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Using perceptions of data accuracy and empirical weighting of information: Assessment of a recreational fish population

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Recreational fisheries management is often compromised by limited information of variable quality from several sources. We develop a form of catch-age analysis to combine uncertain information from creel surveys, age composition, and mark-recapture estimates of abundance. Four systems are used in weighting annual observations: equal, inverse of squared coefficients of variation (CV-2), perceptions of accuracy, and a combination of the latter two. The model is applied to a humpback whitefish (Coregonus pidschian) population in Alaska and evaluated for model fit, parameter uncertainty, conservative forecasts of exploitable abundance, and biological plausibility. The probability of forecasted stock abundance occurring below a threshold level defined by an agency management plan is evaluated for various recruitment and exploitation scenarios. The perception model is judged to be best with the use of the analytic hierarchy process, a decision-making technique. By incorporating perceptions into fisheries decision-making, beliefs in the accuracy of uncertain information are made explicit. In a conservative context, fishery management decisions should include reducing risk to the stock in the setting of harvest policy and in the selection of the assessment model.
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Using perceptions of data accuracy and empirical
weighting of information: assessment of a
recreational fish population
Margaret F. Merritt and Terrance J. Quinn, II
Abstract: Recreational fisheries management is often compromised by limited information of variable quality from sev-
eral sources. We develop a form of catch-age analysis to combine uncertain information from creel surveys, age com-
position, and mark–recapture estimates of abundance. Four systems are used in weighting annual observations: equal,
inverse of squared coefficients of variation (CV–2), perceptions of accuracy, and a combination of the latter two. The
model is applied to a humpback whitefish (Coregonus pidschian) population in Alaska and evaluated for model fit, pa-
rameter uncertainty, conservative forecasts of exploitable abundance, and biological plausibility. The probability of fore-
casted stock abundance occurring below a threshold level defined by an agency management plan is evaluated for
various recruitment and exploitation scenarios. The perception model is judged to be best with the use of the analytic
hierarchy process, a decision-making technique. By incorporating perceptions into fisheries decision-making, beliefs in
the accuracy of uncertain information are made explicit. In a conservative context, fishery management decisions
should include reducing risk to the stock in the setting of harvest policy and in the selection of the assessment model.
Résumé : La gestion des pêches sportives est souvent compromise par le fait que l’information dont on dispose est li-
mitée, de qualité variable, et issue de sources diverses. Nous avons mis au point un mode d’analyse de l’âge à la cap-
ture pour combiner l’information incertaine fournie par les enquêtes auprès des pêcheurs, la composition par âge et les
estimations de l’abondance par marquage–recapture. Nous avons pondéré les observations annuelles à l’aide de quatre
systèmes : égalité, inverse des coefficients de variation carrée (CV–2), perceptions de l’exactitude, et enfin une combi-
naison des deux derniers. Le modèle est appliqué à une population de corégone à bosse (Coregonus pidschian)de
l’Alaska, et nous avons évalué l’ajustement du modèle, l’incertitude des paramètres, les prévisions prudentes de
l’abondance exploitable et la plausibilité biologique. Nous avons évalué la probabilité de voir l’abondance prévue du
stock tomber au-dessous d’un seuil défini par le plan d’un organisme de gestion pour divers scénarios de recrutement
et d’exploitation. C’est le modèle sur les perceptions qui est jugé le meilleur, employé avec la méthode décisionnelle
de hiérarchie multicritère. En intégrant les perceptions dans la prise de décision sur les pêches, on rend explicite les
croyances sur l’exactitude de l’information incertaine. Dans un contexte de gestion prudente, les décisions de gestion
des pêches doivent inclure la réduction du risque pour le stock dans l’établissement de la politique de prélèvement et
dans le choix du modèle d’évaluation.
[Traduit par la Rédaction] Merritt and Quinn 1469
Introduction
Fisheries management is a decision-making process in
which managers must often work with uncertain parameters
in models that may inadequately describe the process and
with information of unknown precision and accuracy. Be-
cause the sources of uncertainty will never be completely re-
moved, fisheries managers have personal perceptions about
various sources of information and the plausibility of esti-
mates, which routinely leads to intuitive judgments in their
decision-making. Perception is a type of inference that oc-
curs when mental images are converted into interpretations,
resulting in beliefs. Perceptions are particularly important in
situations where there is little current knowledge, yet deci-
sions must be made on the basis of best available information.
While there is no question that perceptions play a role in
decision-making (Tait 1988; Badinelli and Baker 1990) and
perceptions are important to fisheries management decisions
(Pearse and Walters 1992), little research has been con-
ducted to quantify the influence of perceptions on solutions
to a fisheries problem. Martin (1979) used subjective proba-
bility estimates to determine the optimum number of units of
gear in salmon drift gillnet fisheries. Geiger and Koenings
(1991) incorporated subjective information about spawner–
recruit parameters based on perception of smolt characteris-
tics and lake system limitations into a Bayesian framework
to evaluate escapement goals for sockeye salmon (Onco-
rhynchus nerka). Weighting according to perceptions is not
commonly found in abundance estimation procedures, but it
Can. J. Fish. Aquat. Sci. 57: 1459–1469 (2000) © 2000 NRC Canada
1459
Received July 16, 1999. Accepted March 2, 2000.
J15253
M.F. Merritt.1Alaska Department of Fish and Game,
Sport Fish Division, 1300 College Rd., Fairbanks,
AK 99701, U.S.A.
T.J. Quinn, II. Juneau Center, School of Fisheries and Ocean
Sciences, University of Alaska Fairbanks, 11120 Glacier Hwy.,
Juneau, AK 99801-8677, U.S.A.
1Author to whom all correspondence should be addressed.
e-mail: peggy-merritt@fishgame.state.ak.us
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is employed in decision analysis (see Srinivasan and
Shocker 1973).
An alternative way of incorporating uncertainty into decision-
making is to use models that explicitly weight by the inverse
of the variance, or equivalently the statistical information, of
the data. Inverse variance weighting is commonly employed
in weighted regression analysis (Steel and Torrie 1960) and
in abundance estimation (following Paloheimo, cited in
Seber 1982) when observations have unequal variances. In
weighted regression models the inverse variances are used as
weights in the residual sum of squares (Abraham and
Ledolter 1983), and inverse variance weighting usually leads
to smaller variances in parameter estimates.
This study examines the influence of perceptions and em-
pirical inverse variances (expressed in relative terms as the
inverse of squared coefficients of variation, or CV–2)onthe
performance of a catch-age model. Perception used here re-
fers to a belief in the accuracy of information. Where harvest
can be sufficiently sampled for age structure and auxiliary
information is available, catch-age analysis (Deriso et al.
1985; Funk et al. 1992) can be a useful analytical tool for
modeling the interaction of the fish stock with a recreational
fishery. We use this tool with the humpback whitefish
(Coregonus pidschian) population in the Chatanika River,
Alaska, which is subject to a recreational spear fishery. The
goal is to account for risk in the threshold harvest manage-
ment of humpback whitefish as specified by an agency man-
agement plan described below. One common measure of risk
in fisheries management involves the probability of an unde-
sirable event (e.g., stock abundance falling to an unaccept-
able level) leading to some consequence or loss (e.g.,
diminished social or economic benefits derived from fishing)
in relation to a reference point (Francis and Shotton 1997).
Four systems for weighting annual observations are exam-
ined: equal (hereafter referred to as the base model), CV–2,
perceptions of accuracy by a knowledgeable individual, and
a combination of perception and CV–2 (hereafter referred to
as the combined model). Equal weighting of data has often
been used in catch-age analysis (Funk et al. 1992; Zheng et
al. 1993).
Model performance is judged using the analytic hierarchy
process (Saaty 1990), a decision-making technique that leads
to a “best” model. Four criteria are identified to judge model
performance: model fit of the data sets, parameter uncer-
tainty, conservative forecasts of exploitable abundance, and
biological plausibility of estimates.
Conservatism in setting harvest policy has been advocated
when there is uncertainty (Clark 1985), but to incorporate
conservatism in the stock assessment method is a concept
that is now receiving more attention. Management agencies
often have competing and uncertain assessments, in which
they usually choose the more conservative one. Thus, a
model with a low probability of dropping below a specified
abundance level would be considered less conservative than
one with a higher probability. The level of acceptable risk is
implicit in the assessment that is most preferred (Saaty
1990). Thus, fisheries managers express their level of risk
acceptance in setting harvest policy and in selecting the
stock assessment method.
We also use the models to assess the probability of stock
abundance declining below a threshold level in the harvest
policy by forecasting exploitable abundance for varying ex-
ploitation and recruitment scenarios using age-structured
models. Here, a harvest policy with a lower probability of
dropping below a prespecified abundance level would be con-
sidered more conservative than one with a higher probability.
Materials and methods
Approach
Information from a typical recreational fishery includes (i) total
harvest, (ii) age composition, and (iii) a survey or index of exploit-
able abundance. In the Application section, we illustrate how un-
certainties in this information can be modeled. We then describe
how these sources of information can be combined into an age-
structured model to provide estimates of abundance, age vulnera-
bility, and exploitation rate. Annual weights for the data sources
come from quantifying a manager’s perceptions of data accuracy or
incorporating a CV–2 scheme.
Data
Total harvest, Hy, in year yand its variance V[Hy] are usually es-
timated using standard sampling survey techniques (e.g., Thomp-
son 1992) by extrapolating sample harvest to total harvest. Age
composition is usually obtained from a simple random sample or a
two-stage length–age protocol (Quinn and Deriso 1999). For sim-
ple random sampling, age samples of size nyare often assumed to
be randomly drawn from a multinomial distribution. The estimated
proportion at age ain year y,,
pay
, is then ,
pay =na,y/ny, where na,y
is the number of fish of age ain the sample. Estimated harvest at
age is then ,
Hay =,
Hp
yay
, where Hyis the estimated harvest from
the creel survey in year y.
A survey or fishery index of abundance is obtained from a par-
ticular sampling strategy or analytical method. Approaches include
catch per unit of effort, transect, mark–recapture, and removal
methods (Seber 1982).
Error structure
It is necessary to describe the error structure of the data in order
to develop a model. Like Kimura (1990), we prefer to use empiri-
cal distributions to model error structure. Creel harvest estimates
are often assumed to be normally distributed. The multinomial
model for age composition error is assumed (Methot 1990). A
common alternative is lognormal error for harvest at age. Addi-
tional approaches to modeling the multinomial distribution include
adding a dispersion parameter (Fukushima et al. 1998) and model-
ing observed age composition with the Dirichlet distribution (Wil-
liams and Quinn 1998). Survey error structures are often normal or
lognormal in distribution; we present an inverse normal distribu-
tion in the application.
Annual weights
Many fisheries managers know the circumstances under which
data are collected and have subjective perceptions of the data’s ac-
curacy. Perceptions can be influenced by observations of experi-
mental design, sampling strategy, environmental conditions under
which sampling is conducted, and expertise of personnel involved.
These perceptions can be quantified by asking the manager to rate
his/her beliefs in the accuracy of annual abundance, harvest, and
age composition estimates using a scale between 0 (inaccurate) and
1.0 (accurate) where increments of 0.1 represent differences in de-
grees of perceived accuracy. By assigning a numerical score to es-
timates each year, data can be weighted (qx) according to their
perceived accuracy.
Alternatively, the weights (qx) can be chosen inversely propor-
tional to the estimated variation of the data. In practice, we have
found that this weighting scheme can effectively give weights near
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0 to large amounts of data when the data vary over orders of mag-
nitude (such as catch data). In this situation, it is sometimes better
to weight by the inverse of the relative variation in the data as
measured by the CV–2. (Various process errors could be added;
however, we were unable to do so due to the limits of the data.) A
combined model for weighting is obtained by multiplying the CV–2
and perception weights together for each annual observation and
then normalizing the result. A naive approach is to use equal
weighting in which each annual estimate has the same weight; this
is the most common situation in practice.
Catch-age models
In catch-age analysis, an age-structured population model is de-
veloped to estimate catch, age composition, and related informa-
tion from population parameters for recruitment, gear selectivity,
fishing mortality, and catchability (Deriso et al. 1985; Quinn and
Deriso 1999). Parameter estimates are often obtained by fitting the
model to data using nonlinear least squares or maximum likelihood
approaches according to some objective function. Auxiliary infor-
mation in the form of indices or survey estimates of abundance is
needed to stabilize parameter estimates.
Without loss of generality, we minimize the objective function.
In the case of our typical recreational fishery, the three objective
function components are the harvest sum of squares Oharvest,the
negative of the log likelihood component for age composition Oage,
and the sum of squares for the survey index Oindex.AgivenOcom-
ponent can be written
(1) OO
x
xx
=å
lq
where lis an overall weight used to scale Ocomponents, qxis a
data-specific weight given to Ocomponents, Oxis a data-specific
sum of squared deviations of observed and estimated values (or re-
lated objective), and xis the data index (such as year, age, or some
combination).
The qs represent the weight of influence given to a subset of
data within an individual Ocomponent; in our study, they are de-
termined annually. Scaling coefficients (l) are weights for the Oxs
from different data sources. Theoretically, scaling coefficients are
ratios of variances between the catch data and another data source
(Quinn and Deriso 1999), although in many situations, these vari-
ances are not known, so must be estimated. Alternatively, scaling
coefficients can be used to bring the Os closer in magnitude
(Merritt 1995) for better parameter estimation or to downweight a
contradictory data source.
Forecasting exploitable abundance
Forecasts are used to evaluate population trends, set catch limits,
and examine the probability of the population falling below some
level. In age-structured models, forecasting is often a three-step
process: (1) decide on a harvest rate or fishing mortality to use,
(2) project abundance at other ages from the most recent abun-
dance estimates from the model, and (3) forecast recruitments in
future years using a deterministic or stochastic model. In step 3,
the deterministic model is usually a constant recruitment, density-
independent (recruitment proportional to spawning biomass), or
spawner–recruit model such as the Ricker or Beverton–Holt. A sto-
chastic model usually combines a deterministic model with a nor-
mal or lognormal error structure. Confidence intervals can be
developed from the bootstrap distribution of forecasts using the
percentile method (Efron and Tibshirani 1993). Alternative recruit-
ment scenarios can be subjected to the same forecasting treatment
and the results summarized in a decision table (Hilborn et al. 1994).
Choosing the best model
Model performance can be judged using the following criteria:
(i) a model should fit the individual data components well with the
residual mean square (RMS) measuring the fit of each model com-
ponent, these fits then being compared pairwise using Lehmann’s
asymptotic test (Lehmann, cited in Criddle and Havenner 1991);
(ii) a model should produce parameter estimates with small vari-
ance and minimal bias; as a summary measure of bias and variabil-
ity, we use the root mean square error RMSE = SD bias
22
+;
(iii) forecasts of exploitable abundance should be conservative; we
evaluate the probability of forecasted exploitable abundance falling
below a threshold level from 1000 bootstrap iterations; and (iv)es
-
timates should be plausible and within the realm of productivity of
the stock.
As these criteria need not support the same model, a way of syn-
thesizing these considerations is needed. The analytic hierarchy
process (Saaty 1990) incorporates quantitative measures and intu-
itive judgments into a mathematical process used to describe the
problem and arrive at a solution. The problem of selecting the best
model is structured as a hierarchy, with goal at the top, criteria to
be considered in subsidiary levels, and models to be evaluated at
the bottom. Weighting of criteria used in judging model perfor-
mance is accomplished prior to fitting the models using the com-
puter program Expert Choice (Expert Choice (1998), Decision
Support Software, Pittsburgh, Pa.). Pairwise judgments of the rela-
tive importance of two criteria are expressed using an inverse posi-
tive ratio scale devised by Saaty (1990), modified in this paper.
The modified scale consists of values 1/5 to 5, with verbal mean-
ings as follows: 1/5 (strongly unimportant or undesirable), 1/3
(moderately unimportant), 1 (equally important), 3 (moderately im-
portant), and 5 (strongly important). Compromise judgments are
interpolated. A scoring system is then set up and model perfor-
mance is rated against criteria. The total score for each model is
calculated by adding the weighted proportions of all criteria for
that model. The total score of each model determines its rank (for
details, see Saaty 1990).
Application
The humpback whitefish population in the Chatanika River,
Alaska, is subject to a recreational spear fishery (Merritt 1995). To
achieve sustained yield, management follows a threshold harvest
policy that is likely to reduce the risk of a stock collapse and to en-
hance long-term productivity (Quinn et al. 1990). The threshold
abundance level of 10 000 humpback whitefish was previously de-
termined heuristically, based on the observed range of historical
abundances and harvests. Higher exploitation rates are allowed as
the population increases: at 10 000 – 15 000 spawners, the allow-
able maximum harvest rate is 10%; at 15 000 – 20 000 estimated
spawners, harvest is held to the “midrange” of between 10 and
15%. When estimated spawner abundance exceeds 20 000 fish,
harvest is allowed to approach the 15% exploitation limit.
Previous assessments involved subjective assimilation of the in-
formation about harvest, age composition, and abundance. We use
catch-age analysis to assimilate the information in order to provide
a firmer quantitative basis for the assessment. We incorporate a
manager’s perceptions about the population, fishery, and historical
sampling of the fish stock and harvest directly into the assessment,
as described below. Forecasting future age distributions of abun-
dance for alternative scenarios about recruitment allows us to eval-
uate exploitation rates specified in the management plan, given
various states of the population.
Data
Three data sources are used: (i) harvest at age estimates from a
creel survey (1986–1992 except 1991 when the fishery was closed),
(ii) estimates of exploitable abundance from mark–recapture exper-
iments (1986–1992 except 1990 when the mark–recapture experi-
ment failed), and (iii) observed age frequencies (1986–1992), ages
3 through 10+ (Table 1).
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Merritt and Quinn 1461
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Error structure
In examining the empirical distributions of the three data
sources, we found no reason to reject the assumption of normality
for creel estimates of harvest or the multinomial assumption for
age composition data. The mark–recapture experiment followed
the Petersen design of single release, single recapture (Seber
1982). Performing a bootstrap procedure showed that during sev-
eral years, distributions of abundance are skewed due to low num-
bers of recaptures. The inverse transformation of the mark–
recapture estimates provides a good approximation to normality, in
accord with theory (Seber 1982).
Annual weights
To incorporate perceptions about data accuracy, we asked the
fishery manager to rate the quality of annual abundance, harvest,
and age composition estimates on a scale of 0 to 1. For example, a
low score (0.2) was assigned to the 1989 abundance estimate (Ta-
ble 2) because the manager perceived that the sampling strategy did
not meet the assumption of closure in the mark–recapture estimator.
Empirical sampling variances differed interannully due to large
changes in the magnitudes of harvest and fish abundance and in-
creasing efficiency in creel survey and mark–recapture study de-
signs over time. The most weight in the harvest data was placed on
the estimate in 1992 because its CV was the smallest, compared
with prior years’ harvest estimates (Table 2). Because the CV–2
weighting scheme gave a weight for harvest in 1992 greater in
magnitude than in 1986 and 1987, this gave less weight to harvest
in earlier years. Similar differences in annual harvest weights occur
in the combined model because its weighting scheme is the prod-
uct of the CV–2 and perception weights (Table 2).
Catch-age models
We used an Excel spreadsheet form of the catch-age analysis ap-
proach of Deriso et al. (1985), developed in Funk et al. (1992). The
specific form of the catch-age model used in this paper is given in
Merritt (1995). The four weighting models are applied to the
humpback whitefish data set in accord with the principles de-
scribed above. The overall objective function comprises the sum of
squares for the harvest estimates, the negative of the log likelihood
multinomial component for age composition, and the sum of
squares component for the inverses of exploitable abundance from
the mark–recapture experiment.
A bootstrap procedure is used to obtain standard errors for pa-
rameter estimates (Efron and Tibshirani 1993), wherein age com-
position is sampled with replacement from the multinomial
distribution, and residuals are resampled and added to predicted
values for the harvest and survey components. The bootstrap pro-
cedure is repeated 1000 times for each of the four models.
Forecasting exploitable abundance
The four models are used in the forecasting procedure to exam-
ine their conservatism and to evaluate the management plan.
Starting with estimated abundances by age in 1993 for each boot-
strap replication (except for age 3, which must be forecasted), the
exploitable abundance is forecasted for years the 1993–1998. Per-
centile ranges at the 2.5 and 97.5% levels are generated to approxi-
© 2000 NRC Canada
1462 Can. J. Fish. Aquat. Sci. Vol. 57, 2000
Age (years)
Year Harvest (SE)aAbundance (SE)b345678910+
1986 2528 (914) 14 906 (3172) 0 0.205 0.356 0.164 0.151 0.096 0.027 0
1987 4577 (926) 28 165 (3434) 0.028 0.290 0.357 0.201 0.079 0.026 0.015 0.004
1988 3572 (293) 41 211 (5155) 0.005 0.156 0.421 0.284 0.099 0.026 0.009 0
1989 3835 (491) 17 322 (1655) 0 0.013 0.215 0.441 0.202 0.086 0.023 0.020
1990 956 (34) 0.010 0.050 0.257 0.330 0.221 0.083 0.023 0.025
1991 0 15 313 (2078) 0.029 0.038 0.058 0.149 0.251 0.208 0.129 0.137
1992 393 (9) 20 180 (1633) 0.011 0.064 0.056 0.075 0.184 0.229 0.176 0.204
aThe fishery was closed in 1991.
bNo mark–recapture abundance estimate was generated in 1990.
Table 1. Estimates of harvest, exploitable abundance, and proportion by age for humpback whitefish, Chatanika River, 1986–1992.
Year HarvestaAbundancebAge composition
Base model
1986 1 1 1
1987 1 1 1
1988 1 1 1
1989 1 1 1
1990 1 0 1
1991 0 1 1
1992 1 1 1
CV–2 model
1986 0.003 0.148 0.018
1987 0.010 0.450 0.516
1988 0.062 0.429 0.367
1989 0.024 0.724 1.000
1990 0.250 0 0.009
1991 0 0.362 0.214
1992 1.000 1.000 0.450
Perception model
1986 0.6 0.7 0.6
1987 0.7 0.7 0.6
1988 0.8 0.7 0.6
1989 0.6 0.2 0.6
1990 0.9 0 0.6
1991 0 0.8 1.0
1992 0.9 0.8 1.0
Combined model
1986 0.002 0.130 0.018
1987 0.008 0.394 0.516
1988 0.055 0.375 0.367
1989 0.016 0.181 1.000
1990 0.250 0 0.009
1991 0 0.362 0.357
1992 1.000 1.000 0.750
aThe fishery was closed in 1991.
bNo mark–recapture abundance estimate was generated in 1990.
Table 2. Weights of influence (q) assigned to each annual objec-
tive function component (harvest, mark–recapture abundance, and
age composition estimates) for the four models used in estimat-
ing abundance of humpback whitefish in the Chatanika River.
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mate the amount of forecast variability in total annual exploitable
abundance.
Two recruitment scenarios are examined: constant and a density-
independent relationship:
(2) Rt=aENt–3
Constant recruitment is based on a random sample of recruitment
during 1986–1992. The density-independent relationship between
recruits and spawning stock size is uncertain due to only four data
points, but the variation about the relationship is included in the
forecast using lognormal error. The constant recruitment scenario
is more optimistic than the density-independent one because higher
recruitments estimated for 1986–1988 are used (but not in the density-
independent scenario because there is no corresponding lagged
estimate of exploitable abundance).
Exploitation rate as described in Merritt (1995) is examined at
0, 10, and 15% because these are the average rates of harvest spec-
ified in the management plan.
Choosing the best model
In our judgment, first, the model should fit the data sources well,
second, one desires parameter estimates that are of low bias and
variability, third, estimates of exploitable abundance should be
conservative, and fourth, parameter estimates must be plausible.
Our judgement matrix is
For example, model fit is 1.5 times as important as uncertainty and
2.0 times as important as conservatism, in our opinion. Although
we chose the values based on our beliefs, we have made those be-
liefs explicit. The normalized eigenvector for weighting criteria is
w=
Fit
Uncertainty
Conservatism
Plausibility
0386
0277
.
.
0196
0141
.
.
é
ë
ê
ê
ê
ê
ù
û
ú
ú
ú
ú
The potential values that could be obtained for each crite-
rion are rated for their desirability (Table 3). For example,
small RMSs of the model are strongly desirable (=5),
whereas large RMSs indicate poor model fit and are strongly
undesirable (=1/5).
Results
Estimated parameters
Age-specific gear vulnerability curves show lower vulner-
ability until older ages (8+ years) for the CV–2 and com-
bined models, as compared with the base and perception
models. Ages at 50% vulnerability for the CV–2 and com-
bined models are 5.3 and 5.1 years, respectively, and 4.7
years for both the base and perception models. As a result,
the CV–2 and combined models are less conservative be-
cause they predict less effect of harvesting.
All models generate plausible estimates of exploitable
abundance. Estimated exploitable abundance increases from
15 000 – 18 000 fish in 1986 to 29 000 – 34 000 fish in
1988, similar to the trend observed in mark–recapture exper-
iments (Fig. 1). While exploitable abundance estimated by
catch-age models is similar between 1988 and 1989, in con-
trast, a precipitous decline in exploitable abundance is esti-
mated in mark–recapture experiments. Thus, relative to
catch-age models, the mark–recapture experiment underesti-
mated abundance in 1989. Strong recruitment as evidenced
in age composition just prior to 1989 (Table 1) argued for a
higher abundance estimate than observed in the mark–recap-
ture experiment. The models account for this recruitment
(Fig. 1), whereas the mark–recapture experiment in 1989 esti-
mated abundance too low because the assumption of closure
was not satisfied. Declines in exploitable abundance esti-
mated by catch-age models occur in 1990–1992.
Average exploitation for catch-age models in 1986 is esti-
mated at about 14–16%. The catch-age models suggest that
the empirical information slightly underestimates exploita-
tion in 1987 and 1988 and greatly overestimates exploitation
in 1989 (because there was no population estimate in 1990,
no exploitation rate could be calculated). The discrepancies
in average exploitation rates between the empirical data and
the catch-age models are due largely to exploitable abun-
dance estimates. For example, the empirical data overesti-
mates average exploitation in 1989 largely due to the
underestimation of exploitable abundance by the mark–
recapture experiment during that year (see Fig. 1).
Strong age 3 recruitments appear in 1986 and 1987 in all
models that contribute to high exploitable abundances in
1988 and 1989. Age composition data (Table 1) show simi-
lar strong recruit years in 1986 and 1987. Subsequent de-
clines in age 3 recruits are reflected in a declining trend in
exploitable abundances. For all models, recruitment appears
© 2000 NRC Canada
Merritt and Quinn 1463
Model fit Uncertainty Conservatism Plausibility
Model fit 1 1.5 2.0 2.5
Uncertainty 1 1.5 2.0
Conservatism 1 1.5
Plausibility 1
Value of a criterion Rating
Model fit: RMSa
Best fit 5
Worst fit (significant at p< 0.01) 1/5
Parameter uncertainty: RMSEb
Lowest error (value is <70% of 1.0) 5
Moderate error (value is 70–85% of 1.0) 3
Highest error (value is >85% of 1.0) 1/5
Forecast: EN (p< 10 000 fish) at 15% exploitationc
Conservative, p³0.15 5
Moderate, 0.10 < p< 0.15 3
Optimistic, p£0.10 1/5
Plausibility of EN for 1986–1992d
10 000 – 40 000 fish 5
<10 000 fish 1/5
>40 000 fish 1/5
aSignificant from zero based on Lehman’s test.
bTotal values are normalized among models so that the highest error is
assigned 1.0. Model totals are then proportionally compared with 1.0 for
each parameter.
cProbability is based on 1000 bootstraps for each of six years (1993–
1998) for a total of 6000 chances that EN will be below a threshold of
10 000 fish (see Table 4); the greater the probability, the more
conservative the stock assessment model. The constant recruitment
scenario is presented.
dExpert judgment based on probable productivity of stock and historical
performance.
Table 3. Rating of criterion values.
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to be linearly related to exploitable abundance for the years
1989–1992 (Fig. 2 shows this relationship for the perception
model).
Catch-age models
Model fit of the combined and CV–2 weighting schemes in
the early years of harvest is poor (Fig. 3), possibly evidence
of misspecified model dynamics; the qweighting schemes in
these models place the least emphasis on harvest in 1986
and 1987. Sums of residuals are greatest in the combined
and CV–2 weighting schemes. Lehmann’s test showed that
the RMSs of the harvest data set for all model pairs were
significantly different (p< 0.01) except the base–perception
pair. Thus, the base and perception models are preferred
over the combined and CV–2 models with regard to the har-
vest data set. RMSs in the abundance and age composition
data sets do not significantly differ among the models; no
model is preferred over another with regard to RMSs in the
abundance and age composition data sources.
RMSE in exploitable abundance, exploitation rate, and re-
cruitment is greater for the combined and CV–2 models than
for the base and perception models (Fig. 4). Thus, the q
weighting scheme in the base and perception models pro-
vides parameter estimates with the most precision and least bias.
Forecasting exploitable abundance
The choice of a sustainable exploitation rate is greatly de-
pendent on the recruitment scenario. With constant recruit-
ment, fish abundance tends to remain steady beginning in
1996, but at diminishing levels with increasing exploitation
(Fig. 5). With density-independent recruitment, the stock
does not remain above the minimum threshold of 10 000 fish
over the forecast period at an exploitation of 10% or greater
(Fig. 5). The width between the 2.5 and 97.5 percentile
ranges increases over time with constant recruitment, dem-
onstrating greater uncertainty in the expected long-run fore-
cast by 1998.
Results are summarized in a decision table (Table 4),
which provides the manager information about the probabili-
ties of being below the threshold for each exploitation strat-
egy under various abundance and recruitment scenarios, as
obtained by tabulations of the bootstrap abundance forecasts.
Choosing the best model
Overall model ratings (derived from weighted scores) vary
depending on the criterion (see Table 5), and without a tech-
nique for synthesizing all aspects involved in selecting the
best model, the selection process would be arbitrary. The
perception model ranks highest overall (Tm= 0.337), based
on rank order among criteria and weights of importance as-
© 2000 NRC Canada
1464 Can. J. Fish. Aquat. Sci. Vol. 57, 2000
Fig. 1. Estimates and bootstrapped 2.5 and 97.5 percentiles (vertical lines) for exploitable abundance of humpback whitefish from
catch-age models by year, compared with mark–recapture estimates. , base; ,CV
–2;, perception; , combined. Boxes are the
mark–recapture 95% CI.
Fig. 2. Relationship between exploitable and prefishery abun-
dance of age 3 recruits lagged 3 years for the perception model
(r2= 0.494). ,Y;, predicted Y).
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signed to criteria (Fig. 6). Because plausibility was the same
for all models, the superiority of the perception model is due
to its excellent fit of the harvest data, relatively low parame-
ter uncertainty, and low forecast risk.
Discussion
Rather than accept uncertain data as equally reliable, or
entirely discount these data, this study explores a methodol-
ogy for explicitly weighting observations according to prior
perceptions of accuracy. Our methodology is not a full
Bayesian treatment as in McAllister and Ianelli (1997). The
Bayesian uses probability theory to interpret the indefinite-
ness of one’s opinions, arguing that point values of belief are
actually interval valued. However, surrounding the point es-
timate of belief with an interval does not necessarily bring
greater realism to the model outcome. In situations where an
exact value of belief can be numerically expressed with con-
fidence, then a briefer form of the full Bayesian analysis,
such as presented in this paper, will bring economy to the
assessment method. Explicit weighting to incorporate prior
beliefs narrows the field of possibilities and is simple in its
execution. Instead of modeling using a probability distribu-
tion, we use explicit weights based on perception of data ac-
curacy, which are placed on annual estimates of harvest, age
composition, and auxiliary mark–recapture estimates of abun-
dance. Both heuristic and Bayesian methods seek to deal with
subjectivity explicitly in minimizing estimation error.
In situations where information has been collected under
varying circumstances and is uncertain, and there is an indi-
vidual (or group) familiar with the assessment program,
perception-weighted models may offer advantages over other
weighting schemes. One disadvantage of including percep-
tions of several people in the weighting scheme is that hu-
man choice behavior in comparing among data sources or
annual estimates is affected by many biases (Hogarth, cited
in Poyhonen et al. 1994). In combining individual percep-
tions, it is assumed that their scales of comparison are con-
sistent (see Keeney and Raiffa 1976). In this study,
perceptions were elicited from one fishery manager who was
familiar with all aspects of the entire assessment program. A
long data series may necessitate the combination of numeri-
cal perceptions from several individuals.
Effects of weighting schemes on model results
Weights (q) attached to annual Ocomponents represent
uncertainty about the precision and (or) accuracy of the data.
Selection of annual weights of influence is based on prior
© 2000 NRC Canada
Merritt and Quinn 1465
Fig. 3. Residuals for the (a) abundance and (b) harvest data sets
and (c) likelihood components for age composition from catch-
age models by year. There was no abundance estimate in 1990
and no harvest in 1991. , base; ,CV
–2;, perception; ,
combined.
Fig. 4. Root MSE in estimated (a) exploitable abundance, (b)ex
-
ploitation rate, and (c) recruitment from catch-age models by
year. Solid line, perception; short-dashed line, base; long- and
short-dashed line, combined; long-dashed line, CV–2.
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© 2000 NRC Canada
1466 Can. J. Fish. Aquat. Sci. Vol. 57, 2000
Fig. 5. Forecasted exploitable abundance for humpback whitefish at two recruitment scenarios (constant, density independent) and three
exploitation rates (0, 0.10, and 0.15) for the perception model. The dashed line is the 95% CI.
Constant recruitment Density-independent recruitment
Assessment model m=0 m= 0.10 m=0.15 m=0 m=0.10 m=0.15
Base 0.054 0.092 0.120 0.342 0.455 0.505
CV–2 0.065 0.102 0.117 0.261 0.341 0.389
Perception 0.087 0.132 0.155 0.421 0.544 0.591
Combined 0.125 0.196 0.199 0.429 0.524 0.568
Table 4. Probability that forecasted exploitable abundance during 1993–1998 is below the minimum threshold
of 10 000 fish, by assessment model, for two recruitment scenarios and three exploitation strategies.
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information regarding empirical evidence (variance esti-
mates) or personal perceptions. Weights associated with
each source of information (l) reflect attempts to bring these
Os closer in magnitude for use in the nonlinear minimization
procedure. The process for selecting ls is more arbitrary in
nature than that for qs. Further research on lscaling coeffi-
cients is needed for robust weighting systems when annual
weightings are used.
The RMSs are significantly different among models for
the harvest data set. In both the CV–2 and combined models,
the fit to the observed data in the early years of harvest is
poor. The least weight in these models is applied to the har-
© 2000 NRC Canada
Merritt and Quinn 1467
Fig. 6. Hierarchy of criteria and models showing the weight of importance for the criteria (Wk), the weighted proportion of the total
score by model (WkPk,m), and the sum of each model’s score (Tm) using the analytic hierarchy process.
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vest data set in 1986 and 1987 because those years of large
harvests were associated with the highest variances. The
slight emphasis of the qweighting scheme in years when
harvests were high resulted in possibly misspecified model
dynamics for the CV–2 and combined models. Incorporating
perception resulted in conservative estimates of exploitable
abundance under both the constant and density-independent
recruitment scenarios because the highest weights of influ-
ence are placed on years (1991 and 1992) in which ages 3–5
fish comprise the smallest proportion of the population (0.13
for both years).
Because it is the magnitude of the weights that contributes
to how a particular weighting scheme will affect the results,
generalizations about the merits of variance or perception
weights, or a combination of the two, are premature. Thus,
several weighting schemes should be explored in applica-
tions. In this application, the perception model produces the
most conservative strategy to avoid overfishing humpback
whitefish in the Chatanika River. Decisions based on percep-
tions of data accuracy are always present to some extent in
empirical analyses. This study makes these perceptions ex-
plicit and quantifies their influence on a given model solu-
tion. There are limited alternatives to the manager.
Discarding data from flawed studies is inefficient, arguing
that there is no signal in the data, only noise. Alternatively,
if suspect data are accepted as equal to reliable data, the in-
fluence of the “sound” observations is diluted.
Choosing the best model
Because the true state of the population is not generally
known, choosing a stock assessment model is not simple.
The analytic hierarchy process enabled us to determine the
relative importance of four criteria (model fit, parameter un-
certainty, conservative forecasts of exploitable abundance,
and biological plausibility) that we considered crucial to
model selection. It is typical for risk of the harvest strategy
to be evaluated and a preference given to a conservative har-
vest policy (Food and Agriculture Organization of the
United Nations 1995). The idea that one should prefer an es-
timation procedure because it results in a low estimate of
abundance or biomass is a concept that is receiving more at-
tention. The advice to managers contained in Table 4 is
valuable because it presents the probability of the stock be-
ing below threshold, depending on recruitment scenario and
harvest strategy. Selecting the stock assessment model with
lower biomass estimates, and hence with higher probabilities
of being below a fixed abundance level, establishes a pre-
cautionary approach to forecasts of exploitable abundance.
In a conservative context, fishery management decisions
should include considerations of risk to the stock in the set-
ting of harvest policy and in the selection of the assessment
model.
It is instructive to note that the risk measure is interpreted
differently in these two situations. In evaluating harvest pol-
icy, within a particular stock assessment model, a policy
with a lower probability of dropping below the threshold is
more risk averse because the model is considered correct in
the evaluation. In evaluating a suite of stock assessment
models, a policy with a higher probability of dropping below
the threshold is more risk averse because one is accounting
for the uncertainty in the model being wrong.
Conservatism is not the only criterion that should go into
model selection and harvest policy. In our case, the com-
bined model had the highest conservatism in terms of the
probability of future abundance being below the threshold
but was the lowest ranked model because of its poor fit to
the harvest data and most uncertain parameter estimates
compared with other models.
Implications for management
We examined the consequences of several exploitation
rates on the humpback whitefish population under a thresh-
old harvest policy by using forecasts of exploitable abun-
dance. A variety of outcomes could result from the current
management policy based on the recruitment scenarios ex-
amined. From these outcomes, the probability that stock
abundance will be below a threshold level of 10 000 fish is
presented up to 6 years into the future. Exploitable abun-
dance is sustained above the threshold of 10 000 fish with
constant recruitment. Not surprisingly, the probability of
abundance falling below the threshold increases with exploi-
tation rate and if density-independent recruitment is present.
Thus, a sustainable recreational fishery for humpback white-
fish under the current management policy is when periodic
strong year-classes occur. Otherwise, the density-independent
scenario shows that the possibility of a nonsustainable har-
vest is high.
One feature of catch-age analysis is that current popula-
tion abundance estimates can vary substantially as more data
become available (Parma 1993). Thus, the strength of catch-age
analysis may not be so much in providing current informa-
tion for management, but rather as a tool for incorporating
weights, such as perceptions, into a structured framework to
quantify the risks to the population associated with various
harvest strategies. Because the catch-age model developed in
this study conforms to the error structure of data collected
for recreational fish assessment, it is possible that the model
can have wide application in recreational fisheries problems.
Acknowledgments
Partial funding for this research was provided through the
© 2000 NRC Canada
1468 Can. J. Fish. Aquat. Sci. Vol. 57, 2000
Model (m)
Criterion (k) Base CV–2 Perception Combined
Model fit
Harvest 5 1/5 5 1/5
Exploitable abundance 5 5 5 5
Age composition 5 5 5 5
Parameter uncertainty
Exploitation rate 5 3 3 1/5
Exploitable abundance 5 3 5 1/5
Recruits 3 1/5 5 1/5
ConservatismaEN(p<
10 000), m= 15% 33 5 5
Plausibility 5 5 5 5
aThe constant recruitment scenario is presented. Results are similar
for the density-independent recruitment scenario.
Table 5. Scores (rk,m) by criterion (k) and model (m) used in
finding the weighted overall rating (Tm) of each model.
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Alaska Cooperative Fish and Wildlife Research Unit, Uni-
versity of Alaska Fairbanks, and the Sport Fish Division,
Alaska Department of Fish and Game. We thank John H.
Clark and James B. Reynolds for encouragement and sup-
port. Keith Criddle introduced us to the analytic hierarchy
process and provided guidance in decision theory. Linda
Brannian and Fritz Funk offered advice regarding the appli-
cation of a spreadsheet approach to catch-age analysis. We
thank Jie Zheng, Fritz Funk, Milo Adkison, Philip Symons,
and Martin Dorn for helpful comments on the manuscript.
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