Estimation of weight-length 

 relationships from group 

 measurements 



William H. Lenarz 



Tiburon Fisheries Laboratory 



National Marine Fisheries Service, NOAA 



3 1 50 Paradise Drive. Tiburon, CA 94920 



Catch sampling provides data 

 that are basic to fisheries re- 

 search and is often an important 

 component of research budgets. 

 Samplers typically select fish ran- 

 domly, measure length, remove 

 ageing structures, and determine 

 sex for each individual. In many 

 schemes for sampling commercial 

 (e.g., Sen, 1986; Tomlinson, 1971) 

 and survey catches (e.g., Gun- 

 derson and Sample, 1980), sample 

 weight is needed to expand the 

 sample results to the total catch. 

 Individual weights are usually 

 not needed to satisfy the main 

 objectives. Often only the aggre- 

 gate weight of the sample is taken 

 to save time, and if at sea, to 

 avoid difficult logistics. While 

 sampling costs are easily justified 

 by program objectives, scientists 

 frequently use the data for addi- 

 tional research. 



Investigators often use weight- 

 length relations to study possible 

 correlations between condition of 

 fish and environmental factors or 

 population density (e.g., Pat- 

 terson, 1992). A literature search 

 revealed only two previous devel- 

 opments of methods of estimating 

 weight-length relations from 

 samples of individual lengths and 

 aggregate weights (WLRAW). 

 Cammen (1980) used a general 

 nonlinear regression program 

 from the BMDP package (Dixon, 

 1983) as a WLRAW method. He 

 tested the method with simulated 

 data and compared the results of 

 regression using unweighted ob- 

 servations to using observations 

 weighted by the inverse of sample 



weights, and with various esti- 

 mates made when individual 

 weights were known. Since the 

 data were simulated, assuming a 

 multiplicative error term, it would 

 have been more appropriate to 

 use the inverse of sample weight 

 squared for weighting. The non- 

 linear method produced good fits 

 to the simulated data, and 

 weighted parameter estimates 

 were closer to the true values 

 than unweighted estimates. 

 Damm ( 1987) developed two non- 

 linear WLRAW methods. One 

 method is a biased approxima- 

 tion, and his report indicated that 

 the other method did not always 

 produce estimates of the param- 

 eters. 



In this note I describe a new 

 WLRAW method, compare it with 

 Cammen's method, explore error 

 term characteristics, and describe 

 bootstrap estimates of confidence 

 limits of estimates. The methods 

 of Damm (1987) were not studied 

 because his biased approximation 

 method requires as much calcula- 

 tion as my new method and his other 

 method does not always work. 



Methods 



The relation between expected 

 weight and length of an indi- 

 vidual fish is usually assumed to 

 be the power equation, 



E(W l ) = alf l 



Where V^ = weight of fish i, 

 a - parameter. 



(1) 



L, = length of fish i, 



p = parameter. 

 For the new WLRAW method I 

 modeled the weight-length rela- 

 tionship as 



W, 



flK) 



+ £, 



(2) 



J i=i 



where W = 



L. = 



e. = 



T = 



average weight of 

 fish in sample j, 

 number of fish in 

 sample j, 

 length of fish i in 

 sample j, 

 error term for 

 sample j, 

 1,  • • , T, 

 number of 

 samples. 



I assumed that error was additive 

 because under field conditions 

 much of the error was due to lim- 

 its to the accuracy in measure- 

 ment of sample weights. Because 

 the dependent variable in Equa- 

 tion 2 was a sample average, its 

 variance should contain a compo- 

 nent which is proportional to the 

 inverse of n . Thus in the new es- 

 timation procedure, I weight each 

 observation by n to stabilize the 

 variance. I made the assumption 

 that, after weighting by sample 

 size, error was random and inde- 

 pendent of,/. 



The new method treated esti- 

 mation of parameters of (Eq. 2) as 

 a separable least-squares problem 

 (Seber and Wild, 1989). For a trial 

 value of (3 (P'l, y was calculated 

 for each sample, 



/,=<2X>/», 



(3) 



With the new notation, Equation 

 2 becomes 



W- = a y ; + e 



j~ 



(4) 



Manuscript accepted 16 August 1993 

 Fishery Bulletin 92:198-202 (1994) 



198 



