Relative Abundance and
Catch per Unit Effort
Wayne A. Hubert and Mary C. Fabrizio
■ 7.1 INTRODUCTION
Knowledge of the abundance of fish in a stock is a component of the information
used in management of fisheries (Ney 1999). Abundance estimates are used along
with data on age and length composition and weight–length relations to make judg-
ments regarding the status of fish stocks. Many methods have been developed to
estimate the numerical abundance of fish in a stock including counts within iso-
lated segments of a water body, mark and recapture, and removal methods (Chap-
ter 8; Van Den Avyle and Hayward 1999). However, in many freshwater fisheries
these methods require more time and money than can be allocated to the assess-
ment. In these cases, fisheries managers use indices of abundance to estimate rela-
tive abundance of fishes (Fabrizio and Richards 1996; Hubert 1996; Ney 1999).
The most common indices of relative abundance are computed from catch per
unit effort (C/f ) data for samples from a fish stock (Fabrizio and Richards 1996;
Hubert 1996; Ney 1999). A C/f index is defined mathematically as
C/f = qN, (7.1)
where C is the number of fish caught, f is the unit of effort expended, q is the
catchability coefficient or probability of catching an individual fish in one unit of
effort, and N is the absolute abundance of fish in the stock. When numerical
abundance cannot be estimated, fisheries scientists often use C/f to make judg-
ments about the abundance of fish in a stock.
Effort (f) is computed in many ways depending on the sampling gear and habi-
tat in which the target species resides. Units of effort may include individual sets
or hauls with a gear, the volume or area of habitat sampled, or the temporal dura-
tion of sampling. With passive gears (Hubert 1996), such as gill nets and trap nets,
effort is generally expressed in terms of the standard “set” with a specific piece of
279
7
280 Chapter 7
gear. For example, a gill-net set might involve placing the net on the bottom over-
night for 12 h, and the net may be 100 m long and 2 m high, constructed of 2.54-
cm-square-mesh monofilament netting, and have a float line and a lead line. With
active gears (Hayes et al. 1996), such as trawls, effort is often described in terms of
the duration or length of the haul at a given boat speed. Similarly, effort with
seines is often quantified relative to the area or length of shoreline over which the
seine is pulled. With small larval fish trawls or push nets (Kelso and Rutherford
1996), the volume of water filtered is often computed, and C/f is expressed as the
numbers captured per unit of water filtered. The C/f of electrofishing samples
(Reynolds 1996) is often described in terms of the number of fish caught in a
given amount of time (minutes or hours) or length of shoreline sampled.
7.1.1 General Applications in Freshwater Fisheries
Applications of C/f to assessment of stocks of freshwater fish occur in both sport
and commercial fisheries. A stock is a group of fish or other aquatic animals that
can be treated as a single unit for management purposes (Lackey and Hubert
1976). A stock is generally considered to be a self-contained and self-perpetuating
population of a single species with no mixing from the outside and within which
biological characteristics and impact of fishing are uniform. This definition is
accurate when applied to populations in small lakes and impoundments. How-
ever, the geographic boundaries of many freshwater fish stocks are vague and
unknown, as in streams, rivers, large reservoirs, or large lakes. Consequently, de-
fined areas and not biological populations are sometimes used as the manage-
ment units.
Catch per unit effort data are commonly used to monitor or assess stocks when
the boundaries of the populations are unknown. Sport fisheries are often assessed
by sampling with active or passive gears (Hayes et al. 1996; Hubert 1996) or by
surveying recreational anglers and sampling creeled fish (Malvestuto 1996). Com-
mercial fisheries are often assessed using onboard or port-side sampling of the
catch (Fabrizio and Richards 1996), but sampling protocols with active or passive
gears are also used. Commercial fishery sampling programs are often used to
estimate the catch of species in a fishery and the amount of fishing effort (Gillis
and Peterman 1998). All of these sampling approaches can generate C/f data
that can be used to assess temporal and spatial trends of fish stocks.
7.1.1.1 Monitoring of Stock Abundance over Time
One of the earliest applications of C/f data in inland waters was a description of
annual changes in relative abundance of sport fish in Clear Lake, Iowa from 1947 to
1968 based on gill-net data (Carlander 1953; Bulkley 1970). Similarly, the temporal
patterns in relative abundance of prey fishes in Lake Michigan from 1973 to 1993
have been described using C/f data from trawl sampling (Fabrizio et al. 2000). Also,
cyclic patterns in abundance of yellow perch in an oligotrophic lake have been
described using C/f data (Sanderson et al. 1999). Many similar monitoring pro-
grams have been conducted by state, provincial, and federal management agencies.
Relative Abundance and Catch per Unit Effort 281
Most commonly, time series of C/f data are used to assess the efficacy of fisheries
management actions, such as the response of largemouth bass and bluegill popula-
tions to the removal of excess vegetation in lakes (Pothoven et al. 1999). Monitor-
ing of C/f is also conducted to determine declines or increases in abundance of
rare species, such as Atlantic sturgeon in the Hudson River (Peterson et al. 2000).
Other C/f monitoring efforts may be used to evaluate restoration efforts, such as
those for lake trout in Lake Superior (Hansen et al. 1995). Similarly, the response
of fisheries to introductions of exotic species can be assessed using C/f, as has been
done for Lake Erie fishes relative to the appearance of zebra mussels in the lake
(Trometer and Busch 1999).
Several measures of annual variation in C/f have been developed to predict
the future abundance of fish or the quality of a fishery. For example, C/f of small
yellow perch in trawls in the southern portion of Lake Michigan has been used to
predict the future abundance of fish acceptable to anglers (Shroyer and McComish
1998). Similarly, C/f of walleye in gill nets during the fall can be a predictor of
angler catch rates the following summer (Isbell and Rawson 1989). Also, C/f of
age-0 fish has been used as a predictor of recruitment of age-1 fish of some species
in reservoirs (Willis 1987; Sammons and Bettoli 1999).
7.1.1.2 Evaluation of Spatial Distribution Patterns within Stocks
Another common application of C/f data is the evaluation of spatial distribution
patterns or patchiness of fish within a stock. For example, C/f data have been
used to describe spatial distributions of fishes in large (Ward et al. 2000) and
small (Hi and Lodge 1990) lakes, as well as reservoirs of various sizes (Hubert and
O’Shea 1992; Van Den Avyle et al. 1995; Michaletz and Gale 1999). Habitat asso-
ciations of fishes may be identified using C/f data obtained from different habi-
tats in both lentic (Irwin et al. 1997; Sammons and Bettoli 1999) and lotic (Jack-
son 1995; Johnson and Jennings 1998) waters. Seasonal patterns in fish distributions
have also been described using C/f data. For example, seasonal abundance of
fishes in tributaries to the Missouri River has been described in this manner
(Braaten and Guy 1999). Fisheries scientists also use C/f data to ascertain the
effects of habitat mitigation efforts on the spatial distribution of fishes (e.g., Moyer
et al. 1995; Chipps et al. 1997).
7.1.1.3 Assessment of Stocks Relative to Other Stocks
Comparison of fish stocks in two or more water bodies based on C/f data obtained
by standard fish sampling protocols has also been applied by freshwater fishery
managers. For example, among biologists managing small impoundments there
is general consensus that electrofishing C/f is a good measure of largemouth bass
abundance (Flickinger et al. 1999).
7.1.1.4 Surveys
Surveys are sometimes conducted in which C/f data are used to describe the fish
assemblage in a water body. However, the catchability coefficient (q) with a par-
ticular gear differs among species, so the actual composition of a fish assemblage
282 Chapter 7
is generally not well represented by C/f data. Nevertheless, researchers have at-
tempted to consider the effects of differential encounter probabilities, fish size,
fish swimming speed, and retention probabilities of a specific gear to provide a
better indicator of actual assemblage composition. For example, Spangler and
Collins (1992) made such adjustments to C/f data from gill nets to describe fish
assemblages in different portions of Lake Huron. Parsley et al. (1989) computed
capture efficiencies for small fishes sampled with beach seines to achieve a better
estimate of assemblage structure in a reservoir.
7.1.2 Underlying Assumptions
An underlying assumption of using C/f as an index of abundance is that the num-
ber of fish captured is proportional to the amount of effort expended. When a
population is closed, one unit of sampling effort removes a fixed proportion of
the total population (Seber 1982). As the population declines in abundance, the
number of animals captured by one unit of effort declines. This simple linear
relation between C/f and abundance has been extended to research and moni-
toring surveys such that C/f data are typically treated as a measure of abundance.
However, when the assumption of a linear relation fails, C/f can be a misleading
indicator of stock abundance.
7.1.2.1 Density As an Index of Abundance
The classic catch equation expresses catch as a proportion of abundance, and this
proportion varies with the amount of effort:
C = fq (N/A), (7.2)
where C is catch, f is fishing effort, q is (constant) catchability, N is abundance,
and A is the area in which the stock occurs (Gulland 1969). This equation can be
re-arranged to C/f = q(N/A), so if catchability is known, C/f is a measure of fish
density (N/A).
Assumptions of this model are (1) the population is in equilibrium (i.e., birth,
recruitment, and immigration rates are balanced by death and emigration rates);
(2) units of effort (such as individual trap or net sets) operate independently
(one unit of fishing gear does not interfere with other units); (3) q is constant
throughout the sampling period; and (4) every individual in the stock has the
same probability of capture (Seber 1982). The fourth assumption concerns the
spatial distribution of fish and is met when fish are uniformly distributed within
the boundaries of the stock. Additionally, when sampling of fish within a stock is
without replacement (i.e., live fish are not returned to the water), it is assumed
that the effects of such removals are negligible.
7.1.2.2 Constant Catchability
Technically, the constancy of the catchability coefficient (q) determines how well
C/f serves as an index of abundance (Gulland 1969). The assumption of equal
Relative Abundance and Catch per Unit Effort 283
capture probability for each fish in the population implies that fish are uniformly
distributed in space and that all occupied areas are accessible to the gear and are
randomly sampled. However, neither fishing effort nor fish are uniformly distrib-
uted (Paloheimo and Dickie 1964). Even when effort is uniform, such as in re-
search studies using standardized sampling methods, variation in catchability arises
when changes occur in the spatial distribution of fish. It has long been recognized
that C/f data reflect changes in animal distributions as often as they reflect changes
in abundance (Paloheimo and Dickie 1964). Changes in fish distribution (and
availability to the gear) may occur vertically (e.g., changes in the thermocline
affecting the vertical distribution of fish) or horizontally (e.g., different habitats
are occupied such that the proportion of a population occurring outside the sur-
vey area changes). Catch-per-unit-effort data are further confounded when changes
in spatial distribution occur concurrently with changes in abundance. For example,
at low abundance a relatively greater proportion of Atlantic cod were found in
shallow regions outside a survey area and were unavailable to the trawl, thereby
reducing catchability during times of low abundance (Swain et al. 1994).
Care must be taken to restrict interpretations of C/f estimates to the portion of
a stock actually sampled. For example, when fish in a stock are spatially distrib-
uted among exploited and unexploited regions, and individuals move from an
unexploited to an exploited region, C/f estimates from the exploited segment
are not a good measure of total stock abundance (Sampson 1991). Catch-per-
unit-effort estimates from the exploited region are representative of the entire
stock only when the rates of movement between the two regions are random.
Effects of shifts in distribution on C/f estimates have been recognized for some
time and have been incorporated into equilibrium models of production for ex-
ploited fisheries (Die et al. 1990).
Variations in catchability decrease the accuracy of C/f estimates as indices of
abundance. Catchability can vary with size, sex, or other intrinsic characteristics
of fish. Catchability can also vary with time of day, season, sampling site, water
temperature, dissolved oxygen levels, or other environmental features that may
affect the ability of the gear to capture fish or the distribution of fish relative to
the gear (i.e., availability).
There are several ways to address departures from the constant catchability
assumption. One approach is to stratify sampling of a stock according to the fea-
ture of interest. For example, when catchability varies with fish length, then an
estimate of catchability for a stock is really the average q for all fish in the stock. As
long as the length structure of the fish in the stock does not change, the average
q will be a reasonable estimate for the entire stock. However, the length structure
of fish in a stock is generally not constant, and catchability may best be estimated
separately for individual length-classes in the sample (Seber 1982). Another ap-
proach is to adjust C/f data to account for changes in extrinsic factors such as
changes in fishing power (Kimura 1981). For example, trawlers with larger en-
gines consistently catch more fish and have greater fishing power than do trawlers
with smaller engines, all other things being equal (Gulland 1977). Still another
approach is to estimate catchability independently, for example, from a tagging
284 Chapter 7
study that yields estimates of stock abundance through time (e.g., Paloheimo 1963).
Whichever approach is taken, the factors affecting catchability should be mea-
sured and used to adjust C/f data.
Numerous relationships between catchability and environmental factors have
been found by examining correlations between C/f and these factors. For in-
stance, swimming speed of fish generally decreases at lower temperatures and fish
become more vulnerable to capture by a trawl. At the same time, the spatial distri-
bution of fish in a stock may change as temperature decreases, thereby changing
their availability to the gear. However, although C/f may be significantly corre-
lated with environmental conditions, catchability may not be affected by those
factors (Swain et al. 2000). Careful examination and appropriate experimental
designs are needed to understand the nature of the relationship between
catchability and environmental factors.
7.1.2.3 Validation of Assumptions
The distinction between density and abundance is often overlooked, but the key
to understanding their difference is the validity of the constant area assumption.
Typically, the area occupied by a stock is assumed to remain constant. As abun-
dance changes, the expectation is that density will also change, and C/f estimates
will remain proportional to abundance. Although the proportional relation be-
tween C/f estimates and abundance is convenient, it is not universal (e.g., Crecco
and Savoy 1985). In some cases, as abundance increases, fish may increase their
spatial distribution and spread into adjacent nonsampled areas.
In other cases, C/f may exhibit “hyperdepletion” in relation to abundance
(Hilborn and Walters 1992). In this situation, the rate of change for C/f is higher
than it is for abundance. This relation is observed when C/f decreases faster than
abundance because the most vulnerable animals are captured first, leaving behind
less vulnerable individuals (Ricker 1975; Miller 1990; Hilborn and Walters 1992).
An opposite effect is “hyperstability,” which occurs when C/f remains high even
as abundance decreases (Hilborn and Walters 1992). This relationship occurs
when the search for fish is highly efficient, effort is concentrated in areas of high
densities, and the fish remain concentrated as abundance declines (Hilborn and
Walters 1992). This has been observed among commercial (Rose and Kulka 1999)
and recreational (Peterman and Steer 1981) fisheries. Aggregation of fish in a
small portion of the stock’s boundaries during a period of declining abundance is
termed hyperaggregation (Rose and Kulka 1999). For example, anglers experi-
enced high catchabilities of Chinook salmon during periods of low riverine abun-
dance because both fish and anglers were concentrated in small areas of the river
(Peterman and Steer 1981).
The assumption of constant catchability has been investigated for commercial
fisheries because of known changes in fishing efficiency associated with vessel
power, learning by crews, and technological improvements in commercial fleets
through time (Fabrizio and Richards 1996). These factors increase catchability
and introduce systematic error in C/f data. Thus, long-term C/f data from com-
mercial fisheries must be adjusted prior to computation of C/f estimates. Without
Relative Abundance and Catch per Unit Effort 285
such adjustments, increased catchability leads to overestimation of stock abun-
dance from commercial fishery statistics. Variations in fishing power may also char-
acterize research surveys when more than one crew, vessel, or unit of gear is used
(Munro 1998). Often, such variations must be explored experimentally to derive
conversion coefficients (e.g., Pelletier 1998). When C/f data are adjusted for dif-
ferences in catchability there is a tendency to overestimate the variance of C/f ;
thus, Munro (1998) developed a method to determine when adjustments are
warranted.
The assumption of independence of fish-sampling units (i.e., no interference
of one unit of gear with another) has been considered in a few studies. Interfer-
ence among highly aggregated gill nets has been documented (Rose and Leggett
1989). Among anglers, interference is commonly observed when total effort is
high (e.g., during holidays when crowding can lead to lower catchabilities; Ricker
1975). In general, data are insufficient to determine how frequently interference
may occur (Gillis 1999). Simulation studies show that when stock abundance is
low, C/f estimates can fail to reflect abundance even under low levels of interfer-
ence (Gillis and Peterman 1998).
In some instances, mark–recapture experiments can be conducted to estimate
stock abundance (N) and relate these estimates to C/f data to obtain an estimate
of the catchability: q = (C/f )/N. For example, electrofishing catchability has been
related to largemouth bass abundance estimated by mark–recapture methods in
small impoundments (Hall 1986) and lakes (Coble 1992).
■ 7.2 SAMPLING DESIGN
The importance of sampling design cannot be overemphasized when using C/f
estimates as an index of stock abundance. Catch per unit effort can vary widely
because fish distributions are patchy and fish exhibit spatial and temporal varia-
tion in their distribution and activity patterns. Mean C/f estimates often have
high variance (Peterman and Bradford 1987; Allen et al.1999), thereby introduc-
ing uncertainty when using C/f to assess differences in stock abundance. Thus,
sampling designs that minimize variation in C/f should be used. For example, in
an effort to reduce the variation in C/f , fisheries scientists often sample with the
same gear, in the same locations, and at the same time each year when assessing
annual changes in abundance of fishes (e.g., Fabrizio et al. 2000).
Generally, sampling designs are developed to minimize variation in C/f that is
due to factors other than the true abundance of fish. Sampling locations and
times are selected based on knowledge of the life history, movement, and habitat
associations of a species (Pope and Willis 1996). Substantial literature is dedi-
cated to identifying where and when to sample different species. For example,
Mero and Willis (1992) assessed seasonal variation in gill-net catches of walleye
and sauger from Lake Sakakawea, North Dakota, to determine when C/f was high-
est and the coefficient of variation of the C/f data was lowest. Similarly, variation
in C/f data for largemouth bass sampled by electrofishing is minimized when
sampling only shoreline areas (McInerny and Cross 2000).
286 Chapter 7
A study is dependent on the objectives of the fisheries scientist, and objectives
must be clearly identified. For example, objectives may be to (1) define annual
trends in abundance of walleye in a prairie lake, (2) evaluate changes in relative
abundance of channel catfish in a river in response to implementation of a mini-
mum-length limit, or (3) determine effects of shoreline restoration efforts on the
relative abundance of largemouth bass in a small impoundment. Each objective
may require a different sampling design dependent not only on the question but
also on the species and type of water body.
The experimental designs described in Chapter 3 have the potential of being
used in studies in which C/f estimates are the response variable. Simple random
sampling is generally not appropriate when C/f estimates are used to assess fish
stocks because low precision generates C/f data that are too variable to detect
trends or differences that may occur. Within the fisheries literature, we have found
no examples of simple random sampling where C/f estimates were the response
variable, but it is possible that situations may occur for which such a design may be
applicable, particularly in small water bodies with homogeneous habitat features.
There is a strong tendency among fisheries scientists to use stratified random
sampling designs, especially when assessing temporal trends in C/f. Four general
reasons for using stratified random sampling to assess fish stocks are (see Cochran
1977) (1) calculation of C/f statistics may be required for different portions of a
stock, such as different bays within a large lake; (2) sampling constraints may
necessitate using different sampling methods in different areas, such as trawling
in offshore areas and beach seining in nearshore areas of a large lake; (3) stratifi-
cation may result in a gain in precision of C/f estimates for the whole stock; and
(4) administrative convenience may require stratification in different areas, such
as different states or provinces around one of the Great Lakes. Michaletz and
Gale (1999) provide an example of the application of stratified random sampling
where C/f estimates were used to assess both spatial and temporal patterns of
abundance (also, see example in section 7.5.2).
A systematic sampling design is another approach that may be considered for
studies based on C/f data. In this approach, sampling begins at a randomly se-
lected site or time and continues at equally spaced locations or time intervals.
Systematic sampling may be used effectively in rivers to gather information on the
relative abundance of organisms along a gradient of environmental conditions
(Karr 1999). Although several estimators exist for the variance of the mean from
a systematic sample, all estimators require data from replicated systematic samples
(Cochran 1977). The variance of the mean from a single systematic sample may
be estimated, but estimators studied to date are biased and inconsistent (Skalski
et al. 1993). For example, when mean abundance estimates are obtained from
hydroacoustic surveys, systematic designs may (Simmonds and Fryer 1996) or may
not (Jessop and Harvie 1990; Skalski et al. 1993) yield highly precise estimates of
abundance. In general, systematic sampling provides less precise estimates of the
mean than does stratified random sampling (Cochran 1977) and should be con-
sidered only when the objectives of the study are not compromised by the lower
precision of systematic sampling estimators or when preliminary analyses indicate
Relative Abundance and Catch per Unit Effort 287
that lower sampling costs associated with systematic designs outweigh the need
for precision. Systematic sampling designs are probably most useful when com-
bined with stratified random sampling in a two-stage approach (Schweigert et al.
1985; Chapter 3). Additional sampling designs may be applicable to assessment of
C/f (see Chapters 2 and 3).
■ 7.3 ASPECTS OF SAMPLING EFFORT
Design and construction of a sampling gear and factors associated with its opera-
tion contribute to variations in gear efficiency. For instance, catch rates for traps
are affected by soak time and the type of bait used to attract animals (Miller 1990).
In addition, gear efficiency may be affected by the interaction of captured ani-
mals and the gear itself (the saturation effect) and may vary according to life
stage of the target species (Miller 1990). When conducting research or monitor-
ing, care must be taken to standardize gear, not just in terms of the design and
construction, but also in terms of the operation. Standardization also pertains to
techniques used by operators of the gear. It is widely recognized that even though
using the same gear, some operators can obtain a higher catch than others. By
standardizing gear design, construction, and operation, fisheries scientists mini-
mize variation in catchability and C/f data.
Often, preliminary sampling is necessary to identify factors associated with varia-
tion in catchability of a target species. A good example of an informative prelimi-
nary analysis is described in Bernard et al. (1991). They examined diurnal changes
in catchability, optimal baiting strategies, optimal soak duration, and hoop-net
size effects (among other factors) on the efficacy of hoop nets for capturing bur-
bot in Alaskan lakes. These results were used to design surveys of stocks of burbot
in 15 Alaskan lakes (Bernard et al. 1993).
7.3.1 Selectivity and Saturation
Gear performance is species and habitat specific (Choat et al. 1993). For example,
light-trap selectivity for larval fish sampling depends on the attraction of different
species to light, and not all taxa are equally phototaxic (Choat et al. 1993). En-
counter rates of some species or sizes can be increased by deploying the gear in
appropriate habitats and exploiting behavioral differences among species or life
stages. However, encounters with gear do not necessarily result in capture. Fish
are captured when they encounter the gear and are also retained. The probability
of retention is termed selectivity. With some gear, retention of organisms will vary
with mesh size and the likelihood of extrusion through the mesh. Body size, body
shape, and pressure exerted by fish across the net mesh are three factors that
determine the likelihood of extrusion.
Gear saturation is another factor affecting gear efficiency and catchability. Satu-
ration occurs when the present catch reduces the potential for additional catch
by reducing the number of new captures, increasing escapement, or both (Miller
1990). As a gear becomes saturated, the likelihood of capturing additional animals
288 Chapter 7
decreases. Good examples of saturation effects can be found for gill nets (the
presence of entangled fish may scare other fish away), longlines (as more fish are
captured, the number of vacant hooks decreases, and eventually no additional
fish are caught), and baited pots or traps (captured animals deplete the bait or
discourage other animals from entering the trap). In traps, reduced entry is thought
to be due to intimidation by trapped organisms using odor, posture, or sound to
prevent entry of additional animals into the trap (Miller 1990). Longlines are
notoriously prone to the effects of saturation and interspecific competition for
hooks. Under these conditions, time fished is not a good indicator of true effort.
An extreme example occurs when all hooks bear fish at time t but the longline is
retained in place until t + i; in this case, C/f is biased low because true effort was
(over) estimated by t + i. New methods have been developed to more accurately
estimate effort associated with longlines based on time to capture as measured by
fish-activated timing devices placed on each hook (Somerton and Kikkawa 1995).
7.3.2 Sampling Issues Specific to Gear Types
7.3.2.1 Passive Gears
Passive gears rely on the movement of organisms, either schooling or more di-
rected migrations such as spawning, to bring organisms in contact with the gear
(Hubert 1996). Schooling behavior creates density differences that affect the esti-
mation of relative abundance. About one-fourth of teleosts are obligate schoolers
and exhibit schooling behavior throughout their life, and about half of all teleost
species school as juveniles (Shaw 1978). Schooling increases the vulnerability of
fish to capture by fishing gear. Increased vulnerability of individuals in schools
leads to less time expended in capturing fish, thus leading to biased C/f esti-
mates. For example, catch rates with passive gears may be higher when environ-
mental factors cause fish movements and increase their vulnerability to capture
(Rose and Leggett 1989).
Because the effective area fished by passive gear is impossible to measure, ef-
fort is measured in terms of soak time. Although it may seem that longer soak
times should produce greater catches, in fact, as soak time increases, the gear may
become saturated and catch per unit of time will decrease. At this point, C/f does
not provide an index of relative abundance (Hansen et al. 1998). For traps and
pots, as soak time increases, the catch actually may decrease as more organisms
escape than enter (Miller 1990). For baited longlines, saturation begins to occur
as the odor concentration from baited hooks decreases (Sigler 2000).
The relation between catch and soak time is specific for particular gear types
(Miller 1990) and must be determined experimentally. When designing experi-
ments or surveys involving traps, Miller (1990) suggests the following: (1) deter-
mine the relation between catch and soak time; (2) ensure that catch rates are
uniform throughout the study area; (3) standardize bait quantity and quality; (4)
standardize time of setting and hauling; (5) standardize trap spacing; (6) maintain
Relative Abundance and Catch per Unit Effort 289
traps in good repair; and, if following an experimental protocol, (7) randomize
sampling spatially and temporally within strata.
With other passive gears such as gill nets and drift nets, catchability may be
related to the visual acuity of fish, which, in turn, is affected by turbidity, light
intensity, or other environmental conditions (e.g., Cui et al. 1991). Light intensity
varies over a daily cycle, but lunar phase also affects light intensity. In addition,
the visibility of the net depends on the color of the mesh.
Catchability of passive gears may be affected by changes in activity of fish asso-
ciated with light levels. For example, most decapods are more active during dawn,
dusk, or generally at night, and catchability increases during these times (Miller
1990). Some fish are also more active at night, increasing their vulnerability to
passive gear. However, this relation of increased activity and light intensity may
change seasonally. For example, burbot are nocturnal in spring and summer but
diurnal in the fall (Bernard et al. 1993).
7.3.2.2 Active Gears
Active gears often have different catchabilities depending on light intensity. This
may be due to diel vertical movements of fish or reduced visibility (Walsh 1991;
Casey and Myers 1998; Korsbrekke and Nakken 1999). Catchability may also be
affected by the ability of fish to escape, and that ability depends on the behavior
of individuals during herding and capture (Godø et al. 1999).
Electrofishing is a highly effective active sampling gear for fish in streams
and littoral zones of lakes (Reynolds 1996). Electrofishing tends to be more
effective for larger fish and for species that float at the surface when stunned.
Some species exhibit relatively low catchabilities to electrofishing gear. For in-
stance, benthic fishes exhibit low catchabilities because the likelihood of seeing
immobilized individuals is low, whereas other pelagic species avoid the electric
field (Bohlin et al. 1989), thereby reducing their catchability. Additionally, in-
creasing water levels can reduce electrofishing catchabilities in rivers (Bohlin et
al. 1989). Standardization of electrofishing techniques is important when using
C/f as an index of abundance.
7.3.3. Standardization of Effort
The appropriate units for measuring effort for a given gear can vary depending
on the target species and habitat sampled. For example, electrofishing C/f is usu-
ally reported as catch per minute (e.g., Tillma et al. 1998), especially for highly
abundant species or life stages such as age-0 bluegills in Midwestern lakes. At
times, electrofishing C/f may be reported as catch per hour (e.g., Paragamian
1989; Miranda et al. 1996), but this usually occurs when the species of interest is
rarely captured. In some cases the shoreline of a lake or reservoir may serve as the
sampling unit in an electrofishing survey and C/f is measured as catch per area
(e.g., fish per 100 m
2
of littoral zone) or catch per length of shoreline (e.g., fish
per 100 m).
290 Chapter 7
Fishing power should be standardized to maintain constant catchability. For
example, C/f estimates from trawl surveys can change with vessel speed, so vessel
speed must be constant. With all types of sampling, fisheries scientists must con-
sider how procedural changes may affect fishing power. For example, electrofishing
equipment is typically standardized by using constant voltage or constant amper-
age. Variations in electrical power (wattage) have caused 12–15% increases in
variation in electrofishing catch rates (Burkhardt and Gutreuter 1995).
7.3.3.1 Multiple Gears
In some instances it is desirable to use two or more gear types to sample organisms
and to combine C/f estimates from the different gears (Seber 1982). However,
care must be taken to ensure that catchability of each gear remains constant.
Although catchability may be constant through time for one gear, it may not be
for another (e.g., gill nets and trap nets for Atlantic cod, Rose and Leggett 1989).
At times it may be possible to calibrate C/f from multiple gears, but patterns of
variation in C/f may be due to behavioral characteristics of the species under
study (Methven and Schneider 1998).
7.3.3.2 Effects of Seasonal and Daily Variation
Catch per unit effort can change seasonally due to variations in recruitment, growth,
and mortality, but such changes may not be the same for all species or for a given
species in all habitats (Pope and Willis 1996; Richards et al. 1996). For example,
seasonal variation in C/f for the virile crayfish was observed in Minnesota lakes
but not for the northern clearwater crayfish in streams (Richards et al. 1996). For
some species, catchability may increase temporarily during a particular season as
fish increase activity levels in response to environmental factors such as tempera-
ture and photoperiod (e.g., Bernard et al. 1993; Braaten and Guy 1999; Gordoa et
al. 2000). Also, seasonal patterns in C/f may not be observed every year due to
climate variability (Gordoa et al. 2000).
When collecting C/f data over time to examine trends, care must be exercised
to sample at appropriate times if seasonal variation in density exists. In the case of
sampling during a seasonal spawning migration, annual C/f data will reflect rela-
tive abundance only if the seasonal timing of the migration remains the same
from year to year (Fréon and Misund 1999). Thus, when sampling migratory spe-
cies, the timing within the run is critical. For example, two-thirds of the annual
emigration of Chinook salmon smolts occurred during a new or waning moon
(Roper and Scarnecchia 1999). If using C/f data to compare abundance of a
species from various areas, then care must be exercised to sample the areas of
interest during the same time. For example, a comparative survey of bluegill abun-
dance in Minnesota lakes found that data collected at different times of the year
should not be compared because about 40% of the variation in C/f was explained
by day of the year (Cross et al. 1995).
Daily or circadian variations in C/f are well known (e.g., Walsh 1991). Such
variations may be related to the visual acuity of fish or diel vertical movements in
lentic systems (Stoner 1991). Similarly, electrofishing catchability may (Paragamian
Relative Abundance and Catch per Unit Effort 291
1989; Dumont and Dennis 1997) or may not (Maceina et al. 1995; Van Zee et al.
1996; Dumont and Dennis 1997) increase at night depending on the target spe-
cies or the habitat in which the species occurs (Kessler 1999).
7.3.3.3 Consideration of Life History and Behavior
Catchability may be affected by life history or physiological stage of the target
species. This is illustrated among decapod crustaceans. Typically, the increased
activity levels of decapods during warm temperatures increase their vulnerability
to capture in baited traps, but their vulnerability ceases during molting (e.g., Somers
and Green 1993; Richards et al. 1996). Similarly, catchability of the American
lobster in traps decreases to near zero during molting, and because males and
females may molt at different times of the year, sex-specific catchability varies
(Miller 1990). In Minnesota streams, catchability of northern clearwater crayfish
in baited traps was highest between molting periods when animals were actively
feeding (Richards et al. 1996). Thus, behavioral changes associated with life his-
tory events should be taken into account when interpreting C/f estimates.
Some of the most effective fishing gears use the behavioral responses of organ-
isms to maximize encounter rates and retention. A pertinent example is how ol-
factory cues can be used to elicit behavioral responses of fish to enhance encoun-
ter probabilities. For example, when Gerhardt and Hubert (1989) baited hoop
nets, the C/f of channel catfish was doubled during the postspawning period.
Some species or life stages are photopositive, so gear catchability can be in-
creased by using light lures at night. While lighted traps and other nets may in-
crease nocturnal catches of certain fishes or life stages, the phase of the moon
may interact with catchability if fish activity varies with lunar phase. For example,
Rooker et al. (1996) found that nocturnal catches of larval fishes increased sig-
nificantly when lighted lift nets were used during the new moon.
The presence of predators or competitors may influence catchability. For
example, in Ontario lakes, crayfish catchability in baited traps declined in the
presence of rock bass and smallmouth bass and with increasing numbers of co-
occurring crayfish species (Collins et al. 1983; Somers and Green 1993). These
affects were noted only in lakes with relatively high abundance of predatory fishes
(Collins et al. 1983).
Habitat preferences and behavior of organisms contribute to variation in C/f
(Fréon and Misund 1999). Juvenile and adult fish may be distributed in areas vary-
ing in depth (Hubert and Sandheinrich 1983; Bernard et al. 1993). Individuals of
some species may segregate spatially on the basis of sex (Miller 1990). If a substan-
tial portion of a stock occupies a habitat that is inaccessible to the sampling gear,
then the proportion available to the gear is likely to vary through time depending
on environmental factors that alter habitat selection (Fréon and Misund 1999). For
example, tidal currents in the Barents Sea have been shown to influence the verti-
cal distribution of cod and haddock such that they are available to bottom trawls
only during periods of low or decreasing tidal currents (Michalsen et al. 1996).
Habitat preferences of fish are sometimes exploited to enhance catchability.
For example, some species prefer areas with cover and fisheries scientists may
292 Chapter 7
purposefully sample in these areas. Often, nets are set or gear is towed in areas
likely to contain fish, and the sampling locations are not truly standardized or
random. This type of selection mimics the manner in which commercial fisheries
operate by locating areas with potentially high densities of fish and fishing only in
these areas. If the objective is to compare changes through time in an impound-
ment or lake, then such judgment sampling may be appropriate (Hubbard and
Miranda 1988). When sampling for largemouth bass, electrofishing in areas near
weed beds, stump fields, or flooded timber in the littoral zone would constitute
judgment sampling (Hubbard and Miranda 1988). As long as the judgment sam-
pling sites are constant over time (i.e., permanent sampling sites), this approach
can yield an efficient means to assess temporal trends in relative abundance within
a given water body.
7.3.3.4 Consideration of Gear Efficiency in Different Habitats
The efficiency of a given gear can vary substantially among habitat types. For
example, electrofishing efficiency can vary widely among habitat types. In habi-
tats with low water clarity (transparencies less than 1 m) and depths greater than
a few meters, electrofishing is not very efficient (Bohlin et al. 1989). For example,
Dewey (1992) reported that in turbid, highly vegetated waters, electrofishing was
less efficient than were other gears because low visibility and entanglement of fish
in the vegetation reduced capture efficiency.
7.3.4 The Need to Minimize Variance and Bias
One of the most common approaches to increasing the precision of C/f estimates
is to increase the number of samples. Assuming the sampling design is appropri-
ate (see section 7.2; Chapters 2 and 3) and catchability is constant, increasing
sample size will likely increase precision. However, factors affecting catchability
must remain constant during the sampling period. For instance, if catchability
varies greatly with light intensity and samples are collected throughout a 24-h
period without regard to this factor, then an increase in the number of samples
may not improve precision. We recommend that variation in catchability be stud-
ied with respect to factors that influence the magnitude of C/f estimates includ-
ing those that influence availability of animals to the gear and vulnerability to
capture. Once these factors are known, then the value of increasing the number
of samples can be determined.
In stratified random sampling designs the optimal sampling plan may not in-
volve equal sampling among all strata, but rather optimal sampling intensity may
vary according to stratum size. Minimizing the variation of C/f data is particularly
important when these data are used to evaluate changes in relative abundance.
Trends in abundance may be difficult to discern or detect when the data are highly
variable (see Box 7.1).
The considerations we discussed to maximize precision of C/f estimates are
not exhaustive, and additional considerations should be made. For example,
only fully recruited age-classes should be considered in deriving C/f estimates;
Relative Abundance and Catch per Unit Effort 293
otherwise, recruitment variability will induce variation in catchability (Seber 1982).
A single gear will not capture all components of a stock in proportion to their
abundance, so a key piece of information is the selectivity of the gear for various
life stages or length-classes of the target species. In addition, all habitats inhabited
by a species will not be equally sampled. Sampling should be conducted during a
time when factors affecting catchability are similar if C/f data are used to com-
pare across time or space (Richards and Schnute 1986; Miller 1990). A good ap-
proach is to focus the unit of study and define it properly, and then consider gains
in precision through replication.
Box 7.1 Detection of Changes in Relative Abundance with Highly Variable Catch
per Unit Effort (C/f ) Data
The time series data below illustrate the effects of highly variable C/f data on the ability to detect
relative abundance changes for a hypothetical fish population that is declining through time. The
first column is the year; the second column, population abundance (N), shows a decline of 5% each
year; the third column gives C/f as 0.001N; the forth column shows C/f varying randomly by 5% or
10% above or below 0.001N; and the fifth column is C/f varying randomly by 20% or 40% above or
below 0.001N.
Table Times series data for a hypothetical fish population.
Year Population abundance (N) C/f C/f ± 5% or 10% C/f ± 20% or 40%
1 10,000 10.0 10.5 6.0
2 9,500 9.5 8.6 13.3
3 9,025 9.0 8.1 5.4
4 8,573 8.6 7.7 10.3
5 8,145 8.1 7.7 4.9
6 7,739 7.7 6.9 4.6
7 7,351 7.4 8.1 5.9
8 6,983 7.0 7.7 5.6
9 6,634 6.6 7.2 9.2
10 6,302 6.3 6.0 8.8
11 5,987 6.0 6.3 8.4
12 5,688 5.7 5.4 3.4
13 5,404 5.4 5.7 6.5
14 5,133 5.1 4.8 3.1
15 4,877 4.9 5.4 6.9
A significant correlation (r = 0.64; P < 0.001) is observed between N and the C/f ± 5 or 10% measure-
ment error, but the correlation (r = 0.33; P = 0.23) between N and the C/f ± 20% or 40% measure-
ment error is not significant. Note that with the C/f ± 20% or 40% measurement error the C/f in year
15 exceeds the C/f in year 1.
Both levels of measurement error used in this example are within the range of what may be
encountered in the field when sampling fish populations and obtaining C/f data. This illustrates
how C/f measurement error can mask changes in actual abundance of fish populations.
294 Chapter 7
7.3.5 Assessment of Sample Sizes Prior to Initiation of Sampling
A power analysis allows the researcher to determine the level of effort (i.e., sample
size) necessary to detect a change of a predetermined magnitude given a mea-
sure of the variability in the factor of interest (see Box 7.2). For instance, the
number of trap-nights necessary to detect a 25% change in relative abundance of
bluegill can be determined using an estimate of the variance of the mean C/f. In
general, power analysis requires the assumption that C/f data follow a normal
distribution. If the C/f data are not normally distributed, it becomes important to
identify a transformation that yields an approximately normal distribution (Gryska
et al. 1997). An example of statistical power analysis applied to C/f data from
electrofishing samples is given in Paller (1995). In general, many samples will be
necessary to detect small (<20%) differences among means, but when C/f is low,
an even greater number of samples is necessary (Paller 1995).
■ 7.4 STATISTICAL ANALYSIS
A common approach to analysis of C/f data has been to compute means and
assume normal distributions of the data. However, the frequency distributions of
C/f data are seldom normal. This is not surprising because C/f is a ratio estimator
having catch and effort as random variables (Cochran 1977). Testing hypotheses
regarding C/f generally involve application of statistical tests that assume the vari-
ables have a continuous scale of measure, the data exhibit a normal frequency
distribution, and standard deviations are independent of the mean. Statistical
analyses that require these assumptions can lead to reductions in power and mis-
leading results when the assumptions are not met. Nonparametric statistical pro-
cedures have less restrictive assumptions regarding distributions, but it is difficult
to assess the magnitude of difference between treatments or change over time
based on nonparametric procedures.
7.4.1 Normalization of C/f Distributions
The shapes of C/f sample distributions can vary widely and may include normal
frequency distributions and negative binomial distributions. It is common for C/f
distributions to have standard deviations that are about equal to the mean, to be
positively skewed (Moyle and Lound 1960), and to have standard deviations that
increase proportionally with the mean—indications of distributions that are not
normal. Among 703 published studies on larval fish abundances estimated from
replicated sampling, Cyr et al. (1992) found many positive relationships between
the variance and the mean, indicating that C/f frequency distributions were not
normal in many studies. It has been suggested that the shape of C/f sample distri-
bution changes with fish abundance (see Hubert 1996). At very high fish densi-
ties, C/f data may be normally distributed, but as fish densities decline, the mode
shifts to the left and the distribution becomes skewed to the right. At relatively
Relative Abundance and Catch per Unit Effort 295
Box 7.2 Power Analysis Assessment of Sampling Effort
Preliminary sampling of channel catfish in two hypothetical small impoundments is conducted
with traps in early summer to obtain C/f data as the first step in establishing an annual monitoring
program to assess temporal variation in mean C/f. In each reservoir, 20 traps are set at randomly
selected locations, left overnight, and retrieved the following day. The following C/f data (i.e., fish/
trap-night) and statistics are obtained for each impoundment. Each C/f value is transformed as
log
10
(C/f + 1) to assess the effects of data transformation on C/f statistics and estimates of needed
sampling effort.
Table Catch per unit effort data and summary statistics for channel catfish in two hypothetical
impoundments.
Net set and
Impoundment A Impoundment B
summary statistic C/f log
10
(C/f + 1) C/f log
10
(C/f + 1)
1 0010.301
2 0010.301
3 0020.477
4 0020.477
5 0030.602
6 0030.602
7 0030.602
8 0040.699
9 1 0.301 4 0.699
10 1 0.301 4 0.699
11 1 0.301 4 0.699
12 1 0.301 5 0.778
13 2 0.477 5 0.788
14 2 0.477 5 0.788
15 3 0.602 6 0.845
16 3 0.602 6 0.845
17 7 0.903 8 0.954
18 9 1.000 10 1.041
19 12 1.114 12 1.114
20 20 1.322 20 1.322
Mean 3.1 0.385 5.4 0.731
SD 5.2 0.422 4.4 0.254
Observation of the data and the summary statistics suggests that the C/f data for Impoundment A
are highly skewed and substantially depart from a normal distribution; furthermore, the logarith-
mic transformation did little to affect the shape of the distribution. For both forms of C/f data the
standard deviation exceeds the mean. The C/f data for Impoundment B are less severely skewed,
and a logarithmic transformation reduces the standard deviation and creates a frequency distribu-
tion that more closely resembles a normal distribution.
Power analysis allows definition of the required sampling effort to determine specified changes in
mean C/f at predetermined levels of significance (␣) and power (1 –) to guard against type I
(Box continues)
296 Chapter 7
(rejecting the null hypothesis of no difference when it is true) and type II (failing to reject the null
hypothesis of no difference between the means when it is false) errors (Brown and Austen 1996;
Gryska et al. 1997). Power analysis is conducted by computing needed sampling effort at various
levels of significance, power, and detectable effect sizes. Means and variances of C/f data from
previous or preliminary sampling periods are used in the computations. The detectable effect size
is the specified difference in two means when the null hypothesis is rejected at a specified ␣ and 
(Cohen 1969). For example, if the mean C/f is 3.1 fish/trap-night (as was observed for Impoundment
A during the preliminary sampling) and the fisheries scientist specifies that the desire is to detect a
change in C/f in either direction of 10% or more, the detectable effect size is 0.31 fish/trap-night.
The desire is that the null hypothesis would be rejected if future sampling means differ from the
preliminary sampling mean by 0.31 fish/trap night or more. How much sampling effort is needed to
detect such a difference at various probabilities of type I and type II error?
Calculations are performed using the formula of Snedecor and Cochran (1989):
n = 2 (z
␣
+ z

)
2
(s
2
/d
2
);
n = number of samples needed;
z
␣
= standard normal deviation for the probability of a type I error at a given level of probability
(significance);
z

= standard normal deviation for the probability of a type II error at a given level of probability
(power = 1 – );
s = standard deviation of the preliminary C/f data; and
d = the detectable effect size as an absolute number.
Standard normal deviations, or z scores, are easily obtained from tables in reference books or
programs in various statistical software packages.
An example computation is conducted using the logarithmic transformation of C/f data from
Impoundment B because it most closely resembles a normal distribution and yields the smallest
estimates of needed sampling effort. The mean log
10
(C/f + 1) is 0.731 and SD (s) is 0.254. If it is
specified that the detectable effect size is 10% of the mean, or 0.0731, ␣ is 0.05, and  is 0.10, then
z
␣
= 1.65, z

= 1.28, and
n = 2 (1.65 + 1.28)
2
(0.254
2
/0.0731
2
) = 207 trap nights.
It is unlikely that the needed sampling effort could be achieved by practicing fisheries scientists as
part of a routine monitoring program.
If the specified criteria are relaxed, lesser amounts of sampling effort are needed. For example, if it is
specified that the detectable effect size is 20% of the mean, or 0.146, ␣ is 0.10, and  is 0.20, then
n = 2 (1.28 + 0.84)
2
(0.254
2
/0.146
2
) = 27 trap nights.
Although substantially less sampling effort is needed, the magnitude of change in C/f that would
occur before that change is detected is doubled, and the probabilities of both type I and type II
errors are doubled.
Box 7.2 (continued)
Relative Abundance and Catch per Unit Effort 297
low fish densities, the most frequent catch is no fish and the distribution is likely
to approximate a negative binomial probability (Power and Moser 1999). Because
most fish stocks occur in relatively low densities and have patchy spatial distribu-
tions, C/f sample distributions that resemble negative binomial probability distri-
butions are fairly common.
The negative binomial distribution is widely recognized as a descriptor of ani-
mal distribution patterns, and it has been argued that the negative binomial distri-
bution is a reasonable probability distribution for the overall description of C/f
data (Moyle and Lound 1960; Power and Moser 1999). Often, C/f data are char-
acterized by a high frequency of zeroes (Bannerot and Austin 1983; Power and
Moser 1999), and occasionally one or more C/f values are excessively large, thereby
exerting excessive influence on the arithmetic mean (Pennington 1996;
Kappenman 1999). The variance of the mean C/f is often large; thus, it is difficult
to discern if mean C/f estimates differ among groups or over time using paramet-
ric statistical testing (e.g., a t-test or analysis of variance [ANOVA]; Bannerot and
Austin 1983). In addition, if the mean C/f is small and the variance is large, the
probability of observing zero catches will be high (Power and Moser 1999) if the
fisheries scientist assumes that data are from a normal probability distribution.
For these reasons, mean C/f calculated from data distributed as a negative bino-
mial distribution may not provide a reasonable statistic for comparison of samples.
Because the occurrence of negative binomial distributions of C/f data have
been recognized, logarithmic transformations (y = loge [x + 0.001] or y = log
10
[x +
0.001]) have been applied frequently in an attempt to normalize distributions
(Bulkley 1970; Bagenel 1972) but with quite variable success. It has become com-
mon practice to apply logarithmic transformations to C/f data prior to conduct-
ing statistical tests and to assume that the transformation sufficiently normalizes
the distribution so that test assumptions are not grossly violated. Fisheries scien-
tists who follow this practice seldom carry out statistical tests to determine if nor-
mal distributions are achieved by the transformation. It is our experience that
logarithmic transformations of C/f data seldom yield a normal distribution but
can reduce the variance relative to the mean (see Box 7.2). Other transforma-
tions of C/f data have been applied in attempts to normalize the distributions
(Shroyer and McComish 1998), but none was found to have wide success.
7.4.2 Appropriate Sample Statistics
Fisheries scientists occasionally use statistics other than the arithmetic mean to
describe C/f distributions, primarily the geometric mean, median, and the fre-
quency of occurrences of the target species among samples.
The back-transformed mean of the logarithmically transformed C/f data is called
the geometric mean (Sokal and Rohlf 1981) and is used by fisheries scientists as a
measure of central tendency for C/f data (Craig and Fletcher 1982; Hamley and
Howley 1985; Hansen et al. 1995). It seems to be a logical expression of C/f when
the data are logarithmically transformed for analysis. However, because the scale
298 Chapter 7
is not familiar to many, it is difficult to grasp the magnitude of difference or change
using the geometric mean.
The median of the C/f data distribution has an equal number of observations
on either side of it (Sokal and Rohlf 1981) and also has been used by fisheries
scientists as a measure of central tendency (Moyle and Lound 1960; Moyer et al.
1995). Moyle and Lound (1960) provided a method for computing confidence
limits around median C/f estimates.
Another statistic applied to C/f data is based on enumeration of the frequency
of occurrence of the target species among individual units of effort (Bannerot
and Austin 1983; Counihan et al. 1999). If the frequency distribution of C/f data
resembles a negative binomial, Bannerot and Austin (1983) suggested comparing
the frequency of zero catches, which they found was a less biased index of abun-
dance than mean C/f . The frequency of zero catches was more responsive to
changes in stock abundance than mean C/f for a marine fishery (Bannerot and
Austin 1983). Similarly, Counihan et al. (1999) stated that an index based on the
proportion of individual units of effort when a target species is captured may have
advantages over mean C/f because it is robust to biases and errors in sampling
and insensitive to extremely high C/f values. Presence–absence indices generate
proportional data that can be analyzed for differences among groups or over time
(Sokal and Rohlf 1981). This approach circumvents issues of normal distributions
associated with using the mean C/f and statistical tests requiring the assumption
of normal C/f frequency distributions.
7.4.3 Bootstrap and Jackknife Techniques
Bootstrap and jackknife techniques are used to answer the same question: how
precise is a particular statistic? These techniques can provide estimates of preci-
sion of C/f statistics (Dixon 1993). These techniques release fisheries scientists
from the restrictive assumption that C/f data conform to a normal frequency dis-
tribution (Krebs 1989). Because both techniques compute a standard error for a
statistic, they allow us to compute t-tests.
Bootstrap and jackknife techniques can be applied to statistics computed from
C/f data, including the arithmetic mean, geometric mean, and median. They
provide measures of the precision of the statistics and enable statistical compari-
sons of two samples. For statistics that are bounded in range (such as any of the
three C/f statistics mentioned above, which are always greater than or equal to
zero), these techniques may work more satisfactorily if the data are subjected to
logarithmic transformation. Programs for bootstrap and jackknife routines are
available, but some computational shortcuts may yield erroneous results. We rec-
ommend that fisheries scientists who want to apply these techniques work with
statisticians to develop programs appropriate for their applications. It should be
noted that the bootstrap and jackknife techniques will not usually yield the same
answer (Dixon 1993), and there is no agreement on which technique is “better”
for analysis of C/f data (Krebs 1989).
Relative Abundance and Catch per Unit Effort 299
Some examples of using bootstrapping to obtain estimates of precision from
C/f data can be found in the fisheries literature. In one example, Kimura and
Balsiger (1985) applied bootstrapping to estimate the precision of C/f data for
sablefish captured in pot gear off the Pacific Coast. Estimates of precision (i.e.,
coefficient of variation) of C/f data were obtained from different sampling areas
and water depths and were then used to develop recommendations for the num-
ber of locations that should be sampled. Kimura and Balsiger (1985) also used the
bootstrap technique to compute Z-statistics to estimate the statistical significance
of observed differences in C/f between years within specified locations and depths.
Similarly, Stanley (1992) used bootstrapping to estimate the variance and confi-
dence limits for C/f data from four trawl fisheries along the Pacific Coast, and this
information was then used to estimate the number of hauls needed to estimate
mean C/f at ±25% of the actual rate 80% of the time (␣ = 0.2). Bernard et al.
(1993) applied bootstrapping to an assessment of burbot in Alaskan lakes. The
bootstrap procedure was used to generate an empirical sampling distribution from
which the variance was estimated for individual lakes. More recently, Smith (1997)
developed bootstrap confidence limits for groundfish trawl survey estimates of
mean C/f .
7.4.4 Comparison of Two Samples
In a classical statistical approach, a comparison of two samples is undertaken by
testing the equality of means. Assuming the observations are selected randomly
from a normal frequency distribution, the arithmetic mean provides a measure of
central tendency. Because the mean is computed from a sample (and not the
entire population), the uncertainty of the mean can be measured by the variance,
another characteristic of the distribution. Under these conditions, a comparison
of two samples is fairly simple: estimates of the variance are used to calculate
standard errors, and confidence intervals around the means are constructed.
However, as mentioned earlier, C/f data generally violate many of the assump-
tions of classical statistical approaches. For example, using traditional statistical
approaches (t-tests and analysis of covariance) and logarithmically transformed
data, only order-of-magnitude differences in larval fish abundance could be de-
tected among sampling areas or time periods (Cyr et al. 1992).
To compare mean C/f values from different sampling locations or across time,
an estimate of the variance (hence, standard error) is needed. Although some in-
vestigators advocate estimating the variance with regression methods (e.g., regress-
ing catch on effort, Smith 1980) or jackknifing the variance of the ratio (Smith
1980), these approaches assume a linear relation between catch and effort. Because
this assumption is not often met, we recommend seeking alternate approaches.
One such approach uses maximum-likelihood methods to estimate C/f and its vari-
ance from the bivariate distribution of catch and effort (Richards and Schnute 1992).
Recently, Power and Moser (1999) applied an approach based on the assump-
tion that the distribution of the catch data follows a negative binomial and variances
300 Chapter 7
need not be homogeneous. Their generalized linear model permits comparison of
catch rates among two or more samples and allows catch rates to vary as linear (or
nonlinear) functions of exogenous variables. Generalized linear models share the
same structure as general linear models (GLMs), but unlike general linear models,
generalized linear models are not constrained by the assumption of normality. Us-
ing bootstrap simulations, Power and Moser (1999) demonstrated that the linear
model with negative binomial errors performed better than the t-test in detecting
differences between the means of two samples; furthermore, this was true when the
t-test was applied to either raw data or logarithmically transformed data.
7.4.5 Analysis of Variance
7.4.5.1 Comparisons Based on Blocked Designs
When testing hypotheses based on ecological experiments, particularly field ex-
periments, experimental units are sometimes grouped together in blocks. The
purpose of this is to identify groups of similar experimental units so there is more
similarity within blocks than among blocks. Blocks are not randomly assigned (as
are treatments) but are either intrinsic characteristics of the experimental units
(e.g., year-classes of a particular species, where the species is the experimental
unit) or arbitrary segments of the experimental unit (e.g., 0.5-ha area of sandy
bottom, where sections of sandy areas are the experimental units in a lake; Newman
et al. 1997). In fisheries fieldwork, the blocking factors most likely to be encoun-
tered are areas fished (e.g., station or individual lakes), length-classes, or age-
classes. The purpose of blocking is to improve the precision of the estimator. The
example in Box 7.3 demonstrates how blocking can improve the ability to detect
effects associated with factors of interest to fisheries scientists.
7.4.5.2 Other Analysis of Variance Models
Catch-per-unit-effort data are frequently collected from surveys and are intended
to measure changes in abundance of fish in a stock. Depending on the design,
ANOVA can be used to analyze these data under certain assumptions and con-
straints. Because surveys typically consist of repeated measures (e.g., C/f is esti-
mated from a set of fixed or random stations through time), a repeated-measures
ANOVA could be used to analyze such data (see Maceina et al. 1994). This ap-
proach accommodates temporal autocorrelation among observations—that is, it
explicitly accounts for the fact that two observations taken closely apart in time
will likely be correlated, and the correlation is likely to decrease as observations
further apart in time are considered. The repeated-measures approach is recom-
mended when (1) the study includes only fixed effects, (2) the data are balanced,
and (3) the variance–covariance structure of the data conforms to a restrictive form
(i.e., compound symmetry; Neter et al. 1996). Some adjustments exist for incorpo-
rating random effects into a repeated-measures ANOVA and for relaxing the as-
sumption of compound symmetry (Neter et al. 1996). However, in many fishery
surveys, C/f measures are repeated not just across time but also through space. In
these cases, a different approach must be taken to accommodate correlations in
Relative Abundance and Catch per Unit Effort 301
Box 7.3 Illustration of Blocked Design in One-Way Analysis of Variance (ANOVA)
An investigator wishes to determine if the abundance of bluegill differs among vegetated and
nonvegetated areas of lakes. Bluegills are sampled using trap nets set within a vegetated and a
nonvegetated area in each of eight lakes. In this study, the vegetation (presence or absence) is a
fixed effect, the blocking factor is lake, and the response is the C/f of bluegill (fish/trap-night). Using
the following hypothetical data set, we show how ignoring the blocking factor (lake) can lead to
erroneous conclusions about the effect of vegetation on the relative abundance of bluegill.
Table Catch per unit effort data for bluegills.
Lake
Vegetation A B C D E F G H
Absent 63 45 30 50 80 67 48 55
Present 7056426887754962
Program
The following SAS program is employed.
data bluegill;
input lake $ vegetation $ cpue;
lines;
[input data]
proc glm data=bluegill;
class lake vegetation;
model cpue=lake vegetation;
title ‘One-Way ANOVA, Block Design’;
prog glm data=bluegill;
class vegetation;
model cpue=vegetation;
title ‘ANOVA without Blocking’;
run;
Results
Table Results of one-way ANOVA with block design. The number of observations in the data set is
16. Abbreviations are as follows: mean square error (MSE); coefficient of variation (CV); and sum of
squares (SS).
Class Level Information
Class Levels Values
Lake 8 A B C D E F G H
Vegetation 2 absent present
Analysis of Variance
Source df SS Mean square F-value P > F
Model 8 3339.000 417.375 34.20 0.0001
Error 7 85.438 12.205
Corrected 15 3424.438
R
2
0.975 Root MSE 3.494
CV 5.903 C/f mean 59.188
(Box continues)
302 Chapter 7
Analysis of Variance (continued)
Source df Type I SS Mean square F-value P > F
Model 8 3339.000 417.375 34.20 0.0001
Lake 7 3023.938 431.991 35.39 0.0001
Vegetation 1 315.063 315.063 25.81 0.0014
Table Results of ANOVA without blocking. The number of observations in the data set is 16.
Class Level Information
Class Levels Values
Vegetation 2 absent present
Analysis of Variance
Source df SS Mean square F-value P > F
Model (vegetation) 1 315.063 315.063 1.42 0.253
Error 14 3109.375 222.098
Corrected total 15 3424.438
R
2
0.092 Root MSE 14.903
CV 25.179 C/f mean 59.188
Interpretation
A cursory examination of the data reveals that within a lake, bluegill C/f is higher in vegetated areas
than in nonvegetated areas. However, on closer inspection, we see that bluegill C/f in nonvegetated
areas of lakes A and F was just as high as C/f in vegetated areas of lakes H and D. Results from the
blocked ANOVA indicate that vegetation significantly affects the mean relative abundance of
bluegills (vegetation is a significant factor in the ANOVA). In addition, the relative abundance of
bluegills varied significantly among lakes (lake is a significant factor in the ANOVA). Thus, by
blocking the design and accounting for lake to lake differences in the relative abundance of
bluegills, the investigator was able to examine the effect of vegetation on bluegill abundance
within lakes. Incidentally, if the investigator had randomly sampled one set of lakes (say lakes L, M,
N, O, P) to estimate bluegill C/f in vegetated areas, and another set of lakes (say lakes Q, R, S, T, U) to
estimate bluegill C/f in nonvegetated areas, then lake would not be a blocking factor because the
treatment (vegetated versus nonvegetated) was randomly applied across lakes. In our blocked
design, the treatment (vegetated versus nonvegetated) occurred in the same lake; because of this,
the response to the two treatment levels was measured from the same lake. In this respect, data
from a blocked design could also be analyzed as a paired t-test (see Sokal and Rohlf [1981] for
further discussion).
In the second analysis, blocking is ignored, and it is not possible to detect the effect of vegetation
on bluegill abundance. This occurs because in the unblocked design, the variation associated with
lakes is considered part of the error term (compare the sums of square error terms from both
models). By ignoring the blocking effect, we would interpret the results of this ANOVA in the
following manner: on average across all lakes sampled, variation in relative abundance of bluegills
is not affected by the presence of vegetation.
Box 7.3 (continued)
Relative Abundance and Catch per Unit Effort 303
space in addition to serial correlations in time. Fabrizio et al. (2000) demonstrated
how the mixed-model procedure (MIXED) in SAS (SAS Institute 1998) can be
used to study changes in fish abundance from a complex repeated-measures de-
sign. Procedure (PROC) MIXED was used to fit a linear model with correlated
errors to a 20-year time series of catch data from Lake Michigan, but the approach
is applicable to other fixed-station fishery surveys. The linear model used by Fabrizio
et al. (2000), y = X + e, is similar to a GLM (one fit with SAS’ PROC GLM) except
that in the GLM, the vector e is a vector of independent random variables, and in
the linear model with correlated errors, e is a vector of possibly correlated random
errors with covariance matrix R (Littell et al. 1996). In this notation, y is a vector
of observations, X is a matrix of fixed effects values, and  is a vector of fixed
effects coefficients. Another difference between the two models is that instead of
simply modeling the mean and a single variance of y (the GLM), the mean, vari-
ance, and covariance of y are modeled in the linear model with correlated errors.
Procedure MIXED is a flexible approach that works well for unbalanced data.
It can be used to fit a variety of models, including mixed models that contain both
fixed effects and random effects: y = X + Z + e, where y, X, , and e are defined
as before, Z is the design matrix (usually a matrix of 0s and 1s), and is a vector of
random effects parameters (Littell et al. 1996). To use PROC MIXED, the investi-
gator must identify and specify the type of the variance–covariance structure that
defines the error term of the model. Version 6.12 of SAS offers about 20 options
for the structure of the variance–covariance matrix, and Wolfinger (1996) dis-
cusses useful variance–covariance structures for models fit to repeated-measures
data. A new procedure, PROC NLMIXED, available with SAS version 7 and higher,
can fit nonlinear models using likelihood-based methods (Wolfinger 1999).
7.4.6 Nonparametric Alternatives to Analysis of Variance
Because nonparametric tests do not require the assumption that the data come
from a normal distribution, these tests have been recommended when assump-
tions of parametric statistics cannot be met. Standard parametric tests such as the
t-test and ANOVA have nonparametric counterparts that can be easily implemented
once the raw data are rank transformed. For example, the counterpart to the t-
test is the Mann–Whitney U-test or Wilcoxon’s rank-sum test. However, it should
be noted that the null hypothesis of a nonparametric test is not equivalent to that
tested by the analogous parametric method. In the case of the Mann–Whitney or
Wilcoxon test, the null hypothesis is that two distributions are identical. Nonpara-
metric tests will detect differences not just in central tendency but also differ-
ences in the spread or shape of distributions (Johnson 1995). When performing a
nonparametric comparison of two samples, a significant test result provides no
information on whether the difference is due to the mean, variance, shape, or
some other characteristic of the distribution (Johnson 1995).
Although nonparametric techniques obviate the need for normally distributed
data and appear to be well suited for analysis of C/f data, relations between C/f
and other variables must still be considered. For example, Richards and Schnute
(1986) found that C/f data had to be standardized prior to applying a Kruskal–
304 Chapter 7
Wallis test when evaluating the effects of sea surface condition, time of day, and
tidal phase on C/f. These data were standardized by working with observations
from a restricted time (only when sea surface conditions were calm) when catches
were thought to be most reliable.
With nonparametric tests investigators still must apply a priori significance levels
to tests and consider the trade-offs between type I and type II errors. Because some
fisheries scientists perceive the power of nonparametric tests to be low, larger alpha
levels (e.g., ␣ = 0.10 or 0.15) are sometimes used in significance testing. However,
nonparametric tests often have as much power as their parametric counterparts.
7.4.7 Time Series Analysis
Fish abundance measures that are estimated repeatedly through time are typi-
cally examined for patterns of change through time by means of regression analy-
sis. The relation between observations close in time may be similar—that is, values
in a given year may be influenced by values in the previous year. This relationship
generally decreases with increasing time intervals. Such data are said to exhibit
positive autocorrelation. The presence of autocorrelation (or serial dependence)
in fish abundance data compromises statistical interpretation of correlation and
regression analyses that may be undertaken to relate changes in fish abundance
to environmental or biological variables (Pyper and Peterman 1998). The reason
is that most parametric statistical tests assume independence (correlation equals
0) of observations. Hypothesis tests on autocorrelated data require adjustments
to the degrees of freedom to reflect the lack of independence among observa-
tions (Pyper and Peterman 1998).
For predictive modeling or exploratory analyses, autocorrelated data must be
transformed. Several transformations have been used with autocorrelated data,
including smoothing, first-differencing, and prewhitening. A smoothed data se-
ries results from the computation of a series of weighted averages from nearby
points. A simple smoothing technique is the running average (moving average).
Smoothing is an effective transformation for removing high-frequency variation,
which appears as rapid changes over short time scales. An example of high-
frequency variation is measurement error. Sometimes it may be desirable to re-
move the signal associated with slow, long-term changes from a time series of C/f
data. These changes are typical of low-frequency variation and first-differencing
or prewhitening may be used to transform the data series (Pyper and Peterman
1998). In first-differencing, the observation at time t – 1 is subtracted from the
observation at time t. Prewhitening is typically applied when the analyst wishes to
relate C/f data to one or more environmental variables. For example, to deter-
mine if the pattern of variation in C/f is associated with the pattern of variation in
temperature data, the time series of temperature data are modeled with an ap-
propriate time series model. That model is then applied to the series of C/f data
to prewhiten the C/f series. Model identification, parameter estimation, and
diagnostic checking procedures are beyond the scope of this chapter but are
Relative Abundance and Catch per Unit Effort 305
well described in Box and Jenkins (1976). Additional information on smooth-
ing, first-differencing, and prewhitening is available in Pyper and Peterman (1998),
along with an effective method for adjusting the degrees of freedom for statistical
testing of autocorrelated data.
It should be noted, however, that the decision to employ any of the transforma-
tions should be taken with extreme caution. Pyper and Peterman (1998) point
out that if low-frequency variation is removed from a time series of data, the effect
of a slowly changing variable on the dynamics of the population will be difficult to
detect, but the effect of a quickly changing variable (high-frequency variability)
will be well detected if it is the dominant source of covariation. When the domi-
nant source of covariation is low frequency, Pyper and Peterman (1998) recom-
mend adjusting the degrees of freedom because this approach has greater statis-
tical power.
7.4.8 Assessment of Relationships between C/f and Other Variables
Regression analysis is commonly applied to C/f data to make predictions. For
example, Isbell and Rawson (1989) found that C/f of walleye captured in experi-
mental gill nets was a predictor of angler catch rates in western Lake Erie. Mean
C/f in gill nets was used as the predictor variable, and mean C/f among anglers
was used as the response variable. Similarly, Shroyer and McComish (1998) pre-
dicted the future C/f of quality-length (>200 mm total length) yellow perch based
on C/f of stock-length (>130 mm total length) yellow perch in trawl samples in
Indiana waters of Lake Michigan.
Regression analysis has also been used to predict C/f of fish from various habi-
tats. For example, Johnson and Jennings (1998) assessed the habitat associations
of small fishes around islands in the upper Mississippi River based on C/f as an
index of abundance. They predicted C/f from measures of habitat. Similarly, Irwin
et al. (1997) assessed the habitat associations of age-0 largemouth bass along the
shoreline of a large reservoir. Regression analysis was used to determine if mea-
sured habitat features accounted for variation in C/f of age-0 largemouth bass
among 43 discrete shoreline sections.
These approaches are fairly straightforward when habitat variables are static
characteristics of the environment. However, when habitats or environmental con-
ditions are dynamic (such as salinity and water temperature), it is advisable to
remove the high frequency (<24 h) variation of these dynamic physical variables
prior to using such data in a regression (Rose and Leggett 1989).
Fishery scientists have long recognized the problem of using C/f as a predictor
variable in regression analysis (see Ricker 1975) because the predictor variable is
assumed to be known accurately. This is an untenable assumption with C/f data
that are fraught with measurement errors. Measurement errors in C/f data are a
good example of what statisticians call the “errors in variables” problem. In general,
errors in variables tend to flatten the probability density function and increase dis-
persion; such changes lead to upwardly biased variance estimates and downwardly
306 Chapter 7
biased estimates of the mean (Chesher 1991). Ricker (1975) demonstrated how to
use functional regression analysis to estimate regression parameters from C/f data,
but Ricker’s approach is now considered ad hoc (Hilborn and Walters 1992). A
good review of the fisheries-related work on the errors in variables problem is
provided in Hilborn and Walters (1992). Although errors in variables are unavoid-
able in fisheries modeling, the magnitude of the bias associated with the errors in
variables problem can be investigated using Monte Carlo simulation techniques
(Hilborn and Walters 1992). When modeling the stock–recruitment relation, even
small measurement error (mean = 0 and SD = 0.2) can lead to erroneous conclu-
sions about the nature of the relationship (Walters and Ludwig 1981). Several
other approaches have been proposed to work with C/f data subject to the errors
in variables problem, including techniques for data containing measurement er-
ror in both the dependent and independent variables (e.g., Richards and Schnute
1986; Kimura 2000).
Most linear models, including regression analysis, do not address extra-Poisson
or extra-binomial variation, and, because of this, such models may not provide
reliable confidence intervals or significance tests for parameters of interest (Casey
and Myers 1998). As discussed in section 7.4.3, the jackknife or bootstrap ap-
proach may be useful in estimating precision of regression parameters. Another
approach is to use simulation modeling. In addition to these techniques, the ran-
domization approach may be used to estimate confidence intervals for a regres-
sion parameter, particularly if the standard significance levels or standard errors
of the parameter estimates are not reliable (Casey and Myers 1998).
Detection of trends in C/f data has been recently pursued with regression
tree methods (Watters and Deriso 2000). Observed trends were ascribed to
changes in catchability and to actual changes in abundance. This regression
tree application required estimation of 139 parameters for 30 years of monthly
data on bigeye tuna from the Pacific Ocean, so it is not likely to be appropriate
for short time series. Regression trees are useful in examining the interaction of
factors such as area (e.g., latitude–longitude grids) and time (e.g., specific
months) and may be more parsimonious (fewer parameters) than GLMs or spa-
tially explicit models that account for variations in environmental conditions
(Watters and Deriso 2000).
■ 7.5 INTERPRETATION AND APPLICATION OF C/f STATISTICS
Monitoring changes in fish stock abundance through time is a costly activity un-
dertaken typically by federal and state agencies. The goal of these surveys is to
provide long-term information on the status of species so that changes in abun-
dance can be detected. These surveys require an investment in gear and person-
nel as well as an institutional commitment to multiyear support. Analysis of data
collected from monitoring surveys is often difficult due to the nature of the C/f
data, which may also reflect the vagaries of the weather and the reliability of the
equipment and gear.
Relative Abundance and Catch per Unit Effort 307
7.5.1 Example of a Temporal Monitoring Program
In this section, we illustrate how to assess patterns of change in C/f over time. We
use mean C/f of bay anchovy from two regions of a mid-Atlantic estuary as an
example (see Box 7.4). The C/f data were collected from fixed stations along a
salinity gradient in the estuary and are considered repeated measures. To begin
analysis, components that represent the treatment structure and those represent-
ing the design structure must be designated so that the appropriate statistical
model can be identified. In general, the treatment structure refers to the compo-
nents of the experimental design whose effects are of interest. In a temporal
monitoring program, this typically includes effects of time and may also include
effects of other factors such as region (in this example), habitat type, or habitat
manipulation. Design structure components are elements necessary to conduct
or construct the experiment and assist in addressing the components of the treat-
ment structure. Randomization and blocking are two examples of design struc-
ture elements. Statistical testing is focused on components of the treatment struc-
ture and generally not on the design structure. In this example, we use PROC
MIXED to examine changes in mean C/f between two regions in the estuary and
through time. The treatment structure consists of two fixed effects: region (bay
versus river) and time. The design structure incorporates a random component
(i.e., stations, which are nested within regions).
In a typical repeated-measures design, the response is measured from the same
subject multiple times. In this temporal monitoring example, the “subjects” are
stations from which C/f data were sampled; the repeated measures are C/f . To
designate stations as the experimental units, we must restrict our inferences to
the two regions sampled and assume that the C/f data from the two regions are
uncorrelated. (Technically, because our treatment [region] cannot be applied to
each station, the stations are not independent experimental units, and further
investigation of the dependency among stations may be pursued prior to model-
ing. However, this line of inquiry is beyond the scope of this example and will not
be illustrated here.)
Although PROC GLM provides several approaches for analyzing repeated-
measures data (using split-plot or multivariate approaches), we advise against using
this SAS procedure to analyze fisheries survey data such as these. The split-plot
and multivariate approaches to analysis of repeated measures are useful for analy-
sis of some ecological data, but in those cases, the experimental units are typically
groups of organisms or samples that can be randomly assigned a treatment. In
addition, the split-plot repeated-measures design has been used to analyze fisher-
ies data obtained at various points in time before and after an experimental ma-
nipulation, where the manipulation affects all possible sample sites within a water
body (e.g., to test the effects of vegetation removal on length structure of fish in a
small lake; Maceina et al. 1994). The split-plot approach is well suited to surveys of
individual water bodies from which replicates are taken and treatments can be
applied to each experimental unit (station and replicate). The split-plot approach
308 Chapter 7
Box 7.4 Analysis of C/f Data from a Temporal Monitoring Program
Bay anchovies were sampled with an otter trawl every 6 months from November 1996 through May
2000 for a total of eight sampling periods. Trawl tows were taken at 13 randomly selected sites in
the river and 11 randomly selected sites in the bay. A single 5-min tow was completed at each
station during each sampling period, the area swept was calculated, and C/f was computed as the
number captured per unit of area swept. The analysis was designed to address two questions. Did
mean C/f change through time? Did mean C/f differ between the river and bay?
The data set contained 192 otter trawl samples, but 134 samples contained no bay anchovies.
Because the C/f data contained many zeros, had a large coefficient of variation, and had a variance
that exceeded the mean, the C/f observations were transformed as log
e
(x + 0.0001). River sites were
coded as 0 and bay sites as 1 in a variable termed region. Sampling periods were coded 1 through 8
in a variable termed time. Sampling sites were coded 1 to 13 in the river and 1 to 11 in the bay in a
variable termed station.
We first identified the type of covariance matrix that best described the random component (i.e.,
stations sampled repeatedly) in this study. The keyword TYPE identifies the covariance matrix in the
repeated statement of PROC MIXED. Because the C/f data are repeated measures, temporal
correlation among samples from a site may occur with the correlation decreasing as the time
interval increases. If the decrease is exponential, then the covariance structure can be modeled
using a first-order autoregressive structure (TYPE = AR[1]). If the correlations are equal across time
intervals, the covariance can be modeled using a compound symmetric structure (TYPE = CS). For
maximum flexibility in modeling the correlations, an unstructured covariance matrix may be
specified (TYPE = UN). We fitted these three covariance structures to the data and compared the
model fit using a likelihood-based criterion, Akaike’s Information Criterion (AIC; Littell et al. 1996).
The best model provides the smallest AIC, which is reported in a SAS output under “Fit Statistics.”
We found TYPE = AR(1) provided the best description of the covariance structure for these data.
Procedure MIXED was used to test for region and time effects. Before fitting the model to the data
we examined the interaction plot to determine the relation between region and time.
Program
The following SAS program was employed.
/* Plot the mean data through time for each region — Interaction Plot */
proc univariate noprint data=cpue.anchovy;
var lcpue_a;
by region t;
output out=anc_out mean=ybar;
proc plot data=anc_out;
plot ybar*t=region/box;
proc mixed data=cpue.anchovy;
class region station t;
model lcpue_a = region t region*t / outp=predict1;
random station(region) / s;
/* This is the error term for testing the region effect */
repeated / subject=station type=ar(1);
ods listing exclude solutionr;
ods output solutionr=randsoln;
title ‘Proc Mixed Results for Anchovy Data’;
/* Assess residuals for approximate normality at the whole plot (region)
level. Actually, these are estimated random effects. */
proc univariate data=randsoln plot normal;
var tvalue;
probplot tvalue / normal;
title ‘Residuals — Estimated Random Effects’;
Relative Abundance and Catch per Unit Effort 309
/* Assess residuals for approximate normality at the subplot (time) level. */
proc univariate data=predict1 plot normal;
var resid;
probplot resid / normal;
title ‘Residuals — From Predicted’; run;
In the PROC MIXED statements, note that region, station, and time are class variables. The model
statement includes region, time, and the region × time interaction, which we suspect may be
significant. The output option (outp=) in this statement specifies that the output SAS data set is
called predict1. All the modeled effects are fixed effects. The single random effect identified in the
random statement was station nested within region. The s option in the random statement
requests estimation of the solution, which will be used to evaluate normality of random effects. The
repeated statement defines the subjects of this repeated-measures analysis, which were stations.
Two statements are included to control the output delivery system (ods). The first (listing exclude
solutionr) suppresses the listing of the model estimates for each of the 24 random effects. The ods
output statement places the random effects estimates in a SAS data set (randsoln). Portions of the
output from the PROC MIXED analysis are given below.
Results
Table Portion of results for mixed-model ANOVA of bay anchovy C/f. Abbreviations are first-order
autoregressive structure (AR[1]); time (t); numerator (Num); and denominator (Den).
Covariance Parameter Estimates
Covariance Parameter Estimates
Covariance parameter Subject Estimate
Station (region) 0.5511
AR(1) Station 0.2283
Residual 4.4531
Type 3 Tests of FIxed Effects
Source Num df Den df F-value P > F
Region 1 22 2.81 0.1077
t 7 154 7.01 <0.0001
Region*t 7 154 2.79 0.0093
Interpretation
The interaction plot indicated that mean C/f may be changing differently in the two regions,
hinting that an interaction of time and region may occur (see Quinn and Keough 2002). The output
reveals the estimate of the variance of the mean C/f among stations nested within regions (0.5511).
The correlation coefficient (0.2283) indicates a relation in C/f between adjacent 6-month sampling
times. The F-test identifies a significant interaction of region and time (region*t, F value = 2.79, P > F
= 0.0093), as suggested by the interaction plot, making interpretation of region and time effects
difficult to assess. The interaction indicates that processes contributing to changes in mean C/f
differ in the river and bay. Options for further analysis may be to test for the presence of trends in
C/f for each region separately or omit the distinction between regions and assess the collective
data set for temporal trends.
310 Chapter 7
cannot be applied to our example data set because each whole plot (station in bay
or river) would need to contain both treatments (regions). Furthermore, for some
moderately complex designs, the GLM procedure is known to compute incorrect
standard errors. When analyzing repeated-measures data with PROC GLM, the
analyst should be aware of potential problems that occur with missing data, espe-
cially when the random statement is used or when modeling multivariate con-
trasts (Littell et al. 1996). We note that for most field studies, missing data are
common due to experimental failures, weather-related loss of sampling opportu-
nity, and other unplanned problems. Procedure MIXED was developed to ad-
dress some of the limitations of PROC GLM for modeling data from experiments
that incorporate random components either in the design structure, treatment
structure, or both.
7.5.2 Example of an Assessment of Spatial Patterns
Fisheries scientists are frequently faced with the need to identify spatial patterns
in fish distributions. Given the target species and type of water, an approach may
entail the use of C/f . A hypothetical example of an assessment of spatial distribu-
tion patterns of a fish stock based on C/f data may be illustrated by means of
yellow perch in a stratified Midwestern lake (Box 7.5). The fisheries scientists
wanted to know if the midday depth distribution patterns of yellow perch differed
between June and August. A stratified random sampling design was used. The
strata were the two sampling months and five sampling depths (2.5, 5, 10, 15, and
20 m). Within each depth stratum, three redundant locations were randomly se-
lected for sampling during each sampling month. Gill nets were set perpendicu-
lar to the shore 1 h before midday and retrieved 2 h later. Fish/net/h was used as
an index of yellow perch relative abundance. Two-way ANOVA (i.e., PROC GLM)
was used to assess variation in C/f among months and sampling depths (see Box
7.5). No significant difference in mean C/f was found between June and August,
but mean C/f differed significantly among sampling depths. Sampling month
and depth exhibited no significant interaction, indicating that patterns in the
depth distribution of yellow perch were similar in June and August. The mean C/f
was greatest at the 10-m sampling depth during both months. The data suggest
that yellow perch are most abundant between 5 and 15 m with lower numbers
near shore (2.5 m) and at 20 m.
7.5.3 Example of the Use of a Regression Estimator
A common problem encountered by fisheries scientists is the need to identify
relations between fish abundance and habitat features. Such relationships help
define habitat features needed by a species, determine habitat quality, or define
the likely responses of aquatic organisms to improvement or degradation of habi-
tat (Orth and White 1999; Summerfelt 1999). Because experiments involving
manipulation of habitat are difficult, time-consuming, and expensive to conduct,
Relative Abundance and Catch per Unit Effort 311
Box 7.5 Assessment of Depth Distribution Patterns of Yellow Perch Based on C/f Data
Data on C/f (fish//net/h) of yellow perch captured with gill nets in a Midwestern lake were obtained
during midday at five depths during 2 months with three randomly selected sites sampled at each
depth during each month. A two-way ANOVA was used to assess effects of sampling depth and
month as well as the interaction between the two.
Table Data on C/f of yellow perch in a Midwestern lake. Three sites were sampled at each of five
depths during two dates (June = 1 and August = 2).
Month Depth (m) C/f Month Depth(m) C/f
12.522 2.50
12.542 2.52
12.572 2.50
15.062 5.010
1 5.0 10 2 5.0 10
15.0122 5.09
1 10.0 8 2 10.0 13
1 10.0 12 2 10.0 40
1 10.0 33 2 10.0 46
1 15.0 10 2 15.0 6
1 15.0 10 2 15.0 5
1 15.0 17 2 15.0 12
1 20.0 0 2 20.0 0
1 20.0 2 2 20.0 0
1 20.0 3 2 20.0 0
Program
The following SAS program was employed.
data yelperch;
input month depth catch;
cards;
[input data]
proc glm;
class month depth;
model catch=month depth month*depth;
proc sort;
by month depth;
proc means mean stderr;
by month depth;
var catch;
Results
The ANOVA indicated that C/f varied significantly among sampling depths, but no significant
difference occurred among sampling months and no interaction occurred.
(Box continues)
312 Chapter 7
fisheries scientists often rely on a regression analysis to make inferences on the
relationships between fish abundance and habitat features. Many examples exist
in the literature in which a regression analysis was used to identify habitat features
that might be related to fish abundance, particularly as measured by C/f (Irwin et
al. 1997; Tillma et al. 1998; Braaten and Guy 1999). Cause and effect relations
between measured habitat features and C/f cannot be proven by means of a
Table Two-way ANOVA of C/f data for yellow perch. There were 30 observations.
Class Level Information
Class Levels Values
Month 2 1 2
Depth 5 2.5 5 10 15 20
Analysis of Variance
Source df SS Mean square F-value P > F
Model 9 2659.633333 295.514815 5.48 0.0008
Error 20 1079.333333 53.966667
Corrected total 29 3738.966667
R
2
0.711328 Root MSE 7.346201
CV 76.25814 Catch mean 9.633333
Source df Type I SS Mean square F-value P > F
Month 1 9.633333 9.633333 0.18 0.6772
Depth 4 2249.800000 562.450000 10.42 <0.0001
Month*depth 4 400.200000 100.050000 1.85 0.1582
The mean C/f and SE for each sampling month and depth are given below so that comparisons can
be made between months.
Table Mean C/f and SE for yellow perch data.
June August
Depth (m)Mean C/f SE Mean C/f SE
2.5 4.33 1.45 0.66 0.66
5.0 9.33 1.76 9.66 0.33
10.0 17.67 7.75 33.00 10.15
15.0 12.33 2.33 7.67 2.19
20.0 1.66 0.88 0 0
Box 7.5 (continued)
Relative Abundance and Catch per Unit Effort 313
regression analysis, but substantial insight and predictive capabilities can be gen-
erated if studies are designed properly and analyses are conducted carefully.
We provide a hypothetical example to illustrate application of a regression de-
sign to examine habitat quality when C/f data are used to index fish abundance.
In this example, fisheries scientists wanted to identify habitat features that may
affect the abundance of age-0 smallmouth bass in shoreline areas and to develop
the ability to predict abundance from measured habitat features. Box 7.6 con-
tains a hypothetical data set for age-0 smallmouth bass and demonstrates how
habitat features along the shoreline of a small natural lake may be associated with
C/f of age-0 fish. Twenty sites representing the range of shoreline habitats were
selected from the periphery of the lake. At each site, a 50-m segment was sampled
between the shoreline and the 1-m-depth contour in late July. The mean bottom
slope, proportion of the bottom composed of gravel–cobble substrate, and pro-
portion of the bottom covered by aquatic macrophytes were measured at each
site. Over each 50-m segment, one pass was made at night with a boat-mounted
electrofishing unit, and all age-0 smallmouth bass captured during the pass were
counted. The C/f (number/50 m of shoreline) was used as an index of age-0
smallmouth bass abundance at each site.
Pearson’s correlation coefficients were computed to assess relations among the
three habitat features. A significant correlation was found between the propor-
tion of gravel and the bottom slope indicating that these two independent vari-
ables may be redundant measures of the same ecological feature, which may or
may not be important to age-0 smallmouth bass. Linear regression analysis was
next used to evaluate relationships between C/f and each habitat feature. A log
10
(x
+ 1) transformation of the C/f data was made to improve the linear relation. Gravel
accounted for significant variation in C/f (see Box 7.6). When C/f was trans-
formed, the coefficient of determination (r
2
) increased and the probability (P)
that the relation was due to chance declined, indicating a more linear relation-
ship. Vegetation, which was not correlated with gravel, did not account for addi-
tional variation in C/f when included in a multiple-regression model with gravel.
Based on the high coefficient of determination (r
2
= 0.84), the relation between
gravel and log
10
(C/f + 1) would be judged as a good predictor of age-0 small-
mouth bass abundance in shoreline areas. However, fisheries scientists using this
model should note that cause and effect relations were not defined by the regres-
sion model. In this case, it is likely that small gravel is a suitable spawning sub-
strate for smallmouth bass and that age-0 fish are abundant where spawning was
concentrated not necessarily because gravel is a needed habitat feature for age-0
fish. The model should be tested with several independent data sets before it is
used for management decisions.
■ 7.6 SUMMARY
When assessing temporal or spatial trends in fish stocks, freshwater fisheries sci-
entists often use C/f as an index of relative abundance. Underlying assumptions
associated with the relationship between C/f data and actual population abundance
314 Chapter 7
Box 7.6 Regression Analysis to Assessment of Habitat Features when C/f Data
Are Used As the Response Variable.
This hypothetical problem focuses on defining the habitat features affecting the densities of age-0
smallmouth bass around the shoreline of a natural lake in the Midwestern United States.
Table Data: for 20 sites sampled along 50-m segments of shoreline of a Midwestern natural lake in
late July. The mean bottom slope, proportion of the bottom composed of gravel–cobble substrate,
and proportion of the bottom covered by aquatic macrophytes were measured at each site, and
one pass was made at night with a boat-mounted electrofishing unit for age-0 smallmouth bass.
C/f (fish/50 m) Gravel (%) Vegetation (%) Slope (%)
0007.3
5 5.2 10.1 1.5
1 0.3 1.1 2.2
10 5.5 13.6 1.2
12 6.7 7.3 1.7
0 0.8 10.9 5.3
1 0.1 10.0 8.3
3 1.9 0 3.0
25 8.3 0 1.5
0 0.5 5.0 4.3
98 11.1 1.8 1.5
2 0.9 4.0 4.3
15 6.6 1.3 1.1
60 10.0 11.0 1.4
0 3.0 0 5.9
1 1.0 7.7 8.0
7 5.9 1.0 1.3
1 1.0 4.0 4.4
0 4.0 0 8.7
5 4.3 7.6 1.4
The data were entered into a spreadsheet, and the C/f data were transformed [log
10
(C/f + 1)]
to create a second response variable. Correlations were assessed among the habitat features to
avoid inclusion of redundant variables in regression models. Simple linear regressions were
computed between each of the three habitat features and each of the two measures of relative
abundance.
Program
The following SAS program was employed.:
data bass;
input cpue logcpue gravel vegetation slope;
gravveg=gravel*vegetation;
cards;
[...input .data...]
proc corr;
var gravel slope vegetation;
proc reg;
model cpue=gravel;
model cpue=slope;
model cpue=vegetation;
model logcpue=gravel;
model logcpue=slope;
model logcpue=vegetation;
Results
The correlation analysis indicated that the proportion of gravel and the shoreline slope were
negatively correlated (r = –0.649, P = 0.002).
Relative Abundance and Catch per Unit Effort 315
Table Pearson’s correlation coefficients (r), (n = 20) for habitat variables and the probability of a
greater |r| under the null hypothesis that rho = 0.
Gravel Slope Vegetation
Gravel 1.00000 0.03204 –0.64916
0.8933 0.0020
Slope 0.03204 1.00000 –0.09309
0.8933 0.6963
Vegetation –0.64916 –0.09309 1.00000
0.0020 0.6963
The regression analyses indicated that the strongest linear relationship between relative abun-
dance and a measured habitat feature occurred between log
10
C/f and the proportion of gravel
(r
2
= 0.837, P < 0.0001):.
Table Regression analysis of log
10
(C/f + 1) of smallmouth bass and the proportion of gravel.
Analysis of Variance
Source df SS Mean square F-value P > F
Model (gravel) 1 5.93149 5.93149 92.59 <0.0001
Error 18 1.15315 0.06406
Corrected tTotal 19 7.08464
R
2
0.8372 Root MSE 0.25311
Adjusted R
2
0.8282 Dependent mean 0.66490
CV 38.06690
Parameter Estimates
Variable df Parameter estimate SE t-value P > |t|
Intercept 1 0.04083 0.08608 0.47 0.6410
Gravel 1 0.16189 0.01682 9.62 <0.0001
Interpetation
The model that used the untransformed C/f as the response variable and gravel as the dependent
variable was significant (<0.0001), but the amount of variability in C/f accounted for by gravel was
substantially less (r
2
= 0.636).
Multiple-regression models were computed with both the proportion of gravel and the proportion
of vegetation as habitat variables, as well as with the interaction term (gravveg = gravel × veg). The
following SAS program was employed:.
Proc reg;
model logcpue=gravel vegetation;
model logcpue=gravel vegetation gravveg;
Results
Neither the proportion of vegetation nor the interaction term was significant. Slope was not
included in the multiple-regression model because it was significantly correlated with the propor-
tion of gravel. Therefore, the regression analysis suggests that the relative abundance of age-0
smallmouth bass can be predicted from the proportion of gravel along shoreline areas.
316 Chapter 7
must be considered, or C/f can be a misleading indicator of abundance. While
there are several assumptions to be considered, the assumption of constant
catchability may be the most critical and commonly violated. Substantial effort
should be made to assure constant catchability in management assessments and
research designs. In order to minimize uncontrolled sources of variation (error)
in C/f, stratified random and systematic sampling designs are commonly used.
Such designs incorporate standardization of gear and effort and identification of
sampling times and locations. Assessing the extent of variability in C/f with a par-
ticular design and identifying the sampling effort required to detect changes over
time or to detect differences among sampling sites through preliminary sampling
are necessary components of management and research efforts.
A major problem in the application of C/f sampling data is that the distribu-
tion is seldom normally distributed. Negative binomial distributions are common
among C/f data sets, but they cannot be assumed to occur. A variety of descriptive
statistics have been used to characterize the distribution of C/f data, but none are
universally applicable. The power of classical statistical methods is substantially
reduced when C/f data are incorrectly assumed to be normally distributed. Fur-
thermore, changes in C/f over time or among different locations may not be
detectable when, in fact, differences in fish abundance exist. However, recent
applications of general linear models and mixed models that incorporate tempo-
rally and spatially autocorrelated errors into C/f analyses provide substantial prom-
ise for more powerful analyses. Similarly, advances in regression analyses beyond
classic least-squares regression are providing better descriptors of relations be-
tween C/f and other variables. The historic and emerging statistical methods de-
scribed in this chapter have utility in management and research; however, users
of these techniques are advised to seek consultation of professional statisticians to
assure that the most appropriate analytical methods are used and to avoid mis-
leading results or interpretations.
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