792 



Fishery Bulletin 100(4) 



L =a+Rh, 



110) 



where L = the total length; and R = the otolith radius and rep- 

 resents the independent variable (assumed to be measured 

 with out error); 2 ) functional regression ( model II ), which has 

 the identical formula as Equation 10 but does not assume an 

 independent variable (i.e. both L and R are measured with 

 error); 3) WeibuU cumulative function (WeibuU, 1951), 



K 



and 4) a third order quadratic equation 



L =c + {d^R^) + (e.yR;-) + (f^R/) 



(11) 



(12) 



These four functions were fitted with the SAS NLIN pro- 

 cedure (SAS, 1988). 



Ten combinations of the back-calculation formula and 

 OR-TL fitting procedures were used (Table 1). Methods 1 

 through 9 were simple derivations from a standard regres- 

 sion equation and required only a fitting of the parameters 

 and substitution into the equation (Bagenal, 1978). Meth- 

 od 10 however used a derivation of the WeibuU distribution 

 function. In this method, the parameter defining the as- 

 ymptotic limit of the function tK) was modified by LJL as 



[KiLJL^)\ 



1-exp 



(13) 



where L = the theoretical length of the fish according to 

 its otolith radius as predicted by the fitted OR- 

 TL WeibuU function; and 

 L = the actual length at capture. 



If, for instance, the actual length of the fish was less than 

 the theoretical length iLJL is less than 1 ). the parameter 



K was corrected downward and subsequent back-calcula- 

 tions for that fish were inade according to its own individ- 

 ual trajectory (Fig. 2). In this way, LJL was calculated for 

 each individual fish in the same way that the slope of the 

 Fraser-Lee back-calculation equation was estimated for 

 each fish. These ten combinations were used for all avail- 

 able annuli and then repeated by using the last annulus 

 only, for a total of twenty different methods. 



As a measure of bias, the back-calculated length at age 

 2 was regressed on the age of the fish from which the 

 estimate came (source age). In this way. for instance, a 

 strong "Lee's phenomenon" (the phenomenon that back- 

 calculated lengths for a given age group become smaller as 



