666 



Fishery Bulletin 92(3), 1994 



neously with o? and ol, or simultaneously with 

 a\,c\ and p could not be estimated because of over- 

 parameterization. Estimates of parameters, derived 

 from linear regression of Equation 1 by using least 

 squares method — equivalent to maximizing the like- 

 lihood function 



n r ,,-i [Y,-E[Y\X 3 ),f 



L= I~I 2/roj ~e i^f 



and from maximizing the likelihood function 



L = n|2/ro-oX 



1 = 1 



[Y,-EIY\X 3 ] 



are given in Tables 2 and 3 respectively. Estimates 

 in both tables are very similar between sexes for each 

 species and between species, roughly with a species- 

 wide /j, =-10.89, fi 2 =2.99, o t =-0.006638 and 

 6 2 =0.014932. Thus, while V[Y\X :i \ can be treated 

 approximately as a constant, as is usually assumed 

 in previous applications, it does change quadratically 

 with X Q . 



Discussion 



Peters ( 1983) observed a large amount of variability 

 in most allometric relationships and recognized a 

 need to identify independent variables of general bio- 

 logical interest other than size. The general model 

 presented in this study takes into account both body 

 size and parameter variability among individual ani- 

 mals in allometric predictions. A major problem in 

 allometry is that allometricians are more apt at pro- 

 viding a statistical description of a new data set than 

 at using their data for hypothesis testing (Peters, 

 1983). This tendency has led to a plethora of only 

 slightly different allometric equations, none of which 

 can be rejected objectively. Our general model or any 

 of its special cases would form a basis for intrataxal 

 or intertaxal generalizations by treating some of 

 those estimates of allometric parameters as 

 intrataxal or intertaxal variations, hence providing 

 a means for a general "house cleaning" in allometry. 



Incorporating more independent variables in allo- 

 metric modelling may explain more variability in the 

 dependent variable, but it may result in a loss of a 

 basis for comparison between, and manipulation of, 

 allometric equations, such as allometric cancellation 

 (Calder, 1984). The model presented above conforms 

 exactly with conventional allometry and maintains 

 commensuration by its estimated parameter means. 



Specification of error structures in allometric mod- 

 els is an essential part of allometric modelling. Er- 

 rors for Equation 1 are often assumed to be normally 



distributed with a constant variance, say a 2 . Sev- 

 eral other interpretations arise from V[Y"IX 3 ] in that, 

 for estimation purposes, cr 2 can be interpreted by any 

 combinations of terms on the right-hand side of Equa- 

 tion 5. These and other alternative interpretations 

 may pose problems for some applications. Thus, er- 

 ror structures of an allometric model must be speci- 

 fied cautiously. 



There was no gain in precision or accuracy in esti- 

 mates of allometric parameters in length and weight 

 relationships of some fishes from considering indi- 

 vidual variability of allometric parameters. Both 

 Equation 1 and Equations 4 and 5 with of or of, 

 alone give an equally adequate description of weight 

 at length data from all 14 species of groundfish con- 

 cerned. Overparameterization occurred in cases of 

 Equations 4 and 5 simultaneously with a 1 and o|, 

 or simultaneously with of, oJ , and p, and, as a re- 

 sult, not all parameters could be estimated from our 

 data. The overparameterization lent further support 

 to this conclusion. Also, although a 1 and a 2 can be 

 estimated separately for each species, they are ei- 

 ther equivalent to model error or take such small 

 values (Tables 2 and 3) that V[yiX 2 ] can be treated 

 effectively as constant. Finally, when interpreting 

 regression results from various cases of the general 

 model, it should be noted that all other variability 

 will be confounded with, and added to, that of allom- 

 etric parameters. Our data sets are of moderate sizes 

 (Table 1) and many others of similar size could be 

 expected to behave similarly. Individual variability 

 of allometric parameters probably has a negligible 

 effect on allometric predictions in length and weight 

 relationships of certain fishes. Thus, our work sup- 

 ports the common use of Equation 1 to model in- 

 traspecific length and weight relationships in those 

 fishes. However, all parameters in Equations 4 and 

 5 may be estimable simultaneously for length and 

 weight relationships, as well as for other allometric 

 relationships, if larger data sets or higher taxonomic 

 levels, or both, are used. 



A key assumption in our model is that the inde- 

 pendent characteristic, L, (e.g. length) has little mea- 

 surement error relative to the dependent character- 

 istic, W (e.g. weight). Theoretically, this may not be 

 the case. However, we believe that our model will 

 provide good approximations for many allometrically 

 scaled phenomena, such as length and weight rela- 

 tionships in certain fishes. For other allometric phe- 

 nomena, alternative formulations, such as those of 

 Pienaar and Ricker (1968), Saenger (1989), Seim and 

 Saether ( 1983), and Shoesmith ( 1990) may be useful. 



V | Y I X 3 ] is a function of the independent variable 

 whenever there is individual variability in X 9 or in 

 X, andX,,. If this is not taken into account in regres- 



