Ws Equation Research Articles

Proposed Standard Weight (Ws) Equations and Standard Length Categories for Redbreast Sunfish and Spotted Sunfish

AbstractWe developed standard weight (Ws ) equations for Redbreast Sunfish Lepomis auritus and Spotted Sunfish L. punctatus using the traditional regression‐line‐percentile (RLP) technique and the empirical‐percentile‐linear (EmP‐L), and the empirical‐percentile quadratic (EmP‐Q) method. Individual length and weight data for 175,541 Redbreast Sunfish from 8 to 31 cm TL across 315 populations and 66,851 Spotted Sunfish from 7 to 21 cm TL across 130 populations were used in the final analysis. These fish were collected from 1975 to 2021 by various agency and university personnel from most known populations across the species' native and introduced ranges in North America. We present the EmP‐L‐derived Ws equations for the Redbreast Sunfish as log10W = 3.1218 × log10L − 5.0018 and the Spotted Sunfish as log10W = 3.3343 × log10L − 5.3739, where W is weight (g) and L is total length (TL, mm). Equations from all methods exhibited length‐related bias, but the EmP‐L equations had a median bias of 0–0.6 for the length range that comprised 99.7% (for 8–24 cm TL) of Redbreast Sunfish and 99.0% (for 7–18 cm TL) of Spotted Sunfish in our data set. Regardless of the equation, the most bias was found at the upper end of the length spectrum, so we do not recommend these equations for Redbreast Sunfish ≥30 cm TL or Spotted Sunfish ≥20 cm TL. We also developed standard length categories for Redbreast Sunfish with minimum lengths for five standard length categories of 8, 13, 18, 20, and 25 cm TL for stock, quality, preferred, memorable, and trophy sizes and for Spotted sunfish 5, 10, 13, 15, and 18 cm TL, respectively. Development of a Ws equation and standard length categories will aid biologists in assessing condition and size structure of wild Redbreast Sunfish and Spotted Sunfish populations and those being propagated for supplemental stocking into areas with depressed native populations.

Proposed Standard Weight Equation and Standard Length Categories for Lake Chubsucker

AbstractStandard weight equations have been proposed for a variety of fishes, including many nongame species. Here, we developed standard weight (Ws) equations for Lake Chubsucker Erimyzon succetta using the traditional regression‐line‐percentile technique and the newer empirical‐percentile method. Length and weight data for 19,727 Lake Chubsuckers from 8 to 42 cm TL and 129 populations were used in the final analysis. These fish were collected from 1977 to 2020 by various agency and university personnel from most known populations across the species’ range in North America. This species is considered vulnerable, imperiled, or extirpated in several Midwestern and northern states, but it appears to be secure in the Southeast. The resulting Ws equations were log10W = −5.0962 + 3.1125 × log10L and log10W = –5.9735 + 3.8828 × log10L −0.1701 × (log10L)2 for the RLP and quadratic empirical‐percentile method techniques, respectively, where W is weight (g) and L is total length (TL, mm). The regression‐line‐percentile equation␣exhibited a greater length‐related bias especially at the ends of the length spectrum, so we recommend the quadratic empirical‐percentile equation␣for this species. We also developed standard length categories for assessing fish population size structure. We propose minimum lengths for five standard length categories of 11, 18, 24, 28, and 35 cm TL for stock, quality, preferred, memorable, and trophy sizes, respectively. Development of a Ws equation␣and standard length categories will aid biologists in assessing condition and size structure of wild Lake Chubsucker populations and those being propagated for supplemental stocking into trophy bass ponds for native forage or in areas with depressed native populations. Additionally, by evaluating this nongame species that occupies the same environments as higher profile game fish species, a more holistic approach of evaluating the overall health of systems can be examined.

Proposed Standard Weight (Ws) Equation and Length Categories for Utah Chub

AbstractCondition indices, such as relative weight (Wr), provide a simple method for comparing length–weight relationships among populations. However, no standard weight (Ws) equation has been developed for Utah Chub Gila atraria, a species of important management focus in the Intermountain West. We obtained length–weight data for 30,541 Utah Chub from 24 populations in Idaho, Montana, Utah, and Wyoming. We used the regression line percentile (RLP), linear empirical percentile (EmP), and quadratic EmP methods to develop average (50th percentile) and above average (75th percentile) Ws equations. Additionally, Froese’s method was used to develop another Ws equation for Utah Chub. Length‐related biases were detected in Ws equations developed using the RLP, 50th percentile quadratic EmP, and Froese methods. The linear EmP Ws equations did not exhibit length‐related biases for the 50th and 75th percentiles. We propose using the 75th percentile linear EmP Ws equation for Utah Chub between 90 and 410 mm TL. The EmP 75th percentile equation was log10(Ws) = −4.938 + 3.031·log10(TL), where Ws is weight in grams and TL is in millimeters. The English equivalent of this equation is log10(Ws) = −3.335 + 3.031·log10(TL), where Ws is weight in pounds and TL is in inches for 4–16‐in Utah Chub. Additionally, we propose that minimum TLs of 100 mm (4 in; stock), 200 mm (8 in; quality), 250 mm (10 in; preferred), 300 mm (12 in; memorable), and 380 mm (15 in; trophy) be used to calculate proportional size distribution (PSD) indices. Better understanding Utah Chub populations using Wr and PSDs will aid managers in assessing management strategies (e.g., biological controls) focused on Utah Chub.

Proposed Standard Weight (Ws) Equations for Arctic Grayling

AbstractRelative weight (Wr) is an index that is commonly used by fisheries managers to evaluate the condition within and among fish populations. Managers also use Wr comparisons to assess condition dynamics among species. Rather than using population‐level mean Wr for comparisons, standard length categories are frequently used to evaluate mean Wr differences among length categories. To establish Wr values for a species, a species‐specific standard weight (Ws) equation is required. Regression‐line‐percentile (RLP) and empirical‐percentile (EmP) methods were used with length–weight data that were collected from 50 Arctic Grayling Thymallus arcticus populations in the United States and Canada. The RLP method yielded a Ws equation of log10Ws(g) = −5.241 + 3.083 log10TL(mm) and the EmP method of log10Ws(g) = −5.279 + 3.096 log10TL(mm). These equations were developed to calculate relative weights (Wr) for fish measuring 150 mm total length (TL) and greater. Subsequent analysis did not reveal significant systematic length‐related bias in the application of either Ws equation. Furthermore, differences in Ws calculations between the methods were at most 2.2% over the lengths that we observed. Our findings highlight the relative importance of using a robust data set when calculating Ws equations over the method used to calculate Ws. We suggest that both equations are valid and managers should report which Ws equation was used.

Standard Weight Equation for Brook Trout in Southern Appalachian Mountains Streams

AbstractBrook Trout Salvelinus fontinalis in southern Appalachian Mountains streams of the United States occur at the southernmost portion of their native range, and occupy small, isolated, and low-productivity headwater streams. The existing standard weight (Ws) equation is applicable only to Brook Trout > 120 mm total length (TL), but many individuals in the region are smaller than this minimum size threshold due to their habitat characteristics. Here, we developed a new Ws equation for Brook Trout in southern Appalachian Mountains streams using length–weight data on 72,502 individuals. The weighted quadratic empirical-percentile method minimized length-related bias in relative weight compared to the regression-line-percentile and weighted linear empirical-percentile methods. The proposed Ws equation was: log10W = −3.364 + 1.378 × log10L + 0.397 × (log10L)2, where W was weight (g) and L was TL (mm). The new equation characterized body condition of Brook Trout in southern Appalachian Mountains streams more accurately than the existing equation.

Length-weight, length-length and empirical standard weight equations for Capoeta baliki, Capoeta sieboldii and Chondrostoma angorense, three endemic cyprinid species of northwestern Anatolia

In this study, length and weight data for three endemic fish species of North-western Anatolia (Capoeta baliki, Capoeta sieboldi and Chondrostoma angorense) were collected throughout their distribution ranges and used to estimate length-length, total length (TL) – weight and empirical standard weight (Ws) equations. The obtained Ws equations were: log10Ws = -6.743 + 4.768 log10 TL - 0.437 log10 TL2 (TL range: 7-28 cm) for C. baliki; log10Ws= - 5.966 + 3.833 log10TL - 0.181 log10 TL2 (TL range: 7-36 cm) for C. sieboldii; log10Ws= - 10.017 + 7.402 log10TL - 0.971 log10TL2 (TL-range: 7-24 cm) for C. angorense. For C. angorense, these data represents the first reference on length-weight for the species. In addition, a new maximum length was reported for C. baliki.

Proposed Standard Weight (Ws) Equation and Standard Length Categories for Shoal Bass

AbstractStandard weight (Ws) equations have been proposed for a variety of fishes, including many of the black basses Micropterus spp. We developed Ws equations for Shoal Bass M. cataractae by using the traditional regression‐line‐percentile (RLP) technique and the relatively new empirical‐percentile (EmP) method. Length and weight data for 9,058 Shoal Bass from 15 populations were used in this analysis. These fish were collected from 1978 to 2017 by various agency and university personnel from most of the known populations across the species’ range in Alabama, Florida, and Georgia. The resulting Ws equations were log10W = 3.243·log10L − 5.490 for the RLP technique and log10W = 1.225·log10L + 0.415·(log10L)2 – 3.052 for the quadratic EmP technique, where W is weight (g) and L is TL (mm). Unlike the quadratic EmP equation, the RLP equation exhibited a slight length‐related bias on each end of the length spectrum, so we recommend the EmP technique as the preferred equation for this species. To assess fish population size structure, we also propose minimum lengths for five standard TL categories of 18, 28, 35, 43, and 51 cm for stock, quality, preferred, memorable, and trophy sizes, respectively. Development of a Ws equation and standard TL categories will aid biologists in evaluating and monitoring Shoal Bass populations as part of continuing conservation efforts.

Quantile Regression Estimates of Body Weight at Length in Walleye

AbstractQuantile regression is a method of estimating fish weight at length for alternate portions of a probability distribution, but it is an approach that has not received much attention in fisheries literature. Quantile regression can provide estimates of any quantile of weight at length without bias (including the 75th quantile, which was often the focus of standard weight [Ws] equations), and this is more advantageous than previously defined Ws equations derived from linear or quadratic regression methods. The goal of this study was to demonstrate the utility of quantile regression as a tool to assess fish weight at length at various portions of the probability distribution without bias using Walleye Sander vitreus as a case study. Quantile regression models at the 75th quantile were developed for three randomly selected Walleye populations from Georgia and South Dakota and compared with a large (N = 33,589) reference population. Bootstrap resampling procedures indicated that only one population from the state of Georgia had an intercept and slope similar to the reference population. For the one population that had similar intercept and slope to the reference population, predictions of weight at various lengths still fell below the 95% CIs for predicted weights of the reference population, suggesting that slight differences in intercepts and slopes in allometric relationships can result in predicted weights that still differ at some lengths. Predicted weights of Walleye derived from the 10th, 25th, 50th, 75th, and 90th quantiles were used to demonstrate how individuals and populations may be compared at different management targets. Overall, this study demonstrates the relative ease with which quantile regression may be used to compare fish body condition between populations without bias.

Length–weight and length–length relationships for three endemic cyprinidspecies of the Aegean region (Turkey) with proposed standard weight equations

Empirical standard weight equations and length-weight and length-length relationships were estimated for Barbus pergamonensis, Capoeta bergamae, and Ladigesocypris irideus, three endemic cyprinid species of Turkey that are still not widely studied. To the best knowledge of the authors, empirical standard weight equations for the three species and L-W parameters for L. Irideus are given here for the first time. In this study, length and weight data for B. Pergamonensis, C. Bergamae, and L. Irideus were collected throughout the watercourses of Mugla Province and were used to develop species-specific Ws equations. The results were for B. Pergamonensis (total length [TL] range: 6-22 cm) log10Ws= -5.713 + 3.718 log10TL - 0.166 (log10TL)2, for C. Bergamae (TL range: 8-29 cm) log10Ws= -5.224 + 3.311 log10TL - 0.083 (log10TL)2, and for L. Irideus (TL range: 6-10 cm) log10Ws= -13.298 + 12.116 log10TL - 2.471 (log10TL)2. This study also reported new maximum TLs for the three species, together with species-specific length-weight and length-length equations.

Development of empirical standard weight equation for Pursak chub Squalius pursakensis, an endemic cyprinid species of Northwest Anatolia

Indices of condition enable the evaluation of the well-being of fish, with the assumption that heavier fish of a given length are in a better condition. Relative weight (Wr) is one of these indices; it is calculated by comparing the actual weight of a specimen with the ideal weight of a specimen in good physiological condition of the same length from the same species, i.e. standard weight (Ws). In this research, length and weight data over the distribution range for Pursak chub Squalius pursakensis, an endemic species distributed in the Sakarya and Porsuk drainages in Northwest Anatolia (Turkey), were used to compute a Ws equation by means of the empirical percentile (EmP) method. The Ws equation obtained was log10 Ws = -4.657 + 2.614 log10 TL + 0.127 (log10 TL)2, and the total length range of application was 80-340 mm. Since the EmP Ws equation was not influenced by length variation, the use of this equation to compute the relative weight (Wr) for S. pursakensis throughout its area of distribution is suggested.

Empirical Standard Weight Equation for the Aegean Chub Squalius fellowesii, an Endemic Freshwater Fish Species of Western Anatolia, Turkey

AbstractData on length and weight of Aegean chub Squalius fellowesii, an endemic species distributed in the Aegean drainages of Anatolia, Turkey, were used to calculate a standard weight (Ws) equation by means of the empirical percentile (EmP) method. The resulting Ws equation was log10 Ws = −3.801 + 1.783·log10TL + 0.329·(log10TL)2. Over the range of application (70–220 mm TL), the EmP Ws equation was not biased by length, and the use of this equation is suggested as a way to calculate the relative weight (Wr) of Aegean chub throughout the species’ distribution area.Received June 12, 2012; accepted July 27, 2012

Proposed Empirical Standard Weight Equation for Brook Chub Squalius lucumonis

AbstractRelative weight (Wr) is an index of condition that enables evaluation of the well‐being of fish by comparing the actual weight of a specimen with the ideal weight of a specimen of the same species and the same length that is in good physiological condition (i.e., the standard weight [Ws]). This index was primarily developed to assess the status of sport fishes. Recently, however, many authors have encouraged the use of this index as a fisheries assessment tool for nongame species as well, especially those that are endangered or threatened. Length and weight data on brook chub Squalius lucumonis, an Italian endemic species listed as endangered by the International Union for the Conservation of Nature, were collected across its area of distribution and used to compute a standard weight (Ws) equation by means of the empirical percentile method. The Ws equation thus obtained (log10 Ws = –7.75 + 5.75 log10[total length {TL}] – 0.66 [log10TL]2; TL range of application = 90–210 mm) was not biased by length and is recommended as a way to compute Wr for brook chub.Received October 11, 2011; accepted January 12, 2012

Erratum: Proposed Standard Weight (Ws) Equation and Standard Length Categories for Suwannee Bass (volume 30:983–988)

Please make the following corrections in a recent issue of this journal: Volume 30(4), August 2010: “Proposed Standard Weight (Ws) Equation and Standard Length Categories for Suwannee Bass,” by Timothy F. Bonvechio, Kimberly I. Bonvechio, and Richard L. Cailteux, pages 983–988. Page 983. The third sentence of the Abstract should read as follows: The resulting Ws equations were log10 W = −5.455 + 3.260 × log10 L and log10 W = −0.106 − 1.363 × log10 L + 0.995 × (log10 L)2 (where W is weight in grams and L is TL in millimeters) for the RLP and quadratic EmP techniques, respectively. Page 985. The second sentence of the second paragraph of Results and Discussion should read as follows: The resulting standard weight equations were determined as for metric units (where W is weight in grams and L is TL in millimeters) and for English units (where weight is in pounds and TL in inches). Page 985. The sixth sentence of the second paragraph of Results and Discussion should read as follows: The EmP-Q standard weight equations for the species are and for metric and English units, respectively.

Development of Standard Weight Equations for Caribbean and Gulf of Mexico Amphidromous Fishes

AbstractWe collected and compiled length and weight information from four countries and one commonwealth to develop standard weight (Ws) equations for three amphidromous fish species native to the Caribbean and Gulf of Mexico regions: mountain mullet Agonostomus monticola (N = 9,768 individuals, 52 populations), river goby Awaous banana (N = 1,847 individuals, 62 populations), and bigmouth sleeper Gobiomorus dormitor (N = 2,983 individuals, 53 populations). Linear and quadratic Ws equations for three quartiles (25%, median, 75%) are presented for these three species. The length–weight relationship from eight lentic bigmouth sleeper populations was significantly different from that of lotic populations, reflecting higher weights of juvenile fish (<70 mm total length) in lentic environments. Thus, independent Ws equations were developed for lotic populations of bigmouth sleepers. Ws equations were not developed from lentic bigmouth sleeper populations alone due to the low number of applicable populations caused by life history constraints; the equation from combined lentic and lotic populations is suggested for application to lentic bigmouth sleeper populations. These morphometric relationships for amphidromous fishes may improve the ability to assess existing and potential sport fisheries and allow ecological assessment based on fish condition.

Proposed Standard Weight (Ws) Equation and Standard Length Categories for Suwannee Bass

AbstractWe developed standard weight (Ws) equations for Suwannee bass Micropterus notius using the traditional regression‐line‐percentile (RLP) technique and the newer empirical‐percentile (EmP) method. Length and weight data for 2,137 Suwannee bass at least 11 cm total length (TL) collected from all 10 populations within the species' narrow geographic range were used to develop the equations. The resulting Ws equations were log10 W = −5.548 + 3.303 × log10 L and log10 W = −1.014 − 0.583 × log10 L + 0.828 × (log10 L)2 (where W is weight in grams and L is TL in millimeters) for the RLP and quadratic EmP techniques, respectively. The RLP‐derived Ws equation exhibited a small (<4%) bias that varied by length‐group but that was insignificant for management purposes. To assess fish population size structure, we also propose minimum lengths for five standard length categories of 15, 25, 30, 35, and 40 cm TL for stock, quality, preferred, memorable, and trophy sizes, respectively. Urban development and the recent introduction of nonnative species into some of these systems pose threats to the conservation of this rare black bass. Thus, the development of a Ws equation and standard length categories will aid biologists in evaluating and monitoring Suwannee bass populations as part of continuing conservation efforts.

Assessing Length‐Related Bias and the Need for Data Standardization in the Development of Standard Weight Equations

AbstractThe recently developed empirical percentile (EmP) method, a technique for deriving standard weight (Ws) equations, putatively reduces the length‐related biases that often plague such equations. To determine whether the EmP method is superior to the regression line–percentile (RLP) method in reducing length‐related biases, we developed new Ws equations by applying both methods to two morphologically distinct species, walleye Sander vitreus and black crappie Pomoxis nigromaculatus. We also investigated diagnostic approaches to provide quality control for weight–length data. We evaluated the new Ws equations with filtered independent data to determine which equation reduced length bias the most. We suggest a protocol for evaluating length‐related bias using an independent data set. Our results showed that for randomly selected walleye populations, the RLP method did not have any length‐related biases when relative weight (Wr) was plotted as a function of length. However, the Wr values calculated from the EmP Ws equations were length biased when the latter were applied to those same populations. Both methods generated Ws equations that were length biased when Wr was plotted as a function of length for black crappies. Further, the absolute difference in Wr between the RLP and EmP methods indicates that there is little difference between the methods as far as their relevance to management is concerned. Based on these results, we believe that revising existing Ws equations using the EmP method is unnecessary and that the RLP technique should remain the standard for developing Ws equations pending the development of an approach that clearly eliminates methodological length bias.

Biases with the Regression Line Percentile Method and the Fallacy of a Single Standard Weight

AbstractThe regression line percentile (RLP) method for computing standard weight (Ws) equations produces third‐quartile estimates that tend to be biased downward for long fish as a result of extrapolation acting in concert with systematic variation in the distribution of length–weight relationships among fish populations. Parameter mismatch in the development and use of Ws equations (i.e., third quartile for development of Ws equations and use of mean relative weight Wr for application and testing) induces artifactual biases when testing equations and leads to the appearance of reduced fish condition when applied to longer fish. Consistent interpretation of mean Wr against a single standard (e.g., third quartile, median, or any other) is not usually possible due to nonconstant variation in the distributions of mean Wr among species and among length‐classes for a given species. I recommend that future computations of Ws use quadratic equations for all three quartiles to enable consistent and more informative interpretation of mean Wr.

Ruffe Length–Weight Relationships with a Proposed Standard Weight Equation

AbstractRuffe Gymnocephalus cernuus length–weight data from 141 data sets were obtained from a variety of waters across Europe and the Laurentian Great Lakes for summarization and development of standard weight (Ws) equations. The mean slopes of the length–weight relationships from all populations and for all percentiles except the 95th percentile were not different from 3.0, suggesting that ruffe growth is generally isometric. The Ws equations developed using the regression line percentile, linear empirical percentile (EmP), and Froese's methods exhibited length‐related biases. The quadratic EmP Ws equations for the 75th and 50th percentiles did not exhibit length‐related biases and thus can be used to compute the relative weight of ruffe. The EmP 75th percentile Ws equation was log10(Ws75) = −2.5800 + 0.6210·log10(total length, TL) + 0.6073·[log10(TL)]2, and the 50th percentile Ws equation was log10(Ws50) = −3.3524 + 1.3969·log10(TL) + 0.4054·[log10(TL)]2 when constrained to ruffe between 55 and 205 mm TL. We propose that the minimum TL for a five‐cell length categorization system used to compute stock indices for ruffe be 55, 90, 120, 140, and 175 mm TL. These results provide a method for computing relative weight indices for typical (50th percentile) and above‐average (75th percentile) ruffe; this method will allow comparisons of body condition across time, among habitats, among bodies of water, and among length categories.

Empirical Percentile Standard Weight Equation for the Blacktail Redhorse

AbstractWe generated a standard weight (Ws) equation for the blacktail redhorse Moxostoma poecilurum by use of the empirical percentile (EmP) method. Length and weight data were compiled from 3,098 individuals representing 60 populations across five states in the southeastern USA. These data were subsequently used to create the Ws equation. The species' total length (TL) constraints (130–450 mm) were used to produce the final Ws equation: log10 Ws = −5.182 + 3.087 · log10TL. No length‐related bias could be detected from the equation, indicating that it is a strong Ws equation. A separate data set from 42 populations and 179 individuals was used to validate the Ws equation. We recommend use of this equation for assessment of blacktail redhorse population dynamics, and we contend that creation of EmP Ws equations for other nongame fishes would be useful in management of fish communities as a whole.

A Proposed Standard Weight Equation for Blue Suckers

AbstractLength and weight data were obtained from 46 populations of blue suckers Cycleptus elongatus (N = 8,811 fish) across the species' geographic distribution to develop a standard weight (Ws) equation. The Ws equation was derived using the empirical percentile technique applied to fish between 240 and 809 mm total length (TL). We calculated a quadratic model and a linear model to derive Ws. The proposed 75th‐percentile quadratic equation (metric units) is log10 Ws = −2.980 + 0.977 log10TL + 0.461(log10TL)2, where Ws is expressed in grams and TL is expressed in millimeters. The proposed English‐equivalent 75th‐percentile quadratic equation is log10 Ws = −3.354 + 2.273 log10TL + 0.461(log10TL)2, where Ws is expressed in pounds and TL is expressed in inches. The 75th‐percentile linear equation (metric units) is log10 Ws = −6.301 + 3.456 log10TL, and the proposed English equivalent is log10 Ws = −4.104 + 3.456 log10TL.