134 



Fishery Bulletin 101(1) 



turbing each parameter ±1% and calculating the resulting 

 effect on biomass. 



In some instances, variance estimates (i.e. squared stan- 

 dard errors) and covariances were extracted directly from 

 computer output produced by the SAS (1987) procedures 

 PROC REG and PROC NLIN. In the case of linear regres- 

 sions, these are the usual estimates of these quantities 

 (e.g. Draper and Smith, 1981), whereas for nonlinear re- 

 gression, these are asymptotic variances and covariances 

 (e.g. Seber and Wild, 1992). In addition, following the as- 

 sumptions of normal-based regression, the residual error 

 parameter estimates (ct^,,,^,) used in bias adjustments are 

 independent of other regression parameters (i.e. covari- 

 ances are zero) and n&l^_.Ja^^^ has a chi-square distribu- 

 tion with k degrees of freedom, where k is the degrees of 

 freedom associated with (Tj,^^^. (e.g. for a linear regression 

 k=n-2). This chi-square distributed random variable, i.e. 



no:. 



2k, 



was then used to estimate the variance of the mean square 

 errors (Larsen and Marx, 1981 ) 



var(o--.,. 





x2k. 



51 5.2 53 5,4 5.5 56 5.7 58 5.9 

 LoQe (total length [mm]) 



Figure 2 



Fit of ordinary least-squares regression to log^.-transformed 

 weight and length data from shortbelly rockfish sampled 

 by trawl in February-March 1991. 



covariances among the von Bertalanffy parameters for 

 males and females. 



For these cases, a generalized version of the delta 

 method was used to estimate covariances (and variances): 



This straightforward approach to obtaining variances 

 and covariances is appropriate when a suite of param- 

 eters estimated by a regression procedure is based on 

 a data set that could reasonably be expected to be inde- 

 pendent of data sets used to estimate other parameters, 

 and when values of other estimated parameters were not 

 used as "known" constants during a regression or estima- 

 tion procedure. In two cases, however, estimates of other 

 parameters were treated as known constants during a 

 regression or estimation procedure. These cases were 1) 

 estimation of the mean and standard deviation of the 

 seasonal distribution of age-0 larvae, and 2) estimation of 

 von Bertalanffy growth equation parameters. In the case 

 of the seasonal distribution of age-0 larvae, parameter 

 estimates depended upon the natural mortality rate esti- 

 mated for larvae (Z). Hence, variances for these param- 

 eters calculated directly from the regression procedure are 

 conditional on the estimated mortality rate. Imprecision 

 associated with estimating this rate adds to variances of 

 these parameters, and influences their covariance. Like- 

 wise, for von Bertalanffy growth, the weight "data" used 

 were calculated on the basis of the estimated relationship 

 between weight and length. Hence regression estimates 

 of variance and covariances for the von Bertalanffy 

 parameters are conditional on the parameters for the 

 weight-length relationship. Uncertainty associated with 

 the estimates of the weight-length parameters influences 

 the unconditional variances and covariances of the von 

 Bertalanffy parameters, sets up covariances between the 

 two suites of parameters, and because the same weight- 

 length relationship was used for both sexes, also sets up 



cov[g(.v,y),/!(.v,£)] 



^^cov|.r,,x^ 



dx, dx, 



(Seber, 1982, p. 9). Hereg and /; represent regression pro- 

 cedures by which two specified parameters are estimated, 

 X represents a set of shared values (assumed known 

 parameters and data) used by the two regression proce- 

 dures, y and z represent distinct data or parameters with 

 no covariance among them, and partial derivatives are 

 calculated as described above. 



Results 



Population weight-specific fecundity and sex ratio 



The weight (W) of a fish is equated to its length (TL) by 

 use of the power function, i.e. 



W = aTUK 



where n and /i are fitted parameters (Ricker, 1975). 



In our study, weight (gni) and TL (mm) data were collected 

 from 352 fish sampled during the two adult trawl cruises. 

 The (lata from these fish were log -transformed and fitted 

 by linear regression I Fig. 2, Table 2). The fit was good, as 

 indicated by a high r'~ value (0.94), and the distribution 

 of residuals did not deviate from normality (P=0.109). To 

 predict weight at lengtli on the unlransformed scale, a 



