FISHERY BULLETIN: VOL. 80. NO. 4 



Y = 



such that xn — 1 for all i and 



Xi2 = -a - 1/2) A 



[yi] is the vector of dependent vari- 

 ables with elements 



m-i 



2 2 r, 



y, = In* 



i=i 



di] 



X r sij + 2 1 r, 



(12) 



rfii 



Here IT is a matrix of statistical weights, and m,- 

 is the number of cohorts for which recapture sta- 

 tistics are available in the ith postrelease inter- 

 val. The symbol T denotes the matrix transpose. 

 If sufficient recaptures are obtained to analyze 

 each cohort separately (say r. tJ > 10) but a com- 

 mon shedding rate is assumed, several alterna- 

 tive approaches are available. First we can treat 

 the separate releases as partial replicates of the 

 same experiment and construct the dependent 

 variables as the logarithms of the geometric 

 means of individual statistics for each cohort. 

 Thus to estimate lnp and L we use Equation (11) 

 as before but now set 



A second approach when the shedding rates 

 are constant and the v v - are sufficiently large is to 

 treat returns from each of the m releases sepa- 

 rately and then average the individual estimates. 

 Thus the overall estimate of L, for example, 

 would be 



L = X.wj Lj 



where Lj is the estimated slope from the linear 

 regression of 



Vv 



2r di 



In 



r S y + 2r dij 



on r,. Here the individual estimate of L from the 

 j'th cohort is given a weight Wj inversely propor- 

 tional to its relative variance. In practice we sub- 

 stitute the statistic 



Wi 



-2 



m 

 Z o 



M l 



y< = X In 



2r, 



dij 



„ + 2 ^ 



(13) 



Although as an estimator of ln/c, Equation (13) 

 usually has slightly greater negative bias than 

 Equation (12), such bias is negligible and the 

 approach taken in Equation (13) has the advan- 

 tage that statistical weights may be calculated 

 empirically for cases where w ; > 2. In particu- 

 lar, define the tth diagonal element of W as 



2-i 



Wu — rriiim, — 1) 



where y, is given by Equation (13), and let u% = 

 for i =&j. When some of the m, are equal to 1, then 

 the w n may be computed using the delta method 

 as 



w-n = m, 



m I / 



T S y{r s jj -- r dtJ ) 

 *dij\ r sij + 2r dlJ ) 



-i 



on the assumption that the r d ij and r S ij are comple- 

 mentary binomial variables. 



o'i being the estimated variance of L, computed 

 in the jth regression. For the regression analysis 

 itself, appropriate statistical weights for the y v - 

 would be proportional to 



~-2 _ 



r,i,j (r slJ + 2r d ij) 2 



Finally, the variance of L may be estimated as 



= I 



3=1 



* rl ~2 

 Wj o~ 



[jl \| 



A third approach is to assume that the set of 

 regression estimates from the m cohorts are sam- 

 pled from an underlying but unspecified stochas- 

 tic process which, with respect to the estimation 

 of Type II shedding rate, has mean L and vari- 

 ance o\. The regressions of y tJ on r, are unweight- 

 ed, and empirical estimates of L and a~ are given 

 very simply by 



L = 



2 Lj/m 



and 



~ 2 

 °1 



2 (Lj - Lf/m(m 

 j=i / 



- 1), 



694 



