days through the IATTC season closure date (1 

 January through 26 March); trip-type 2 included 

 all vessel days approximately corresponding to 

 the "last trip" (27 March through 4 July); and 

 trip-type 3 contained all vessel days after 4 July. 

 This stratification of time was used to preclude 

 overlapping trips. When a trip crossed a stratum 

 boundary, it was assigned to more than one strat- 

 um. For example, for a trip lasting from 1 March 

 to 5 May, the days from 1 March to 26 March 

 would be assigned to trip-type 1, and days from 

 27 March to 5 May assigned to trip-type 2 (Table 

 1). 



and 



-2 



cov(X,Y ) 

 X { Y, 



where s, 2 



s u = 



nx v - xy 

 j_ 



Wj — 1 



XiYa- Yf 



Hi — 1 



S(X U - X,) ( Y tJ - Y { ) 



Table 1.— Layout of stratification of vessel trips (sets) for kill- 

 per-day and kill-per-set methods. 



Kill-per-day method with maximum 30 strata: 15 for each gear type 



NMFS Trip 1 Trip 2 



vessel 1 Jan -26 Mar 27 Mar -4 July 



class open 1 regulated 



Trip 3 



5 July-end of fishing 



open 2 regulated 



I 



III 



Kill-per-set method with maximum 32 strata 



IATTC 

 vessel 

 class 



Successful sets 



Unsuccessful sets 



Inside CYRA Outside CYRA Inside CYRA Outside CYRA 



'Trips 

 trips." 

 Hast 



not subject to IATTC season closure Most of them were "last 

 trips" or trips made outside of CYRA 



The statistical formulation for dolphin mortal- 

 ity estimation according to a stratified ratio esti- 

 mator is as follows (Cochran 1977): 



For the ith stratum, i = 1 /, 



let N, = total number of vessel trips 

 n t — number of observed trips 

 Xij = kill for the jth observed trip j = 1, . . ., 



71 i 



Y,j = days-at-sea for the jth observed trip 



r, = kill-per-day 

 s r , = the approximate sample standard error 



of r, (kill-per-day) 

 Mj = total number of days-at-sea 



T, = estimated total kill 



then r, = 



XXij 



I 



XYa 



j 



(1) 



cov(X,Y) 



n , • — 1 



r,M, 



T = (IT,) = Zr,M, 



Sr 2 = ZM? S* 



The ratio estimator (r,) and its approximate 

 sample variance (s, 2 ,) are unbiased only under 

 certain conditions (Cochran 1977). Alternative 

 variance formulas have been suggested to cor- 

 rect the bias (Royall and Eberhardt 1975; Royall 

 and Cumberland 1981). We chose the commonly 

 used variance formula (s 2 ) in our procedure be- 

 cause the results of a simulation study showed 

 that the bias of the ratio estimator and the ap- 

 proximate variance is negligible (Lo 3 ). The simu- 

 lation study was based upon the empirical dol- 

 phin mortality data collected in 1977. 



Beginning on 30 June 1976, NMFS observers 

 radioed their mortality counts to a shore base 

 each month (starting in 1977, estimates were 

 made biweekly). Data from this source were used 

 to estimate cumulative mortality and to pro- 

 ject the date when the annual quota would be 

 reached. 



Combined Kill-Per-Day and Kill-Per-Ton Method 



The method using combined kill-per-day and 

 kill-per-ton was developed to project at the end of 



3 Lo, N. C. H. Simulated results of a commonly used ratio 

 estimator applied to incidental dolphin mortality by U.S. tuna 

 purse seiners in the eastern tropical Pacific. Manusc. 

 Southwest Fisheries Center La Jolla Laboratory, National 

 Marine Fisheries Service, NOAA, La Jolla, CA 92038. 



398 



