146 



Fishery Bulletin 94(1). 1996 



tion of the mixture of the components in the distri- 

 bution. Following the idea of the trimmed mean and 

 the idea of the mode, one is able to define a central 

 location which is located around the mode with a 

 certain amount of spread. A central value can be es- 

 timated from this spread and used to represent the 

 mean length for the cohort at time i. The estimated 

 mean length from several length intervals around 

 the mode should be more robust than the mode in 

 representing the growth of a cohort, especially when 

 the length distribution is skewed, which is often the 

 case for small-size samples. In this study, an ap- 

 proach using the mean length, estimated from the 

 central part of the distribution, for estimating the 

 parameters of a seasonal von Bertalanffy growth 

 equation was proposed and applied to green tiger 

 prawns, Penaeus semisulcatus, in Kuwait waters. 

 With the proposed method, which might be termed 

 the Central Location Measure, one is able to apply 

 the methods of the linear or nonlinear least squares 

 in order to estimate the growth parameters with 

 variances and covariances and thus to study the sta- 

 tistical differences in growth performance between 

 sexes and between cohorts. 



Materials and methods 



The seasonal growth model and the fitting 

 technique 



Various versions of the seasonal growth models which 

 incorporate some parameters defined by season into 

 the von Bertalanffy model (von Bertalanffy, 1938) 

 have been proposed (Ursin, 1963; Pitcher and 

 MacDonald, 1973; Cloern and Nichols, 1978; Pauly 

 and Gaschiitz 1 ). Among these models, Pauly- 

 Gaschiitz's model is the most widely used: 



L t =L„ 



1-exp 



-K\ t-t B + — sin2)i(J-i.) 

 2n 



(1) 



where L t = the length at age t; L^ is the asymptotic 

 length; K = the intrinsic growth rate; t {} = the age at 

 which the length of the animal is 0; and t B and C are 

 the parameters denning the seasonal growth patterns. 

 The seasonal growth model (Eq. 1) was fitted by 

 using the nonlinear least squares method with SAS 



1 Pauly, D., and G. Gaschiitz. 1979. A simple method for fit- 

 ting oscillating length growth data, with a program for pocket 

 calculators. Int. Coun. Explor. Sea, Council Meeting 1979/G:24. 

 26 p. 



Proc NLIN (SAS, 1992). A SAS program for fitting 

 the seasonal growth model with the method of the 

 nonlinear least squares is available upon request. A 

 multivariate test, Hotelling's T 1 statistic (Johnson and 

 Wichern, 1992), was used to test the hypothesis of no 

 differences in growth between two populations, i.e. H tj : 

 /}j=/3 2 , if the assumption of the equality of the two co- 

 variance matrices holds; otherwise, an alternative sta- 

 tistic, T'*, for unequal covariance case (Johnson and 

 Wichern, 1992; Hanumara and Hoenig, 1987 ) was used: 



r* = (ft-A0(A+A0 (0i-A>) 



(2) 



where p and D j = the vectors of the estimated growth 

 parameters and the estimated covariance matrix of 

 the growth parameters, respectively, for population 

 i. The null hypothesis, H Q : fi x =fi , is rejected if 

 T v > Xa-.p , where p is the dimension of multinormal 

 populations. The hypothesis of equality of??; covari- 

 ance matrices, H : £j = Z 2 = ••• =£,„> was tested by 

 using an approximate chi-squared statistic MC l 

 modified from Morrison ( 1990, p. 297), for which 



m m 



M = £(*,-l)ln|S|-Xk-i)m|S,| 



(=i 



i=i 



and 



c -i = 1 . 2p-+3p- 



6(p+l)(w-D 





£(n,-l) 



i i I 



where n, is the sample size for population i; In indi- 

 cates natural logarithm; S ; (the estimate of £,■) is set 

 as S =n D in this study as suggested by Hanumara 

 and Hoenig (1987); S is the pooled estimate of the 

 common covariance matrix of the estimated growth 

 parameters; and \S\ represents the determinant of 

 the matrix S. 



Bernard (1981) and Hanumara and Hoenig* 1987) 

 discussed the application of Hotelling's T l in com- 

 paring the growth offish; however, the statistics pro- 

 posed by Bernard ( 1981 ) and Hanumara and Hoenig 

 ( 1987 ) were different by a constant multiple, because 

 they treated the covariance matrix of the growth 

 parameters differently. In this study, the estimated 

 covariance matrix of the growth parameters has the 

 same form as that proposed by Hanumara and 

 Hoenig ( 1987), therefore the latter was used. 



The approach to obtaining length-at-age 

 data 



To estimate the parameters in model 1 (Eq. 1), one 

 needs a set of length-at-age data. The method pro- 



