Dorn: Environmental covariates of Merluccius productus growth 



263 



where eats'^NCO, WatgO^). The additional sub- 

 scripts in this equation are: t for year, s for 

 sex, and i and j to index different environ- 

 mental variables (e.g., Xi and Xj). The case 

 weights, Wats, ^1"^ determined by the sum of 

 the estimation variance for the two length-at- 

 age estimates used to calculate the growth 

 increment. 



In this regression model, environmental 

 variables can enter as either intercept or slope 

 terms. An intercept coefficient affects go and 

 indicates a constant percent change in the 

 growth increment regardless of age. Slope 

 coefficients affect gi and provide flexibility 

 for a varying percent change in the growth in- 

 crement with age. Together these two types 

 of coefficients, intercept and slope, cover a 

 wide range of different ways that environmen- 

 tal conditions can effect growth at different 

 ages. Note also that in this formulation, it is 

 possible to use indicator variables to param- 

 eterize growth differences between different 

 constituent groups of the population; for ex- 

 ample, sex differences or geographic differ- 

 ences in growth. This model resembles a linear 

 ANOVA model proposed by Weisberg (1986) 

 to analyze back-calculated fish lengths, though 

 he does not use von Bertalanffy growth to scale 

 the annual growth increments. 



The general procedure for fitting a nonlinear 

 regression model in Ratkowsky (1983) was 

 followed using the PAR algorithm in the 

 BMDP statistical package for estimating a non- 

 linear regression model using weighted least- 

 squares (Dixon 1983). Mean-square error was 

 estimated by fitting a full model consisting of 

 the coefficients go and gi , and separate inter- 

 cept and slope coefficients for all environmen- 

 tal covariates (temperature, upwelling, bio- 

 mass, recruitment strength) assessed in the 

 analysis. Mean-square error was estimated by 

 dividing the residual sum of squares for this 

 model by the degrees of freedom. A full model 

 should account for all the explainable variabil- 

 ity, so that the residual error gives an estimate of 

 mean-square error. A P-value of < 0.05 was established 

 as the criteria for statistical significance. Because of 

 the presence of negatively-valued growth increments 

 due to measurement error, it was not possible to take 

 the logarithm of the growth increment and analyze the 

 model using linear regression. 



The analysis with this model uses the change in mean 

 length of an age-group from the early period of the 

 fishery (April-June) of one year to the early period of 

 the following year. Geographic strata are not used in 



Age 



Figure 2 



Families of asymptotic von Bertalanffy growth curves parameterized 

 by g„, the initial growth increment, and g, , the exponential dechne in 

 the annual grovrth increment with age. All curves were constrained to 

 pass through 20 cm at age 1. 



the analysis because the migratory nature of the coastal 

 population of Pacific whiting would make any conclu- 

 sions regarding regional growth patterns impossible 

 to defend. It is assumed that the annual increment in 

 growth from one spring to the next is due to conditions 

 prevalent during the summer season of active growth. 

 Although growth increments could be studied for 

 shorter time-periods, this was considered inappropriate 

 for our study because of possible lags between envi- 

 ronmental conditions and the growth response of the 

 fish. In addition, the fishery estimates of length-at- 



