496 



Fishery Bulletin 105(4) 



Similarly, the probability of a fish, tagged at age a, 

 being caught in the unobserved component of the fishery 

 at age i, and having its tag returned, is 



(1-5,>,A, i = a 



l-5,)S,-S,_iM,A, i>a 



(2) 



Thus, the probability of a fish, tagged at age a, not 

 being recaptured by age I from either component is p^ = 

 1- p° , - p'^ ,. Here, and below, a dot in the subscript 

 denotes summation over the index it replaces. 



For tags released at age a, the numbers of returns at 

 ages a to / from the observed component (i?° ,, j = o, . . . , 

 /) and unobserved component (R"^ ,, ; =a, . . . , 7), plus the 

 number not returned by age I from either component 

 (R'^=N^ - R\ , ~ R"^^), are multinomial with probabilities 

 given by Equations 1, 2 and p'^, respectively. Thus, the 

 likelihood equation for the returns from tags released 

 at all ages is the product of multinomials, given by 



R° .. R" . 



Lr-yY[[p:)''''-Y[' ?:,)"■'<?:.. 



(3) 



where 



N„\ 





Note that y is a constant that can be left out when maxi- 

 mizing the liklehood. 



Next, consider the catch component of the model. 

 Recall that no age information is obtained for the un- 

 observed catches; therefore only catch-at-age data from 

 the observed component are available for inclusion in 

 the model. The probability of an age-1 fish from the 

 cohort of interest subsequently being caught at age i in 

 the observed component of the fishery is 



5^11^ i = 1 



(4) 



If the numbers offish from the cohort of interest that 

 are caught at ages 1 to 7 in the observed component of 

 the fishery (0"^, j=l, . . . , 7) are known accurately, then 

 these numbers, along with the number of fish from the 

 cohort not caught by age 7, are multinomial and have 

 probabilities given by Equation 4. Usually, however, 

 the numbers of fish caught at each age are not known 

 precisely because the ages are estimated either from 

 lengths or from annuli in hard parts (the estimates 

 will be more accurate in the latter case, but will still 

 contain uncertainty). We assume the aging error of the 

 age i catch has a Gaussian distribution with mean 

 (i.e., no bias) and a variance if^. 



Rather than modeling the catch data with both mul- 

 tinomial process error and Gaussian aging error, which 



would require a fairly complex approach, we approxi- 

 mated the distribution of the catch of age / fish in the 

 observed component, C'[, as Gaussian with overall vari- 

 ance a'~ = )f^ + T~, where t~^=P^ji''^{1-jt°^}, is the multinomial 

 variance component. The aging error, unless negligible, 

 will tend to dominate the process error when the cohort 

 size is reasonably large (> 100,000 individuals), as would 

 be expected in most commercial fishery situations. For 

 example, if the coefficient of variation (CV) of the aging 

 error is 0.10, the cohort size is 100,000 and the prob- 

 ability of catching an age / fish (in either the observed 

 or unobserved component of the fishery) is 0.10, then the 

 ratio of the aging error variance to the process error 

 variance is ~10 when the proportion of the catch in the 

 observer component is 0.10, and it is -50 when the pro- 

 portion of the catch in the observer component is 0.50. 

 Thus, assuming that the C']'s are independent be- 

 tween ages, the likelihood for the observer catch data 

 is 





2a. 



—ic- 



E(C°)\ 



(5) 



where £(C°)=Pi.T°;. 



When only a single cohort of fish is being modeled, 

 the assumption that the catch data are independent 

 between ages (i.e., years) should be reasonable in most 

 situations. First, the correlation in the multinomial 

 errors will be close to zero when the size of the cohort 

 is much larger than the size of the catch (as would be 

 expected in most fisheries). Second, the aging errors 

 should be uncorrelated between years provided sam- 

 pling and aging data are collected each year. However, 

 in some situations, particularly where age is being esti- 

 mated from a growth curve, covariance in the estimates 

 between years may exist and should be accounted for. 

 Furthermore, if more than one cohort is being modeled, 

 then catch data from multiple ages within the same 

 year will enter the model, and aging errors within a 

 year will be correlated across ages. The level of corre- 

 lation, and thus the degree to which the independence 

 assumption is violated, will depend on the specifics of 

 the situation, such as how many age classes are pres- 

 ent in the year's catch. When the correlation is strong, 

 a more sophisticated approach for modeling the catch 

 data may be required. 



The overall likelihood for the combined recapture and 

 catch data can be obtained by multiplying likelihoods 

 (Eqs. 3 and 5) together: 



 ij p X i-^f' . 



(6) 



In a tagging experiment with A consecutive release 

 years, estimates can be obtained, at most, for A-1 natu- 

 ral mortality-rate parameters (regardless of the number 

 of recapture years) because information for estimating M^ 

 comes from the differential between the expected returns 

 at age i+\ offish released at age ; and those released at 

 age ; + l. One option is to assume that M,=M^_j for i>A, 



