLOW PERCH DYNAMICS IN SOUTHWESTERN LAKE MICHIGAN DURING

1986-2002

Introduction

Yellow perch Perca ﬂavescens is an ecologically and economically important
species in Lake Michigan (Wells and McLain 1972). Yellow perch are native to Lake
Michigan, play an important role in near-shore energy cycling and transfer (Evans 1986),
and have provided a ﬁshery on Lake Michigan since the late 18003 (Wells and McLain
1972; Wells 1977). Yellow perch is the only native species in Lake Michigan that has
supported a commercial ﬁshery continuously during the last century (Baldwin et al.
1979), although the ﬁshery has only continued in Green Bay since 1998. During the
19803 and 19903, the recreational ﬁshery harvested more yellow perch than any other
species in Lake Michigan (Bence and Smith 1999).

In southern Lake Michigan, yellow perch abundance underwent periodic
ﬂuctuations during 1934-1964, and declined greatly during the 19603 (Francis et a1.
1996). The decline in yellow perch abundance in the 19603 coincided with a large
increase in alewife Alosa pseudoharengus abundance, and therefore alewife interference
with yellow perch reproduction (either through competition or predation) was considered
the primary cause of the decline (Wells 1977). However, exploitation was also
considered a contributing factor to the overall decline and the primary cause of the
decline of adults (Wells 1977). Prior to 1969, all the states bordering Lake Michigan

(Indiana, Illinois, Michigan, and Wisconsin) had commercial ﬁsheries for yellow perch

10

(Baldwin et al. 1979). In 1969, the state of Michigan was the ﬁrst to close their
commercial ﬁshery (Wells 1977). During the 19703, yellow perch populations in
southern Lake Michigan began to recover (Wells and Jorgenson 1983), and abundance
was high during the 19803 with strong year-classes in 1980 and 1983-1988 (Jude and
Tesar 1985; Makauskas and Clapp 2000). Abundance declined to low levels during the
19903 with a series of weak year-classes during 1989-1997 and 1999-2000. As yellow
perch abundance declined, the sex ratio became skewed toward males, which may have
been caused by intense ﬁshing mortality targeted on large females (Madenjian et al.
2002). The selective removal of large females may have led to further declines in yellow
perch recruitment. As the abundance of yellow perch declined in southern Lake
Michigan during the mid to late 19903, commercial ﬁsheries in Indiana, Illinois, and
southern Wisconsin were restricted to smaller quotas (Francis et a1. 1996), and were
eventually closed during 1996-1997; these ﬁsheries remain closed. Stricter regulations
were also imposed on the recreational ﬁshery with reductions in daily bag limits
implemented in all states during 1996-1998, the incorporation of a slot size limit in
Illinois during 1997-2000, and seasonal closures of the ﬁshery (Francis et al. 1996).
Reproductive failure has been implicated as the primary cause of the population collapse
(Francis et al. 1996; Heyer et al. 2001; Marsden and Robillard 2004), but the role of
ﬁshing in the collapse has not been rigorously investigated.

Our objectives were to estimate ﬁshing mortality rates and abundance of yellow
perch in Wisconsin and Illinois waters of southwestern Lake Michigan during 1986-2002
to determine the contribution of ﬁshing to the collapse of yellow perch in southern Lake

Michigan. We also wanted to determine if ﬁshing mortality rates exceeded the maximum

11

that could be supported, and to integrate diverse sources of data to get the best estimates
of recruitment and population size. Our approach was to ﬁt age-, size-, and sex-
structured population models to ﬁshery and survey data. No previous population model-
based stock assessments have been conducted for yellow perch in southern Lake
Michigan. Similar age-structured assessments have been applied to lake trout Salvelinus
namaycush (Sitar et al. 1999), lake Whiteﬁsh Coregonus clupeaformis (Ebener et al. in
press), walleye Sander vitreus (Deriso et al. 1988), and yellow perch (Lake Erie Yellow
Perch Task Group 2001) in other areas of the Great Lakes.
Methods

We implemented statistical catch-at-age models (detailed description in Appendix
A) for yellow perch in southwestern Lake Michigan (Figure 1.1). Statistical catch-at-age
models are age-structured models that follow cohorts of ﬁsh over time and consider the
catch-at-age data to be measured with error (Megrey 1989). Such models consist of
population and observation submodels, where the model parameters are estimated by
ﬁtting the models to data (Megrey 1989). Our assessment models contained annual time
intervals and considered the period from 1986 to 2002, and ages 2 through 9 (age 9 was
an aggregate age class that included all ﬁsh age 9 and older). We began our models in
1986 because recreational ﬁshery data were not available for earlier years. During model
development, we tested the effect of sequentially changing the aggregate age class lower
(down to age 6) and results were similar to those we report. Our models also contained
two ﬁsheries, recreational and commercial, and a ﬁshery independent gillnet survey. Our
models produced estimates of ﬁshing mortality rates, abundance, biomass, and spawning

stock biomass (SSB). We deﬁned SSB as the biomass of mature females in the

12

population and calculated this based on a length-based maturation curve derived outside
our model ﬁtting process (see Appendix A).

Our assessment model was age-, size-, and sex-structured. In statistical catch-at-
age models, relative vulnerability (i.e., selectivity) to the ﬁsheries is usually modeled as a
time-invariant function of age (Quinn and Deriso 1999). However, this assumption does
not appear to be reasonable for yellow perch, because these ﬁsheries are highly size
selective (Kraft and Johnson 1992) and yellow perch size-at-age has changed
substantially over time (Marsden and Robillard 2004). Also, yellow perch show sexually
dimorphic growth, with females growing faster and to larger sizes than males, which is
suspected to cause higher ﬁshing mortality rates for females (Wells and Jorgenson 1983;
Madenjian et al. 2002). We modeled selectivity of the ﬁsheries and surveys as functions
of length and allowed growth to change over time with a time-varying von Bertalanffy
growth model (Szalai et a1. 2003). We accounted for temporal variations in growth by
allowing the von Bertalanffy parameters to change in accord with random walk
submodels (see Appendix A). Our approach allowed the relative vulnerability of
different age-sex categories of yellow perch to change over time as their mean length-at-
age changed, even though relative vulnerability was a constant function of length that did
not differ between the sexes (Methot 1990; Hampton and Fournier 2001). We also
included a different selectivity pattern to capture changes in recreational ﬁshery
selectivity during 1997-2000 when a slot size limit was implemented in Illinois. We
assumed a time-, sex-, and age-invariant natural mortality rate, M, of 0.37, which was

consistent with estimates of M for yellow perch in Indiana waters of southern Lake

13

Michigan (Allen 2000) and with values used for stock assessments of yellow perch in
Lake Erie (lake Erie Yellow Perch Task Group 2001).

As well as allowing for changes in the relative vulnerability of different ages in
response to changes in growth, our model allowed for temporal changes in the
vulnerability of the most selected size of yellow perch, so that the ﬁshing mortality
imposed by a given amount of ﬁshing effort could change over time. As for the growth
model this was done by having ﬁshery catchability parameters vary according to random
walk models (see Appendix A).

Genetic analyses have found that yellow perch in the southern basin of Lake
Michigan form a single genetic stock (Miller 2003). However, our approach implicitly
assumed that there was no net migration for either of the model areas (Illinois, and
Wisconsin WM-4 to WM-6; Figure 1.1). We believe this assumption is a reasonable
approximation because preliminary tagging data suggest that the median dispersal
distance for adult yellow perch in southwestern Lake Michigan was relatively low ( < 30
km; D. Glover, University of Illinois at Urbana-Champaign, personal communication).
Also, Horns (2001) attributed differences in growth patterns among yellow perch stocks
in southern Lake Michigan to geographic segregation. Evidence from physical current
modeling studies suggests that genetic structure of the yellow perch population of
southern Lake Michigan may be caused by mixing during the larval stage (Beletsky et al.
2004).

Model Fitting
We took a Bayesian approach to obtain posterior probability estimates for the

parameter values and quantities of interest such as ﬁshing mortality rates, abundance,

14

biomass, and SSB. We ﬁtted our models to commercial yield, recreational harvest,
commercial length frequency, recreational length frequency, commercial effort,
recreational effort, mean length-at-age in the survey, age composition of the survey by
sex, total survey CPE by sex, and survey length composition by sex. The objective
function contained 11 additive components for the Wisconsin model and 12 additive
components for the Illinois model (Appendix A). Each component represented a type of
data or a speciﬁed informative distribution (i.e., prior distribution) for parameters.
Variations in catchability and growth model parameters according to random walks were
included as components. We estimated 149 parameters for the Wisconsin model and 151
parameters for the Illinois model. We used Markov Chain Monte Carlo (MCMC)
simulations with a Metropolis-Hastings algorithm to estimate posterior probability
intervals (the Bayesian analog of conﬁdence intervals) of several model parameters and
estimates (Otter Research Limited 2000). We ran the MCMC chain for 2,000,000 steps,
sampling every 250 steps, and discarded samples from the initial 250,000 steps as a burn
in period, which reduces the effect of starting values on the MCMC results (Gelman et al.
2004). We determined that the length of our burn in period was long enough by
separating the MCMC chains (of the objective function) into several smaller chains and
comparing the distributions of these blocks (Gelman et al. 2004); the distribution of each
block was nearly identical to the other blocks.

We assumed that total catch for all ﬁsheries was median-unbiased, and that the
coefﬁcient of variation (CV) of the catches was constant for each ﬁshery (i.e., we
assumed lognormal errors). We set the CV for the commercial ﬁshery by assuming that

recorded yield was accurate to within approximately 10% in Illinois and 20% Wisconsin

15

95% of the time. The CV for the recreational ﬁshery was set to approximately 10%
based on estimates of the CV from the Wisconsin recreational ﬁshery during 1998-2001
(Wisconsin Department of Natural Resources [WDNR], unpublished data). Independent
estimates for the CV of the Illinois recreational ﬁshery were not available. The CV3 of
survey CPEs and effective sample sizes of the age and length compositions of the surveys
and recreational and commercial ﬁsheries were estimated using an iterative approach
where we adjusted the assumed initial CVs and effective sample sizes of the objective
function components to match the residual variance (McAllister and Ianelli 1997).
Effective sample sizes for survey age composition determined by otoliths or anal ﬁn
spines were weighted ﬁve times higher than those determined by scales because scale
aging is thought to be a less accurate method of aging yellow perch (Baker and
McComish 1998; Robillard and Marsden 1996; Wisconsin Department of Natural
Resources, unpublished data). For the Illinois model, we set the CVs of the random walk
deviations for commercial and recreational catchability to about 25%. For the Wisconsin
model, we used the same CV for recreational ﬁshery catchability, but used a higher CV
of about 40% for commercial catchability because, based on the large amounts of
unreported catch, we thought the commercial effort data were less accurate for Wisconsin
than for Illinois. For the Wisconsin model, we set the CVs to about 5% for the random
walk deviations for the L,o and K parameters of the growth model because mean length-
at-age of the older age groups rarely changed rapidly from year to year. In contrast, we
set the CV of the random walk deviations for mean length-at-age 2 to 10% because mean
length-at-age 2 showed more variation from year to year than older ages. Using the same

CV values for L00 and K in the Illinois model as in the Wisconsin model resulted in poor

16

convergence. Therefore, we set the CV3 on Loo and K to about 2.5% to further constrain
the growth model for Illinois, but the CV for deviations in mean length-at-age 2 was the
same as the Wisconsin model.
Sensitivity Analyses

We performed sensitivity analyses to determine the effects of some of our
assumptions on the results of the analysis. To test the sensitivity of the model estimates
to the weighting factors for each data source, we increased and decreased the weighting
factors for each data source ﬁve-fold and reﬁt the models. We also tested the sensitivity
of our estimates to our assumed value of M by increasing and decreasing M by 20% and
reﬁtting our models. We then evaluated sensitivity of the model estimates to the change
by comparing model estimates of abundance, biomass, and mean ﬁshing mortality rates
for females and males age 4 and older in 2002 to those obtained with the baseline
weighting factors and natural mortality rate. Also, because of large suspected amounts of
unreported commercial harvest in Wisconsin during 1989-1992, we tested the effects of
three levels (one to three times the reported amount) of commercial harvest during those
years on our results.
Data

Commercial yield and effort were estimated from mandatory bimonthly reports
submitted by commercial ﬁshermen. In some cases, these reports were validated by law
enforcement ofﬁcials, but underreporting may have been a large problem, especially in
Wisconsin. The exact magnitude of underreporting is unknown, but during 1990-1992
commercial yield in Wisconsin was underreported by at least 44%, which law

enforcement ofﬁcials documented during a multi-year sting operation (W DNR,

l7

unpublished data). Two commercial ﬁshermen indicted for unreported harvest testiﬁed
that unreported harvest was two to three times reported harvest. Wisconsin implemented
a commercial quota for yellow perch in the summer of 1989, so there was less incentive
for commercial ﬁshermen to underreport prior to 1989. For observed commercial yield
in Wisconsin during 1989-1992, we added the reported commercial yield and the veriﬁed
illegal yield and multiplied the number by two. In Illinois, unreported commercial
harvest was thought to be relatively low (Illinois Department of Natural Resources
[IDNR], unpublished data). Length frequency estimates of the commercial catch were
collected by dockside monitoring. Sampling did not occur for most lifts.

Creel surveys were conducted by the Wisconsin DNR and the Illinois DNR to
estimate recreational ﬁshery harvest, effort, and composition of the harvest (details in
Austen et a1. 1995). Creel clerks visited access points and interviewed anglers to
determine target species and angler effort. Anglers’ catches were examined for species
composition and length frequency.

Graded-mesh gillnet surveys were conducted in Wisconsin (2.54-7.62 cm stretch-
measure with 0.64 cm increments) in the winter and in Illinois (254-889 cm stretch-
measure with 1.27 cm increments) in June of each year to obtain ﬁshery independent
relative abundance data. Nets were set overnight in the same locations each year at
multiple depths. CPE was measured as the number of yellow perch per 30.5 m gillnet.

The length of each ﬁsh was measured, and the age composition of the catch was
estimated by estimating ages for a randomly chosen subsample and applying the
subsequent age-length key to the length frequency. Ages were estimated by counting the

annuli on scales during 1986-1999 in Wisconsin and 1986-1993 in Illinois. However,

18

this method was found to be fairly unreliable (Robillard and Marsden 1996; Baker and
McComish 1998; WDNR, unpublished data). Therefore, Illinois estimated ages of ﬁsh
by counting annuli in otoliths during 1994-2002, and Wisconsin estimated ages of ﬁsh by
counting the annuli in anal ﬁn spines during 2000-2002. Ages estimated by different
readers of spines and otoliths agreed 86% of the time (W DNR, unpublished data).
Rams
Model Fits

Most of our data sources contained relatively large amounts of contrast and our
models produced reasonable ﬁts to all data sources. Fishery and survey catch was
relatively high in the beginning of our time series and decreased to low levels during the
mid 19903. Our models predicted observed commercial yield and recreational harvest
within 5% of observed values in most years (Figure 1.2). For total survey CPE, our
models produced the same declining trend as was observed, but produced lower
predictions of survey CPE than was observed in most years prior to 1991 (Figure 1.2).
This may be due to decreases in survey catchability caused by increases in water clarity
since the colonization of Lake Michigan by zebra mussels Dreissena polymorpha.
Relative differences between observed and predicted survey CPE tended to be larger than
ﬁshery catch residuals (especially for the Wisconsin survey); this result is not surprising
given that survey CPE had relatively high CV3 and that CV3 were higher for the
Wisconsin survey than for the Illinois survey. Mean age in the survey was relatively
stable during 1986-1992, increased during 1992-1997, and decreased thereafter (Figure
1.3). Deviations between model predictions and observations of mean age in the survey

were usually less than 15%. Mean length in the recreational ﬁshery and surveys

19

increased during 1986-2002, but did not show a trend for commercial ﬁsheries (Figure
1.4). Predicted mean length was usually within 10% of the observed value for the
commercial ﬁshery and surveys and within 5% of observed values for the recreational
ﬁshery (Figure 1.4). Predicted mean length of females in the Illinois survey during 1986-
1992 was lower than observed values and may be low because the survey mainly targets
mature ﬁsh; after 1990, a smaller proportion of females were immature.
Model Estimates

Model estimates of mortality rates were generally higher for females than males,
and were higher during the mid-19803 through the mid-19903 than in the late 19903 and
after (Figure 1.5). In Wisconsin, the commercial ﬁshery was the predominant source of
ﬁshing mortality until the commercial ﬁshery was closed, and in Illinois, the recreational
ﬁshery was the predominant source of ﬁshing mortality. Estimated instantaneous ﬁshing
mortality rates for females age 4 and older exceeded 1.0 in most modeled years prior to
1996 in Wisconsin waters and averaged 1.16, which corresponds to an annual mortality
rate of about 69%. In Illinois, estimated ﬁshing mortality rates were not as high as in
Wisconsin, although total mortality rates averaged about 0.92 (annual mortality rate of
about 60%) for females age 4 and older during 1986-1997. In Wisconsin during 1986-
1996, instantaneous total mortality rates for males age 4 and older averaged 0.67 (annual
mortality rate of about 49%), and in Illinois during 1986-1997, instantaneous total
mortality rates averaged 0.57 (annual mortality rate of about 44%). Until severe
restrictions were placed on commercial and recreational ﬁsheries (1996-1997), ﬁshing
was the predominant source of mortality for female yellow perch age 4 and older in

Wisconsin and Illinois. After the ﬁsheries were considerably restricted in 1996 in

20

Wisconsin and 1997 in Illinois, ﬁshing mortality rates declined substantially and natural
mortality was the predominant source of mortality.

Model estimates of recruitment in Illinois and Wisconsin showed similar patterns,
with recruitment generally higher in Illinois than in Wisconsin (Table 1.1; Figure 1.6).
Recruitment was relatively high during 1984-1989 and was substantially lower than
19803 levels thereafter, except for the 1998 year-class. The largest year-class during the
19803 was in 1988 and the largest year-class during the 19903 was in 1998. Model
estimates of average recruitment of the 1984-1989 year-classes were 13 times higher in
Illinois and 23 times higher in Wisconsin than the estimated average recruitment of the
1990-1997 year-classes. Recruitment was not strongly related to stock size and yellow
perch produced weak year-classes across a wide range of stock size (Figure 1.6).

Estimated abundance of yellow perch in Wisconsin waters of southwestern Lake
Michigan increased from 1986 to 1990, and then decreased from 1991 to 2002 except for
a small increase in 2000 (Figure 1.7). Estimated abundance of yellow perch in Illinois
waters declined from 1986 to 2002, except during 1990 and 2000. In 2002, yellow perch
abundance was approximately 8% of 1986 abundance in Wisconsin and approximately
20% of 1986 abundance in Illinois. Model estimates of relatively high abundance
throughout the 19803 resulted from high estimated recruitment during that period.
Abundance decreased drastically during the 19903 because recruitment declined and
ﬁshing mortality rates were relatively high.

Changes in estimated biomass were smaller than changes in abundance; estimated
biomass in 2002 was approximately 74% of 1986 biomass in Wisconsin and 123% of

1986 biomass in Illinois (Figure 1.7). Estimated biomass showed somewhat different

21

trends over time than abundance because the age structure of the population changed and
growth rates increased. In 1986, the population was composed of mostly age-2 and 3
yellow perch. In 2002, the majority of the population was age-4 and substantially larger
at a given age due to faster growth.

Patterns of estimated SSB were similar to patterns of biomass (Figure 1.7).
Model estimates of SSB increased during 1986-1992 in Illinois and during 1986-1991 in
Wisconsin, and decreased until the late 19903. Estimated SSB increased greatly during
1997-2002 in Illinois and during 1999-2002 in Wisconsin. In 2002, SSB was at its
highest level since the early 19903 and was 346% and 854% of 1986 levels in Illinois and
Wisconsin, respectively. The large increase in SSB during 1999-2002 was due to the
relatively good recruitment of the 1998 year-class, low ﬁshing mortality rates, and rapid
growth and maturity of females. We estimated that spawning stock biomass per recruit
(SSB/R) was approximately 0.46 kg in Wisconsin and 0.44 kg in Illinois in 2002. We
compared these SSB/R values to scenarios without ﬁshing mortality, and estimated that
2002 SSB/R was approximately 84% of the unexploited scenario in Wisconsin and 87%
of the unexploited scenario in Illinois. In contrast, SSB/R during 1986-1995 was
approximately 0.03 kg (18% of the unexploited scenario) in Wisconsin and 0.06 kg (33%
of the unexploited scenario) in Illinois. These dramatic differences in SSB/R occurred
because ﬁshing mortality rates were much lower during 2002 than during 1986-1995 and
yellow perch were growing faster, and therefore maturing at younger ages, during 2002
than during 1986-1995.

Females grew faster and to larger sizes than males (Figure 1.8); the mean length-

at-age of females at all ages older than age-2 were higher than males of the same age.

22

Estimated mean length-at-age remained relatively stable during 1986-1994 and increased
substantially during 1994-2000. During 2000-2002, mean length-at-age decreased
slightly, but was still higher than during the 19803 and early 19903. In Wisconsin, yellow
perch were generally smaller at a given age than in Illinois.

Selectivity patterns of the recreational ﬁsheries in Wisconsin and Illinois were
quite similar to one another when no length-based regulations were in effect (Figure 1.9).
Commercial selectivity patterns were also similar. This latter result was not surprising
because the scarcity of biological data for the Illinois commercial catch had led us to
assume an informative prior for the selectivity parameters, based on the results of the
Wisconsin assessment (see Appendix A). Due to differences in selectivity of the
commercial and recreational ﬁsheries, yellow perch recruited to the recreational ﬁshery at
smaller sizes than to the commercial ﬁshery. Selectivity of the Illinois recreational
ﬁshery changed substantially when a slot size limit was implemented during 1997-2000.
In Illinois during 1997-2000, average mortality rates for males age-4 and older were
slightly higher than for females due to the selectivity pattern of the recreational ﬁshery.
Selectivity patterns in the survey were substantially different between Illinois and
Wisconsin. Differences in selectivity patterns are likely attributable to differences in the
surveys such as mesh sizes of assessment gillnets and time of year of the survey.
Sensitivity Analyses

The models were somewhat sensitive to changes in the assumed CV3 and
effective sample sizes for the different data sources (Table A4). The Illinois model was
slightly less sensitive to these assumptions than the Wisconsin model. In general, ﬁve-

fold changes in the weights for each data source usually resulted in less than 15%

23

changes in mean ﬁshing mortality rates, abundance, and biomass. Weights that resulted
in increased estimates of mean ﬁshing mortality rates usually resulted in decreased
estimates of abundance and biomass. The Illinois model was most sensitive to changes in
the CV and effective sample sizes associated with females caught in the survey and the
effective sample size of the length composition from the recreational ﬁshery. The
Wisconsin model was most sensitive to CV and effective sample size associated with
males caught in the survey and the CV for catchability of the commercial ﬁshery.
Increasing M by 20% resulted in higher model estimates of average ﬁshing mortality
rates and lower estimates of abundance and biomass. The Illinois model was less
sensitive to our assumed value of M than the Wisconsin model; Illinois model estimates
changed approximately 12% and Wisconsin model estimates changed approximately
47%.

The Wisconsin model estimates of abundance, biomass, and mean ﬁshing
mortality rates were also somewhat sensitive to the different levels of commercial harvest
(Table 1.2). When we ﬁt the model using only reported yield, model estimates of
abundance and biomass in 2002 were more than 20% lower than the baseline (2x reported
during 1989-1992) scenario, and estimates of mean ﬁshing mortality rates were about
27% higher than baseline estimates. Under the 3x reported yield scenario, abundance and
biomass were about 20% greater than the baseline scenario, but mean ﬁshing mortality
rates were about 17% lower than the baseline.

Discussion
The decline in abundance of yellow perch in southwestern Lake Michigan during

the 19903 was likely caused by a combination of recruitment failure and relatively high

24

ﬁshing mortality rates, and our results are consistent with other authors’ descriptions of
the decline. During 1989-1994, yellow perch larvae were abundant shortly after
hatching, but recruitment to age 0 in the fall was poor, which has led some researchers to
propose that at least the initial decline in recruitment was not due to ﬁshing (Francis et al.
1996; Robillard et al. 1999; Marsden and Robillard 2004). Our results also indicated that
several successive year-classes failed despite relatively high SSB. However, after 1994,
the relative abundance of yellow perch larvae was less than 10% of the relative
abundance during the early 19903, which may indicate that SSB had decreased to low
enough levels to limit recruitment (Francis et al. 1996; Marsden and Robillard 2004). We
estimated that between 1991 and 1996 yellow perch SSB in Wisconsin declined almost
94% and between 1992 and 1997 yellow perch SSB in Illinois declined almost 90%. The
resultant low SSB may have prolonged the period of poor reproduction.

The decline of yellow perch SSB in southern Lake Michigan would probably not
have occurred at such a rapid pace if ﬁshing mortality rates had been lower. We
projected dynamics for the 1986 through 1996 period using our estimated recruitment
time series and age-based selectivity estimates, while changing the overall level of F.
Our projections indicated that SSB in 1996 would have been more than ﬁve times higher
than our model estimates in Wisconsin and nearly twice as high in Illinois if ﬁshing
mortality rates for fully selected ages and sexes had been equal to the natural mortality
rate (0.37) during 1986-1997. While our simple projections do not account for
compensatory changes that might have occurred if ﬁshing mortality had been lower, we
believe they do illustrate that high ﬁshing mortality rates on adult females were a

substantial contributor to the rapid decline in SSB that occurred. An alternative

25

hypothesis to the effect of ﬁshing yellow perch population dynamics is that natural
mortality decreased concurrently with restrictions on the ﬁsheries. In a supplemental
analysis (detailed results not reported), we explored this possibility by adding one more
estimated parameter to each model that allowed natural mortality to change from one
level for the 1986-1996 period to another for 1997 and after. The estimated changes in M
were opposite in sign for the Wisconsin and Illinois models and were much less than the
estimated changes in ﬁshing mortality for these periods.

The declines of yellow perch abundance in southern Lake Michigan were similar
in the 19603 and 19903, and recruitment failures of several successive year-classes may
be likely in the future. In the early 19603, yellow perch suffered a recruitment failure
(Wells 1977) similar to the recruitment failure observed in the early 19903 (Robillard et
a1. 1999; Marsden and Robillard 2004). The recruitment failure in the 19603 was
preceded by an increase in abundance during the late 19503 (Wells 1977), which was
similar to the increase in abundance during the late 19803 (Francis et al. 1996). Adult
abundance had decreased rapidly by the mid-19603 due to intense ﬁsheries (Wells 1977).
Yellow perch growth was slow during the 19503 (Wells 1977) and the 19803 (Marsden
and Robillard 2004). Extremely high ﬁshery catches preceded both declines in
abundance. However two major differences in the Lake Michigan community exist
regarding exotic species: alewife abundance in Lake Michigan was extremely high
during the 19603 compared to relatively low alewife abundance in the 19803 and 19903,
and zebra mussels were absent from Lake Michigan in the 19603, but their abundance
was high in the 19903 (Madenjian et al. 2002). Because the reproduction failure in the

19603 was associated with extremely high levels of alewife abundance, the decline in

26

recruitment was blamed on alewife (Eck and Wells 1987). Schroyer and McComish
(2000) found a negative correlation between alewife abundance and yellow perch
recruitment in Indiana waters of Lake Michigan during 1988-1997, but little direct
evidence of alewife preying upon yellow perch larvae has been observed in southern
Lake Michigan (Dettmers et al. 2003). Also, alewife abundance during the 19903 was
substantially lower (perhaps more than 20 times lower) than during the mid-19603
(Madenjian et a1. 2002), the period when alewife interference with yellow perch
recruitment was originally proposed as a cause for yellow perch reproduction failure.
Marsden and Robillard (2004) suggested that declines in yellow perch recruitment may
be exacerbated by changes in the ecosystem due to zebra mussel colonization, and

J anssen and Leubke (2004) found that poor recruitment was correlated with the presence
of zebra mussels in Indiana waters of Lake Michigan. Indeed, zebra mussels can alter the
composition of the zooplankton community (MacIsaac et al. 1992), which may decrease
food supplies for larval yellow perch. However, yellow perch recruitment did not
collapse after invasion of zebra mussels in Oneida Lake (Mayer et al. 2000) or the
western basin of Lake Erie (Tyson and Knight 2001).

Based on several reference points, yellow perch likely experienced overﬁshing in
southwestern Lake Michigan during 1986-1996. Beverton (1998) recommended the use
of the F 95 reference point (F at which yield is 95% of maximum sustainable yield) to
sustainably manage ﬁsheries. A rough estimate of F 95 is usually around M for medium-
lived species (Beverton 1998), which would be approximately 0.37 for yellow perch in
southern Lake Michigan. Others have argued that M should be an upper bound on ﬁshing

mortality rates that maximize yield (Deriso 1982, Quinn and Deriso 1999). Fishing

27

mortality rates for adult females were well above M in Illinois (1-2 times M) and
Wisconsin (2-4 times M). A number of US. marine commercial ﬁsheries are managed to
keep ﬁshing mortality below levels that would reduce SSB/R below a set percentage of
the unﬁshed situation (F x%), and typical percentages have been in the 35% to 45% range
(Quinn and Deriso 1999). In Wisconsin and Illinois, F was higher than F 35% during
1986-1996.

Regulation changes likely helped to substantially reduce ﬁshing mortality rates.
In 1996 in Wisconsin, the commercial quota was set to zero and a daily bag limit of ﬁve
yellow perch per angler was implemented for the recreational ﬁshery (reduced from 50 to
25 in 1995). When these policies were introduced, ﬁshing mortality decreased
noticeably. Recreational effort decreased, but may not have been a direct consequence of
the implemented bag limit. When stricter bag limits were implemented in some inland
Wisconsin lakes for walleye, anglers preferred to ﬁsh in lakes that had less restrictive bag
limits (Beard et al. 2003). In Illinois in 1995, the recreational daily bag limit was reduced
from no limit to 25 yellow perch per angler. In 1997, the commercial quota was reduced
to zero and a daily bag limit of 15 yellow perch per angler and a slot size limit of 8-10 in
(ﬁsh within this range could be kept) were implemented for the recreational ﬁshery.
Mortality rates also declined substantially in Illinois, as they did in Wisconsin;
commercial effort was reduced to zero, and recreational ﬁshing effort decreased
noticeably. Also, the slot size limit caused the recreational ﬁshery selectivity to change
so that average ﬁshing mortality rates were higher for age-4 and older males than for age-

4 and older females.

28

We did not incorporate age-estimation error into our model and this may bias our
estimates of recruitment and mortality rates. Our results likely underestimate the amount
of variability in recruitment because age-estimation error tends to blend strong and weak
year classes together (Richards and Schnute 1998). Speciﬁcally, our estimates of
recruitment of the 1989 and 1990 year-classes are probably high because of age-
estimation error associated with the 1988 year-class. However, our estimates of
recruitment are consistent with external estimates of year-class strength from age-0
assessments (Pientka et a1. 2003). Our mortality rate estimates are likely biased low for
the beginning of the time series when ages of yellow perch were estimated from scales.
Younger yellow perch tended to be aged as older when ages were estimated from scales
(Robillard and Mardsen 1996; Baker and McComish 1998; Wisconsin DNR, unpublished
data) and the overrepresentation of older ﬁsh in the data is most likely interpreted by the
model as an indication that older ﬁsh were more abundant. Annual mortality rates in the
late 19703 in Indiana and Illinois were estimated to be about 70% for males age-3 and
older and substantially higher for females age-3 and older (Wells and Jorgenson 1983).
These mortality rate estimates are similar to our estimates for Wisconsin in the late 19803
and for Illinois in the mid-19803.

Yellow perch growth may be density dependent and may also have increased due
to zebra mussel colonization. Patterns of growth during 1986-1998 resembled growth
during 1954-1979 for yellow perch in southern Lake Michigan. Yellow perch growth
may have been density dependent during 1986-2002 and 1954-1975 (Wells 1977). We
found similar growth patterns in Wisconsin and Illinois; growth was relatively slow when

yellow perch were at high abundance and growth was fastest at low abundance.

29

However, growth during 1999-2002 (low abundance) was the fastest observed for yellow
perch in southern Lake Michigan during the last ﬁve decades. This increased growth
coincided with substantial changes in yellow perch habitat due to colonization by zebra
mussels. Thayer et al. (1997) found increased adult yellow perch growth associated with
zebra mussels in ponds enclosures and Tyson and Knight (2001) found increased growth
of age-2 and age-3 yellow perch in the western basin of Lake Erie after zebra mussel
colonization; these increases in growth were attributed to increased food availability.
However, Mayer et al. (2000) found no increase in adult yellow perch growth associated
with zebra mussel colonization in Oneida Lake.
Management Implications

Since 1998, recruitment has continued to be poor in southern Lake Michigan
except for the 2002 year-class (Pientka et al. 2003; Clapp and Dettmers 2004; Fitzgerald
et al. 2004). Success of the 1998 year-class has renewed pressure on the agencies to
implement less restrictive regulations. Based partially on development of the models
described here, the Lake Michigan Yellow Perch Task Group recommended that
regulations remain unchanged for the time being. The models we developed will
continue to be used to monitor changes in the population and to advise managers.

Overexploitation of yellow perch has not previously been considered a likely
hypothesis for the decline of yellow perch in southern Lake Michigan (Francis et al.
1996). However, we found that SSB had reached very low levels by the mid-19903 and
intense ﬁshing likely compounded the rapidity of the decline in SSB. Although exotic
species or climatic changes may have affected recruitment, ﬁshing mortality rates during

the late 19803 and early 19903 probably were above levels that would be sustainable over

30

the long term. Therefore, management of yellow perch in Lake Michigan should focus
on limiting ﬁshing mortality and be ﬂexible to adjust to future recruitment failures.
Despite poor recruitment, SSB has increased to its highest point since the early 19903 in
Wisconsin and Illinois. This is partly a response to extensive management actions taken
by Wisconsin and Illinois, which have reduced ﬁshing mortality rates. However,
relatively few year-classes are represented in the population and future increases in
biomass and SSB will depend upon relatively strong recruitment of future cohorts to the

adult population.

31

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34

 

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35

 

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36

Table 1.1. Model estimates of yellow perch abundance-at-age (in thousands) during

1986-2002 in Illinois and Wisconsin waters of southwestern Lake Michigan.

 

 

Age

Year 2 3 4 5 6 7 8 9+
Illinois

1986 9,674 1 1,417 2,082 769 284 105 39 14
1987 9,598 6,682 7,518 862 146 33 10 4
1988 ‘ 6,807 6,629 4,518 3,715 247 32 6 2
1989 7,255 4,701 4,457 2,567 1,245 67 8 2
1990 17,535 5,011 3,180 2,759 1,370 545 30 4
1991 5,432 12,110 3,322 1,837 1,383 633 231 15
1992 2,521 3,726 8,070 1,949 969 694 31 1 1 16
1993 444 1,718 2,401 4,930 1,090 526 371 224
1994 22 302 1,070 1,370 2,691 573 270 295
1995 190 15 188 598 727 1,434 298 284
1996 325 127 9 102 310 378 747 292
1997 1,153 216 70 4 48 144 175 476
1998 130 787 143 45 3 31 93 420
1999 879 89 529 96 30 2 21 342
2000 8,91 1 599 59 349 63 20 l 240
2001 38 6,144 404 40 235 43 14 163
2002 38 26 4,139 265 26 153 28 1 15

 

37

Table 1.1. Continued.

 

Wisconsin

1986

1987

1988

1989

1990

1991

1992

1993

1994

1995

1996

1997

1998

1999

2000

2001

2002

1&863
fi083
5L757
1L438
1L935
fi221
L237
310
102
83
60
289
128
373
3,115
29

29

1L702
ALOSO
iiSlO
(1045
51826
1L23O
21603
853
214
71
57
42
200
88
258
2J47

20

1A90
'L922
2L776
2L353
IL994
IL786
<L964
2315
572
144
49
39
29
138
61
175

L405

300
741
4L246
L676
L250
ZLOZO
L665
1L227
L327
336
85
33
27
20
93
41

111

101
106
195
L718
694
530
682
627
L043
579
158
54
22
18
13
62

26

4O

21

20

71

578

287

161

225

285

408

231

96

35

14

11

39

16

19

204

78

48

96

108

153

139

63

22

47

34

33

53

122

170

147

107

77

53

 

38

Hd~*

Table 1.2. Model estimates of abundance (N; 10003), biomass (B; 1000 kg). mean rate of
ﬁshing mortality for females age-4 and older (F4+ females), and mean rate of ﬁshing
mortality for males age-4 and older ( F4, males) for 2002 under three scenarios of

unreported commercial harvest in Wisconsin waters of southwestern Lake Michigan

during 1989-1992.

 

 

N B 174+ females F4+ males
Reported 1,33 1 280 0.095 0.077
2 X Reported 1,690 356 0.075 0.060
3 X Reported 2,020 427 0.062 0.050

 

39

 

 

Figure 1.1. Map of Lake Michigan statistical districts with modeled areas shaded. WM
indicates Wisconsin waters, IL indicates Illinois waters, IN indicates Indiana waters, and

MM indicates Michigan waters.

4o

Illinois Wisconsin

 

 

 

 

 

 

 

2 600 — 2 600 _
0 500- o 500-
9 53
— 400- — 400‘
.2 a
2 300 - T3 300 - 0
I) o C
E
8 1001,:i : 5 100-
0 - 0 _
1986 1991 1996 2001 1986 1991 1996 2001
Year Year
it 131 2,500 _
S ‘5
2,000 -
:1! :E
'5 3 1,500 -
= s
'3 a; 1.000 -
9 9 50° 1%
° 3
g . H, a; 0 Tjrfwjm
1986 1991 1996 2001 1986 1991 1996 2001
Year Year

 
    
     

Survey CPE
N -h g on
8 8 8

 

 

 

Survey CPE
N ab g on
o 8 8 8

 

00° .. .. .
0 v.......9....r4. ...r,....
1986 1991 1996 2001 1986 1991 1996 2001
me' ‘er

Figure 1.2. Model ﬁts to commercial yield (1000 kg), recreational harvest (10003), and
gill net survey catch-per-effort (CPE; number per 30.5 m) in Illinois and Wisconsin
waters of southwestern Lake Michigan during 1986-2002. Model predictions are

represented by solid lines and observed values are represented by dots.

41

Illinois Wisconsin

1

Female

Female

1 I
O

 

MeanAge
o-tmwhmaauooco
MeanAge
O‘NQAOIOINCDCO
1 .144

1

 

 

 

 

1 988 1 993 1 998 1986 1 991 1 996 2001

Vac-v

Year

8- Male 8- Male .o'

Mean Age
Mean Age

 

 

 

 

 

o Tﬁ T ﬁl T I I rfT Y 7 I 1

1 988 1 993 1 998 1 986 1 991 1996 2001

Year Year

Figure 1.3. Mean age of yellow perch caught in gill net surveys in Illinois and Wisconsin
waters of southwestern Lake Michigan during 1986-2002. Lines represent model

predictions and dots represent observed values.

42

.I

Illinois

 

 

 

 

 

 

 

 

30‘ I
Commercral
3’st
O 0
g 20-
15 7 .ﬁ . , , , .
1986 1991 1996
Year
301 .
Recreational
0
ii 25-1 e 9
3 ' .
g 20!
15 .W4
1986 1991 1996 2001
Year
35-
Female Survey
t 3°“
.11 25 .
C
N
g 20-
15 r..ﬁr....,....,.
1986 1991 1996 2001
Year
30
1 Male Survey
E 25 r
3
15.,..,r...,..1.,.

 

1 986

1 991 1 996

Year

2001

 

Mean Length Mean Length

Mean Length

 

 

 

 

 

 

 

 

Wisconsin
30 ‘ I
Commercnal
25 — e
M
20 -
15
1986 1991 1996
Year
30 q I
Recreational
25 -
20i
15rr.t,....,.4.4,.
1986 1991 1996 2001
Year
35 4
Female Survey
30 _
25 j
20 -
15'.,4.f.,..
1986 1991 1996 2001
Year
30 ..
Male Survey

 

 

 

1996
Year

Figure 1.4. Mean length of yellow perch caught in the commercial and recreational

ﬁsheries and gill net surveys in Illinois and Wisconsin waters of southwestern Lake

Michigan during 1986-2002. Lines represent model predictions and dots represent

observed values.

43

Illinois Wisconsin

  
 
   

  
    

1.6 . 1.6
3 1 4 Female lRecreational g 14 Female IRecreatlonal
E 1.2 a Commercial E 1.2 E! Commercial
- ' IM ‘ - l M
Q 08 a 0.8
3 3
g 016 § 0.6
g 0'4 g 0.4
0.2 0.2
g 0 g o
N m (D N (I) N
88§a§§a§§ s§§a§§a§s
1- 1- v- v- 1- v- N N ,— ,_ ,— ,_ F 1— 1- (\l (\I
Year Year
‘3 1'6 M I ‘3 1.6 M I
0: ae m 4 ae _
E I: I Recreational 5' 1'2 I Recreatlonal
a ' :1 Commercial :3 ' Cl Commercial
§ 1 IM 2° 1 IM
., 0.8 a
3 3
o 0
° 4»
C C
9 a
5 s
5 s

   

Figure 1.5. Model estimates of average instantaneous mortality rates for yellow perch
age-4 and older in Illinois and Wisconsin waters of southwestern Lake Michigan during

1986-2002.

20,000 ,

 

 

l I Illinois
uWisconsin
15,000
on!
c
o
5
-5 10,000
I-
5
5,000 - H
0 i ? *Ijﬁm v 4-i‘r-'vl=*_*-='r 1? "T7l
Q? 03’ ‘8” OP 05" 9“ 4° 9% 0°
,9 ,9 .9 .9 .9 ,9 .9 .9 ,9
Year-class
20,000 i
. 0 Illinois 1986-1989
0 Illinois 1990-2000
‘E 15'0“) “ oWisconsin1986-1989
g o oWisconsin 1990-2000
g 10,000 -
L. O O O
3 0
I:
5.000 ~ 19980 I ’
year-c ass
04/ .
O
0 ‘W—4ﬁ—‘F—F—l
o 100 200 300 400 500 600 700
SSB

Figure 1.6. Model estimates of yellow perch recruitment (10003) in Illinois and
Wisconsin waters of southwestern Lake Michigan for the 1984-2000 year-classes and

estimates of recruitment plotted against yellow perch spawning stock biomass (SSB;

1000 kg).

45

 

Abundance
N 8 J?-
o O

_L
0

Illinois

 

0711,,
1986

ZJXX)-
1,500'~

1,000-

 

Biomass

500

1991

ITWYIIIIYIII

1996 2001
Year

 

0 (‘7.1 1
1986

1,000W
800i
6007
400-

SSE!

 

200

1991

I r f‘rr I I 1

1996 2001
Year

 

O ,r i
1986

r1rrrrlrrvr11

1991

1996 2001

Year

Abundance
'8 8

—L
0

Wisconsin

  

 

01m

1986

2JXX)-
1,500 -

1.000 4

Biomass

500

YWIIIII

1991 1996
Year

2001

 

 

0 .
1986

1,000 -
800 ~

SSE!

 

rrrtrrrrlrfr

1991 1996

ff!

2001

Year

  

  

 

 

 

 

Y 7 I

 

 

' l

1991 2001

1996
Year

Figure 1.7. Estimated abundance (10005), biomass (1000 kg), and spawning stock

biomass (SSB; 1000 kg) of yellow perch age-2 and older in Illinois and Wisconsin waters

of southern Lake Michigan during 1986-2002. Error bars represent 95% probability

intervals (the Bayesian analog of conﬁdence intervals).

46

 

Illinois

—0— Female
mom Male

    

Mean length at age 5 (cm)

 

 

 

 

 

15 I I I f I I I I I I I I I I I
1986 1991 1996 2001
Year
i c n in

35 _ WS 0 s
A —o—Female
g .....o Male
ID
0
oz
I
’6
5
a:
5
I:
8
E

15 I I T I T I I I r I I I I I I

1986 1 991 1 996 2001
Year

Figure 1.8. Model estimates of mean length at age 5 Illinois and Wisconsin waters of

southwestern Lake Michigan during 1986-2002.

47

    
  
  

 

 

 

 

 

1.0 - Commercial . .
lllanlS
0.8 - \ — Wisconsin
.4?
.5 0.6 -
8
-q—, 0.4 -
a)
0.2 -
0.0 -
8 18 28 38
Length (cm)
1'0 " Recreational F , ,
/ i — — lllmons
a 0-8 " \ - — — Illinois Slot
jg 0,5 _ \\ -—— Wisconsin
0
{ll-3 0.4 -
a)
0.2 -
0.0 -
8 18 28 38
Length (cm)
1 '0 - Survey / ’ ‘—
0.8 -
3:“
g 0.6 -
8
3 0.4 — Illinois
‘0 / — Wisconsin
0.2 - /
0.0 “WW—[Tm
8 18 28 38
Length (cm)

Figure 1.9. Model estimates of selectivity of the commercial ﬁshery, recreational ﬁshery,

and survey in Illinois and Wisconsin during 1986-2002.

48

CHAPTER 2
PERFORMANCE OF TIME-VARYING CATCHABILITY ESTIMATORS IN
STATISTICAL CATCH-AT-AGE ANALYSIS

Introduction

Statistical catch-at-age analysis (SCA) is used to provide estimates of absolute
abundance, recruitment, and ﬁshing mortality for many ﬁsheries throughout the US. and
the rest of the world (National Research Council [NRC] 1998; Quinn and Deriso 1999).
In contrast to virtual population analysis and its variants, SCAs generally assume that
ﬁshing mortality rate-at—age can be modeled as a function of a year effect and an age
effect (selectivity). This approach allows statistical estimation where ﬁshery catch-at-age
data are assumed to contain some amount of measurement error (Megrey 1989; Fournier
and Archibald 1982). These models require catch-at-age data as well as an index of
abundance; other data sources can also be included in the model (Deriso et al. 1985).
Under many conditions, SCA provides more accurate estimates of stock size and other
important management quantities than other stock assessment techniques (NRC 1998,
Punt et al. 2001, Radomski et al. in press)

Many SCAs use ﬁshery catch per effort (CPE) as an index of relative abundance,
and thus assume that ﬁshery CPE is proportional to abundance (Quinn and Deriso 1999).
However, violations of this assumption can cause SCA models (and other stock
assessment models) to produce biased estimates (NRC 1998). Time-varying catchability
has been documented in a wide range of ﬁsheries, spanning commercial and recreational
ﬁsheries and freshwater and marine systems. In some cases, catchability may change

with abundance or the area inhabited by a stock (e.g., Peterman and Steer 1981; Winters

49

 

and Wheeler 1985; Crecco and Overholtz 1990; Harley et al. 2001), environmental
effects (Ziegler et al. 2003), or due to changes in ﬁsherman behavior or gear (Hilbom and
Walters 1992). Interactions between population size, stock area, and ﬁsher behavior can
lead to hyperstable ﬁshery CPE, where CPE remains high despite decreases in abundance
(Hilbom and Walters 1992; Harley et al. 2001). Hyperstable CPE in combination with a
stock assessment model that does not account for this can lead to overestimated stock size
and catch limits (NRC 1998).

Methods have been developed to account for time-varying ﬁshery catchability,
but there is little consensus about best practices in this area (e. g., Fournier and Archibald
1982, Fournier 1983; Methot 1990, Fournier et al. 1998, NRC 1998). Generally, ﬁshery
effort or CPE data are ignored if an independent survey is available for a stock (NRC
1998). However, in many ﬁsheries, survey data are not available and ignoring ﬁshery
effort data is not an option (NRC 1998). Likewise, ignoring ﬁshery effort data may
decrease the accuracy and precision of SCA estimates in some cases because ﬁshery CPE
may be informative about changes in relative population size or survey data may be poor.
Our objective was to determine how well different methods of estimating time-varying
catchability performed within an SCA framework. Speciﬁcally, we tested four
estimation models to determine how well they performed in scenarios where catchability
changed over time.

Methods

We used Monte Carlo simulations to compare how four different methods of

estimating ﬁshery catchability within an SCA model performed when models were

confronted with different data generating scenarios. Our data generating models included

50

 

ﬁve cases where catchability changed abruptly or gradually over time and where
catchability was explicitly a function of population abundance. Our data generating
models also contained three levels of ﬁshing mortality and three levels of survey
measurement error. While the general inﬂuence of ﬁshing mortality level and survey
measurement error on the performance of SCA methods is well understood (e.g., Bence
et al. 1993), we included these factors to determine whether they act to change the
relative performance of different approaches of modeling time-varying catchability. We
generated 1000 datasets for each scenario (45 total scenarios). For data sets that included
survey data, we ﬁt each data set with four different models that made different
assumptions regarding ﬁshery catchability; catchability was modeled as white noise, a
random walk, density dependent, or catchability was effectively estimated as a free
parameter for each year. This last method ignores any information contained in ﬁshery
CPE or effort. For data sets that did not include survey data we used the ﬁrst three of
these estimation methods. Each scenario used the same sets of random numbers.

All models contained 15 years of data and eight age classes with the last age class
representing all ﬁsh that age and older. Data generating models were based on
commercial ﬁsheries for lake Whiteﬁsh (Coregonus clupeaformis) in the upper Great
Lakes. Symbols and equations deﬁning the data generating models and estimation
models are presented in Tables 2.1 and 2.2. Equations are referred to in the text as eq.
Tx.y, where x is the table number and y is the equation number within Table x. To avoid
redundancy, equivalent quantities and parameters in estimation and data generating
models are not differentiated except when they both appear in the same equation, in

which case estimated quantities are denoted with a caret above the symbol.

51

 

Data Generating Model

The data generating model described the population dynamics and created data
sets of total ﬁshery catch, the age composition of the ﬁshery catch, and in some scenarios
total survey CPE and the age composition of the survey. For the population dynamics,
we used an age-structured model that followed cohorts over time. Recruitment
(abundance at age 1) was generated from a lognormal distribution with a coefﬁcient of
variation (CV) of 100%. Numbers-at-age in the ﬁrst year were calculated assuming a
stable age distribution with lognormal errors, where recruitment and mortality rates prior
to the ﬁrst year of the simulation were on average the same as in the ﬁrst year (eq.
T2.2.1). Cohorts were tracked over time by applying a simple exponential mortality
model (eq. T2.2.2a); the last age class was treated as representing all ﬁsh age 8 and older
(eq. T2.2.2b). Biomass each year was the sum of age-speciﬁc abundance and mean
weight-at-age (eq. T2.2.3).

We used a separable model to generate ﬁshing mortality rates. The total mortality
rates were determined by the natural mortality rate and age-speciﬁc ﬁshing mortality
rates (eq. T2.2.4). M was held constant across ages and years at 0.25. The instantaneous
ﬁshing mortality rate was a function of catchability, ﬁshing effort, and age-speciﬁc
selectivity (eq. T2.2.5). We used three levels of ﬁshing mortality where F at fully
selected ages was approximately 2M (high), M (medium), and 0.5 M (low). We allowed
ﬁshing mortality to change over time by allowing effort to change (Figure 2.1) and by
incorporating several processes of time-varying catchability (see below). For a given

level of ﬁshing mortality, each of the models used the same effort series and each effort

52

series had the same amount of contrast in absolute terms. The selectivity pattern for the
ﬁshery was dome shaped to simulate a gill net ﬁshery (Figure 2.2).

We included ﬁve models for time-varying catchability, which incorporated a
range of possible ways that catchability could vary over time. The logc of catchability
was modeled as white noise to simulate a ﬁshery where catchability varied from year to
year about a constant mean (eq. T2.2.6), perhaps due to environmental effects, but where
year-to-year deviations were not correlated. We also included four treatments that had
varying amounts of autocorrelation: ﬁrst order autoregressive (ARI), density dependent,
linear increase, and abrupt change. The ARI process was also on the loge-scale and
could mimic catchability changes from many sources (eqs. T2.2.7a, T2.2.7b), such as
density dependent catchability or correlated environmental effects. We set the correlation
(p) of the AR] process to 0.9 and the CV (08) to 0.16. Density dependent catchability
followed a power relationship where catchability declined with increasing abundance (eq.
T2.2.8; Paloheimo and Dickie 1964). Because each of the different levels of ﬁshing
mortality had different average levels of abundance, we used three sets of parameters (a
and ,6) to deﬁne the density dependent power function, one for each level of ﬁshing
mortality. In the linear increase scenario, catchability increased linearly over time (eq.
T2.2.9), which could represent learning by ﬁshers or increases in gear efﬁciency. In the
abrupt change scenario, catchability was constant until year eight of the time series and
increased to a higher level where it remained for the rest of the time series (eq. T2.2.10).
This scenario simulated the adoption of a more efﬁcient technology by the ﬁshery. All
models were parameterized to have the same expected catchability (over the time series)

and similar variances of logcqf. We achieved this by simulating data sets and adjusting

53

the catchability parameters until the mean and variance of catchability were the same as
in the white noise case. We used a value of 0.2 for the standard deviation of the log of
catchability as the standard for all other catchability models. This value is similar to
estimates of the CV of catchability for commercial ﬁsheries in New Zealand (Francis et
al. 2003), but was less than median values of the CV of ﬁshery CPE estimated by Harley
et al. (2001) for International Council for the Exploration of the Sea ﬁsheries of 0.4-0.8,
which should be an upper bound.

Fishery catch was calculated with the Baranov catch equation (eq. T2.2.13; Quinn
and Deriso 1999). We multiplied total catch by a lognormal measurement error to
calculate observed ﬁshery catch (eq. T2.2.14); the measurement error CV for ﬁshery
catch was 10%. Observed age compositions were generated by drawing a sample from a
multinomial distribution of size n (100 for the ﬁshery) with proportions equal to the
expected catch-at-age in the ﬁshery. Survey CPE-at-age was calculated as the product of
survey catchability, abundance, and survey selectivity (eq. T2.2.15), and observed survey
CPE was the product of total survey CPE and a lognormal measurement error (eq.
T22. 16). Our simulation model contained three levels of survey quality with differing
levels of measurement error: good CV=O.25, poor CV=1.0, and no survey. Catchability
of the survey was constant over time. Observed survey age compositions were generated
by drawing a random sample from a multinomial distribution of size 75 with proportions
equal to the expected CPE at age in the survey.
Estimation Model

The estimation models were largely the same as the simulation models except for

how catchability was estimated and how numbers-at-age in the ﬁrst year and recruitments

54

 

were handled. Common parameters among models included N1,1...N15,1 (Recruitment),
N13. . .N1,3 (numbers-at-age in the ﬁrst year), and s”. . .S7_f (ﬁshery selectivity); models
with surveys also included s,,_,...sz, (survey selectivity) and q, (survey catchability).
Numbers-at-age in the ﬁrst year and recruitment for each year were estimated as
parameters during the model ﬁtting process. After the ﬁrst year and age, abundance-at-
age followed a standard exponential mortality model with the last age representing all
ﬁsh that age and older (eqs. T2.2.2a, T2.2.2b).

The total mortality rate (ZN) was the sum of M and F ,3 (eq. T2.4); M was
assumed known at 0.25 (the true value from the simulation models). Fishing mortality
followed a separable model for all of our estimation models. Fishery and survey
selectivities were estimated as individual parameters by setting selectivity at the oldest
age-class to one. Estimation models contained four methods of estimating catchability:
white noise, random walk, density dependent, and no catchability (directly estimating
ﬁshing mortality) with survey data. The ﬁrst estimation model allowed ﬁshery
catchability to vary with white noise about a constant mean (eq. T2.2.6). The second
estimation model allowed ﬁshery catchability to vary according to a random walk (eq.
T2.2.16). The third estimation model allowed catchability to be a density dependent
function (eq. T2.2.8). The density dependent model did not contain any random
deviations. In our fourth estimation model, we estimated the ﬁshing mortality rate for
fully selected age classes as a parameter, and then applied the estimated ﬁshery
selectivity to calculate age-speciﬁc ﬁshing mortality rates (eq. T2.2.l7). This method
does not use ﬁshery effort as a data source. The estimation models also predicted

proportions of ﬁshery and survey catch-at—age.

55

Model Fitting and Convergence

We ﬁt the models using a likelihood-based approach where we used a numerical
search to ﬁnd parameter values that minimized our objective function. The objective
function was the sum of the likelihood components and each component was the negative
of the log—likelihood for a single data source or a penalty related to time-varying
catchability (eq. T2.3.1).

Our estimation models assumed lognormal distributions of errors for total catch
for the ﬁshery (eq. T2.3.2) and survey CPE (eq. T233) and multinomial distributions for
age compositions of the ﬁshery (eq. T2.3.4) and the survey (eq. T2.3.5; Fournier and
Archibald 1982). Effective sample sizes and CVs of the ﬁshery and survey catch and age
compositions were set to their true values from the generating models. The likelihood
components for survey CPE and age composition were only included in models that
included survey data.

For estimation models that used ﬁshery effort as a data source, ﬁshery CPE was
not explicitly modeled. Instead, ﬁshing mortality was an explicit function of effort, and
catch was linked to abundance and ﬁshery effort by estimating the catchability
coefﬁcient. We assumed lognormal deviations for catchability in the white noise (eq.
T2.3.6) and random walk (eq. T2.3.?) estimation models. The CV for the white noise
catchability was set to the true expected value, which was 0.2 for all data generating
models. For the random walk model, we set the CV to 0.165, the CV that on average
created a time series with a sample CV of 0.2. This component in the objective function
can be thought of as a penalty that produces a shrinkage estimator (in the Frequentist

case) or as a Bayesian prior and penalizes large deviations from mean catchability (for

56

the white noise model) or large year-to-year deviations (in the random walk model).
Estimation models that contained density dependent catchability or ignored effort data

did not contain likelihood component 2 5.

We minimized the objective function iteratively using an efﬁcient quasi-Newton
implementation in AD Model Builder software that takes advantage of automatic
differentiation (Otter Research Limited 2000). We minimized the objective function in
stages, where the initial stages were penalized if the model estimates deviated from the
expected average ﬁshing mortality rates under each scenario (early stages can be viewed
as providing starting values for subsequent stages). This constraint was removed for the
ﬁnal stage of ﬁtting and therefore did not penalize ﬁnal model estimates. Iterative
adjustment of the parameters terminated when the maximum gradient of parameters with
respect to the objective function was less than 0.0001 , or more than 1000 function
evaluations had occurred. We denoted any terminated parameter estimates where the
maximum gradient component was less than 0.0005 as converged, based on trial
investigations after the completion of the simulations that used different parameter
starting values.

Evaluation of Estimation Model Performance

In stock assessments, estimated quantities in the last year are often most important
for forecasting and management. Therefore, we evaluated estimation model performance
by calculating the relative error (RE) of estimated biomass in the last year.

_ estimated — true

 

(2. 1) RE
true

We report only results for stock size measured in biomass. Other common

measures of stock size (e.g., measures of exploitable abundance) showed similar patterns

57

and estimates of fully selected F or exploitation rate reflected similar but inverse patterns
(i.e., if estimated biomass was higher than the true value, estimated F was usually lower
than the true value and vice versa). We evaluated systematic over or under estimation
using the median of the relative error (MRE). If MRE equals zero, half of the estimates
are higher than the true value and half are lower than the true value. Throughout the rest
of the paper we use the term unbiased as meaning median unbiased (i.e., MREs near
zero). We also compared estimation model performance using the median of the absolute
values of relative error (MARE), which indicates the width of the distribution of RES if
the median is zero. In situations where the RES are either all (or mostly) positive or
negative, the MRE will equal the MARE. We compared relative performance of the
estimation models by calculating the difference of their MAREs and report these
differences as percentages because the units of MARE are percent. We used MRE and
MARE instead of mean relative error and root mean squared error because mean values
were heavily inﬂuenced by several cases with large relative errors (>100). We checked
whether these outliers represented false convergence by restarting the estimation with
different starting values. Convergence was veriﬁed and obtained the same parameter
estimates.
Results

All estimation models performed best in situations with high ﬁshing mortality and
low survey CV and worst in cases with low ﬁshing mortality and no survey (Table 2.4,
Table 2.5). The performance of a given estimation model depended on the level of
ﬁshing mortality, survey quality, and data generating model. In almost all cases,

estimation models that made use of both survey CPE and ﬁshery effort outperformed

58

models that used only ﬁshery effort or survey CPE. Performance of the estimation model
that ignored ﬁshery effort data was independent of the underlying catchability model that
generated the data and was only a function of survey quality and ﬁshing mortality. The
estimation model that ignored effort data was relatively unbiased (MRE near zero) in all
cases, but the MARE was often signiﬁcantly higher than for estimation models assuming
white noise and random walk catchability and the relative performance of this method
was highly dependent on survey quality. For the other estimation models, the results can
be separated into two categories: ones where all estimation models were relatively
unbiased (white noise, autoregressive, and density dependent) and ones where some
estimation models had substantial bias (linear increase and abrupt change). Although the
density dependent estimation model was relatively unbiased in many cases, it performed
relatively poorly overall because it did not converge for 15-35% of the simulated data
sets that did not contain density dependent catchability; the other estimation models
usually failed to converge less than 1% of the time. This lack of convergence likely
occurred because the two parameters describing density dependent catchability were
confounded with one another (i.e., many combinations of a and 0 could produce equally
good ﬁts) for many data sets, and thus the optimization procedure could not ﬁnd a unique
best solution. Because of problems with convergence in most cases, we did not believe
that the density dependent estimation model was a viable candidate for most situations.
White Noise, First Order Autoregressive, and Density Dependent

In cases where the data generating models contained white noise catchability, ﬁrst
order autoregressive catchability, or density dependent catchability, all estimation models

produced relatively unbiased estimates of biomass in the last year (i.e., MREs near zero;

59

 

Table 2.4), with the most biased estimation model in these scenarios having an MRE of
only -6.1% (random walk estimation model ﬁtting density dependent generation model
with low mortality and no survey). There were larger differences in precision among the
estimators and this was reﬂected in MARE and the tightness of the distributions of
relative errors (Table 2.4). For cases where the estimation model was the same as the
generating model (white noise and density dependent), the estimation model that matched
the generating model performed best (i.e., had the lowest MARE and tighter
distributions). In the case of the ARl data generating model, the random walk model
performed best in most cases. Differences in MARE among estimation models that
modeled catchability as white noise, a random walk, or ignored ﬁshery effort were
usually less than 5% for cases with good surveys (Figure 2.3; Figure 2.4). However,
MAREs of random walk and white noise estimation models were 7-30% lower than
estimation models that ignored ﬁshery effort in cases with a poor survey. Differences in
estimation model relative performance were largely accounted for by differing
performance of random walk and white noise catchability models because the
performance of the estimation model that ignored ﬁshery effort data was relatively
constant for a given level of ﬁshing mortality and survey quality. White noise and
random walk models were most accurate in cases with white noise catchability,
somewhat less accurate for cases with density dependent catchability, and least accurate
in cases with ARI catchability.
Linear Increase and Abrupt Change

The white noise and random walk estimation models were biased in cases where

catchability increased linearly or changed abruptly, but the amount of bias depended on

60

survey quality, ﬁshing mortality rate, and data generating model. The MREs of biomass
in the last year for estimation models with white noise and random walk catchability were
above zero in all cases, indicating a positive bias (Table 2.5). The positive bias seen in
our simulations undoubtedly reﬂects the direction of change in catchability built into our
simulations, where the estimation models did not fully account for the increase in ﬁshery
catchability. Neither the white noise nor the random walk estimation models performed
well in cases with no survey, trending catchability, and low mortality. The amount of
bias was highest in cases where ﬁshery catchability changed abruptly and ﬁshing
mortality rates were low and decreased as the level of ﬁshing mortality increased and as
survey quality improved.

Although the random walk estimation model was biased, it usually had a lower
MARE than our other estimation models, but performance relative to the other estimation
models depended on the treatment. In cases with a good survey, the MARE of the
estimation model that ignored ﬁshery effort and the MARE of the random walk
estimation model were within 5% of one another (Figure 2.3). However, in cases with a
poor survey, the random walk model usually had MAREs 10-20% lower than the
estimation model that ignored ﬁshery effort. The estimation model that ignored ﬁshery
effort data only outperformed the random walk model in the scenario with an abrupt
change in catchability and low ﬁshing mortality. The estimation model that ignored
ﬁshery effort and the random walk estimation model clearly outperformed the white
noise estimation model in these cases and had MAREs 12-50% lower than the white

noise estimation model (Figure 2.4).

61

Discussion

Often stock assessment scientists will not use or will substantially downweight
(i.e., specify an arbitrarily large CV) ﬁshery effort or CPE data in an SCA if a ﬁshery
independent index of abundance is available for a given stock. Indeed, the NRC (1998)
recommended that ﬁshery dependent indices of abundance should be ignored if an
independent index of abundance is available based on the results of their simulations.
However, our results argue against automatically ruling out the use of ﬁshery dependent
indices of abundance when a survey is present. In cases where the survey CV is large,
we believe that use of ﬁshery dependent indices is justiﬁed if they are believed to contain
information on stock size. Of course ﬁshery effort should be adjusted for known changes
in ﬁshing efﬁciency, and the estimation model should allow for ﬂexible changes in
catchability over time, as was the case for our random walk estimator. The reliability of
ﬁshery effort data may be suspect in some ﬁsheries and, in these cases, it may make
sense to ignore ﬁshery effort. Using methods that do not allow for trends in catchability
can lead to severely biased SCA estimates, and modeling ﬁshery catchability as white
noise (which is often done) may not provide the necessary ﬂexibility for models to
accurately depict system dynamics. Also, there may be a tendency to overstate the
precision of survey data and understate the precision of ﬁshery data in SCAs, which is
what Francis et a1. (2003) found for assessments of many New Zealand commercial
ﬁsheries.

Our recommendations are contrary to NRC (1998), because we evaluated a wider
range of structural models for time-varying ﬁshery catchability within SCAs, but our

results yield similar insights for the cases they explored. In the NRC (1998) study,

62

ﬁshery catchability increased over time combined with density dependence; their survey
had a CV of 30% (near the level of our “good” survey). Also, the NRC (1998) study
mainly included SCA estimation models that contained white noise models for
catchability or ignored ﬁshery effort data (see Restrepo (1998) for details of models used
in the NRC (1998) study). The exception was one estimation model where fishery
catchability was modeled as a mixture of random walk and white noise processes (Ianelli
and Fournier 1998). However, the CV of the white noise term was large relative to the
CV of the random walk term (Ianelli and Fournier 1998), which likely caused the model
to perform similarly to a white noise model. Similar to the results of NRC (1998), we
also found that that SCA models that ignored ﬁshery effort data outperformed SCA
models that modeled ﬁshery catchability as white noise in cases with trending
catchability.

Independent survey indices of abundance or relative abundance are extremely
important for obtaining accurate SCA estimates, especially in situations with low ﬁshing
mortality. Our results agree with the NRC (1998) recommendation to use survey data if
they are available. In our study, estimation models that utilized ﬁshery effort data and
survey data (even with a CV of 100%) outperformed models that used only ﬁshery effort
data, especially in cases where catchability trended over time and ﬁshing mortality was
not high.

It is important to standardize effort series to remove catchability trends to as large
an extent as possible. Our experiments showed that SCA estimates were most biased
when trends or abrupt changes in ﬁshery catchability occurred and that all our estimation

models performed reasonably well in cases where catchability did not trend over time.

63

 

Trending ﬁshery catchability is probably common. Many mechanisms could lead to
trends in ﬁshery catchability, such as increasing power of the ﬁshery, increasing
aggregation of ﬁsh stocks and ﬁshers, or trending recruitment dynamics and density
dependent catchability. Salthaug and Aanes (2003) presented a method to correct CPE
for the spatial distribution of ﬁshing effort, which has been shown to affect catchability
(Winters and Wheeler 1985; Rose and Kulka 1999). Also, improvements in vessels, and
other ﬁsher behaviors can be accounted for either by preprocessing (e.g., analyzing CPE
data to estimate mean CPE by accounting for vessel characteristics and spatial and
temporal patterns of ﬁshing) ﬁshery data or by integrating the standardization process
into the stock assessment model (e. g. Maunder 2001; Maunder and Starr 2003; Maunder
and Punt 2004). The procedure of simultaneously standardizing catch and effort data and
ﬁtting the stock assessment model can lead to improved estimates over the two—step
approach of standardizing catch and effort data and then ﬁtting the assessment model
with the standardized values (Maunder 2001; Maunder and Langley 2004).

Our results probably provide a best-case view of the performance of SCAs when
faced with time-varying catchability and may exaggerate the accuracy of all estimation
models used in our study. Except for the catchability aspect, the structure of the
estimation models was correct (i.e., the same as the data generating model). In reality, it
is likely that M may vary among years and ages and that the data analyst will not know
the true M. Fishery selectivity may vary over time, which can cause biased estimates
from SCA models if it is not accounted for (Radomski et al. in press). Likewise, our
models did not contain trends in survey catchability over time or correlation with changes

in ﬁshery catchability, which could cause models that used survey indices of abundance

to generate less accurate estimates. Lastly, our data generating models contained a
survey with an asymptotic selectivity pattern, which allows SCA models to produce more
accurate estimates than other survey selectivity patterns (Bence et al. 1993).

While our results favored the random walk model in general, this was not true
under all circumstances. We recommend that data analysts ﬁt multiple stock assessment
models with different assumptions about time-varying catchability. One may be able to
then determine the best catchability model using a Bayesian framework, where each of
the catchability models we ﬁt is a special case of a “full” model (McAllister and Kirchner
2002; Gelman et al. 2004). For instance, the estimation models that ignore effort data,
use white noise, or a random walk are all special cases of a ﬁrst order autoregressive
process (eq. T2.8). In the case of white noise, the correlation coefﬁcient (p) equals 0. In
the case of random walk, p equals 1. And in the case of ignoring effort data, the CV of
the random deviations (a) is inﬁnity. Thus, one possible procedure would be to allow
catchability to follow a ﬁrst order autoregressive process and estimate the p and 0
parameters. If the CVs of the other likelihood components are speciﬁed, these
parameters (p and a) may be estimable and this method could lead to better SCA
estimates of parameters and uncertainty. Alternative approaches would be to select
among our special case models using the deviance information criterion (Spiegelhalter et
al. 2002) or other measures that account for both goodness of ﬁt and model complexity,
or to average over the alternative models using Bayesian Model Averaging (McAllister

and Kirchner 2002). These are topics warranting future research.

65

References

Bence, J. R., A. Gordoa, and J. E. Hightower. 1993. Inﬂuence of age-selective surveys
on the reliability of stock synthesis assessments. Can. J. Fish. Aquat. Sci. 50:
827-840.

Crecco V. A., and W. Overholtz. 1990. Causes of density-dependent catchability for
Georges Bank haddock Melanogrammus aegleﬁnus. Can. J. Fish. Aquat. Sci. 47:
385-394.

Deriso, R. B., T. J. Quinn 11, and P. R. Neal. 1985. Catch-age analysis with auxiliary
information. Can. J. Fish. Aquat. Sci. 42: 815-824.

Fournier, D. A. 1983. An analysis of the Hecate Strait Paciﬁc cod ﬁshery using an age-
structured model incorporating density dependent effects. Can. J. Fish. Aquat.
Sci. 40: 1233-1243.

Fournier, D., and C. P. Archibald. 1982. A general theory for analyzing catch at age data.
Can. J. Fish. Aquat. Sci. 39: 1195-1207.

Fournier, D. A., J. Hampton, and J. R. Sibert. 1998. MULTIFAN-CL: a length-based,
age-structured model for ﬁsheries stock assessment, with application to South
Paciﬁc albacore, Thunnus alalunga. Can. J. Fish. Aquat. Sci. 55: 2105-2116.

Francis, R. I. C. C., R. J. Hurst, and J. A. Renwick. 2003. Quantifying annual variation
in catchability for commercial and research ﬁshing. Fish. Bull. 101: 293-304.

Harley, S. J ., R. A. Myers, and A. Dunn. 2001. Is catch-per-unit-effort proportional to
abundance? Can. J. Fish. Aquat. Sci. 58: 1760-1772.

Hilbom, R., and C. J. Walters. 1992. Quantitative ﬁsheries stock assessment. Chapman
and Hall, New York.

Maunder, M. N. 2001. A general framework for integrating the standardization of catch
per unit effort into stock assessment models. Can. J. Fish. Aquat. Sci. 58: 795-
803.

Maunder, M. N., and A. D. Langley. 2004. Integrating the standardization of catch-per-
unit-of-effort into stock assessment models: testing a population dynamics model
and using multiple data types. Fish. Res. 70: 389-395.

Maunder, M. N., and A. E. Punt. 2004. Standardizing catch and effort data: a review of
recent approaches. Fish. Res. 70: 141-159.

Maunder, M. N ., and P. J. Starr. 2003. Fitting ﬁsheries models to standardized CPUE
abundance indices. Fish. Res. 63: 43-50.

66

 

McAllister, M, and C. Kirchner. 2002. Accounting for structural uncertainty to facilitate

precautionary ﬁshery management: illustration with Namibian orange roughy.
Bull. Mar. Sci. 70: 499-540.

Megrey, B. A. 1989. Review and comparison of age-structured stock assessment models
from theoretical and applied points of view. In Mathematical analysis of ﬁsh
stock dynamics. Edited by E. F. Edwards, and B. A. Megrey. American Fisheries
Society Symposium 6: 8-48.

Methot, R. D. 1990. Synthesis Model: an adaptable framework for analysis of diverse
stock assessment data, p. 259-277. In Proceedings of the Symposium on
Application of Stock Assessment Techniques to Gadids. Edited by L. Low.
International North Pacific Fisheries Commission Bulletin 50.

National Research Council (NRC). 1998. Improving ﬁsh stock assessments. National
Academy Press. Washington DC.

Otter Research Limited. 2000. An introduction to AD Model Builder version 4 for use
in nonlinear modeling and statistics. Otter Research Ltd., Sidney B.C.

Peterman, R. M., and G. J. Steer. 1981. Relation between sport-ﬁshing catchability
coefﬁcients and salmon abundance. Trans. Am. Fish. Soc. 110: 585-593.

Punt, A. E., A. D. M. Smith, and G. Cui. 2002. Evaluation of management tools for
Australia’s south east ﬁshery 2. how well can management quantities be
estimated? Mar. Freshw. Res. 53: 631-644.

Quinn, T. J ., H, and R. B. Deriso. 1999. Quantitative Fish Dynamics. Oxford University
Press, New York.

Radomski, P. A., J. R. Bence, and T. J. Quinn 11. In press. Comparison of virtual
population analysis and statistical kill-at-age analysis for a recreational kill
dominated ﬁshery. Can. J. Fish. Aquat. Sci.

Rose, G. A., and D. W. Kulka. 1999. Hyperaggregation of ﬁsh and ﬁsheries: how catch-
per-unit-effort increased as the northern cod (Gadus horhua) declined. Can. J.
Fish. Aquat. Sci. 56(Suppl. 1): 118-127.

Salthaug, A., and S. Aanes. 2003. Catchability and the spatial distribution of ﬁshing
vessels. Can. J. Fish. Aquat. Sci. 60: 259-268.

Spiegelhalter, D.J., N. G. Best, B. P. Carlin, and A. van der Linde. 2002. Bayesian

measures of model complexity and ﬁt. Journal of the Royal Statistical Society
Series B 64:583-639.

67

 

Winters, G. H., and J. P. Wheeler. 1985. Interactions between stock area, stock
abundance, and catchability coefﬁcient. Can. J. Fish. Aquat. Sci. 42: 989-998.

Ziegler, P. E., S. D. Frusher, and C. R. Johnson. 2003. Space-time variation in
catchability of southern rock lobster Janus edwardsii in Tasmania explained by
environmental, physiological and density-dependent processes. Fish. Res. 61:
107-123.

68

Table 2.]. Symbols and descriptions of variables for data generating and estimation

 

 

models.
Symbol Description Value (if needed in the
data generating model)
E Average recruitment 1,000,000
N Abundance by age and year
y,a
By Biomass
Zy a Total instantaneous mortality rate by age and
year
Fy a Instantaneous ﬁshing mortality rate by age
and year
M Instantaneous natural mortality rate 0.25
Saf Fishery age-speciﬁc selectivity See ﬁgure 2.2
3a S Survey age-speciﬁc selectivity See ﬁgure 2.2
Ey Fishery effort See ﬁgure 2.1
Fishe catchabilit
4y, f 1')’ Y
q Survey catchability 0.0001
s
21' f Mean ﬁshery catchability 0.05
C Expected ﬁshery catch-at-age
y,a

69

Expected survey catch-at-age

 

Iy’a
5 Observed total ﬁshery catch
y
1" Observed total survey catch
y
Proportion of catch-at-age in ﬁshery
“Ma-f
Proportion of catch-at-age in survey
“y,a,s
Wa Mean weight at age 0.16, 0.45, 0.82, 1.2, 1.55,
1.86, 2.11, 2.3
6y Deviations for white noise catchability
6y Deviations for ﬁrst order autoregressive
catchability
my Deviations for random walk catchability
a, ,6 Parameters for density dependent catchability 175, 0.53; 90, 0.49; 35,
(low, medium, high) 0.42
a, b Parameters for linear increase in catchability 0.032, 0.00225
q 1 , q; Parameters for abrupt change in catchability 0.0402, 0.0598
f Fishing intensity by year
y
p Correlation parameter for autoregressive 0.9
catchability
a}, CV for recruitment variation 1.0

70

 

Fishery measurement error CV

Survey measurement error CV

CV for white noise catchability deviations

CV for autoregressive catchability deviations

CV for random walk catchability deviations

0.1

0.25; 1.0

0.2

0.16

0.165

 

71

Table 2.2.

Data generating and estimation model equations.

 

 

Population model equations Application
(T2.2. 1) a -1 Generation
_ _ )3 Zl,a +7a 2
N =Re “-1 ;7~N(0,0')
La 7
(T2.2.2a) — Z Both
: e y’a
y + l, a + 1 y, a
(T2.2.2b) — Zy 7 — Zy 8 Both
Ny+ 1,8 = y,7" +Ny,8e
(T2.2.3) B = 2N w Both
y y,a a
a
(T2.2.4) Z a = M + F Both
(T215) F = q E s BOII'I
y,a y y a
Catchability model equations
(T2.2.6) White noise Both
lo :10 - +5 ;5 ~N(0,0'2)
ge qy.f ge qf y y 5
(T2.2.7a) First order autoregressive Generation
_ 03
loge ql,f ~ N loge qf,-1———2—
‘P
(T2.2.7b) Generation

loge qy+l,f = loge qf +p(loge qy,f —loge c7)+ey

6 ~ N(0,0'2)
y e

72

(T2.2.8)

(T2.2.9)

(T2.2.10)

(T2.2.11)

(T2.2.12)

(T2.2.13)

(T2.2.14)

(T2.2. 15)

(T2.2.16)

Density dependent

_ -ﬂ
qy.f 'aNy

Linear increase

qy.f =a+b(y)

Abrupt change

q1 ify<8
qy.f —q2ify28

Random walk

_ . ~ 2
loge qy+1,f —loge qytf +a)y,a)y N(0,0'w)

Freely estimate f3, (ignore ﬁshery effort)
F = s

Ma fy a. f
Observation model equations

-e y,a )N
y,a Z y,a

T
.. _ y . ~
C —e Ecyﬂ’ry N(0,0'T)

y
I =q s N '

y,a sa y,a
I =e I ;v ~N0,0'
, 2;... , < .9

Both

Generation

Generation

Estimation

Estimation

Both

Both

Both

Both

 

73

Table 2.3. Objective function equations for statistical catch-at-age analysis simulation

study. Equations T2.3.3 and T2.3.5 were only used in estimation models that considered

survey data. Equations T2.3.6 and T2.3.7 were only used in estimation models that

modeled ﬁshery catchability as white noise or a random walk respectively.

 

(T2.3.l)

(T2.3.2)

(T2.3.3)

(T2.3.4)

(T2.3.5)

(T2.3.6)

(T2.3.7)

L=Z€i
i

i :

~ A 2
1 2(loge(cy>—loge(cy))

_1__

2
207 y
1 ~ A 2

£2 =—§-Z(loge(ly)-loge(ly))

20’” y

(’3 :_nf§§uy,a,f loge(uy,a,f)

(4 z—nszzuy,a,s loge(uy,a,s)
ya

1 3 2
e=—zl l
5 20'2y y

4

1 2
e=——z(o)
5 20'3y y

Objective function

Fishery catch

Survey catch-per-effort

Proportion at age in the ﬁshery catch

Proportion at age in the survey catch

White noise catchability

Random walk catchability

 

74

 

Table 2.4. Simulation results for statistical catch-at-age estimation model performance in
cases where data generating models included white noise catchability (W N), ﬁrst order
autoregressive catchability (AR), and density dependent catchability (DD). Shown are
median relative error (MRE) and median of the absolute values of relative error (MARE)
for estimated biomass in the last year (year 15) from four statistical catch-at-age
estimation models: white noise (WN), random walk (RW), density dependent (power
relationship; DD), and freely estimated F at maximum selectivity (i.e., not ﬁtted to
ﬁshery effort data; FF). Data generating models included three levels of ﬁshing mortality
(high [F=2M], medium [F=M], and low [F =0.5M]), and 3 levels of survey precision
(good [CV=25%], poor [CV=100%], and no survey). Estimation models with the lowest

MARE for each treatment are indicated in bold.

 

 

 

Estimation Model

MRE ME.
g-Model Mortality Survey WN RW DD FF WN RW DD FF
WN low good -0.005 0.009 0.007 0.017 0.196 0.212 0.259 0.216
WN low poor -0.006 -0.010 0.016 0.018 0.214 0.258 0.284 0.493
WN low none 0.006 -0.019 0.000 0.231 0.255 0.319
WN medium good 0.001 0.009 0.019 0.024 0.133 0.155 0.191 0.170
WN medium poor 0.012 0.002 0.031 0.036 0.156 0.213 0.219 0.400
WN medium none 0.018 0.006 0.035 0.183 0.234 0.233
WN high good 0.012 0.015 0.015 0.024 0.103 0.1 19 0.143 0.146
WN high poor 0.01 1 0.001 0.025 0.046 0.124 0.166 0.149 0.308
WN high none 0.020 0.017 0.041 0.141 0.185 0.179
AR low good -0.004 0.004 0.022 0.000 0.271 0.222 0.356 0.226
AR low poor 0.009 -0.025 0.030 -0.001 0.355 0.335 0.434 0.498
AR low none 0.020 -0.015 0.01 1 0.409 0.419 0.456
AR medium good 0.008 0.006 0.006 0.031 0.21 1 0.164 0.302 0.169

75

AR

AR

AR

AR

AR

DD

DD

DD

DD

DD

DD

DD

DD

DD

medium
medium
high
high
high
low

low

low
medium
medium
medium
high
high

high

poor
none
good
poor
none
good
poor
none
good
poor
none
good
poor

110116

0.002

0.031

0.007

0.004

0.018

-0.019

-0.025

-0.025

-0.026

-0.023

-0.020

-0.010

-0.012

0.004

-0.045

-0.031

0.009

-0.022

-0.037

0.022

-0.006

-0.061

0.009

-0.031

-0.047

0.005

-0.028

-0.030

0.034

0.028

0.029

0.029

0.025

0.006

0.009

0.007

0.024

0.023

0.018

0.027

0.020

0.022

0.014

0.030

0.019

0.013

0.046

0.027

0.039

0.031

0.037

0.297

0.332

0.174

0.236

0.271

0.244

0.342

0.401

0.177

0.252

0.295

0.137

0.193

0.218

0.278

0.340

0.131

0.196

0.236

0.233

0.324

0.384

0.163

0.249

0.292

0.121

0.178

0.213

0.365

0.373

0.251

0.301

0.311

0.195

0.234

0.273

0.139

0.177

0.199

0.114

0.130

0.155

0.412

0.148

0.314

0.243

0.514

0.185

0.418

0.146

0.329

 

76

 

Table 2.5. Simulation results for statistical catch-at-age estimation model performance in
cases where data generating models included linearly increasing catchability (LI) and an
abrupt increase in catchability (AC). Shown are median relative error (MRE) and median
of the absolute values of relative error (MARE) for estimated biomass in the last year
(year 15) from four statistical catch-at-age estimation models: white noise (W N), random
walk (RW), density dependent (power relationship; DD), and freely estimated F at
maximum selectivity (i.e., not ﬁtted to ﬁshery effort data; FF). Data generating models
included three levels of ﬁshing mortality (high [F =2M], medium [F =M], and low

[F =0.5M]), and 3 levels of survey precision (good [CV=25%], poor [CV=100%], and no

survey). Estimation models with the lowest MARE for each treatment are indicated in

 

 

 

bold.
Estimation Model
MRE MARE
q-Model Mortality Survey WN RW DD FF WN RW DD FF
LI low good 0.353 0.050 0.437 0.013 0.353 0.187 0.437 0.234
LI low poor 0.762 0.419 0.647 0.044 0.762 0.419 0.647 0.501
LI low none 0.751 0.635 0.642 0.751 0.635 0.642

LI medium good 0.345 0.074 0.488 0.029 0.345 0.150 0.488 0.174

LI medium poor 0.671 0.299 0.653 0.032 0.671 0.302 0.653 0.390

LI medium none 0.716 0.429 0.647 0.716 0.429 0.648
LI high good 0.322 0.072 0.482 0.031 0.322 0.121 0.482 0.140
LI high poor 0.551 0.172 0.598 0.033 0.551 0.189 0.598 0.280
LI high none 0.605 0.21 l 0.61 1 0.605 0.213 0.61 1
AC low good 0.571 0.165 1.386 0.018 0.571 0.255 1.386 0.242
AC low poor 1.154 0.655 2.145 0.055 1 . 154 0.655 2.145 0.491
AC low none 1.755 1.413 2.462 1.755 1.413 2.462

AC medium good 0.371 0.088 0.730 0.029 0.371 0.168 0.730 0.171

77

AL:

AK:

AI:

AI:

AC

Ineduun
rnedhun
lngh
high

lﬁgh

poor
none
good
poor

none

(1699

(1842

(1291

(1497

(1550

(1282

(1472

(1038

(1084

(1103

1.107

L207

(1488

(1641

(1620

(1039

(1030

(1019

(1699

(1842

(1291

(1497

(1550

0.300

(1472

0.114

(1144

11154

L107

L207

(1488

(1641

(1620

(1390

(1139

(1280

 

78

 

    

 

 

12 _ -———Medium
10 -
t: 8‘
0 ,AL
5 6- ,/’/ \‘\~
” “
4" o "o . \s
4 ‘1 ’/” .. o 0’ ~ .. ..‘ \s\~
2 "‘ .90.. '°. . ..\0..-.
o O T j I W F ..1
0 5 10 15
Year

Figure 2.1. Effort series used for high, medium, and low ﬁshing mortality rate scenarios
in the data generating models. The average ﬁshing mortality rates for fully selected age
classes were approximately 2M for the high scenario, M for the medium scenario, and

0.5M for the low scenario.

79

Selectivity
.0 .0 .O
A a: co _.
l I 1 J

 

0.2 4

 

Figure 2.2. Fishery and survey selectivity patterns used in the data generating model.

80

0.3 a

 

 

 

oWNIAFl
ADD OLI O
02" cAC A 2
q) I
- t
s— 0.1“
O
s o 0 ’
D
33" 0 r ‘ .
<
2
-0.11
O
-0.2

LG MG HG LP MP HP
Treatment

Figure 2.3. Relative performance of the estimation model that ignores ﬁshery effort
versus the random walk estimation model measured by the difference of median of the
absolute value of the relative errors (MARE). Positive values indicate that the estimation
model that ignored ﬁshery effort data had a larger MARE than the random walk
estimation model and vice versa. Data generating models are indicated by the symbol
shape: WN - white noise, AR — autoregressive, DD — density dependent, LI — linear
increase, and AC — abrupt change. Two letters identify each treatment: the ﬁrst letter for
level of ﬁshing mortality (L - low, M — medium, H — high) and the second letter for level

of survey quality (G — good, P — poor).

81

051 oWNIAFl .

 

 

ADD <>L|
0'4“ .AC 3 . .
. <> 0 e
8 0.3" o
c
93
g 0.2~ O Q 8
0
pg 01- 0
< I I I
2 o a a 5 ' 4 1 t t F
-0.1-
-0.2

 

LG MG HG LP MP HP LN MN HN
Treatment

Figure 2.4. Relative performance of white noise versus random walk estimation model
measured by the difference of median of the absolute value of the relative errors
(MARE). Positive values indicate that the white noise estimation model had a larger
MARE than the random walk estimation model and vice versa. Data generating models
are indicated by the symbol shape: WN — white noise, AR — autoregressive, DD -— density
dependent, LI — linear increase, and AC — abrupt change. Two letters identify each
treatment: the ﬁrst letter for level of ﬁshing mortality (L — low, M - medium, H —- high)

and the second letter for level of survey quality (G - good, P — poor, N — none).

82

CHAPTER 3
PERFORMANCE OF BAYESIAN MODEL SELECTION IN STATISTICAL CATCH-
AT-AGE ANALYSIS
Introduction

Development of a ﬁshery stock assessment often involves ﬁtting alternative
models and using what is thought to be the best among them to provide management
advice. The “best” model is often selected by ad hoc criteria with unknown performance
characteristics. Model selection is an area of importance because estimated quantities
important for management, such as exploitable biomass, can be extremely sensitive to
model structure (McAllister and Kirchner 2002). Common uncertainties in statistical
catch-at-age (SCA) model structure include stock-recruitment relationships, selectivity
functions, and assumptions linking ﬁshery catch with abundance and effort (McAllister
and Kirchner 2002). In some cases, results from several models will be reported to
managers, but quantitative estimates of the relative likelihood a particular model being
most “correct” are typically not provided (McAllister and Kirchner 2002).

Model selection has been applied to SCA models, but previous applications have
been limited in the types of models that could be compared. Helu et al. (2000) evaluated
performance of Akaike’s Information Criterion (AIC; Akaike 1973) and Schwartz’s
Bayesian Information Criterion (BIC; Schwartz 1978) to assess model selection in SCA
models and found that AIC and BIC both performed well by selecting the candidate
model that was the same as the data-generating model in most of their scenarios.
Unfortunately, although AIC or BIC may perform well in some cases, their

implementation is problematic when models differ in their random effects or hierarchical

83

 

structures because the number of parameters in these models is not easy to determine
(Bumham and Anderson 2002). Therefore, to be able to compare structurally complex
SCA models requires alternative model selection approaches that can account for random
effects and priors on parameters.

The Deviance Information Criterion (DIC) has been developed relatively recently
to select among complex hierarchical models where the number of effective parameters is
not readily apparent (Spiegelhalter et al. 2002). Much like AIC and BIC, DIC selects
among models by trading off goodness of ﬁt and model complexity. DIC is a
generalization of AIC and reduces to AIC in the case of a model with diffuse priors
(Spiegelhalter et al. 2002). DIC is particularly applicable to models with random effects
or hierarchical structure because it estimates the effective number of parameters rather
than requiring the user to provide this. Unlike BIC, DIC does not depend on the number
of data points directly in its calculation.

Although DIC has been applied in many studies (e.g., Zhu and Carlin 2000; Barry
et al. 2003), relatively few studies have evaluated the performance of DIC model
selection (Spiegelhalter et al. 2002; Cardoso and Tempelman 2003; Kizilkaya and
Tempelman 2003; Berg et al. 2004; Kizilkaya and Tempelman 2005; van der Linde
2005). In general, these studies found that DIC usually selected the correct model (i.e.,
the model that generated the data) from the set of candidate models and that the estimated
number of effective parameters seemed reasonable for their given models.

Bayes factors are another method to compare models that can account for random
effects and hierarchical structure (Gelman et al. 2004). Fournier et al. (1998) used

posterior Bayes factors (an approximation to Bayes factors; Aitkin 1991) to estimate

84

“weight of evidence” of one model over another (Lavine and Schervish (1999) showed
that weight of evidence is not quite an accurate description of Bayes factors). However,
like AIC and BIC, posterior Bayes factors also require the number of parameters as an
input to the calculation. McAllister and Kirchner (2002) estimated Bayes factors for
several competing assessment models of Namibian orange roughy (Hoplostethus
atlanticus) using the sampling-importance resampling algorithm. To date, there are no
published studies of ﬁshery stock assessments that have evaluated the performance of
model selection or model averaging based on Bayes factors. However, in complex
models, such as SCA models, Bayes factors can be difﬁcult to calculate and sensitive to
priors (Kass and Raftery 1995; Lavine and Schervish 1999; Han and Carlin 2001).

My objectives were to determine if using DIC or an approximation of Bayes
factors as model selection criteria resulted in choosing an appropriate model structure and
level of complexity. Also, I wanted to evaluate whether using formal model selection
methods provided more accurate estimates of important ﬁshery management quantities,
such as ﬁshing mortality rate and biomass in the last year. To achieve these objectives, I
designed a simulation study and challenged the model selection criteria with three
estimation models and three scenarios of data accuracy and time-varying catchability.
Methods

I evaluated whether using DIC and approximate Bayes factors to select among
SCA model variants provided more accurate estimates of quantities used for management
than an approach of using a single model structure in all cases. My data-generating
models contained three basic scenarios, which differed in their relationship between

ﬁshing mortality and observed effort. These scenarios included (1) modeling ﬁshery

85

catchability as white noise, (2) modeling ﬁshery catchability as increasing a constant
amount each year, and (3) treating ﬁshing mortality as unrelated to observed effort. I
chose these data-generating scenarios because previous results indicated that the relative
performance of different estimation models was likely to change over this range of
conditions. Three different estimation models were ﬁtted to each of the 30 datasets (ten
from each scenario). These estimation models contained different assumptions regarding
ﬁshery catchability; (1) catchability was modeled as white noise, (2) as a random walk,
and (3) where catchability was effectively estimated as a free parameter for each year.
This last method ignores any information contained in ﬁshery effort data.

All models contained 15 years of data and eight age classes with the last age class
representing all ﬁsh that age and older. Data-generating models were based on
commercial ﬁsheries for lake Whiteﬁsh (Coregonus clupeafonnis) in the upper Great
Lakes. Symbols and equations deﬁning the data-generating models and estimation
models are presented in Tables 3.1 and 3.2. Equations are referred to in the text as eq.
Tx. y, where x is the table number and y is the equation number within Table x. To avoid
redundancy, equivalent quantities and parameters in estimation and data-generatin g
models are not differentiated except when they both appear in the same equation, in
which case estimated quantities are denoted with a caret above the symbol.
Data-generating Model

The data-generating model described the population dynamics and created data
sets of total ﬁshery catch, the age composition of the ﬁshery catch, total survey CPE, the
age composition of the survey, and ﬁshery effort. To model population dynamics, I used

an age-structured model that followed cohorts over time. Recruitment (abundance at age

86

l) was generated from a lognormal distribution with a coefﬁcient of variation (CV) of
100%. Numbers-at-age in the ﬁrst year were calculated assuming a stable age
distribution with lognormal errors, where recruitment and mortality rates prior to the ﬁrst
year of the simulation were on average the same as in the ﬁrst year (eq. T3.2.1). Cohorts
were tracked over time by applying a simple exponential mortality model (eq. T3.2.2a);
the last age class was treated as representing all ﬁsh age 8 and older (eq. T3.2.2b).
Biomass each year was the sum of age-speciﬁc abundance and mean weight-at-age (eq.
T3.2.3).

I used a separable (i.e., ﬁshing mortality was the product of an age effect and a
year effect) model to generate ﬁshing mortality rates. The total mortality rates were
determined by the natural mortality rate and age-speciﬁc ﬁshing mortality rates (eq.
T3.2.4). M was held constant across ages and years at 0.25. The instantaneous ﬁshing
mortality rate was a function of catchability, ﬁshing effort, and age-speciﬁc selectivity
(eq. T3.2.5). I allowed ﬁshing mortality to change over time by allowing ﬁshery effort to
change and by incorporating two processes of time-varying catchability (see below).

The overall level of ﬁshing mortality varied among simulations. This was
accomplished by multiplying the baseline effort (Figure 3.1) by a Uniform(1,2) number
selected for each simulation. The baseline effort series was designed to produce an
average level of F for fully selected ages approximately equal to M. Thus, this procedure
led to F for fully selected ages varying among simulations between M and 2M. For the
white noise catchability and linearly increasing catchability scenarios observed effort
equaled true effort. For the scenario with uninforrnative effort, the observed effort series

was drawn as uniform random numbers between the minimum true effort (effort in year

87

1) and the maximum true effort (effort in year 8). The selectivity pattern for the ﬁshery
was dome shaped to simulate a gill net ﬁshery (Figure 3.2).

I included two models for time-varying catchability, which caused SCA models to
have variable performance (chapter 2). The loge of catchability was modeled as white
noise to simulate a ﬁshery where catchability varied from year to year about a constant
mean (eq. T3.2.6), perhaps due to environmental effects. In the second scenario,
catchability increased linearly over time with a small amount of white noise error (eq.
T3.2.9), which could represent learning by ﬁshers or increases in gear efﬁciency. Both
models were parameterized to have the same expected catchability (over the time series)
and similar variances of logeqf. I achieved this by simulating data sets and adjusting the
catchability parameters until the mean and variance of catchability were the same as in
the white noise case. I used a value of 0.2 for the standard deviation of the loge of
catchability. This value is similar to estimates of the CV of catchability for commercial
ﬁsheries in New Zealand (Francis et al. 2003), but was less than median values of the CV
of ﬁshery CPE estimated by Harley et al. (2001) for International Council for the
Exploration of the Sea ﬁsheries of 0.4-0.8, which should be an upper bound on the CV of
catchability.

Fishery catch was calculated with the Baranov catch equation (eq. T3.2.13; Quinn
and Deriso 1999). I multiplied total catch by a lognormal measurement error to calculate
observed ﬁshery catch (eq. T3.2. 14); the measurement error CV for ﬁshery catch was
about 0.1. Observed age compositions for the ﬁshery catch were generated by drawing a
random sample from a multinomial distribution of size 200 with proportions equal to the

true proportions of catch-at-age in the ﬁshery. Survey CPE-at-age was calculated as the

88

product of survey catchability, abundance, and survey selectivity (eq. T3215), and
observed survey CPE was the product of total survey CPE and a lognormal measurement
error (eq. T3.2. 16).

As was the case for average ﬁshing mortality, survey quality varied randomly
among simulated datasets. This was accomplished by selecting the measurement error
CV for each simulation from a Uniform(0.2,0.8) distribution. These levels of survey CV
were selected because they provided contrast in performance of several estimation
models in chapter 2. Catchability of the survey was constant over time. Observed survey
age compositions were generated by drawing a random sample from a multinomial
distribution of size 150 with proportions equal to the true pr0portions of CPE at age
calculated from eq. T3.2.15.

Estimation Model

The estimation models were largely the same as the simulation models except for
how catchability was estimated and how numbers-at-age in the ﬁrst year and recruitments
were handled. Common parameters among models included N1,,...N15,1 (Recruitment),
N12. . .Nm (numbers-at-age in the ﬁrst year), and s 1 J. . .s7 J (ﬁshery selectivity), s”. . .37”,
(survey selectivity) and q, (survey catchability). All models had 52 unique estimated
parameters. Parameterization of the models to reduce correlations among parameters is
described in Appendix B. Numbers-at-age in the ﬁrst year and recruitment for each year
were estimated as parameters during the model ﬁtting process. After the ﬁrst year and
age, abundance-at-age followed a standard exponential mortality model with the last age

representing all ﬁsh that age and older (eqs. T3.2.2a, T3.2.2b).

89

The total mortality rate (2”,) was the sum of M and F M (eq. T2.4); M was
assumed known at 0.25 (the true value from the simulation models). Fishing mortality
followed a separable model for all of my estimation models. Fishery and survey
selectivities were estimated as individual parameters by constraining the log of the age-
speciﬁc selectivities to sum to zero. This method was used to reduce correlations among
selectivity parameters. Estimation models contained three methods of estimating
catchability: white noise, random walk, and no catchability (directly estimating ﬁshing
mortality). The ﬁrst estimation model allowed loge ﬁshery catchability to vary with
white noise about a constant mean (eq. T3.2.6). The second estimation model allowed
loge ﬁshery catchability to vary according to a random walk (eq. T3.2. 16). In my third
estimation model, I estimated the ﬁshing mortality rate for fully selected age classes as a
parameter, and then applied the ﬁshery selectivity to calculate age-speciﬁc ﬁshing
mortality rates (eq. T3.2.17). This method does not use ﬁshery effort as a data source.
The estimation models also predicted proportions of ﬁshery and survey catch-at-age.
Model Fitting and Convergence

I ﬁt the models using a Bayesian approach as implemented in AD Model Builder
version 6.0.2 (Otter Research Ltd. 2000). The objective function was the sum of the
likelihood components and priors. Each component was the negative of the log-
likelihood for a single data source or an informative prior related to time-varying
catchability (eq. T3.3.1). My estimation models assumed lognormal distributions of
errors for total catch for the ﬁshery (eq. T332) and survey CPE (eq. T333) and
multinomial distributions for age compositions of the ﬁshery (eq. T 3.3.4) and the survey

(eq. T3.3.5; Fournier and Archibald 1982). Effective sample sizes and CVs of the ﬁshery

90

and survey catch and age compositions were set to their true values from the generating
models.

For estimation models that used ﬁshery effort as a data source, ﬁshing mortality
was an explicit function of effort and catch was linked to abundance and ﬁshery effort by
estimating the catchability coefﬁcient. I assumed lognormal deviations for catchability in
the white noise (eq. T3.3.6), and random walk (eq. T3.3.7) estimation models. The
standard deviation for the white noise and random walk catchability deviations (on the
logc scale) was assumed known at 0.2, which was approximately equal to the expected
standard deviation in the data-generating models. This component in the objective
function is a prior and penalizes large deviations from mean catchability (for the white
noise model) or large year-to—year deviations (in the random walk model). Note that

priors 8 5 contained the constants for the likelihood function so that the priors were

comparable to compare approximate Bayes factors. I placed uninformative uniform
priors on common parameters among models and these priors were the same in each
model.

The AD Model Builder implementation of Markov Chain Monte Carlo (MCMC)
includes ﬁrst estimating the maximum likelihood parameter estimates and asymptotic
variance-covariance matrix, then using the estimated parameters as starting values for the
MCMC chain. The Metropolis-Hastings algorithm sampled from a scaled multivariate
normal distribution with variances and covariances proportional to the asymptotic
variance covariance matrix. Iran the MCMC chain for each model for 5,000,000 cycles
and saved values from every 100‘h cycle. To estimate the precision of the DIC estimates,

I estimated the variance of a shorter chain (as a minimum estimate for my cases) using

91

the “brute force” method of Zhu and Carlin (2000), which involves running many parallel
MCMC chains, estimating DIC, and then estimating the variance of DIC from the parallel
chain estimates. The MCMC chains were divided into subchains of 500,000 cycles to
estimate the variance of DIC for a chain of that length. I dropped the initial 100,000
cycles of each chain as a burn in period, which reduces the effect of starting values on the
MCMC estimates (Gelman et al. 2004). In some cases, the models did not converge to a
stable mixing distribution for at least 1,000,000 cycles. In these cases, I used a burn in
period of 1,500,000 cycles. I then estimated the variance of DIC estimates from the ten
subsamples (seven in the cases with long burn in periods) for each chain. If the MCMC
chain has converged to a stable mixing distribution, this method should provide the same
result as running ten independent chains.
DIC Calculations

DIC, like other information-theoretic information criteria, trades off a measure of
model ﬁt (estimated deviance) and a measure of model complexity (effective number of

parameters; Spiegelhalter et al. 2002).

DIC=D+pD

The average deviance, D , for model j is an estimate of model adequacy and is estimated

by

5.:

J —2loge p(data | 0c)

1

(3|.—
HMO

C

where C is the number of MCMC cycles saved minus the burn in, and log, p(data I 0,)

was the natural logarithm of the likelihood function (Spiegelhalter et a1. 2002). Like with

AIC and BIC, smaller DIC values indicate better models. I estimated the effective

92

number of parameters as the difference between the average deviance and the deviance

evaluated at the maximum likelihood parameter estimates,

pD =D—D(0

ML) '

Normally, the effective number of parameters is estimated as the difference between the
mean deviance and the deviance evaluated at the mean of the parameter vector, which is
estimated by the mean parameters from the MCMC chain (Spiegelhalter et al. 2002).
However, Spiegelhalter et al. (2002) noted that other measures of the central tendency,
such as the mode or median of the parameters could be used. DIC differences calculated
using the maximum likelihood estimates were usually within 0.1 DIC units of DIC
differences calculated with the mean of the parameters from the MCMC chain. I also
attempted a third method of estimating the effective number of parameters, which used V2

of the variance of the deviance chain values to approximate the effective number of

parameters (Gelman et al. 2004),

 

1 C D0 13 2
pD_2(C—1)C_Z__1( (c.j)_ 1')

This method performed poorly (large DIC variance) and almost always estimated more
parameters for the model than the actual number of parameters in the models.
Approximate Bayes Factors

The probability that model M,- is the best in a set of candidate models can be
approximated by

lMi)

p(M )= pildataIOMLlp. (OML
. IMi) ’

l gpi (data I OML )pi(0ML

93

where pildata 10 l is the likelihood evaluated at the maximum likelihood estimates of

ML

the parameters, and pl. (0 | M i) is the prior for the parameters conditional on model i

ML
(Hilbom and Mangle 1997). Throughout the rest of the paper, this method for estimating
the posterior probability that a model is the best in the set of candidate models is referred
to as approximate Bayes factors. However, the term “approximate Bayes factors” in the
model selection literature usually refers to using BIC differences to approximate Bayes
factors (Kass and Raftery 1995).

Evaluation of Estimation Model Performance

I determined how often the correct structural model was selected, even though
there was not a truly correct model in the scenario with a linear increase in catchability or
in the uninformative effort scenario. In the white noise case, the white noise estimation
model was correct. In the linear increase case, the random walk model was considered
the correct model because it tended to perform better than other models in this scenario
(chapter 2) and because it is designed to allow for gradual changes. In the case with
uninforrnative effort data, the model that ignored ﬁshery effort data was considered the
correct model.

In stock assessments, estimated quantities in the last year are often most important
for forecasting and management. Therefore, I evaluated estimation model performance
by calculating the relative error (RE) of estimated biomass and average ﬁshing mortality
(for ages 4-8) in the last year.

estimated — true
RE =

 

true

94

I evaluated systematic over or under estimation using the mean of the relative error
(MRE). I also calculated the mean square relative error (MSE), which summarizes the
variance and bias of model predictions. If the bias of the estimates is zero, MSE equals
the variance of the estimator.

Results

Most of the MCMC chains appeared to have converged to their stable mixing
distribution within 10,000 cycles. However, in several cases, the MCMC routine
required nearly 1,500,000 cycles as a burn in period. The estimated standard deviation
for DIC from a chain of 500,000 cycles was about 0.3. This indicates that, for a chain
length of 500,000 cycles, DIC differences less than one are probably not important. For
the 5,000,000 cycle chains, effective sample sizes were usually greater than 19,000 (from
an actual sample size of 49,000 saved cycles) and DIC estimates should have lower
standard deviations than from a substantially smaller chain.

Estimates of the effective number of parameters, pp, were generally less than the
actual number of estimated parameters, 52. The effective number of parameters for the
estimation model with random walk catchability was the lowest with a mean of 47.7 and
a range of 47.0—48.4. The estimation model with white noise catchability had the second
fewest effective parameters with a mean of 48.8 (range 47.4-50.0). The estimation model
that freely estimated ﬁshing mortality for each year had the most effective parameters
with a mean of 52.4 (range 51.4-54.1), which was quite close to the true number of
estimated parameters.

DIC usually selected the correct model. However, DIC differences between the

best model and the other models were usually less than seven, except in the

95

uninformative ﬁshery effort scenario, indicating that the evidence was not
overpoweringly in favor of the model with the lowest DIC (Figure 3.3). In the white
noise catchability case, DIC selected the white noise model (correct model) eight out of
ten times. In the two times when the white noise estimation model was not selected, the
white noise catchability model and the random walk catchability model were quite close
in terms of DIC scores (<0.7). In the linear increase catchability scenario, the random
walk model was selected in nine out (of ten cases. In the only case where DIC did not
choose the correct model, the white noise estimation model was chosen and the
difference in DIC scores was less than 1.0. In the uninformative effort scenario, the
model that ignored ﬁshery effort was always selected.

Model selection using approximate Bayes factors performed somewhat differently
than DIC model selection and always selected the correct model in the case of white
noise catchability (between 80-99% probability). However, approximate Bayes factors
only selected the correct model six out of ten times for scenarios with a linear increase in
catchability or uninformative ﬁshery effort data. In the scenario where effort data were
uninformative, approximate Bayes factors selected the white noise model twice and
random walk model twice. In the scenario where catchability increased linearly,
approximate Bayes factors did not choose any model strongly; posterior model
probabilities were between 55 and 91% for the best model, and only the white noise
model was selected in cases where the random walk model was not. The posterior model
probabilities for the estimation model that ignored ﬁshery effort were always less than

0.1% in both scenarios with informative effort data.

96

In general, using Bayesian model selection helped to choose relatively accurate
models. Models selected using either DIC or approximate Bayes factors had smaller
MSEs than always using any single model (table 3.3, ﬁgure 3.4). DIC model selection
slightly outperformed approximate Bayes factors, but the difference was probably not
signiﬁcant because of the small sample size.

Discussion

In general, DIC and approximate Bayes factor model selection produced better
point estimates of biomass and ﬁshing mortality in the last year on average than relying
on any single model. However, the best DIC or approximate Bayes factor model did not
always produce the best estimates of biomass and ﬁshing mortality rates in the last year.
Indeed, DIC and approximate Bayes factors only selected the model with the lowest
relative errors in ﬁshing mortality or biomass in the last year between 7% (for DIC and
ﬁshing mortality) and 14% (approximate Bayes factors and biomass) of the time. Helu et
al. (2000) also found that incorrect models often produced more accurate estimates of
biomass in the last year than the structurally correct model in their study of AIC and BIC
model selection for SCA models.

DIC model selection seems to perform well in cases where increased model
complexity is warranted, but may not perform as well in determining when less
complexity is warranted. Kizilkaya and Tempelman (2005) found that DIC strongly
selected their model with heteroskedastic residual variances when residual variances were
heteroskedastic, but did not strongly select the simpler model when variances were
homoskedastic in linear mixed models and generalized linear mixed models. This is

similar to my results where DIC fairly strongly selected the model that ignored ﬁshery

97

effort data (the most complex model) in cases where ﬁshery effort data were
uninformative, but did not strongly select the models with fewer effective parameters
when ﬁshery effort data were informative. Indeed, Spiegelhalter et al. (2002) and van der
Linde (2005) suggested that DIC does not provide a large enough penalty for model
complexity for models with exponential family likelihoods. This family of distributions
includes the normal and multinomial distributions that I used in the objective functions of
my estimation models. However, increasing the penalty term for DIC would increase
selection of simpler models in cases where a more complex model may be warranted.

Although MCMC methods can be quite time-consuming, calculating DIC should
not be prohibitive in terms of time, given current levels of computer speed. In general,
estimation models took about 1.5-2 hours to run 5,000,000 cycles on a computer with 2.8
gHz processors (Intel Xeon). These times are probably overestimates because I ran these
models longer than was necessary (in most cases) to ensure convergence and to estimate
the variance of DIC estimates for shorter chains. However, models that are structurally
more complex or have more data (i.e., more years or age classes) will require longer run-
times.

In most cases, model averaging provides superior predictive performance than
using only the best model selected by DIC (or some other method) because estimates
from a single model ignore uncertainty in model selection (Hoeting et al. 1999; Bumham
and Anderson 2002; Bumham and Anderson 2004 and references therein). Therefore, I
calculated model average estimates of biomass in the last year with the approximate
Bayesian posterior model probabilities and posterior model probabilities derived from

DIC differences (by adapting the method of Bumham and Anderson (2002) for AIC).

98

Both methods of model averaging had slightly larger MREs (about 1%) and slightly
smaller MSEs (0.1-0.3%) than using only the “best” model. Differences in performance
between the best model and the model average were probably slight because the best
models were the same as or quite similar to the data-generating models. However, in real
world applications it is unlikely that the estimation models will be as similar to the data-
generating reality as was the case in this study. Therefore, model average estimates may
provide a larger increase in performance than in this study. Interestingly, using DIC
differences to estimate model probabilities and average model results seemed to perform
reasonably well, although Spiegelhalter et al. (2002) describe this as an area requiring
more research.

Certainly DIC and approximate Bayes factors are not exhaustive tools for model
selection. Factors such as model plausibility, sensitivity, and examination of residual
patterns should also be considered when choosing among models. However, DIC does
show some promise for helping select among stock assessment models even when models

are quite similar.

99

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Table 3.1. Symbols and descriptions of variables for data-generating and estimation

 

 

models.
Symbol Description Value (if needed in the
data-generating model)
E Average recruitment 1,000,000
A
Ny,a bundance by age and year
Biomass
By

Zy,a

Fy,a

qyf

U11

Total instantaneous mortality rate by age and
year

Instantaneous ﬁshing mortality rate by age

and year

Instantaneous natural mortality rate 0.25

Fishery age-speciﬁc selectivity See ﬁgure 3.2
Survey age-speciﬁc selectivity See ﬁgure 3.2
Fishery effort See ﬁgure 3.1
Fishery catchability

Observed ﬁshery effort

Survey catchability 0.0001

Mean ﬁshery catchability 0.05

102

CM

[y,a

“y,a.f

“y,a.s

Expected ﬁshery catch-at-age

Expected survey catch-at-age

Observed total ﬁshery catch

Observed total survey catch

Proportion of catch-at-age in ﬁshery

Proportion of catch-at-age in survey

Mean weight at age 0.16, 0.45, 0.82, 1.2, 1.55,
1.86, 2.11, 2.3

Deviations for white noise catchability

Deviations for linear increase catchability

Deviations for random walk catchability

Parameters for linear increase in catchability 0.032, 0.00225

Fishing intensity by year

Standard deviation for loge recruitment 1.0
variation
Standard deviation for loge ﬁshery 0.1

measurement error

Standard deviation for loge of survey 0.2-0.8

103

measurement error

Standard deviation for loge catchability
deviations for white noise

Standard deviation for log, catchability
deviations

Standard deviation for log, random walk

catchability deviations

0.2

0.05

0.2

 

104

Table 3.2.

Data-generating and estimation model equations.

 

 

Population model equations Application
(T3.2.1) a - 1 Generation
— Z Z + 7
_ a - 1 1, a N O 2
N1, a Re ’ 7 ~ ’ 07
(T3.2.2a) - 2 Both
_ y ,a
— e
y + l, a + l y, a
(T3.2.2b) — Zy 7 — Zy 8 Both
Ny+1,8 = y,7e +Ny,8e
(T3.2.3) B = Z N w Both
y y,a a
a
(T324) Z = M + F Both
(T3.2.5) F : q E 3 Both
y,a y y a
Catchability model equations
(T3.2.6) White noise Both
log q =log c7 +6 ;5 ~N(0,0’2)
e y. f e f y y 5
(T3 .2.9) Linear increase Generation
q =a+b(y)+£ :8 ~N(O,0'2)
y. f y y 8
(T3.2.1 1) Random walk Estimation
log q =log q +0) :0) ~N[0,0'2)
6’ y + 1. f e y. f y y w
(T3.2.12) Freely estimate fy (ignore ﬁshery eﬁort) Estimation

105

(T3.2.13)

(T3.2. 14)

(T3215)

(T3216)

F
y,a

=fyscuf

Observation model equations

 

F a -z
= y, (l—e y,a)N
y,a Z y,a
y,a
Ty
Cy=e 2Cy,a,r ~N(0,aT)
a
[y,azqssa y,a
I =e I ;v ~N0,cr
y Ey, y ( v)

106

Both

Both

Both

Both

Table 3.3. Objective function equations for statistical catch-at-age analysis simulation

 

study.
(T3.3.1) L=Z£i Objective function
i
(T3.3.2) 2 Fishery catch
€1=—2(loge (Cy )—loge (C y))
20'2 y
(T333) 2 Survey catch-per-effort
l2=—Z(loge (1y )—loge (I y))
20'2 y
(T3.3.4) (, ) Proportion at age in the fishery
l =—n 10 u
3 f2§“y.a,f ge y,a.f
y catch
(T335) [4 =—n 22“ log (a ) Proportion at age in the survey
s y,a,s e y,a,s
y a catch
(T3.3.6) 2 White noise catchability
e =-n yelog J2 7m q+—z(5)
5 2 y
0' y
q
T3.3.7) 2 Random walk catchability
( €5=—ny loge J2 7:0' q+—Z(c’b )
20'2 y y
q
(T3.3.8) ( 1 ] Freely estimateF
I =—n log —
5 y e 20

 

107

Table 3.4. Mean relative error (MRE) and mean square error (MSE) of models selected

using deviance information criterion (DIC), approximate Bayes factors (ABF), only white

noise catchability estimation model (WN), only random walk catchability estimation

model (RW), and only using the estimation model that estimated ﬁshing mortality for

each year independent of effort (FF).

 

 

DIC ABF WN RW FF
Biomass MRE 0.182 0.189 0.279 0.203 0.246
Biomass MSE 0.077 0.087 0.151 0.092 0.159
Fishing Mortality MRE -0.028 -0.037 -0.111 -0.016 0.005
Fishing Mortality MSE 0.046 0.056 0.058 0.066 0.080

 

108

Fishery effort

 

 

Year

Figure 3.1. Baseline effort series used in data-generating models.

109

 
 
  

 

 

—e—-Fishery
0'8 4 ---o--- Survey
2'
g 0.6 ‘
Ti 0.4 -
(D
0.2 1
O T I I n 1
0 2 4 8

Age

Figure 3.2. Fishery and survey selectivity patterns used in data-generating models.

110

FF vs. RW
FF vs. WN
FlW vs. FF
RW vs. WN
WN vs. FF
WN vs. RW

N
01
l
DIODOD

DIC difference
a
l
Cll
O

 

 

T I I
UE WN

Data generating model

01
I
D
_. ’
C as».
.-
as

Figure 3.3. Deviance Information Criterion (DIC) differences among models.
Differences from the best model for each data set are shown. Data-generating models are
indicated by WN for white noise catchability, LI for linear increase in catchability, and
UE for the case where observed effort data were uninformative. Estimation model
comparisons are indicated by X vs. Y (legend), where Y is the hypothetical best
estimation model for the scenario. Positive DIC differences indicate that the model Y is

better than model X. Points are randomly jittered to reduce overlap.

111

Relative Error

Relative Error

1.5-

 

 

 

Biomass
*
1.0-
0.5-
0.0-
’0-5 W l l l I l
ABF DIC FF RW WN
Estimation Method
1.0- . . .
Fishing Mortality
0.51
0.0-
-0.5-
'1 .0 l I I I l l

 

white noise catchability.

ABF DIC FF RW WN
Estimation Method

Figure 3.4. Box plots of relative error of estimates of biomass and average ﬁshing
mortality in year 15. The middle line indicates the median, the box indicates the
interquartile range, and the whiskers indicate the 95% quantile range. Estimation
methods are indicated by ABF for approximate Bayes factors, DIC for deviance
information criterion, FF for the estimation model that freely estimated F, RW for the

estimation model with random walk catchability, and WN for the estimation model with

112

APPENDD( A

Appendix A describes the yellow perch assessment models and additional results
for chapter 1.
Description of Yellow Perch Models

The population submodel predicted how yellow perch numbers-at-age and size-at-
age changed over time, while the observation submodel predicted observed quantities
given the predicted dynamics. Symbols used in the population and observation
submodels are in Table A.1, and equations for these submodels are in Table A2. We
used the posterior likelihood to determine the best ﬁt parameters.
Population Submodel

Total recruitment (deﬁned as age-2 numbers) at the start of each year was
estimated as a free parameter, and the sex ratio at recruitment was assumed to be 1:1 (eq.
A.2.1). Numbers-at-ages 3 and 4 for each sex in the ﬁrst year (1986) were also estimated
as parameters. Numbers at ages 5-9+ in 1986 were calculated based on an assumption
that each of those cohorts had the same abundance at age-4 as was estimated for age-4 in
1986 and suffered an estimated mortality rate that was sex speciﬁc (Wisconsin) or the
same for both sexes (Illinois) (eq. A22 and eq. A.2.3). We used this approach because
sample sizes for ages ﬁve and above were low and these cohorts were not observed for
many subsequent years. For Illinois we used a common mortality parameter for both
sexes because sexes were aggregated in the Illinois survey data for 1986-1988. These
assumptions about numbers-at-age in the ﬁrst year have a relatively small effect on

model estimates, because there were few old yellow perch in 1986.

113

Abundance-at-age of these cohorts were then tracked over time by applying age-
and sex-speciﬁc mortality rates (eq A.2.10). Biomass was simply the product of the
number of ﬁsh in a given length bin and their length-speciﬁc weight summed over sexes,
ages, and lengths. Spawning stock biomass (SSB) was calculated using only females and
a time-invariant maturity schedule based on length, which we estimated by ﬁtting a
logistic function to maturity-at-length data from Indiana waters of Lake Michigan (Ball
State University, unpublished data) outside the model ﬁtting process.

Total mortality rate for a given age and sex was the sum of the natural mortality
rate and the age-, sex-, and year-speciﬁc ﬁshing mortality rates for the two ﬁsheries
(recreational and commercial) (eq. A.2.5). Fishing mortality rates at age for a sex were
calculated as a weighted average of the length speciﬁc ﬁshing mortality rates, with
weights equal to the proportion of ﬁsh that were a given age, sex, and length (eq. A26).

The age speciﬁc rates were calculated from length speciﬁc ones. For each
ﬁshery, ﬁshing mortality rates for a given length bin of yellow perch for the commercial
and recreational ﬁsheries was the product of catchability, effort, and selectivity, and the
log of catchability followed a random walk (eq. A27) and therefore was year-speciﬁc
for each ﬁshery. We modeled selectivity as constant functions of length, based on the
midpoint for each length bin. Note that the ﬁshing process inﬂuences ﬁsh in the same
length in the same way, irrespective of their sex or age. We used a double logistic
function to model the dome-shaped selectivity pattern (Quinn and Deriso 1999) for the
commercial gill net ﬁsheries (Kraft and Johnson 1992) and for the Illinois recreational

ﬁshery during 1997-2000 when a slot limit was in effect (eq. A.2.8). For the Illinois and

114

Wisconsin recreational ﬁsheries (except for the Illinois ﬁshery during 1997-2000), we
modeled the selectivity pattern with an asymptotic logistic function (eq. A.2.9).

Growth was modeled using a stochastic von Bertalanffy growth model, where the
parameters were allowed to vary over time (Szalai et al. 2003). For 1986, mean length at
age (for the beginning of the year) was calculated assuming these ﬁsh had lived under

constant growth conditions with all cohorts starting with mean length-at-age 2 as in 1986,

and experiencing constant Leo and K pre-1986 values (eq. A.2.4). Mean length-at-age 2

was equal for males and females, but changed over time with a random walk (eq. A.2.14).
For years after 1986, mean length-at-ages 3-8 were equal to the mean from the previous
age and year plus the increments from the von Bertalanffy model (eq. A.2.11). The same
model was used to estimate the mean length for the aggregated age-9 and older group, but
this was based on a weighted average of growth expected for age-8 and age-9 ﬁsh, with
weights determined by the contribution of the two ages to this group in the next year (eq.
A2. 12). To estimate mean length at age in the fall, ﬁsh were grown for 8/10th of the year
(eq. A.2.13). Like length at-age-2, asymptotic mean length and the Brody growth
coefﬁcient also changed over time with with a random walk (eq. A.2.14), which were
modeled separately for males and females. The modeled length composition for a given
age was normally distributed with a mean predicted by the von Bertalanffy equation. The
proportion in each one cm length bin was calculated from the corresponding standard
normal cumulative distribution function ((1) ) (eq. A.2.15). The standard deviation of
each normal distribution was the product of the mean length-at-age and an age and sex-
speciﬁc coefﬁcient of variation (CV). We used a hockey stick function to describe how

the CV decreased with increasing age for ages 2 to 5, and then remained constant after

115

age 5. This pattern of decreasing variation in length-at-age with increasing age is
common to many teleost ﬁshes (Bowker 1995), and the CVs we used were based on
observed variation of length-at-age (WDNR, unpublished data).

Observation Submodel

Catch-at-length (in numbers) for the commercial and recreational ﬁsheries was
calculated with the Baranov catch equation (eq. A216 and eq A2. 17). Commercial
catch calculations used numbers-at-length calculated from numbers-at-age reassigned to
length categories based on the fall distribution of length-at-age whereas recreational catch
calculations were based on spring length distributions. This is an approximation that is
intended to account for the fact that the two ﬁsheries are prosecuted at different times
during the year (commercial ﬁshery centered in the fall, recreational ﬁshery in the spring
and summer), that ﬁsh grow during the year, and that ﬁshery selectivity is length-based.
Total catch in numbers was simply the sum over length bins of catch-at-length.
Commercial yield was calculated by multiplying catch-at-length by weight-at-length
(from fall lengths) and summing over length categories.

Catch per effort (CPE) at-length and sex for the survey were calculated as the
product of catchability, selectivity, and numbers-at-length (eq. A218). Catchability of
the survey was sex-speciﬁc for Illinois, but the same for males and females in Wisconsin,
because of differences in survey design between the two surveys. We modeled survey
selectivity using the same logistic function of length used for recreational ﬁshery
selectivity (eq, A.2.9). Total CPE by sex for the survey was the sum over lengths of the
length-speciﬁc survey CPEs. CPE at-age and sex for the survey was calculated as the

product of the survey catchability, numbers at age and sex, and the age— and sex-speciﬁc

116

survey selectivity (given by a weighted sum of length speciﬁc selectivity values) (eq.
A.2.19). For each year proportions of the catch for the ﬁsheries and the survey falling
into each length bin and proportions of the survey catch for each age were calculated for
comparison with observed proportions.

Model predictions of mean length-at-age seen in the survey were calculated by
taking the modeled population length distribution at age and adjusting it for the estimated
survey selectivity (eq. A.2.20).

Likelihood Equations

Our objective function was the posterior negative log-likelihood, A = Z I i , with
i

individual negative log-likelihood components and priors (dropping some ignored

constants) given by l i . Our point estimates minimized this function. One set of

components had the general form:

I.
I

1 2
2)];XJ. (A1)

Where Xj is an assumed standard normal variate and j is an index distinguishing the terms

being summed for the i‘h component. These likelihood components were based on an
assumed independent normal (mean length-at-age) or lognormal distribution (ﬁshery total
catch or survey total catch per unit effort) for deviations between observed quantities and

model predictions or an informative normal prior distribution for random walk errors (for

mean length-at-age 2, L00, K, and catchability for the commercial and recreational

ﬁsheries) and for two parameters of the Illinois commercial ﬁshery selectivity function

(Table A3). We used an informative prior for two of the four Illinois commercial ﬁshery

117

selectivity parameters because the observed length composition of the Illinois

commercial catch contained relatively few measurements, and we based these priors on

the point estimates and standard errors for of the same parameters from the Wisconsin

model. Small constants were added to observed and predicted values (for the lognormal

distributions) to reduce the inﬂuence of very small values (Hampton and Fournier 2001).
An additional set of components took the general form:

ll. =-anZZuT’k loge(uT’k +c) (A2)
k yT

based on our assumption that multinomial distributions led to the observed proportions at
length and age for all data sources for which there were observations. This included a
component for the ﬁshery length compositions, and components for the survey length and
age compositions. The outer sum is over categories of data (k), which were ﬁsheries 1
and 2 (for the ﬁshery length compositions) and sexes (for survey age and length
compositions), and the inner sum was over types (7) of ﬁsh within a category and year
(lengths bins or ages). Small constants (c = 0.0001 for length compositions and c =
0.001 for age compositions) were added to likelihood functions to reduce the effect of
small proportions during model ﬁtting (Fournier and Archibald 1982).

For completeness we note that for parameters other than those with the normal
priors described above, we assumed uniformly distributed priors on the scale they were
estimated. These priors did not enter explicitly into the objective function because they

were implemented by placing bounds on the allowed parameter range during estimation.

118

Table A. 1. Symbols representing parameters, data, and calculated quantities for

assessment models.

 

 

4f

M}

LG

Ly,2

Parameter Deﬁnition
Indicator Variables
a Age-class; 2-9+
y Year; 1986-2001
I Midpoint of each length bin; 8-38 cm
G Sex; male or female
f Fishery; commercial = 1, recreational = 2 or, survey = 3
Estimated Parameters
Ry Recruitments for each year
N Numbers at age in 1986 for ages 3 and 4
1986,a, G
Mortality rate for the ﬁnal ﬁve age classes in the ﬁrst year
in it G
q f Catchability

Parameters for logistic and double logistic selectivity functions

Asymptotic length

Brody growth coefﬁcient

Mean length-at-age 2

Rate of natural mortality time-, sex-, and age-invariant

119

7L0

)nG

y,f

y,a,l,G
Fy.a.l,G.f
py,a,l,G
Ny,a,l,G
Ny,a,l,G

Ly,a.G

l)

y,a.G
Ly,a.G

Sf

)

y,a,l,G

“y,a,l,G,f

Random walk deviations for mean length-at-age 2
Random walk deviations for L“

130

Random walk deviations for K “G

Random walk deviations for catchability

Calculated Quantities

Total instantaneous mortality rate

Instantaneous rate of ﬁshing mortality

Proportions-at-length for each age

Numbers-at-age, length in the beginning of the year, and sex in year y
Numbers-at-age, length in the fall of the year, and sex in year y

Mean length-at-age in population in beginning of year

Model predicted mean length-at-age measured by survey

Mean length-at-age in population in fall

Selectivity

Survey index of abundance

Model prediction of proportions of catch-at-age, length, and sex

120

y.l.f

E)

N

1

y,a,G
y,a,l,G
“y,a,l,G,f
y.l.f =2

y,l.f=1

Model prediction of catch
Model predicted commercial yield (kg)

Likelihood Weighting Components
Sample size of ﬁsh aged for the mean length-at-age likelihood function
and effective sample size for age and length compositions

CV for ﬁshery catches
Standard deviation for mean length-at-age 2 random walk deviations

Standard deviation for L“ 0 random walk deviations
Y.

Standard deviation for K110 random walk deviations

Standard deviation for ﬁshery catchability random walk deviations

Standard deviation for commercial selectivity prior for Illinois

Data

Observed mean length-at-age in the survey

Observed CPE in the survey

Observed proportions at age and length in the ﬁsheries
Harvest (numbers) in the recreational ﬁshery

Yield (kg) in the commercial ﬁshery

121

Fishery effort

Weight-at-length

Number of ﬁsh aged by age year and sex

Mean parameter for the prior of commercial selectivity function for
Illinois

Instantaneous rate of natural mortality (age- and sex-independent)

 

122

Table A2. Equations for population and observation submodels.

Population submodel

Recruitment, initial abundances at age, initial mean length at age
R

- (a — 4)ZinitG
Ny=1986 a G =Ny=l986 a=4 Ge ;a>4, Wisconsin (A.2.2)
—(a—4)Zinit ' ' 2 3
Ny=1986,a,G =Ny=1986,a=4,Ge ,a>4,Illrnors (A. . )

 

 

 

 

 

 

 

 

Ly: 1986,a+1,G = Ly = 1986,a,G +
— A.2.4
L _L 1_e Ky=prel986,G ( )
°° y=l986,a,G
y=pre1986,G
Mortalityrates
2
2110:“+ Z Fy.a.G.f (A25)
f=l
Fy’a’G’f :§pyaa.l,GFY.l,f §py,a.l,G=l (AH26)
Eyf
F .___ E = ’ A.2.7
y.l.f qxf y.fsl.f qy+1,f qy,fe ( >
f V l
51f = 1 1- 1 (A28)
’ - A (0—1 ] - 1 (l)—,t ]
4.
\l+e [l’f 3’f A 1+e 2,f f)
1
51f = (A.2.9)
’ — ,1 (l)-/l ]
1+e 1’f 2’f

123

Population and length-at-age dynamics

 

 

 

 

 

 

 

_ y,a,G
Ny+1,a+l,G _Ny,a,Ge (A.2.10)
L 2L +(L —L )(1—e—Ky’G) (A211)
y+l,a+l,G y,a,G coy G y,a,G ° '
—K
_ _ #3
Ny,a=8,G Ly,a=8,G+(LooyG Ly,a=8,G)1 e
Ly+1,a=9,G = N +N +
y,a=8,G y,a=9,G
—K
_ _ x0
Ny,a=9,G Ly,a=9,G+[Looy G Ly,a=9,G]l e
(A.2.12)
Ny,a=8,G+Ny,a=9,G
—O.8K
- G
= —L l—e y,
Lyﬂ, G Ly,a,G+(L°°yG y,a,Gx ) (A.2.l3)
6 7
L +12: 28 y L00 :1” 8 LG
y ’ y’ y+l,G y,G
a7
= ”9 A.2.14
Ky+1,G y,Ge ( )
r \ r \
(l+1)-L l—L
p alG= y’a’G -<I> ——y’“’G (A.2.15)
y, 9 9 0,1 0.1
K a,G } K (1,0 )

124

Observation submodel

F -Z
A )5],le yl '

C =-————— 1— , N
y.l,f=1 Zyl ( e )3 y,l,G
F —Z

,. y,l.f=2 yl
C =___1_ ’ N
fo=2 Zyl ( e )3 LLG

A

5,1,0 = qG,f = 3‘z,f = 3Ny,l,G

A

[y,a,G = qG,f =3Ny,a,G§‘1,f = 3py,a,l,G

A
~

Fm =3Py,a,z,G(’)

 

y,a,G =
§3Lf=3

125

(A.2.16)

(A.2.17)

(A.2.18)

(A.2.19)

(A.2.20)

Table A3. Speciﬁcation of terms for normal and lognormal negative log-likelihood

components (see equation Al).

 

Standard normal variate Squared variates summed over

these indices

 

~ _ .2. y, a, G
(Ly,a,G Ly,a,G)/(ay,a,G/ my,a,G)
(logeWy,f=l-10gewy,f=1)/Uf=1 y

(loge(Cy’f=2)-loge(Cy,f=2))/0'f=2 y

_ " ,G
(loge(ly,G) loge(1y,G))/df=3 y
6y/0'6 Y

,6
7y,G/Uy,G and my, Glow, G y
e /0' y,f
, e
y f f

j.j<3

 

 

126

Table A4. Results of sensitivity analyses of changes of weights of data sources in the
objective function for yellow perch catch-at-age models for Illinois and Wisconsin waters
of southwestern Lake Michigan. Differences from baseline estimates are displayed as

percentages. Baseline model estimates of abundance (N; 10005), biomass (B; 1000 kg),

mean ﬁshing mortality for females age-4 and older (R4 + females), and mean ﬁshing

mortality for males age-4 and older ( f4 + males) for 2002 are displayed for comparison.

In two cases the model’s parameter estimates failed to converge to values that minimized

the objective function and these are denoted by NC.

 

Illinois Baseline Adjustment N B F
4 + 4 +

”1'1

value factors
females males

 

 

Baseline 4,790 81 8 0.058 0.025
Commercial yield 0.0025 5 -3.1 -3.1 3.3 3.2
Commercial yield 0.0025 0.2 0.7 0.7 -0.8 07
Commercial catchability 0.06 5 -16.1 -16.6 12.8 5.8
Commercial catchability 0.06 0.2 41.7 43.1 -30.5 -28.7
Commercial length 32 5 -10.2 -10.9 12.8 5.8
Commercial length 32 0.2 -2.0 -0.7 0.4 5.0
Recreational harvest 0.01 5 2.3 2.0 -5.4 -6.7
Recreational harvest 0.01 0.2 -0.7 -0.6 3.4 4.1
Recreational catchability 0.06 5 3.5 2.9 -0.5 -2.7
Recreational catchability 0.06 0.2 -7.6 -6.2 3.4 9.5
Recreational length 367 5 -24.1 -24.4 34. l 7 .6
Recreational length 367 0.2 17.6 1 1.5 -15.1 -12.1

127

Survey CPE, females
Survey CPE, females
Survey female ages
Survey female ages
Survey female lengths
Survey female lengths
Survey CPE, males
Survey CPE, males
Survey male ages
Survey male ages
Survey male lengths

Survey male lengths

Female Loo

Female LC>0

Female K

Female K

Male Loo

Male Lco

Male K

Male K
Length-at-age 2
Length-at-age 2
M

M

0.19

0.19

27

27

61

61

0.22

0.22

53

53

58

58

0.0006

0.0006

0.0006

0.0006

0.0006

0.0006

0.0006

0.0006

0.01

0.01

0.37

0.37

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

0.2

1.2

0.8

30.7

-32.6

17.2

-10.1

25.1

-21.8

4.3

-4.9

17.0

3.2

5.3

-1.4

-l9.1

0.0

-O.2

-3.2

6.1

12.8

29.3

-31.0

11.0

17.0

-l7.1

4.2

2.7

-13.9

2.1

4.5

-l.2

—4.7

-15.2

5.7

-11.5

16.3

-22.8

45.8

7.2

-20.4

23.2

4.0

5.4

7.3

6.2

16.5

-3.8

2.7

-4.4

1.8

5.4

19.6

1.0

17.2

-15.8

-25.8

53.0

~30.7

20.5

-248

40.1

-4.5

6.4

21.5

14.0

15.3

30.5

-1.9

-6.4

0.9

37.8

-l.8

0.5

5.0

 

128

 

Wisconsin

 

 

Baseline 1,690 356 0.075 0.060
Commercial yield 0.0125 5 -3.7 -3.8 4.2 3.8
Commercial yield 0.0125 0.2 3.9 4.0 -4.0 -3.8
Commercial catchability 0.16 5 25.8 25.2 -20.3 -21.2
Commercial catchability 0. 16 0.2 -22.8 -22.1 28 .2 31 .4
Commercial length 43 5 -7.9 -7.7 8.3 9.0
Commercial length 43 0.2 5.8 2.4 -0.8 -8.3
Recreational harvest 0.01 5 4.0 3.7 6.0 5.5
Recreational harvest 0.01 0.2 -2.0 -1.9 0.6 0.7
Recreational catchability 0.06 5 6.6 6.3 -7.8 —8.2
Recreational catchability 0.06 0.2 -16.1 -15.4 27.1 28.7
Recreational length 141 S -1.4 3.2 -3.3 5.7
Recreational length 141 0.2 -7.7 -8.5 9.4 7.8
Survey CPE, females 1.06 5 4.0 3.8 -3.8 39
Survey CPE, females 1.06 0.2 -4.2 -3.8 4.0 4.6
Survey female ages 31 5 13.6 10.7 -10.0 -7.9
Survey female ages 31 0.2 -17.9 -17.2 21.8 20.1
Survey female lengths 45 5 NC NC NC NC
Survey female lengths 45 0.2 0.2 —0.1 0.6 -0.1
Survey CPE, males 0.92 5 25.0 24.6 -20.1 -20.6
Survey CPE, males 0.92 0.2 -41.3 -40.9 72.0 74.]
Survey male ages 50 5 NC NC NC NC
Survey male ages 50 0.2 -33.3 -33.3 50.6 52.5

129

Survey male lengths

Survey male lengths

Female Loo

Female L<>0

Female K

Female K

Male Loo

Male Loo

Male K

Male K
Length-at-age 2
Length-at-age 2
M

M

63

63

0.0025

0.0025

0.0025

0.0025

0.0025

0.0025

0.0025

0.0025

0.01

0.01

0.37

0.37

0.2

0.2

0.2

0.2

0.2

0.2

1.2

0.8

20.0

6.6

-1.2

-0.4

-l.3

13.7

-18.7

-34.3

49.4

14.4

8.9

-1.0

-1.0

0.3

13.0

~18.3

-2.0

1.8

0.3

-35.0

50.8

-11.9

0.2

1.5

0.9

0.6

-11.8

23.2

2.0

-1.7

-1.2

7.4

59.0

-36.1

-19.6

1.5

0.1

1.7

0.2

-13.1

24.8

2.0

-1.9

0.0

6.0

58.3

-35.5

 

130

Recreational

 

 

 

 

 

03):: - - -c- . - Wisconsin
' 1, —o—lllinois
g 0.025 9‘
E 0.02 -
3 0.015 -
(U
o 0.019
0.005 4
ofritlvlrrrrrrrrr
1986 1991 1996 2001
Year
Commercial
:2 1 o ...o... Wisconsin 7' 3::
1'4 " +lllinois 0'14
g ' , x :0'12'E
1’3 -o.1 3
g - 0.08 §
- 0.06 8
E . \ . . -0.04='
0.21 °"°"°.....°‘ o 40.02
0 1 1 1 1 lﬁ+1 1 1 1 0

 

 

6°6@69099\&99¢9909é‘
9 .9 .9 .9 .9 9 .9 .9 .9 .9 .9 9

Year

Figure A. 1. Estimated catchability coefﬁcients for Wisconsin and Illinois recreational

and commercial ﬁsheries in southwestern Lake Michigan during 1986-2002.

131

APPENDIX B

Appendix B describes the parameterization of estimation models used in chapter 3
to reduce correlations among parameters. The MCMC algorithm I used was very
sensitive to parameter correlations greater than about 0.8. Under these conditions, the
MCMC algorithm mixed very poorly and produced very “sticky” MCMC chains (i.e.,
chains with high autocorrelation). Therefore, I reparameterized aspects of the models to
reduce these correlations. All parameters described below were estimated on the log
scale. Two groups of parameters were highly correlated within each group: parameters
that determined overall scale of population size, and selectivity parameters for the ﬁshery
and survey. Parameters that determine the overall scale of the population size included,
in this case, mean recruitment, mean abundance at age in year 1, ﬁshery catchability (or
mean F in the model that ignored ﬁshery effort data), and survey catchability. In order to
minimize correlation among these parameters, I parameterized the model by estimating
the log, of mean recruitment and a deviation from this for each of these other “scale-
setting” parameters. The other parameters that had high correlations were the selectivity
at age for the ﬁshery and the survey. To reduce these correlations, the models were
parameterized to estimate deviations from a mean loge selectivity that was forced to equal
zero. This constraint serves to make the selectivity parameters identiﬁable and not
confounded with the associated catchability (for ﬁshery or survey), in the same way that
the more usual approach of setting selectivity to 1.0 for a fully selected age (e.g.,

Fournier and Archibald 1982) does.

132

RA

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