Animal Ecology Publications Animal Ecology

1996

Back-Calculation of Fish Length from Scales:Empirical Comparison of Proportional MethodsClay L. PierceNational Biological Service, cpierce@iastate.edu

Joseph B. RasmussenMcGill University

William C. LeggettQueens University - Kingston

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Recommended CitationPierce, Clay L.; Rasmussen, Joseph B.; and Leggett, William C., "Back-Calculation of Fish Length from Scales: Empirical Comparisonof Proportional Methods" (1996). Animal Ecology Publications. 2.http://lib.dr.iastate.edu/aecl_pubs/2

Back-Calculation of Fish Length from Scales: Empirical Comparison ofProportional Methods

AbstractWe compared three proportional back-calculation methods for scales using data sets for pumpkinseedsLepomis gibbosus and golden shiners Notemigonus crysoleucas from 10 southern Quebec lakes, and wevalidated back-calculations by comparing them with observed lengths at lime of annulus formation. Ordinaryleast-squares regression (OR) was compared with geometric mean regression (GMR) for describing body-scale relationships. Although minor differences were detected in body-scale regressions among lakes, poolingdata across lakes yielded linear bodyscale relationships with very high r2. Differences between OR and GMRbody-scale relationships were negligible in both species. Likewise, all back-calculation methods producedequivalent results. Back-calculated lengths generally corresponded well with observed lengths in allpumpkinseeds age-classes and in golden shiners older than 1 year. Observed lengths were often greater thanback-calculated lengths for age-1 golden shiners. Our results, indicating little or no difference among methods,contradict recent reviews claiming substantial disagreement among methods. Tighter body-scale relationshipsin our data sets than in previous studies appear to explain this contradiction. We suggest that light body-scalerelationships are attainable for many species, obviating concern over which proportional back-calculationmethod is chosen.

Keywordsanulus formula, ordinary least-squares regression, geometric mean regression

DisciplinesAquaculture and Fisheries | Natural Resources Management and Policy | Terrestrial and Aquatic Ecology

CommentsThis article is from Transactions of the American Fisheries Society 125 (1996): 889, doi:10.1577/1548-8659(1996)125<0889:BCOFLF>2.3.CO;2.

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Back-Calculation of Fish Lengthfrom Scales: Empirical Comparisonof Proportional MethodsClay L. Pierce a , Joseph B. Rasmussen b & William C.Leggett ca National Biological Service, Iowa Cooperative Fish andWildlife Research Unit , Department of Animal Ecology 7Science II, Iowa State University, Ames, Iowa, 50011, USAb Department of Biology McGill University , 1205 Avenue Dr.Penfield , Montreal, Quebec, H3A 1B1, Canadac Department of Biology , Queens University , Kingston,Ontario, K7L 3N6, CanadaPublished online: 09 Jan 2011.

To cite this article: Clay L. Pierce , Joseph B. Rasmussen & William C. Leggett (1996)Back-Calculation of Fish Length from Scales: Empirical Comparison of ProportionalMethods, Transactions of the American Fisheries Society, 125:6, 889-898, DOI:10.1577/1548-8659(1996)125<0889:BCOFLF>2.3.CO;2

To link to this article: http://dx.doi.org/10.1577/1548-8659(1996)125<0889:BCOFLF>2.3.CO;2

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Transactions of the American Fisheries Society 125:889-898. 1996€> Copyright by the American Fisheries Society 1996

Back-Calculation of Fish Length from Scales:Empirical Comparison of Proportional Methods

CLAY L. PIERCENational Biological Sendee, Iowa Cooperative Fish and Wildlife Research Unit

Department of Animal Ecology, 7 Science II, Iowa State University. Antes, Iowa 50011, USA

JOSEPH B. RASMUSSENDepartment of Biology, McGill University

1205 Avenue Dr. Penfteld. Montreal. Quebec H3A I B I . Canada

WILLIAM C. LEGGETTDepartment of Biology, Queens University

Kingston. Ontario K7L 3N6, Canada

Abstract.—We compared three proportional back-calculation methods for scales using data setsfor pumpkinseeds Lepomis gibbosus and golden shiners Notemigonus crysoleucas from 10 southernQuebec lakes, and we validated back-calculations by comparing them with observed lengths atlime of annulus formation. Ordinary least-squares regression (OR) was compared with geometricmean regression (GMR) for describing body-scale relationships. Although minor differences weredetected in body-scale regressions among lakes, pooling data across lakes yielded linear body-scale relationships with very high r2. Differences between OR and GMR body-scale relationshipswere negligible in both species. Likewise, all back-calculation methods produced equivalent results.Back-calculated lengths generally corresponded well with observed lengths in all pumpkinseedsage-classes and in golden shiners older than 1 year. Observed lengths were often greater thanback-calculated lengths for age-1 golden shiners. Our results, indicating little or no differenceamong methods, contradict recent reviews claiming substantial disagreement among methods.Tighter body-scale relationships in our data sets than in previous studies appear to explain thiscontradiction. We suggest that light body-scale relationships are attainable for many species,obviating concern over which proportional back-calculation method is chosen.

Growth is an important aspect of the ecology ods were used to estimate lengths. Some of thisand life history of fish, and quantification of variety in methodology is clearly warranted by cir-growth is frequently a crucial part of fisheries re- cumstance. For instance, species exhibiting non-search and management (Summerfelt and Hall linear body-scale relationships require a different1987; Weatherley and Gill 1987). Back-calculation method than those whose body-scale relationshipof lengths from scales is a widely used approach is linear. Back-calculations based on otoliths orfor estimating the growth history of individual fish other structures present at hatching may require aand characterizing the growth of fish populations different method than those based on scales, which(Jearld 1983; Carlander 1987; Busacker et al. typically appear at some time after hatching1990). Back-calculation of lengths from scales re- (Schramm et al. 1992). However, much of this di-lies on recognition of annual growth markings (an- versity in methods reflects historical inertia, dis-nuli) on scales to calculate an estimated body agreement on the theoretical merits of differentlength associated with each annulus. Body lengths approaches, and ignorance of some of the availableestimated in this way make up a growth history, techniques (Francis 1990). The lack of conformityfrom which growth rate can be inferred. in back-calculation also introduces unnecessary

Since first being introduced by Lea (1910) near- confusion and perhaps reduces confidence in back-ly a century ago, many back-calculation methods calculation results because it is unclear how close-have been proposed and used (Francis 1990). The ly results of different methods correspond,recent review by Francis (1990) covered 54 studies The purpose of this study was to empiricallypublished since 1978 that used back-calculation, compare several back-calculation methods forFive forms of body-scale relationships were used, scales applied to common data sets and to validatefour kinds of regression were used to describe back-calculations by comparing them with ob-these relationships, and six back-calculation meth- served lengths at time of annulus formation. We

889

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890 PIERCE ET AL.

used similar data sets for two species, one withctenoid and the other with cycloid scales. The datasets were collected at the same time by consistentmethods in 10 lakes.

We have not attempted an exhaustive compar-ison of all available methods and variations. Sev-eral authors (Carlander 1981; Gutreuter 1987;Francis 1990; Schramm et al. 1992) have refutedthe general class of techniques known as "regres-sion" methods on both theoretical and empiricalgrounds. Use of this approach apparently persistsdue to its intuitive simplicity and to ignorance ofits shortcomings. In this study we focused on thegeneral class of techniques known as "propor-tional" methods. Specifically, we examined theFraser-Lee method because of its popularity (Bus-acker et al. 1990; Francis 1990) and two lesser-known proportional methods because of their pro-fessed theoretical merits (Francis 1990). Inaddition, we compared ordinary least-squares re-gression with geometric mean regression (Ricker1992) for calculating the intercept parameter usedin the Fraser-Lee equation.

MethodsStudy lakes a fid species.—We conducted this

study in 10 lakes located in the Eastern Townshipsregion of southern Quebec. Locations, littoral fishcommunities, and other characteristics of theselakes were described by Pierce et al. (1994 andreferences cited therein). Pumpkinseeds Lepomisgibbosus and golden shiners Notemigonus cryso-leucas are common and widely distributed littoralzone fishes in North America (Scott and Crossman1973; Lee 1981). Together, they account for 30%of the littoral zone fish biomass and are among themost abundant littoral species in our study lakes(Pierce et al. 1994). Pumpkinseeds have ctenoidscales; golden shiners have cycloid scales.

Fish sampling.—Using beach seines as de-scribed by Pierce et al. (1990), we sampled pump-kinseeds and golden shiners from each of the 10lakes in spring (9 May to 20 May) and again inlate summer (8 September to 22 September) of1988. These samples were collected as part of alarger sampling effort to quantify the biomass andcommunity characteristics of all littoral fish spe-cies in these lakes (Pierce et al. 1994), as well asgrowth (C. L. Pierce, unpublished) and condition(Liao et al. 1995) of pumpkinseeds and goldenshiners.

Captured fish were anesthetized immediately in2-phenoxyethanol, put on ice, and frozen within afew hours. In the laboratory, length-stratified ran-

dom subsampling yielded at least 50 fish of eachspecies from most combinations of lake and sam-pling date. The fish were individually weighed(wet) to the nearest 0.01 g on an electronic balanceand measured (total length) to the nearest milli-meter. A few subsamples contained fewer than 50fish, reflecting low abundance on the correspond-ing sampling date. Scale samples for age andgrowth analysis were collected from each fish insubsamples. Pumpkinseed scales were taken at thetip of the depressed left pectoral fin; golden shinerscales were taken above the lateral line dorsal tothe tip of the depressed left pectoral fin.

Aging, scale measurement, and back-calcula-tion.—Growth histories of individual fish in sub-samples were determined by aging and back-cal-culation of lengths at previous ages from scales(Busacker et al. 1990; Francis 1990). Ten or morescales per fish were cleaned and mounted betweenglass slides; large, opaque scales were impressedon acetate slides. All scales on slides were viewedwhen ages were assigned to fish, and a single read-er did all aging. Scales from 30 fish of each specieswere viewed by a second reader without knowl-edge of age assignments from the primary reader,and age assignments were in 100% agreement.Ages assigned by reading scales corresponded wellwith length-frequency distributions.

Anterior radii and interannular distances on 10scales per fish were measured to the nearest 0.01millimeter by using a dissecting microscope (25 Xmagnification), drawing tube, and computerizeddigitizing tablet as described by Frie (1982). Theseradii are hereafter referred to as scale lengths. Re-generated or otherwise distorted scales were notmeasured, resulting in fewer than 10 replicatescales measured from some fish. Replicate mea-surements were then averaged for each fish, pro-viding precise estimates of scale lengths for back-calculations (Newman and Weisberg 1987).

With the data on individual fish body lengthsand mean scale lengths described above, we thenback-calculated body lengths at previous ages us-ing the three primary proportional back-calcula-tion methods reviewed by Francis (1990) and ex-amined the effect of using an alternative body-scale regression as advocated by Ricker (1992).Whereas all fish from subsamples were used ingenerating body-scale relationships, fish judged tobe older than 7 years were omitted from back-calculations to avoid potential errors from incor-rect aging of older fish. Scale edge incrementsfrom fish collected in spring were also omittedfrom back-calculations (see below). Because 6-

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BACK-CALCULATION OF PISH LENGTH FROM SCALES 891

and 7-year-old fish were relatively few and oftenmissing in subsamples, we restricted this evalua-tion of back-calculation methods to fish aged 1through 5.

Back-calculation methods included the widelyused Fraser-Lee formula,

Li = c + (L(. - c) (Sf/Sc),

a formula reflecting a scale-proportional hypoth-esis (SPH) credited to Hile (1941),

Lf = -(alb) + \LC + (alb)] (S/ASt),

and a formula reflecting a body-proportional hy-pothesis (BPH) credited to Whitney and Carlander(1956),

LI = [(c + dSi)l(c + dSc)]Lc;

LI = back-calculated fish body length at age /, L(= fish body length at capture, S,- = mean scalelength at annulus /, Sc = mean scale total length,c = intercept from the regression of body lengthon mean scale length, a = intercept from the re-gression of mean scale length on body length, b- slope from the regression of mean scale lengthon body length, and d = slope from the regressionof body length on mean scale length. Alternativebody-scale regressions, which supply the interceptor c parameter, were evaluated for the Fraser-Leemethod only and included the widely used ordi-nary least-squares regression (OR) and geometricmean regression (GMR). Detailed descriptions ofthe formulae and rationale of these methods canbe found in Francis (1990) and Ricker (1992).

Validation of back-calculated lengths.—As astandard for comparison among the different back-calculation methods, we used direct measurementsof body length at annulus formation obtained fromthe spring collections in each lake. These collec-tions were made when water temperatures aver-aged 13.6°C among lakes (range, 9.2-16.0°C) andnew annuli were not yet evident on scales. Thus,the observed body lengths from these collectionsrepresent a known standard for comparing back-calculated body lengths of fish at correspondingages. Statistical comparisons were limited to fishof the same cohort and age as recommended byFrancis (1990). In other words, observed bodylengths of fish of a given age sampled in the springwere compared with corresponding back-calculat-ed body lengths from similar aged fish sampled inlate summer. Accuracy of the different back-cal-culation methods was judged by how closely theymatched known body lengths at time of annulusformation.

Statistical analyses.—Data were analyzed withlinear regression, geometric mean regression, anal-ysis of variance (ANOVA), and analysis of co-variance (ANCOVA). Potential differences inbody-scale relationships among lakes were eval-uated for each species by subjecting body lengthsto ANCOVA with mean scale length as the co-variate. Similar ANCOVAs were run for each com-bination of species and lake to test for potentialdifferences due to sampling period. Linear regres-sions of fish body length on mean scale length,mean scale length on body length, and geometricmean regressions of body length on mean scalelength were performed on the entire data set foreach species to generate pooled body-scale rela-tionships.

Patterns of length at age differed widely amongspecies and lakes, making single analyses com-bining the effects of lake, age, and back-calcula-tion method unwieldy. Therefore, potential differ-ences among back-calculation methods were eval-uated separately for each combination of speciesand lake by using 2-way ANOVAs with interac-tions. These ANOVAs tested the effects of back-calculation method and age on back-calculatedbody lengths. The four back-calculation methodsevaluated were Fraser-Lee with OR, Fraser-Leewith GMR, SPH, and BPH.

A similar approach was used in testing for dif-ferences between back-calculated and observedbody lengths; separate 2-way ANOVAs with in-teractions were performed for each combination ofspecies and lake. These ANOVAs tested the effectsof length estimate type and age on body lengthestimates. The length estimate types evaluatedwere observed body lengths and Fraser-Lee (OR)back-calculations.

All analyses were performed on untransformeddata according to the GLM and REG proceduresof SAS (SAS Institute 1988). Significance testshad an alpha of 0.05.

ResultsBody-Scale Relationships

Subsamples from the 10 lakes yielded 1,049pumpkinseeds and 1,085 golden shiners for cal-culation of body-scale relationships. The AN-COVAs testing the effects of lake on body-scalerelationships showed similar results for both spe-cies; interactions of lake X mean scale length weresignificant (pumpkinseeds: F = 20.03, P = 0.0001;golden shiners: F = 11.36, P = 0.0001). Thesesignificant interactions, due in part to very large

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892 PIERCE ET AL.

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175 -

^ 150 -

E

O)125-

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Pumpkinseeds——— OR Une?= 0.98P< 0.00001n = 1049----GMRLine

50 100 150 200 250Body Length (mm)

0 1 2 3 4 5 6

Mean Scale Length (mm)FIGURE I.—Pumpkinseed body-scale relationships pooled from 10 southern Quebec lakes. Plotted points are

body lengths and mean scale lengths of individual fish. Solid lines (OR) were generated by ordinary least-squaresregression; dashed line (GMR) was generated by geometric mean regression. Regressions of body length on meanscale length (main graph) estimated parameters used for back-calculations based on the Fraser-Lee model and thebody-proportional hypothesis. The regression of mean scale length on body length (inset graph) estimated parametersused for back-calculations based on the scale-proportional hypothesis.

sample sizes, imply slightly different trajectoriesfor body-scale relationships among lakes. How-ever, the tight fits of the pooled data in Figures 1and 2 suggest that lake-specific regressions are un-necessary for characterizing the body-scale rela-tionships for these two species. The average dif-ference between lake-specific and pooled OR in-tercepts was 1.7 mm for pumpkinseeds and 4.9mm for golden shiners.

The ANCOVAs testing the effects of samplingperiod on body-scale relationships also showedsimilar results for the two species. Nine of the 10ANCOVAs for each species indicated no signifi-cant difference in body-scale relationship betweenthe two sampling periods. The sampling period Xmean scale length interaction was significant (F =5.71, P = 0.019) for Roxton Pond pumpkinseeds,indicating slightly different body-scale trajecto-ries among sampling periods. The sampling periodmain effect was significant (F = 4.73, P = 0.031)for Roxton Pond golden shiners; body length av-eraged 2.6 mm greater for a given scale length inthe spring than in late summer.

Because of the very minor differences amonglake-specific regressions, the general lack of sea-sonal effects, and the excellent fit of the pooledregressions, pooled body-scale regressions wereused to generate the intercepts (c parameter in Fra-ser-Lee, a parameter in SPH) and slopes (b pa-rameter in SPH, d parameter in BPH) used in back-calculations. These pooled body-scale relation-ships were linear for both species and explainedvery high percentages of variance (Figures 1,2).

For pumpkinseeds, the intercepts (c) generatedby OR and GMR regression of body length onmean scale length were very similar, 23.2 and 22.3respectively. Corresponding golden shiner OR andGMR intercepts, 19.1 and 16.6 respectively, werealso similar but not quite as close as the pump-kinseed estimates. Slopes (d parameter used inBPH) generated by OR and GMR were also verysimilar, 36.3 versus 36.7 for pumpkinseeds and49.8 versus 51.1 for golden shiners. The similarityof OR and GMR regressions for both species canbe seen by the nearly superimposed lines in Fig-ures 1 and 2.

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BACK-CALCULATION OF FISH LENGTH FROM SCALES 893

225

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o>125-

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Golden Shiners——— OR Line^ = 0.95P< 0.0001n=1085- - - - GMR Line

50 100 150 200 250Body Length (mm)

0 1 2 3 4 5

Mean Scale Length (mm)FIGURE 2.—Golden shiner body-scale relationships pooled from 10 southern Quebec lakes. Details are the same

as those for Figure 1.

Comparisons among Back-Calculation MethodsSubsamples from the 10 lakes yielded 990 pump-

kinseeds and 1,050 golden shiners, or roughly 100fish per species per lake, for back-calculation ofbody length at previous ages. Omission of all fishgreater than 5 years old and of I-year-old fish fromspring collections caused these totals to be slightlylower than those used in generating body-scale re-lationships. Mean body lengths estimated by thefour back-calculation methods were very similar forboth species and across the 5 age-classes and 10lakes examined (Figures 3, 4). For an individualfish, differences among body lengths back-calcu-lated by the four methods for a given age typicallyvaried by 1 mm or less. These very small absolutedifferences among methods were not statisticallysignificant in any of the 20 ANOVAs (P » 0.05for both method main effect and method X ageinteraction in all ANOVAs). For the two speciesand 10 lakes we studied, all four back-calculationmethods gave essentially the same results.

Comparisons of Back-Calculated Lengthswith Observed Lengths

Back-calculated body lengths corresponded wellwith observed body lengths in most cases, al-

though there were some exceptions (Figures 3, 4;Table 1). For pumpkinseeds, back-calculated andobserved body lengths matched very closely inmost lakes and age-classes; observed lengths av-eraged either slightly higher or lower than back-calculated lengths in a few cases. Only one pump-kinseed ANOVA indicated a significant effect oflength estimate type (Table 1). In Lake Massa-wippi, observed lengths were greater than back-calculated lengths for age-1 pumpkinseeds, where-as back-calculated lengths were greater for olderfish (Table 1; Figure 3).

Back-calculated and observed body lengthswere somewhat more divergent in golden shinerpopulations, both in magnitude (Figure 4) and sta-tistically (Table 1). Six of the 10 golden shinerANOVAs indicated significant differences be-tween back-calculated and observed lengths (Table1). In four lakes—d'Argent, Hertel, Magog andRoxton Pond—observed lengths were greater thanback-calculated lengths for young golden shiners,whereas back-calculated lengths were equal orgreater than observed lengths for older fish (Table1; Figure 4). Although not statistically significant,a similar pattern of interaction among length es-timate type and age is suggested in the Lake Mem-

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894 PIERCE ET AL.

200 -qBrompton 200 -T Magog

0)

Ii

D Observed LengthsFraser-Lee, ORFraser-Lee, GMRSPHBPH

Age (yr)FIGURE 3.—Mean (+ half the 95% confidence interval) back-calculated and observed total lengths for pump-

kinseeds aged 1-5 in 10 southern Quebec lakes. Back-calculated lengths (shaded bars) were generated by one offour methods. Observed lengths (open bars) were obtained from fish collected in the spring at time of annulusformation. Key is in the upper right panel.

phremagog golden shiner data (Figure 4). Ob-served lengths were consistently greater than back-calculated lengths in golden shiners from LakeBrome (Figure 4; Table 1).

Although differing in magnitude and occurringin different lakes, the differences between back-calculated and observed body lengths followed asimilar age-related pattern in both species (Figure5). Differences were greatest for 1-year-old fish,

averaging roughly 10% for pumpkinseeds and25-30% for golden shiners. In nearly all caseswhere this difference was evident, observedlengths were greater than back-calculatedlengths. For fish older than 1 year, differenceswere much lower and fairly consistent betweenthe two species, averaging roughly 5%. The signsof these differences in older fish were more vari-able; observed lengths were less than or greater

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BACK-CALCULATION OF FISH LENGTH FROM SCALES 895

^E,nc-J

S.5

200150100500

200150100500

200150100500

200150100500

200150100500

Brompton 200 -T Magog

FIGURE 4.-shiners aged

Age (yr)-Mean (+ half the 95% confidence interval) back-calculaied and observed total lengths for golden1-5 in 10 southern Quebec lakes. Details are the same as those for Figure 3.

than back-calculated lengths with nearly equalfrequency.

DiscussionDespite the importance and widespread use of

back-calculation in studies of fish growth, there re-mains an abundance of different approaches in com-mon use, considerable confusion surrounding cer-tain methods, and little agreement on which methodis best (Francis 1990; Ricker 1992; Schramm et al.

1992). In the most thorough review to date, Francis(1990) described in detail all the proportionalmethods and attempted to classify them theoreti-cally based on hypotheses implied in their for-mulation. Francis (1990) concluded that the SPHand BPH methods are preferable to others becauseof their solid theoretical bases, and rejected theFraser-Lee method because of its lack of a clearunderlying hypothesis regarding the body-scalerelationship. Francis also rejected use of GMR on

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BACK-CALCULATION OF FISH LENGTH FROM SCALES 897

theoretical grounds. Francis (1990) went on to la-ment the paucity of validation in back-calculationapplications and used several published data setsto argue the point that using different back-cal-culation methods can result in large differences inestimated lengths. Ricker (1992) strongly sup-ported the use of GMR and the Fraser-Lee method,both largely on theoretical grounds.

We believe our empirical results shed light upontwo important questions regarding back-calcula-tion. The first is, "Does back-calculation estimategrowth history accurately?" Our comparisons ofback-calculated body lengths with observed bodylengths address this question and serve as a vali-dation for the back-calculations. Secondly, "Whichback-calculation method is best?" For the pro-portional methods that we evaluated, our compar-isons tested for differences among back-calcula-tion methods and for correspondence with ob-served body lengths. Previous synthetic reviewsof back-calculation methods (Francis 1990; Ricker1992) focused largely on theoretical analyses ofvarious methods. Strengths and weaknesses in-ferred on theoretical grounds were then illustratedwith data sets exhibiting much more variability(e.g., the mean r2 for body-scale relationships inFrancis' Table 2 is 0.83) than our data sets (Figures1, 2). We do not dispute the theoretical conclusionsof these reviews, nor do we contest the empiricalevidence used to support them. However, we doquestion the source of high variability in these datasets in light of our finding of much tighter body-scale relationships for two species having differentscale types and with data pooled across 10 lakes.Perhaps the older data sets were length-truncatedor otherwise biased toward certain size-classes.Differences in true body-scale relationship vari-ability among species are difficult to separate frompotential variability introduced by sampling bias(Francis 1990; Ricker 1992) or errors in measuringboth the fish (Gutreuter and Krzoska 1994) andscales (Hirschorn and Small 1987; Newman andWeisberg 1987) in different studies. We speculatethat by obtaining a large and unbiased sample ofthe population, taking careful measurements offish body length to the nearest millimeter, usingcurrently available digitizing pad or video imageanalysis technology for accurately and preciselymeasuring scales to the nearest 0.01 millimeter,and averaging measurements from multiple scales,much better body-scale relationships could be ob-tained for most species than have been reported inthe past. We further speculate that this level ofdetail would result in negligible differences among

back-calculation methods for other species, as wehave demonstrated for pumpkinseeds and goldenshiners.

Our results suggest that the Fraser-Lee, SPH, andBPH methods all give equivalent results when basedon body-scale relationships that are linear and havehigh r2 values. Furthermore, the choice of OR orGMR for generating body-scale relationships ap-pears not to affect resulting estimates. Francis(1990) suggested back-calculating lengths by bothSPH and BPH methods and interpreting the differ-ences as rough estimates of back-calculation error.In our study, these differences were typicallyaround 1 mm regardless of which methods werebeing compared. Given all the difficulties in ob-taining unbiased samples and the relative impre-cision inherent in many field measurements, we be-lieve that this level of back-calculation error wouldbe more than acceptable in most applications.

Although our back-calculated body lengths gen-erally corresponded well with observed bodylengths, there were several exceptions that raisequestions about our method of validation. The mostconsistent disagreement was with age-1 fish, forwhich observed lengths were frequently greaterthan back-calculated lengths. This pattern closelyresembles the well known "Lee's phenomenon"(Lee 1920), which is generally attributed to back-calculation error, differential mortality, biased sam-pling, or some combination thereof (Ricker 1969,1992). We doubt that our discrepancies resultedfrom back-calculation error, because they did notoccur in all lakes and they were more prominentfor golden shiners than for pumpkinseeds. We sus-pect that the most likely cause of these discrepan-cies was either missing the appropriate spring sam-pling time of annulus formation for young fish insome lakes or biased sampling in the spring suchthat the smaller age-1 fish were underrepresented.Due to their fusiform bodies, the smallest age-1golden shiners may have escaped through the 6-mmmesh of our seine. Alternatively, differential habitatuse or other behavioral differences may have intro-duced unknown biases into our samples. Becauseof problems such as these, Francis (1990) recom-mended comparing back-calculated lengths withobserved lengths for individual fish as the preferredmethod of validation, although conceding that thiswas far more difficult than validation based on com-parisons among groups.

Despite these potential problems, our back-cal-culated body lengths typically differed from ob-served body lengths only by about 5% for fisholder than age 1, indicating that back-calculation

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reasonably estimates length at age for these spe-cies. Because this type of validation is far morepractical than validation based on individual fishin most field populations, we suggest that it is areasonable approach. However, we agree withFrancis (1990) that obtaining unbiased samples isextremely important when this method is used, thattiming of sampling for observed lengths is critical,and that interpretations must be made with caution.

AcknowledgmentsWe thank Brian Walker, Lesley Pope, and Mich-

el Simard for assistance in both the field and lab-oratory, Carolyn Hunt, John Kalas, HongshengLiao, and Barb Thomas for help digitizing scales,and Kerri Shaneberger and Melissa Veylupek fordrafting figures. Comments from Pat Braaten,Chris Guy, Matt Herbert, Mark Pegg, Dave Wahl,and the Kaskaskia Biological Station fish ecologygroup improved the quality of this paper. Supportwas provided by the Natural Sciences and Engi-neering Research Council of Canada, the Fondspour la Formation de Chercheurs et TAide a laRecherche, and the Donner Foundation. The seniorauthor dedicates this paper to Kenneth D. Carland-er, whose insightful work on the age and growthof fish and years of service to the fisheries pro-fession have been inspirational.

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Received March I I , 1996Accepted July 8. 1996

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