cpue standardisation and the construction of indices.pdf

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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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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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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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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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= 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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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


9/19


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)


10/19


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


11/19


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


12/19


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.


13/19


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


14/19


(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


15/19


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


16/19


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-


17/19


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


18/19


(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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