Abstract.— Recent papers have 

 provided new insights into the prob- 

 lem of estimating von Bertalanffy 

 growth parameters from tag- 

 recapture data. In particular, the in- 

 consistency and bias of Fabens' 

 (1965) estimates appear to have been 

 addressed by James (1991). Using 

 simulation, we examine the pattern 

 of bias associated with different er- 

 ror assumptions for Fabens' esti- 

 mates, weighted Fabens' estimates 

 proposed by James, and a robust 

 method also proposed by James. Our 

 results corroborate James' finding 

 that his robust estimates can be sig- 

 nificantly less biased than other 

 methods. We then apply these esti- 

 mators to tag-recapture data ob- 

 tained for sablefish Anoplopoma fim- 

 bria found in the Gulf of Alaska and 

 off the U.S. west coast, and Pacific 

 cod Gadus macrocephalus found in 

 the eastern Bering Sea. These spe- 

 cies are difficult to directly age, so 

 tag-recapture data provide welcomed 

 independent estimates of growth pa- 

 rameters and an indirect method of 

 validating age-determination crite- 

 ria. The von Bertalanffy parameter 

 estimates using tag-recapture data 

 and James' method were most simi- 

 lar to estimates calculated directly 

 from length-at-age data. 



Estimating von Bertalanffy growth 

 parameters of sablefish 

 Anoplopoma fimbria and Pacific 

 cod Gadus macrocephaius using 

 tag-recapture data 



Daniel K. Kimura 

 Allen M. Shimada 

 Sandra A. Lowe 



Alaska Fisheries Science Center National Marine Fisheries Service. NOAA 

 7600 Sand Point Way NE. Seattle. Washington 98 1 1 5-0070 



Manuscript accepted 22 January 1993. 

 Fishery Bulletin, U.S. 91:271-280 ( 1993). 



In its most common form, the von 

 Bertalanffy growth curve has three 

 parameters (L,,, K, t ). Conventional 

 interpretation of these parameters is 

 that L, is asymptotic size, K de- 

 scribes the growth rate, and t de- 

 scribes the age at length-0. Fabens 

 (1965) was apparently the first to 

 show how least-squares could be used 

 to estimate two of these parameters 

 (L_, K) from tag-recapture data. If at 

 least one additional value of length- 

 at-age was known, then t could be 

 estimated, and hence all values of 

 length-at-age could be estimated di- 

 rectly from recapture data without 

 recourse to direct ages from indi- 

 vidual specimens. Besides providing 

 growth-curve estimates, such a pro- 

 cedure would seem to provide an in- 

 direct validation of ageing criteria, 

 since estimated growth parameters 

 from length-at-age data and tag- 

 recapture data could be compared. 



However, comparisons of growth 

 curves estimated using Fabens' 

 method and ordinary length-at-age 

 data seemed to provide evidence that 

 Fabens' method provided biased pa- 

 rameter estimates. Following the sug- 

 gestion of Chapman (1961) and oth- 

 ers, Sainsbury (1980) showed how 

 individual variability [i.e., the possi- 

 bility that each fish has different val- 

 ues of (L., K)], could lead to bias in 

 population parameter estimates. 



Francis (1988a) argued from a 

 sampling point of view that von 

 Bertalanffy parameters calculated 

 from length-at-age and tag-recapture 

 data could be different. He also ar- 

 gued that average growth parameters 

 for individual fish may not describe 

 the population growth curve (Francis 

 1988b). Mailer & deBoer (1988) 

 showed mathematically and with 

 simulation that Fabens' estimates, 

 and related estimates of Kirkwood & 

 Somers (1984), could be inconsistent. 

 Kirkwood (1983) suggested combin- 

 ing length-at-age and tag-recapture 

 data into a single likelihood model, 

 but did not address the problem of 

 bias caused by the interpretation of 

 tag-recapture data. 



A recent paper by James (1991) 

 suggests that bias in Fabens' esti- 

 mates could arise from bias in the 

 estimating functions. James shows 

 how weighted least-squares can be 

 used to derive improved estimates 

 for Fabens' method under the usual 

 "observational error" assumption. 

 James also gives distribution-free es- 

 timates for a more general model in 

 which, in addition to the usual ob- 

 servational error, L . itself is allowed 

 to vary among individual fish. Al- 

 though James' estimates appear to 

 be consistent and less biased than 

 the least-squares estimates, they also 

 appear to be less efficient, usually 



271 



