Fishery Bulletin 100(1) 



mation will be necessary. One way to generate more in- 

 formation is to have a third independent reader (Walter, 

 1984). With three readers, there are seven essential pa- 

 rameters; 7i'ii;^"-'''\r^'^\i,''-"" and p. There is also 2^ - 1 = 7 

 df, so that all the parameters are estimable. Estimation 

 is most commonly done by the method of maximum likeli- 

 hood. 



If readings are assumed to be independent among read- 

 ers and among otoliths, the likelihood function is 



i = H,\V ) = H,\V*:H,VV 



This likelihood function must be maximized numerically 

 and methods for this computation will be discussed later 

 If more than three readers are used, there are extra de- 

 gi-ees of freedom that can be used to assess goodness-of-fit. 



For example, with four readers there will be nine param- 

 eters with 15 df leaving 6 df for goodness-of-fit. Pearson chi- 

 square or likelihood ratio G'-^ tests would both be applicable. 

 Another way to generate additional information was 

 proposed by Hui and Walter ( 1980). Suppose there are two 

 or more strata with different hatchery proportions in each 

 strata. For example, catch could be stratified temporally 

 or spatially. If it is assumed that ;r||||| and /Ty^iw remain 

 constant over strata, then a solution for just two readers 

 may be obtained. For example, if there are two readers and 

 two strata, then there are six parameters; 'rH|H"'>'i'w|w' > 

 Pj, and p.,, with 2(2'^ - 1) = 6 df Increasing the number of 

 strata increases the degrees of freedom; e.g. three strata 

 for two readers gives 3(2^ - 1) = 9 df for 7 parameters. The 

 likelihood function for two readers and S strata is 



fin ni^^'^^iH'^^^+'i-^. 



(1) 12) 1" 

 'Iw'TjIWl 



g=l (=H,W_/ = H.W 



Table 2 



Examples from cross-classification data generated as 

 expected counts from a sample of 1000 otoliths based on 

 different accuracy rates for identifying hatchery fish < tt,., | ,^ I 

 and wild fish (/Twiw' under different mark proportions tp). 

 The examples used illustrate differences between obsei"ved 

 agreement IP, i and chance-corrected agi-eement U') under 

 different underlying conditions. 



A third way to supply additional information is to take 

 a Bayesian approach (see "Discussion" section). By speci- 

 fying prior distributions of the model parameters, unique 

 estimates can be obtained (Joseph et al., 1995). 



A critical assumption in the above models is that read- 

 ings are independent. Specifically, the reading of each oto- 

 lith by a given reader is independent of any other reading 

 by the same reader, and each reading by various readers 

 on a given otolith is independent given the true state of 

 the otolith. In principle, the latter assumption may be dif- 

 ficult to meet especially if all readers examine the same 

 otolith. The fact that the otolith is not prepared indepen- 

 dently by each reader could induce a dependence among 

 the readers. Also, variability in the readability of the mark 

 due to the marking process can induce a dependence. Such 

 dependence can bias the estimators of n and p (Vacek, 

 1985). Note that this latter assumption of independence is 

 also required for v. 



One remedy for the problem of dependence due to prepa- 

 ration is to require independent preparations. This however, 

 requires additional otoliths and with only two otoliths per 

 fi.sh, this would limit the number of readers to two. But 

 in practice, this may not be a large concern. Typically, the 

 second reader has the option to provide additional process- 

 ing effort to the first otolith or, if needed, to process the 

 second otolith. In almost all cases additional preparation 

 is not done and readers feel they are able to extract suf- 

 ficient information about the presence or absence of a mark 

 from each other's preparations. In addition, reader accura- 

 cy rates obtained by LCM do not appear to vary systemati- 

 cally with the reading order, which also suggests that prep- 

 aration-induced dependency is not a significant factor 



Dependency associated with variability in the appear- 

 ance of the mark may be harder to address. A general so- 

 lution is to model the dependence with additional param- 

 eters (e.g. Vacek, 1985; Qu et al., 1996; Yang and Becker, 

 1997; Qu and Hagdu; 1998; Albert et al., 2001). Modeling 

 dependence requires either more readers or more strata. 

 These modeling approaches are complicated and are cur- 

 rently evolving (see Albert et al., 2001). Alternatively, ad- 



