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Fishery Bulletin 90(2). 1992 



Each of these geographic regions encloses a center of 

 Pacific whiting abundance and a concentration of 

 fishing activity (Dorn and Methot 1990). Three time- 

 periods were also defined as strata: (1) early (April- 

 June), (2) middle (July- August), and (3) late (Septem- 

 ber-November). These time-periods divide the fishing 

 season into three roughly equal parts. Over the years 

 1978-88, 27.9% of the catch came from the early time- 

 period, 47.4% came from the middle time-period, and 

 24.7% came from the late time-period. In compiling the 

 length-at-age estimates for the spatial and temporal 

 strata, all data collected within that strata were ag- 

 gregated and assumed to originate from random sam- 

 pling of the catch within that strata. 



Some of the detrimental effects of ageing error bias 

 and low sampling intensity of uncommon age groups- 

 common problems in analyzing fishery length-at-age 

 data— can be reduced if the precision of the length- 

 at-age estimates is known. A delta-method variance 

 estimator of length-at-age for a two-phase sampling 

 plan was derived and implemented for the U.S. fishery 

 samples. Details of this estimator and a procedure for 

 combining the length-at-age from different strata are 

 described in the Appendix. 



Two general methods of analyzing the growth in 

 length of fish have been used widely in fishery research. 

 The first method interprets individual observations of 

 length-at-age or mean length-at-age in the population 

 by fitting asymptotic growth curves, most typically the 

 von Bertalanffy (or monomolecular) growth curve 

 (Boehlert and Kappenman 1980, Kimura 1980, Shep- 

 herd and Grimes 1983). Using this technique to study 

 environmental effects on growth on an annual scale is 

 difficult because growth curves summarize the growth 

 history of a year-class or a population over the lifespan 

 of the organism. One approach to generalizing growth 

 curves is to include seasonal environmental effects on 

 growth. An example of this is the work of Pauly and 

 Gaschiitz (1979); they incorporated a sine wave in the 

 von Bertalanffy growth curve to model the seasonal 

 growth cycle. 



The second common approach to analyzing growth 

 data is analysis of variance (ANOVA). Factorial de- 

 signs have been used to investigate regional growth 

 variability (Francis 1983, Reish et al. 1985). Multiple 

 linear regression is often used to examine the effect 

 of the environment or population density on growth 

 (Kreuz et al. 1982, Ross and Almeida 1986, Peterman 

 and Bradford 1987). A factorial ANOVA of length 

 using age, year, region, sex, and time-period as fac- 

 tors is reported in the Results. It should be recog- 

 nized, however, that analysis of variance does not 

 account for changes in asymptotic growth, except by 

 fitting interaction terms that tend to obscure the 

 analysis. It is used in this paper only as an explora- 



tory technique to identify the sources of variability in 

 length. 



Because asymptotic growth is a universal feature of 

 fish growth, a model to examine the effect of the 

 environment on growth should account for this char- 

 acteristic. At the same time, such a model must be 

 general enough to allow for covariates to influence 

 annual growth. To meet this objective, a simple exten- 

 sion of the asymptotic von Bertalanffy growth model 

 was developed. The model has a framework similar to 

 analysis of covariance, in that it allows for the pos- 

 sibility of differences in growth between constituent 

 subgroups of the population and differences in growth 

 due to the influence of population density or environ- 

 mental covariates. 



The von Bertalanffy growth model for the mean 

 length la of a year-class at age a is given by 



la = U(l-e-K(a-a„)), 



where 1^^^ is the asymptotic maximum length, k is a 

 growth coefficient, and ao is the hypothetical age at 

 length zero. Subtracting the length at age an- 1 from 

 the length at age a gives the first difference of this 

 equation, the annual growth increment from age a to 

 age a-i-l, 



la.l - la = l„,(l-e-K)e-Ma-a„). 



Defining go = ln[l^(l-e-'^)], and gi= -k, a simple ex- 

 pression for annual growth is obtained: 



'a+l 



la = exp(go + gi(a-ao)). 



As might be expected, the parameter a^ becomes 

 redundant in this model for annual growth, since it is 

 confounded with the parameter go . One possibility is 

 simply to drop it from the equation. Another alterna- 

 tive, and the one used in this analysis, is to. use ao to 

 scale chronological age to some initial age for which 

 the growth model is intended to apply. In the Pacific 

 whiting data, there are growth increments from age 

 1 to age 2, so ao is set to 1. In this parameterization, 

 structural growth coefficients, go and g] , describe 

 simple elements of asymptotic growth: exp(go) is the 

 annual growth increment at age ao, and gi is the 

 exponential decline in the annual growth increment 

 (Fig. 2). 



To assess the effect of an environmental covariate, 

 X, this model is augmented with an additional coeffi- 

 cient for that environmental variable, 



la+l t+l s - lats = exp[go + ^ go, Xj 



i 



+ (gi + Z gij Xj)(a-ao)] + eats, 



j 



