A simple generalized model of 

 allometry, with examples of length 

 and weight relationships for 

 1 4 species of groundfish 



Yongshun Xiao 

 David C. Ramm 



Fisheries Division, Department of Primary Industry and Fisheries 

 GPO Box 990. Darwin NT 0801. Australia 



Allometry is a set of relations be- 

 tween an animal's characteristics 

 and its body size, and is applied in 

 many branches of biological sci- 

 ences including ecology, physiology, 

 and morphology (Peters, 1983; 

 Calder, 1984; Schmidt-Nielsen, 

 1984; Bookstein et al., 1985; Reiss, 

 1989). Allometry is represented by 

 the power function, \y = pj*o , 

 where W is a characteristic of an 

 animal (e.g. body weight), L is its 

 body size, and A and X 9 are its al- 

 lometric parameters. To determine 

 an allometric relationship for a par- 

 ticular characteristic, the power 

 function is usually, albeit at times 

 inappropriately, double log-trans- 

 formed into a simple linear equation. 



Y = X,+X,X,, 



(1) 



with Y = log(W), X, = log(A), and 

 X 3 = log(L), and is then fit to data 

 from different individuals. 



Use of allometry in this way as- 

 sumes constancy of X ; and X,, in 

 Equation 1. While both allometric 

 parameters may be treated ap- 

 proximately as constants in certain 

 applications, the assumption may 

 be violated for a wide variety of bio- 

 logical phenomena because of ge- 

 netic, phenotypic, and/or behav- 

 ioral variability among individual 

 animals. In fact, Mosimann and 

 James ( 1979) have concluded that 

 X. y varies spatially in the Florida 

 red-winged blackbird, Agelaius 



664 



phoeniceus. Variability in X, is also 

 implied in Reiss' ( 1989) hypothesis 

 thatX., contains phylogenetic infor- 

 mation and is less variable 

 intraspecifically than inter- 

 specifically. Peters (1983) convinc- 

 ingly demonstrated interspecific 

 variation in X 9 and computed its 

 mean and standard deviation for 

 metabolic rates scaled to body sizes 

 across many animal taxa. Variabil- 

 ity inX ; has not been examined but 

 is certainly implied in the compre- 

 hensive appendices of Peters' 

 ( 1983 ) book on the ecological impli- 

 cations of body size and in Reiss' 

 ( 1989) monograph on the allometry 

 of organismic growth and reproduc- 

 tion. X ; may be strongly negatively 

 correlated with X., for length- 

 weight relationships in fish (e.g. 

 Caillouet, 1993). 



Variability inX ; andX 9 may have 

 major implications in the widely 

 used allometric equation because it 

 represents a fundamental concept 

 in biology (Peters, 1983). In this 

 paper, we generalize Equation 1 by 

 explicitly incorporating variability 

 in and correlation between, X ; and 

 X 9 , and study the consequences of 

 such variability and correlation in 

 allometric predictions. The gener- 

 alized model is demonstrated by 

 using length and weight relation- 

 ships for 14 species of groundfish 

 of the families Centrolophidae, 

 Haemulidae, Lethrinidae, Lutjan- 

 idae, Nemipteridae, and Synodon- 



tidae from northern Australian 

 waters. 



Model 



Suppose that a joint probability dis- 

 tribution of Xj and X 9 conditional 

 on X,j could be formed for a group 

 of animals, with each individual 

 having its own pair of allometric 

 parameters which it retains 

 throughout its life, and that values 

 of pairs of allometric parameters 

 are serially independent. The value 

 of Y for the z'th individual with al- 

 lometric parameter pair (X ; , X, ) at 

 X 3 is 



y, = x ; , + x 2 ,x :l . 



For a group of animals selected ran- 

 domly from the population, the ex- 

 pected value of Y at X. { is 



E\Y\X 3 ] = E[X 1 + X 2 X :1 ] (2) 



with variance 



V[Y\X 3 ] =v\x, +A',A',| (3) 



= E[Y 2 \X :l ]-E\Y\X 3 f 



= E[iX l + X,X l r]-ElX, +A-.,.Y ( |' J 



Given information on howX ; and 

 X., vary, one can develop Equations 

 2 and 3. X, may closely follow a 

 normal distribution for metabolic 

 rate of animals scaled to body size 

 (Peters, 1983), being strongly nega- 

 tively correlated with X, for length- 

 weight relationships in fish (e.g. Cail- 

 louet, 1993). We will assume below 

 that Xj and X., follow a joint nor- 

 mal distribution, i.e. (X ; ,X 9 ) ~ N 

 ^j.i 1 4i 2 \a 1 1 ,a 2 \p) with mean p ( , and 

 variance of of X., and correlation 

 coefficient p. Under general condi- 

 tions, the sum (or average) of a 

 number of random variables is ap- 

 proximately normally distributed, 

 and such approximation can be 

 quite good even if that number is 

 relatively small. The above assump- 



Manuscript accepted 25 February 1994. 

 Fishery Bulletin 92: 664-670 ( 1994). 



