Brodziak and MiRlis Variation in life history parameters of Miaostomus paaficus 



663 



and Fournier, 1980; Ratkowsky, 1983) because its para- 

 meters were simpler to interpret and because it exhibited 

 better statistical properties than other parameterizations 

 of the von Bertalanffy model. This form was 



Ut,) = L +(L,„,^ - L 



(1- 



d-c'- 



(1) 



where 

 L ,L 



min^ max 



t, 



and c 



sample; 



(„„„ and t„,„_^ 



^nun and L„,„^. 



age of the /''^ ; 



parameters; 



denote the youngest and oldest ages 



observed in the length-at-age sample; 



denote the predicted lengths at ages 



'„„„ and /„,„^.; and 



independent and identically distributed 



normal error terms with zero mean and 



constant variance e, - NiO, d-). 



asymptotic standard error estimates were used owing to 

 the large sample size. 



As found in previous research, some sexual dimorphism 

 in growth was expected, but there was no information 

 about geographic variation in growth. A likelihood-ratio 

 test (Kimura. 1980) was applied to determine whether 

 growth curves differed by sex or by INPFC area. This test 

 compared two hypotheses: H^, the hypothesis of identical 

 growth parameters between sexes or among areas; and 

 //[, the hypothesis of different growth parameters. The 

 test statistic {x~) was 



X- = -N\og 



o 



(2) 



where a,? and a. 



sample estimates of residual variance 

 for growth curves estimated under H^ 

 and Hy 



There is a one-to-one relationship between parameters of 

 the alternative form and the standard von Bertalanffy 

 model (see Ratkowsky, 1983). Residuals from estimated 

 growth models were tested for normality with the Sha- 

 piro and Wilk (1965) test. Standard errors of parameter 

 estimates were computed by using the conditional boot- 

 strap with 1000 bootstrap replicates (Efron and Tibshi- 

 rani, 1993), except for the pooled-sex analysis, where 



The likelihood-ratio test was applied to male and female 

 samples from all INPFC areas and was then separately 

 applied to male and female samples for paired adjacent 

 INPFC areas. When there was no difference between adja- 

 cent areas, samples were grouped and the process was 

 repeated. Standard errors of parameter estimates of the 

 final groups were computed by using the conditional boot- 

 strap with 1000 replicates. 



