NOAA Technical Memorandum NMFS 

SAMPLING COMMERCIAL ROCKFISH 

LANDINGS IN CALIFORNIA 

A. R. Sen 

Department of Statistics and Applied Probability 

Mivefsity Q# Alberta 

Edmonton, Alberta, Canada 

NOAA-TM-NMFS-S WFC-45 

U.S. DEPARTMENT OF COMMERCE 

Malcolm Baldrige, Secretary 

National Oceanic and Atmospheric Administration 

John V. Byme. Administrator 

National Marine Fisheries Service 

William G. Gordon, Assistant Administrator for Fisheries 
This report was prepared for the National Marine Fisheries Service, 

Southwest Fisheries Center, by A. R. Sen under cooperative agreement 

83-ABH-00025 with the Hurnboldt Foundation, Hurnboldt State University. 

The statements, findings, conclusions and recommendations herein are those 

of the author and do not necessarily reflect the views of the National 

Marine Fisheries Service. Norman J. Abramson, Tiburon Laboratory, 

Southwest Fisheries Center served as Technical Representative for this 

contract. 
CONTENTS 

PAGE 

1. INTRODUCTION 

2. DATA COLLECTION 

3. DESIGN OF THE SURVEY 

4. COLLECTION OF REPRESENTATIVE SAMPLE DATA FROM LANDINGS AND 

CLUSTERS 

5. ESTIMATION PROCEDURE 

5.1 Notation 

5.2 

5.3 

5.4 Estimation Based on Post-Stratification 

5.4.1 Cluster Size 

Estimates of Mean, Total and Errors 

Estimation Based on Random Categories 

I 

2 

3 

8 

9 

11 

13 

14 

5.5 General Case: Unequal Cluster Size 16 

5.5.1 Random Categories 

5.5.2 Post-Stratification into Sort Groups 

16 

17 

5.5.3 Cluster Weight Constant Within Sort Group 20 

5.5.4 21 

5.6 Comparison of Methods. Simple Random Sampling vs. 

Estimation of Ratio to Another Variable 

22 

25 

27 

29 

30 

Post-Stratification 

6. OPTIMUM SAMPLING AND SUB-SAMPLING FRACTIONS 

6.1 Cost Function 

6.2 Optimum Allocation: Survey Data 

6.3 Optimum Allocation: Experimental data 

6.4 Variance Components: Species-sex-age and length Groups 

7. RELATIVE EFFICIENCY OF ESTIMATORS USING POST-STRATIFICATION 

36 

40 
(ii) 

CONTENTS - Continued 

8. RELIABILI'IY OF THE ESTIMATES: COEFFICIENT OF VARIATION 

8.1 

8.2 

8.3 

Coefficient of Variation (c.v.) of Species Catch by Port-Year 

Groups 

Coefficient of Variation (c.v.) of Species Weight by 

Port-Year-Sort Groups 

Coefficient of Variation of Speciesby Sex-Age for Port-Year 

Groups 

9. RELIABILITY OF THE ESTIMATES: SAMPLE SIZE 

9.1 Sample Size for Species 

9.2 Sample Size for Species 

9.3 Sample size for Species 

Groups 

9.4 Sample Size for Species 

10. AGE COMPOSITION 

10.1 Number of Strata 

10.2 Strata Boundaries 

Catch by Port-Year Groups 

Weight by Port-Year Sort Groups 

Catch by Sex and Age far Port-Year 

10.3 Optimum Allocation Plan 

11. S-Y AND RECOMMENDATIONS 

Catch by Ports on Quarterly Basis 

ACKNOW L€DGEMENTS 

LITERATURE CITED 

44 

44 

46 

46 

62 

63 

65 

66 

66 

79 

81 

82 

a3 

87 

93 

94 
1. INTRODUCTION 

This study was undertaken during 1983 (May 15-October 15) under 

agreement between the present author, the Humboldt State University 

Foundation and the Department of Commerce, National Oceanic and Atmospheric 

Administration (NOM), National Marine Fisheries Service Tiburon Laboratory. 

The purpose of the investigation was in the first place to determine if 

the present sampling plan for the estimation of parameters such as age 

and species composition of California rockfish landings is workable; 

i.e., if the operational procedures being utilized during the execution 

of the plan meet its requirements. If not, whether a revised plan could 

be designed which would conform to operationally feasible sampling procedures. 

It was also intended to obtain, under certain assumptions, the precision 

of the estimates of total catch of rockfish by species and by sex-age 

groups stratified by type of fishery and port and time of landing in 

California north of Point Arguel1o;'and compare these with the precision 

of alternative estimators with a view to find the most efficient estimator. 

Finally, it was required to determine the optimum sample size for estimating 

the parameters as accurately as possible within the usual limitations of 

Ibudget and personnel. 

The fishery can be broadly divided into two types--commercial and 

sport. Commercial fishing can be classified into fishing by trawlers 

and by nontrawlers. 

party boats, from piers, and from shore other than piers. 

Sport fishing is generally done by party and by nonIn view of the greater importance and complexity of the sampling and 

operational problems olved in commercial fishing, the present study 

will be confined to an examination of the problems outlined above in the 

commercial fishery and to the important species , Widow (Sebastes entometas) 
2 

Bocaccio (Sebastes pawispinis) and Chilipepper (Se3uste.s goodei) . It is 

proposed to take up the study of sport fishing as a separate project at a 

later date. 

We will now briefly review the data collection process and the 

principal features of the current design to understand the extent to which 

the assumptions made at the data collection and estimation stage are 

justified and the changes suggested in the sampling plan when the 

assumptions are not met. 

2. DATA COLLECTION 

Rockfish are being landed at 14 ports on the California coast. 

Of these, three cater only to commercial fishing, four to sport ffshing 

and seven to both sport and commercial fishing. 

are grouped into six port groups with a sampler (six in all) assigned to 

each--Eureka, Fort Bragg, Bodega Bay, San Francisco, Monterey and Morro Bay. 

In the current plan a sampler is expected to visit each port in his 

The ten commercial ports 

jurisdiction at least once a week for gathering information- m sport and 

commercial fishing. The samplers employed hold temporary jobs which 

range from 4 to 24 months in duration. 

Trawlers ~ke trips varying in length from one to eight days. These 

vessels maintain logbooks to keep records of area fished and appropriate 

catch for each tow. Sampling by tow is generally not feasible since the 

sampler has to be on board during haul time. 

estimates of fish being rejected and returned to the sea are obtained since 

this would involve collection of discarded fish from randomly selected tows 

For the same reasons no 

within sampled trips. Also, sampling by area of catch is not practicable 

unless all tows are in a limited area and the sampler is on board at haul time. 
3 

3. DESIGN OF THE SURVEY 

A two-stage stratified random sampling plan with port-month group as 

stratum and boat trips within a stratum as first-stage sampling units was 

adopted. To take advantage of the sorting at sea provided by market 

categories, the first-stage sampling units are post-stratified into sort 

groups and at least one cluster of a given weight is subsampled within 

each sort type. 

pounds cluster is taken when sampling small fish like Sebastes wora, 

Cluster size is either 25 or 50 pounds. Twenty-five 

Sebastes s&coZa or Sebastm diplopma or any time small rockfish are 

landed such that there would be more than 20 fish in the fifty pounds 

(lbs.) cluster. In all other cases 50 lbs. standared cluster size is 

selected. 

weight, which are recorded along with sex, total length and otoliths from 

A cluster is next separated by number of each species and its 

Sebastes and SebastoZobus. 

The instructions are to "sample all market categories from a boat, 

and from as many boats as possible and select: 

(i> 1 duster per 20,000 Ibs. of widow rockfisb fanired by each boat, 

up to 4 clusters, 

(ii) 1 cluster for all other species, if less than 5,000 lbs. landed and 

(iii) 2 clusters for all other species if more than 5,000 lbs. are 

lauded. 

sampling another boat. " 

The second cluster should not be taken if this precludes 

Besides, the sampler has to obtain from the skipper the total weight 

of each category of a sampled landing, as would be clear from the discussion 

in Section 5.6. 
4 

I 

4. COLLECTION OF REPRESENTATIVE SAMPLE DATA FROM L;LNDINGS AND CLUSTERS 

Owing to the uncertainty of arrival times and varying unloading 

procedures, no objective method is available to ensure random sampling of 

the trips. 

sampling except when the landing is transshipped immediately due to inclement 

When the vessels return to port, they are usually available for 

weather OF lack of processing facilities, uncooperative buyers, or due to 

unscheduled deliveries at short notice. It is, however, not unreasonable 

to assume that a representative set of sample landings can be obtained from 

a port-month stratum. 

Although rockfish are landed by sort groups which are mostly 

determined by market agreement based on size, composition and condition of 

the catch, the number of sorts per delivery can not be predetermined. This 

number would vary from delivery to delivery and from dealer to dealer. 

There are no guarantees that a complete boat sample, covering clusters from 

each sort group can be taken on any sampling day and some of the categories 

may be missed in sampling. 

categories are 

(i.e., early morning or late nights) 

shipped before the sampler could arrive at the spot 

sorts may be quite small and there may be a buyer at the dock waiting for 

the fish to be taken away; and 

the other sort/s will have either been processed or shipped away 

(iv) the sampler may be prevented from taking a sample from another sort 

since the skipper may not like some of his fish being cut. 

generally, happen at ports where either processing facilities may Rot be 

Some of the possible reasons for missing the 

(i) when landingwould not occur during regular hours 

one of the sorts may have already been 

(ii) often one of the 

(iii) while the sampler is working on a sort, 

and finally 

This may, 

I adequate, and a large portion of the landings are shipped, or are bought by 
5 

local merchants immediately after landing. 

to sample from all the categories of a sample landing as required in the 

current plan would cause appreciable bias and consequent loss in efficiency 

The question arises if failure 

in the current estimates of species nwnberand its weight. 

is proposed to examine this as well as the feasibility of the current plan 

based on actual data collected in the past. 

It 

It may be pointed out that the current technique of selecting a 

cluster (box) of fish as second-stage sampling unit was rightly preferred to 

the 'grab technique' based on the assumption of random selection of fish 

by the sampler since in practice the potential of personal bias of the sampler 

could be considerable. Tomlinson (1971) feels that a sampler 

may have a tendency to choose fish with certain qualities and thus may 

introduce procedural bias. Tomlinson sees no way to avoid the conclusion 

that "a simple random sample of individual fish is operationally impracticable." 

The selection of a representative cluster would depend upon whether samples 

after sorting on the vessel come from bins, strap boxes or off conveyor belts. 

At Monterey they are mostly unloaded into metal bins which are either placed 

in a large cooler or transferred to a conveyor belt for transport to the 

fillet line. Buyers from small markets occasionally select fish from the 

top of bins. Hence, to avoid bias, it is preferable to select the cluster 

from the conveyor belt which exposes unsorted fish from the lower portion of 

the bin. I noticed this practice being rightly followed at Monterey during 

However, where small-market buyers do not buy fish, 

ed from a bin. Where many bins are present, a 

systematic sample'of two clusters one from each separated in rime and 

preferably from the beginning and end of the trip should be selected. It may 
6 

be pointed out that an efficient method of subsampling is eo assign a number 

to all individuals in the populationband select from it a sample of the 

required size with the aid of a table of random numbers. 

procedure is not feasible since it is too expensive and time-consming. 

Where fish are graded on a conveyor belt before they enter the plant (e.g., 

Fields Landing at Eureka), the sampler should try to intercept the landings 

prior to secondary sorting or obtain separate weights for each sub-sort 

category. 

followed. 

In practice, this 

Whichever method is relatively easier in practice should be 

As has been pointed out earlier, bias may also arise through personal 

selection of fish from within a cluster. 

number of clusters with too few fish per cluster (e.g., 25 lbs. cluszer for 

medium and large fish) a cluster will, on the average, contain more of big 

fish. Thus, although the procedure will reduce the sampling error, it will 

tend to increase considerably the non-sampling error due to selection bias 

which could be serious. Sometimes, the top few fish in a bin are selected 

and put there to impress small buyers. 

be avoided by taking all the fish in a cluster (e.g., 50 lbs.) from one side 

so that the sample measured consists of all the fish originally to one side 

of the box. At Montere)., Eureka (where sampling of cluster is either from 

conveyor belt or from bins) and at San Francisco (where clusters are generally 

selected from strap boxes] we observed the samplers rightly selecting the 

clusters (without looking at the fish] from one side in a cylindrical section 

thus collecting fish of different sizes and kinds over the section. 

If the sampler were to select a 

The resulting bias in selection can 

To summarize, random sampling of boat trips is not practicable owing 

to the uncertainty of their arrival times and the most reasonable assumption 
7 

is that the boats arrive at a port during a month in a random order. 

Selection of a random and representative cluster (box) of fish from a boat 

trip would depend to a large extent on the expertise and experience of the 

sampler.- Hence, the need to have permanent staff at least at the important 

In general, selection of a cluster for a market category should be 

done before any presorting is done at the port either on conveyor belt or 

in bins or from strap boxes; it is felt that fish landed from strap boxes 

are likely to be subjected to greater sorting by length, etc. than ones from 

bins. 

time, e.g., at the beginning and termination of loading from a box, bin or 

Clusters should be selected for each market category separated in 

off conveyor belt. 

side of a box, including fish from the top all the way down to the bottom, 

and fish selected should not be seen in the process. 

As far as practicable,.selection should be made from one 

For obtaining reliable, comprehensive and complete information on 

population characteristics, it is necessary that good relationships be 

maintained by the sampler with both the skipper and the buyer. 

depends to a large extent on the experience of the sampler gained in the 

course of his field work over years. 

for permanent staff at least at the important ports which have too many 

problems to handle within a short time. 

This 

This emphasizes again, the need 

5. ESTIMATION PROCEDURE 

e problem of estimating the total catch of a given species 

for a port-month stratum. 

istics are straight forward and can be obtained by substituting the value 

The formulas for estimation of other characterof the characteristic for the catch of the 'species. 
5.1. Notation 

Let 

N = total number of trips, 

n = number of randomly sampled trips, 

W = total weight of fish caught from all trips, 

fii = 

8 

weight of fish caught on trip i, 

weight of sort j caught in trip i, 

number of clusters sampled from sort j of trip i, 

Li 1 Wij where Li is number of sorts in trip i, 

i J 

number of the species in cluster k from sort j of trip i, 

total number a€ the species caught from sort j oftrip i, 

total number of the species caught from all trips, 

AJ I Yijk /mij = unbiased estimate of Pij , k 

weight of the kth cluster from the jth sort of the 

ith trip, 

average weight of all clusters sampled, 

constant (say) , 

'i -= - W estimate of Mi. 
9 

5.2. Estimates of Mean, Totpl and rirrors 

nn n 

yR L 1 wiYi / 1 wi = ratio estimate of mean 7 

i i 

catch per cluster 

n n 

xc w. .y.. / 1 w 

1J 1J j ij j 

and 

w- (2) $ - YR = ratio estimate of total catch Y 

The above estimates are biased and are based on the assumption that the 

average weight Fer cluster is a constant. 

more efficient than the corresponding unbiased estimates(as would be seen 

The estimates' are consistent and 

and take into accounr the size of the landing. These satisfy the 

important requirement that trips larger in size should receive higher 

weights at the estimation stage. In view of the lack of information (in 

advance) on the size of the landing, the approach to sample trips proportionate 

to trip size is not practicable. 

Approximate estimates of V(? ) and V(yR) are respectively given by R 
10 

In practice, both M and 

2 

2i S" 

Mi are not known. These are estimated by 

iii = Wi/W and 

will be subject to high errors, when Wi*s are highly variable and n 

is small. should be replaced by the true values N, wherever these are 

available after the season. 

The above estimators which were recommended for use in the current 

design are, however, not workable since these pose serious operational 

problems in data collection from all the sort types within sample landings 

as was obvious from an examination of basic data for 1982 available with 

the California Fish and Game. 

during January 1, 1982 to September 10, 1982 are reproduced to illustrate 

the point .(Table 1) 

Data for ten sample landings for Eureka 

The sampler failed in all cases to sub-sample from more than one 

category in the samples (trips) where the landing weight from a boat trip 

comprised more than one category. The reasons for failure to collect the 

data were discussed in Secthn 4. 
11 

Sample No. 

(Boat trip] 

1528 

1529 

1530 

1535 

1536 

1537 

1541 

1531 

1533 

1534 

TABLE 1. Distribution of Landing Weights From All Categories 

and From the Sampled Category for Eureka for 1982 

No. of Cluster: 

Sampled 

1 

1 

2 

1 

1 

1 

1 

E 

1 

3 

for the 

Sampled 

(3) (4) (5) 

269 

250 

26 9 

269 

269 

269 

269 

250 e 

2 50 

269 I 

26,550 24,176 

4,133 445 

54,213 58,239 ' 

20,511 15,987 

35,022 14,661 

20,757 20,705 

15,812 8,436 

E, 975 1,810 

16,055 1,075 

3 

65,837 65,837 

51 

44 

1 04 

51 

49 

54 

50 

52 

53 

145 

- 

I I W i 

ik W 

41 k' 

k 

51 

44 

52 

'5 1 

49 

54 

50 

52 

53 

48 

5.3. Estimation Based on Random Categories 

Assuming that the clusters sampled from a category from within a sampled 

trip are simple random mples from all possible clusters in the trip, more 

- 

of y and flR of Y are respectively given by '1R 

valid ratio estimates 
W= I,, = (=)Y1 

W 

1 $. Yijk k3 

c i mij 

where Ti = and Wi is the total landing weight from all catego

J 

for the ith boat trip in the sample (W = 1 i Wi). In practice, the samp

would tend to sample from a category which is preponderant and is accessibl

Hence, the estimate may be slightly biased, though its contribution to the 

total error will be negligible, since this would occur at the second stage 

of sampling. 

Equations (3) and (4) will be replaced by 

n 

and 
13 

- n 

m = 1 mi/"; 

N 

1 Wi = Total of all rockfish landings in the port-month stratum. 

i=l 

W = 

5.4. Estimation Based on Post-Stratification 

We will now consider an operationally feasible plan based on poststratification by sort groups, 

category from a boat trip at a port during a month, these can be classified 

into the categories into which the port-month group has been post-stratified. 

Thus,. when a sampler'sub-samples from a 

This method is almost as precise as proportional'stratified sampling if 

within each port-month stratum (nj 2 4) 

is, selected for each category, and [b) the landing weights are available 

by sort groups after the season to serve as weights at the estimation stage. 

The procedure has four advantages over the one currently being used: 

(a) a minimum of four landings 

(i) it would provide estimates by market categories for each port-month 

stratum, 

categories, 

from another category would preclude its selection, and 

(ii) it would account for the bias in estimation due to missing 

(iii) would enable sampling from another boat when sub-sampling 

(iv) finally it 

would be more efficient than the current method. The analysis of the data 

will show that is is preferable to sample as many boat trips as possible. 

Recall that 
14 

= pounds of sort j caught in trip i, in a *i j 

port-month stratum, 

w = 1 Wij = pounds of sort j caught in a port-month ji 

stratum. 

5.4.1. Cluster Size 

Assume as before (4.2) 

= constant, i.e., i jk W 

cluster weight for all trips, sorts within trips is a constant and is 

estimated by 

In practise, a 50 lbs. cluster should be within 50 2 5 lbs. in weight 

i.e., individual cluster weights should be between 45 and 55 lbs. at 

the maximum and care should be taken to ensure that a cluster should be as 

close to 50 lbs. in weight as possible; for a 25 lbs. cluster in small 

fish category, e.g., rose fish, individual clusters should be between 

and 27 lbs. at the maximum. For j = 1,2,. . . ,L sorts we have ratio 

estimate of the mean per cluster and of total given by 

23 
In a few cases, where 2 or more categories are sub-sampled from the same trip, 

it is reasonable to assume that the sub-samples from the categories are 

independent since the categories are likely to be different in their species 

composition and weights. 

in some bias 

If the assumption is incorrect, this may result 

in the estimates of error which will not be substantial. 

Further 

and 

where v(f-) and ~(7.) can be obtained as in (3) and (4) by 3 3 =2 

m and W for N, n, Mi, 2ij' ij 

m and W respectively. Since ni would occur in the 

j substituting N , nj, Mij, yij, yj, s j -= 2 

Yy YR, S2i' i J 

denominaior, n > 0. For efficient estimates R > 4 (see page 13). 

j jEmphasis on "25 lbs. weight for small rock fish including widows and 

SO lbs. 

samples (boat trips) resulting in violation of the assumption of equal 

cluster weight required in the current method of estimation of parameters. 

Formulas for the estimation of parameters and their errors have been developed 

for the general case of variable cluster weight in Section 5.5.2. 

exact formula for the estimates of error variance are rather complicated, 

for all other clusters" led in some cases to wide variability among 

Since 
14 

approximate workable expressions are provided. 

are provided in Section 5.5.3 for the more practical case when cluster weight 

can be treated as a constant within a sort group. 

However, exact expressions 

5.5. General Case: Unequal Cluster Size 

We will consider an estimating procedure, when clusters of variable 

weights are used and it may not be possible to take two 

when sampling landings from sorts of small fish like rose fish or small 

widows either because of small landings or buyer not cooperative to let 

the sampler select more than one cluster. 

50 Pbs. cluster 

5.. 5.1. Random Categories 

Assume that the weight per cluster var.ies over boat trips but remains 

the sane within a trip, i.e., 

As discussed above, the cluster weight may vary more among sort groups but 

is approximately a constant within a sort group, so that in practise 

will be the weight of a cluster for the 

w 

j 

jth sort group. 

Consider the two ratio estimators 
17 

m m 

W m ik ik ik 

- 1 f -=- 'ik 1 

rik where r = - im 

where 

,. 

and 

In this study we wiil use the estimator ?,, since only a small 

sample is selected from each boat trip and also since R A is likely to be i 

more stable than Fi. 

v(?,,) and ~(7,~) can be obtained similar to (6) and (7) using 

(16) and (17) and noting wi varies over samples. 

5.5.2. Post-Stratification into Sort Groups 

Assume as be 

w 

to size sorting of widow and rock fish in small fish categories. 

varies with i, as observed among cluster weights during 1977 and due i 
18 

th 

For any sort category (say j 1, let 

where 

- .- = y JWij iJ 

= estimated number of fish (per pound) 

of a species for the j th category from the ith boat in the sample and 

If n is small compared to N j and if the same subampling strategy is applied 

to each of the n 

second-stage units, 

j 

sampled landings, we have, ignoring contribution due to 

j 

approximately where z = 6 W and x ij = W ij . (20) reduces to ij ij ij 

A 

Another estimator of V(R.) is given by J 
19 

where 

and 

Thus 

calculating instead of 6 as in (19). 

is obtained by omitting trip i from the sample for sort j and 

ij 

ij ij 

For estimating mean and total for a sort j for a species 

Also, 

or 

v(2j) = w2v j 1 (fi j ) 

where v (6 ) and v (i ) are given by (21) and (22). 1j 2j 

For estimates of total over all sort groups 
20 

assuming that sub-samples from two or more categokies from within a boat-trip 

sample are independent. 

5.5.3. Cluster Weight Constant Within Sort Group L 

It is more reasonable to assume that cluster weight would be 

approximately a constant within a sort group (e.g., rose fish category) but 

would vary among sort groups. 

total for a species are given by 

If so, the estimates of 

where 

the cluster mean and 

and 

L is the number of strata with the sample mean weight 

th j of clusters in the j group. 

It follows that 

where 

W 

Aj = -$- 

! 'j J 
21 

5.5.4. Estimation of Ratio to Another Variable 

Estimation of the ratio of number of fish in a age, sex or agesex group to the total 

obtained as a ratio of 

Thus, if Ti and Ti 

fish in an age-sex and 

number for the species for a sort group can often be 

two variables in a two-stage sampling procedure. 

represent respectively the mean of the number of 

sex group respectively, we have 

where R = Z/Y, 

given age) in the jth cluster of the ith trip. 

zij is the number (or weight of a given species for a 

22 -- 2 (l-fl) n n wi (Ti - RY~) 

c 2 (n-1) v(B) 5 

i (1 wii;i) n 

c1-fzil (36) 

W. 22 

WiYi 

n n 

+ i 'i z! (; "_ ) 'dZi 

i 

and 

22 2n 2- 2 M. s 

* 

(l-fzil (37) 

N d2i 

i i 

N (1 -z) n Mi(zi - ifi) 

+-C m n- 1 n '('Rd n c 

i-1 

A n 

m C Wi , fZi =-' , fl=-- 

m i xzi. - Ryi.) 2 

-- and N and ai n 

N W 

2 where sd = 1 

Mi are given by (4a). 

Mi m -1 2i j=l i 
22 

stratification by categories and (2) using post-stratification by categories 

(Table 3) show that the estimates of mean catch of widow were estimated with 

less than 8 percent C.V. by method (2) at both Eureka and Montercy 

though the precision was considerably less using method (1). Again, using 

method (2) both Chilipepper and Bocaccio were estimated with higher 

precision (13.9 and 10.3 respectively) at Monterey though at Eureka the 

precision was .somewhat lower. The estimated C.V. for Chilipepper at Eureka 

5.6. Comparison of Methods. Simple Random Sampling Versus Post Stratification 

In this section we will compare the efficiencies of the estimators (5) 

based on random categories, i.e., clusters selected from all the sort groups 

from within a sampled trip with (10) based on post-stratification of a 

port-year stratum into sort groups. 

The analysis will be based on 1982 data (January-September) for Eureka 

and Monterey for which total landing weights by categories were available 

from the California Fish and Game. 

not large enough to provide estimates with any reasonable degree of accuracy. 

The sample size by individual months was 

The distribution of the samples (Table 2) by categories show that the 

number of landings are too few for certain categories. This number should be 

raised by either increasing the number of sample landings within each category 

for a porr-month group or by decreasing the number of strata by suitably 

combining some of the categories or both. 

The coefficient of variation (c.v.) of mean catch per cluster for 

the species based on the methods (1) random categories, i.e., ignoring 

based on the same number of samples (40) as at Monterey was 21.55 which 

was much higher than the corresponding figure (13.92) at Monterey. The 

c.v.'s for short-spine thorny head not shown in the table were much lower 

by method (2). 
23 

r 

Categmy Mean Cluster wt Landing wt Landing wt Number in l’n SampIe Crpi’J in PopuIation (w) Sample [Boat trips) ’ 

250 47.71 241,696 1,150,164 29 

16 253 47.96 177,165 1,607,824 

2 50 00 3,655 144,177 26 2 

6 269 50.80 395,137 1,352,110 

1 32.50 10,330 104,070 270 

Total 827,983 4,358,345 54 

TABLE 2. Distribution of Landing Weights (lbs.) in the 

Sample and Population by Categories During 

January 1 to September 30, 1982 

Eureka 

IL Category Mean Cluster wt I 

2 

21 

6 

88 

26 2 50.50 4,390 1,903,258 

269 51.49 694,924 5,519,313 

270 27,83 1,249 57,895 

Total 855,340 . 10,301,688 
24 

I Sample Coefficient of Variation 

Size (0) Species 

Method 1* Method 2** 

Widow (54 1 18.31 6.62 

(2336) 

Chilipepper c54 3 15.68 13.92 

(2320) 

Bocaccio 

(2334) 

64 3 12.57 10.32 

TABLE 3. Coefficient of Variation (in Percent) of Mean Catch by 

Species for Eureka and Monterey based on the two Methods 

During January 1 to September 30, 1982 

Eureka 

Species Sample Coefficient of Variation 

Size (%I 

Method 2** 

Widow 

(2316) 

Chili pepper 

(2320) 

Bocaccio 

(2334) 

7.33 

32.12 

24.40 

I 1 

Monterey 

* 

Method 1. Based on random categories. 

Method 2. 

** 

Based on post-stratification by categories. 
25 

Thus the estimates of catch of a species based on post-stratification 

by market categories were generally more efficient than those based by 

ignoring such stratification. 

The coefficient of variation for the species at Eureka and Monterey 

during 1982 by sex-age groups for which the number of sample landings was 

greater than urequal to ten (Table 4) show that in almost all cases method (2) 

is more precise than method 

species in a particular sex-age group. 

(1) for estimating the number of fish of a 

In summary, method (2) based on post-stratification by sort groups 

is preferred to method (1) for estimating the number of a species 

. based on random categories; also, method (2) is recommended for 

estimating the number of a species in a sex-age group. 

6. OPTIMUM SAWLING AND SUB-SAMPLING FRACTIONS 

The instructions on the number and distribution of clusters to be 

selected from sample landings in the current plan are not specific enough 

in the absence of information on the between and within sample component 

of variation in species number, weight for obtaining the optimum allocation 

for a given cost. 

function and the optimum number of clusters for a given cost using both 

the survey data from sample landings during 1978 (when four clusters were 

mostly available per sample) and data based on an experimental study 

undertaken during 1983 at some of the ports. 

In the following sections, we will obtain the cost 
I -3 

TABLE 4. Coefficient of Variation (in Percent) of Mean Catch per Cluster 

by Species for Important* Age-Sex Groups based on the two Methods 

for Eureka and Monterey During 1982 (January 1 - September 30) 

Eureka 

Widow 

(2316) 

C.V. (%) 

Age Sex Method 1 Method 2 (years) 

12 M1 15.49 14.84 

7 M 19.71 18.83 

16 F 15.82 16.08 

12 F 15.14 14.10 

7 F 13.50 10.94 

Chilipepper 

(2320) 

C.V. (%) 

Sex Method 1 Method 2 Age ~ 

(years) 

13 

12 

F 39.98 24.89 

F 34.77 31.21 

26 

Mon t erey 

Widow 

(2316) 

C.V. (%> 

Age Sex Method 1 Method (years) 

13 

12 

9.. . 

F 39.98 24.2

F 35.16 20.4

F 31.22 27.0

Chilipepper 

(2320) 

C.V. (%) 

Age Sex Method I Method (years) 

9 F 18.48 7.6

8 F 27.84 9.0

7 F 22.09 9-8

6 F 25.67 8.8

* 

Age-sex groups for which primary sampling units (landings) are - > 10 
27 

C.V. (%) Age Sex Method 1 Method 2 (years) 

TABLE 4. (Continued) Coefficient of Variation (in Percent)of Mean Catch 

Per Cluster by Species for Important Age-Sex Groups Based on the 

Two Methods for Eureka and Monterey During 1982 (Jan. 1 - Sept. 30) 

C.V. (%) 

Age Sex Method 1 Method 2 (years) 

Eureka 

Bocaccio 

Mont erey 

Bocaccio 

6 M 30.10 19.82 

7 M 30.74 31.91 

M 39.22 41.71 

F 35.87 32.45 

F 22.65 22.80 

7 M 27.46 12.45 

6 M 23.23 9.75 

5 M 23.27 14.38 

4 M 25.35 10.65 

7 F 24.34 10.06 .. 

6 F 23.98 8.03 

5 F 22.32 9.65 

4 F 27.23 12.34 

6.1. Cost Function 

From equations (6) and (7) we have 

2 -- 2 (l-fl) n wi (yi-ylR) (1-7~) n - 

VGIR) I n 1- (62 n-1 (38) 

n f m. 

i and m = - 

-nm - 1 

- wi 

n Wi n' 

- A C Wi 

where W = - - f2=-C- n 

* 
28 

Putting 

and 

we have 

for the simple cost function (ignoring travel costs between sample landings) 

- 

C = cln + c nm 2 

where cl is the average cost (in minutes). per boat trip dne totransport, 

contact and delay in making a contact, 

minutes) of data collection (e.g., identification of species, sex, length, 

otoliths, etc.) per cluster and C is the total cost involved in visiting 

the primary sampling units (boat trips) and collecting data from the 

c 2 is the average cost (in 

n 

boats with an average of zclusters per boat sampled. From data collected 

at Tiburon as a result of the cooperative program between California Dept. 

of Fish and Game and the National Marine Fisheries Service we have 

c = 111.80 minutes, c 2 = 58.3 minutes 1 
1 

2 

C 

so that c = 1.9177 = 2 approximately. However, from some more recent 

data collected we have approximately 

Actually, the components of 

29 

c1/c2 = 3 

c and c, were estimated at 1 L 

Activity 

Transport 

Contact 

Delay (off-loading, etc.) 

Data Collecti'on 

Species * 

Sex, length 

Otolith 

Preparation Time 

Percent Mean (in minutes) 

50 81.7 

5 8.7 

13 

. 6%. 

7.7 

5.8 

10.8 

7.7 

32.0 

- 

21.4 

111.8 minutes . 

Mean (in minutes) 

14.0 

10.6 

19.7 

14.0 

58.3 

- 

* 

excluding samples dominated by single species. 

6.2. Optimum Allocation - Survey nata 

at the different ports on the basis of 

survey data collected by California Fish and Game department based on its 

cooperative prog 

estimating the mean or total catch of a species at a port during a season. 

opt 

We will now estimate m 

with the National Marine Fisheries Service for 
30 

It would be seen (Tables 5 and 6) that the oprimm number of clusters 

per boat in the sample for estimating species number or species weight was 

unity in almost all cases. Since a minim of two clusters is needed to 

provide an estimate of the component of cluster variation required for 

, obtaining an estimate of error, it is recommended that two clusters per 

primary sampling unit (boat trip) be randomly selected. In practice, it is 

preferable to select a systematic sample of clusters separated in time. 

This will provide an unbiased estimate of V(FR), if fl is small. However, 

if fl is not negligible, the estimate of the error will be slightly biased. 

6.3. Optimum allocation: Experimental data 

We will now consider the case where data-from an experimental study 

using 

of the between and within cluster variation EO throw further light on the 

4 clusters per sample were analysed to find the relative contribution 

problem. 

Four clusters each of approximately 50 lbs. in weight (in a few 

cases when the total fish caught was small in weight, clusters of 25 lbs. 

were selected) were selected at random from a sort group of a sample 

landing (boat trip) on different landing days of June and July 1983 at 

MorroBay, Monterey, Moss Landing, Sam Francisco, Fort Bragg and Eureka. 

Details of the method of random sampling are given in the section on data 

collection. A cluster is next separated by number of each species and its 

weight which are recorded along with sex. 

The proportion of the number of each species to the total for all 

species in a cluster from a sample landing was next analysed using the 

arc sine transformaxion. Where the total number n in a cluster was < 50, 

a zero proportion was counted as 1/4n and a 100% proportion as 
31 

TABLE 5. Optimum Values of. m 

by Sort Groups for Different Variance and Cost Ratios 

for Estimating Species Number per Cluster 

Eureka: 1978 

- 

mOpt Species 2 2 m 'b sW 

Sort n (number) 

c1/c2 = 2 c1/c2 = 3 

Bocaccio 

Chilipepper 

250 25 1.80 3.01 2.16 3.86 

250 13 24.45 3.13 1.92 0.52 

4.73 

0.64 

Widow 250 11 59.49 8.71 2.46 0.56 0.68 

Monterey: 1978 

Bocaccio 253 31 95.15 4.20 1.97 0.63 0.77 

Chilipepper 

Widow 

253 33 43.71 4.16 1.94 0.45 0.55 

253 12 22.38 4.66 2.00 0.68 0.84 

San Francisco: 1978 

Bocaccio 253 20 17.99 5.74 2.30 0.86 1.05 

Chilipepper 253 15 10.79 14.70 2.65 2.31 2.82 

Fort Bragg: 1978 

Bocaccio 253 86 14.81 2.37 1.26 0.61 0.74 
32 

TABLE 6. Optimum Values of m for'Estimating Species Weight per Cluster 

by Sort Groups for Different Variance and Cost Ratios 

Eureka: 1978 

Species 

lwei ght 1 

2 

b Sort n S m 

opt 

c1/c2 = 2 cl/cz = 3 

Chilipepper 

Widow 

250 12 181.64 18.67 2 0.47 

250 11 231.15 48.09 2.4 0.67 

0.57 

0.83 

Monterey: 1978 

Bocaccio 253 36 63.17 34.56 2 1.23 

Chilipepper 253 33 86.45 21.31 1.94 0.75 . 

1.50 

0.92 

Widow 253 12 27.07 37.08 2 2.95 3.61 

San Francisco: 1978 

Bocaccio 253 20 117.27 62.43 2.3 1.18 1.44 
33 

(n--)/n 1 before transforming tb angles. This transformation was done to 4 

improve the equality of variance in the angles. The F values based on 

the analysis of variance of the angles at the ports for the species 

Chilipepper, Widow and Bocaccio are shown in Table 7. 

The F values for variation due to samples are large in almost all 

the cases though those due to the variation between clusters within samples 

are much too small (assuming that all the within-cluster variation is 

binomial). 

per sample to estimate mean number of a species per cluster with a high 

degree of precision. 

Chilipepper at Morro Bay will be given by 

This suggests that we should have more samples and fewer clusters 

Thus, the optimum value of the number of clusters for 

assuming c = cln + c h. 2 

In this case from the analysis of variance table we have 

s = = 7.3485 

W 

Hence 

- 7*3485 zz if c1/c2 = 2 

= 0.6002 

= 1 approximately. 

opt 17.3133 m 

In practice, we sh 

of total variance. 

= 2 to provide estimate of within component *Pt 
Mono Bay 1983 

Source 

Samples 

Clusters 

34 

TABLE 7. Analysis of Variance in Angles 

Chi 1 ipepper Widow Bocaccio 

DF MS F P MS F P MSF P 

4 1307 24.2 <0.001 72 10.3 <0.001 444 14.21 ~0.

14 54 (within samples) 

7 31 

Within Clusters 330 101 63 67 

Monterey 1983 

Source Chi 1 ipepper Widow Bocaccio 

DF MS F P MS F P MS F P 

Sampl es 4 1308 8.9 <0.001 449 15.2 <0.001 1671 12.6 <0.0

Clusters 

(within samples) 

15 147 30 132 1.7 <0.05 

Within Clusters 437 82 61 80 

San Francisco 1983 . 

Source Chilipepper Widow Bocaccio 

DF Ms F P MSF P MSF P 

Samp 1 es 4 254 7.7 <0.005 5.6 233 

C 1 us t er s 15 33 5.7 118 

(within samples) 

Within Clusters 259 160 68 159 
35 

TABLE 7. (Continued) Ana 1 y s i s of Variance in Angles 

Fort Bragg 1983 

Source 

Samples 

Chilipepper 

8 1512 9.1 

DF MS F 

Widow 

2.9 <0.025 1416 

F P MS 

Bocaccio 

F P 

6.3 <0.001 

P 

<o. 001 

Ms 

55 

Clusters 

(within samples) 24 165 1.8 <0.05 18 223 2.3 <0.01 

Within Clusters 606 93 40 95 

Eureka 1983 

Source 

Samples 

F P Ms 

5 556 30.7 <0.001 124 

DF Ms F P Ms F P 

11.9 <0.001 590 12-3 ‘0.001 

13 18 C1 us t ers 

(within samples) 

Within Clusters 230 338 

10 

82 

49 

167 
36 

6.4. Variance components: Species- age- sex and length groups 

We will now obtain the relative contribution of variation in lengthjage 

of fish due to sample landings, clusters within landings to within-cluster 

variation for a given species within a sort group. This will, incidentally, 

throw light on the extent to which sorting 

between and within samples for a given market category. 

by length and or age is done 

It would be seen (Tables S(a) to 8(i)) that the variation due to 

length and age was 

within samples. For widow most of the variation was due to sample landings 

at the centres (for which adequate data was available); also, the variation 

generally high among sample landings compared to that 

between clusters was small and was of the same order as that within clusters. 

It would be seen that the optimum number of clusters was 2. 

Further, most of the significant variation for the species Bocaccio/ 

Chilipepper 

in a few cases, however, (e.g., Chilipepper at San Francisco 1979, 

Monterey 1979; Bocaccio at Monterey 1979) the variation between clusters 

was significant but not consistent. 

was due to samples and was uniformly high for length and'age; 

The variation in length was generally high among sample landings than 

in age €or the species under study. 

On the whole, both the variation in species as well as in length-age 

for a species was high among sample landings relative to that between clusters 

within landings. Hence, for precise estimation of species number, it is 

recommended that data be collected from a large number of sample landings 

and from few clusters within each of the landings. 

To summarize the optimum plan for efficient estimates of number and 

weight of a species is to have two clusters of 25 lbs. each from sample 

landings in 'small fish' category (e.g.,rose fish) and two of 50 lbs. each 
37 

TABLE 8. Two-Level Nested ANOVA of 

with Unequal Sample Sizes 

Length/Age for a Species 

by Ports and Years 

(a) Analysis of Variance (ANOVA) for Widows at Bodega Bay, 1981. 

AGE LENGTH 

Source DF MS F P MS F P 

Samples 22 87.24' 8.724 <0.001 197.72 4.700 <0.005 

Clusters 

(w i thin s amp1 es ) 12 10.00 1.197 QO.25 42.07 1.314 1.0.20 

Within Clusters 636 8.352 32.01 

(b) Analysis af Variance (ANOVA) ~QZ Chilipepper at. Fort. Rragg, 1979. 

AGE LENGTH 

Source DF MS F P MS .F P 

Samples 37 44.02 3.12 ~,0.008 172.13 2.10 1.0.05 

11 14.12 1.80 C1 us t ers 

(within samples) 81.78 2.36 ~0.01 

Within Clusters 508 7.85 34.65 

(c) Analysis of Variance (ANOVA) for Widow at Eureka, 1979. 

AGE LENGTH 

Source DF MS F P MS F P 

Samples 15 34.45 4.75 co.005 37.86 3.09 <0.025 

13 7.25 1.19 0.35 12.27 1.43 q~0.18 Clusters 

(within samples) 

Within Clusters 320 6.09 8.58 

Only variation between samples among lengths was significant at Eureka 

during 1978. 
38 

TABLE 8. (Cont.) Two-Level Nested AWVA of &ength/Age for a 

Species with Unequal Sample Size by Ports and Years 

Source 

Samples 

ANOVA for - Chilipepper at San Francisco , 1979. 

AGE LENGTH 

DF Ns F P DF MS F P 

9 23.92 1.46 110.10 9 136.17 2.82 $0.025 

C lusters 15 16.36 2.29 ~0.01 16 48.22 1.7.8 $0.05 

Within Clusters 185 7.14 214 27.08 

(el ANOVA for ChilipeE at Monterey , 1978. 

AGE LENGTH 

Source DF MS F P DF MS F P 

Samples 38 32.10 8.04 ~0.001 42 221.42 7.21 <0.001 

C lusters 29 3.99 1.16 ~0.25 33 30.71 1.28 %0.15 

Within Clusters 536 3.43 617 24.07 

( fl ANOVA for Chilipepper at Monterey , 1979. 

AGE LENGTH 

Source 

Samples 

Clusters 

DF MS F P DF MS F P 

43 31.74 4.05 <0.001 48 145.20 4.02 <0.001 

39 7.84 1-80 %0.001 44 36.10 1.43 50.035 

Within Clusters 859 4.35 971 25.25 

.. 
39 

TABLE 8. (Cont.) Two-Level Nested ANOVA of Length/Age for a 

Species with Unequal Sample Size by Ports and Years 

ANOVA for Bocaccio at San Francisco, 1979. 

AGE LENGTH 

F P 

Source DF MS F P DF MS 

Samples. 10 84.97 6.95 <0.001 10 317.88 6.98 <0.001 

Clusters 15 12.23 1.20 %0.30 

Within Clusters 225 10.20 

16 45.55 0.80 sO.75 

227 57.11 

(h l ANOVA for Bocaccio at Monterey, 1978. 

AGE LENGTH 

Source DF MS F P DF MS . 

F P 

Sampl-es 37 49.17 14.32 <0.001 41 1167.33 14.43 <0.001 

Clusters 33 3.43 1.33 %0.10 37 80.91 1.38 ~0.06 

Within Clusters 579 2.59 645 58-79 

(il ANOVA for Bocaccio at Monterey, 1979. 

AGE LENGTH 

Source DF MS F P DF MS F P 

Samples 45 59.07 9.36 <0.001 50 918.70 11.74 <0.001 

Clusters 41 6.31 1.48 ~0.025 46 78.27 1.25 %0.01 
4

in all other categories; for small landings of rockfish like widows, one or 

two clusters of 50 lbs. each should be taken depending upon availability. 

For a boat landing more than one important category, the sampler should subsanrple from as many of such categories as possible; additional sort groups 

should not be sampled at the cost of gathering less than clusters from 

a sort group. 

should be made to cover the other landing after completing the requirements 

of the first lading. 

2 

If there is more than one landing on a given day, attempt 

If a sort group is infrequently landed, sampling should be directed 

towards the infrequent sort group, as long as the number of landings for 

the sort group is less than four per month. 

7. RELATIVE EFFICIENCY OF ESTIMATORS USING POST-STRATIFICATION 

We will now compare the efficiency of three important estimators 

using different estimation methods but the same selection procedure with 

post-stratification by sort groups. 

For any sort group, the unweighted mean of the unit means is given 

I 

by 

- where yi is the simple mean of the species number (or weight) per cluster 

for the sort from the ith sample landing and n is the number of 

landings at a port-year stratum. - 

for 

yRl. 

Consider the ratio estimator 

a sort group given by 
41 

which is the same as (31). 

Consider a more general estimator. (ratio) than (44) 

where is given by (18) and is an estimate of average weight per 

cluster. 

It would be seen that all the above estimators are biased. The first 

estimatar is simpler The other two are to work out but may have large bias. 

ratio estimators. 

The unbiased estimator 

n 

i=l 

- 

(46) 

- N 

Mi Yi =- "i n~ 

cannot be used since Mi and N are unknown. The estimate of v(el) is 

given by 

A 

y(p) = (--- 1w 2 lGn 1 ii 2 

n W) 'b + n w 4 (q-F)'2i 

Between Within 

- * 

(47) n- 1 

W 

(1 - El n +[ 2 W c 

(3ias) 

.-__l___ ~ 
I 

Also 

where n 

m 

nm - ni 

W' 1 iR 

n 

f2 = ii !(e) /n , w = 1 c WiR / c mi 

C*i 

fl = - 

42 

where v2(k) is given by (221, 

Formula (49) is easy to work out but is approximate and the 

approximation is close if fl is small. 

The coefficient of variation of mean number of Bocaccio, Chilipepper 

andwidow per cluster for different categories by port-year groups and the 

three estimators, y' 

(44) and (45) are highly efficient. In all cases, the contribution 

of bias to total variance of (43) was considerably high. Further, 

estimator (44) is somewhat more efficient than (45), taking into account 

the fact that v(7 ) is likely to be under estimate since it does not 

take into account the within component of variance. 

results was obtained for c.v.'s of mean weight per cluster for the 

- - 

yR1 and 7, (Table 9) show that the estimators 

2 

R2 

The same trend of 

species. 

The empirical evidence supports strongly the use of the estimator 

- = against y' and 7R . We will in the following section present some 

yRl 2 

of the coefficient of variationsfor estimating 

(b) number of a species in a particular age-sex group and 

(a) number of a species 
TABLE 9. Coefficient of Variation (in Percent) of Mean Number of Bocaccio, 

Chilipepper 

Different Categories for the Three Estimators 

and Widow per Cluster by Port-Year Group and for 

Port 

San Francisco 

Fort Bragg 

Bocaccio 

- 

Year Category Sample Size 

1978 253 20 10.24 

1978 250 86 7.36 

1979 250 46 25.70 

13.51 11.64 

16.21 8.14 

47.73 27.10 

Mont erey I978 253 31 17.93 12.03 19.51 

1979 253 51 7.36 8.95 9.53 

Eureka 1978 250 25 26.00 40.11 29.84 

1982 250 59 25.24 39.22 28.03 

Ctpklipepper 

Eureka 1978 250 13 34.52 37.66 42.33 

Widow 

Mont erey 1978 250 12 43.47 111.20 68.29 

Eureka 1978 250 11 27.81 72.69 33.90 
44 

(c) weight-of a species per cluster by port-year groups; the results will 

also be presented for all ports combined for 1981 (for which datawere 

available) since this has important management implications. 

8. RELIABILITY OF THE ESTIMATES: COEFFICIENT OF VARIATION 

8.1. Coefficient of Variation (c.v.) of Species Catch by Port-Year Groups 

The components of variance, i.e., between and within sample landings 

and the coefficient of variation of the ratio estimate (10) of average . 

number of a species per cluster 

L L 

'2R I3 j 

- = 1 w.?./C wj 

are presented in Tables 10 to 10.2 by Port-year groups. 

obtained on the assumption that the cluster weight is a constant for all 

trips and sorts within trips for the species Widow? Chilipepper and Bocaccio. 

A more realistic assumption would be to use .y,, 

weight is a constant within a sort group but varies over sort groups. 

Although this change will not affect the over-all conclusions, it is 

recommended that f, and ~(7~~) 

The estimates were 

- 

and thus assume that cluster 

be used in such situations. 

For 1982 representative data were available for only 2 important 

port groups Eureka and Monterey and for the period January to September. 

The variation between sample landings was invariably high in all cases and 

the variation within landings was mostly negligible suggesting that for 

obtaining efficient estimates of total number of a species we should take 

more landings and too few clusters from each landing. 
45 

During 1982 the C.V. of the number of Widows per cluster at Eureka 

summed over the sort groups was reasonably low (7.33) when 88 sample 

landings were used in the study. 

Chilipepper the precision was low though for Bocaccio the C.V. was lower 

(24.40) compared to Chilipepper (32.12, not shown in the table). This 

is because Bocaccio was more abundant being available in 45 of the 88 

landings compared to only 18 landings for Chilipepper. For Yellowtail 

(results not shown), the C.V. was as high as 54 percent. Here again, 

only 18 out of the 88 sample landings contained the species. For some 

of the sort groups the sampled clusters from the landings did not contain 

For the other two species Bocaccio and 

any a€ the rackfish under study (e.gz, the sample with the sort groups 262, 

270 

probability, estimation made by eliminating such sort groups and selecting 

more samples from other sort groups should result in increased over-all 

did not have any Widows). If this is known apriori with a high 

efficiency . 

"he estimated c.v.'s at Monterey (1982) were reasonably accurate, 

being respectively 6.62, 13.92 and 10.31 for Widow, Chilipepper and 

Bocaccio. Although the total number of sample landings was smaller (54) 

compared to that at Eureka 

measured in terms of the proportion of landings containing the species. 

(88), the species were more abundant at Monterey 

During 1981 the c.v.'s varied considerably between species among 

port groups. The estimate of total catch for each species based on 232 

landings for 'all ports' obtained by combining the port groups Monterey, 

Sari Francisco, Fort Bragg, MorroBay, and Eureka was sufficient enough for 

management purposes (c.v.'s ranging between 7 to 11.6 percent). However, 

for Yellowtail (not shown in table) the overall C.V. was high (25%), 

presumably because only 31 out of the 232 landings had Yellowtail. 
46 

8.2, Coefficient of Variation of Species Weight by Port-Year Sorr Groups 

The coefficients of variation (c.v.) 

1978 are given by sort groups (Table 10.3). 

of mean weight per cluster during 

We have used 7, as the estimate of mean cluster weight within a 

sort group which is a reasonable assumption to make. 

estimating corresponding mean number of a species per cluster are shown in 

brackets in the last column of the table which shows that for the same sample 

size, the estimates of species weight are generally more efficient than the 

corresponding estimates for species number. 

The c.v.'s for 

The estimated c.v.'s of weight are higher for Chilipepper and 

Bocaccio at Eureka than at Monterey which is due to Monterey being abundant 

in the species so that a larger proportion of th-e samples contained the spe 

The C.V. for Widow at Eureka based on 11 landings was rather high (24.79) 

during 1978 and is a suspect in view of the small sample size on which the 

estimate is based. However, during 1980 (based on 54 landings not shown 

in the table) it was as low as 

catch was 2.2 percent. This latter figure agrees with estiwates during 

1981 and 1982 as discussed in the preceding pages. 

higher precision (c.v. = 7.45) for Bocaccio than other centres. 

2 percent and the corresponding figure for 

San Francisco gave a 

8.3. CoefEcient of Variation of Species by Sex-Age for Fort-Year Groups 

The c.v.'s along with the between, within and total component of 

variance of mean catch by sex-age for a species (1981-82),are shown in 

Tables 10.4 to 10.6. Sex-age groups for which c.v.'s were greater than 

25% are not shown in the table. 

In general, the estimates of cluster mean for a species by sex-age 

groups are less precise than for the species as a whole without such 
47 

classification. Thus for Widow at Eureka and Nonterey (1982),the c.v.'s 

were estimated at 7.33 and 6.62 respectively (Table lo), though the 

corresponding figures for a specific sex-age group (Table 10.4) were 

considerably higher. 

groups showed lower c.v.'s (higher precision) than the corresponding 

figures for estimating the species number. 

For Chilipepper and Bocaccio only a few of the sex-age 

The coefficients of variation for estimating the number of Bocaccio 

for various Port-year groups were lower than for 

Bocaccio (Table 10.7). 

need a larger number of sample landings for estimating the total catch of 

4- year-old female 

Thus, to ensure equal precision we would generally 

a species in a given sex-age group than for estimating the catch for the 

species as a whole. This is further discussed in the section on sample size. . 

The c.v.'s for estimating total catch of a species vary considerably . 

among port-year groups. Thus during 1982, both Chilipepper and Bocaccio were 

estimated with much higher precision at Monterey than at Eureka although the 

estimates at the latter port were based on a larger sample size 

at Monterey 

than Bocaccio. 

at a port will depend on how important the 

(88) than 

(54). This is because Monterey was more abundant in the species 

Hence, the reliability of the estimates of a catch of a species 

port is with regard to the species. 

The combined estimates for 'all ports' for 1981 based on 232 sample 

landings were reliable enough for estimating total catch of a species 

ranging between 7 to 11.6 percent) and even for some of the dominant sexage groups 

and 5- year-old male and female Bocaccio) for which the c.v.'s were 

reasonably small. 

(c.v.'s 

(12-year-old male and female Widows, &yea+old female Chilipepper 
48 

TABLE 10. Coefficient of Variation (in Percent) of Mean Number 

. 

Species 

of Species Per Cluster Using 72R at 

Eureka - 1982 (January- September) 

(88) 

Var i anc e 

Between Within - Total C.V. (%I 

Widow 0.5695 

Chi 1 ipepper 0.0361 

Bocaccio 0.0184 

0.5717 7.33 0.0022 

0.0002 0.0363 

0.0080 0.0185 24.40 

Monterey - 1982 (January- September) 

Variance 

Between - Within - Total 

Species 

C.V. (%I 

Widow 

Chilipepper 

Bocaccio 

0.2210 6.62 

0.0978 13.92 

0.1408 10.31 

0.2078 0.0132 

0.0937 0.0041 

0.1354 0.0054 

'Figures within brackets show sample landings on which estimates are based. 
TABLE 10.1. Coefficient 

Species per 

Species 

Widow 

Chilipepper 

Bocaccio 

of Variation (in Percent) of Mean Number 

- 

Cluster Using yzR by Port-Year Groups 

MOITO Bay - 1981 

Variance 

Between Within 

0.3317 

- Total 

0.0000 0.3317 

1.1329 0.0000 1.1329 

0.8489 0.0000 0.8489 

49 

of 

C.V. (%) 

14.21 

12.28 

Fort Bragg - 1981 

(44) 

Species Variance 

Between Within - Total C.V. (%) 

Widow 0.4385 0.0000 0.4385 - 

Chilipepper 0.3219 0.0000 0.3219 14.70 

Bocaccio 0.4658 0.0000 0.4658 14.36 

Eureka - 1981 

(101) I 

Species Variance 

Between Within Total C.V. (%) 

Widow 0.1582 0.0003 0 ..1585 3.45 

Chilipepper 0.0004 0 0000 0.0024 - 

Bocaccio 0.1176 0.0000 0.1176 - 
TABLE 10.1. (Continued) Coefficient of Variation (in Percent) of Wean 

Number of Species per Cluster Using 3,, by Port-Year Groups 

Monterey - 1981 

(34) 

Species Variance 

Between - Within - Total . C.V. (%> 

Widow 0.1659 0.0004 ' 0.1663 - 

Chilipepper 

Bocaccio 

0.7093 20.98 

0.5965 12.14 

0,7091 0.0002 

0.5959 0.0006 

San Francisco - 1981 

Species Variance 

Between - Within - Total C.V. (%> 

0.6552 0.0000 

2.6615 0.0000 

0.6552 23.64 

2.6615 19.45 

Widow 

Chilipepper 

Bocaccio 0,6304 0.0000 0.6304 - 
51 

TABLE 10.2. Coefficient of Variation (in Percent) of Mean Number 

of Species per Cluster Using 3,, for 

Species 

Widow 

Chilipepper 

Bocaccio 

All Ports* - 1981 

Var i ance 

Between Within 

0.2625 0.0001 

0.0949 

0.1103 

0.0000 

0.0000 

* 

Includes the port-groups Monterey, San Francisco, 

Eureka. 

Total C.V. (%) 

0.2626 6.99 

0.0949 11.60 

0.1103 10.72 

Fort Bragg, Morro Bay and 
52 

TABLE 10.3. Coefficient of Variation [in Percent) of Mean Weight of a 

Species per Cluster Using 

- T6R by Port-Year-Sort Groups 

Eureka - 1978 

Species - Sort Sample Landings Var i an c e C.V. (%

Between Within Total 

Widow 250 

Chilipepper 250 

Bocaccio 250 

11 

13 

25 

24.79 25.1717 0.5668 35.7385 (27.81

20.4407 0.0665 20.5071 - 

21.81 1.6132 0.9451 2.5583 (26*00

. Monterey - 1978 

Species - Sort Sample Landings Variance C.V. [%

Between Within Total. 

Widow 253 

Chi 1 ipepper 253 

Bocaccio 253 

12 

33 

31 

33.2973 0.2329 33.5302 - 

13.86 2.5313 0.0204 2.5517 (16.63

13.84 10.4338 0.0375 10.4713 f17,931 

San Francisco - 1978 

Species - Sort Sample Landings Variance C.V. (%

Between Within Total 

Bocaccio 253 20 5.6482 0.0447 5.6929 7 -45 

1. Dash (-) indicates that the corresponding c.v.'s are greater than 25%. 

2. Figures within brackets ( ) show the corresponding c.v.'s for estimating 

mean number of a species. 
53 

TABLE 10.4. Coefficient of Variation (in Percent) and Components of Variance 

of Mean Catch per Cluster During 1982 (January- September) by 

Species-Age-Sex Groups Using 7 at Eureka 2R 

- Sex 

M 

M 

M 

M 

M 

F 

F 

F 

F 

F 

F 

F 

M 

F 

F 

M 

Age (in years) 

16 

12 

11 

9 

7 

18 

17 

16 

15 

12 

11 

7 

8 

19 

14 

5 

Between 

0.0013 

0.0080 

0.0050 

0.0010 

0.0042 

0.0019 

0.0013 

0.0083 

0.0045 

0.0190 

0.0027 

0.0051 

0.0022 

0.0037 

0.0126 

0.0050 

Widow’ (88) 

Variance 

Within 

0.0002 

0.0003 

0.0002 

0.0002 

0.0002 

0.0003 

0.0004 

0.0006 

0.0002 

0.0014 

0.0002 

0.0008 

0.0001 

0.0001 

0.0003 

0.0000 

Total 

0.0015 

0.0083 

0.0052 

0.0012 

0.0044 

0.0022 

0.0017 

0.0089 

0.0047 

0.0204 

0.0029 

0.0059 

0.0023 

0.0038 

0.0129 

0.0050 

C.V. (%) 

20.37 

14.84 

18.77 

19.92 

18.83 

19.68 

14.81 

16.08 

20.88 

14.10 

19.70 

10.94 

23.04 

23.22 

22.20 

24.14 

Chilipepper 1 (88) 

F 13 0.000s 0.0000 0.0005 24.89 

F 11 0.0006 0.0000 0.0006 23.03 

Bocaccio 1 (88) 

M 6 0.0004 0.0002 0.0004 19.82 

F 5 0.0002 0.0000 0.0002 22.65 

M 5 0.0034 0.0000 0.0034 31.91 

F 5 0.0002 0.0000 0.0002 22.65 

1. M -t Male; F -t Female. 

2. C.V. - < 25% are shown. 
54 

TABLE 10.4. Coefficient of Variation (in Percent) and-Components = of Variance 

of Mean Catch per Cluster by Species-Age-Sex Using y2R at 

Monterey During 1982 (January- September) 

Widow' (543 

Variance 

petween - Within - Total C.V. (%) s_ 

sex Age (in years) 

0.000s 0.0015 18.11 

0.0013 0.0024 25.38 

24.29 

20.49 

11 0.0010 

10 0.0011 

13 0.0102 

F 12 0.0090 

M 

M 

F 0.0007 0 0109 

0.0003 0.0019 21 .oo 

0 0006 0.0096 

F 11 0.0016 

Variance 

(in years) Between Within - Total C.V. (%I 

- Sex Age 

0.0002 18.36 

0.0000 0.0000 16.14 

0 * 0000 0.0019 12.30 

9 0.0002 0.0000 M 

M a 0.0009 

M 7 0.0019 

M 5 0.0002 

M 3 0.0001 

11 0.0010 

10 

0.0000 0.0007 15.55 M 6 0.0007 

0.0000 0.0002 15.67 

0.0000 0.0001 22.56 

0.0001 0.0006 21.40 

0.0001 0.0011 24.90 

16.61 

7.63 

F 13 0.0005 

F 

F 0.0006 ' 0.0001 0.0007 

F 9 0.0023 0.0002 0.0025 

F 8 0.0028 0.0003 0.0031 9.00 

F 

F 

F 

F 4 0.0007 

0.0034 9.81 

0.0007 8.86 

0.0002 21.87 

0.0001 0. oooa 18.34 

7 0.0033 0,0001 

6 0.0006 0.0001 

5 0.0002 0.0000 
55 

TABLE 10.4. (Continued) Coefficient of Variation (in Percent) and 

Components of Variance of Mean Catch per Cluster by SpeciesAge-Sex Using f2R at Monterey During 1982 (January-September) 

- 

1 Bocaccio (54) 

- Sex 

M 

M 

M 

M 

M 

Age Variance 

Between Within - Total C.V. (%) (in years) 

10 0.0004 0.0000 0.0004 23.22 

9 0.0004 0.0000 0.0004 19.39 

8. 0.0002 0,0000 0.0002 7.43 

7 0 . 0006 0.0001 0.0007 12.45 

6 0.0015 0.0001 0.0016 9.75 

M 5 0.0122 0.0001 0.0123 14.38 

M 4 0.0032 0.0001 0.0033 10.65 

M 3 0.0005 0.0000 0.0005 20.28 

F 15 0.0002 0.0000 0.0002 24.64 

F 13 0.0008 0.0001 0.0009 21.03 

F 12 0.0002 0 0000 0.0002 8.96 

F .11 0.0001 0.0001 0.0002 6.90 

F 10 0.0003 0.0000 0.0003 13.57 

F 9 0.0003 0.0001 0.0004 16.24 

F 8 0.0003 0.0001 0.0004 10.55 

F 7 0.0047 0.0003 0.0050 10.06 

F 6 0.0013 0.0002 0.0015 8.03 

F 5 0.0066 0.0003 0.0069 9.65 

F 4 0.0022 0.0002 0.0024 12.34 

F 3 0.0005 0.0001 0.0006 12.41 

'Coefficient of variation - e 25%. 
56 

Of 

at 

TABLE 10.5. Coefficient of Variation (in Percent) and Components Qf Variance - 

' Mean Catch per Cluster by Species-Age-Sex Groups Using y2R 

Ports During 1981 

Mom0 Bay 

Chilipepper (40) 

Variance 

C.V. (%) 

c_ 

Total 

Sex 

(in years) Between - Within - 

0.0000 0,0191 24.16 

0.0000 0.0139 25.84 

0.0000 0.0939 20.41 

0.0000 0.0592 16.00 

0.0000 0.0367 22.77 

M 8 0.0191 

F 11 0.0139 

F 9 0.0939 

F 8 0.0592 

F 7 0.0367 

Bocaccio (40) 

- Variance 

C.V. (%I 

Sex k 

Total 

0.0000 0.0026 25.43 

(in years) Between Within - 

M 6 0.0026 

PI1 5 0.0608 0.0000 0.0608 11.99 

0.0000 0.0051 21.92 

0. Q. oaw 12-67 

F 6 0.0051 

F 5 8. f3%W 

Fort Bragg 

Chi 1 ipepper (44) 

- 

Sex k Variance 

(in years) Between Within Total 

- 

C.V. (%) 

F 9 0.0196 0.0000 0.0196 25.00 

Bocaccio (44) 

Variance 

C.V. (%I _e 

Total 

Sex Age 

Id 5 

T: 5 0.0558 

(in years) Between . Within 

0.0866 - 0.0866 22.27 

- 0.0558 18.39 
57 

TABLE 10.5. (Continued) Coefficient of Variation (in Percent) and 

Components of Variance of Mean Catch per Cluster by SpeciesAge-Sex Groups Using :,, at Ports During 1981 ’ 

Mont erey 

Chilipepper (34) 

- Sex 4ge Variance 

Between Within - Total C.V. (%) (in years] 

F 8 0.0154 0.0000 0.0154 25.80 

F 7 0.0276 0.0000 0.0276 18.39 

Bocaccio (34) 

M 5 0.0310 0.0003 0.0313 13.30 

F 5 0.2065 0.0005 0.2070 17.88 

San Francisco 

Widow (13) 

- Sex Age Variance 

Within - Total C.V. (%) (in years) Between 

M 12 0.0247 0.0000 0.0247 24.33 

M 7 0.0056 0.0000 0.0056 25.58 

F 14 0.0002 0.0000 . 0.0002 14.93 

Chilipepper (13) 

M 10 0.0123 0.0000 0.0123 24.10 

M 9 0.0341 0.0000 0.0341 22.44 

M 5 0.0083 0.0000 0.0083 23.03 

F 9 0.0949 0.0000 0.0949 24.36 

F 7 0.4019 0.0000 0.0000 21.75 

F 6 0.0118 0.0000 0.0118 19.84 

Bocaccio (1 3) 

F 5 0.0373 0.0000 0.0373 21.25 
TABLE 10.5. (Continued) Coefficient of Variation (in Percent) and 

Components of Variance of Mean Catch per Cluster by Species- - 

Age-Sex Groups Using yZR at Ports During 1981 

Eureka 

Widow (101) 

- Sex 

M 

M 

F 

F 

F 

F 

F 

F 

F 

Age (in years) 

12 

11 

17 

16 

15 

14 

13 

12 

11 

Between 

0.0148 

0.0221 

0.0074 

0.0048 

0.0110 

0.0142 

0.0171 

-0.0316 

0.0074 

Variance 

- Within 

0.0003 

0.0004 

0.0002 

0.0000 

0. Lp0o-T 

0.0001 

0.0001 

0.0009 

0 * 0001 

Total 

0.0151 

0.0225 

0.0076 

0.0048 

Q,OIIl 

0.0143 

0.0172 

0.0325 

0.0075 

- C.V. (%> 

14.04 

20.46 

15.52 

12.88 

20.98 . 

15.94 

13.56 

10.96 

14.03 
59 

TABLE 10.6. Coefficient of Variation (in Percent) and Components of Variance 

of Mean Catch per Cluster by Species-Age-Sex Group Using the 

Ratio Estimate i2, for 

All Ports* - 1981 

Widow (232) 

_c Sex %= 

Between (in years) 

M 13 

M 12 

M 11 

M 8 

M 7 

F 17 

F 16 

F 15 

F 14 

F 13 

F 12 

F 11 

F 8 

0.0091 

0.0070 

0.0092 

0.0012 

0 a 0029 

0.0036 

0.0028 

0.0045 . 

0.0061 

0.0081 

0.0145 

0.0035 

0.0026 

Variance 

Within 

0.0000 

0.0001 

0 0001 

0.0000 

0.0001 

0.0001 

0.0000 . 

0.0001 

0.0000 

0.0000 

0.0002 

0.0000 

0.0000 ' 

Total 

0.0091 

0.0071 

0.0093 

0.0012 

0.0030 

0.0037 

0.0028 

0.0046 

0.0061 

0.0081 

0.0147 

0.0035 

0.0026 

C.V. (%) 

20.22 

13.94 

20.59 

21.48 

24.24 

18.04 

15.57 

20.69 

17.58 

15.23 

12.09 

15.42 

23.02 

Chilipepper (232) 

M 8 0.0010 0.0000 0.0010 21.14 

M 7 0.0016 0.0000 0.0016 25.50 

F 12 0.0009 0.0000 0.0009 24.43 

F 11 0.0011 0.0000 0.0011 19.75 

F 10 0.0012 0.0000 0.0012 24.80 

F 9 0.005s 0.0000 0.0055 17.34 

F 8 0.0036 0.0000 0.0036- 15.81 

F 7 0.0077 0.0000 0.0077 19.61 

F 6 0.0006 0.0000 0.0006 17.80 

* 

Includes the port groups Monterey, San Francisco, Fort Bragg, Morro Bay 

and Eureka. 
60 

TABLE 10.6. (Continued) Coefficient of Variation (in Percent) and 

Components of Variance of Mean Catch per - Cluster by SpeciesAge-Sex Group Using the Ratio Estimate yzR for 

All Port§ - 1981 

Bocaccio (232) 

Variance 

C.V. (%) 

- 

Total 

0.0000 0.0008 22.94 

0.0000 0.0127 14.29 

0.0000 0.0019 22.78 

0.0000 0.0018 18.70 

0. moo 0.0122 1-1.28 

0.0000 0.0003 . 25.30 

Sex 432 

(in years) Between - Within - 

M 6 0.0008 

M 4 0.0019 

F 5 0.6122 

M S 0.0127 

F 6 0.0018 

F 4 0.0003 
61 

TABLE 10.7. Coefficient of Variation (in Percent) of Mean Catch per Cluster 

of Bocaccio and of Four-Year-Old Female Bocaccio by PortYear Groups 

Port 

San Francisco 

- 

Fort Bragg 

Monterey 

1 Eureka 

Bocaccio 4Year OldFemaleBocaccio 

Year 

1978 

- 

1978 

1979 

1978 

1979 

1981 

1982 

1978 

Sample Size 

20 

86 

46 

51 

34 

54 

25 

C.V. (%) 

11.59 

7.36 

27.10 

19.55 

9.53 

12.14 

10.31 

30.03 

Sample Size C.V. (%) 

20 39.14 

86 15.00 

46 70.81 

27 36.69 

45 18.53 

34 32.93 

54 12.34 

25 53.05 

1982 88 24.40 88 35.35 

'c.v. for 1981 (not shown) are high for species and considerably higher for 

4 year-old females. 
62 

9. RELIABILITY OF THE ESTIMATES: SAMPLE SIZE 

This section will deal with the number of landings required to ensure 

a specified level of C.V. for estimating the total catch (or weight) at 

a port during a year; sample sizes for obtaining quarterly estimates will 

also be indicated. 

We will consider C.V.'S of 10, 15 and 20 percent for estimating 

total catch of a species (and weight) and of 10 and 20 percent for 

estimating catch in a particular sex-age group. Sample size to ensure 

5 percent C.V. was generally too high for estimating species catch with 

I the present staff; even a 10 percent C.V. could not be realized in some ~ 

cases, e.g,, Chilipepper and Bocaccio at Eureka, Widow at Fort Bragg and 

Morro Bay during 1981. Sample size to estimate the catch in a species-sexage group with a 10 percent C.V. was even high in a number of cases and 

l 

a twenty percent C.V. is recombended in such cases. 

For a population, the sample size needed to ensure a coefficient 

of variatiaa e of estimated total number (or weight) of a species is 

given by 

n 

0 

2 . Nc 

n= 2 2%- 

0 

n 

where no = (c/e) 2 

where c is the population coefficient of variation, N, the total number 

2 of landings in the pulation. As a first approximation we take no = (de) , 

by substituting an advance estimate of c and knowing e, the desired C.V. 

of the sample estimate. 
63 

In practice, N is not known and is estimated by 

It is essential we have a good estimate of €4. 

fi on a large sample. In the absence of such an estimate of N or when fi 

One possibility is to base I 

is less than no, no adjustment is possible as in (51) and no is our 

best choice. 

each weight group, N will be an unbiased estimate of N. 

If the number of landings is proportional to the number in 

A 

Sampling within sort groups can not be realized in practice since no 

sampling frame by sort groups is available in a port-time stratum. However, 

an improved estimate of the population coefficient of variation (and hence 

of the sample size) can be obtained by using a weighted mean of the variances 

by sort--groups at the postdstratification stage after the samples are selected. 

Also, the estimate of sample size will be more precise when a minimum number 

of samples per sort group (i.e. 4 per month) is selected at each port during 

a month as recommended elsewhere in this report. 

9.1. Sample Size for Species Catch by Port-Year Groups 

The sample landings needed for estimating total catch by port-year 

groups (1981-1982) show that, generally speaking, the number needed is very 

high, if a 5 percent C.V. is aimed at in obtaining the estimates. Hence, 

it is decided to present more workable estimates using 10, 15 and 20 

percent. 

A smaller number of landings is needed at Monterey (where the species 

variability is small) than at Eureka for estimating Chilipepper and Bocaccio; 

in fact, the number needed at Eureka during 1981 for estimating Chilipepper 
64 

1 

and Bocaccio with a C.V. of 10 percent is over 1000 sample landings,_ 

which certainly cannot be managed even if the staff were doubled for the 

port-group. 

estimating the species with a IQ percent C.V. are reasonable and 'can be 

managed by the present staff assuming that a minimum of four landings per 

month per sort group are covered at a port. At San Francisco, the sample 

size required for estimating catch of Widow and Chilipepper with a ten percent 

C.V. are 56 and 41, and it should be possible for the present staff to 

complete 

be used with caution since the estimates of population c.v.'s are based on too 

few landings (13). 

At Fort Bragg and Mono Bay, the sample landings required for 

this task working at the desired speed. The estimates should, however, 

For 'all ports' which comprised the port-groups Monterey, San Francisco, 

Fort Bragg, Morro Bay and Eureka, the sample landings (not shown in the table) 

needed to 

c.v.) with 5 percent C.V. is about 50 percent more than the sample landings 

estimate the catch of Widows (using 1981 data to estimate population 

(232) used in estimating with 7 percent. C.V. For Bocaccio and Chilipper, 

the sample size needed to estimate with 10 percent C.V. is slightly more than 

used for *all ports', which the present staff should be able to manage. 

with a minimum of 4 

where, the total number of samples needed for 'all ports' is estimated at 432 

In fact, 

samples per sort per port-month group as recommended elselandings per year which should provide annual estimate of total catch of Widow 

for 'all ports' with less than 5 percent C.V. and of Bocaccio and Chilipper with 

c.v.*s ranging between 5 and 10 percent. Additional staff will be needed 

at least at the important port groups (e.g., Eureka) to provide more reliable 

estimates for the latter species. 
65 

9.2 Sample Size for Species Weight by Port-Year Sort Groups 

The sample size needed for estimating total weight (Table 11.2) is 

based on the analysis of data for 1978 for Eureka, Monterey and San Francisco 

and for 2 sort groups. The sort groups and sample sizes on which estimates 

of population coefficient of variation are based are the same as those for 

estimating coefficient of variation (Table 10.3). 

The number of landings needed for estimating weight of Widow at 

Eureka (using 1978 data to estimate population c.v.) with a 10 percent C.V. 

was high compared to those using 1981 and 1982 data. As stated earlier this 

estimate may not be reliable enough, because of the small sample (eleven 

landings) on which it is based. 

estimate population C.V. a sample size of 54 would provide an estimate of 

Using 1980 data (figures not shown) to 

weight with a 2 percent C.V. and of species catch with a 2.2 percent C.V. 

It is, therefore, possible for the present staff to estimate annual weight 

of Widow landings at Eureka with a five percent C.V. 

The sample size needed to estimate weight of Chilipepper and Bocaccio 

at Monterey during a year with a ten percent C.V. is between 50and 60 

landings which, again lies within the workload of existing staff as per recommendations of landings for the port group during a year. 

The sample size needed for estimating total weight of a species for 

a port-year group is generally less than that for estimating total catch. 

For example, the number of landings needed to estimate with a ten percent 

C.V. the catch of Bocaccio at Eureka, Monterey and San Francisco (using 

1978 data) are 109, 87 and 21 respectively as against 86, 53 and 11 for 

estimating total weight. 

section that the variation (c.v.) 

.-. 

This supports the conclusion in the earlier 

among samples for species weight is 

~ 
66 

I 

less than that for less than species catch (Table 10.3) for the ports considere

in this study. 

9.3. Sample Size for Species Catch by Sex and Age for Port-Year Groups 

The number of landings at a port during a year needed for estimating 

catch of a species in a sex-age group (Tables 11.3 - 11.4) with a ten percent 

C.V. is 

for some of the dominant sex-age groups. 

old female Widow at Eureka, eleven-yeapold male Widow at Monterey, sixgenerally high and cannot be managed by the present staff except 

These are seven-and twelveyear 

seven- eight-year-old male and female Chilipepper and Bocaccio at Monterey, 

and five-year-old male and female Bocaccio at Morro Bay. 

For !all ports' the annual catch for mostly dominant sex-age groups 

e.g., 12-year-old male and female Widow, 8-year-old female Chilipepper and 

5-year-old male and female Bocaccio were estimated with a ten percent 

For all other sex-age groups the estimates 

percent C.V. or more) and the current staff is inadequate for obtaining 

C.V. 

are subject to high error (20 

estimates with a oreasonable degree of accuracy. 

9.4. Sample Size for Species Catch by Ports on Quarterly Basis 

We have seen that for port-year estimates for species catch (or 

weight) sample sizes should be generally based on a ten percent 

for sex-age groups on a 20 percent C.V. except where these are available 

in abundance when a smaller sample size might serve the purpose. 

ports', estimates during a year 

i.e., 5 to 7 percent C.V. 

C.V. and 

For 'all 

will be measurable with smaller error 

For quarterly estimates, therecommended sample size can at best 

provide estimates of catch with 20 percent C.V. assuming that the 
67 

population C.V. 

estimates. 

quarter we would need more samples and hence additional staff, at least at 

would be approximately the same for quarterly and annual 

For estimating species catch with a smaller error during a 

the important port groups. In view of the very low precision of quarterly 

estimates it is recommended that either these be not published or published 

with the note that their use is restricted to serve merely as indicators 

ratha than provide a measure of reliability for species catch. 

To sum up, the present staff should be able to provide estimates of 

total catch of species by 'all ports' during a year with a C.V. between 5 

and 7 percent and at port-year level with a C.V. of 10 percent. Sample size 

for estimating total weight would be somewhat smaller to provide the same level 

of accuracy. 

To estimate species catch (or weight) by sex and age at the port-year 

level with a reasonable degree of accuracy, we would need more samples and 

hence extra staff for their collection, at least at the important port groups 

such as Eureka and Monterey. 
48 

TABLE 11. Sample Size Needed for Specified C.V. Levels for Estimating 

Total Catch of a Species by Port-Year Groups 

Eureka - 1982 (January - September) 

(88) 

Species Number of Landings Required For 

10% C.V. 15% C.V. 20% .c.v. 

Widow 43 20 12 

Chilipepper 

Bocaccio 

4 89 292 187 

351 191 117 

Monterey - 1982 (January - September) 

(54) 

Species 

Widow 

Chilipepper 

Bocaccio 

Number of Landings Required For 

10% C.V. 15% C.V. 20% C.V. 

22 11 6 

77 40 24 

47 24 13 

'Figures in brackets show ' sample landings ' . 
69 

TABLE 11.1. Sample Size Needed for Specified C.V. Levels for Estimating 

Total Catch of a Species by Port-Year Groups 

Morro Bay - 1981 

(401 

Species Number of Landings Required For 

20% C.V. 

87 

10% C.V. 15% C.V. 

Widow 565* 251* 

Chilipepper 

Bocaccio 

60 31 18 

48 24 4 

Fort Bragg - 1981 

(44) 

Species ~ Number of Landings Required For 

10% C.V. 15% C.V. 20% C.V. 

2 74 189 

83 39 23 

Widow - 

Chilipepper 

Bocaccio 80 38 22 

Eureka - 1981 

(101) 

Species Number of Landings Required For 

10% C.V. 15% C.V. 20% C.V. 

Widow 12 5 3 

Chilipepper - - 803* 

Bocaccio - 845* 283 

'Figures in brackets show 'sample landings'. 

'Dash (-) indicates over 1,000 landings. 

* 

Unadjusted estimates. 
70 

TABLE 11.1. (Continued) 

Estimating Total Catch of a Species by Port-Year Groups 

Sample Size Needed for Specified C.V. Levels for 

Monterey - 1981 

(34) 

Species Number of Landings Required For 

10% C.V. 15% C.V. 20% C.V. 

Widow 200 120 77 

Chilipepper 111 

Bocaccio 45 

58 34 

21 13 

San Francisco - 1981 

113) 

Species Number of Landings Required For 

10% C.V. 15% C.V. 20% C.V. 

Widow 56 28 17 

Chilipepper 

Boc ac c i 0 

41 20 11 

102 59 37 

Species 

All Ports* - 1981 

(232) 

Number of Landings Required For 

10% C.V. 15% C.V. 20% C.V. 

49 28 

75 

65 

Widow 107 

Chilipepper 272 130 

Bocaccio 2 37 112 

'Figures in brackets show 'sample landings'. 

* 

Includes the port groups Monterey, San Francisco, Fort Bragg, Morro Bay, 

and Eureka. 
71 

TABLE 11.2. Sample Size Needed for Estimating Total Weight of Catch 

of a Species for Specified C.V. (%) Levels 

Eureka - 1978 

Species - Sort Number of Landings Required For 

10% C.V. 15% C.V. 20% C.V. 

250 49 27 15 

Chilipepper (13) 250 233* 77 44 

Widow (11) 

Bocaccio (25) 250 86 45 27 . 

Monterey - 1978 

Species - Sort Number of Landings Required For 

10% C.V. 15% C.V. 20% C.V. 

Widow (12) 253 208 105 67 

Chilipepper (33) 253 56 26 15 

Bocaccio (31) 253 53 18 14 

Species 

San Francisco - 1978 

- Sort Number of Landings Required For 

10% C.V. 15% C.V. 20% C.V. 

Bocaccio (26) 253 11 5 3 

* 

Unadjusted estimates 

Number within brackets show the number of landings for estimating population C.V. 
72 

TABLE 11.3. Sample Size Needed for Estimating Total Catch of a Species 

for Specified C.V. (%) Levels by Age, Sex, Port and Year Groups 

Eureka - 1982 (January - September) 

Sex 

M 

M 

M 

M 

M 

F 

F 

F 

F 

F 

F 

F 

- 

Age (in years) 

16 

12 

11 

9 

17 

18 . 

7 

16 

15 

12 

11 

7 

Widow (88) 

Number of Landings Required For 

10% C.V. 20% . c .v . 

271 84 

164 46 

240 72 

262 80 

24 1 73 

258 79 

163 46 

187 54 

282 88 

150 42 

258 79 

95 25 

Chilipepper (88) 

F 13 

F 11 

360 

324 

120 

105 

Bocaccio (88) 

M 6 261 SO 

M 5 4 86 185 

F 5 316 102 
73 

TABLE 11.3. (Continued) 

a Species for Specified 

and Year Groups 

Sample Size Needed for Estimating Total Catch of 

C.V. (%) Levels by Age, Sex, Port 

Monterey - 1982 (January - September) 

Widow (54) 

Number of Landings Required For 

Sex Age 

M 11 109 

(in years) 10% C.V. 20% C.V. 

38 

67 

62 

47 

49 

- 

M 10 346* 

F 13 319* 

F 12 126 

F 11 129 

Chi 1 ipepper (54) 

38 

64 19 

90 30 

M 9 109 

M 8 34 

31 

M 7 

N 5 

M 6 89 30 

56 

51 

65 

M 3 140 

F 13 132 

F 11 335* 

98 33 

38 11 

44 12 

36 10 

F 10 

F 9 

F 8 

F 7 

F 6 

F 5 135 

F 4 106 

Not adjusted 

28 8 

53 

39 

* 
TABLE 11.3. (Continued) Sample Size Needed for Estimating Total Catch of 

a Species for Specified C.V. (%) 

and Year Groups 

Levels by Age, Sex, Port 

Monterey - 1982 (January - September) 

Bocaccio (54) 

- Sex 

M 

M 

M 

M 

'M 

M 

M 

M 

F 

F 

F 

F 

F 

F 

F 

F 

F 

F 

F 

Age (in years) 

10 

9 

8 

7 

6 

5 

4 

3 

15 

13 

12 

11 

10 

9 

8 

7 

6 

5 

4 

Number of Landings Required For 

10% C.V. 20% C.V. 

291" 

118 

27 

65 

43 

80 

50 

i 24 

328* 

130 

37 

24 

88 

95 

49 

46 

31 

42 

63 

58 

43 

7 

19 

12 

25 

14 

46 

64 

49 

10 

6 

23 

32 

14 

13 

9 

11 

19 
75 

TABLE 11.4. 

- Sex 

M 

F 

F 

F 

F 

Sample Size Needed for Estimating Total Catch of a Species for 

Specified C.V. (%) Levels by Age, Sex, Port and Year Groups 

Age (in years) 

8 

11 

9 

8 

7 

Mono Bay - 1981 

Chilipepper (40) 

Number of Landings Required For 

10% C.V. 20% C.V. 

46 

‘5 2 

233* 

267* 

97 35 

70 

109 

23 

42 

Bocaccio 

M 6 

M 5 

F .6 

F 5 

(40) 

259* 

46 

105 

50 

51 

13. 

40 

37 

Fort Bragg - 1981 

Chilipepper (44) 

Number of Landings Required For 

- Sex Age 

(in years) 10% C.V. 20% C.V. 

F 9 194 62 
76 

TABLE 11.4. Sample Size Needed for Estimating Total Catch of a Species for 

Specified C.V. (%) Levels by Age, Sex, Port and Year Groups 

Monterey - 1981 

Chilipepper (34) 

- Sex Age 

(in years) 

F 

F 

a 

7 

Number of Landings Required For 

10% C.V. 20% C.V. 

148 

112 

49 

27 

Bocaccio (34) 

M 5 53 14 

F 5 87 2s 

San Francisco - 1981 

Widow (13) 

- 

Number of Landings Required For Sex Age (in years) 10% C.V. 20% C.V. 

M 

M 

F 

ET 

7 

14 

59 

63 

26 

E8 

19 

7 

Chilipepper (13) 

M 10 57 18 

M 9 51 15 

M 5 54 16 

F 9 59 18 

F 7 49 14 

F 6 42 12 

Bocaccio (13) 

F 5 48 14 
77 

TABLE 11.4. (Continued) 

a Species for Specified 

and Year Groups 

Sample Size Needed for Estimating Total Catch of 

C.V. [%) Levels by Age, Sex, Port 

Eureka - 1981 

Widow (101) ' 

Number of Landings Required For - Sex Age 

(in years) 10% C.V. 20% C.V. 

M 12 155 . 47 

M 11 

F 

F 

F 

F 

F 

F 12 

F 

263 92 

17 180 56 

16 92 40 

15 271 96 

14 188 59 

13 147 43 

11 155 47 

103 29 

All Ports* - 1981 

Widow (232) 

Number of Landings Required Far 

- Sex Age 

(in years) 10% C.V. 20% C.V. 

13 657 231 

12 373 107 

M 

M 

M 11 674 ' 221 

8 713 238 

7 833 294 

17 558 174 

16 445 132 

M 

M 

F 

F 

F 

F 

F 

F 

F 

F 

15 678 222 

14 537 165 

13 430 126 

12 293 82 

11 439 130 

8 781 268 
78 

TABLE 11.4. (Continued) 

a Species for Specified C.V. (%) Levels, by Age, Sex, Port 

and Year Groups 

Sample Size Needed €or Estimating Total Catch of 

All Ports* - 1981 

Chilipepper (232) 

- Sex Age (in years) 

M 

M 

F 

F 

F 

F 

F 

F 

F 

8 

7 

12 

11 

10 

9 

8 

7 

6 

Number of Landings Required For 

10% C.V. 

699 

886 

84 1 

636 

857 

527 

456 

630 

54 7 

20% C.V. 

‘231 

321 

298 

204 

306 

161 

136 

202 

169 

Bocaccio (232) 

37% 

388 

771 

588 

259 

877 

26 7 

112 

264 

185 

72 

316 

* 

Includes the port groups Monterey, San Francisco, Fort Bragg, Morro Bay 

and Eureka. 
79 

10. AGE COMPOSITION: DOUBLE SAqLING 

For most theoretical population work and for management purposes, 

the knowledge of the age composition is important to predict the status 

of the stock in future years. 

fishing it requires considerable time and expense to age each individual in 

the sample. This can be done from otoliths with a fair degree of accuracy. 

The length of a fish gives a good guide to its age. 

measurements may be done relatively quicker thana. age determination, the 

age composition is often easily obtained by double sampling using a large 

sample of length measurements and a relatively few age determinations as a 

sub-sample from the large Sample (Fridyikssm, 

When sample sizes are large, as in commercial 

Because many length 

1934). 

Ketchen (1950) improved over Fridriksson's approach by sub-sampling 

from large samples sorted into length categories or strata which provided 

more accurate results for age groups at the extremities of the distribution. 

Kutkuhn (1963) mentions the limitations of this (age-length key) 

approach except in situations where price differentials may demand sorting 

of landings by size criteria. 

the age-length key approach will almost always give biased estimates. 

Following the method due to Tanaka (1953) in which stratification occurs 

after sub-sampling for age, Kutkuhn estimated absolute age composition of 

California salmon landings by port-month groups. 

procedure is not effective unless the 

costlier than the length sample, 

Westrheim and Ricker (1978) point out that 

He showed that the sampling 

age sample is at least five times 

Mackett (1963) found double sampling more efficiknt th'an simple 

random sampling with fixed sampling costs for estimating relative 

composition of Pacific albacore landings. 

age 
80 

Southward*(l976) found that a srample of otoliths proportional to 

the length frequency of sampled fish from each port was preferable to 

fixed sample size procedure %or estimating age composition of Pacific Halibut. Kirmrra (1977) arrived at the same conclusion as Southward by following 

a somewhat different 

In a personal 

sampling to estimate 

by allocating effort 

strata. 

approach. 

cornmication (1983),Lenarz examines the use of double 

the age composition of fish landings for a fixed cost 

between length and age sampling and among length 

We will present in this report some of the important statistical 

considerations in sampling for estimation of age composition with illustration from recent Widow rock fish data from the 

/ 

California coast. 

The previous studies have shown that since aging from otoliths of 

each individual fish in a sample is more expensive than an easily measured 

quantity such as length, it may pay to measure the length of each individual 

and then either (a) choose a random sub-sample from the whole sample, or 

(b) stratify the sample according to length classes and choose a sub-saniple 

from each class for age determination. 

This technique is profitable only if the correlation between 

the length and age is fairly high for use of ratio or regression estimates ' 

or stratification. 

In the construction of length strata for selection of the sub-sample 

the following additional questions arise. 

(i) How many strata, 

decide boundaries between strata and 

(iii) determination of the allocation plan Fong strata 

for deriving maximum gain from double sampling. 
81 

10.1 Number of Strata 

I present below the value of V(y )/V- as a function of L , st Y 

number of strata using the linear model 

where Y 

y = a + Bx + E 

is the length and x the age of a fish and 

Data Set 

L 1 2 3 

2 0.6041 0.5114 0.4747 

3 0.5308 0.4209 0.37Tri 

4 0.5052 0.3892 0.3434 

.5 0.4933 0.3746 0.3276 

Type of Data 

Set &tea* X Y Source 

age length 

(years) (cm) 

1. Female widow rockfish (328) 1980 1980 California Fish 

landed at Eureka 

during April - December (1980) 

and Game and 

Tiburon Laboratory 

2. Female widow rockfish (444) 1981 1981 I1 11 

landed at Eureka 

during January - September (1981) 

3. Female widow rockfish (532) 1982 1982 11 11 

landed at Monterey, 

San Francisco and Bodega Bay 

during January - March (1982) 

.2 

= [> + (1 - ,211 (52) 

V(Y1 
1 

where F 

and L is the number of strata. 

P > 0.95 there is hardly any gain due eo stratification if L exceeds 6. 

Increasing 

reduction in sample size in different strata. 

data set 3 for which r2 = 0.7004 and lowest for set 1 €or which 

p2 = 0.5278. 

is the corfelation between length land age in the unstratified populat

It can be shown that with this model unless 

L would however, result in somewhat lower precision owing to 

The improvement is highest for 

10.2 Strata Boundaries 

Using the length-age data on for 1982 

for San Francisco and the rule based on the cumulative of Jf(y: 

(Cochran 2977 , pp. k27-kB), where y- denotes length in centi239 female widow rockfish 

meters, the nearest available points for 

Stratum 

1 

Boundaries (36 - 47) 

Intervals on 

em 

18.70 

Stratum 

Boundaries 

Intervals on 

cum ?fF 

1 2 

(36 - 43) (44-47) 

cm cm 

9.30 9.40 

2 and 4 strata are 

2 

(48 - 55) 

cm 

23.72 

3 4 

(48-51) (52-55) 

cm cm 

16.25 7.47 
83 

It turns out 

as for young and old widow. 

length-age which has linear regression with high correlation. The correlathat the overleaf division point is approximately the same as 

This should give efficient stratification for 

will be moderate when a number of strata are used. 

For length-age data (1981) based on 444 female widow rock fish landings 

at Eureka, the boundaries using 2, 3, and 4 strata are 

Stratum 

1 

Boundaries (31.5 - 46) 

Intervals em 

on cum fi 17.70 

2 

(46.5 - 55) 

cm 

29.01 

Stratum 

.1 2 3 

(49.5 - 55) 

cm 

Boundaries (31.5 - 46) 

Intervals on 

em 

(46.5 - 49.0) 

cm 

cum rn 17.70 13.12 15.89 

Stratum 

1 

Boundaries (31.5 - 43) 

Interval em 

oncum ? r 9.34 

2 3 

(43 5-46) 616.5-49 

cm cm 

8.36 13.12 

(49.5-55) 4 

em 

15.9 

10.3 Optimum Allocation Plan 

We have seen in the earlier section that one of the requirements 

of double sampling is that the correlation between the length and age of fish 

should be high. In fact it is known that double sampling-is more efficient 

single sampling (when the first sample is measured for age alone) for the 

same cost if 

p* > 4(c/c') 

2 (53) (1 + c/c ') 
84 

where P 

are respe 

is the correlation between length and age of fish and c and c' 

tively the costs of aging and measuring a fish. 

Assuming that the average cost of aging fish (including small and large) 

is 6 minutes and of measuring is 1.2 minutes (estimates based on measurements by W. Lenarz of Tiburon Laboratory). 

We have from (53) 

p2 > 0.5555 

or p > 0.7453 

2 For the three data sets (p.81) the values of P are respectively 0.5278, 

0.6515 and 0.7004, so that e31 is approximately satisfied. However, 

neither P nor c/~, are large enough to suggest that double sampling will be 

much more efficient than single.sampling. On the other hand, with possible 

improvement in economics of aging techniques 

single sampling could be used with advantage. 

c/c' would be smaller so that 

We will illustrate the use of double sampling by analysing 1981 lengthage data to estimate the proportion of female widow rock fish in age group 

eleven at Eureka based on a sample of 444 fish. 

Consider the 3 length strata h = 1, 2, 3 with stratum boundaries based 

on quadratic fits of length on age (Figs, D2), 31.5 - 43 cm, 43.5 - 49 cm and 

49.5 - 55 cm. [Note this is different than boundaries based on length only.) 

Co = 1.2 mins, C = 3.8 mins, C2 = 3.8 mins., Cg = 8 mins. 1 

w = 0.3896 

s = 0.1503 

3 

3 

w = 0.5451, 

s = 0.4966, 

2 

2 

w = 0.0653, 

s = 0.1825, 

1 

1 
85 

x 

L 
86 

where wf, w2 and w3 are the proportion of fish in the sample and co is 

the cost of measuring a fish, cl, c2, c3 are respectively the costs of aging 

them in the 3 length groups. From Cochran (1977, p. 331) we have 

= 1.1225/c* 

where p,, is the estimated proportion and c* = B(con + Cchnh) = 3179.79 

with nl-==-14, n2 = 120 and n3 = 48, n = 444. 

The efficiency of double sampling with respect to single sampling is given by 

v. SRS (p)/v min (pst) = 120 

i.e. double sampling is 20 percent more efficient.than single sampling. 

Our studies have shown that the best length-age relationship (logarithmic 

or quadratic) do not change significantly if every other fish is sub-sampled 

systematicarly over time. Hence the sampEe size for estimation of age distribution in a port-year group can be reduced appreciably (without loss in 

efficiency) by selecting fish for age from length-time strata with the number 

in each cell proportional to the product of the number of fish and standard 

deviation of age estimated from earlier studies conducted in the region. 
11. 

1. 

2. 

3. 

4. 

5.1 I 

5.2. 

5.3. 

5.4. 

87 

SUMMARY AND RECOMMENDATIONS 

Random sampling of boat trips is not practicable owing to the 

uncertainty of their arrival times and the most reasonable assumption is that boats arrive at a port during a month in a random order 

(pp. 6-71 

The current design of post-stratification of a boat sample and subsampling clusters from each sort type is not feasible (unless the 

present strength of staff responsible for data collection is doubled) 

since some of the categories may be missed in sampling (pp. 4, 11). 

Estimates of species catch (Widow, Chilipepper and Bocaccio) and by 

sex and age based on post-stratification of each sample landing by sorts 

at a port during a month are less efficient than ones based on poststratification by sort types of arrivals at port-month level (Tables, 

pp. 24, 26 and 27). 

Post-stratification of kzndings by sorts is recommended by port-monklt groups

As far as pmcticable, setection of a cluster for- a? market category 

shouM be done before any presorting is done at the port either 

from bins, strap boxes or off conveyor belts. 

CZusters shouZd be selected from the conveyor beZt where smaZZ-market 

buyers would select fish from the top of bins. 

For Zarge-market buyers, clusters should be se Zected systemticaZZy, 

separated in time from the beginning and end of Zoading. 

A cZuster shoutd be selected from one side (of a box atZ the uay 

dam to the bottom) without looking at fish being seZected in the 

process. 
6. 

7. 

8. 

9. 

10. 

10.1. 

Selection of a random and representative cZuster (box) of fish 

from Q boat trip wuZd depend to a large extent on the expertise 

CLllCl ezperiencs of the sampler. Hence, the need to have permanens 

staff at least at the important ports to buiZd up the experience. 

Variation (within sort group) in length and age for a species was 

considerably higher among sample landings than among clusters within 

landings: also, variation among clusters was not significant compared 

to variation within clusters (Tables pp. 37-39). 

some presorting is done in the boats before landing. 

Using the cost function c = c 1 n + c 2 nm where c1 is the average 

cost and n and m the number 

of sampled boats and clusters per boat (p.28), the optimum number 

of clusters per sampled boat for a fixed cost for a sort type is 

This suggests that 

- 

c2 , the cost of data collection, 

. 

two (pp. 29-35). 

Emphasis on 

widows and 50 1bs.for all others" in the instructions on data 

collection led in several cases to wide variability among samples 

and hence to biased and inefficient estimates of species catch owing 

"25 1bs.cluster weight for small rock fish including 

to violation of the assumption of equal cluster weight required in 

the current method of estimation of parameters (pp. 8,151. 

TWO clusters of 25 lbs. each in "smaZZ fishrt in the pose fish 

category and 50 Zbs. each for all others shouZd be selected 

fyom sample Wings. 

For small tandings of widms one or tu0 clusters of 50 Ibs. each 

should be se Zected depending upon ami lability (pp. ,16,40). 

i"he mrent instructions on the selection of cZusters shouZd be 
89 

modified according lg . 

11 .I. The prinoipat contribution of this Feport is that a minimum 0' 

11.2. 

11.3. 

11.4. 

12. 

4 Zandings should be savpled for each category from a port-month 

stratwn ;.e., approxdmtezy one per week rp. 13) and Wo clusters 

(as in reconanendation 10 above) of 50 Zbs. for 25 Zbs.) each shouZd 

be savpled to provide port-year estimates of species catch (or 

weight) with a reasonable degree of accuracy. Effort shou Id be 

directed to savpling more Zandings for an important category as 

judged by prior knowledge of the share of its landing to total 

landing weight for the port-month group. 

For a bout Zanding with more than one important category, the sampZer 

should subscnple from as many of such categories as possible; 

additiona2 sort groups. should not be sampled at the cost of gathering 

less than 2 clusters from a sort group. 

If there is more than one Zanding on a given day, attempt should be 

made to cover the other Wing after completing the requirements of 

the firat kmd;ng (p. 40). 

If a sort group is infrequentZy landed, scmrpZing should be directed 

tooards the infrequent sort group, as long as the nwnber of landings 

for the infrequent sort group is less than four per month (p. 401. 

Formulas for the estimation of parameters and their errors have been 

developed for the general case of unequal cluster size (section 5.5.2) 

and for the more practical case when cluster weight can be treated 

as a constant within a sort type (section 5.5.3). 
13. 

14.1. 

14.2. 

14.3. 

14.4. 

The efficiency of the ratio estimator [pp. 41, 42, equation, (441, 

based OR post-stratification by sort groups at port-year level Kas 

compared with two others (theunbiased estimator and the ratio 

estimator based on Jack Knife). 

indicates that the ratio estimator of total catch (32) of a species 

has the smallest coefficient of variation and is, therefore, mst 

efficient amongst others for estimating species catch (and weight). 

The empirical evidence [Table 9) 

The between component of total variation was considerably higher 

than the within component which was almost negligible (Tables 

pp. 48-59) indicating that for obtaining efficient estimates of 

the parameters, a large number of landings with few clusters 

(two per sort group) per landing should be selected. 

The coefficient of variation (c.v.) 

varied considerably among port-year groups. 

The c .v. 's for species weight by port-year sort groups were 

slightly mra1le-r than &e- c-crl.~-e.j.porr&mg C.P. &s €or species catch 

for species catch (or weight) 

(p. 52, Table 10.3). 

The c.v.'s for species catch by sex-age group are generally 

higher than for species without such classification. 

Widow at Eureka and Monterey (1982), the c.v.'s were 7.33 and 6.62 

(Table 10) respectively as against higher figures for sex-age groups

Similarly, estimates of 4-year-old Bocaccio (Table 10.7) were 

subject to very high error compared to estimates of Bocaccio by 

port-year groups under similar conditions. 

Thus, for 
91 

14.5. 

15.1. 

The c.v.'s of combined estimates for species catch by sex-age group 

for "All ports (Monterey, San Francisco, Fort Bragg, Morro Bay and 

Eureka)" were reasonably small for some of the dominant sex-age 

groups (Table 10.6). 

The sample size required to estimate species catch during a year 

at "All ports" level with a 5 to 7 percent C.V. can be managed by 

the present staff with a work load as in recommendations (10) and 

(11) above; at port-year level the accuracy will be somewhat lower 

i.e., 10 percent (p. 70). 

15.2. Sample size for estimating total weight would be somewhat smaller 

than for estimating total catch to ensure the same level of 

accuracy (p. 71, Table 11.2). 

. 15.3. Species catch (or weight).by sex and age at the 'All port-year' 

level can be estimated with a 20 percent C.V. (except f.or some 

of the dominant species); to estimate with a 10 percent c.v., we 

would need mor'e staff, at least at the important ports. 

To -&mate total catch of species by sex and age at- pur+year 

level with a 10 percent c.v., we would certainly need additional 

staff at almost all the port groups. 

Quarterly estimates of catch (or weight) can be obtained by the 

present staff with a 20 percent c.v., assuming that the population 

C.V. would be approximately the same for quarterly and annual estimates. 

In view of the very low precision of quarterly estimates these can 

at best serve as indicators. 

14.4. 

16. 
17. For estimating proportion of fish in an age group by d6uble sampling 

a sample of 444 Widow fish landed at Eureka during 1981, was 

divided into 3 lenth strata and optimum allocation for age was 

adopted within strata. 

in eleven year age group was 20 percent more efficient than single 

It showed that the estimate of proportian 

sampling for age when average cost for aging is about six 

that for length measureinent. 

times 

With possible improvements in technology, single shpling may, 

however, prove more efficient than at present. 

Double sampling for age with the first sample stratified by tength 

and time is reconmended for estimation of age composition. 

For obtaining reliable and valid estimates of population churuzferist

18. 

19. 

it is essential that good relationship be maintained by the s-kr 

with both the skipper and the buyer. This would depend to a Zage 

extent on the experience of the sampler gained in the course of 

his field work. 

at least ut the important port groups which have too mavly problems 

to be handled within a short time. 

This emphasizes the need for having permanent ssaff 
93 

ACKNOWLEDGEMENTS 

Acknowledgements are due to the valuable assistance by employees of 

Tiburon Laboratory during the period of field work (May - September 1983). 

My special thanks are due to William Lenarz who provided me with information 

on the problems related to rock fish landings on the California coast, 

to Candis Cooperriderand to Mark Allen of Humboldt State University for the 

computations done on the data collected, to the field staff of the California 

Department of Fish and Game responsible for collection of relevant'data and 

to Norman Abramson, Director of the Laboratory for all the assistance rendered 

to me during my work at the Laboratory. 

My thanks are also due to Vivian Spak of the Department of Statistics 

and Applied Probability, University of Alberta for assistance in typing the 

report. 
94 

LITERATURE 

Cochran, W.G. 1977. Sampling Techniques, 

CITED 

third edition, Wiley and Sons, New York

Fridsikssan, A. 1934. On the calculation of age-distribution within a stock 

of cod by means of relatively few age determinations as a key to 

Mer. 86: 1-14. 

Ketchen, K.S. 1950. Stratified 

Trans. Am. Fish. SOC. 79 

Kimura, D.K. 1977. Statistical 

subsampling for determining age distributions. 

; 205-212. 

Assessment of the Age-Length Key. J. Fish. 

measurements on a large scale. Rapp. P.V. Reun. Cons. Int. Explor. 

Res. Board. Can. 34; 317-3~4. 

Kutkuhn, J.H. 1963. Estimating absolute age composition of California salmon 

landings. California Department of Fish and Game, Fish Bulletin 

No. 120; 47 p. 

Mackett, D.J. 1963. A Method of Sampling the Pacific Albacore (Thunnus Germo) 

Catch for Relative Age Composition F.A.O. Fisheries Reports. No. 6, 

 YO^. 3; pp. 1355-1366. 

Quinn 11, Best, E.A., Bijsterveld & R. McGregor. Sampling Pacific Halibut 

Landings for Age Compostion: History, Evaluation and Estimation. 

International Pacific Halibut Commission, P.O. Box 95009, Univ. Station, 

Seattle, Washington, 98105, U.S.A. 

Southward, G.M. 1976. Sampling landings of halibut for age composition. 

Int. Pacific Halibut Commission Sci. Rep. 58; 1-31. 

Tanaka S. 1953. Precision of age-composition of fish estimated by double 

sampling method using the length for stratification. Bull. Japan. 

SUC. Sci. Fish., Vol. 19, no. 5, pp. 657-670. 
95 

Tomlinson, Patrick 

Biometrics 

K. 1971. Some Sampling Problems in Fishery Work. 

27, Sept. pp. 631-41. 

Westrheim, S.J., and W.E. Ricker. 1978. Bias in 

estimate age-frequency 

35: 184-189. 

distributions. J. 

using 

Fish. 

an age-length key to 

Res. bard. Can.