NOTE Xiao and Ramm: A simple model of allometry for groundfish 



665 



tion would be at least approximately valid because both X t and X. 2 can be regarded as the sum (or average) of 

 numerous (e.g. genetic, phenotypic, and behavioral ) random components. Analogous models may be developed for 

 other probability distributions. Under that assumption. Equations 2 and 3 become, respectively, 



E[Y\X 3 ] = E[X,+X 2 X 3 ] 



and 



\\ -r 



(Xj + x 2 X 3 )e 2 " 



"';-^' 't.-p : . l' 



dxjdx 2 



(4) 



V[Y\X 3 ] = V[X,+X 2 X 3 ] 



E[Y 2 \X 3 ]-E[Y\X 3 ] 2 



E [( X, + X 2 X 3 ) 2 ]-E[X,+ X 2 X 3 f 



a 2 + 2o 1 o 2 pX 3 + o 2 2 X 2 , 



(5) 



Thus variability in, and correlation between, X t and 

 X 2 only affect V[Y\X 3 ].V[Y\ X 3 ] increases linearly 

 with p from (a, - o.,X 3 f at p = -1 through o 2 + o 2 X 2 

 at p = to (o, +o 2 X 3 ) 2 at p = 1. It quadratically 

 decreases with o,, o 9 , and Xo to a minimum 



T 2 X 3 



of a 2 X^(l-p z )>0 at ct 



-o 2 pX 3 , crjd-p") at 

 a., = -a,p/A 3 ana crju-p i at X 3 =-a lp la 2 , re- 

 spectively, and finally increases unboundedly, under 

 the constraint that o l ,o. I , andX 3 > 0. However, ifX 1 

 and X are both deterministic (o t =0,p = 0), 

 V[Y\X 3 ] = 0. 



If Xj is random (c? >0) and X is deterministic 

 (<r 2 2 = 0,p = 0),V[Y \X 3 ) = a* HXris random (a\ > 0) 

 and Xj is deterministic (07 = 0,p = 0), V17 I X 3 ] = 

 otX 2 . Finally, ifX ; andX, are random but independent 

 (a] > 0,ct| > and p = 0)" V[Y \X 3 \= a 2 +a 2 X 2 . 



Data and parameter estimation 



Data on fish weight at length were collected from 

 Australia's continental shelf in the Timor and Arafura 

 Seas (9-14°S, 127-137°E) from 20 October to 16 De- 

 cember 1990 as part of the Northern Territory De- 

 partment of Primary Industry and Fisheries' pro- 

 gram assessing commercial fish stocks. Of 240 sta- 

 tions allocated randomly within a depth range of 20- 

 200 m, 199 were successfully sampled with a Frank 

 and Bryce trawl net (headline height, 2.9 m; wing 

 spread, 14.4 m; door spread, 60.1 m) at a speed of 

 1.54-2.06 rrvs -1 . Nearly 48 tonnes of fish' represent- 

 ing about 483 species in 119 families were caught 

 during sampling. A representative subsample of in- 

 dividuals of 14 species, mostly of commercial fish, of 

 the families Centrolophidae, Haemulidae. Lethrin- 

 idae, Lutjanidae, Nemipteridae, and Synodontidae 



were frozen immediately on board, returned to the 

 laboratory, thawed, sexed, measured (fork length) to 

 the nearest 1 mm, and weighed (wet weight) to the 

 nearest 1 g with an electronic balance (Mettler, 

 PC4000). For each of the 14 species, data on indi- 

 vidual wet weight at length were pooled across all 

 stations and fit to all cases of Equations 4 and 5 for 

 females, males, and mixed sexes. Parameter esti- 

 mates indicated by hats ( A ) were obtained by linear 

 regression for Equation 1 by using SAS regression 

 procedure (SAS Institute Inc., 1985) and by maxi- 

 mizing the general likelihood function, 



L = u\2nV[Y\X,] + a 2 } 



, = lL J; 



for all other models by using the simplex algorithm 

 of SYSTAT nonlinear regression procedure (Wilkin- 

 son, 1989). We included a model error term, a 2 , in 

 the likelihood function to show that, in this case, it 

 is compounded with a 2 and is hence equivalent to 

 a 2 and a 2 + a 2 for estimation purposes. For this rea- 

 son, we treated both error components collectively 

 as ' a 2 ' during model fitting and result presentation, 

 unless otherwise stated. 



Results 



Some statistics offish length and weight data used 

 in this analysis are given in Table 1. We attempted 

 to fit data for mixed sexes (both sexable and 

 unsexable individuals included) and males and fe- 

 males (with unsexable juveniles excluded) of each of 

 14 species of groundfish to all cases of Equations 4 

 and 5. However, parameters could be estimated for 

 models with o 2 or o 2 only; those in models simulta- 



