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Fisheries Research 70 (2004) 209–227
CPUE standardisation and the construction of indices of stock abundance in a spatially varying fishery using
general linear models
Robert A. Campbell∗
CSIRO Division of Marine Research, GPO Box 1538, Hobart, TAS 7001, Australia
Abstract
Construction of annual indices of stockabundancebasedon catchand effort dataremains central to manyfisheries’ assessments.
While the use of more advanced statistical methods has helped catch rates to be standardised against many explanatory variables,
the changing spatial characteristics of most fisheries data sets provide additional challenges for constructing reliable indices of
stock abundance. After reviewing the use of general linear models to construct indices of annual stock abundance, potential
biases which can arise due to the unequal and changing nature of the spatial distribution of fishing effort are examined and
illustrated through the analysis of simulated data. Finally, some options are suggested for modelling catch rates in unfished strata
and for accounting for the uncertainties in the stock and fishery dynamics which arise in the interpretation of spatially varying
catch rate data.
© 2004 Elsevier B.V. All rights reserved.
Keywords: Standardisation of catch rates; General linear models; Stock abundance indices; Spatial distribution of fishing effort; Modelling
uncertainty
1. Introduction
Despite an ongoing debate about the nature of there-
lationship between catch rates and underlying resource
abundance (e.g. Harley et al., 2001), the interpretationof catch and effort data, and the construction of in-
dices of resource abundance based on these data, re-
mains an integral part of the stock assessment process
∗ Tel.: +61 3 6232 5368; fax: +61 3 6232 5012.
E-mail address: [email protected].
for many fisheries. While complex age-based stock as-
sessment models are used routinely, indices of resource
abundance based on an analysis of commercial catch
and effort data are usually required to calibrate these
models. This is particularly the case for major oceanicpelagic fish stocks, where the large spatial extent of
the fisheries usually precludes any attempt to conduct
fishery-independent surveys of stock status.
Early methods which made use of temporal changes
in catch rates to measure annual changes in rela-
tive stock abundance were based on the principal as-
sumption that catchability either remained constant
0165-7836/$ – see front matter © 2004 Elsevier B.V. All rights reserved.
doi:10.1016/j.fishres.2004.08.026
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210 R.A. Campbell / Fisheries Research 70 (2004) 209–227
over the entire fleet, or that the nominal effort could
be adjusted to account for the differences in rela-
tive vessel efficiency (Beverton and Parrish, 1956;
Gulland, 1956; Robson, 1966). These methods werewidely adopted and routinely used in fisheries as-
sessments despite the simplicity of their assumptions.
This was despite early evidence that the assumption
of constant (or adjusted) catchability was often vio-
lated. For example, Garrod (1964) and Gulland (1964)
pointed out that variation in the catchability may re-
sult not only from the differences in fishing power
among vessels, but from differences in vulnerability
to the gear, changes in seasonal and spatial patterns
of both the fishing effort and the stock, and changes
in stock abundance itself. Indeed, the persisting re-
liance of stock assessments on the estimation of an-
nual abundance indices based on the use of commercial
catch-rate data is perhaps somewhat surprising given
the continued concern about the failure of the under-
lying assumptions (e.g. Paloheimo and Dickie, 1964;
Rothschild, 1972; Radovich, 1976; MacCall, 1976;
Clark and Mangel, 1979; Ulltang, 1980; Winters and
Wheeler, 1985). More recently, however, the advent of
high speed computing and the use of more advanced
statistical methods (e.g. general linear models, general
additive models) has allowed the inclusion of more fac-
tors in the standardisation process and has helped toovercome some of the more obvious failures of the ear-
lier methods. Whether by design or necessity, the use
of abundance indices based on catch and effort data has
continued to be integralto the assessment of fish stocks.
However, other issues apart from changes in ves-
sel catchability also influence the ability to construct
reliable indices of stock abundance. The dramatic de-
clines in the abundance of northern cod, accompanied
by equally dramatic changes in the distribution of the
stock and fishing effort (Atkinson et al., 1997), illus-
trate some of these issues. For example, it has beennoted that the ‘fleet was fishing a smaller and smaller
area of ocean’ and ‘the fishermen were catching more
fish perhour than thescientists because they were going
to warmer patches where they knew cod were congre-
gating. The research vessel, on its random course, was
encountering empty ocean’ (Anon, 1995). Others have
also concluded that the decline was ‘due to a high and
rapidconcentrationoffishingeffortonapopulationthat
. . . had shown a pronounced shrinkage of its distribu-
tion’ (Avila de Melo and Alpoim, 1998). There are ob-
vious lessons to be learnt from the cod experience, not
the least of which is the correct interpretation of catch
and effort data from a commercial fleet which does not
cover the spatial extent of the stock adequately, and themore general issue of whether data from a commercial
fishery intent of maintaining high catches reflects stock
abundance (Salthaug and Aanes, 2003).
In light of the continued widespread use of catch and
effort data, it is important that fisheries scientists and
managershave a goodunderstanding of the relationship
between catch rates and indices of fish abundance and
the factors that may unduly influence this relationship.
In this paper we review the basis for this relationship
and the manner in which annual indices of stock abun-
dance are constructed based on the widely used gen-
eral linear models approach. Potential biases that can
arise in annual abundance indices due to the unequal
spatial distribution of fishing effort and when changes
or contractions take place in the spatial distribution of
the fishery are then illustrated using information from
the fishery for southern bluefin tuna (Thunnus mac-
coyii) and an analysis of simulated data. Finally, some
options are suggested for modelling catch rates in un-
fished strata and for accounting for uncertainties in the
stock and fishery dynamics, which arise in the inter-
pretation of spatially varying catch rate data.
2. Basic equations
2.1. CPUE as a measure of stock abundance
The relationship between catch rates (CPUE) and
stock abundance is based on the catch equation which,
at a first order approximation, relates the number of
fish in the catch, C , fishing effort, E , and average fish
population density, D, on the fishing grounds:
C = qED (1)
where q is a fixed constant of proportionality known
as the catchability coefficient and is related to the effi-
ciency of the fishing gear. From this equation:
CPUE =C
E= qD =
qN
A(2)
where N is the number of fish on the fishing grounds
and A the spatial area of the fishing grounds. It fol-
lows that changes in CPUE are due either to changes
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R.A. Campbell / Fisheries Research 70 (2004) 209–227 211
in the stock density (or number of fish on the fishing
grounds) or to changes in the catchability coefficient.
If the changes in q can be accounted for, then the re-
maining changes in CPUE can be related to those instock density. This is the basic idea underlying what is
known as the standardisation of catch rates.
The concept of abundance needs some elaboration,
however. Of particular importance is the related con-
cept of availability. The following definitions were pro-
posed by Marr (1951):
Abundance is the absolute number of individuals in a
population. Availability is the degree (a percentage) to
which a population is accessible to the efforts of a fish-
ery. Apparent abundance is the abundance as affectedby availability, or the absolute number of fish accessi-
ble to the fishery.
From these definitions, if B represents the true abun-
dance and N measures the apparent abundance, then:
N = aB (3)
where a represents the availability or proportion of the
total stock available to the fishery. Substituting into Eq.(2) and rearranging gives:
B =N
a=
A · CPUE
aq(4a)
and
CPUE =aqB
A(4b)
If one is to use changes in catch rates alone as a mea-
sure of changes in stock abundance over time, then
one must assume that both the catchability together
with the availability of fish remain constant over time
(or at least aq remains constant). At best, the varia-
tions induced by the fishery on the size of the available
population must be large relative to those caused by
fishery-independent factors. When variations in avail-
ability are also large, the problem of relating changes
in abundance to changes in fishing efficiency becomes
increasingly difficult.
2.2. Standardisation of CPUE and the use of
general linear models
It is usual practice to model the expected catchrate using a multiplicative model when standardising
catch rates. An observed catch rate is related to the
expected catch rate under a standard set of conditions,
multiplied by a number of factors, which correct
for the non-standard conditions. For example, the
model for the expected catch rate in a spatial-temporal
region using a given type of gear and under cer-
tain environmental conditions can be expressed
as:
E(CPUEijkeg) = aeqgDijk
= (Eeao)(Ggqo)(Y iQj RkDo)
= (Y iQj RkGgEe)aoqoDo (5)
where Y i is the effect of the ith year relative to a stan-
dard year, Q j the effect of the jth quarter relative to a
standard quarter, Rk the effect of the k th fishing region
relative to a standard fishing region, Gg the effect of the
gth gear-type relative to a standard gear-type, E e the
effect of the eth environment relative to a standard en-
vironment, qo the value of the catchability coefficient
for the standard gear-type, ao the value of the avail-ability parameter for the standard environment, and
Do the density of fish in the standard spatio-temporal
region. Gg and E e can be thought of as standardising
catchability and availability while Y i, Q j and Rk standardise the density in different spatio-temporal
strata.
The values of the parameters in the above equation
need to be estimated from the observed catch rates.
The standard method for parameter estimation, and
those used here, follow Garvaris (1980) and Allen and
Punsley (1984) and involves the use of a general linearmodel (GLM). Traditionally, it was assumed that the
modelled catch rates had a log-normal distribution
(Beverton and Holt, 1957), so that the transformed
variable z = log(CPUE) has a normal distribution. In
this case, the model becomes a linear model, and one
can make use of the classical regression/ANOVA type
of analyses to fit it to the transformed data (Draper
and Smith, 1981). Alternative model structures, where
the response variable can belong to any member
of the exponential family of distributions (Crosbie
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212 R.A. Campbell / Fisheries Research 70 (2004) 209–227
and Hinch, 1985; McCullagh and Nelder, 1989;
Dobson, 1990; SAS Institute Inc., 1993), are also
possible, though there is no loss of generality in the
following work in adopting the traditional log-normalmodel.
From Eq. (5), the linear equation can be expressed
as:
E(zijkeg) = E(log CPUEijkeg)
= log(Y i) + log(Qj ) + log(Rk) + log(Ee)
+log(Gg) + log(aoqoDo)
= µo + yj + qj + rk + ee + gg (6)
where µo = log(aoqoDo), yi = log(Y i), etc. The
above model is known as an effects model since each of the explanatory variables can be seen as representing
the effect of an applied treatment (e.g. region, gear-
type, etc.). As well as these ‘main’ effects, variables,
which model an explicit functional form of an effect
(e.g. polynominal, trigonometric), or the influence of
interactions among different effects, can also be in-
cluded.
An undesirable consequence of using the logarithm
of the catch rates in the above model is that an adjust-
ment is needed to accommodate any zero catch rate
observations. The usual practice is to add a small con-stant to the calculated catch rate for all observations,
i.e. CPUE in Eq. (6) is replaced by the adjusted catch
rate, adjCPUE = CPUE + δ. The value of δ is some-
what arbitrary (it is commonly set equal to 1), but Xiao
(1998) indicatesthat very small values of δ (e.g. 10−100)
should be avoided because of the way log(δ) behaves as
δ approaches zero. Simulation testing suggests that set-
ting δ equal to 10% of the mean overall catch rate used
in the analysis may minimise any bias resulting from
adjusting the catch rate in this manner (Campbell et al.,
1996). However, when many fishing operations result
in catches of zero or one fish, the delta method (Lo et
al., 1992) or Poisson models are perhaps more appro-
priate. The use of the negative binomial error structure
to model the predicted catch also avoids the need to
adjust the observed catch values.
2.3. Construction of indices of stock abundance
The expected value of the standardised log
(adjCPUE) for the ith year, jth quarter and k th region
can be found by setting the value of the standardis-
ing parameters for the catchability and availability
factors to zero (e.g. ee = 0 and gg = 0). For a model,
which includes a year × region interaction, thisgives:
E[log(adjCPUEijkoo)] = µo + yj + qj + rk + (yr)ik
(7)
Given that log(adjCPUEijkoo) has a mean µ and
variance σ 2, and the distribution of CPUE is indeed
log-normal, then the expected value of the corre-
sponding standardised adjusted catch rate is given by
(Aitken et al., 1989):
E(adjCPUEijkoo)
= exp(µ + 12 σ 2)
= exp(µo + yj + qj + rk + (yr)ik) exp(12 σ
2)
= aoqoDo exp(yj + qj + rk + (yr)ik ) exp(12 σ
2),
E(CPUEijkoo) = CPUEo exp(yj + qj + rk
+(yr)ik + 12 σ
2) − δ (8)
where CPUEo = aoqo Do is the standardised catch rate
in the reference spatio-temporal stratum. Note that
inclusion of δ in Eq. (8) can result in the expectedcatch rate being less than zero. In these cases, the
expected catch rate should be set to zero.
From Eq. (4a), a relative index of abundance, B ijk ,
for the size of the fish population in the ith year, jth
quarter and k th region can be obtained by multiplying
the standardised catch rate by the size, Ak , of the region
fished, i.e. Bijkoo = Ak E (CPUEijkoo). The total index of
abundance for a season is then obtained by summing
over all regions of the fishery. An annual index of abun-
dance for the ith year, I i, can then be obtained by taking
theaverageover allseasons in that year. Either thearith-metic or geometric mean can be used, the latter being
scale invariant:
I i =1
NS
NS j =1
NRk=1
AkE(CPUEijkoo)
(9a)
I i = NS
NS j =1
NRk=1
AkE(CPUEijkoo)
(9b)
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R.A. Campbell / Fisheries Research 70 (2004) 209–227 213
where NS and NR are the number of seasons (quarters)
and regions used in the analysis, respectively. Seasonal
weightings can be used if desired. Finally, a relative
index which relates the average abundance in year i tosome reference year can be calculated as:
I i,ref =I i
I ref
For an analysis based on Eq. (8), the annual index of
abundance (using the arithmetic mean) is given by:
I i = aoqoDo exp
yi +
σ 2
2
× 1
NS
NS j =1
NRk=1
Ak exp(qj + rk + (yr)ik)
−NRδ
For those models where the interaction term is not used
and the sizes of the areas fished remain constant across
years, the term in the square brackets is constant for
all years. If one also ignores the term involving the
constant δ, the relative index is then reduced to:
I i,r = exp(yi − yr) (10)
In this situation, the year effects alone are the indicesof abundance. However, for models which incorporate
interaction terms incorporating the year effect, the ex-
pression for the relative index is more complicated,
involving the sum over a number of other explanatory
variables.
The formulation of the annual abundance index (Eq.
(9)) highlights the fact that the index is the product of
the density of fish within several spatial areas and the
sizes of those areas. The common practice of reducing
the index to a function of the year effect alone runs the
risk of ignoring information on the spatial dynamicsof the fishery which may be relevant to the underlying
dynamics of the stock and hence the correct interpreta-
tion of the catch and effort data. Additionally, Eq. (9)
will give indices of total stock abundance only if the
spatial extent of the fishery coincides with, or is greater
than, the spatial extent of the stock. Otherwise, the in-
dex of abundance pertains only to that portion of the
stock that is found on the fishing grounds. Uncertainty
will remain as to the size of the stock beyond the area
fished.
3. Influence of an unequal spatial distribution
of fishing effort
The equations defined in the previous section madeno assumptions about the spatial characteristics of
the fishery. However, a fish population is usually dis-
tributed with variable densities across its stock range
and consequently the spatial distribution of fishing ef-
fort is also usually highly variable. The fishery is usu-
ally divided into a number of regions, and estimates of
stock density are obtained for each region to account
for this spatial heterogeneity. For example, consider a
regulatory area of total size A consisting of R sepa-
rate regions. The total abundance, N , across the entire
area can be related to the regional abundances, N r , and
densities, Dr , by:
N =
Rr=1
N r =
Rr=1
Ar Dr
where Ar is the size of region r . Ideally the spa-
tial distribution of the resource within each region
should be reasonably homogeneous and the corre-
sponding estimates of density should be based on a
random sample from across each region. Assuming
that the catchability coefficient q is constant across
all regions (and availability in each region is unity),
and using the relationship between the mean den-
sity and the mean of the individual observed catch
rates CPUEri in each region given by Eq. (2), we
have:
N =
Rr=1
N r =
Rr=1
ArCPUEr
q
=1
q
R
r=1
Ar
nr
nr
i=1
CPUEri (11)where nr is the number of CPUE observations in the r th
region. Hence, the average of the regional catch rates
weighed by the size of each region gives an unbiased
estimate of the total stock abundance across the entire
regulatory area (Quinn and Hoag, 1982).
In terms of the structure of a GLM, the above situ-
ation can be described by:
P r = E(CPUEr) = µ + Rr
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214 R.A. Campbell / Fisheries Research 70 (2004) 209–227
where µ is a constant, Rr and Pr are the region-effect
and the predicted catch rate in the r th region respec-
tively, and where, for illustrative purposes, we have
used an additive model for CPUE instead of a mul-tiplicative model. Values of the parameters µ and Rr can be found from a least-squares fit, which minimises
the sum of squares of the differences between the nr observed ( Z ri) and predicted catch rates across all re-
gions. There is no loss of generality by setting R1 = 0
(in which case parameter µ gives the expected catch
rate in region 1), so that:
µ = Z̄1 and
Rr = Z̄r − Z̄1 where Z̄r =1
nr
nri=1
Zri
Using the predicted value of the catch rate within each
region, the population index for the total area is found
to be:
I =
Rr=1
ArP r
q=
Rr=1
Ar(µ + Rr)
q=
Rr=1
Ar Z̄r
q
That is, the index is proportional to the weighted mean
of the average observed catch rate in each region, and
according to Eq. (11), this gives an unbiased estimate
of the total population abundance over the entire regu-
latory area.
An interesting feature of the above result is that
an unbiased estimate of abundance is obtained even
though the number of observations in each region may
not be equal. However, a further example will show
that this is not the case when one considers more than
one year of observations. Again, consider observations
from NY years for a fishery divided into NR regions
with n yr catch rate observations in the r th region dur-ing the yth year. We fit the following model to these
observations:
P yr = E(CPUEyr) = µ + Y y + Rr
For simplicity, we consider the situation where NY =
NR = 2 and again, with no loss of generality, we set
Y 2 = R2 = 0. The following least-squares solution is
found:
ωµ =
1
n21+
1
n12+
1
n11
Z̄22
+
1
n11 (Z̄21 + Z̄12 − Z̄11),
ωY 1 =
1
n22+
1
n12
(Z̄11 − Z̄21)
+
1
n11+
1
n21
(Z̄12 − Z̄22),
ωR1 =
1
n22+
1
n21
(Z̄11 − Z̄12)
+ 1
n11+
1
n12 (Z̄21 − Z̄22)where
ω =
1
n11+
1
n12+
1
n21+
1
n22
and
Z̄yr =1
nyr
nyri=1
Zyri
Based on the above model, the population index in the
first year, given that each region is of area A, is given
by
I 1 =A(µ + Y 1 + R1)
q+
A(µ + Y 1 + R2)
q
=A(2µ + 2Y 1 + R1)
q
=
A[Z̄11(2/n12 + 1/n21 + 1/n22)
+Z̄12(2/n11 + 1/n21 + 1/n22)
+Z̄21(1/n11−1/n12) + Z̄22(1/n12−1/n11)]
ωq
Unlike the previous example, the predicted abundanceindex is no longer proportional to the weighted mean
of the observed catch rates across each region, but is
instead dependent in a complex manner on the number
of observations in each region. Only in the special sit-
uation where the n yr are all equal does the total index
equal the weighted mean of the regional estimates. A
consequence of thisresult is thatmisleading differences
between the indices of annual population size can be
obtained. To illustrate this point, consider the example
where Z̄11 = Z̄22 = 5, Z̄12 = Z̄21 = 10, n11 = n21 = n22
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R.A. Campbell / Fisheries Research 70 (2004) 209–227 215
= 50, n12 = 200 and A = q = 1. Based on these values,
the sum of the catch rates across the two regions is the
same for both years (15) but based on the results of the
least-squares fit we obtain I 1 = 17.3 and I 2 = 15.The above examples illustrate that when the number
of observations in each spatio-temporal stratum varies
(i.e. the data set is unbalanced) the relative indices of
annual abundance based on the parameter estimates
obtained from a GLM may be biased. This is due to
the fact that equal weight is given in the estimation
procedure to each observation instead of giving equal
weight to each region, as demanded by Eq. (11). Thus,
the annual indices based on the least-squares fit will be
biased to favour those regions with the most number of
observations.
A weighted least-squares approach needs to be used
to obtain an unbiased index of population abundance.
For example, if one weights each observation by the in-
verse of the number of observations in the correspond-
ing stratum, one obtains the corresponding solution:
µ = 14 (3Z̄22 + Z̄21 + Z̄12 − Z̄11),
Y 1 = 12 (Z̄11 +
Z̄12 − Z̄21 − Z̄22),
R1 = 12 (Z̄11 +
Z̄21 − Z̄12 − Z̄22)
From this result we obtain I 1 = A(Z̄11 + Z̄12) andI 2 = A(Z̄21 + Z̄22) as desired. Note that a similar re-
sult is also obtained when the full model is fitted to the
data (i.e. when a year × region term is included). In
this case, the number of parameters equals the number
of spatio-temporal strata and parameter estimates can
be found which are independent of the relative number
of observations in each region. The single year model
described previously was a simple (if trivial) example
of this situation. The annual indices are not equal to the
weighted means of the catch rates in each region when
the regions are of different sizes. This indicates thatwhen constructing indices of abundance based on the
results of a GLM analysis, regions of equal size should
be used.
The weighting to be given to each observation to
achieve an unbiased annual abundance index is not
unique; indeed, any weights that satisfy the condition
that the sum of the individual weights given to each
observation in each region is the same for all regions
will ensure that all regions are treated equally. For the
observations within each region, the weight assigned
to each observation will itself ensure the importance of
that observation. A suitable weighting factor for each
observation may be based on the corresponding effort
of that observation divided by the total effort in thatregion. This would be most appropriate in situations
where aggregated catch rate observations are used in
the analysis. An example of such a weighting factor,
which also takes into consideration the fact that the
logarithm of the catch rates is taken, is described by
Punsley (1987). A consequence of a weighted estima-
tion procedure is that the squared residuals from the
model are no longer χ2-distributed, thus invalidating
the F -test used to determine significance among differ-
ent models. Again, Punsley (1987) suggests an alter-
native approach.
4. Potential biases in a spatially contracting
fishery
The previous section considered the influence on pa-
rameter estimation of an unequal distribution of fishing
effort across regions. However, for many fisheries the
number of regions fished each year can also vary. Such
changes may be influenced by perceived shifts in the
distribution of the fish and/or changes in spatial man-
agement arrangements. Increased knowledge about thespatial distribution of the resource and technical im-
provements in the ability to find and target those areas
having greater abundance will also influence the dis-
tribution of fishing effort. These changes may occur
on both large- and fine-spatial scales and, as shall be
demonstrated, each has ramifications for the manner in
which catch and effort data need to be interpreted. The
importance of any contraction in the spatial distribution
of effort will be magnified if fishing effort concentrates
into regions with generally high catch rates. An exam-
ple of fishery in which this has occurred is that forsouthern bluefin tuna.
4.1. Spatial changes in the fishery for southern
bluefin tuna
Southern bluefin tuna (SBT) have a widespread but
patchy distribution, which is reflected in the spatial dis-
tribution of the fishing effort for this species. Like other
tunas, SBT also tend to form transient aggregations in
areas where oceanic thermal features favour local en-
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216 R.A. Campbell / Fisheries Research 70 (2004) 209–227
Fig. 1. Statistical areas used to provide the coverage of the fishery for southern bluefin tuna.
richment. The Japanese longline fishery for southern
bluefin tuna has undergone remarkable changes since
its inception as a major fishery in the early 1950s.
Shingu and Hisada (1971) outline the changes between
1957 and 1969, during which the area exploited by this
fishery expanded by about nine-fold. The statistical ar-
eas for the fishery are shown in Fig. 1.No major new fishing grounds for SBT have been
discovered since 1971. However, possible shifts in fish-
ing effort to favourable areas and other changes in the
spatial distribution of effort since 1971 have created
problems for the interpretation of catch and effort data.
The number of 1◦-squares fished each year within sta-
tistical areas 4–9 is shown in Fig. 2. There have been
substantial contractions in the spatial distribution of ef-
fort in most areas since 1971. For example, the number
of 1◦-squares fished in area 7 has more than halved
since 1975. These changes are concurrent with thelarger scale changes in the amount of effort being ex-
pended within each statistical area, as well as changes
in the percentage of the total effort within each area
(Tuck et al., 1996).
The changes in the spatial distribution of fishing
effort within statistical area 7 illustrate the nature of
some of those across the entire fishery. The 1◦-squares
fished each year were ordered by the amount of fishing
effort (number of hooks), and the cumulative percent-
age of the total annual effort expended in each decile of
the squares fished each year was then calculated. Theseresults were then averaged over each 5-year period
between 1970 and 1994, and used to plot cumulative
effort against cumulative area fished for each 5-year
period (Fig. 3a). During 1970–1974, on average 94%
of fishing effort occurred in only 50% of the 1◦-squares
fished, with 44% in the top 10%. This pattern of spatial
aggregation is repeated in all subsequent periods.
There appears to be little change during 1970–1984,
but the proportion of the effort expended in the top
10% of squares increased substantially after this time,
reaching 68% during 1990–1994. This increase in the
level of aggregation appears to have been a feature of
the fishery since 1986 and may be a response to the
introduction of total catch quotas in the mid-1980s.
The relationship between the distribution of effort
and stock density (using catch rates as a proxy) was
analysed to investigate the extent of targeting of areasof higher stock densities; cumulative effort was plotted
against area fished after ranking the 1◦-squares by catch
rate (Fig. 3b). For 1971–1974, on average 56% of the
effort was expended in the top 50% of squares fished
and only 8% in the top 10% of 1◦-squares indicating
that there was not very effective targeting of the areas
with higher nominal catch rates. However, the extent
of spatial targeting increased substantially over time.
This reached its greatest extent during the 1985–1989
when ∼78% of the effort was targeted in the top 50%
of 1◦-squares and 27% in the top 10%.It is evident that spatial targeting of fishing effort
has always been a feature of the SBT longline fishery
withinarea 7. Whilethe tendency to targeta greater pro-
portion of the fishing effort at areas with higher catches
rates will unduly weight the average catch rate across
this region, changes in the level of targeting over time
will also influence the relationship between changes in
average catch rates and corresponding changes in the
abundance. An appropriate spatial structure is neces-
sary for the interpretation of the catch and effort data to
account for these changes. However, a more intractableproblem is accounting for the change in abundance in
those regions no longer fished. A procedure adopted
for dealing with this problem is described in Campbell
(1998).
4.2. Simulations
A range of indicative longline data sets were gen-
erated to examine the consequences of changes in the
spatial distribution of fishing effort and the influence
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R.A. Campbell / Fisheries Research 70 (2004) 209–227 217
Fig. 2. Number of 1◦-squares fished by Japanese longliners in each of the SBT statistical areas 4–9 (1969–1995).
of strata with no data on the estimation of annual abun-
dance indices using GLMs. As a reference case, the
simulated fishery was divided into a number of spatial
areas and catch and effort data were generated uni-
formly across these areas over five years. These data
were then standardized to obtain an index of relative
abundance for the fishery. Variations on the reference
case were then explored to ascertain the impact on the
calculated abundance index of spatial aggregation and
contraction of the fishery over time. The sensitivity of
the resulting abundance index to differences between
the underlying spatial scale used to generate the catch
and effort data for the fishery and the spatial scale
assumed in the standardising model was also inves-
tigated.
The simulated fishery was deemed to consist of 200
grids of equal size. Multiple catch rate observations
were generated for each grid using the model:
log(CPUEik) = Y i + Gik (12)
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218 R.A. Campbell / Fisheries Research 70 (2004) 209–227
Fig.3. Cumulative effort vs. cumulative areafished (bothexpressed as a percentageof the respective annual totals) for the SBT fishery in statistical
area 7 during each 5-year period between 1970 and 1994, after ordering the 1 ◦-squares fished by (a) decreasing effort and (b) decreasing catch
rates.
where Y i and Gik parameterize the logarithm of the
catch rate for the ith year and k th spatial grid respec-
tively. Associated catch and effort data were also gen-
erated for each observation as follows:
effort = 1500 + 200 · N int[10 × U (0, 1)],
catch =effort × CPUE
1000
where N int( ) is the nearest integer function and U (0,1)
is a randomly generated number from the uniform dis-
tribution on 0–1, i.e. effort was given in increments of
200 hooks between 1500 and 3500.
A different value of Gik was generated for each grid
each year to mimic the changes in the annual spatial
distribution of the fish population. The impact of dif-
ferentspatial distributions of fishing effort on thecalcu-lation of the annual indices of relative stock abundance
was investigated by changing the number of observa-
tions in each grid. The spatial grids were grouped into
eight regions each consisting of 25 grids to investigate
changes in the spatial distribution of the resource over
a larger spatial scale than the grid, and to investigate
the influence of changes in the distribution of the re-
source occurring on a finer-spatial scale than that used
in the standardisation model. The catch rates in each
grid in each region were then given a similar range of
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R.A. Campbell / Fisheries Research 70 (2004) 209–227 219
Table 1
Parameter values used in the model to generate the catch rate data.
Thevalues for Y i refer tothe 5 years,and the valuesfor Gk referto the
eight spatial regions used to group the spatial grids in the simulated
fishery
Index value Y i Gk
1 1.8 U (0,1)
2 1.6 U (0,1)
3 1.4 U (0,2)
4 1.2 U (0,2)
5 1.0 U (0,3)
6 U (0,3)
7 U (0,4)
8 U (0,4)
values. The parameter values used to generate the catchrate observations in each grid for the eight regions each
year are given in Table 1.
For the reference case (scenario 1), ten catch rate
observations were generated annually for each grid,
though a grid was only fished if a number generated
from U (0,1) was greater than 0.5. This mimics a fishery
where fishing effort is relatively randomly distributed
across the grids within each region but is relatively ho-
mogeneous across all regions for all years. Three vari-
ants of this reference case were then considered. For
scenario 2, the number of sets in the ith year and k thgrid, nik , was changed each year based on the following
conditions:
• if 0
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220 R.A. Campbell / Fisheries Research 70 (2004) 209–227
Fig. 4. The average distributions of cumulative effort vs. cumulative area fished (after ordering by catch rate) for each of the four scenarios.
A description of the data, model and weighting used ineach analysis is given in Table 3.
4.3. Results
The results of the simulations are summarized by
the means of the relative indices across the 30 data
sets for each scenario (Fig. 5). As expected, when the
spatial distribution of fishing effort was random across
the grids in each region (i.e. scenario 1) the indices of
abundance for all nine analyses gave an unbiased trend
in the stock abundance. On the other hand, when allgrids were fished but the distribution of fishing effort
favoured higher catch rate grids (scenario 2), the un-
weighted fine-scale analyses led to a biased trend in
relative abundance (GLMs 1 and 3) with the resulting
index under-estimating the true decrease in abundance
over time. Only with an appropriate weighting was the
true annual trend realised (GLMs 2, 4 and 5). The re-
sults for GLMs 4 and 5 illustrate that the weighting
scheme used need not be unique. The weights used
were scaled by the total number of observations to
Table 3Structure of the nine GLM analyses
GLM Data Model structure Weighting
1 Finescale E (CPUEijk) = Y i + Gk None
2 Finescale E (CPUEijk) = Y i + Gk Wt =
N obs /( N YG nik)
3 Finescale E (CPUEijk) = Y i + Rj None
4 Finescale E (CPUEijk) = Y i + Rj Wt =
N obs /( N YG nik)
5 Finescale E (CPUEijk) = Y i + Rj Wt = N obs /
( N YR N grids Rij nik)
6 G-aggregated E (CPUEijk) = Y i + Gk None
7 G-aggregated E (CPUEijk) = Y i + Rj None
8 G-aggregated E (CPUEijk) = Y i + Rj Wt = N YG / ( N YR N grids Rij )
9 R-aggregated E (CPUEijk) = Y i + Rj None
The following notation is used. Data: Finescale – use of set-by-set
data, G-Aggregated – catch and effort data aggregated at the grid
level, and R-Aggregated – data aggregated at the regional level.
Model structure: Y i – the effect of the ith year, Rj – the effect of the
jth region, and Gk – the effect of the k th grid. Weighting: N obs – total
number of observations across all years, N YG – number of year-grid
combinations, N YR – number of year-region combinations, N grids Rij – number of grids in the jth region in the ith year, and nik – number
of observations in the kth grid in the ith year.
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R.A. Campbell / Fisheries Research 70 (2004) 209–227 221
Fig. 5. Indices of relative abundance based on alternative GLM analyses of the data for each of the four scenarios. The structure of the GLM
analysis is indicated in the title of each figure by the level of data aggregation (Finescale, Grid or Region), the area-effect used in the fitted model
(Grid or Region), and whether or not the analysis was weighted.
preserve the scale of the parameter estimates obtained
from the unweighted analyses. The correct relative in-
dex was also obtained whether one used a grid- or
regional-scale model for the GLM analysis. The cor-
rect relative index was also obtained under scenario 2
for the analyses based on data aggregated at the grid
level (GLMs 6, 7 and 8). For this scenario the weight-
ing assigned to the aggregated data is not required
because the same number of grids is fished in all re-
gions and there is only a single observation per grid.
A biased index was obtained, however, for the analysis
on data aggregated at the regional level (GLM 9) be-
cause the catch rate calculated for each region became
increasingly weighted over time by the higher pro-
portion of observations in the grids with higher catch
rates.
All indices were biased for scenario 3, with the bias
being greater for the unweighted fine-scale analyses
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222 R.A. Campbell / Fisheries Research 70 (2004) 209–227
(preference for fishing grids with higher catch rates
and an increased likelihood of low catch rate girds not
being fished in later years). The bias was, however,
slightly less severe for the weighted regional analysisusing the grid-aggregated data (GLM 8). The bias in all
analyses is due to the unrepresentative sampling of the
catch rates in the grids fished across each region. The
bias increases under scenario 4, where whole regions
are not fished in the last 2 years.
5. Modelling uncertainty in unfished spatial
strata
The results of the previous section indicate that
when the spatial distribution of a fishery contracts over
time, the indices of stock abundance based on the
results of a GLM analysis may become biased. The
essence of the problem stems from the fact that there
are no data for those areas which are not fished and,
as such, the data which are used in the analyses are
incomplete and not totally representative of the total
spatial distribution of the stock. For scenarios 3 and 4
there was an increasing lack of data from areas with
low catch rates so that the mean annual catch rates be-
came increasingly upwardly biased. If such a trend is
carried forward in time, then the temporal change in theresource abundance will be under-estimated. The issue
of unfished strata is a more general one than the spa-
tially contracting fishery example used here, though the
results of this example indicate that without a careful
interpretation of the assumptions underlying the anal-
ysis of catch and effort data, misleading trends in stock
abundance can result (Walters, 2003).
There are two options for overcoming the problems
inherent with data with missing strata. First, we can
undertake an analysis of that spatial subset of the data
commonto allyears. However,this approach is likelytoresult in too much useful information being discarded,
and the resulting index notbeing indicative of the entire
stock. However, it is often useful to define a core spa-
tial and temporal extent to the fishery which eliminates
marginal strata seldom fished or where catch rates are
persistently low (Campbell et al., 1996). The alterna-
tive is to define an appropriate spatial coverage of the
fishery and model the likely catch rates in those strata
for which there are no observations. While statistical
methods have been developed for the interpolation of
spatial data(e.g. kriging)and smoothingtechniques can
be used (Kulka et al., 1996), a simple alternative pro-
cedure is developed here to model appropriate catch
rates for the missing strata. A rationale for this ap-proach is that one can model the catch rates in the
areas bypassed by the fishery under explicit assump-
tions concerning the spatial dynamics of the stock and
the fleet. Furthermore, it is possible to bracket much of
the uncertainty associated with the analysis of spatially
incomplete data by adopting a range of assumptions.
However, depending on the scale of spatial analysis
which is possible (e.g. region- or grid-based) two dif-
ferent levels of modelling the catch rates in missing
strata are possible. Each is considered in turn.
5.1. Region-scale analysis
An estimate of the standardised catch rate in each
year-region stratum is first obtained by fitting the fol-
lowing model to the data:
E[log CPUE] = µ + YRij
+other standardising effects
where µ is the intercept and YRij parameterizes theinteraction between the effects in the ith year and jth
region. The expected value of the standardised catch
rate in each region is then:
CPUEstdij = exp(µ + YRij )
An abundance index Bij for each region is then calcu-
lated by multiplying the standardised catch rate for the
region by an estimate of the spatial extent of the stock
in that region. The number of grids in each region is
used for this purpose. However, the number of grids
fished in a region can change from year to year so it is
necessary to make some assumptions about the spatial
extent of the stock in each region in each year. Here we
assumethat the spatial extent of the stock in each regioneach year either coincides only with those grids fished,
N obsij , (i.e. there are no fish in grids not fished) or the
maximum number of grids fished in that region across
all years, N maxj (this is equivalent to assuming that the
grids fished in any year randomly sample the stock in
that region). Calling these the B-zero and B-avg indices
respectively, we have
B-zeroij = N obsij CPUEstdij ,
B-avgij = N maxj CPUEstdij
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R.A. Campbell / Fisheries Research 70 (2004) 209–227 223
Where there are no observations for a whole region,
the index for that region is modelled following the pro-
cedure:
• Find the maximum regional index for each year,Bmaxi
• For each year, calculate the relative index for each
fished region, Brelij = Bij / Bmaxi• For each region, calculate the average relative index
Brelj across those years when all regions were fished.
• For those regions with no observations, the likely
catch rate is set equal to the multiple of the aver-
age relative index for that region and the maximum
regional index for that year, Bmodij = Bmaxi Brelj
The total index for a given year is then the sum of the
regional indices across all regions.
5.2. Grid-scale analysis
An estimate of the standardised catch rate in each
year-grid stratum is first obtained by fitting the follow-
ing model to the data:
E[log CPUE] = µ + YGik
+other standardising effects
where µ is the intercept and YGik parameterizes the
interaction between the effects in the ith year and k th
grid. The expected value of the standardised catch rate
in each grid is then:
CPUEstdik = exp(µ + YGik)
As before, there are several options for modelling
the standardised catch rate for those grids within each
region that are not fished, and the abundance index Bijfor each year and region is then given by the sum of the
observed and modelled standardised catch rates across
all grids in each region. The B-zero and B-avg indices,
defined previously, are now given by
B-zeroij =
N obsj k=1
CPUEstdik ,
B-avgij =N maxj
N obsij
N obsij k=1
CPUEstdik
Given the finer spatial scale of the analysis, several
other indices may also be defined. First, one can define
the B-min index, which is similar to the B-avg index
but assumes that the catch rates in those grids, which
are not fished are, on average, equal to the minimum of
the catch rates in the fished grids. Alternatively, one candefine the B-target index which assumes that the spatial
extent of the stock remains the same for all years and
is equivalent to the maximum number of grids fished
in any year, but assumes that the grids fished in any
year coincide with those grids with the highest catch
rates (i.e. there is prefect targeting). The catch rates in
those grids not fished each year are then modelled by
the tail of the average distribution of catch rates across
the maximal extent of grids fished, i.e. for each region:
(a) For each year, sort the standardised catch rates in
the grids fished in descending order and find themaximum standardised catch rate, CPUEmaxij .
(b) Calculate the relative index for each grid:
CPUErelir =CPUEstdir
CPUEmaxij r = 1, . . . , N obsij
(c) Calculate the mean relative index for each grid
CPUErelr across those years when all grids are
fished in that region (i.e. when N obsij = N maxj ).
(d) The expected standardised catch rates for those
grids with no observations are then modelled as:
CPUEmodir = CPUEmaxij CPUErelr
r = N obsij + 1, . . . , N maxj
An abundance index for the region can then be
defined as:
B-targetij =
N obsij k=1
CPUEstdik
+
N maxj k=N obsij +1
CPUEmodik
Finally, the annual abundance indices are calcu-
lated by summing across the regional indices for
each year. An index for a region with no observa-
tions can be modelled as in Section 5.1.
5.3. Results
Annual abundance indices were calculated for each
of the 30 data sets for scenarios 3 and 4, using both the
region- and grid-scale analyses described above. The
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224 R.A. Campbell / Fisheries Research 70 (2004) 209–227
Fig. 6. Comparison of the relative indices of abundance based on modelling of catch rates in unfished spatial strata for various hypotheses
concerning the spatial distribution of the stock and fishing effort.
mean of the calculated indices across all data sets for
each scenario, relative to the value of the corresponding
index for the last year, were then calculated (Fig. 6).
For the region-scale analyses, the B-avg index was
similar to, if slightly worse, than the indices calculated
using the GLM analyses in Fig. 5. However, for sce-
nario 3 the region-scale annual B-zero and B-avg in-
dices bracketed the true annual index. While it is usefulto obtain a set of indices which bound the true index,
the difference between the B-zero and B-avg indices is
so large that considerable uncertainty remains as to the
true state of the stock over time. On the other hand, for
scenario 4 both indices under-estimated the true state
of the annual index.
The grid-scale annual B-zero and B-avg indices
bracketed the true annual index for scenarios 3 and 4.
However, as before, considerable uncertainty remained
regarding the true value of the index in any year. On the
other hand, both the B-min and B-target indices were
considerably closer to the true annual index, with the
B-target index being more accurate. This result is due to
the fact that the assumptions used in constructing these
latter indices more closely represent the true dynamics
of the stock and the fishery.
The results presented here are limited to five ways
of constructing indices. This should, however, in noway limit the nature or the number of indices which
can be constructed. Indeed, the nature of the stock and
effort assumptions used to construct the various indices
should be based on an understanding of the actual stock
and fishery dynamics of the fishery being analysed
(Campbell and Tuck, 1996). For example, one could
use the results concerning the levels of targeting of high
catch grids each year to weight the effort assumption
in the B-target index for the SBT fishery. Alternatively,
the existing individual indices could be combined us-
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R.A. Campbell / Fisheries Research 70 (2004) 209–227 225
ing different weights in different years. For example,
Hearn and Polacheck (1996) used the concept behind
the density-dependenthabitat model of MacCall (1990)
as a basis for constructing an index that assigns differ-ent annual weights to two indices based on the B-zero
and B-avg indices.
6. Discussion
Commercial catch and effort data continue to be re-
lied upon to estimate annual indices of stock abundance
in the absence of fishery-independent data. While
GLMs and other statistical techniques have improved
our ability to standardise such data, problems still per-
sist. While some of these problems relate to the choice
of the most appropriate model and error structure, and
the absence of data on factors which are likely to in-
fluence catch rates, there are more general problems of
deciding whether catch rate data from a fishery under-
going changes in the spatial allocation of fishing effort
can, in fact, reflect stock abundance.
The analyses in this paper have illustrated the
manner in which biases can enter into the estimates
of annual stock abundance due to the unbalanced and
changing spatial distribution of fishing effort. While
these biases generally relate to changes in the spatialcharacteristics of the fishery (either for the stock or
the fishing effort), biases can also arise due to a lack
of spatial detail in the analyses due to inappropriate
model structures or the use of too coarse a spatial
level of data aggregation. While the potential for such
biases is generally acknowledged, the manner in which
these biases arise in the GLM analyses commonly
used to model catch and effort data, and how they
can be dealt with, do not appear to be generally
appreciated.
The issue of an unequal spatial distribution of fish-ing effort and the preferential targeting of areas with
higher catch rates across the spatial areas used for a
GLM analysis can be corrected for by an appropriate
weighting. However, additional biases and uncertain-
ties arise due to missing observations, i.e. the areas of
the fishery which are not fished. The extent to which
the fishing grounds are known to overlap the spatial ex-
tent of the stock becomes increasingly uncertain when
there is a spatial contraction of the fishery over time.
The characteristics of the stock in areas not fished pre-
viously similarly remain uncertain for an expanding
fishery.
Given these uncertainties, it is usually not possible
to calculate a single reliably unbiased index of stock abundance. Instead, it may be preferable to calculate a
number of indices based on modelling the likely catch
rates in those areas not fished using various assump-
tions about the spatial distribution of both the stock and
the fishing effort (i.e. concerning the presence or not of
fish in the areas not fished and the targeting practices
of the fishers). Support for or rejection of the assump-
tions underlying the calculation of the various indices
can then be based on a spatial analysis of the data for
the fishery itself and/or an understanding of the deci-
sion rules used by fishers to allocate fishing effort spa-
tially, the behaviour observed in other fisheries or from
the ecological considerations. For example, changes in
the spatial range of a fish population may be consis-
tent with observations from other animal populations
and with the theory of density-dependent habitat selec-
tion (MacCall, 1990). Spatial contractions in a fishery
to areas with high catch rates may also be consistent
with economic practices associated with competitive
quotas.
An advantage of constructing a number of indices
based on modelling the likely catch rates in those areas
not fished using various assumptions about the spatialdistribution of both the stock and the fishing effort is
that any subsequent assessment can make use of a range
of indices which explicitly incorporate a full range of
uncertainty about the data instead of relying only on
a single CPUE-based tuning index (Polacheck et al.,
1996). Indeed, given the lack of independence among
catch rate observations within (and perhaps between
adjacent) strata, which results in an over-estimation of
the number of degrees of freedom in a GLM analysis,
the true uncertainty associated with any single abun-
dance index is usually under-estimated.Ultimately, the interpretation of catch rates and the
construction of indices of stock abundance should be
based on an understanding of the dynamics underlying
the spatial distribution of both the stock and the fish-
ing effort, and preferably on the relationship between
them. In many fisheries, this will entail the need for
surveys to understand and reduce the uncertainties in
the spatial characteristics of the stock in those areas
presently unfished. An experimental fishing program
within the SBT fishery was undertaken for this purpose
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226 R.A. Campbell / Fisheries Research 70 (2004) 209–227
(Anon, 1996). There will also be a need to more fully
understand the decision rules for the targeting prac-
tices of individual fishing vessels. Furthermore, in or-
der to overcome the potential biases which can resultfrom using catch and effort data from a fishery with a
high degree of spatial targeting, analysis of commercial
catch and effort data to obtain annual indices of relative
stock abundance should be carried out at the finest spa-
tial scale possible. For the SBT longline fishery this is
likely to be at the 1◦ level. However, for fisheries such
as purse seines, which are based on targeting aggrega-
tions, the level of spatial analysis may need to be much
finer (Clark and Mangel, 1979).
Finally, while this paper has focused on the prob-
lems with the construction of indices of stock abun-
dance based on the analysis of commercial catch and
effort data in a spatially varying fishery with an un-
certain stock and effort dynamics, many other factors
influence our ability to interpretcommercial catch rates
as indices of stock abundance. Many of these fac-
tors are well known (e.g. Gulland, 1964; Paloheimo
and Dickie, 1964; Hilborn and Walters, 1992) and in-
clude improvements in the operational and technologi-
cal aspects of the fishery, changes in environmental and
oceanographic conditions, together with the influence
of economic- and management-related decisions, all
of which may change catchability and availability overtime. Attempts to document these processes and im-
prove our understanding of how these factors influence
catch rates need to remain a high priority for fisheries
research.
Acknowledgements
Natalie Dowling, Yongshun Xiao and André Punt
are thanked for editorial comments on an earlier draft,
while the suggestions of an anonymous reviewer arealso acknowledged.
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