Estimating Fish Body Condition with Quantile Regression

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We used quantile regression to compare the body condition of walleye Sander ... alewives in 1988, walleye body condition improved for all fish at all lengths (the ...
North American Journal of Fisheries Management 28:349–359, 2008 Ó Copyright by the American Fisheries Society 2008 DOI: 10.1577/M07-048.1

[Article]

Estimating Fish Body Condition with Quantile Regression BRIAN S. CADE*

AND

JAMES W. TERRELL

U.S. Geological Survey, Fort Collins Science Center, 2150 Centre Avenue, Building C, Fort Collins, Colorado 80526-8118, USA

MARK T. PORATH Nebraska Game and Parks Commission, Fisheries Division, 2200 North 33rd Street, Lincoln, Nebraska 68503, USA Abstract.—We used quantile regression to compare the body condition of walleye Sander vitreus and white bass Morone chrysops before (1980–1988) and after (1989–2004) the establishment of alewives Alosa pseudoharengus in Lake McConaughy, Nebraska. Higher quantiles (percentiles ¼ 100% 3 quantiles [0, 1]) of weight (W) at the same total length (TL) were indicative of better body condition in an allometric growth model that included separate slopes and intercepts for the before and after groups. All quantiles of walleye weights by TL increased in the years after alewife introduction, ranging from 1.01 to 1.12 times weights in the years before alewife introduction, with greatest increases for the lower (,0.50) quantiles and greater TLs. Quantiles up to 0.25 (the lowest 25th percentiles) of white bass weights were reduced in years after alewife introduction for TLs less than 300 mm, ranging from 0.78 to 0.98 times weights in the years before alewife introduction. However, quantiles greater than or equal to 0.50 (the upper 50th percentiles) of white bass weights increased for all TLs, ranging from 1.01 to 1.06 times the pre-1988 weights. A three-group analysis, which improved the model fit for longer white bass, indicated a reduction (0.80–1.0) in white bass body condition across all TLs in the first 2 years (1989–1990) after alewife introduction, whereas body condition actually improved (1.02–1.12) across all TLs in later years (1991–2004). Thus, after the establishment of alewives in 1988, walleye body condition improved for all fish at all lengths (the greatest improvement occurring among fish in poorer condition), whereas white bass body condition was initially reduced for all fish at all lengths for 2 years and improved in subsequent years. The approach that we developed for comparing fish body condition before and after a management action in Lake McConaughy could be applied to other weight–length data sets typically evaluated with relative weight indices.

Body condition frequently is used to assess the health or physiological well-being of fish populations as an indirect measure of the effects of various biotic and abiotic factors, including various fisheries management actions (Murphy et al. 1991; Blackwell et al. 2000). For example, comparing fish body condition has been used to assess changes in prey availability (Anderson 1990; Murphy et al. 1990, 1991; Porath and Peters 1997). Porath et al. (2003) suggested that changes in body condition might more reasonably be interpreted as a measure of predation success. Fish of greater weight at the same length are considered to have better body condition and thus greater utilization of prey. Typically, comparisons are actually made with relative weight (Wr), a condition index, where Wr ¼ (W 4 Ws) 3 100 and Ws (standard weight) is from a standard weight equation for a given species (Wege and Anderson 1978). Standard weight equations are commonly based on the 75th percentile of allometric * Corresponding author: [email protected] Received March 16, 2007; accepted June 21, 2007 Published online March 6, 2008

growth relationships, W ¼ b0 TLb1 (TL ¼ total length), estimated among some selected number of populations of a fish species (e.g., Murphy et al. 1990; Brown and Murphy 1991). The regression-line-percentile (RLP) method commonly is used to estimate standard weight equations to provide Ws (Murphy et al. 1991; Blackwell et al. 2000), but other estimation methods (Gerow et al. 2005; Lee and Sampson 2005) also could be used. The essence of the relative weight approach is that differences in weight at different lengths are accounted for by a single standard weight equation for a given species. Numerous statistical concerns have been raised about estimating the equations for Ws, determining the length-related biases associated with comparisons of relative weights (Wrs) derived from standard weight equations, and interpreting relative weight as if it actually measured body condition (Cone 1989, 1990; Brenden et al. 2003; Gerow et al. 2004, 2005; Hansen and Nate 2005). There is nothing particularly special about using the 75th percentile of a weight–length relationship as the standard for computing relative weights (Cone 1989; Springer and Murphy 1990; Murphy et al. 1991). The 75th percentile is a useful

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standard indicating plumper than average fish (Murphy et al. 1991; Blackwell et al. 2000) although other percentiles are equally reasonable. We demonstrate that estimating all percentiles, or more generally quantiles indexed by s 2 [0, 1] (e.g., s ¼ 0.90 is the 90th percentile), of fish weight as a function of total length, QW(sjTL), provides sufficient information to comprehensively examine and compare changes in fish body condition based solely on the previously stated concept; that is, fish with greater weight at the same length are in better body condition. Higher to lower quantiles of weight at a given length provide an ordering of higher to lower body condition for evaluating effects of management actions on individual fish populations. Quantile regression (Koenker and Bassett 1978; Koenker 2005) was used to obtain estimates of the quantiles of weight as an allometric function of length. The standard weight equations were thus not required to compare body condition among relevant groups of fish within a population. Differences were expressed in readily interpreted multiples associated with various percentiles of weight estimated from an allometric growth model, an approach consistent with previous recommendations to compare weight– length regression models among groups with analysis of covariance (ANCOVA) (Carlander 1969:14; Cone 1989, 1990). The choice of relevant groups (or populations) of fish to compare depends on the questions being asked by managers and researchers. In this study we compared walleye Sander vitreus and white bass Morone chrysops body condition in the years before (1980–1988) and after (1989–2004) the establishment of alewives Alosa pseudoharengus in Lake McConaughy, Nebraska, by comparing quantiles of fish weight as a function of length. Our analyses were made to estimate differential effects of prey addition on local populations of walleyes and white bass, expanding on the relative weight comparisons made by Porath et al. (2003), using nine additional years of post-introduction data. Our approach for evaluating changes in fish body condition is similar to methods now routinely used to evaluate human health and economic status by comparing human growth characteristics based on estimates of percentiles or quantiles from regression models (e.g., Cole 1988; Wei et al. 2006). Methods Study area and field sampling.—Details of the study area and sampling of fish are provided in Porath et al. (2003). Briefly, walleyes and white bass are the predominant piscivorous sport fishes pursued by anglers on Lake McConaughy, a 13,770-ha irrigation reservoir on the North Platte River, Nebraska. Adult

alewives introduced into Lake McConaughy from 1986 to 1988 rapidly reproduced and became the most abundant prey species. Alewives were the dominant item in the diet of walleyes captured in 1995 (Porath and Peters 1997). Fish populations were surveyed in Lake McConaughy annually from 1980 to 2004 by gill netting. Captured fish were measured to the nearest millimeter (TL) and weighed to the nearest gram. Statistical model and analysis.—We used quantile regression to obtain estimates of the quantiles of weight as a function of length because it does not require assuming a parametric form for the error distribution (Koenker and Bassett 1978; Koenker 2005). Cade and Noon (2003) provide a primer on quantile regression for biologists, and Terrell et al. (1996), Dunham et al. (2002), and Zoellick and Cade (2006) provide example applications of quantile regression for modeling fish habitat quality. The essence of quantile regression is that the usual single-mean function estimated in a linear or nonlinear model is replaced with a family of functions across all or a selected subset of quantiles on the interval [0, 1]. This provides a comprehensive view of how all parts of the distribution (from the center to the extremes) of the response (weight) change conditional on the predictor variables (TL and grouping variables). The mean function of conventional regression models for p predictors, E(yjX) ¼ b0 þ b1X1 þ    þ bpXp þ e, with l(e) ¼ 0, can be regarded as an average across all the quantile functions, Qy(sjX) ¼ b0(s) þ b1(s)X1 þ    þ bp(s)Xp, with 0  s 1 (Bassett et al. 2002). The term for the error distribution, e, in the mean regression does not appear in the quantile regression model as all the random variation in the probability distributions is accounted for by variation in the quantiles indexed by s. If the quantile functions have heterogeneous slopes, i.e., b1(s) 6¼ b1, . . . , bp(s) 6¼ bp, as must be true any time the probability distributions are not independent and identically distributed (e.g., variances changing as a function of the predictors), then the quantiles contain information that has been lost in the mean function, which is an average across different effects from lower to higher quantiles. Heterogeneity of variances that are a concern when comparing weight–length relations among groups with least-squares regression estimates of means (Brenden et al. 2003) are naturally accommodated by quantile regression. Quantile regression can provide consistent estimates for a wide variety of distributional forms by directly modeling all the quantiles of the probability distribution and does not rely on approximating the probability distribution by estimating a few moments of an assumed parametric form (e.g., l and r2 for a normal distribution). The estimated quantiles are the inverse of

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the empirical cumulative distribution functions, which can be considered nonparametric maximum likelihood estimators (Wasserman 2006). Estimates for quantile regression can be obtained by minimizing an asymmetrically weighted loss function of absolute deviations (the method we used), by maximum likelihood estimation assuming an asymmetric Laplace distribution, and by Bayesian procedures based on the aforementioned likelihood (Koenker and Machado 1999; Yu et al. 2003; Koenker 2005). Allometric models of weight conditional on TL were estimated separately for walleyes and white bass. Following Porath et al. (2003), we considered fish measured during 1980–1988 as pre–alewife introduction and those measured during 1989–2004 as post– alewife introduction. By including an indicator variable for these two groups of fish and the interaction of this indicator variable with TL, our regression model allowed both the intercept and exponent parameters of the allometric growth model to differ between fish captured in the years before and after alewife introduction. This is an expanded ANCOVA-like model where the assumption of equal slopes among groups is not made, but instead separate slopes are explicitly estimated. The following allometric model is nonlinear in the parameters, where the parameters b2 and b3 are multiplicative differences between the intercept and exponent for fish in the years after alewives relative to the intercept (b0) and exponent (b1) for fish in the years before alewife introduction: QW ðsjTL; IÞ ¼ b0 ðsÞ  TLb1 ðsÞ  10b2 ðsÞI  TLb3 ðsÞI ; ð1Þ where W is weight in grams, TL is total length in mm, and I is an indicator variable taking the value 0 for years before and the value 1 for years after alewife introduction. So the terms b0 ðsÞTLb1 ðsÞ describe the quantiles of weight as functions of length for fish captured in years before alewife introduction (I ¼ 0), and 10b2 ðsÞ TLb3 ðsÞ describe the quantiles of the ratio of post– to pre–alewife introduction weights at given lengths for fish captured in the years after alewife introduction (I ¼ 1). We graphically explored the lack of fit to the allometric model for the two groupings of fish by examining how fish weights were distributed among the estimated quantiles by length across years. There was evidence of lack of fit for white bass with TL greater than 300 mm in the post–alewife introduction years. Model fit was improved by including two indicator variables and their interactions with TL, one indicator for the differences between the periods before (1980–1988) and 1989–1990 after alewife introduction and another indicator for the differences between the

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preintroduction period and 1991–2004. This sixparameter model allowed for separate slopes and intercepts for white bass in 1980–1988, 1989–1990, and 1991–2004. We present results of this white bass model to complement analyses of the two-group model that corresponds to comparisons originally made by Porath et al. (2003). Although there is a nonlinear quantile regression estimator (Koenker and Park 1996) that could be used to estimate equation (1) directly, inferential methods for linear quantile regression (Koenker 2005) currently are better developed. Thus, we estimated equation (1) by making it linear via log10 transformation, as is commonly done in fisheries applications (Murphy et al. 1990). The resulting equation is Qlog10 W ðsjTL; I Þ ¼ log10 b0 ðsÞ þ b1 ðsÞlog10 TL þ b2 ðsÞI þ b3 ðsÞIlog10 TL:

ð2Þ

This quantile regression formulation of the linear model allows a wide variety of homogeneous and heterogeneous patterns of weight as a function of length to be estimated. Furthermore, back-transforming to the nonlinear allometric form provides valid quantile estimates in the original scale because the quantile estimators are equivariant to linear or nonlinear monotonic transformations of the dependent variable, that is, Qlog10 W ðsÞ ¼ log10QW(s), something that is not true for the mean regression function, which is biased by back-transformation from nonlinear transformations like the logarithmic transformation (Koenker 2005). The virtue of the quantile regression approach is that the quantiles of weight given TL provide a detailed description of the variation in fish body condition (higher quantiles represent better body condition) in a statistical model that does not assume that changes in weight will be homogeneous across lengths. We used the linear quantile regression function, rq(), available in the Quantreg package for the ‘‘R’’ statistical environment (cran.r-project.org/) to obtain model estimates and used quantile rank score tests to estimate confidence intervals or test hypotheses on parameters. Weighting of the rank scores based on differences within a small interval of quantiles around a selected quantile (local bandwidth) were used to account for heterogeneity in the rank score based confidence interval estimates (Koenker and Machado 1999). Cade et al. (2006) provided a comprehensive evaluation of the quantile rank score test, indicating that valid inferences would easily be obtained from the standard distributional approximations of the rank score tests for s 2 [0.05, 0.95] for our sample sizes of walleyes (n ¼ 6,857) and white bass (n ¼ 2,017) in models with four or six parameters. Cade et al. (2006)

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found that when effective ranks, where effective rank ¼ [(1 s) 3 n]/p for s  0.50 or (s 3 n)/p for s , 0.50, were greater than or equal to 4 that the Chi-square distributional approximation of the quantile rank score test maintained valid coverage levels. Our effective ranks were greater than 16. Because of the large sample sizes, we estimated quantiles by increments of 0.01 from 0.05 to 0.95 (91 estimates) rather than obtaining all possible estimates on the interval s 2 f0.05, 0.95g, which exceeded 16,000 estimates even for the white bass with n ¼ 2,017. Graphs of parameter estimates and 95% confidence intervals were made for s ¼ f0.05, 0.06, . . . , 0.94, 0.95g and estimated functions were graphed for a subset s ¼ f0.05, 0.25, 0.50, 0.75, 0.95g to provide a comprehensive description of changes in quantiles of weight as a function of length. The confidence intervals provide their stated coverage level pointwise by quantile. We also estimated the differences in the predicted quantiles of weight between years before and after alewife introduction and their associated intervals at selected values of TL to provide intervals on the ratio of the differences in the quantiles of weight that were likely to be of interest for fisheries management and to aid interpreting model results. This was done by recentering the intercept of TL by division [log10TLc ¼ log10(TL 4 selected value) ¼ log10TL  log10(selected value)], similar to procedures described in Cade et al. (2005). The confidence intervals provided for b2(s) in b ðsÞ the term 10b2 ðsÞ TLc 3 on such recentered data are then prediction intervals for the ratio of post-introduction weight to pre-introduction weight given TL equal to a specified value, namely, TLc ¼ 1. We provided these predicted quantiles and intervals of weight for TL ¼ f250, 315, 380, 445, 510, 570 mmg for walleyes and TL ¼ f150, 190, 230, 265, 300, 340 mmg for white bass, corresponding to lower, mid, and upper points of TL intervals used to categorize fish as stock- to qualitylength, quality- to preferred-length, and preferred- to memorable-length (Gabelhouse 1984) as in Porath et al. (2003). Upper points for preferred-to-memorable length categories for walleyes (630 mm) and white bass (380 mm) were excluded to facilitate graphical presentation. Example command scripts for conducting our quantile regression analyses in the Quantreg package for ‘‘R’’ are provided in an appendix that appears in the online version of this article only. Results Our allometric quantile regression models provided good fits to the walleye weight–length relationships (Figure 1), but there was some evidence of lack of fit for the allometric model for white bass at TLs greater than 300 mm in the post–alewife introduction years

(Figure 2) for the two-group comparison. The magnitude of the outlying values for the white bass (e.g., low weights at TL . 300 mm) captured from 1980 to 1988 had minimal effect on the quantile regression estimates as the method is relatively insensitive to outliers in the dependent variable (Cade et al. 1999; Koenker 2005). However, scatterplots of weight and length by year indicated that weights at TLs greater than 300 mm had different relationships for white bass captured in 1989– 1990 and 1991–2004. For the purposes of this example we will first discuss the two-group allometric models for white bass and walleyes as they are consistent with the comparisons made by Porath et al. (2003). We will then discuss an alternative three-group model for white bass that had improved model fit for longer fish. The parameter estimates indicated substantial differences in the allometric relationships between the years before and after alewife introduction for the two species. The estimated exponents b1(s) for fish captured in years before alewife introduction varied from approximately 3.30 at lower quantiles to 3.20 at higher quantiles for walleyes (Figure 1) and varied from approximately 3.20 at lower quantiles to 3.10 at higher quantiles for white bass (Figure 2). Heterogeneity of body form across the quantiles was evident because the bands of parameter estimates and their confidence intervals for the exponents b1(s) had a continuous decline in estimates from lower to higher quantiles for the walleyes (Figure 1) and declines in estimates for the white bass for quantiles greater than 0.80 (Figure 2). Homogeneity of body form would have been indicated by flat horizontal bands of estimates and their confidence intervals for the exponents b1(s) across quantiles. The multiplicative differences between the pre– and post–alewife introduction years, b2(s) and b3(s), varied more from lower to higher quantiles for white bass (Figure 2) than for walleyes (Figure 1), even switching signs. Again, changes in allometric equations from before to after alewife introduction were not constant across quantiles for either species as indicated by estimates and confidence intervals for b2(s) and b3(s) that were not flat horizontal bands (Figures 1 and 2). Although 95% confidence intervals for b2(s) and b3(s) often overlapped zero for white bass (Figure 2), rank score tests on the joint effect of these two parameters indicated nonzero differences (F . 4.49; df ¼ 2, 2015; P , 0.01) between the weight–length relationships before and after alewife introduction for all but the 0.25 quantile (F ¼ 1.631; df ¼ 2, 2015; P ¼ 0.20). The joint effect of these two parameters differed (F . 4.60; df ¼ 2, 6855; P , 0.01) for all quantiles for walleyes. Estimates for walleye in the post–alewife introduction years were higher across all quantiles and TLs than for years

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FIGURE 1.—The upper panel shows the relationships between weight and total length (TL) for 6,857 walleyes captured in 1980–1988 (squares) and 1989–2004 (circles). The estimated equations (from the highest to the lowest quantiles) are as follows: ˆ (0.75jTL, I) ¼ 105.59 TL3.23100.14I TL0.06I; Q ˆ (0.50jTL, I) ¼ 105.71 ˆ (0.95jTL, I) ¼ 105.42 TL3.18100.12I TL0.05I; Q Q W W W 3.26 0.11I 0.05I ˆ 5.81 3.29 0.11I 0.05I 5.95 ˆ TL 10 TL ; QW(0.25jTL, I) ¼ 10 TL 10 TL ; and QW(0.05jTL, I) ¼ 10 TL3.32100.16I TL0.07I. Values were back-transformed from the linear model (equation 2), which provides separate slopes and intercepts for the years 1980–1988 (before alewife introduction; dashed lines) and 1989–2004 (after alewife introduction; solid lines). The four lower panels show the linear quantile regression estimates and 95% confidence intervals (gray bands) for s 2 f0.05, 0.06, . . . , 0.94, 0.95g, where log10b0(s) and b1(s) are the intercept and slope, respectively, for fish captured in 1980–1988 (I ¼ 0) and b2(s) and b3(s) are the differences in intercept and slope for fish captured in 1989–2004 (I ¼ 1).

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FIGURE 2.—The upper panel shows the relationships between weight and total length (TL) for n ¼ 2,017 white bass captured in 1980–1988 (squares) and 1989–2004 (circles). The estimated equations (from the highest to the lowest quantiles) are as follows: ˆ (0.75jTL, I) ¼ 105.26 TL3.17100.06I TL0.01I; Q ˆ (0.50jTL, I) ¼ 105.35 ˆ (0.95jTL, I) ¼ 105.06 TL3.10100.18I TL0.07I; Q Q W W W ˆ W(0.25jTL, I) ¼ 105.44 TL3.21100.29I TL0.12I; and Q ˆ (0.05jTL, I) ¼ 105.39 TL3.17100.79I TL0.31I. TL3.19100.07I TL0.04I; Q W Values were back-transformed from the linear model (equation 2), which provides separate slopes and intercepts for years 1980– 1988 (before alewife introduction; dashed lines) and 1989–2004 (after alewife introduction; solid lines). The four lower panels show the linear quantile regression estimates and 95% confidence intervals (gray bands) for s 2 f0.05, 0.06, . . . , 0.94, 0.95g where the terms are defined as in Figure 1.

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FIGURE 3.—Quantile regression estimates and 95% confidence intervals (gray bands) for s 2 f0.05, 0.06, . . . , 0.94, 0.95g of the predicted weights of 6,857 walleyes as a ratio of after (1989–2004) to before (1980–1988) the introduction of alewives, conditional on total length, in the allometric model (equation 1).

before alewife introduction (Figure 1), but the pattern of differences was more complicated for white bass (Figure 2). The effects of the parameter differences between the walleye and white bass models are best summarized in this multiplicative model by the ratio of post– to pre– alewife introduction weights predicted by quantiles for selected TLs. These ratios for walleyes exceeded 1.0 for nearly all quantiles at all TLs, ranging from 1.01 to 1.05 at TL equal to 250 mm to 1.05–1.10 for TL equal to 570 mm, with higher ratios for lower quantiles (Figure 3). This indicates that body condition improved the most for poorer condition walleyes although body condition improved for all walleyes at all TLs after the establishment of alewives. Contrast this with the pattern of estimates for white bass, where the ratios of post– to pre–alewife introduction weights were less than 1.0 for quantiles less than 0.25 (the lower 25th percentiles) for white bass having TLs less than 300 mm and were as low as 0.80 at the lowest quantiles at the shortest TLs (Figure 4). The ratios were not different from 1.0 for quantiles less than 0.25 for white bass having TLs greater than or equal to 300 mm. The ratios were greater than 1.0 for quantiles greater than or equal to 0.50 (the upper 50th percentiles) for white bass with TLs greater than 150 mm, ranging as high as 1.05 (Figure 4). This indicates that body condition actually

became worse for poorer condition white bass of shorter lengths after the establishment of alewives, whereas body condition improved for average to better condition white bass similarly across all TLs, but not as much as it improved for walleyes. Interpretation of body condition changes in white bass from before to after alewife introduction was complicated by the lack of fit we detected for the fish captured after alewife introduction. This lack of fit was due in part to forcing common allometric relationships across all years after alewife introduction. The quantiles of the allometric relationship of white bass weight and length for 1991–2004 were greater than those for 1989–1990 (F  3.124; df ¼ 2, 298; P  0.045). We, thus, estimated a three-group (1980–1988, 1989–1990, and 1991–2004) model for white bass that had less lack of fit. The predicted ratios of the weights in 1989–1990 to those in 1980–1988 as a function of TL ranged from 0.80 at the lower quantiles to 1.0 at the higher quantiles for shorter TLs (230 mm) but fell within the narrow range 0.88–0.90 across all quantiles at higher TLs (265 mm). The predicted ratios of the weights in 1991–2004 to those in 1980–1988 ranged from 1.02 at the lower quantiles to 1.10 at the higher quantiles for shorter TLs and from 1.12 at the lower quantiles to 1.05 at the higher quantiles for higher TLs. Thus, it appears that white bass of all lengths had

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FIGURE 4.—Quantile regression estimates and 95% confidence intervals (gray bands) for s 2 f0.05, 0.06, . . . , 0.94, 0.95g of the predicted weights of 2,017 white bass as a ratio of after (1989–2004) to before (1980–1988) the introduction of alewives, conditional on total length, in the allometric model (equation 2).

reduced body condition in the first 2 years after alewives were established and in subsequent years actually had improved body condition. Discussion Our comparisons, made with a range of quantiles, clearly support the interpretation that the body condition of walleyes improved with a 5–10% increase (corresponding to ratios of 1.05–1.10) in weight at TLs greater than 250 mm after the establishment of alewives in Lake McConaughy, the greatest increases occurring in the lower quantiles of weight, which were associated with poorer condition. Improvements in the body condition of the poorer condition walleyes would seem to be one of the most desirable management outcomes indicating walleyes were successfully taking advantage of the augmented prey availability. That body condition increased for all quantiles is additional evidence that alewives were a prey providing greater net energy to walleyes in Lake McConaughy than other prey species such as gizzard shad Dorosoma cepedianum, threadfin shad D. petenense, emerald shiner Notropis atherinoides, and rainbow smelt Osmerus mordax. Our quantile regression analyses of walleye body condition were consistent with the direction of the effects estimated in the relative weight analyses of Porath et al. (2003) who found greatest increases in

mean relative weight (preintroduction ¼ 85, postintroduction ¼ 98) after the establishment of alewives for preferred- to memorable-length (TL ¼ 510–630 mm) walleyes, with significant increases at shorter lengths. However, proportional increases in quantiles of weight at a given TL, provided by keeping group differences as part of the allometric model, are a more readily interpreted measure of improvement in body condition than are additive changes in relative weight, an index of body condition based on transforming weight by a multiplicative standard weight equation. A difference in mean relative weight of 13 (i.e., 98  85) does not directly provide information on increases in walleye weight at a given length because relative weight is a unitless index and weight–length relations are multiplicative. Our estimates that 90% (0.05–0.95 quantiles) of the walleyes had a 5–10% increase in weight at length convey changes relative to the original units used to quantify weight. In addition, our quantile regression analyses provide a description of the range of body conditions most affected by a management action. In our example, weight of poorer condition walleyes (the lower quantiles) improved the most (10% increase in weight at length). Our quantile regression analyses of white bass body condition provided a detailed description of changes in body condition that were somewhat similar to the

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relative weight analyses of Porath et al. (2003). Those authors found negative effects on relative weight associated with the establishment of alewives for stock- to quality-length (TL ¼ 150–230 mm) and preferred- to memorable-length (TL . 300 mm) categories and positive effects on quality- to preferred-lengths (TL ¼ 230–300 mm). Our two-group quantile regression analyses indicated 1–20% decreases for the lower 25th percentiles of white bass weight at TLs less than 300 mm and 1–5% increases in the upper 50th percentiles of weight at TLs greater than 150 mm. Our comparisons indicated neutral to improved body condition for white bass with TLs greater than 300 mm (preferred to memorable lengths) after the establishment of alewives, in contrast to the decreased mean relative weights found by Porath et al. (2003). However, our three-group analysis had better model fit for white bass with TLs greater than 300 mm and indicated a reduction in white bass body condition across all TLs in the first 2 years (1989–1990) after alewife introduction, and improved body condition across all TLs in later years (1991–2004). White bass may have been competing with alewives for zooplankton and invertebrate food sources more than feeding on the alewives initially (Porath et al. 2003). Competition with alewives may have been reduced in later years as the white bass population declined dramatically within Lake McConaughy in the years following the establishment of alewives (Porath et al. 2003), allowing larger, better condition white bass to actually benefit from alewives as prey items. We limited our comparisons of body condition with quantile regression estimates of weight as a function of length to those that directly addressed the original management question addressed by Porath et al. (2003): Have white bass and walleye body condition changed in a multiyear period after alewife introduction? A more complex model that would allow separate weight–length relations among each of the 25 years surveyed (requiring a model with p ¼ 50 parameters) could be estimated by including the necessary indicator variables if this was required for detecting effects of various management actions. Essentially, any predictor variables that might be incorporated into a regression model might reasonably be included as a basis for comparing differences in body condition defined by the quantiles of weight as a function of length. However, sample sizes have to be sufficient within each group to get reasonable estimates at all quantiles selected. For example, fewer than 100 walleyes were captured in some years that Lake McConaughy was surveyed, so it would be difficult to obtain precise estimates for 0.05–0.95 quantiles across all years although reasonable estimates might still be obtained for a shorter

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interval of quantiles, such as 0.25–0.75. Calculations of effective ranks, as in Cade et al. (2006), can be used to provide some guidance on minimum sample sizes required to maintain valid rank-score-based confidence intervals for different quantiles in models with a specified numbers of predictor variables. While confidence interval coverage may be correct, interval lengths may still be long indicating poor precision of estimates when the number of model parameters p becomes large relative to sample size n for a given quantile s. The number of white bass captured in 1991–2004 (n ¼ 191) and 1989–1990 (n ¼ 109) were an order of magnitude less than in 1980–1988 (n ¼ 1,908), but were sufficient to permit reasonably precise estimates for 0.05–0.95 quantiles (effective ranks . 16) for the six-parameter model, allowing separate slopes and intercepts to be determined for the three time periods. Nonlinear forms that are more flexible (e.g., generalized additive models using b-splines) than the multiplicative allometric model also are available for estimating quantiles of weight as a function of length. Although we did not explore those options here, it is important to recognize that quantile regression estimates can be obtained for many variants of linear and nonlinear models (Koenker 2005). We also note that the quantiles of weight as a function of length could be estimated in fully parametric linear or nonlinear regression models based on complex transformations to parametric error distributions (Rigby and Stasinopoulos 2005), although we do not recommend this approach. Unlike estimates for the mean, estimates for the quantiles are extremely sensitive to assumptions about the parametric form of an error distribution (Cade and Noon 2003; Koenker 2005). The use of the quantiles of weight given length as a basis for directly comparing the body condition of two groups of fish bypasses the use of a standard weight equation as a proxy for comparing the fish populations of interest before and after a specific management action. Using the standard weight equation to calculate relative weights amounted to transforming weights at a given length in each group (before and after alewife introduction) by a constant value and, thus, had no effect on the estimated differences between groups as a function of length. Because relative weights were not used to make our comparisons, we avoided the statistical complexities required to account for the sampling variation of relative weights that incorporates sampling variation in the standardized weight equations (e.g., Brenden et al. 2003; Brenden and Murphy 2006). Other important departures between our statistical analyses and those of Porath et al. (2003) were that we estimated group differences in a multiplicative

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rather than an additive model and used TL as a continuous rather than as a categorical predictor (e.g., stock- to quality-length). Our estimated exponents for walleye weight–length relationships for quantiles less than 0.90 were significantly higher than the 3.180 estimated by Murphy et al. (1990) that Porath et al. (2003) used to compute relative weights. Similarly, the estimated exponents for white bass weight–length relations for quantiles less than 0.80 exceeded and statistically differed from the estimate of 3.081 in the standard weight equation for white bass (Murphy et al. 1990). Cone (1989, 1990) previously noted problems with comparing relative weights based on assuming a common exponent for the allometric weight–length relationships when that assumption was not reasonable, i.e., lack of constant body form. Our quantile regression estimates indicate that not only were there substantial differences in exponents among populations (Lake McConaughy fish differing from those used to compute the standard weight equation), but also there was considerable heterogeneity in exponents among the quantiles within a fish population, as indicated by the estimates for b1(s)  b3(s) that were not constant across quantiles (Figures 1, 2). As the exponents for most of the quantiles of the walleye and white bass weight–length relationships at Lake McConaughy were not similar to the exponents given by the published standard weight equations for the species, dividing weights of Lake McConaughy fish by standard weights from these equations to compute relative weights only confounded interpretations of changes in body condition, as described by Cone (1990), because body form was not constant. Our comparisons of predator body condition benefited from the large sample sizes of walleyes and white bass collected from Lake McConaughy over multiple years. Similar benefits would accrue to characterizing body condition of other fish populations sampled intensively as part of research projects. Management biologists evaluating fish populations in a single water body may have insufficient sample sizes to effectively develop an extensive quantile analysis of body condition and typically evaluate body condition indices by calculating relative weight for individual fish from a standard weight equation developed from multiple populations of the fish species of interest. If the same data sets used to develop the standard weight equation for a species were appropriately pooled and weighted (e.g., to give equal weights to each population regardless of sample size), a family of reference quantile equations could be developed and used in the same way as the single standard weight equation. With such an approach an individual new fish of given TL and W could be referenced to a specific quantile s

for the QW(sjTL) from the reference quantile equations. For example, if we used our post–alewife introduction quantile regressions for walleyes as reference equations and we had measurements for a new walleye with W equal to 2,600 g and TL equal to 600 mm, we would determine that the fish is close to the 91st percentile (excellent body condition) because the predicted QW(0.91jTL ¼ 600 mm) equals 2,602 g and the predicted QW(0.90jTL ¼ 600 mm) equals 2,586 g. It would be possible to average, estimate intervals, or otherwise summarize the distribution of the quantile estimates at specific TLs across any grouping of length–weight data by populations to provide reference values relevant to the management question of interest. A family of reference quantile equations could be made available digitally or standard tables of predicted quantiles (e.g., 0.05–0.95) by length (e.g., by 1-cm increments) could be published for use by management agencies. One needs to think very clearly about the management decisions that will be based on changes in fish condition before deciding on the appropriate groupings of fish weight–length relationships to model. Our analysis of Lake McConaughy predatory fish emphasized differences in time. However, indicator variables can be incorporated in quantile regression models of weight on length to evaluate differences either in time or space, or both, depending on the management questions of interest. If many indicator variables are required, it may be possible to obtain smoothed shrinkage estimates for them in a random-effects model for quantile regression (Koenker 2004). Additional predictor variables related to environmental and climatic factors or management actions also might be incorporated in models to evaluate the effects of multiple factors on the relationship between fish weight and length. Acknowledgments We thank Travis Brenden, Kevin Pope, Joan Trial, Douglas Watkinson, and two anonymous referees for constructive comments on drafts of this manuscript. The fish weight–length data were collected in studies funded by the Nebraska Game and Parks Commission under the Federal Aid in Sport Fish Restoration program. References Anderson, R. O. 1990. Comments: Properties of relative weight and other condition indices. Transactions of the American Fisheries Society 119:1051–1052. Bassett, G. W., Jr., M.-Y. S. Tam, and K. Knight. 2002. Quantile models and estimators for data analysis. Metrika 55:17–26.

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