789 



Abstract— I sinuilali>d somatic fjrowth 

 and accompanyiiit,' otolith growtli using 

 an individvial-l)as('(l bioenorgotics model 

 in order to examine the performance ol' 

 several back-calculation methods. Four 

 shapes of otolith radius-total length re- 

 lations (OK-TL) were simulated. Ten dif- 

 ferent back-calculation equations, two 

 different regression models of radius- 

 length, and two schemes of annulus 

 selection were examined for a total of 

 20 different methods to estimate size at 

 age from simulated data sets of length 

 and annulus measurements. The ac- 

 curacy of each of the twenty methods 

 was evaluated by comparing the back- 

 calculated length-at-age and the true 

 length-at-age. The best back-calculation 

 technique was directly related to how- 

 well the OR-TL model fitted. Wlien 

 the OR-TL was sigmoid shaped and all 

 annuli were used, emploving a least- 

 squares linear regression coupled with 

 a log-transformed Lee back-calcula- 

 tion equation (y-intercept corrected) 

 resulted in the least error; when only 

 the last annulus was used, employing a 

 direct proportionality back-calculation 

 equation resulted in the least error. 

 When the OR-TL was linear, emploving 

 a functional regi-ession coupled with the 

 Lee back-calculation equation resulted 

 in the least error when all annuli were 

 used, and also when only the last an- 

 nulus was used. If the OR-TL was ex- 

 ponentially shaped, direct substitu- 

 tion into the fitted quadratic equation 

 resulted in the least error when all 

 annuli were used, and when only the 

 last annulus was used. Finally, an 

 asymptotically shaped OR-TL was best 

 modeled by the individually corrected 

 WeibuU cumulative distribution func- 

 tion when all annuli were used, and 

 when only the last annulus was used. 



An evaluation of back-calculation methodology 

 using simulated otolith data 



Michael J. Schirripa 



Hatfield Marine Science Center 

 Northwest Fishenes Science Center 

 2030 SE Marine Science Drive 

 Newport, Oregon 97365 5296 

 E mail address Michael Schirripaia'noaa gov 



Manuscript accepted 29 May 2002. 

 Fish. Bull. 100:789-799(20021. 



The average rate of growth of an indi- 

 vidual fish in a population is critical to 

 age-based stock assessments. The aver- 

 age rate at which the fish within the 

 stock increases in weight ultimately 

 determines the level of effort required 

 to extract a desired yield from the stock 

 as a whole (Ricker, 1975). Furthermore, 

 current conservation standards (Gul- 

 land and Boerema, 1973; Goodyear, 

 1993) are dependent upon the rate of 

 individual growth. Thus, errors in the 

 estimation of growth can lead to erro- 

 neous advice to fishery managers con- 

 cerning the present and possible future 

 status of a population. 



By far the most common method of 

 estimating fish growth rate is by esti- 

 mating the age of individual fish from 

 calcified structures (scales, otoliths, 

 spines, etc.; but for this study, however, 

 otoliths were considered the represen- 

 tative hard structure) and with the 

 subsequent assumption that these fish 

 are an unbiased representation of size 

 at that age. Growth is then described 

 as the change in weight or length over 

 some unit of time. To standardize age 

 at which size is estimated, or to obtain 

 length-at-age data on ages not included 

 in the sample, back-calculation tech- 

 niques are often employed to estimate 

 a fish's size at a previous age (Bagenal, 

 1978). The process of back calculation 

 can be broken down into three steps: 

 verification of the periodicity of annulus 

 formation, establishment of an otolith 

 radius-total body length (OR-TL) rela- 

 tion, and the estimation of size at the 

 time of annulus formation. In this study, 

 I used simulations to examine how the 

 establishment of the OR-TL relation 

 and the form of the back-calculation 

 equation used may influence growth 

 rate estimates made from otoliths. 



The back-calculation process as- 

 sumes that somatic growth is directly 

 related to otolith growth (Bagenal, 

 1978). This assumption is usually vali- 

 dated through the demonstration of a 

 relationship between the otolith radius 

 and body length by a least-squares 

 regression of body length on otolith 

 radius. A variation of this technique 

 uses a functional (model II) regression, 

 based on the assertion that neither 

 body length nor otolith radius are truly 

 independent (i.e. measured without 

 error) (Ricker, 1973, Laws and Archie, 

 1981). Uncertainties can enter this pro- 

 cess from several sources. For example, 

 incomplete data can make it difficult 

 to discern if this relationship is linear. 

 Furthermore, using regression to esti- 

 mate beyond the range of the data is 

 not recommended. Estimating beyond 

 the range of the data can become a 

 problem when back-calculating to very 

 early ages that are not represented 

 in the sample. Furthermore, several 

 studies have found that otolith growth 

 and somatic growth can be uncoupled 

 (Mosegaard et al., 1988; Reznick et al, 

 1989; Secor and Dean, 1989; Wright 

 et al., 1990; Milicich and Choat, 1992; 

 Secor and Dean, 1992). Hales and Able 

 (1995) found that changes in somatic 

 growth accounted for only half of the 

 variation in otolith growth. This un- 

 coupling of somatic and otolith growth 

 rates challenges the assumption that 

 back-calculation is based on. 



The question of what is the proper 

 back-calculation equation to use is a 

 question that has received considerable 

 attention. Bagenal (1978) discussed 

 three separate methods and suggested 

 that a combination of methods might 

 be helpful in some cases. Francis 

 ( 1990) presented an in-depth review of 



