Gerow a New Method to Compute Standard-Weight Equations That

8/18/2019 Gerow a New Method to Compute Standard-Weight Equations That

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North American Journal of Fisheries Management 25:1288–1300, 2005 [Article] Copyright by the American Fisheries Society 2005DOI: 10.1577/M04-196.1

A New Method to Compute Standard-Weight Equations That

Reduces Length-Related Bias

KENNETH G. GEROW* AND RICHARD C. ANDERSON-SPRECHER

Department of Statistics, University of Wyoming,

Laramie, Wyoming 82071-3332, USA

WAYNE A. HUBERT

U.S. Geological Survey, Wyoming Cooperative Fish and Wildlife Research Unit, 1

University of Wyoming, Laramie, Wyoming, 82071-3166, USA

Abstract.— We propose a new method for developing standard-weight (W s) equations for use in

the computation of relative weight (W r ) because the regression line–percentile (RLP) method often

leads to length-related biases in W s equations. We studied the structural properties of W s equations

developed by the RLP method through simulations, identified reasons for biases, and comparedW s equations computed by the RLP method and the new method. The new method is similar to

the RLP method but is based on means of measured weights rather than on means of weights

predicted from regression models. The new method also models curvilinear W s relationships not

accounted for by the RLP method. For some length-classes in some species, the relative weights

computed from W s equations developed by the new method were more than 20 W r units different

from those using W s equations developed by the RLP method. We recommend assessment of

published W s equations developed by the RLP method for length-related bias and use of the new

method for computing new W s equations when bias is identified.

A variety of indices of body condition based on

length and weight measurements have been de-

veloped for fish. Body condition indices are used

to describe samples from fish populations and havebecome important tools for fisheries managers

(Anderson and Neumann 1996; Blackwell et al.

2000). Body condition indices should be free of

length-related biases (i.e., any systematic tendency

to over- or underestimate body condition with in-

creasing length) to enable accurate comparisons of

samples from different fish populations and as-

sessments of temporal trends of individual fish

populations (Murphy et al. 1990; Anderson and

Neumann 1996; Blackwell et al. 2000). The rel-

ative condition index (K n; Le Cren 1951) was de-veloped to overcome length-related trends in Fulton-

type condition factors (i.e., K and C ; Anderson and

Neumann 1996). When the concept of K n as orig-

inally proposed by Le Cren (1951) was expanded

into the concept of statewide standards for Ala-

bama fishes (Swingle and Shell 1971), the poten-

* Corresponding author: [email protected] The Unit is jointly supported by the U.S. Geological

Survey, the University of Wyoming, the Wyoming Game

and Fish Department, and the Wildlife Management In-stitute.

Received November 18, 2004; accepted April 13, 2005

Published online August 29, 2005

tial for length-related biases first developed. Con-

sequently, relative weight (W r ) was developed as

a body condition index (Wege and Anderson

1978). Relative weight of an individual fish is de-fined as its measured weight (100) divided by a

standard weight (W s) for fish of that species and

length. Wege and Anderson (1978) defined stan-

dard weight to be the 75th percentile of the weights

of a given species within specified length incre-

ments. This choice results in ‘‘above-average con-

dition’’ becoming the standard against which to

compare fish. A W r equal to 100 for a given fish

indicates that the fish is at the 75th percentile of

mean weights for the species at that length.

Different techniques have been used to modelthe relationship between W s and length in order to

establish a simple expression of standard weight

(Blackwell et al. 2000). Murphy et al. (1990) in-

troduced the regression line–percentile (RLP)

method for computing W s equations, a technique

that has become standard among fisheries biolo-

gists (Anderson and Neumann 1996; Blackwell et

al. 2000). A W s equation should yield approxi-

mately the 75th percentile of mean weights among

populations of the target species for fish of all

lengths within the range of applicable lengths if no length-related biases in W s are present (Murphy

et al. 1990). If no biases are present, variation in

W r across length increments in a sample from a


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1289REDUCED LENGTH-RELATED BIAS IN STANDARD-WEIGHT EQUATION

FIGURE 1.—Linear regression lines depicting log10weight versus log10 length relationships for 10 fish pop-

ulations from the flathead catfish data set demonstrating

the bow-tie effect. We used the two most extreme re-

gression lines in the data set and randomly selected eightothers from the data set.

FIGURE 2.—(A) Linear regression lines of log10 weight

versus log10 length relationships for samples from two

similar flathead catfish populations; (B) the same rela-

tionships plotted in the original measurement scales. The

vertical lines in both panels depict the upper bounds of measured lengths in the fish populations. The divergence

of the lines above the upper bounds of measurement

illustrates the effects of extrapolation beyond the data

set.

given population can be attributed to changes in

body condition.

Gerow et al. (2004) found length-related biases

in W s equations developed by the RLP method for

several species. The combination of linear regres-

sion and extrapolation contribute to biases in W sequations developed by the RLP method. When

an array of samples from fish populations across

the geographic distribution of a species are assim-ilated for the computation of a W s equation using

the RLP method, the samples from fish populations

with the lowest predicted weights for short fish

have the highest predicted weights for long fish

(and vice versa). We call this the bow-tie effect

and illustrate it (Figure 1) using a subset of sam-

ples from populations of flathead catfish Pylodictis

olivaris used in the development of a published W sequation (Bister et al. 2000). A biological paradox

occurs in all weight–length data sets that include

samples from several populations of a species and

is largely an artifact of least-squares methodology.

Among simple linear regressions, predictions from

regression lines at the upper range of the predictor

variable are negatively correlated with predictions

from regression lines at the lower range of the

predictor variable. For input variables that are pos-

itive values, this relation is most readily under-

stood by the fact that the least-squares intercept

coefficient and the least-squares slope coefficient

are forced algebraically to be negatively correlat-

ed. The random accident that gives one sample a

steeper-than-average slope is the same accidentthat tends to give that sample a lower than average

intercept. Some of the variation from sample to

sample among fish populations in weight–length

relationships is due to habitat quality, but random

variation certainly contributes to the bow-tie ef-

fect.

More critical is the RLP method’s violation of

standard warnings against extrapolation when ap-

plying regression methods. In the case of the RLP

method, the very high r 2 values (typically 0.95)

in individual regressions of log10 weight versus

log10 length for samples from fish populations sug-

gest that extrapolation is not a problem. However,

slight differences in fits to the data on the log10scale (Figure 2A) yield large differences in pre-

dicted weights when the regressions are back-

transformed and extrapolations are made over the

deemed applicable length range (Figure 2B). For

example, estimated weights from the extrapolated

models differ by as much as 50% at 1,000 mm of

length among samples from two flathead catfishpopulations. The maximum length of fish in many

populations is relatively short because of limiting

environmental conditions (i.e., low food resources,


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1290 GEROW ET AL.

low water temperatures, poor habitat, or compe-

tition), and fish in these populations tend to have

low weights and to grow slowly. Extrapolation of

weight–length relationships for fish in such pop-

ulations to an ‘‘applicable length range’’ for thespecies causes the RLP third quartiles for long fish

to be biased downward. Data for long fish in such

populations do not exist for biological reasons.

There are many instances in which biologists

use estimated values from a regression model in-

stead of measured values. For example, foresters

use measurements of diameter at breast height to

predict tree biomass (Avery and Burkhart 1994).

However, it is not logical to estimate values when

they are already measured, as is done in the RLP

method where both length and weight measure-

ments are made on sampled fish.Length-related biases in W s equations developed

by the RLP method (Gerow et al. 2004) prompted

our interest in development of a new method for

computing W s equations. The new method uses

only empirical data and is designated the EmP

method. The EmP method is based on quartiles of

measured mean fish weights (not weights esti-

mated from regression models) in a given length-

class among sampled fish populations. Since quar-

tiles are not estimated from modeled means, there

are no modeling artifacts such as the bow-tie effectimpacting resulting W s equations. We compared

the performance of the RLP and EmP methods by

means of simulations using 15 data sets from

which W s equations had been developed by the

RLP method.

Methods

Because definitions differ for statistical popu-

lations and fish populations, we specify statistical

populations and fish populations throughout. Con-

sequently, in the RLP method the statistical pop-

ulation of interest is the population of regression-

estimated mean weights from all possible fish pop-

ulations (Murphy et al. 1990). In the EmP method,

the statistical population of interest is the popu-

lation of actual mean weights (by length-class)

from all possible fish populations.

W s estimators.—For each species, there is a

length range judged to be suitable for the appli-

cation of W r . That range is divided into J 1-cm

length-classes, each with midpoint L j ( j 1,. . . ,

J ). Length–weight measurements on fish from I

fish populations comprise the data. Let W ˆ i,j (i 1,. . . , I )be the log10 weight at L j estimated from

a log10 weight on log10 length simple linear re-

gression for the data from fish population i.

The RLP method comprises three steps: (1)

compute W ˆ i,j for all combinations of i and j; (2)

compute the 75th percentile (third quartile) of W ˆ i,jfor each 1-cm length increment, denoted as

Q j(W ˆ

i,j); and (3) regress Q j(W ˆ

i,j) against log L j toobtain the W s equation for that species. The stan-

dard weight at length L j is W s( L j).

The EmP method is similar to the RLP method

in the use of third quartiles, but the computation

of the standard-weight equation differs. The EmP

method comprises these three steps: (1) let W ij(measured, not modeled) be the sample mean of

log10 weights at length L j from fish population i in

each of the J 1-cm length-classes, (2) compute the

third quartile Q j(W ˆ ij) in each length-class, and (3)

regress Q j(W ˆ ij) against log10 L j using a weighted

quadratic model. A more detailed description ap-

pears in the Appendix.

The primary difference between the two meth-

ods is that the EmP method uses the third quartiles

of measured mean weights instead of estimated

weights. The use of measured weights suggests

modeling curvature in W s relationships via qua-

dratic regression, which was our choice. We also

chose an applicable length range of the resultant

W s equation based on the structure of the data set.

For the EmP method, the applicable range is de-

termined by the length-classes for which at leastthree fish populations contribute mean weights

(that being the smallest sample size that allows

estimation of a quartile).

Estimating quartiles from estimated mean

weights.—When using the EmP method, there is

a positive bias in the usual estimator of third quar-

tiles (Q3), and the extent of bias depends on sample

size (i.e., the number of fish populations contrib-

uting data for a given length-class). The usual es-

timator of Q3 is computed as:

count/(sample size 1).

An improved form for small sample sizes was sug-

gested by Blom (1958) and is computed as:

(count 0.375)/(sample size 0.25).

Specifically, for the third quartile, the Blom esti-

mator from a sample of n observations on some

variable Y is:

˜ Q (F F ) Y (1 F F )Y 3 1 2 [F 1] 1 2 [F ]2 2

Y (1 F F )(Y Y ),[F ] 1 2 [F 1] [F ]2 2 2

where:


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3 3 3 11F n , F round n ,1 2 4 16 4 16

and Y [ j ] is the jth value among the ordered (small-

est to largest) observations. The Blom estimatorreduces positive bias, especially for small samples.

Another source of upward bias of third quartile

estimates, which we call distortion bias, arises

from the fact that the distribution of estimated

means is more variable than the distribution of true

means. This bias decreases with the number of fish

contributing to each mean because estimation var-

iance decreases with sample size.

Simulations.—We conducted 17 sets of simu-

lations to compare properties of W s equations com-

puted by both EmP and RLP methods. Two pre-

liminary simulations were conducted using a data

set for lentic brown trout Salmo trutta (Hyatt and

Hubert 2001a). One set of simulations was gen-

erated for each of 15 kinds of fish using data sets

from which W s equations had been computed using

the RLP method: black crappie Pomoxis nigro-

maculatus (Neumann and Murphy 1991), brook

trout Salvelinus fontinalis (Hyatt and Hubert

2001b), bull trout Salvelinus confluentus (Hyatt

and Hubert 2000), flathead catfish (Bister et al.

2000), golden trout Oncorhynchus aguabonita

(Hyatt and Hubert 2000), Arctic grayling Thy-mallus arcticus (Hyatt 2000), kokanee Oncorhyn-

chus nerka (Hyatt and Hubert 2000), lentic brown

trout (Hyatt and Hubert 2001a), lotic brown trout

(Milewski and Brown 1994), redear sunfish Le-

pomis microlophus (Pope et al. 1995), shovelnose

sturgeon Scaphirhynchus platorynchus (Quist et al.

1998), splake Salvelinus fontinalis S. namaycush

(Hyatt and Hubert 2000), walleye Sander vitreus

(Murphy et al. 1990), white crappie Pomoxis an-

nularis (Neumann and Murphy 1991), and yellow

perch Perca flavescens (Willis et al. 1991). Anothersource of ‘‘variation’’ in the contributed data lies

in the definition of length. For all fish, length

means total length, with the exception of the shov-

elnose sturgeon for which fork length is measured.

The preliminary simulations were designed to

answer several questions. First, is the curvature

that the EmP method apparently accounts for an

artifact of the method or is it real? Second, if the

true log10 weight versus log10 length relationships

are straight lines, will the resulting W s relationship

be better modeled as a straight line or as a curvedline? Third, if there is curvature, would use of

quadratic regression on the RLP quartiles be a use-

ful improvement over a linear equation? Finally,

what is the effect of sampling and distortion bias

in the quartile estimators on W s equations?

We conducted the preliminar y simulations using

the lentic brown trout data set (Hyatt and Hubert

2001b) because it represented a ‘‘typical case’’ andthere were sufficient data upon which to build sim-

ulations. There were 49 fish populations in the data

set, with more than 3,000 fish varying from 120

to 750 mm. Data were ‘‘collected’’ over the 120–

750 mm range of lengths, and random error (from

a normal distribution) was added to the true re-

lationship whenever an observation was made. A

simulated population of ‘‘true’’ length–weight re-

lationships was established by performing log10weight versus log10 length regressions for the sam-

ples from each fish population.

The classical form assumed for this relationshipis a straight line function. We examined that as-

sumption for the length–weight data sets in our

study. If a straight line relationship is correct, one

would expect only a small number of data sets for

which a quadratic regression was significant;

among those that are significant, there ought to be

about as many with positive curvature as with neg-

ative curvature. Of the 999 data sets that we ex-

amined, 336 (about one-third) showed significant

curvature. We chose to use a quadratic model for

our simulations. That way, our ‘‘truth’’ mimickeda realistic situation. For data sets where curvature

is not present, the quadratic coefficient would be

about zero; otherwise, not.

Using these relationships, the ‘‘true’’ W s equa-

tion was calculated using the EmP method with

both the usual and Blom estimators of the third

quartile, the RLP method, and the RLP method

with a quadratic fit to the regression line percen-

tiles. With the truth defined, we simulated samples

with small (5), medium (20), and large (50) num-

bers of fish populations with a small (1), medium

(10), and large (20) numbers of fish in each length-

class in each fish population and all combinations

of both factors.

For each of the 15 simulations on specific kinds

of fish, true log10 weight versus log10 length re-

lationships were computed by fitting a quadratic

regression to the data from each fish population.

A sample structure similar to the one in the original

data set was formed for each of kind of fish. One

of the original fish populations was randomly cho-

sen and the fish lengths observed therein were used

to generate a sample from one of the artificial pop-ulations, random error added to the regressions.

This was repeated with replacement until we had

a simulated sample that was similar in size and


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1292 GEROW ET AL.

FIGURE 3.—Residuals from fitting a straight-line mod-

el to EmP (see text) and regression line–percentile (RLP)

quartiles against log length. Length-related curvature is

evident for both EmP and RLP quartiles, suggesting theneed to account for curvilinearity in the development of

standard weight (W s) equations. Less curvature is dis-

played by RLP quartiles because they have been ‘‘lin-

earized’’ to some extent by the linearly modeled RLP

means.

structure to the original. These data were used to

calculate W s equations using the EmP and RLP

methods, and the RLP method with a quadratic fit.

The process was repeated 100 times in each sim-

ulation. We visually assessed plots to compare rel-ative biases among W s equations computed by the

EmP and RLP methods for each of the 15 kinds

of fish.

Assessing variability of estimated regression

lines.—We measured the precision of estimated

regression lines using mean absolute relative de-

viation (MARD). To compute the MARD, we av-

eraged coefficients of 100 simulated lines and this

served as our estimate of expected value. We then

computed the absolute value of the difference be-

tween an estimated regression line and its expected

value at each of the length-class midpointsthroughout the applicable range of lengths. Each

difference was divided by its expected value and

multiplied by 100 to scale it to a percentage. The

average MARD is a measure of the precision of

the W s equation computed by the EmP method. We

regressed MARD against several measures of sam-

pling effort: (1) number of fish populations, (2)

total number of fish in the study, (3) average num-

ber of fish per fish population, and (4) average

number of fish per length-class (total number of

fish in the sample divided by number of length-classes [FpC]). These regressions provided insight

into sample-size effects when computing W s equa-

tions using the EmP method.

Comparison of W r among methods.—We as-

sessed length-related differences in W r with W sequations computed by the EmP and RLP methods

by using a W r equal to 100 computed by the EmP

method as the standard for comparison, based on

its performance in the simulations. Comparisons

of W r values and applicable length ranges were

made using the five standard length categories

(i.e., stock, quality, preferred, memorable, and tro-

phy) proposed for each of the 15 kinds of fish

(Willis et al. 1993). Note that in this paper, all RLP

equations were calculated using contributed data.

Our equations may differ slightly from published

equations because published equations were pro-

duced with data that had been ‘‘cleaned’’ for qual-

ity control. The contributed data may or may not

have been the data set used in the computation of

the published equation.

Results

Preliminary Simulations

Preliminary simulations showed that W s rela-

tionships were curved even when the original log10

weight versus log10 length relationships were

straight lines (Figure 3). The curvature in the re-

siduals (Figure 3) indicated that the straight line

model for W s in the RLP method had a distinct

lack of fit to RLP-modeled quartiles. The lack of fit did not appear large on the log 10 scale, but back-

transformed differences were as large as 15%.

When the true relationships were linear, the RLP

third quartiles appeared to have a V-shaped rela-

tionship with log10 length (Figure 3), as suggested

by the bow-tie effect (Figure 1). The apparent cur-

vature in the EmP quartiles was greater than that

in the RLP quartiles, suggesting suppression of

curvature in the modeled means used in the RLP

method. This curvature in the relationship between

quartiles of mean weights and log10 length moti-

vated our use of quadratic regression to form the

EmP–W s equations.

When using the EmP method, the standard third-

quartile estimator led to W s equations that were

biased upward compared with those based on the

Blom estimator (Q̃ 3). This source of bias shrank

with increasing numbers of fish populations in

each length-class. Our simulation used data from

50 simulated fish populations. With a single fish

per length-class (Figure 4A), the W s equation using

Q̂ 3 was about 2% higher than the line based on Q̃ 3,

but with 20 fish per length-class (Figure 4B) thedifference was negligible. Hence, our use of Q̃ 3 as

an estimator of third quartiles for the EmP method

was supported by the outcome of the simulations.


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FIGURE 4.—Plots of W s equations standardized to ‘‘truth’’ from simulations with 50 fish populations and either

(A) 1 or (B) 20 fish in each 1-cm length-class. Plots include lines computed by the EmP method using the usual

third quartile estimator (EmP) and the Blom estimator (Blom) and lines computed by the RLP method using linear

(RLP) and quadratic (RLPQ) regression. Note the overlap and convergence of the EmP and Blom lines near W s

100 at 20 fish per length-class as well as the curvature of the RLP and RLP Q lines.

Distortion bias in W s equations computed by the

EmP method occurred in estimates of quartiles of

means when the means were estimated with a smallamount of data. With only one fish per length-class

in each of 50 fish populations, the distortion bias

of W s equations was about 12% (Figure 4A), but

dropped to about 1% when there were 20 fish per

length-class (Figure 4B).

We found that the RLP method using a linearfit to the RLP quartiles (RLP L) or a quadratic fit

(RLPQ) always underestimated truth for short and

long fish, and overestimated truth for midlength


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1294 GEROW ET AL.

fish (Figure 4A, B). The RLPQ W s equations per-

formed better than those computed by the RLP Lmethod, but neither performed as well as the equa-

tions from the EmP method. The equations from

the EmP method were always close to truth withinthe length ranges with sufficient data (Figure 4B).

Species-Based Simulations

Relationships between MARD and the four mea-

sures of sampling effort were strongest (r 2 0.77)

between log10(MARD) and log10(FpC). Figure 5A

shows a plot of average MARD versus FpC, and

Figure 5B shows a linear regression between

log10(MARD) and log10(FpC). The averages of

MARD for splake, bull trout, and golden trout

were higher than for all the other kinds of fish,

and these three species had the smallest FpC val-ues. These results indicate insufficient data to sup-

port a W s equation for these three species.

To attain a W s equation that is within 5% of truth

over its entire applicable length range, our simu-

lations suggest that there must be an average of

least 50 fish in each length-class (Figure 5). This

could be attained with a large number of fish pop-

ulations and a relatively small number of samples

from each fish population, or by a large number

of samples from a smaller number of fish popu-

lations. For example, the W s equation for goldentrout was based on data from 16 fish populations,

a total of 532 fish. Assuming the applicable length

range included 40 1-cm length-classes, a total sam-

ple of 2,000 fish would be needed to produce an

equation within 5% of truth. If a limited number

of fish populations was used (say the 16 used in

the computation of the original W s equation by the

RLP method), that would require increasing sam-

ple size to an average of 125 fish per fish popu-

lation. On the other hand, if the observed 33 fish

per fish population is a reasonable sample size

from the point of view of minimizing impact on

fish populations, data would be needed from about

60 fish populations. We submit that the FpC by

itself is not sufficient to define sample needs, but

it allows assessment of the interaction of the num-

ber of fish populations sampled and the average

number of fish sampled in each fish population in

order to produce precise W s equations by the EmP

method. Further study is required to better under-

stand the several dimensions of this sample-size

problem.

Simulations with sample structures similar tothose used in the development of the original W sequations by the RLP method for each of the 15

kinds of fish indicated that W s equations computed

by the EmP method were typically closer to the

truth (i.e., W s 100) than those computed by ei-

ther the RLP L or RLPQ methods (Figure 6). The

use of quadratic regression in the EmP method

accounted for curvature in the relationships be-tween the third quartile of mean weights and length

for many of the 15 kinds of fish (Figure 6). The

RLP L method did not capture the curvature, lead-

ing to W s equations that did not accurately estimate

the 75th percentile of mean weight when curvature

occurred. For a few fish, W s equations developed

by the RLP method were within 10% of those by

the EmP method over the length range of appro-

priate data used in the EmP method (note black

crappie, white crappie, redear sunfish, and lotic

brown trout in Figure 6). However, the W s equa-tions developed by the two methods were quite

different for most of the fish we assessed, the dif-

ferences in W s being greatest for very long and

short lengths (e.g., note flathead catfish, shovel-

nose sturgeon, Arctic grayling, and yellow perch

in Figure 6).

To summarize the differences in use of W s equa-

tions computed by the EmP and RLP methods for

the 15 kinds of fish, we plotted W r versus length

with W r equal to 100 computed by the EmP method

serving as the standard for comparison (Figure 7).Note that this comparison is not based on simu-

lations—it is a direct comparison. The simulation

results support the use of the EmP method as a

standard. The lengths of the W r equal to 100 lines

represent the applicable ranges of the W s equations

developed by the EmP method given the available

data sets (Figure 7). For a few species (i.e., redear

sunfish, white crappie, and lotic brown trout), W r values were similar over all lengths. However, for

other fish the differences ranged from modest (i.e.,

walleye, lentic brown trout, brook trout, and ko-kanee) to large (i.e., Arctic grayling, yellow perch,

shovelnose sturgeon, and flathead catfish). Dis-

crepancies were generally largest for fish less than

stock length and for fish greater than memorable

length. Within length ranges from stock length to

the maximum adequately supported by data, the

W r values from the EmP and RLP equations were

relatively similar for several species (i.e., black

crappie, white crappie, redear sunfish, and lotic

brown trout). For most species, there were insuf-

ficient data to support computation of W s equationsby the EmP method for fish of trophy lengths and,

in some cases, for fish greater than memorable

length.


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FIGURE 5.—Panel (A) shows the mean absolute relative deviation (MARD) of W s equations computed by theEmP method versus the mean number of fish per length-class (FpC) for the 15 kinds of fish. Panels ( B) shows the

log10(MARD) versus log10(FpC) relationship for the 15 kinds of fish, illustrating the effect of sample size on the

precision of W s equations.

Discussion

The use of measured weights for estimating

third quartiles in the EmP method led to standard-

weight equations that captured curvature in the

relationship between those quartiles and length.

The use of measured weights was proposed by

Wege and Anderson (1978), but they were not ex-plicit on this point. They used third quartiles of

individual fish weights in each length, not means.

Our use of weights differed from that of Wege and

Anderson (1978) and was more similar to the RLP

method in that we used the mean of weights in

each length-class for each fish population. Our

study supports the contention that the correct sta-

tistical population to consider is a population of

summarized weights from each fish population (in

the sense of Murphy et al. 1990).Our simulations illustrated that W s equations

computed by the RLP method can be seriously

biased, especially for large fish. In some cases,


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1296 GEROW ET AL.

FIGURE 6.—Plots of W s equations (from simulations) standardized to ‘‘truth’’ for the 15 kinds of fish studiedusing data with sampling structure that mimics that actually seen in the data sets for each of the fishes. Lines are

computed using the EmP method, the RLP method (RLP L), and the RLP method with a quadratic fit of the third

quartiles (RLPQ). The lines for the W s equations developed by the EmP method extend only over the applicable

length range (as determined by sufficient data for quartile estimation).

estimates of W s from equations developed by the

EmP and RLP methods differed by more than 20%.

The modeled mean weights used in the RLP meth-

od are sufficiently linearized so that curvature in

the quartiles of modeled weights was less pro-nounced than that in quartiles from measured

means. Thus, simply allowing curvature in the re-

gression modeling of the quartiles in the RLP

method was not sufficient to correct for length-

related bias.

In addition to reducing length-related biases in

W s equations, the EmP method offers an empirical

basis for determining applicable length ranges(and hence avoiding extrapolation issues) that the

RLP method does not. The W s equations developed

by the EmP method should not be used to assess


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FIGURE 6. (Continued).

body condition of length-classes for which there

were inadequate data when the equations were

computed (i.e., length-classes for which there are

fewer than three fish populations contributing data

to W s equation development). For many species,

the applicable length range of the W s equation

computed by the EmP method was notably less

than that computed by the RLP method, but the

EmP method provided W s values for that length

range that have much less length-related bias.Our simulations suggested that reasonably pre-

cise W s equations (i.e., those with a MARD less

than 5%) require sufficient samples (an average of

at least 50 fish in each length-class). Similarly,

distortion bias was essentially eliminated if 20 fish

per length-class were sampled from among 50 fish

populations. These findings have implications

when considering the needed numbers of samples

from fish populations and their distributions across

length-classes when computing W s equations using

the EmP method. Further investigation is war-

ranted to elucidate the best balance between num-

ber of fish populations, number of fish per fishpopulation, and number of fish per length-class

when using the EmP method. Extending length-

class width beyond 10 mm could be acceptable in


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1298 GEROW ET AL.

FIGURE 7.—Plots of RLP–W r values 100 (dashed line), relative to EmP–W r values for 12 kinds of fish studied.

For reference, EmP–W r values are shown set to 100 (solid line). Length categories for each of the fishes are based

on proposed values that have been published for each: stock (S), quality (Q), preferred (P), memorable (M), and

trophy (T). Bull trout, golden trout, and splake were omitted because there were insufficient data to compute

accurate W s equations.


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some instances, but further exploration of this

question is needed.

Management Implications

The findings of Gerow et al. (2004) and the re-sults from our simulations suggest that the time

has come for the formation of a panel of fisheries

scientists with expertise in the development of W sequations to facilitate the assessment of published

W s equations and to guide the development of new

equations. The panel will need to (1) identify W sequations that suffer from sufficient length-related

bias to warrant modification; (2) designate a stan-

dard protocol for constructing data sets for the

computation of W s equations that consider appro-

priate length-classes for a species, the number of

samples from fish populations to be used, and thenumber of sampled fish in each length-class; (3)

guide the development of new W s equations using

standard computational methods; (4) provide for

publication of appropriate W s equations for use in

the assessment of fish stocks and their quality in

an efficient manner, and (5) create and manage a

database of length and weight data for individual

species to enable further assessment of W s equa-

tions. These and other issues (e.g., is the 75th per-

centile the optimal target for management pur-

poses or might we consider another percentile)should be examined by the panel so that a con-

sensus can be reached on the individual W s equa-

tions to be used as standards when assessing body

condition so that fisheries managers may be con-

fident when monitoring and assessing fish stocks

in the future.

Acknowledgments

This research was supported by the University

of Wyoming and the Wyoming Cooperative Fish

and Wildlife Research Unit. The Unit is jointly

supported by the U.S. Geological Survey, the Uni-

versity of Wyoming, the Wyoming Game and Fish

Department, and the Wildlife Management Insti-

tute. K.G.G. thanks Anil Gore, Chair, Department

of Statistics, University of Pune, Pune, India, for

hosting and supporting him during the first half of

a sabbatical year; additional support came from a

Flittie Sabbatical Award from the University of

Wyoming. We are grateful to Michael Brown (lotic

brown trout), Brian Murphy (walleye), Robert

Neumann (black crappie and white crappie), Kevin

Pope (redear sunfish), Michael Quist (flathead cat-fish and shovelnose sturgeon), and David Willis

(yellow perch) for providing length–weight data

sets used in the development of published W s equa-

tions. We thank David Willis and anonymous re-

viewers whose critical insights and suggestions

improved this paper immensely; one in particular

inspired the simulation.

References

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Appendix: The EmP W s MethodBeginning with weight in grams and length in

millimeters,

(1) denote by L j ( j 1,. . . , J ) the log10 trans-

formed midpoints of J suitable (contiguous)

10-mm length-classes, ranging from the ob-

served minimum to the observed maximum

length in the data set;

(2) for length–weight data from I fish populations,

let W ij be the mean of log10 weights for fish

in length-class j from fish population i (i

1,. . . , I );(3) denote by Q̃ 3, j(W i,j) the third quartile of mean

log10 weights in length-class j, computed as

Q (W )3, j i, j

F W (1 F )W i, j [F 1] i, j [F ]2 2

W (1 F )(W W ),i, j [F ] i, j [F 1] i, j [F ]2 2 2

where

3 3F n ,1 4 16

3 11F round n ,2 4 16F F F ,1 2

and W i , j [k ] is the k th value among the ordered

(smallest to largest) means in that length-class

(note that quartiles can be estimated only for

intervals containing data from at least three

fish populations); and

(4) denote by j the number of fish populations

with observed weights in length-class j. The

EmP W s equation is formed by a weighted qua-

dratic regression of Q̃ 3, j(W i,j) upon L j, weight-

ing by j.

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