Eveson et aL: Incorporating fishery observer data into an integrated catch-at-age and multiyear tagging model 497 



but other options, such as assuming M is constant or 

 linear with age, are also possible. Furthermore, there 

 is not enough information in the current formulation 

 to estimate the proportion of fish caught each year in 

 the observed component of the fishery (i.e., the d/s). To 

 estimate this proportion would require knowing the 

 total observer catch in each year, as well as the total 

 overall catch in each year. Rather than bringing these 

 data into the model, we assumed that the total catches 

 are known well enough that the 6,"s can be treated as 

 known without error. Lastly, the aging error variance 

 parameters for the observer catches (i.e., the ir,'s) cannot 

 be estimated reliably and therefore they are assumed to 

 be known without error. In Polacheck et al. (2006), we 

 gave a detailed explanation of why the catch variance 

 cannot be estimated reliably in the BP model, and the 

 same argument applies here. We found, however, that the 

 model results were fairly insensitive to the value used for 

 the catch variance so long as it was in the right ballpark 

 (e.g., within -40% of the true value). The parameters 

 that can be estimated by maximizing Equation 6 are F, 

 and A, for ; = 1 to 7, M, for i = 1 to A-1, and P^. 



As is true when combining any sources of informa- 

 tion, it is important to check that the tag-recapture data 

 and the catch data are consistent. This can be done by 

 maximizing the tag-recapture likelihood (Eq. 3) alone 

 and comparing the mortality-rate estimates with those 

 obtained from the joint likelihood (Eq. 6) (note that the 

 catch likelihood alone is insufficient to yield parameter 

 estimates). If the estimates are significantly different, 

 this result would indicate that the tag-recapture and 

 catch data are inconsistent and should not be combined; 

 doing so would yield average values with little biologi- 

 cal meaning. Instead, the source of the inconsistency 

 should be investigated (i.e., does it stem from problems 

 with the data or with the applicability of the assump- 

 tions in the model?). 



Overdispersion in the recapture and catch data 



In the model a multinomial distribution is assumed 

 for the tag-recapture data. If one (or more) of model 

 assumptions 1 to 3 is violated, then the observed return 

 counts are expected to be more variable than predicted 

 by a multinomial distribution; i.e., to be overdispersed 

 in relation to multinomial data. Polacheck et al. (2006) 

 provided a thorough discussion of possible sources of 

 overdispersion and ways in which it can be accounted 

 for. When overdispersion exists in the return counts, the 

 parameter estimates obtained by using a multinomial 

 likelihood should still be unbiased, but their standard 

 errors, as estimated from traditional likelihood methods 

 (i.e., from the inverse Hessian matrix), would be too 

 small. A number of possible methods for obtaining more 

 realistic standard errors are discussed in Polacheck et 

 al. (2006) and Pollock et al. (2001), one of which is to 

 use bootstrap procedures. 



If overdispersion exists in the recapture data as a 

 result of model assumptions 2 or 3 being violated, then 

 it will also exist in the catch data. That is, the compo- 



nent of the variance in the catch-at-age numbers due to 

 process error will be underestimated by a multinomial 

 distribution. As asserted previously, aging error will 

 generally dominate the multinomial process error in 

 the catch data. This will often still be true when the 

 process error is overdispersed. For example, assume 

 that the process error variance is (p times that of mul- 

 tinomial variance; i.e., T-, = (jpPj:T°(l-jr° ). Then, in the 

 example that was given above for multinomial process 

 error, if (p=3, the ratio of the aging error variance to 

 the process error variance would still be 3.3 (=10/3) and 

 17 ( = 50/3) when the proportion of catch in the observer 

 component is 0.10 and 0.50, respectively. In situations 

 where the aging error dominates, not accounting for 

 overdispersion in the catch data should have little ef- 

 fect on the standard error estimates of the parameter 

 estimates. 



The degree to which the likelihood-based estimates 

 of the standard errors are underestimated by not ac- 

 counting for overdispersion in the tag-recapture data 

 and catch data was investigated through simulations, 

 as described below. 



Simulation methods 



Model performance To evaluate how the model performs 

 in terms of the accuracy and precision of the parameter 

 estimates, a series of Monte-Carlo simulations were con- 

 ducted. The first scenario considered, which we will refer 

 to as scenario 1, involved a single cohort of fish being 

 tagged in five consecutive years starting at age 1 (i.e., 

 at age / in year i for i=l, . . . , 5), and recaptured over 

 the same five years. The number of tag releases was set 

 to be 1000 at each age. Corresponding to the releases at 

 each age, tag returns were generated from the observed 

 and unobserved fishery components by using a Dirich- 

 let-multinomial (D-M) distribution (Mosimann, 1962). 

 The D-M distribution allows for overdispersion in the 

 return counts by modeling the return probabilities as 

 random Dirichlet variables (see Appendix A of Polacheck 

 et al., 2006). It can be parameterized in terms of the 

 return probabilities and an overdispersion factor, (p, 

 that specifies the amount of extra variation in relation 

 to multinomial data. For scenario 1, q? was set to be 3 

 (i.e., three times greater variance than a multinomial 

 distribution). Other parameters were set as follows: 

 F, = 0.15, M, = 0.2, A, = 0.75, and 6, =0.10, for j = l, ... , 5. 

 Catch-at-age numbers (ages 1 to 5) for the observer 

 component of the fishery were generated by using, first, 

 a D-M distribution with Pj = 100,000 and the same cp, 

 6,, Mj, and F, values as for the tag-recapture data. To 

 these catch-at-age numbers, additional Gaussian aging 

 error was added by using a constant CV of v = 0.10 for 

 all ages (i.e., »), = vE(C»)). 



The BPO model was fitted to the simulated tag-re- 

 capture and catch data by maximizing Equation 6. 

 For this process, q> and v were assumed to be known 

 without error, and natural mortality was constrained 

 to be the same at ages 4 and above (i.e., M^=M^; re- 

 call that only four natural mortality parameters can 



