Accounting for individual variability in the von ...

Accounting for individual variability in the von Bertalanffy growth model Can. J. Fish. Aquat. Sci. Downloaded from www.nrcresearchpress.com by Renmin University of China on 06/05/13 For personal use only.

You-Can Wang and Mervyn R. Thomas

Abstract: Estimation of von Bertalanffy growth parameters has received considerable attention in fisheries research. Since Sainsbury (1980, Can. J. Fish. Aquat. Sci. 37: 241-247) much of this research effort has centered on accounting for individual variability in the growth parameters. In this paper we demonstrate that, in analysis of tagging data, Sainsbury's method and its derivatives do not, in general, satisfactorily account for individual variability in growth, leading to inconsistent parameter estimates (the bias does not tend to zero as sample size increases to infinity). The bias arises because these methods do not use appropriate conditional expectations as a basis for estimation. This bias is found to be similar to that of the Fabens method. Such methods would be appropriate only under the assumption that the individual growth parameters that generate the growth increment were independent of the growth parameters that generated the initial length. However, such an assumption would be unrealistic. The results are derived analytically, and illustrated with a simulation study. Until techniques that take full account of the appropriate conditioning have been developed, the effect of individual variability on growth has yet to be fully understood.

RCsurnC : L'estirnation des param&tresde croissance de vsn Bertalanffy r e ~ o i depuis t un certain temps une attention considCrable dans le dornaine de la recherche halieutique. Depuis l'article de Sainsbury (1980, J. can. sci. halieutiques aquat. 37 : 241-247), une bonne partie de ces travaux ont CtC axCs sur la f a p n de prendre en compte la variabilitk individualle dans les paramktres de croissance. Dans notre Ctbade, nous dkmontrons que la mCthode de Sainsbury et ses dkrivds, appliquds h l'analyse des donnCes de rnarquage, ne permettent gCnCralernent pas de bien rendre compte de la variabilitd individuelle de croissance, de sorte que les estimations des paramhes ne son%pas cohCrentes. En effet, le biais me tend pas vers ztro i mesure que la taille de 1'Cchantillon va vers l'infini. Ce biais, similaire 2 celui de la mCthsde de Fabens, vient dba fait que ces mCthodes we prennent pas comme base d'estimation les espkrances esnditionnelles approprikes. De telles mCthodes ne peuvent convenir que dans l'hypothkse oh les pararnktres de croissance individuelle qui produisent l'augmentation de longueur seraient indkpendants des paramktres de croissance qui ont produit la longueur initiale. Une telle hypsth&se est toutefois irrkaliste. Les rCsultats sont obtenus par analyse et illustrCs par une simulation. En 19absencede techniques prenant pleinement en csmpte le conditionnement apprspriC, on ne pourra Clucider parfaitement l'effet de la variabilitC individuelle SUP la croissance. [Traduit par la RCdaction]

Introduction Bertalanff~equation has often been used to The describethegrowthoffishandothermarineanimals-The equation describes the relationship between age ( a ) and length ( l ) as [I]

l(a) = L,(1 - e-""-To')

in which L,, K, To are growth parameters.

Y . 4 . Wang. @SIR0 IPP&P Biometries Unit, CSIWO Division s f Fisheries, P.O. Box 120, Cleveland, Queensland 4 %63, Australia. M.R. Thomas. CSIRB IPP&P Biornetrics Unit, @SIR0 Cunningham Laboratory, St. Lucia, Queensland 4067, Australia.

Tag-recapture is a common method for estimating growth parameters (k,, K ) , especially for species that cannot be aged directly. In tag-recapture studies, the length at capture (l),intervalbetweenreleaseandrecapture(~),andgrowth increment during the time interval (I) are recorded. If no infomation is available for the age at capture, tag-recapture data provide no information to allowestirnation of the parameter To. The parameter To can, therefore, be fixed at zero, without consequences for estimation s f the parameters L, and K (Sainsbury 1980). Fabens (1965) derived an expression for I as a function of & and 8: 8-21 1 = ( L , - l)(1 - elK') Estimation of the growth parameters, based on Eq. 2 and using ordinary least squares, is known as the Fabens method. The Fabens method assumes that the parameters in Eq. 2 are the same for all individuals, i.e., each individual

Can. J. Fish. Aquae. Sci. 52: 1368-1375 (1995). Printed in Canada / HmprimC au Canada

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Wang and Thomas

follows an identical von Bertalanffy growth curve. Many authors have recognised that this assumption is unrealistic (Sainsbury 1980; Kirkwood and Somers 1984; Hampton 1991; Xiao 1994), and that the Fabens method leads to inconsistent estimators. These authors have attempted to modify the Pabens method by incorporating individual variability in the parameters of Eqs. 1 and 2. Generalizations of Sainsbury9s (1980) method continue to receive the attention of researchers (Hampton 1991; Xiao 1994). It is possible to characterize the relative performance of these techniques under a wide range of conditions. The objective of this paper, however, is to show that the individual variability in growth is not accounted for in the method of Sainsbury (or its derivatives), and that they are not markedly different from the Fabens method itself. For the convenience of further discussion and introducing the necessary notation, we first present a brief summary of Sainsbury's (1980) and Kirkwood and Somers' (1984) methods, and then analyse both methods based on the expected increments for an individual and for the population. We show that these methods and their derivatives do not properly take account of individual variability in growth parameters (and are therefore inconsistent). Ironically, these methods would be consistent (asymptotically unbiased) only if the growth parameters for any individual vary randomly over time. We illustrate this with a simulation study. Finally we indicate some possible directions for future research to account for individual variability in growth. Throughout this paper, we will denote Pabens (1965) by Hi, Sainsbury (1980) by S, Kirkwood and Somers (1984) by K&S, Hampton (1991) by H, and Xiao (1994) by X.

Kirkwood and Somers (1984) K&S assume that L, is a normal random variable, but that K is fixed (hence K = Their expressions for the conditional mean and variance of I are

a.

K&S constructed a likelihood function based on a normal model, with mean and variance given by Q s . 4 and 5, and proposed use of maximum likelihood estimation. This method, in fact, is the same as the weighted least squares (weighed by the inverse of the variance). In the special case when t is constant for all individuals, their maximum likelihood approach is equivalent to the Pabens method.

Sainsbury (1980) Sainsbury provided a general expression for the mean and variance of I, conditional on 5 and t, under the assumption that K and L, are independent. He proposed a normal distribution for L, and a gamma distribution for K. This generates the following expressions for the conditional mean and variance of I:

where

Analysis of existing methods S, K&S, and X make the following assumptions in their models: (1) Each animal has its own triplet von Bertalanffy parameters, L,, K, To, that are retained throughout its life. (2) The conditional distribution of K and L, is identical over the age or length range considered. Here L,-and K are regarded as random variables with mems L, and K and variances and o:, respectively. Suppose that the mean length of the population at release is i, with individual lengths represented by the random variable I . The mean increment in length, during the time interval t and conditional on the initial mean length 1 is given by E,[ICt], where the subscript p indicates that the expectation is taken over the entire population. Under assumption 2, we may write:

cri_

The evaluation of these expectations depends on the distributions of K and L,. The methods of S, K&S, H, and X are based on different assumptions about the distribution of (K, L,), hence leading to different expressions for the expectations and variances.

and

Sainsbury develops estimates for the growth parameters based on an approximating normal likelihood, with mean and variance given above. As he points out, the exact distribution of I will not be normal, but the approximation should be reasonable for modest sample sizes (larger than 180). The methods of S and K&S (and their derivatives) assume that Eq. 3 is the expected growth increment for an individual, conditional on length at tag 1 and time at liberty t. In fact, the expectations are taken over the entire population, and they are generally different from the expectation for an individual. We will demonstrate that use of Eq. 3 to represent the conditional growth increment for a given animal fails to take proper account of individual variability, and is responsible for the inconsistency in the techniques of S, K&S, and their derivatives. We now present some analytical results and illustrate the difference between these two expectations.

Can. J. Fish. Aquat. Sci. Vol. 52, 1995


Expected incrment %oran iwdhiduar$ As discussed above, the existing methods rely on the expected increment for the population (a group of animals). We now introduce the counterpart of ~,[lli,t],the expected increment for an individual, which has not been given as much attention as it should have in the analysis of tagging data. To obtain the expected increment for an individual of initial length k, we need to calculate the conditional expectation:

where the subscript i indicates that the expectation is for an individual animal, not over the entire population. In general, evaluation of the expectation in Eq. 6 is complex and requires knowledge of the distributions of age (t) and L,. Inspection of Eqs. 3 and 6 shows that the expectation for the individual increment will be the same as that for the population increment only when K and &, are independent of k. We illustrate tkeir difference for the simple case of constant K (i.e., K = a , where To evaluate Ei[LWll,t],we introduce the quantity p = 1 where A is the age at release. If p and %, are independent (i.e., an animal's age at release is independent of its maximum attainable size), using elementary distribution theory, we may write (Wang et al. 1995): e-KA

where g(x) and$(x) are the density functions of p and L,, respectively. From this equation, assumption 2 cannot, in general, be true for individuals, because the conditional expectation of &, is dependent on l. The difference between Ei[lll,t], and ~ , [ l l i , tis] illustrated in Fig. 1 as a function of initial length. These expectations are calculated by assuming that %, is normally distributed with a mean of 100 and a standard deviation of 10, K is a constant of 8.5, and time at liberty is fixed at 1 year. The age at release A was assumed to be uniformly distributed over four different ranges, (I,2), (0.5, 2.5), ( I -0, 1.51, and (2.0, 2.5). These four distributions correspond to four cases: (1) the length at release has a moderate range with a moderate mean (Fig. la); (2) the length at release has a wide range with a moderate mean (Fig. lb); (3) the length at release has a small range with a moderate mean (Fig. lc); and (4) the length at release has a large mean (Fig. Id). Some simulated increments of individuals are also plotted to represent the ranges and variations of the lengths at release under different assumptions. As we can see, the population expectation, E,[lli.r], is always linear in I, while the expectation for an individual, E~[III;~], shows marked nonlinearity with an intercept around the mean initial length. In the cases of Figs. lb-Id, the expected increment for an individual is almost linear in the observed ramage of the lengths at release. In Fig. lb, the conditioning has little effect on the expected mean because of rather large uncertainties in age, while in Figs. Ic and Id, the

expected increment for an individual increases in the length at release while the expected increment for the population always decreases. Perhaps, Fig. 1 is the more mpresentative of real data. Note that these simulated data are based on the assumption of constant time at liberty. In practice, time at liberty is more likely to vary and thus more variations in the observed length increments will be expected. The expected increment for an individual is smaller (larger) than that for the population for those with smaller (larger) length at release. C.nsequently, overestimation of and underestimation of K will occur for the method of E In general, the difference between K&S, S, and the classical method F is only due to different weighting functions and nonlinearlity in K and, therefore, is not fundamental. So we will expect similar performance from these three methods.

x,

SimIation study Two sets of simulations were performed to demonstrate that ( I ) the estimates of KKS, and S are biased under assumption 1, and (2) the bias diminishes when the growth parameters of individuals are assumed to change from time to time randomly and independently, i.e., assumption 2 holds for all individuals. Subsequently, we will refer to the case of the first simulation study as the model with stable growth parameters and the case of the second study as the model with changing growth parameters. Note that the units for time and length are not given here, because they can be chosen as any sensible time and metric units. For instance, the time units can be year, month, or even a quarter of a year, and the length unit can be centimetres, or 0.5 inch, etc. The unit for K is always the inverse of the time unit. In the first study, we generated data sets by sampling animal age at capture (A) and growth parameters (L,, Length at release was calculated from these random variables. Time at liberty (between release and recapture) was then sampled, and length increment at recapture was finally evaluated from the time at liberty and the original growth parameters. A different sampling procedure was adopted for the second study. Once again, age at release, & ,, and K were sampled; length at release was calculated; time at liberty was sampled; and length increment was calculated from length at release, time at liberty, and the growth parameters L, and K. But this time K and %, were resampled from their distributions. Consequently, the growth parameters used to calculate the growth increment during the interval between release and recapture were independent of those used to calculate growth up to the time of release. Therefore, the values of L,, and K that determine the increment are independent of the value of I. Therefore, the expected increment for the individual (Eq. 6) is the same as that for the population (Eq. 3). Under this circumstance we expect the techniques of F, S, and K&S to perform well. Age at release (A) was sampled from a uniform distribution on (1, %),and time at liberty was sampled as 0.1 plus an exponential random variable with a mean of 1. In

a.


Wawg and Thomas

each study, we examined two cases: (1) only L, varies among individuals while K is a constant, and (2) both L, and K vary among individuals. In both cases, L, was sampled from a normal distribution with a mean of 100 and a standard deviation of 10. In the first case, K was assumed to be a constant of 0.5, and in the second case, K was sampled from a gamma distribution with a mean of 0.5, and a standard deviation of 0.158 ( a = 10, f3 = 0.05, where the probability density function is given by x"-I e-x'B/ (PaI'(a))). One thousand data sets, each with 100 recaptures, were generated in each case s f the two studies. Each data set was analysed using the methods of F, S, and K&S, and 1000 sets of estimated parameters were obtained for each method. Results of both simulations are summarised in Takle 1. Means and standard deviations of the estimates of L, and K are given for each estimation method, based on stable growth parameters (i.e., as in the first set of simulations, not resampled) and based on changing growth parameters (from the second set of simulations, resampled). The growth curve is always determined by the joint effect of L, and K. One of the indices that combine thetwo parameters is the growth p e r f o r m ~ n c eindex A = L,K/2, the instantaneous growth rate at L,/2 (Dall et al. 1990; Milton et al. 1993; see Discussion). This index also takes into_ accougt some of the wide variation in the estimates of L, and K. In the case of gable growth parameters, the estimators for both L, and K show substantial biases. The biases become much larger when both L, and K are variable (240%) compared with the case when only L, is variable. For the index A, the bias is still as high as 20% when only L, is variable, and up to 43% when both L, and K are variable. However, when the growth parameters are resampled, the biases diminish to no more than 4% for all the estimates, and the biases become much smaller when K is a constant. There app_ears_to be no clear evidence of bias for the estimates of L,, K, and A from the resampled data, using the methods of F and S. The conclusion to be drawn from this simulation exercise is that these methods may produce substantial bias when individual variability in growth exists, and the bias is negligible when the individual growth parameters change independently from time to time or when the growth p a r m eters are the same for all individuals.

Discussion In general, an efficient estimation method should make use of length at release and length at recapture equally and take into account the correlation between them for each individual. The correlation is obviously due to the fact that both lengths are generated from the same growth parameters and initial age. The methods we discussed above can be easily modified by expressing the expected increment in terms s f length at recapture rather than length at release. Such a simple modification (all the underlying assumptions remain the same) will result in different estimates, suggesting the inefficiency of these methods. K&S noted that their method is effectively the same as a weighted least squares method while the Fabens method

is the ordinary least squares one. Therefore, it is not surprising that their method is inconsistent (a precise statistical term that for our purposes means that the bias in estimates tend to zero as the sample size tends to infinity). Maller and DeBoer (1988) investigated the performance of the F and K&S techniques and provided some analytical results in the case of constant time at liberty. They also concluded that, in general, no weighted least squares analysis of Eq. 2 (in which the weights do not depend on the unknown age) can produce consistent estimates. The weakness in the approaches of S and K&S is that these methods are based on the population expectation of growth increment as given by Eq. 4, rather than on the expected growth increment for the individual. Figure 1 shows that, in general, there may be marked differences between the population expectation of increment (which is linear in initial length) and the conditional expectation of increment for an individual (which shows marked nonlinearity in initial length). The difference between the expected increment for an individual and for the population may be negligible when its length at release, I, is close t s the mean length of the population at the time of release. However, when the distribution of a g e at release (A) has a small range, the expected increment for an individual becomes an increasing function of k, especially when A is the same for all fish released, the expected increment for an individual will be proportional to I and very different from the expected increment for the population (cf. Fig. lc). It appears that the difference between these two expectations may become ignorable when the distribution of age (length) at release is widely spread out (cf. Fig. lb). This is because conditioning has little effect on the expected increment when the uncertainty in A is so large. Assumption 1 is a fundamental requirement of methods that attempt to account for individual variability in growth parameters. We have seen that this assumption leads to dependence between L, and length at release. That is, for an individual animal the conditional expectation of L, given I is not constant (Eq. 7). The conditional expectation of L, for an individual, in general, is difficult to derive, and it depends on the age distribution of the population. It is, therefore, tempting to base the estimation on the population expected increment, which is easy to derive and does not depend on the age distribution. Merely considering L, and K to be random in the increment Eq. 2 does not properly account for individual variability in growth. Ignoring the conditioning of expected increment on length, 1, is equivalent to assuming that there is no relationship between the value of k and the values of L, and K during the period at liberty, and, hence, no relationship between the values of K and &, at release from which & was generated and those during the period at liberty. In our simulation, we deliberately violated assumption B by resampling values of L, and K, and indeed, only very small biases in the estimates were observed. This is because if we assume that the growth parameters for an individual are independent from time ts time, its expected increment will be the same as that for the pspulation, and these methods will be appropriate. The effect sf

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Cars. J. Fish. Aquat. Sci. Vol. 52, 1995

Fig. I. Expected length increments over 1 year for an individual and the population at different initial lengths. The observed (simulated) individual increments were superimposed. The underline distribution of L, is normal with mean 180 and standard deviation 10 while Pa: is a constant of 0.5. The units for time and length can be chosen at discretion. The age at release is assumed to be uniformly distributed between one and two. The age at release, A, is taken to be uniformly distributed on different ranges: (a) A - U(1.0, 2.8); (b) A U(8.5, 2.5); ( c ) A U(1.8, 1.5); and ( d ) A U(2, 2.5).


-

-

-

L e n g t h a t Release

.,....

Individual Population


Can. J. Fish. Aquat. Sci. Downloaded from www.nrcresearchpress.com by Renmin University of China on 06/05/13 For personal use only. Wang and Thomas

Fig. l (@0az@l~iked).


0 Observed

Length a t Release

1374

Can. J. Fish. Aquat. Sci. Vol. 52, 1995

Table 1. Mean, standard deviation (in parentheses), and relative error (percent) of three methods.

Methods

Error (96)

&,

K

Em,K, and A estimated by

Error (%)

A

Error (% )


Stable growth parameters

Only E, variable Fabens Kirkwood and Somers Sainsbury Both E, and K variable Fabems Kirkwood and Somers Sainsbury

112.7 (9.5) 121.0 (8.3) 121.0 (8.3) 140.4 (35.4) 204.2 (59.1) 204.3 (143.5)

12.7 21.0 21.0

0.36 (0.07) 0.33 (0.04) 0.33 (0.04)

-28.2 -35.0 -34.8

19.9 (2.4) 114.5 (1.4) 19.5 (1.4)

-20.4 -21.4 -21.8

40.4 104.3 1104.3

0.21 (0.06) 0.14 (0.04) 0.15 (0.05)

-57.9 -72.2 -69.9

14.0 (1.9) 13.2 (1.2) 13.6 (1 -5)

-43.9 -47.2 -45.7

-0.5 -0.1

24.9 (2.5) 25.0 (1 .5) 24.8 (1.5)

-0.4 -0.1 -0.4

24.7 (2.6) 25.8 (2.3) 25.0 (2.0)

-1.1 3.0 0.0

Changing growth parameters

Only L, variable Fabens Kirkwood and Sorners Sainsbury Both L, and K variable Fabens Kirkwood and Somers Sainsbury Note: The true values s f with 100 recaptured fish.

100.9 (5.4) 100.3 (3.7) 100-7(3-8)

0.9 0.3 0.7

0.50 (0.07) 0.50 (0.05) 0.50 (0.05)

99.4 (5.1) 87.6 (4.1) 100.4 (4.1)

-0.6 -24 0.4

0.50 (0.08) 0.53 (0.07) 0.50 (0.06)

k,, k9and A are

-0.1

0.2 6.0 0.0

188, 0.5, and 25, respectively. These results are based on 1000 simulations, each

individual variability examined by S and X is therefore only due to the nonliwearlity in growth parameters. In the simulation analysis s f H, the values of I were fixed, which is equivalent to the case of changing growth parameters. Therefore, his analysis is virtually invalid. The conclusion to be &awn here is that the existing methods developed in fisheries research do not account for the individual variability in growth for tagging data. For the more realistic assumption (strong dependence over time sf the growth parameters for an individual) these methods give inconsistent estimates. The size s f the bias does not decrease with sample size, and these methods should not then be used when individual variability in growth exists. S did realise these problems, as pointed out in his &scussion, and his analysis should be regarded as a "first analysis" on individual variability. K&S were also well aware that their method was based on population expectation, which $key regarded as an approximation to the relevant conditional expectation. What is perhaps surprising is how poor the approximation can be. One might argue that growth of fish probably depends on enviro~~mental factors, and hence, it is not absurd to assume changing growth parameters in the growth model when the environmental inputs are unknown. Perhaps it is reasonable to assume that, for each fish, L, is genetically determined and retained throughout its life and K, which measures the growth rate, is either environmentally determined or constant if the environmental factors are stable over the period considered (Clsern and Nichols 1978; McCauley and KiBgour 1990). The two growth parameters, as pointed out by Kingsley (1980), in general, measure t w s different characteristics

of growth, corresponding to the two axes s f the growth curve. Apparently, the objective of the parameter estimation is not to estimate these two parameters but to predict the growth rate. It is possible that biased estimates of E, and K may result in a reasonable fit of the underlying growth curve within the age or length range of interest. Therefore, it is of interest to examine the effect of the estimates of %, and K on predicting the growth rate. In addition to the estimates of &, and K, we also investigated the growth performance index X, the instantaneous growth rate at half the maximum length. This index combines the effect of both parameters and gives a different description of the growth characteristics. Apart from the index X discussed here, the mean lengths at certain ages can also be used to assess the overall growth estimation of a method (Francis 1988). Nevertheless, our simulation study (Table 1) clearly demonstrated the problems in these methods. The problems with these methods are retained by all those approaches which have been derived from them (H and X). A number s f possible generalizations of these techniques could be explored (for example dealing with multiple recaptures). However, these generalizations would suffer from the same problems. We believe that further research should be directed away from generalisation of the S and K&S techniques, and towards fully conditional procedures (Palmer et al. 1991; Wang et al. 1995). These fully conditional techniques are no more difficult to implement than the methods s f S and K&S. Further generalization is possible to incorporate environmental inputs, seasonality, and measurement error. The nonlinear random coefficients model of Palmer et al. (1991) is computationally expensive, but useful for

Wang and Thomas


combining data sets with multiple recaptures. Further investigation through the Gibbs Sampler (Gilks et al. 1994) to reduce the complexity is needed. T h e Gibbs sampler also provides a vehicle for developing more ambitious models, incorporating measurement e m r with autoregressive stmcture and individual variability.

We would Bike t o thank Drs. David Die, David Milton, Andre Punt, Anthony D.M. Smith, and Peter Jones for invaluable comments o n an earlier version of the manuscript and 1.E Somers for helpful discussion in the process of preparing this manuscript. Two referees' comments also improved the presentation of this paper.

References Cloern, J.E., and F.H, Nichols. 1978. A von Bertalanffy growth model with a seasonally varying coefficient. J. Fish. Res. Board Can. 35: 1479-1482. Dall, W.D., B.J. Hill, P.C. Wothlisberg, and D.J. Staples. 1990. Biology of the Penaeidae. h a Advances in the Marine Biology. Vol. 27. Edited by J.H.S. Blaxter and A.J. Southward. Fakns, A.J. 1945. Properties and fitting of the von Bertalanffy growth curve. Growth, 29: 265-289. Francis, W.I.C.C. 1988. Are growth parameters estimated from tagging and age-length data comparable? Can. J. Fish. Aquat. Sci. 45: 936-942. Gilks, W . R . , A. Thomas, and D.J. SpeigeHhalter. 1994.

A language and program for complex Bayesian modelling. Statistician, 43: 169-1487. Hampton, J. 1991. Estimation of southern bluefin tuna Thunnus rnwccsye'i growth parameters from tagging data, using von BertaEanffy models incorporating individual variation. Fish. Bull. U.S. 89: 577-590. Kingsley, M.G.S. 1980. von Bertalanffy growth parameters. Trans. Am. Fish. Soc. 109: 252-253. Kirkwood, G.P., and 1.F. Somers. 1984. Growth of the two species of tiger prawn, Penaeus escudentus and Penaeus semisubcatkss, in the western Gulf of Carpentaria. Aust. J. Mar. Freshwater Res. 35: 703-712. Maller, R.A., and E.S. deBoer. 1988. An analysis of two methods of fitting the von Bertalanffy curve to capture-recapture data. Aust. J. Mar. Freshwater Res. 39: 459-446. McCauley, R.W., and D.M. Kilgour. 1990. Effect of air temperature on growth of largemounth bass in North America. Trans. Am. Fish. Soc. 119: 276-281. Milton, D.A., S.J.M. BBaber, and N.J.F. RawHinson. 1993. Age and growth of three species of clupeids from Kiribati, tropical central south Pacific. J. Fish Biol. 43: 89-188. Palmer, M.J., B.F. Phillips, and G.T Smith. 1991. Application of won linear models with random coefficients to growth data. Biometries, 448: 623-435. Sainsbury, K.J. 1980. Effect of individual variability on the von Bertalanffy growth equation. Can. J. Fish. Aquat. Sci. 37: 24 1-247. Wang, YG., ha. Thomas, and I.F. Somers. 1995. A maximum likelihood approach for estimating growth from tag-recapture data. Can. J. Fish. Aquat. Sci. 52: 252-259. Xiao, Y. 1994. von Bertalanffy growth models with variability in, and correlation between, k and E,. Can. J. Fish. Aquat. Sci. 51: 1585-1590.