258 



Fishery Bulletin 97(2), 1999 



of otolith growth between the last annulus and the 

 otolith margin. If the otolith was collected in the 

 spring or summer and there is a relatively large area 

 of growth along the otolith margin, an age reader 

 would probably decide that the annulus had not yet 

 been deposited and would therefore add 1 to the num- 

 ber of annuli observed on the otolith. However, if a 

 relatively small area of growth is seen, then the 

 reader would probably assume that the annulus had 

 been deposited and assign an age equal to the num- 

 ber of annuli observed. By fall and winter, the cur- 

 rent year's annulus is expected to have been depos- 

 ited, and the fish age would, therefore, equal the 

 number of annuli obsei-ved on the otolith. 



Analysis of ageing errors 



Present stock assessment of sablefish and many other 

 groundfish species in Alaska is based on an age-struc- 

 tured model that attempts to estimate the true age 

 composition of the population (Sigler et al.M, Ageing 

 errors in catch-age models can be accounted for by 

 supplying an ageing-error matrix (Fournier and 

 Ai-chibald, 1982; Methot, 1990; Richards etal., 1992). 

 This matrix defines the probability of assigning a 

 particular age to a fish with a given true age. The 

 results of multiple age determinations from indepen- 

 dent age readings can be used to estimate the age- 

 ing-error matrix (Richards et al., 1992). In practice, 

 a normal distribution of observed age for each true 

 age is assumed and because ageing error tends to 

 increase with age, an increase in the standard de- 

 viation with increasing age is used (Lai and 

 Gunderson, 1987; Methot. 1990). Use of multiple 

 readings to assess ageing errors cannot detect a sys- 

 tematic difference (i.e. bias) between obsei-ved and 

 true ages (Richards et al., 1992). 



We constructed two ageing-error matrices to illus- 

 trate the possible difference in perceived ageing er- 

 ror based on agreement between primary reader and 

 tester and between reader and known age. The first 

 matrix was based on primary-reader and tester ages 

 and the second was based on reader (both primary 

 reader and tester) and known ages. 



Richards et al. ( 1992 ) presented a statistical model 

 for estimating ageing error. We used the "normal 

 model" of Richards et al. ( 1992). For a given true age 

 b, the standard deviation aib) of the observed age is 

 defined by three parameters CTj, cj^, and a such that 



' Sigler, M. R, J. T. Fujioka, and S. A. Lowe. 1997. Sable- 

 fish. In Stock assessment and fishery evaluation report for 

 the groundfish resources of the Gulf of Alaska. North Pacific 

 Fish. Manage. Council, 60.5 W 4th Ave.. Suite 306, Anchorage, 

 AK 99.501. 



aib) 



a, +{Oa - cr,) 



1-e 



-aife-ll 



1 l_e-«''4-ii 



O-j +(0>i - CTj) 



fo-1 



A-1' 



a^O 



a = 



(1) 



The CTj and a^ are standard deviations for the mini- 

 mum and maximum ages, respectively. The param- 

 eter a determines the nonlinearity of the function, 

 where o( b ) becomes linear in 6 as a tends to 0. Given 

 the parameter vector 0=(a], o^, a) and observed 

 age classes a, the age-error matrix q(a |6,0) is de- 

 fined by 



g(a|i>,<I>): 



JO) 



(2) 



where -^^^(0) is the discrete normal density function 

 such that 



-m 



'■ab 



(*) = 



(3) 



In Richards et al. ( 1992), the assumed "true age" for 

 a fish aged by multiple readers is the modal age 

 among multiple readers. We used the mean age 

 rounded to the nearest integer because the mode is 

 not defined for two readings when the ages differ. 

 For the reader and known-age data, the true age is 

 the known age. A value for the maximum age A is 

 required. For the reader and known-age data set, we 

 set A equal to the maximum known age. For the pri- 

 mary reader and tester data set, we set A equal to 

 the maximum assigned age. 



The model of Richards et al. (1992) does not in- 

 clude estimation of bias. The use of known ages al- 

 lows bias to be estimated and incorporated in the 

 ageing-error matrix. By including three additional 

 parameters, Ericksen (1997) generalized the meth- 

 ods of Richards et al. ( 1992) to include estimation of 

 bias for tag recapture and known-age data. As with 

 Equation 1, for a given true age 6, the bias jiib) of 

 the observed age is defined by three parameters, j3j, 

 /3^, and A, such that <I>= (ct,, ct^, a, /3j, (i^. A) and 



; X^O 



A = 



(4) 



