Schirripa: An evaluation of back calculation methodology using simulated otolith data 



797 



Figure 10 



Error in mean length at age estimates using all annuli 

 and the ten back-calculation methods outlined in Table 1 

 for the asymptotically shaped OR-TL relation. SS = sum of 

 squares. 



Figure 11 



Error in mean length at age estimates by using the last 

 annulus only and the ten back-calculation methods outlined 

 in Table 1 for the asymptotically shaped OR-TL relation. SS = 

 sum of squares. 



showed that using the improper back-calculation method 

 can result in a false "Lee's phenomenon" effect. Using 

 only the last annulus reduced this effect with some back- 

 calculation methods in this study, but not all of them. In 

 general, the accuracy of the estimated length-at-age was 

 directly related to how well the particular model fitted the 

 OR-TLrelation, suggesting that the OR-TL model is just 

 as, if not more, important as selecting the appropriate 

 back-calculation model. 



Based on the importance of the fit of the OR-TL model, 

 it follows that the methods used to sample the catch are 

 of equal importance. Nonrandom samples of the catch, or 

 length-based regulations that cause the catch to misrepre- 

 sent the population, will affect the OR-TL regression. For 

 instance, a minimum legal size will artificially truncate 

 the OR-TL relation in samples of the catch and selectively 

 sample faster-growing small fish. This could eliminate the 

 youngest ages from the regression and could necessitate 

 extrapolation of the regression beyond the range of the 

 data. Furthermore, a truncation of the OR-TL regression 

 would positively bias they-intercept and lead to an overes- 

 timation of length-at-age, especially for the younger ages. 



It has been pointed out that univariate statistical 

 models, which assume independence of observations, are 

 generally inappropriate for analysis of otolith increment 

 data (Chambers and Miller, 1995). These authors have 

 suggested that because otolith data constitute multiple 

 measures, perhaps examination of the covariance is more 

 appropriate than the comparison of individual means. In 

 this study, however, I did not seek to emphasize the ex- 



istence of (or lack of) a statistical difference between the 

 true and estimated means sizes. Given the large sample 

 sizes made available through simulation, conclusions of 

 significant differences resulting from any statistical tests 

 can be misleading. More useful, I believe, is the shape, 

 direction, and magnitude of the biases that emerged from 

 each back-calculation method. Consequently, I chose to 

 emphasize the percent error between the true and esti- 

 mated mean size-at-age. Using percent error allows more 

 freedom of interpretation and is not subject to the prob- 

 lems associated with excessively large degrees of freedom 

 of simulated data sets. 



The individually corrected Weibull cumulative distribu- 

 tion function presented here proved to be very flexible and 

 capable of accounting for the individual otolith radius- 

 total length trajectories. This function is very similar to 

 the linear y-intercept corrected back-calculation equation 

 of Fraser-Lee but can accommodate a wide varieties of 

 curvatures. The Weibull equation I reported (Eq. 13) has 

 an origin at x and y of 0: however, a y-intercept term can 

 easily be added to accommodate an OR-TL relation with a 

 nonzero intercept. 



Much of the cohort's diversity in biological attributes 

 was lost within the first few months of the life because of 

 mortalities. By the time the cohort had completed one year 

 of growth, the diversity in biological attributes of the indi- 

 viduals that would ultimately represent the cohort were 

 established. Based on the observations of Secor and Houde 

 (199.5), the establishment of the biological attributes of a 



