evaluated at the ML estimates, and s^ is the mean 

 square error. 



For estimates calculated using method (a), 



FISHERY BULLETIN; VOL 77, NO. 4 



Z' 



dli(ti) 



a/n 



Mti) 



dtr. 



Mh) 



dtr 



Mt,) 



a/<i 



3XN 



evaluated at (i,,k,iol, with s^ = S(l^.K,t„)/ 

 {N-3), N = 1, n^ and /the number of age 

 categories. 



As was previously noted for method (d), it is 

 often advantageous to view this model as being 

 unweighted with transformed variables. From 

 this point of view, y^ = (VnOh is the dependent 

 variable, with expectation E(y,) = {V ni)fji(t^). 

 Under this parameterization, variy^) = cr^ with 



7' = 



^ III 



Mti) 



a^ ' 



Mil) 

 dK 



Mh) 

 dtn 



\/n7 



Mh) 



dL 



Mt,) 

 dK 



Mt,) 

 dto~ 



3X/ 



again evaluated at {L,K,i„), with s„,^ = S„. 

 (Lk,io)/(I-3). 



It is easily verified that Z'Z = Z„, 'Z„ and, there- 

 fore, any differences in the covariance matrix es- 

 timates of parameter estimates must be due to 

 differences in the estimates s^ and s„,^ of (t^. Al- 

 though s^ provides a better estimate than s„,^ in 

 terms of degrees of freedom {N-3 versus 7-3), 

 they should be similar in value, and hence it can be 

 expected that methods (a) and (d) provide similar 

 covariEince matrix estimates of parameter esti- 

 mates. 



This analysis points out that good estimates of 

 the covariamce matrix of pairameter estimates re- 

 quire having sufficient numbers of observations so 

 that a^ is adequately estimated. From this point of 

 view method (a) is superior to method (d). How- 

 ever, method (d) can be modified so that data are 



not completely collapsed (averaged) at each value 

 of the independent variable. Instead, data can be 

 partitioned so that there are several dependent 

 variable averages, and weights, at each value of 

 the independent variable. A similar technique can 

 be applied to method (c). 



Another possibility would be to estimate o^ in- 

 dependent of the LS calculations by pooling the s,^ 

 values. However, this estimate based on pure 

 error would tend to underestimate the true (j^ 

 which will often contain a lack of fit component 

 (see the following section for a discussion concern- 

 ing pure error and lack of fit). 



FAILURE OF ASSUMPTIONS 



There are two ways in which the assumed model 

 can fail: 1) growrth may not follow the von Ber- 

 talanffy curve; or 2) error assumptions may not 

 hold. 



Failure of the von Bertalanffy Curve 



Even when growth follows the von Bertalanffy 

 curve, expected lengths at age from sample data 

 may not. Discrepancies can be caused by bias in 

 sampling, bias in age determination, or size selec- 

 tive survival in the natural population. Because 

 samples tend to be biased toward larger individu- 

 als, age readers tend to under-age older individu- 

 als, and larger individuals of an age group tend to 

 have better survival, these factors may bias the 

 observed size upward for a given age. 



When a number of length specimens are avail- 

 able at each age, a statistical measure of lack of fit 

 (departure from the von Bertalanffy curve) can be 

 calculated using the procedure described by 

 Draper and Smith (1966). For this analysis, it is 



768 



