902 



Fishery Bulletin 101(4) 



Q. 



E 



Month 



Figure 1 



Values of index of completion at time t, (c,. t) forL. vitta with five growth rings during 

 the monthly time intervals from 1 October 1993 (t=Q) to 31 March 1995 (t=\S). Also 

 illustrated are the functions Fj and Fj fitted to (C,, t) and the function P, representing 

 the probability that (C,, t) is most hkely represented by function Fj. The likelihood 

 functions, Aj and A.^ are shown for the values of index of completion when t = 10.4. 



where R, S, and R-iS-R) are the values off corresponding 

 to the 50'*^ , 95*, and S"' percentiles of the logistic function. 

 The probability that the index of completion is associated 

 with Fj, rather than Fj, is calculated as 1-P,. 



Figure 1 illustrates typical values of the index of comple- 

 tion at time t and the functions Fj and Fj representing 

 these points before and after new growth-ring detection. 

 The most likely time at which a new growth ring is detected 

 is given by the value oft when P,=0.5. The likelihood func- 

 tions Aj and Ag are illustrated for t = 10.4 months. When 

 X, j is high, (Cj, t) is likely to lie closest to Fj and when A^ , 

 is high, (c,, t) is likely to lie closest to F2. 



As a point (c,, t) will be associated with either Fj or Fg 

 (but not both), it follows that the likelihood function K is 

 given by 



the probability 

 of the observed 

 deviation 

 from Fj, X 



given that 

 the point is 

 associated 

 with F, 



That is, 



probability 

 that the 

 point is 

 represented 

 hyF, 



the probability 

 of the observed 

 deviation 

 from Fji 

 given that 

 the point is 

 associated 

 withF,. 



X 



probability 



that the 

 point is 

 represented 



byF,. 



K,, = KA + ^^-^-P,^- 



The overall log-likelihood associated with all the observed 

 points (c,, t), , for j = 1 to n in a particular age class is given 

 by 



n n 



^ln(^,,) = ^ln[A,,/^ + A,,(l-P,)]. 



The parameters of the functions Fj and Fj (i.e. a, b, d^, 

 and 1^2' as well as a, R, and S) were estimated separately 

 for each value of the number of rings by maximizing the 

 log-likelihood. 



The value oft = R corresponds to the month where a val- 

 ue of the index of completion is equally likely to be on either 

 Fj or F2; that is, the point where the drop in the index of 

 completion occurs. The value of S and R-iS-R) correspond 

 to the 95'*^ and the 5'^ percentiles for the time at which a 

 new growth ring is likely to be detected, indicating reliabil- 

 ity of the estimate of the time of ring formation t = R. 



Results 



The plots of index of completion versus time reflected this 

 growth pattern in the four species we studied, with the 

 growth rate decreasing as the time of new growth-ring 

 detection approached. 



The temporal pattern of growth of the otolith suggests 

 the index of completion could be modeled with a logistic 

 function 



F{t.a.b,d) = - 



1 -i-exp 



lnl9 



(cl't) 



with the maximum value o = 1, phase shift d, and rate of 

 increase b. 



A characteristic of otolith growth is that the distance 

 between growth rings decreases each year, thus, the rate 

 of increase in the marginal increments will be greater for 

 fish with few rings and less for fish with many rings. On 

 the other hand, the index of completion, being the ratio of 



