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Appendix: Variance estimates of 

 mean length-at-age using a two-phase 

 sampling procedure 



The length and age samples collected by observers 

 in the Pacific whiting fishery are recorded by haul or 

 joint-venture delivery. The position and date of each 

 sampled haul and joint-venture delivery are also re- 

 corded. In compiling the length-at-age estimates for 

 spatial and temporal strata, all the data collected within 

 that strata are aggregated and assumed to originate 

 from random sampling from the catch within that 

 strata. 



Sampling design 



A large initial random sample is obtained from the 

 catch, and the length and sex of each fish is recorded. 

 For the second phase of sampling, a subsample of fixed 

 size is selected for each combination of length category 

 and sex. All fish in these subsamples are aged using 

 otoliths or other ageing structures. 



Notation 



i = 1,. .., I length categories 



j = 1 , . . . , J age categories 



n' = first-phase sample size 



n'i = number of fish of n' in length category i 



qi = probability that a fish is in length category i 



nj = subsample sizes 



njj = number of the subsample taken from length 



category i of age j 

 q'ij = Pi"(j I i)' probability of age j given length i 

 qjj = pr(i|j), probability of length category i given 



age j 

 Pj = pr(j), probability of age j 

 1; = midpoint of \th length category 

 Tj = mean length of age j fish 



To simplify notation, subscripts for males and fe- 

 males are not defined. The variance estimator obtained 

 here is conditional on the first- and second-phase sam- 

 ple sizes. Separate estimates for males and females 

 can be obtained by separating the samples by sex and 

 conditioning on the number of each sex in the first- and 

 second-phase samples. The same variance estimator is 

 appropriate for separate sex estimates. 



Sampling distributions 



Assuming that the first-phase sample size n' is much 

 smaller than the size of the population being sampled, 

 the distribution of n'j can be modeled by the multi- 

 nomial distribution, 



