NOTE Lai: Age-length key to estimate age composition of fish population 



387 



60 80 100 



Total Cost 



160 



Figure 1 



Relationship between minimum loss function value and total 

 cost for the lemon sole (= English sole Pleuronectes vetulus) 

 example. The relationship can be obtained from substituting 

 N' and n,' in Eq. 10 into the third equation in Eq. 10. 



Jinn et al. (1987) although the values of N* and n,* are 

 different. An advantage of the classic double-sampling 

 technique is the explicit solutions of the optimal set of 

 N" and n,\ which reduces computational effort. 



In this paper, the optimal sampling design is for 

 stratified-age subsampling. For random-age subsampl- 

 ing in which the number of age subsamples (n) is ran- 

 domly taken from the entire length sample of size N, the 

 estimated variance and covariance (Kimura 1977) are 



Var(^)=l 



'l A /I A , A ,A 



A 



N 



and 



Cou(p r p k 



A A A A A A 



— +1 — - — 



N 



A 

 PjPk 



N 



These two equations are similar to that of proportional- 

 age subsampling in which n,-nl, is substituted into 

 Eq. 2 and 3. However, the estimated variance and co- 

 variance for proportional-age subsampling are approxi- 

 mate, and those for random-age subsampling are not. 

 The similarities of random- and proportional-age 

 subsamplings can be anticipated because N is a ran- 

 dom sample from a population so that E(1,)=E(N,/N)=1„ 

 and n is randomly taken from N so that E(n/n)=N,/N. 

 This indicates that the size of each n, will be approxi- 

 mately proportional to 1„ i.e., n,/n=N,/N=l, and n=n«l, 

 (Kutkuhn 1963). 



An ALK requires a large random sample of fish 

 from which a length-stratified subsample is collected 

 for ageing. Most fishery data are collected either from 

 surveys in which fish from different tows are sampled 

 or from commercial catches in which fish from differ- 

 ent vessel-trips are sampled. Pooling data over such 

 clusters is necessary because of the cost of data gath- 

 ering. In addition to cluster sampling, fisheries data 

 are frequently stratified into time-area, fisheries (or 

 gears), and sex strata (Kimura 1989). The question is 

 how to make the optimal sampling design of ALK 

 generally applicable. To address this, the following 

 factors must be considered: (1) Need of stratifica- 

 tion, (2) ALK sampling within stratum, and (3) com- 

 bined-strata estimation. 



Westrheim & Ricker (1978) showed the need for 

 stratification. An ALK obtained from a population at a 

 time-interval should not be universally applied to 

 length-frequency datasets from other populations or 

 other time-intervals if growth and survival rates are 

 different among the populations and time-intervals. 

 Therefore, the factors that may result in differences in 

 growth and survival rates should be evaluated, and 

 stratification should account for these factors. 



Current sampling programs (e.g., Doubleday & 

 Rivard 1983, Quinn et al. 1983, Kimura 1989) adopted 

 the strategy in which length-frequency data collected 

 from clustered sampling units (e.g., tows or vessel- 

 trips) within a stratum are pooled, from which a length- 

 stratified subsample is collected for ageing. Southward 

 (1963) evaluated an old method (Southward 1963:12) 

 in which a set of length and age data is collected from 

 each landing of a vessel-trip. Because this old method 

 is not developed from a probability sampling design, 

 the within-vessel variability in fish lengths is assumed 

 to be less than between-vessel variability. Southward 

 (1963) showed that this assumption is not valid and 

 the estimated variances of age composition from this 

 old method are so large that little confidence can be 

 placed in it. 



The length-frequency data pooled over clusters 

 should be a representative sampling of that stratum. 

 Therefore, the weighting factor of each sample should 

 be included in the pooling. Ignoring the weighting fac- 

 tor will bias the estimated age composition (Kimura 

 1989). Quinn et al. (1983) described a sampling-rate 

 method in which a fixed proportion of halibut were 

 sampled from landings >1000 lbs for length data, and 

 then age data were subsampled from the pooled length 

 samples from these landings. All length samples are 

 self-weighted and can be pooled directly. 



Quinn et al. (1983) evaluated the methods of com- 

 bined-strata estimation and found that the "project- 

 and-add" method (total catch-at-age is estimated for 

 each stratum and then the estimates are added over 



