# Weight-Length Relationships

Modeling the relationship between length and weight of a species of fish has been considered a routine analysis for which the results do not warrant publication (Froese 2006) or has been scorned as being of little value (Hilborn and Walters 2001). However, the recent review of methods and the meta-analysis of a large number of length-weight relationships by Froese (2006) demonstrated that a synthetic analysis of length-weight relationships for a species can provide important insights into the ecology of that species.

The relationship between the length and weight of a fish is used by fisheries researchers and managers for two main purposes (Le Cren 1951). First, the relationship is used to predict the weight from the length of a fish. This is particularly useful for computing the biomass of a sample of fish from the length-frequency of that sample. Second, the parameter estimates of the relationship for a population of fish can be compared to average parameters for the region, parameter estimates from previous years, or parameter estimates among groups of fish to identify the relative condition or robustness of the population. By convention, this second purpose is usually generically referred to as describing the condition of the species.

### Weight-Length Data

The required data for examining the length-weight relationship for a sample of fish is measurements of the length ($$L$$) and weight ($$W$$) of individual fish at the time of capture (e.g., Table 1). Any other data about individual fish, such as month or year of capture, are of capture, etc. can also be recorded.

Table 1: Length and weight measurements for a portion of Ruffe from the St. Louis River Harbor, 1992.

 month day year individual length weight 4 23 1992 1 90 9.3 4 23 1992 2 128 32.5 4 23 1992 3 112 19.0 4 23 1992 4 68 4.4 4 23 1992 5 56 2.1 4 23 1992 6 58 2.8 $$\vdots$$} $$\vdots$$ $$\vdots$$ $$\vdots$$ $$\vdots$$ $$\vdots$$

Three types of length measurements are common in the fisheries literature Figure 1. Total length (TL) is the length from the most anterior to the most posterior point with the tail of the fish compressed to exhibit the longest possible length. Fork length (FL) is the length from the most anterior point to the anterior notch in the fork of the tail. For fish without a forked tail, the fork and total lengths are the same. The standard length (SL) is the length from the most anterior point to the posterior end of the caudal peduncle. Total length is the most common measurement in fisheries studies, as this is the measurement used in management decisions such as setting minimum lengths.

Figure 1: Demonstration of total, fork, and standard length measurements on a Bluegill.

Length measurements are often taken with the aid of a measuring board where the measuring “stick” is embedded into the bottom board and one end of this board is fit with a vertical end piece. The fish to be measured is placed on the bottom board such that the anterior point of the fish is against the vertical end piece and the measurement can be read directly from the embedded measuring stick (Figure 2). Length measurements are subject to very little measurement error (Gutreuter and Krzoska 1994).

Figure 2: Field measurements of length (left) and weight with a spring scale (right).

Two common weight measurements are used. The usual body weight is the weight of the fish as it was captured, whereas the dressed weight is the weight of the fish with the gills and entrails removed. Dressed weight is usually only used when measurements are reported from a commercial fishery.

Weight measurements can be made in the field on fresh specimens or in the lab on fresh-frozen specimens. Weight measurements in the field can be taken with tared spring or electronic balances (Figure 2). However, field measurements can be extremely variable due to differences in fish surface wetness, boat movements, wind, and other adverse environmental conditions (Gutreuter and Krzoska 1994). Substantial variability in weight measurements can occur when fish weigh less than 10% of a scale’s capacity (Gutreuter and Krzoska 1994). Thus, multiple sizes of scales should be taken into the field (Blackwell et al. 2000). Wege and Anderson (1978)} suggest that the accuracy of the scale should be 1% of a fish’s body weight for use in relative weight calculations. Weight measurements on frozen fish were roughly 1-9% lighter than the measurements on the same fish when fresh, whereas length measurements were roughly 1-4% shorter on frozen then fresh fish for a variety of species (reviewed in Ogle (2009)).

### Weight-Length Model

The relationship between the length and weight of a sample of fish tends to have two important characteristics. First, the relationship is not linear . This can be explained intuitively by thinking of length as a linear measure and weight as being related to volume. Thus, as the organism adds a linear amount of length, it is adding a disk of volume with a commensurate weight. Second, the variability in weight increases as the length of the fish increases (i.e., the scatter of the points increases from left-to-right in Figure 3. Thus, variability in weight among shorter fish is less then variability in weight among longer fish. Unfortunately, because of these two characteristics, length-weight data tends to violate the linearity and homoscedasticity (i.e., “constant variance”) assumptions of simple linear regression.

Figure 3: Length and weight of Ruffe from the St. Louis River Harbor, 1992.

These characteristics of length-weight data suggest that a two-parameter power function with a multiplicative error term should be used to model the length-weight relationship. Specifically, the model typically used is

$W_{i} = aL_{i}^{b}e^{\epsilon_{i}} \quad \quad \text{(1)}$

where $$a$$ and $$b$$ are constants and $$\epsilon_{i}$$ is the multiplicative error term for the $$i$$th fish. The length-weight model Equation 1 can be transformed to a linear model by taking the natural logarithms1 of both sides and simplifying,

$log(W_{i}) = log(a) + blog(L_{i}) + \epsilon_{i} \quad \quad \text{(1)}$

Thus, with $$y=log(W)$$, $$x=log(L)$$, slope=$$b$$, and intercept=$$log(a)$$, Equation 2 is in the form of a linear model. In addition to linearizing the model, this transformation has the added benefit of making the errors additive and stabilizing the variances about the model (i.e., making the scatter around the line nearly constant for all length measurements; Figure 4). With this linearization and stabilization, the usual linear regression methods can be used to fit the relationship between $$log(W)$$ and $$log(L)$$.

Figure 4: Natural log transformed total length and weight of Ruffe from the St. Louis River Harbor, 1992. Note that the “flaring”" of the values in the lower-left corner of the plot is due to minimum weight limitations of the measuring scale.

It should be noted that, with the example in Figure 4, the variability on the log scale appears greater for small’’ fish. This is because the scale used to measure these fish lacked the required precision to distinguish weights of small fish over a wide length range. It is apparent that fish with a log weight of less than -0.5 should be eliminated from this analysis because of scale imprecision for these fish.

If a fish grows without changing its shape or its density then the fish is said to exhibit isometric growth. In this case, the volume of the fish is proportional to any linear measure of its size. If weight is taken as a surrogate of volume (which requires assuming a constant density) and length as the linear measure, then the modeled relationship between length and weight, Equation 1, will have $$b=3$$ under isometric growth. Isometric growth in fish is rare (Bolger and Connolly 1989, McGurk (1985)). If a fish changes shape or density as it grows, then $$b\neq3$$ in Equation 1, and the fish is said to exhibit allometric growth. If $$b>3$$ then the fish tends to become more “plumper” as the fish increases in length (Blackwell et al. 2000).

A test of whether the fish in a population exhibit isometric growth or not can be obtained by noting that $$b$$ is the estimated slope from fitting the transformed length-weight model. The slope is generically labeled with $$\beta$$ such that the test for allometry can be translated into the following statistical hypotheses:

• $$H_{0}:\beta=3 \quad \Rightarrow H_{0}:$$ “Isometric growth”
• $$H_{A}:\beta\neq3 \quad \Rightarrow H_{A}:$$ “Allometric growth”’ \end{Itemize}

Hypothesis tests regarding model parameters can be obtained with a t-test using

$t = \frac{\hat{\beta}-\beta_{0}}{SE_{\hat{\beta}}}$

where $$\hat{\beta}$$, $$SE_{\hat{\beta}}$$ and the df are from the linear regression results and $$\beta_{0}$$ is the specified value in the $$H_{0}$$. Nearly all statistical packages, R included, print the $$t$$ and corresponding $$p-value$$ for $$H_{0}:\beta=0$$ by default, but not for any hypothesized value other than zero. Thus, the test statistic and p-value for the test of isometry must often be calculated “by hand.”

# Body Condition

Condition is a measure of the physical health of a population of fish based on the fish’s relative plumpness or fatness. Most often condition is computed by comparing the actual weight of a fish to some expectation of weight based on the length of the fish. In other words, measuring the condition of a particular fish is an exercise in determining if it weighs more or less than would be expected based on its length. An overall measure of condition for an entire population is obtained by averaging the condition of all fish in a sample.

The utility of measuring fish conditions was summarized by Blackwell et al. (2000) in the following manner:

Fish condition can be extremely important to fisheries managers. Plump fish may be indicators of favorable environmental conditions (e.g., habitat conditions, ample prey availability), whereas thin fish may indicate less favorable environmental conditions. Thus, being able to monitor fish well-being can be extremely useful for fisheries biologists who must make management recommendations concerning fish populations.

There are at least eight metrics of condition (Bolger and Connolly 1989) of which three are commonly used by fisheries managers (Blackwell et al. 2000). These three measures are introduced below and discussed within the context of four properties that should be evident in all condition metrics (Murphy et al. 1990). Those four properties are (1) consistency – similar statistical properties and meaning regardless of species or length; (2) tractability – analysis by standard statistical methods; (3) efficiency – relative precision from small samples; and (4) robustness – relative insensitivity to variations in the way the data was collected and analyzed.

### Fulton’s Condition Factor

Fulton’s condition factor is calculated with

$K = \frac{W}{L^{3}}*constant$

where the constant is simply a scaling factor that is equal to $$100000$$ if metric units are used (i.e., grams and millimeters) or $$10000$$ if English units are used (i.e., pounds and inches).{^2] Fulton’s condition factor assumes isometric growth.2 If a fish stock does not exhibit isometric growth, which is often the case, then $$K$$ tends to differ depending on the length of the fish, violating the consistency property. Furthermore, comparing $$K$$ between species is problematic because both species would need to exhibit isometric growth for the comparison to be valid. Because of these limitations, Fulton’s condition factor should be avoided.

### LeCren’s Relative Condition Factor

The relative condition factor, introduced by Le Cren (1951), is calculated with

$Kn = \frac{W}{W'}$

where $$W'$$ is the predicted length-specific mean weight for the population under study [Blackwell et al. (2000)}. The average $$Kn$$ across all lengths and species is 1.0 [Anderson and Neumann (1996)}. Thus, $$Kn$$ is consistent across lengths. Bolger and Connolly (1989), however, show that $$Kn$$ comparisons are restricted to species or regions that have the same slope ($$b$$) in the length-weight relationship. Thus, $$W'$$ is generally predicted from length-weight equations developed for a population (perhaps from several years of data) or for a region.

### Relative Weight

The relative weight, introduced by Wege and Anderson (1978), is calculated with

$Wr = \frac{W}{Ws}*100 \quad \quad \text{(3)}$

where $$Ws$$ is a standard weight’’ for fish of the same length. In simplistic terms, a standard weight equation for a particular species is a length-weight relationship designed to predict the 75th percentile3 mean weight for a given value of length.4 Standard weight equations have been developed for a wide variety of species.5 It should be noted that the standard weight equations have been developed for either metric or English unit measurements but that the $$log_{10}$$ rather than the $$log_{e}$$ transformation is used. Blackwell et al. (2000) suggest that regional or population-specific $$Ws$$ equations should NOT be developed. If regional or population-specific analysis is desired then $$Kn$$ should be used.

Recent trends, following the work of Gerow et al. (2005), have resulted in standard weight equations that are quadratic rather than linear. Examples of the use of quadratic standard weight equations can be found in Ogle and Winfield (2009) and Cooney and Kwak (2010). While use of these types of standard weight equations is not illustrated in this vignette, their use is a simple and straightforward modification of what is demonstrated here.

The relative weight measure has become the most popular measure of condition (Blackwell et al. 2000). This popularity is partly due to the fact that relative weight summaries have been used as a surrogate measure of the general “health” of the fish (Brown and Murphy (1991), Neumann and Murphy (1992), Jonas et al. (1996), Brown and Murphy (2004), Kaufman et al. (2007), Rennie and Verdon (2008)}; but also see Copeland et al. (2008)) as well as the environment (Liao et al. (1995), Blackwell et al. (2000), Rennie and Verdon (2008)). Thus, relative weight summaries may be used as an indirect means for evaluating ecological relationships and the effects of management strategies (Murphy et al. (1991), Blackwell et al. (2000)). In addition, Murphy et al. (1990) found the distributions of $$Wr$$ values to typically be symmetric (but not normal). Because t-tests and analysis of variance tests are relatively robust to departures from normality, as long as the distribution is symmetric, typical parametric inferential statistics can be used with $$Wr$$ values.

It should be noted, though, that Gerow (Gerow et al. (2004), Gerow et al. (2005), and Gerow (2010)) has been critical of the idea that traditionally developed standard weight equations produce relative weight values that are not dependent on the length of the fish.

### Comparisons Among Length Categories

A measure of overall condition using $$Wr$$ should not be computed without first determining if the $$Wr$$ values differ across fish lengths (Blackwell et al. 2000). In particular, Murphy et al. (1991) suggested that $$Wr$$ values should first be summarized within the usual five-cell length categories of Gabelhouse (1984).6 Use of the five-cell model can be problematic at times because of small sample sizes in the larger length categories. Thus, other authors have summarized by 25- or 50-mm length categories. The mean $$Wr$$ values in the length categories are then tested with analysis of variance methods to determine if differences exist among the length categories. Adjacent length categories that are statistically equal can then be pooled together. If no statistical differences among categories exist, then all length categories can be pooled and an overall measure of condition for the population can be computed. Relative weights should be reported as whole numbers (Blackwell et al. 2000).

# Calculations in R

Methods for performing these calculations in R are described in Sections 7.1-7.3 and Chapter 8 of Ogle (2016).7

## Reproducibility Information

• Compiled Date: Mon Jan 11 2016
• Compiled Time: 9:17:17 AM
• R Version: R version 3.2.3 (2015-12-10)
• System: Windows, i386-w64-mingw32/i386 (32-bit)
• Base Packages: base, datasets, graphics, grDevices, methods, stats, utils
• Required Packages: FSA, FSAdata, captioner, knitr, dplyr, magrittr and their dependencies (assertthat, car, DBI, digest, evaluate, formatR, gdata, gplots, graphics, grDevices, highr, Hmisc, lazyeval, markdown, methods, plotrix, plyr, R6, Rcpp, sciplot, stats, stringr, tools, utils, yaml)
• Other Packages: captioner_2.2.3, dplyr_0.4.3, FSA_0.8.4, FSAdata_0.3.2, knitr_1.11, magrittr_1.5
• Loaded-Only Packages: assertthat_0.1, DBI_0.3.1, digest_0.6.8, evaluate_0.8, formatR_1.2.1, gdata_2.17.0, gtools_3.5.0, htmltools_0.3, lazyeval_0.1.10, parallel_3.2.3, plyr_1.8.3, R6_2.1.1, Rcpp_0.12.2, rmarkdown_0.9.2, stringi_1.0-1, stringr_1.0.0, tools_3.2.3, yaml_2.1.13

## References

Anderson, R., and R. Neumann. 1996. Length, weight, and associated structural indices. in, Murphy, B.R. and D.W. Willis, editors Fisheries Techniques, second edition, American Fisheries Society, Bethesda, Maryland:447–481.

Blackwell, B. G., M. L. Brown, and D. W. Willis. 2000. Relative weight (Wr) status and current use in fisheries assessment and management. Reviews in Fisheries Science 8:1–44.

Bolger, T., and P. L. Connolly. 1989. The selection of suitable indices for the measurement and analysis of fish condition. Journal of Fish Biology 34:171–182.

Brown, M. L., and B. R. Murphy. 1991. Relationship of relative weight (Wr) to proximate composition of juvenile Striped Bass and hybrid Striped Bass. Transactions of the American Fisheries Society 120:509–518.

Brown, M. L., and B. R. Murphy. 2004. Seasonal dynamics of direct and indirect condition indices in relation to energy allocation in Largemouth Bass micropterus salmoides (Lacepede). Ecology of Freshwater Fish 13:23–36.

Cooney, P. B., and T. J. Kwak. 2010. Development of standard weight equations for Caribbean and Gulf of Mexico amphidromous fishes. North American Journal of Fisheries Management 30:1203–1209.

Copeland, T., B. R. Murphy, and J. J. Ney. 2008. Interpretation of relative weight in three populations of wild Bluegills: A cautionary tale. North American Journal of Fisheries Management 28:386–377.

Froese, R. 2006. Cube law, condition factor and weight-length relationships: History, meta-analysis and recommendations. Journal of Applied Ichthyology 22:241–253.

Gabelhouse, D. W. 1984. A length-categorization system to assess fish stocks. North American Journal of Fisheries Management 4:273–285.

Gerow, K. G. 2010. Biases with the regression line percentile method and the fallacy of a single standard weight. North American Journal of Fisheries Management 30:679–690.

Gerow, K. G., R. C. Anderson-Sprecher, and W. A. Hubert. 2005. A new method to compute standard-weight equations that reduces length-related bias. North American Journal of Fisheries Management 25:1288–1300.

Gerow, K. G., W. A. Hubert, and R. C. Anderson-Sprecher. 2004. An alternative approach to detection of length-related biases in standard weight equations. North American Journal of Fisheries Management 24:903–910.

Gutreuter, S., and D. J. Krzoska. 1994. Quantifying precision of in situ length and weight measurements of fish. North American Journal of Fisheries Management 14:318–322.

Hilborn, R., and C. J. Walters. 2001. Quantitative fisheries stock assessment: Choice, dynamics, & uncertainty. Page 570Second. Chapman & Hall, New York, NY.

Jonas, J. L., C. E. Kraft, and T. L. Margenau. 1996. Assessment of seasonal changes in energy density and condition in age-0 and age-1 Muskellunge. Transactions of the American Fisheries Society 125:203–210.

Kaufman, S. D., T. A. Johnston, W. C. Leggett, M. D. Moles, J. M. Casselman, and A. I. Schulte-Hostedde. 2007. Relationships between body condition indices and proximate composition in adult Walleyes. Transactions of the American Fisheries Society 136:1566–1576.

Le Cren, E. D. 1951. The length-weight relationship and seasonal cycle in gonad weight and condition in the Perch (Perca flavescens). Journal of Animal Ecology 20:201–219.

Liao, H., C.L. Pierce, D. H. Wahl, J. B. Rasmussen, and W. C. Leggett. 1995. Relative weight (Wr) as a field assessment tool: Relationships with growth, prey biomass, and environmental conditions. Transactions of the American Fisheries Society 124:387–400.

McGurk, M. D. 1985. Effects of net capture on the postpreservation morphometry, dry weight, and condition factor of Pacific herring larvae. Transactions of the American Fisheries Society 114:348–355.

Murphy, B. R., M. L. Brown, and T. A. Springer. 1990. Evaluation of the relative weight (Wr) index, with new applications to Walleye. North American Journal of Fisheries Management 10:85–97.

Murphy, B. R., D. W. Willis, and T. A. Springer. 1991. The relative weight index in fisheries management: Status and needs. Fisheries (Bethesda) 16(2):30–38.

Neumann, R. M., and B. R. Murphy. 1992. Seasonal relationships of relative weight to body composition in White Crappie Pomoxis annularis Rafinesque. Aquaculture Research 23:243–251.

Ogle. 2009. The effect of freezing on the length and weight measurements of ruffe (Gymnocephalus cernuus). Fisheries Research 99:244–247.

Ogle, D. H. 2016. Introductory fisheries analyses with R. Chapman & Hall/CRC, Boca Raton, FL.

Ogle, D. H., and I. J. Winfield. 2009. Ruffe length-weight relationships with a proposed standard weight equation. North American Journal of Fisheries Management 29:850–858.

Rennie, M. D., and R. Verdon. 2008. Development and evaluation of condition indices for the Lake Whitefish. North American Journal of Fisheries Management 28:1270–1293.

Wege, G. W., and R. O. Anderson. 1978. Relative weight (Wr): A new index of condition for Largemouth Bass. Pages 79–91 in G. D. Novinger and J. G. Dillard, editors. New approaches to the management of small impoundments. American Fisheries Society.

1. Natural logarithms are used throughout this course and will be referred to simply as “logarithms” and will be abbreviated with “log”.

2. See the length-weight vignette for a brief discussion of isometric and allometric growth.

3. However, see Ogle and Winfield (2009) for an equation that uses the 50th percentile in addition to the 75th percentile.

4. Specific discussion of methods for computing the standard weight equations are discussed in detail in Murphy et al. (1990) and Blackwell et al. (2000).

5. In R, examine the data frame for all known equations

6. The five-cell length categorization scheme of Gabelhouse (1984) was defined in the size structure module.

7. Scripts for these calculations are here.