796 



Fishery Bulletin 100(4) 



9 10 11 12 13 1 



Figure 8 



Error in mean length at age estimates using all annuli 

 and the ten back-calculation methods outlined in Table 1 

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

 squares. 



Figure 9 



EiTor in mean length at age estimates using the last annu- 

 lus only and the ten back-calculation methods outlined m 

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

 sum of squares. 



Exponentially shaped OR-TL relation 



Of the four functions fitted to OR-TL/EXP, the quadratic 

 function resuUed in the highest coefficient of determina- 

 tion (/•'-=0.883); however the coefficient of determination of 

 the Weibull function fit was nearly as high (/■2=0.878). The 

 percent errors, when using all annuli and linear regres- 

 sion, followed a pattern similar to the residuals of the OR- 

 TL relation (Fig. 8). This trend was also evident, although 

 not as strong, when only the last annulus was used ( Fig. 

 9). Using the quadratic function rather than the linear 

 regression to fit the OR-TL relation did the most at remov- 

 ing this bias (Fig. 8 method 9, and Fig. 9 method 19). 



When the OR-TL relation was exponentially shaped 

 and all annuli were used, the least error resulted from 

 direct substitution into the fitted quadratic equation (Fig. 

 8, method 9, SS=0.0580), and the greatest error from us- 

 ing the direct proportionality equation (Fig. 8, method 1, 

 SS=L6711). When only the last annulus was used, the 

 least error resulted from direct substitution into the fitted 

 quadratic equation (Fig. 9, method 19, SS=0.0662), and 

 the greatest error from using direct substitution into the 

 OLS regression equation (Fig. 9, method 12, SS=1.5882). 



Asymptotically shaped OR-TL relation 



Of the four functions fitted to OR-TL/ ASYM, the quadratic 

 equation resulted in the highest coefficient of determina- 

 tion (r-=0.963); however the coefficient of determination 

 of the Weibull function fit was nearly as high (r-=0.958). 



As with the exponentially shaped OR-TL relation, when 

 linear regression was used to model the OR-TL relation, 

 the percent error by age followed the trend of residuals for 

 the residuals for the regression (Fig. 10). Using the last 

 annulus only resulted in generally lower sums-of-squares, 

 especially when the y-intercept was corrected for log 

 transformation of the OR-TL relation used (Fig. 11). 



When the OR-TL relation was asymptotically shaped 

 and all annuli were used, the least error resulted from 

 using the individually corrected Weibull cumulative dis- 

 tribution function (Fig. 10, method 10, SS=0.7388), and 

 the greatest error from using direct substitution in to the 

 OLS regression equation (Fig. 10, method 2, SS=1.9319). 

 Wlien only the last annulus was used, the least error 

 again resulted from using the individually corrected 

 Weibull cumulative distribution function (Fig. 11, method 

 20, SS=0.0516), and the greatest error from using direct 

 substitution in to the OLS regression equation (Fig. 11, 

 method 12, SS=1.9261). 



Discussion 



The most accurate estimates of length-at-age resulted 

 from the best model fits of the OR-TL relation. Even 

 though sampling was random, poorly fitted OR-TL regres- 

 sions resulted in back-calculation tables with obvious 

 "Lee's phenomenon" effects. Ricker ( 1969) pointed out that 

 the use of an incorrect otolith radius-total length relation- 

 ship can result in this effect. Smale and Taylor (1987) also 



