Growth Data: Size Matters Too

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Feb 5, 2018 - evaluated with 4 sampling strategies: random, age-stratified, size-stratified, and random supplemented with additional sampling of large fish.
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Growth Data: Size Matters Too C. Phillip Goodyear 1214 N Lakeshore Drive Niceville, FL 32578. E-mail: [email protected] Abstract Biologically meaningful growth-model parameter estimates require size-age data that are representative of true mean size at each age. That requirement may be more difficult than aging individual fish. The topic was evaluated with 4 sampling strategies: random, age-stratified, size-stratified, and random supplemented with additional sampling of large fish. The size-age data were simulated using published von Bertalanffy (VB) parameters for striped marlin (Kajikia audax) at two levels of mortality and two magnitudes of variability of size at age. Size stratified samples biased estimates of length at age and fitted VB parameters even when only 10% of the samples were non-randomly selected. The magnitude of error was 88% – >250% for L∞ and 50% to 85% for k. Error was not lessened when a stratified subset of large fish accounted for only 10% of an otherwise random sample. Natural mortality (M) predicted with the VB data varied 5.5-fold and could lead to widely disparate views of stock productivity and management policies. Such error potentially corrupts reliability of any analysis that depends on the VB estimates. Potentially non-representative sampling should be routinely described when VB models are first published so investigators requiring biologically meaningful growth can address consequences the additional uncertainty in derivative studies.

INTRODUCTION Growth and natural mortality are essential to understand species ecology and guide vital aspects of fisheries management. Growth is commonly described by a von Bertalanffy (VB) growth curve fitted to observations of size and age, and the model parameters are widely used in ecological and fisheries management models. Current methods are described in a recent AFS monograph (Quist and Isermann 2017). The assignment of age usually requires an analysis of a hard part (scale, otolith, fin ray) collected from an individual. Because of uncertainty in the aging, growth studies usually devote significant effort to describe the method and quantify potential error. In contrast, sizes are measured with high accuracy and precision. However, the important measure of size is not the size of the individual but how well the mean size at age can be estimated (Chih 2009b; Gwinn et al 2010). Growth studies sometimes overlook describing how well sampling methods met these requirements. (e.g., Chen et al. 2012; Dunton et al. 2016) Satisfying the length requirement with sampling is often a more difficult task than estimating age. The deleterious effects of bias from size-selective sampling and fishery induced size-selective mortality have been discussed for more than a century (e.g., Lee 1912; Ricker 1969; Gwinn et al. 2010). Bias occurs when the samples were collected with size-selective gears (Gwinn et al. 2010), stratified by size (Goodyear 1995; Bettoli and Miranda 2001), or when the data exhibit Lee’s phenomenon from a history of population exposure to size-dependent mortality

(Ricker 1969). Because growth causes fish to be larger on average and scarcer as they age, investigators sometimes attempt to include presumably older fish by targeting larger individuals for inclusion in the samples or make size-stratification part of the experimental design (Potts et al. 1998; Hwang and Kim 2008; O’Malley et al. 2017). This strategy is distinct from that recommended by Miranda and Colvin (2017) which estimates size-age composition of a sample by expanding size-stratified subsamples using an age-length key (ALK, Ketchen 1950; Kimura 1977). The properties of the ALK for this purpose are well studied (e.g., Bettoli and Miranda 2001; Chih 2009a; 3009b; Coggins et al. 2013). The method removes the size stratification bias of the subsamples while estimating the size-age composition of the larger sample. However, the expanded data retain any size bias imposed by gear-selectivity or other causes that affected the original larger sample. Although the ALK expansion of length-stratified samples is a preferred method, it requires special attention to the sampling protocol (Miranda and Colvin 2017). Many growth studies that employ size stratification in their sampling design fit the VB equation to the stratified size-age data without further treatment (e.g., Potts et al. 1998; Hwang and Kim 2008; Marshall et al. 2009; GarcíaContreras et al. 2009; Siddons et al. 2016; O’Malley et al. 2017). These factors lead to bias and error in VB parameters that potentially affect every estimate in the published literature to some degree. Assuring that samples are representative of the distribution about the true mean size at age is a daunting task. It is effectively

impossible if Lee’s phenomenon is present since part of the size distribution has not survived to be included in the samples (Ricker 1969). These considerations become important when the growth parameter estimates are later used by investigators needing VB data for subsequent studies. Quite apart from a need to describe species growth, a practical consequence of accurate estimation of growth relates to natural mortality (M). Stock productivity and hence permissible harvests are strongly affected by M, but it is notoriously difficult estimate directly. The ubiquity of data from VB models has provided the basis for modeling M as a function of growth and conversely to calculate M for species where it cannot be estimated using parameters from a growth model (e.g., Pauly 1980; Griffiths and Harrod 2007). Investigators with experience developing or applying fish population models relevant to fisheries management will inevitably encounter VB-based values of M which were adopted in lieu of direct estimates because a value for M was required. A good example is the study of the potential benefits of catch and release that was the source of the VB parameter values used in the current paper (Pine et al. 2008). Once used, the estimate of M gains respectability, and both the method and value are reinforced regardless of potential flaws in rationale or source data. In my experience, a discussion of the accuracy of the original VB parameter estimates in a secondary study that employs them for another purpose would be out of the norm. The present study documents the sensitivity of M to the error in VP parameters estimated from data collected with different sampling protocols. The analysis contrasts random, age-stratified, length-stratified, and mixed random-length stratified sampling schemes using simple simulated data based on published growth for striped marlin, Kajikia audax (Pine et al. 2008). The magnitude of the effect is a strong caution about the reliability of values from studies where the sampling requirements may not have been met or were not sufficiently documented.

abundances at the age anniversary dates. The numbers at each age are apportioned to lengths, Na,l using a normal probability distribution with a specified coefficient of variation (CV) which was assumed constant for each age. For simplicity, variation of numbers and lengths between ages was not included. The important variables that influence the distribution of Na,l are the mean size at age, CV and total mortality. For convenience, I adopted a striped marlin growth model from a published simulation study (Pine et al. 2008). Mean size at age correspond to a VB model with growth coefficients: L∞=221 cm, k = 0.23, t0 = -1.6, and CV of sizes about the mean = 0.1. These VB parameters are the “true” values for purposes of this exercise. Similarly, the value of natural mortality, M=0.38 computed from Pauly’s equation would be the “true” value for the purpose of discussion. Total mortality was arbitrarily set to either Z = 0.38 for natural mortality alone, or Z = 0.57 for total mortality at the fishing mortality at maximum sustainable yield (MSY, FMSY). These values are the same used in Pine et al. (2008). The Na,l matrix was evaluated for ages 1-11, at 1 cm intervals of length to 400 cm with CV= 0.1 (Pine et al. 2008). Analyses were also done with CV=0.2 to study the result of more variable growth which increases the overlap of sizes among individuals of the same ages. This overlap has a confounding influence on estimates of mean length at age whenever sampling is size selectivity. Since this example is only about the central tendencies of alternative sampling schemes, the total sample size for each was set to a large value of approximately10,000 observations (small deviations were caused by stratifications with equal sample sizes per stratum). For each sampling scheme the observations were drawn randomly. The probability of an observation at age and length being sampled was proportional to the relative numbers in the Na,l, (p=Na,l/∑Na,l), as modified by the stratification. No measurement error was added to either length or age, and there were no gear selectivity effects included. Four sampling schemes were evaluated: Random. Samples were drawn randomly from Na,l until the total sample size of 10,000 observations was reached. Neither the length frequency at age or age frequency at length are biased by this sampling strategy. Age. Sufficient random samples were drawn so that 909 observations (10,000/11 ages) were obtained for each age stratum. Observations in each stratum in excess of the stratum quota were discarded. The sampling scheme provides an unbiased sample of the length frequency of each age, but the age frequency at lengths are biased by the sampling scheme. Length. Sufficient samples were drawn to satisfy a length-stratum quota sufficient to meet the overall

METHOD The objective here is only to compare VB parameter estimates and resulting computed values of M from alternative sampling stratification schemes, so the simulation is simple. The size metric was lowerjaw fork length (LJFL, cm). Size-age samples are drawn from an age-length matrix of population relative abundances (Na,l). First, a vector of population numbers Na at age (a) was created for an assumed constant level of total mortality (Z) as Na=1e-aZ. Mortality was constant and not size dependent. Sampling was simulated using the sizes and relative 2

goal of about 10,000 observations. Size strata were in 1 cm increments and ranged from 75 cm to an arbitrary maximum value that was the 4CV greater than the mean length at age 11 (292 cm LJFL for CV=0.1 and 375 cm LJFL for CV=0.2). The CV-based upper limit allowed the same proportion of the size range to be included in both cases (>99.5%). Because the total sample of 10,000 observations was spread over a larger size range for CV=0.2 there were more 1cm length classes for the larger CV and fewer samples per stratum. At CV = 0.1 each stratum had 46 observations, and for CV=0.2 each stratum had 33 observations. This sampling scheme provides an unbiased sample of the age frequency of each length stratum, but the length frequencies at age are biased. Mixed. 90% of the total sample was drawn at random and the remaining 10% used length-stratified sampling of fish larger than L11 (208 cm LJFL) to simulate a random scheme supplemented by directed sampling of large fish. For CV=0.1 this convention resulted in 9000 completely random observations plus 1008 length-stratified observations at 12 per 1-cm stratum between 208 and 292 cm LJFL. For CV=0.1 the supplement added 1002 observations at 6 per stratum between 208 and 375 cm LJFL. Both the age frequency at length and length frequency at age are biased when this scheme is employed. Size frequencies for the observations were determined and mean sizes at each age computed for each combination of CV and total mortality. VB growth equations were fitted using the Microsoft Excel “Solver” tool by minimizing the sums of squares of predicted and observed mean size at age. The fitted parameter estimates were compared with the values assumed to be “true” for the VB equation used to predict the mean lengths. Finally, M for each combination was computed using Pauly’s method (1980) to compare with the estimates in Pine et. al. (2008) from the “true” VB parameters and which was used to compute the age frequencies of the simulated population used herein.

the overlap of lengths within an age. The length frequencies of samples used for the size-age analyses by each sampling strategy for each CV and mortality regime are presented in Figure 2. The length frequencies for the random and mixed strategies are similarly skewed to smaller sizes because of the effects of mortality that reduce the abundance of older, larger fish in the population. For the mixed scheme, the result is similar but the effect of including the 10% allocation to stratified sampling of larger fish is evident in the slight elevation of the right tail of the distribution. The length frequencies at age of the age-stratified sampling are unaffected by mortality and so only data for the two levels of CV are presented in Figure 2 for this strategy. The plot shows the influence of smoothing of length frequencies associated with the larger CV seen in Figure 1. The length frequency of the length-stratified sampling strategy is unremarkable since it is only affected by the length range into which the sampling is stratified, which is larger for CV = 0.2 (Figure 2).

Figure 1. Distribution of sampled length frequencies of age 11 individuals from the age stratified sampling scheme at the two levels of variability of length at age investigated here. The histograms also reflect the size distribution of the simulated populations at age 11.

The true mean sizes at age, and the observed means for each sampling strategy are presented in Figure 3. For clarity, no attempt is made to distinguish among the means for the random and age-stratified strategies which were nearly coincident except for small differences at the oldest ages. There, the sample sizes were reduced by the effects of mortality. In comparison, the scatter of the means for length-stratified sampling were substantially elevated, particularly at older ages. The pattern was even more apparent for the mixed strategy where the random sampling was supplemented by the inclusion of large individuals (Figure 3).

RESULTS The variability in the lengths at age for the two levels of CV assumed for this example are illustrated by the sample length frequencies for age 11 drawn for the age-stratified sample strategy (Figure 1). The length frequencies reflect the underlying normal distributions assumed for the model since the samples are randomly drawn from that distribution. The larger CV flattens the center of the distribution and increases the range of lengths in each age class and consequently

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Figure 2. Length frequencies of the samples of the size structure of the hypothetical population used in the analysis of each sampling strategy (Panels A-D). Panels A and D include data for each combination of total mortality (Z) and coefficient of variation (CV) of size at age. The sample length frequencies for the age-stratified (B) and length-stratified (C) schemes were not influenced by mortality, consequently only the effect of the variability in sizes at age is shown for these two strategies

The values of the VB parameters estimated for each scenario are presented in Table 1 along with the values of M that would be predicted using the fitted parameters using the Pauly equation. The VB model fit the data well with R2 ≥ 0.99 in each case. The high precision obviates any need to explore other options as the fitted models accurately predict the sampled mean size at each age. The differences between the fitted model parameter estimates and the “true” values are presented in Table 2 along with the differences between M calculated with the “true” coefficients and M computed from the fitted VB parameters. Also, predictably, age-stratified sampling out-performed the other schemes, but the random scheme was about equivalent, especially for the lower mortality regime. That result is a consequence of the degradation of the numbers of older fish in the random samples caused by the higher mortality. The effect is small here because of the large sample size, but portends the problem of sample-size requirements encountered by

random sampling in the real world. Both strategies that involved size-stratified sampling performed poorly (Table 2) in spite of fitting the VB model well. Surprisingly, the results with supplemental size-selective sampling of large fish showed greater divergence from the true values than where all samples were collected via the size stratification. The errors in both cases were substantial. The errors in L∞ were also relatively greater at the higher CV. For the mixed-sampling model, the error in L∞ increased from about 100% at CV=0.1 to over 200% at CV=0.2. The numerical values of percent differences between k and its “true” value were relatively smaller than the errors in L∞, but the differences in k/ktrue ratios of the values in Table 1 were quite large for either scheme employing length stratification. The values of M computed from these data varied from 0.075 to 0.42 or a range of more than 5.5-fold. Most of this variability is a consequence of the sampling strategy rather than random sampling error (Table 2). The values of M

Figure 3. Mean sizes at age for each sampling strategy and population size structure.The symbols represent the mean lengths at age for each CV-M group for each strategy. Lines pass through the overall means of the strategy. For clarity, lines representing the means for the random and age stratified strategies are omitted because each closely overlaid the solid line showing the location of the true values.

cannot be compared to a true M because the true value is unknown. The differences from M computed using the “true” VB parameters were inconsistently affected by the total assumed mortality (Table 2). Values were about 18% higher at the higher CV for the strategies that included length-stratified observations.

deviation of measured size from the true size of the fish, but rather its deviation from the population’s unfettered true mean. The mean is an aggregate property that can’t be measured with a device. Exposure of the population to size selective fishing mortality shifts the mean sizes of survivors to a new value (e.g., Ricker 1969; Parma and Deriso 1990). Size-selective sampling, whether by the gear or design, biases the sampled means (Goodyear 1995; Gwinn et al. 2010). For the growth model assumed for this example, a sampling-induced error of a magnitude of 10% of the mean length at age 11 is equivalent to misreading age by more than 4 years. These results illustrate some underappreciated consequences of size-selected data on VB parameter values and derivative estimates of M. Age-stratified sampling would provide unbiased growth data, but usually that strategy is also impossible because it would require knowledge of fish age prior to sampling.

DISCUSSION The VB model predicts population mean size from age and is fitted to paired observations of size and age. Both must be independently estimated, and each has its own properties and requirements. Age is straightforward variable, but because it is often difficult to determine, its estimation is usually described in great detail. In contrast, sizes of individuals are usually measured with great accuracy and precision. However, the growth model predicts population means, not the size of the individual. Consequently, the important “error” is not the 5

Random sampling provides the expected accuracy, but sample sizes sufficient to obtain adequate representation of old fish can be unattainable because mortality causes the number of fish surviving to older ages become increasingly rare. Size-selection is everywhere in real data because of gear characteristics, species availability and experimental designs. The reason why mixing random and sizestratified sampling proved worse than size-stratified sampling alone, seems likely a chance outcome of the particular realization of assumptions, but the possibilities are too complex to visualize. Such sensitivity may partly explain the high variability in VB estimates for various species both within and between studies of the same species (e.g., Potts et al. 1998; Chen et al. 2012). Investigators who have studied the impact of size selectivity on model parameter estimation are keenly aware of problems (Ricker 1969; Gwinn et al. 2010), but even a casual reading of the literature reveals a great heterogeneity of awareness both about the effects on parameter estimation and how it propagates into derivative analyses. The implications of possible VB parameter errors can be extremely important when the data might be employed in analyses that lead to management advice (e.g., Chih 2009b; Gwinn et al. 2010). In the example here, using the rule of thumb that FMSY is a constant multiple of M could lead to 5-fold differences in calculated levels of acceptable catch (Table 1). When evaluated over a large number of taxa with large contrast in scale, Pauly’s results explained much of the variance in M, but his equations are also poor predictors (Griffiths and Harrod 2007). Their utility depends on the precision required of the estimate since the 95% confidence intervals are about 2.5 times the estimated value (Gulland and Rosenberg 1992). The utility of those equations to extrapolate VB parameters to obtain values for M can be seriously further eroded whenever size-selective sampling or other error is in the data used to quantify growth. Striped marlin was picked here because the data necessary to conduct the evaluation were available from a single source in which the authors had also estimated M using Pauly’s method (Pine et al. 2008). Though a value for M was required for their analysis, their finding that catch and release can be a useful management tool for marlins is not dependent on a particular value. The authors noted there was uncertainty in M but, their decision to only use the Pauly estimate may be interpreted by some readers as a (possibly unintended) tacit endorsement of the accuracy of the method and a judgement of the accuracy of its estimate for the species. That interpretation could influence parameter selections by other investigators for future studies either directly or

by inclusion in meta-analyses of M values appearing in peer-reviewed publications. The more often we do it, the more legitimate it seems to be. Melo-Barrera et al. (2003) authored the source paper that estimated VB parameters for striped marlin later adopted by Pine et al. (2008) and used here. Melo-Barrera reported that their size-age data were from a subsample of 399 from a total of 1044 marlin sampled from recreational catches during the study. However, their documentation failed to discuss how the subsampling was perform so it is impossible for readers to tell if the distributions of length at age used to fit the VB equation were affected by size stratification. Such information is important for those hoping to extract parameter estimates that are important inputs in subsequent research projects. However, as in the Melo-Barrera et al. (2003) example, discussion of sampling issues is sometimes insufficient (e.g., Chen et al. 2012; Gervelis and Natanson 2013; Dunton et al. 2016). In those cases, the reader may be tempted to rely on size composition data that may have been presented. For example, strongly size-stratified sampling (Figure 2C) could be easily distinguished from the results of random sampling (Figure 2A). However, length frequencies from random samples might be indistinguishable when compared with strategies that supplement observations with extra effort to include large fish (compare Figure 2A to 2D). Consequently, length frequencies of the growth data are not always reliable indicators of their adequacy for estimating growth. The sampling protocols that use ALK methods to estimate sample size-age distributions would belong to the random sample strategy evaluated here even though the substrata are size stratified. This is because the ALK method is designed to provide an unbiased estimate of the size-age distribution of the sample. The VB equation is then fitted to the estimated sample data rather than the observations for the fish that were actually aged. Also, the analyses here do not directly incorporate biases that accrue from size selective gear effects or historical mortality. However, the effects are the same as those that result from size stratification in that the size-age data deviates from the (biological) population they are intended to represent. The VB parameter estimation would suffer the same effects evaluated here. Methods that have been investigated to correct for size selective mortality (e.g., Vaughan and Burton 1994; Schirripa and Trexler 2000) and gear selectivity (Taylor et al. 2005; Gwinn et al. 2010) have been more successful at describing problems than demonstrating cures. Gwinn et al. (2010) showed that corrections for gear selectivity required precise, accurate knowledge of the selectivity curves otherwise the methods can increase the biases they were intended

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to diminish. At best, robust estimation of selectivity curves is a daunting proposition. The result here suggests the known problems of gear selectivity and size selective mortality can be exacerbated by an experimental design that promotes non-random sampling of sizes at age (e.g. Ricker 1969). Growth models are fitted to data collected using uncorrected size-stratified sampling designs more often than one might expect (e.g., Potts et al. 1998; Hwang and Kim 2008; Marshall et al. 2009; GarcíaContreras et al. 2009; Siddons et al. 2016; O’Malley et al. 2017). This strategy, evaluated in the example here, could add bias that would be probably impossible to eliminate because sample selectivity curves would be unpredictable. When added to real world sizeselectivity of gears, the additional effect of sizestratified sampling would make estimation of biologically meaningful growth coefficients a matter of conjecture. These features need not compromise within-study objectives which can be addressed with appropriate statistical control or experimental design (Nate and Bremigan 2005; Tyszko and Pritt 2017). However, the result will often be VB parameters values that have little biological meaning and which are dangerous to extrapolate. Although it is possible that the magnitudes of sampling effects in the example here are atypical, it is clear that growth parameters are highly susceptible to sampling bias even though models fit the data well. This example also shows that errors in the values can cause important changes in values of M calculated therefrom. Such data are used in all manner of ecological and fish population models, often without knowledge or acknowledgment of the uncertainty in results. There will always be a need for growth and mortality information as cornerstones of analyses and touchstones for evaluating modeling results. Those of us who use published data for subsequent studies should caution about uncertainties that accrue from the original study as well as those added by the new research. However, that will remain difficult until researchers provide similar detail to the protocol used to obtain length frequencies as they do for age determination. Finally, VB models can be fitted using methods other than analysis of size-age data. Examples include mark-recapture experiments and progressions of modes in population lengthfrequencies. Ultimately, a well fitted VB model does not necessarily equate to biological realism. The source and extent of biases should be included with the primary publication and in subsequent reports where the parameter values are used for other purposes.

ACKNOWLEDGEMENTS This study was supported by The Billfish Foundation. I thank W Pine, B. van Poorten, and one anonymous reviewer for comments and suggestion on draft manuscripts. LITERATURE Bettoli, P. W., and L. E. Miranda. 2001. Cautionary note about estimating mean length at age with subsampled data. North American Journal of Fisheries Management 21:425-428. Chen, K. -S., T. Shimose, T. Tanabe, C. -Y. Chen, and C. -C. Hsu. 2012. Age and growth of albacore Thunnus alalunga in the North Pacific Ocean. Journal of Fish Biology. 80: 2328–2344. Chih, C. P. 2009a. Evaluation of the sampling efficiency of three otolith sampling methods for commercial king mackerel fisheries. Transactions of the American Fisheries Society 138:990–999. Chih, C. P. 2009b. The effects of otolith sampling methods on the precision of growth curves. North American Journal of Fisheries Management 29:1519-1528. Coggins, L. G., Jr., D. C. Gwinn, and M. S. Allen. 2013. Evaluation of age–length key sample sizes required to estimate fish total mortality and growth. Transactions of the American Fisheries Society 142:832–840. Dunton, K. J., A. Jordaan, D. H. Secor, C. M. Martinez, T. Kehler, K. A. Hattala, J. P. Van Eenennaam, M. T. Fisher, K. A. McKown, D. Conover, and M. G. Frisk. 2016. Age and growth of Atlantic sturgeon in the New York Bight. North American Journal of Fisheries Management 36: 62-73. García-Contreras, O. E., C. Quiñónez-Velázquez, R. E. Morán-Angulo, and M. C. Valdez-Pineda. 2009. Age, growth, and age structure of amarillo snapper off the Coast of Mazatlán, Sinaloa, Mexico. North American Journal of Fisheries Management. 29: 223-230. Gervelis, B. J., and L. J. Natanson. 2013. Age and growth of the common thresher shark in the Western North Atlantic Ocean. Transactions of the American Fisheries Society. 142: 1535-1545. Goodyear, C. P. 1995. Mean size at age: an evaluation of sampling strategies with simulated red grouper data. Transactions of the American Fisheries Society 124:746-755. Gulland, J. A. and A. A. Rosenberg. 1992. Review of length-based approaches to assessing fish stocks. FAO Fisheries Technical Paper 323. Rome, FAO. 100p. Griffiths, D. and C. Harrod. 2007. Natural mortality, growth parameters, and environmental temperature in fishes revisited. Canadian Journal Fisheries and Aquatic Science 64:249–255.

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Gwinn, D. C., M. S. Allen, and M. W. Rogers. 2010. Evaluation of procedures to reduce bias in fish growth parameter estimates resulting from size-selective sampling. Fisheries Research 105:75-79. Hwang, S. D., and J. Y. Kim. 2008. Age, growth, and maturity of chub mackerel off Korea. North American Journal of Fisheries Management 28:1414–1425. Ketchen, K. S. 1950. Stratified subsampling for determining age-distributions. Transactions of the American Fisheries Society 79:205-212. Kimura, D. K. 1977. Statistical assessment of age-length key. Journal of the Fisheries Research Board of Canada 31:317–324. Lee, R. M. 1912. An investigation into the methods of growth determination in fishes. Publicacion de Circonstance Conseil International pour l’Exploration de la Mer, s1: 3–34. Marshall, M. D., M. J. Maceina, and M. P. Holley. 2009. Age and growth variability between sexes of three catfish species in Lake Wilson, Alabama. North American Journal of Fisheries Management 29:5 1283-1286. Melo-Barrera, F. N., R. Felix-Uraga, and C. QuinonezVelazquez. 2003. Growth and length–weight relationship of the striped marlin, Tetrapturus audax (Pisces: Istiophoridae), in Cabo San Lucas, Baja California Sur, Mexico. Ciencias Marinas 29: 305–313. Miranda, L. E., and M. E. Colvin. 2017. Sampling for age and growth estimation. Chapter 5 in M. C. Quist and D. A. Isermann, editors. Age and Growth of Fishes: principles and techniques. American Fisheries Society, Bethesda, Maryland. Nate, N. A., and M. T. Bremigan. 2005. Comparison of mean length at age and growth parameters of bluegills, largemouth bass, and yellow perch from length-stratified subsamples and samples in Michigan lakes. North American Journal of Fisheries Management 25: 1486-1492. O’Malley, A. J., C. Enterline, and J. Zydlewski. 2017. Size and age structure of anadromous and landlocked populations of rainbow smelt. North American Journal of Fisheries Management 37: 326-336. Parma, A. M., and R. B. Deriso. 1990. Dynamics of age and size composition in a population subject to size-selective mortality: effects of phenotypic variability in growth. Canadian Journal of Fisheries and Aquatic Sciences 47:274-289. Pauly, D. 1980. On the interrelationships between natural mortality, growth parameters, and mean environmental temperature in 175 fish stocks. Journal du Conseil International pour l’Exploration de la Mer 39: 175–192.

Pine, W. E., S. J. D. Martell, O. P. Jensen, C. J. Walters, and J. F. Kitchell. 2008. Catch-and release and size limit regulations for blue, white, and striped marlin: the role of postrelease survival in effective policy design. Canadian Journal Fisheries and Aquatic Science 65: 975-988. Potts J. C., C. S. Manooch, and D. S. Vaughan. 1998. Age and growth of vermilion snapper from the Southeastern United States. Transactions of the American Fisheries Society 127: 787-795. Quist, M. C., and D. A. Isermann, editors. 2017. Age and growth of fishes: principles and techniques. American Fisheries Society, Bethesda, Maryland. Ricker W. E. 1969. Effects of size-selective mortality and sampling bias on estimates of growth, mortality, production, and yield. Journal of the Fisheries Research Board of Canada 26: 479-541. Schirripa, M. J., and J. C. Trexler. 2000. Effects of mortality and gear selectivity on the fish otolith radius-total length relation. Fisheries Research 46: 83-89. Shoup, D. E. and P. H. Michaletz. 2017. Growth Estimation: Summarization Chapter 11 in M. C. Quist and D. A. Isermann, editors. Age and Growth of Fishes: principles and techniques. American Fisheries Society, Bethesda, Maryland. Siddons, S. F., M. A. Pegg, N. P. Hogberg, and G. M. Klein. 2016. Age, growth, and mortality of a trophy channel catfish population in Manitoba, Canada. North American Journal of Fisheries Management 36: 1368-1374. Taylor, N. G., C. J. Walters, and S. J. D. Martell. 2005. A new likelihood for simultaneously estimating von Bertalanffy growth parameters, gear selectivity, and natural and fishing mortality. Canadian Journal Fisheries and Aquatic Science 62: 215– 223. Tyszko, S. M, and J. J. Pritt. 2017. Comparing otoliths and scales as structures used to estimate ages of largemouth bass: consequences of biased age estimates. North American Journal of Fisheries Management 37: 1075-1082. Vaughan, D.S., and M. L. Burton. 1994. Estimation of von Bertalanffy growth parameters in the presence of size-selective mortality: a simulated example with red grouper. Transactions of the American Fisheries Society 123: 1-8.

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Table 1. Values of von Bertalanffy growth coefficients fitted to sample size-age data using alternative sampling schemes and corresponding computed values of natural mortality (M). The fits were to mean lengths for each age in the sampled data. R2 ≥ 0.99 for all fitted equations. The “true” values were L∞=221 LJFL (lower jaw fork length), k = 0.23, and t0 = -1.6. Z=0.38 CV=0.1 t0 L∞ k Random 222.3 0.227 -1.6 Age 223.5 0.220 -1.7 Length 361.8 0.097 -2.2 Mixed 415.9 0.064 -3.4

M 0.380 0.372 0.189 0.140

Z=0.57 CV=0.2 L∞ k t0 222.3 0.226 -1.6 218.6 0.241 -1.5 495.0 0.084 -1.6 774.2 0.033 -3.2

M 0.378 0.397 0.159 0.075

CV=0.1 L∞ k t0 213.7 0.264 -1.3 223.5 0.220 -1.7 346.6 0.112 -1.9 441.6 0.066 -2.9

M 0.424 0.372 0.212 0.140

CV=0.2 L∞ k t0 215.0 0.260 -1.3 218.6 0.241 -1.5 438.8 0.116 -1.1 734.5 0.043 -2.2

M 0.419 0.397 0.203 0.092

Table 2. Percent error [100(estimate-true)/true] in estimated von Bertalanffy growth coefficients L∞, k and to, and percent difference between the natural mortality (M) calculated with the “true” coefficients and M computed from the fitted VB parameters. Z=0.57

Z=0.38

Random Age Length Mixed

CV=0.1 L∞ k t0 0.6 -1.2 0.5 1.1 -4.2 5.4 63.7 -58.0 40.1 88.2 -72.0 112.8

M 1.0 3.0 50.6 63.5

CV=0.2 L∞ k t0 0.6 -1.8 1.3 -1.1 4.8 -4.8 124.0 -63.4 -1.8 250.3 -85.8 101.7

CV=0.1 M L∞ k t0 1.4 -3.3 14.7 -16.3 -3.4 1.1 -4.2 5.4 58.6 56.8 -51.2 20.0 80.4 99.8 -71.2 80.4

9

M -10.4 3.0 44.9 63.5

CV=0.2 L∞ k t0 -2.7 13.0 -16.3 -1.1 4.8 -4.8 98.6 -49.4 -29.2 232.3 -81.2 37.1

M -9.2 -3.4 47.1 76.1