9

NOAA Technical Report NMFS 9

Sampling Statistics in the

Atlantic Menhaden Fishery

Alexander J. Chester

August 1984

u.s. DEPARTMENT OF COMMERCE

National Oceanic and Atmospheric Administration

National Marine Fisheries Service
NOAA TECHNICAL REPORTS NMFS

The major responsibilities of the National Marine Fisheries Service (NMFS) are to monitor and assess the abundance

and geographic distribution of fishery resources, to understand and predkt fluctuations in the Quantity and distribution of

these resources, and to establish levels for optimum use ofthe resources. NMFS is also charged with the development and implementation of policies for managing national fishing grounds, development and enforcement of domestic fisheries regulations, surveillance of foreign fishing off United States coastal waters, and the development and enforcement ofinternational

fishery agreements and policies. NMFS also assists the fishing industry through marketing service and economic analysis

programs, and mortgage insurance and vessel construction subsidies. It collects, analyzes, and publishesstatistics on various

phases of the industry.

The NOAA Technical Report NMFS series was established in 1983 to replace two subcategories of the Technical

Reports series: "Special Scientific Report-Fisheries" and "Circular." The series contains the following types of reports;

Scientific investigations that document long-term continuing programs of NMFS, intensive scientific reports on studies of

restricted scope, papers on applied fishery problems, technical repons of general interest intended to aid conservation and

management, reports that review in considerable detail and at a high technical level certain broad areas of research, and

technical papers originating in economics studies and from management investigations.

Copies of NOAA Technical Report NMFS are available free in limited numbers to governmental agencies, both Federal

and State. They are also available in exchange for other scientific and technical publications in the marine sciences. Individual copies may be obtained from: Publications Services Branch (EIAI 13), National Environmental Satellite, Data, and Information Service, National Oceanic and Atmospheric Administration, U.S. Department of Commerce, 3300 Whitehaven

St., Washington, DC 20235.
NOAA Technical Report NMFS 9

Sampling Statistics in the

Atlantic Menhaden Fishery

Alexander J. Chester

August 1984

u.s. DEPARTMENT OF COMMERCE

Malcolm Baldrige, Secretary

National Oceanic and Atmospheric Administration

John V. Byrne, Administrator

National Marine Fisheries Service

William G. Gordon, Assistant Administrator for Fisheries
The National Marine Fisheries Service (NMFS) does not approve, recommend or endorse any proprietary product or proprietary material

mentioned in this publication. No reference shall be made to NMFS, or

to this publication furnished by NMFS, in any advertising or sales promotion which would indicate or imply that NMFS approves, recommends

or endorses any proprietary product or proprietary material mentioned

herein, or which has as its purpose an intent to cause directly or indirectly

the advertised product to be used or purchased because of this NMFS

publication.
CONTENTS

Introduction .

Brief history of the sampling program .

Structure of the sampiing design .

Basic calculations in theory and practice .

Estimating the mean weight per fish landed .

Present method and biases .

Approximate confidence intervals .

Optimal allocation of sampling resources .

Estimating the total number of fish landed .

Number of fish landed by port-week .

Number of fish landed over ports and weeks .

Estimating the age composition of the catch .

Basic calculations and variance formulations .

Optimal allocation of resources in age composition sampling .

Number of fish at age .

Conclusions and recommendations .

Acknowledgments .

Literature cited .

Appendix .

Figures

1

1

2

4

4

4

6

7

8

8

9

9

9

10

13

13

15

15

16

I. Map showing the locations of menhaden reduction plants in operation along the U.S. east coast during 1955 2

2. Hypothetical population of 81 sets in which the number of fish per set is inversely related to the average weight per fish. . . . 4

3. Relationship between average weight per fish and estimated set size for Port Monmouth, N.J., during 1979 5

4. Relationship between average weight per fish and estimated set size for Reedville, Va., during 1979 . . . . . . . . . . . . . . . . . . . . 5

5. Relationship between average weight per fish and estimated set size for Beaufort, N.C., during 1979 . . . . . . . . . . . . . . . . . . . . 5

6. Theoretical relationship between number of sets sampled and number of fish sampled per set and the variance in average

weight and cost of the sampling program. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

Tables

I. Tests of the hypothesis of random sampling of sets taken inside and outside Chesapeake Bay during 1979 2

2. Test of the hypothesis that the last set of the day is randomly distributed between inside and outside sets when both kinds of

sets are made on a single trip 3

3. Test of the hypothesis ofrandom vessel selection for the port of Reedville, Va., during the 1979 fishing season. . . . . . . . . . . . 3

4. Number of sets sampled, mean weight per fish, standard error of the mean weight per fish, and the ratio-to-size estimate of

mean weight for Port Monmouth, N.J., Reedville, Va., and Beaufort, N.C., for weeks during the 1979 fishing season. . . . . . . 6

5. Survey data collected during the week ending 6/16/7 9 at Port Monmouth, N.J. 7

6. Estimate of the total number of fish landed at Port Monmouth, N.J., during the week ending 6/16/79 with approximate 95%

confidence intervals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

7. Estimate of the total number of fish landed by port-week and for the entire 1979 season. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

8. Total catch per week, weighting factor per week, mean age proportion and variance by age by week, and overall 1979

seasonal mean proportions, variances, and approximate 95% confidence intervals for Port Monmouth, N.J. . . . . . . . . . . . . . . 11

9. Total catch per week, weighting factor per week, mean age proportion and variance by age by week, and overall 1979

seasonal mean proportions, variances, and approximate 95% confidence intervals for Reedville, Va. . . . . . . . . . . . . . . . . . . . . 11

10. Total catch per week, weighting factor per week, mean age proportion and variance by age by week, and overall 1979

seasonal mean proportions, variances, and approximate 95% confidence intervals for Beaufort, N.C.. . . . . . . . . . . . . . . . . . . . 12

11. Estimated age composition for the combined catch of Port Monmouth, N.J., Reedville, Va., and Beaufort, N.C., during the

1979 fishing season. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

12. Estimate of number of age 2 fish landed at Port Monmouth, N.J., during the week ending 6/16/79, with method to calculate

approximate 95% confidence intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

iii

Sampling Statistics in the Atlantic Menhaden Fishery

ALEXANDER J. CHESTERl

ABSTRACT

Atlantic menbaden, Brevoortia tyrannus, tbe object of a mlYor purse-seine fisbery along tbe U.S. east coast,

are landed at plants from nortbern Florida to central Maine. Tbe National Marine Fisberies Service bas sampied

tbese landings since 1955 for lengtb, weigbt, and age. Togetber witb records of landings at each plant, the

samples are used to estimate numbers of fisb landed at eacb age. Tbls report analyzes the sampling design in

terms of probability sampling theory. Tbe design is classified as two-stage cluster sampling, the first stage con·

sisting of purse-seine sets randomly selected from tbe popUlation of all sets landed, and the second stage con·

sistlng of fish randomly selected from eacb sampied set. ImpUclt assumptions of this design are discussed with

special attention to current sampling procedures. Metbods are developed for estimating mean nsb weight,

numbers of fish landed, and age composition of the catcb, witb approximate 95% confidence intervals. Based on

specific results from three ports (Port Monmouth, N.J., Reedville, Va., and Beaufort, N.C.) for the 1979fisbing

season, recommendations are made for improving sampling procedures to comply more exactly with assumptions of the sampling design. Tbese recommendations include adopting more formal methods for randomizing

set and fish selection, increasing the number of sets sampled, considering the bias introduced by unequal set

sizes, and developing methods to optimize the use of funds and personnel.

INTRODUCTION

Since 1955 the National Marine Fisheries Service Laboratory

at Beaufort, N.C., has sampled catches of Atlantic menhaden,

Brevoortia tyrannus, for length, weight, and age at reduction plants

located from Florida to Massachusetts. This information, along

with records of landings, numbers of purse-seine sets, and

estimates by vessel captains of catch per set, has been used by

fishery scientists to investigate overall population structure and

migratory patterns of menhaden (Nicholson 1971a, 1972; June

1972; Nelson et al. 1977), to evaluate the effects of fishing on

menhaden stocks (Nicholson 1971 b; Schaaf and Huntsman 1972;

Schaaf 1979), and to derive management principles for the fishery

(Schaaf 1975; Schaaf et al. 1975). In 1981 the Atlantic Menhaden

Management Board recommended in its Fishery Management

Plan for Atlantic Menhaden that at least 10% of all fish landed be

age 3 or older. Although estimates of the number of fish of each

age landed annually are basic to this management proposal and to

the conclusions of the other cited studies, the overall reliability of

the estimates and the factors affecting sampling efficiency have

not been satisfactorily addressed.

The purposes of this report are to 1) define the basic menhaden

sampling procedure in terms of probability sampling theory, 2)

examine the assumptions of the statistical model and the consequences of their violation, 3) demonstrate methods for calculating

approximate confidence intervals about some commonly

estimated parameters, 4) explore implications of the sampling

design for the optimal allocation of resources to this survey, and 5)

provide suggestions for improving the sampling methodology so

that it complies more exactly with underlying assumptions. I will

also recommend areas requiring further research to document the

validity of assumptions and to develop and present additional

theory.

'Southeast Fisheries Center Beaufort Laboratory, National Marine Fisheries Service, NOAA. Beaufort, NC 28516-9722.

BRIEF HISTORY OF THE

SAMPLING PROGRAM

Preliminary sampling of commercial menhaden landings began

at the Lewes, Del., reduction factory in 1952 (June and Reintjes

1959). This pilot survey was intended to provide information on

length, weight, and age distributions that could be used to develop

an efficient design for sampling the entire fishery. At that time,

processing plants were operating at 14 ports along the Atlantic

coast (Fig. I). Sampling was expanded during the 1953 and 1954

seasons, but was limited to ports from Chesapeake Bay northward.

Sampling was expanded to include ports south of Chesapeake Bay

in 1955, the first year of relatively complete coverage of the

fishery. Although intensive data collection has continued since

then, the number of ports decreased after the decline of the fishery

in the 1960's. Today only six ports remain active. These are

routinely 'sampled by on-site personnel hired on a seasonal basis.

June and Reintjes (1959) gave an explanation of the early

sampling strategy. They verified that fish of similar size and age

tend to school together and concluded from the pilot survey that a

sample size of 20 fish was adequate to estimate the mean length of

fish in a purse-seine set to within ± 2% (P = 0.05). Requirements

for estimating age composition were not considered. Samples

from purse-seine catches were collected directly from the holds of

vessels by shoveling fish from the top of each load into buckets.

Thus, each sample was assumed to represent fish from the last set

made and did not necessarily reflect the composition of the entire

boatload of fish. One hundred fish were sequentially withdrawn

from buckets. A 20-fish sample was then constituted by selecting

every fifth fish. These were measured (mm FL), weighed (g), sexed, and scales were removed for later age determination. Port

samplers were instructed to gather 10 to 15 samples per week at

each port sampled, but this number varied, depending on vessel

activity and areas fished.

During the late 1960's, fishery biologists in the menhaden program concluded from theoretical and field studies that age and size

variability within vessels was smaller than variability among

vessels. They decided that more efficient estimates would be ob-
Figure I.-Map showing the locations of menhaden reduction plants In operation along the U.S. east coast during 1955.

Number of sets sampled

Observed ifj) Expected 11;)

198 162.50

42 77.50

7,223

3,445

Inside sets

Outside selS

Ij G = 2 "i. f; In ( 1; ) = 26.79

X.••,l = 3.87

A) All inside and outside sets taken:

Table I.-Tests of the hypothesis of random sampling of sets taken Inside and

outside Chesapeake Bay (Reedville, Va., port) during 1979.

mation biases. The present sampling procedure will be examined

in light of these assumptions.

Because it is assumed that only the last purse-seine set of the

trip is actually sampled, the boatload cannot be considered the

primary sampling unit. I assume that a set is the fundamental unit

and that sampling the last set of the trip yields a random sample

from the population of all sets landed at a port during a given

week. The selected sets are a true random sample if 1) individual

schools are located and set upon at random, so that, in aggregate,

last sets per trip are a random sample of all sets taken (there must

not be a geographic stratification whereby, for instance, the last

set of the day is always made close to home port); 2) sets are

sampled at random from the total population of scts landed at a

port.

Violation of the first assumption is especially critical if the age

and sizc distribution of fish is regulated in any way, such as by

proximity to shore. The tendency toward nonrandom sampling by

set location was examined for pooled data from the Reedville, Va.,

port. Logbooks from each vessel were used to determine the 1979

total numbers of I) sets taken in Chesapeake Bay for trips when

all sets were made in the bay, 2) sets taken outside Chesapeake

Bay for trips when all sets were made outside, 3) sets taken inside

the bay for trips when fish were caught both inside and outside,

and 4) sets taken outside the bay for trips when fish were caught

both inside and outside. A log likelihood ratio test (0 statistic)

(Sokal and Rohlf 1981) was used to test the hypothesis of random

sampling of sets for various combinations of inside and outside

sets (Table 1). Under the null hypothesis, the ratio of inside to outside sets sampled would be expected to equal the ratio of inside to

outside sets taken. When all inside and outside sets were included,

\ 86" 18"

..-

.. r- ... ) ".

J

(

.!

.,

1- .

- ...,~~ ..-'.

.',....

:. Reject the hypothesis of random sampling.

B) Inside and outside sets from "pure" trips on which either all sets were taken

inside or all sets were taken outside the bay:

Number of sets sampled

Observed ifj) Expected lfI)

tained if the number of vessels sampled was increased and the

number of fish per sample was decreased. Therefore, in 1971,

sample size was decreased to 10 fish and the number of vessels

sampled per week was increased to 20-25. In practice the total

number of samples per week was often less than recommended,

partially because of a decline in the number of landings (sampling

opportunities).

Inside sets

Outside sets

5,320

1,334

159

30

151.11

37.89

STRUCTURE OF THE SAMPLING DESIGN

G = 2.18

:. Do not reject the hypothesis of random sampling.

Number of sets sampled

C) Inside and outside sets from "mixed" trips on which both inside and outside

sets were landed:

:. Reject the hypothesis of random sampling.

24.18

26.82

39

12

Observed ifj ) Expected lfI)

1,903

2,111

Inside sets

Outside sets

G = 17.99

2

For any given port-week combination, the sampling design

outlined above may be classified as two-stage cluster sampling

(also called subsampling) (Cochran 1977), where purse-seine sets

are the primary sampling units of unequal population size and individual fish are the secondary sampling units. Both primary and

secondary sampling units are selected with equal probabilities and

without replacement. The application of probability theory to this

problem, while not completely straightforward, requires random

selection at all stages of sampling and minimal selection and esti-
:. Reject the hypothesis that the final set of the day was randomly distributed

between inside and outside sets.

Table 2.-Test of the hypothesis that the last set of the day Is randomly distributed between Inside and outside sets when both kinds ofsets are made on a single

trip. Data are from Reedville, Va., for the 1979 ftshing season.

the hypothesis of random sampling was rejected (P < 0.05); sets

taken inside Chesapeake Bay were more likely to be sampled.

However, when sets were divided into those from "pure" trips

(trips during which all sets were made either inside or outside the

bay) and those from "mixed" trips (trips during which sets were

made both inside and outside), a possible source for the nonrandom behavior was discovered. When only pure trips were considered, the random-selection hypothesis could not be rejected;

when only mixed trips were considered the sampling distribution

was strikingly nonrandom. Apparently on trips catching fish both

inside and outside the bay, vessels were more likely to make the

last set of the cruise inside, and that is the set available for sampling. This possibility was tested by tabulating the actual number

of last sets made inside and outside the bay and comparing these

with the number of last inside and outside sets expected from the

ratio of total inside to outside sets made on all mixed trips (Table

2). The G-test indicated (P < 0.05) that the last sets were not randomly distributed; i.e., vessels tended to make the last set inside

Chesapeake Bay on their way to home port. This is the likely

reason that inside sets are favored in the sampling process. The

degree of bias introduced to our landing estimates depends on the

degree of size and age difference between inside and outside fish.

For the Chesapeake Bay example, I attempted to estimate the

magnitude and direction of the bias by comparing sampled sets

taken inside and outside the bay during mixed trips. The average

weight of fish in sampled sets was significantly greater outside the

bay (169.92 g) than inside (142.16 g) (t-test, P < 0.05, df = 49). I

calculated that the observed oversampling of inside sets during

1979 may have introduced a 5.4% underestimate of average fish

weight and an overestimate of numbers of fish landed for mixed

trips. This bias would be somewhat ameliorated because only

about 35% of all sets came from mixed trips.

For the second assumption (p. 2), random set selection has been

defined operationally in terms of random vessel selection. It is important to point out, however, that random vessel selection does

not, in and of itself, guarantee random set selection. In situations

where certain vessels routinely land more sets than average and

fish in these sets are significantly older or younger than in sets

landed by other vessels, an unknown bias may be introduced. Such

a situation may be imagined where the vessels of one company

opt to land many small sets of older fish, and the vessels of another

company opt to land a few large sets of younger fish. At any rate,

port samplers have been instructed to sample vessels "randomly,"

i.e., not to follow patterns or show favoritism in the selection of

vessels. This involves an element of judgment on the part of port

samplers and as such may be a possible source of bias in that the

actual method of random selection is not exactly specified

(Williams 1978). As a first approximation, the assumption of ranNumber of final sets made

Plant Vessel No. of No. of times No. of sampling

landings sampled times expected

A 64 6 8.60

65 8 8,74

102 16 13.71

98 6 13,18 -

96 ? 12.91 -

94 8 12.64 -

94 II 12.64

8 95 11 12.77

9 95 16 12.77 +

10 41 5 5.5 I

11 23 2 3.09

12 7 2 0.94

13 4 0 0,54

B 14 71 16 9.55 +

15 76 14 10,22 +

16 63 3 8.47 -

17 60 II 8.07

18 61 12 8,20 +

19 54 2 7.26 -

20 62 13 8.34 +

21 71 16 9.55 +

22 62 9 8.34

23 71 6 9.55 -

24 74 14 9.95 +

25 78 17 10.49 +

f· G 2 !.f. In (-,k ) = 55,89

, Ii

X.•'.l. = 3~.42

:. Reject the hypothesis of random vessel selection.

Table 3.-Test of tbe bypotbesls of random vessel selection for tbe port of Reedville, Va., during tbe 1979 ftshlng season. Pluses and minuses Indicate vessels

where the observed number of samplIng events dlfren from tbe expected by

more than 3.

dom vessel selection was investigated for the port of Reedville,

Va., for the 1979 fishing season (Table 3). Based on the number of

landings made by each vessel, the expected number of sampling

events per vessel was calculated under the assumption of random

selection. The goodness-of-fit test (G statistic) applied to the data

pooled from plants A and B led to a rejection of the hypothesis of

random sampling (P < 0.05). The degree and direction ofselection

bias may be judged for each vessel (Table 3). In general, plant A

vessels were underrepresented with respect to plant B vessels. To

attain a random selection of vessels a more rigorous procedure

should be specified. For example, a port sampler might obtain a

list of all vessels expected to land that day and choose the one(s) to

be sampled by tossing a six-sided die, picking names from a hat,

or other appropriate randomization device. In the development of

statistical methodology for this report, random vessel selection is

assumed.

For each set selected (at random), the port sampler presently

obtains a subsample of 10 fish, which is itself considered to be

randomly chosen from the population of fish constituting the set.

The method of subsample selection has varied over the years. Initially, every fifth fish from a bucketful was selected to make up

the subsample. However, less formalized methods have been

employed since the decision to reduce subsample size from 20 to

10 individual fish. Presently, port samplers try to collect a bucketful of fish from the top of the hold before offloading begins and

pick out the 10-fish sample "randomly" from the bucket. For

practical reasons, such as time constraints or vessel configuration,

this is not always possible, and a sample may be obtained from the

Expected

189.64

210.36

305

95

Observed

1,903

2,111

Inside sets

Outside sets

G = 138.82

X.•,•• = 3.87

3
"drag" or at other points along the offloading procedure. In some

cases a deep hold might require a dip-net collection, and it is even

possible that an occasional sample may result from simply picking

10 fish from the hold. The general instructions have been to

choose fish as randomly as possible within a set, consistent with

safety and practicality, so as to minimize possible bias (Gulland

1966). Here again judgment of the port samplers is involved, and

a more rigorous plan might be specified. A good critique of the

substantial problems inherent in the random selection of fish from

a vessel is given by Tomlinson (1971), who concluded that the

task is operationally impossible. In any case, we assume that any

added bias is small.

BASIC CALCULATIONS IN

THEORY AND PRACTICE

In this section, the basic catch statistics and their derivatives

are explored with respect to problems of estimation and the development of variance formulae. Three main topics are considered. I

begin by discussing how mean weight per fish is currently

calculated for any port-week combination, suggest some potential

difficulties with the method, and estimate the sample variance and

resulting confidence intervals. I then follow the current cooputational method through to arrive at the total number of fish landed

at a plant during a given week and for the entire fishing season,

and derive confidence intervals for the estimates. Finally, I examine the problem of estimating age composition in the catch and

attempt to evaluate the precision with which numbers of specific

age fish landed are now estimated.

In the following discussion, the 1979 menhaden catch records

are examined for three ports. These ports were selected because

they reflect broad geographic differences in age composition, and

because they demonstrate how difficulties encountered in

estimating catch characteristics differ among locations. Port Monmouth, N.J., is representative of northern waters. Landings are

made between late May and early October and are composed of

older (age 2 and older) fish. At Reedville, Va., where the fishing

season extends from mid-May to late October, younger (age 1 and

2) fish are landed. At Beaufort, N.C., there are two fishing seasons.

During the first season (early May to late October) the catch is

composed mainly of age 1 and age 2 fish. During the second

season (November to mid-January) the North Carolina "fall

fishery" takes advantage of impressive aggregations of southerly

migrating fish to land fish of all ages. These include older, spawning fish from northern waters and small, young-of-the-year fish

that move into the ocean from bays and sounds.

In the development of sample computations and variance

estimates, I will make use of the following notation patterned

closely after Cochran (1977):

N number of 3ets made per week per port

n number of sets sampled per week per port

Mi number of fish in the ith sampled set

mi number of fish sampled from the ith set

Yij value obtained from the jth fish sampled from ith set

II = sampling fraction in the first stage (sets)

12 = = sampling fraction in the second stage (fish).

Further notation will be supplied as needed.

4

Estimating the Mean Weight Per Fish Landed

Present method and blases.-In practice the mean weight of

an individual fish landed at a given port during a given week has

been estimated by the following method:

If Yij = the weight of the jth fish from the ith sampled set, then

m

I y" Y,' =. ::JL = average weight of a fish in the ith set, (1) J=I m

n

I y.

and, y =. --'- = average weight of a fish in n sampled sets. (2) ,=1 n

That is, the overall sample mean is simply the average of the mean

weights per sampled set. The estimate is inherently biased because

the sets are composed of unequal numbers of fish (Pope 1956).

This bias arises because the inequality of set size (Mi) affects the

probabilities of including a given fish in a subsample. This bias

may not be appreciable, however, if the set sizes do not vary considerably, if there is no corrt'iation between set size and true mean

weight per fish, or if the number of sets sampled (n) is large

(Sukhatme and Sukhatme 1970).

1'0 illustrate the possible effect of unequal set size on the

estimate of mean weight, consider a hypothetical population of 81

sets (Fig. 2), where the number of fish in a set is an inverse linear

function of the average weight per fish. The example assumes no

subsample variability, I.e., all fish in a set are the same weight,

and any subsample of 10 fish will estimate the true mean per set.

The true population mean in the example is 235.46 glfish. From

this population we now randomly sample 10 sets with equal probability and caiculate two means for each sample. The first mean is

the simple average of the 10 sets sampled, analogous to the y

computed above. The second mean is a ratio-to-size estimate

(Cochran 1977: 303), which is essentially an average weighted by

set size:

3500

3000

... 2500 '"..

...

...

:z: 2000 '"...

...

0

.. 1500 ...

..

;;;)

Z 1000

500

100 200 300 400

FI5H WEIGHT (ll)

Figure 2.-Hypothetlcal population or 81 sets In which the number or ftsh per

set Is Inversely related to the average weight per ftsh.

'.

500
Figure 3.-Relatlonshlp between average weight per ftsh (estimated

from docksIde samples) and estimated set size (from captains'

logbook records) for Port Monmouth, N.J., during 1979.

." 800 C

0 700

...

... 600

'"•...

... 500 %

'"...

... 400 0

•...

• 300

;;)

Z

Q 200 ...

...

e 100

;:

'"... 100

Po,t Monmouth, N.J.

, = -0.24

AVERAGE WEIGHT PER FISH (ll)

Chuapoako lay

This sampling procedure was repeated 25 times and the average y

and YR computed. In the first instance, yo = 295.18 g, while the

weighted statistic cYR) equalled 227.39 g, much closer to the true

population value.

This example demonstrates the potential dangers inherent in

the present method of calculating average weight. Later we will

see that such a bias could, if it exists, result in profound errors in

the estimation of total numbers of fish caught. It is, therefore,

critical to determine the extent to which set size is correlated with

average weight per fish. June (1972) observed a significant

negative correlation (r = -0.60) between mean fork length and

numbers of fish per set for 275 summer purse-seine sets made

during 1955-62. Whether his data are valid for our purposes is

open to question, since the sets are representative of a wide

geographic area and long time interval. For sets taken within any

single port-week combination the correlation may not be so striking. I have examined similar relationships in the three ports over

the 1979 fishing season (Figs. 3-5). The number of fish per set,

computed by dividing the captains' estimated catch per set by the

average weight per fish from dockside samples, was itself plotted

Figure 5.-Relatlonshlp between average weight per ftsh (estimated from

dockside samples) and estimated set size (from captains' logbook records)

for Beaufort, N.C., during 1979.

Figure 4.-Relatlonshlp between average weight per ftsh (estimated

from dockside samples) and estimated set size (from captains'

logbook records) for Reedville, Va., during 1979.

n n

YR = L M· v· / L M· ;=1 I· 1 ;=1 I

...

...

'"•...

...

%

'"

...

... .; 0 c

•...

• 0

;;)

Z

Q

...

...

e

;:

'"...

(3 )

1400

1200

1000

800

600

16

.

c

.2 14

.!

...

... 12 III

•...

... 10 %

III

i6:

... 8

0

•... 6 •:IE

;;;)

Z 4 Q

...

le

:IE 2

;:

III

... 0

___ Plant A, , = -0.25

_ Plant I, , = -0.19

_._ Combin.d, , =-0.20

400

AVERAGE WEIGHT PER FISH (ll)

lIoaufo,t (Fall Fi.h.,y)

, = ·0.40

300

AVEIIAGE WEIGHT PEII FI51l (ll)
There are, therefore, two parts to the variance expression. In

analysis of variance terms, s~ is equivalent to Among Set Mean

close agreement; YR is always included within the 95% confidence

intervals about y.

n

L <Y i-y)2

i=l where s~ =----- n-l

(4)

Approximate confidence intervals.-Turning now to the

problem of calculating confidence intervals about the mean

weight estimate, we will continue to assume equal set sizes ;n the

development of variance functions. If the n primary units (sampled sets) and the m subunits (fish within sets) from each chosen

unit are selected by simple random sampling, Cochran (1977)

gives the sample variance as:

A further verification that the present method of calculating

mean weight is reasonable is a comparison between the simple

average,y, and t~ approximate weighted mean'YR (Table 4). The

computation of YR is limited to Port Monmouth and Reedville

because it requires the captains' daily fishing report estimates of

set size for every sample taken. This knowledge is incomplete for

the port of Beaufort. In almost every instance the two means are in

against average weight. Relationships of this form (Le., a ratio vs.

its denominator) commonly yield spuriously high selfcorrelations, but this effect is reduced here, since the variance in

average fish weight is small compared with the variance in

estimated catch (Kenney 1982). A correlation between set size

and average weight is not strongly evidenced for either Port Monmouth or Reedville (r = -0.24 and -0.20, respectively; neither

significantly differs fro.m zero at P < 0.01). For the Beaufort fall

fishery, however, the correlation coefficient (r = -0.40) is significantly negative and the relationship appears nonlinear. It may be

tentatively concluded that, for most of the fishery, the present

method of calculating average weight is not materially biased by

the implicit assumption of equal set size. For the North Carolina

fall fishery, however, this bias may be larger, and, depending on

the age distribution of the catch, the estimate of average weight

per fish may be overstated. Hence, total numbers landed would be

underestimated.

Table 4.-Number of sets sampled, mean_weight per fish <Y), standard error (SE) ofy, and the

ratlo·to-slze estimate of mean weight GR) for Port Monmouth, N.J., Reedville, Va., and

Beaufort, N.C., for weeks during the 1979 fishing season.

Port Monmouth, N.J. Reedville, Va. Beaufort, N.C.

Week n Y SE YR y SE YR n Y SE YR ending

5/12 3 123.7 17.8

5/19 2 96.0 0.8

5/26 8 142.6 5.2 141.6 3 98.7 5.0

6/ 2 10 125.2 4.8 117.9 5 94.1 4.3

6/ 9 10 137.7 5.0 144.1 4 118.4 7.4

6/16 9 252.2 10.8 254.8

6/23 5 317.6 28.8 308.8 10 130.7 5.8 134.9 4 112.4 3.3

6/30 4 305.9 17.3 306.6 10 126.6 8.3 107.8 5 116.2 2.8

7/ 7 6 266.4 28.8 237.0 10 154.2 21.5 143.4 2 95.6 25.6

7/14 10 146.1 17.6 130.8 2 100.2 3.0

7/21 10 274.4 16.0 265.3 10 148.5 7.8 152.0 4 106.6 3.3

7/28 8 286.6 26.3 259.9 10 168.5 14.6 171.1 3 108.0 3.8

8/ 4 7 322.0 35.0 379.4 8 142.6 9.1 143.5 2 106.2 22.2

8/11 6 318.5 22.8 8 162.1 24.8 129.9 2 111.0 5.8

8/18 6 330.9 6.8 10 154.1 15.8 143.8

8/25 10 155.5 !l.8 154.0

9/ I 4 259.1 6.6 261.6 8 187.6 23.4 187.9 2 78.9 69.8

9/ 8 4 309.3 17.1 314.9 6 112.3 3.0 109.3

9/15 4 382.9 33.4 411.5 8 105.0 6.6 97.8

9/22 6 399.1 41.0 342.6 10 159.3 21.0 160.1

9/29 4 319.6 9.9 317.8 6 96.6 5.7 98.3

10/ 6 4 359.5 22.5 351.2 10 153.7 17.4 135.6 96.0 9.7

10/13 8 107.3 11.5 104.3

10/20 10 96.7 11.1 83.9 4 139.9 10.9

10/27 4 343.5 7.0 8 123.9 28.0 106.0 5 104.3 41.8

II/ 3 10 133.1 16.4 112.4 6 54.5 20.0

11/10 8 97.0 22.5 79.3 7 92.0 32.9

11/17 2 137.0 14.3

11/24 2 382.1 6.7 376.7 11 206.8 9.9

12/ 1 10 193.4 13.9

12/ 8 20 110.3 17.4

12/15 16 114.4 30.8

12122 14 27.8 2.8

12/29

1/ 5 9 18.6 1.2

1/12 4 22.4 2.0

6
Assuming no finite population correction, the variance can be

estimated from the sample means by the equation given above, or

equivalently from the analysis of variance:

In either case, vG) = 116.49, SE = 10.79, and the confidence

intervals are estimated as 252.19 ± 2(10.79) g. (Standard errors

are reported for the three selected ports for weeks during 1979 in

Table 4.)

Square/m, a reflection of the variance among the sampled set

means. In contrast, s~ is identical to the error mean square (MS),

and as such it estimates the amount of variance within a set. The!1

and!2 terms arc sampling fractions for the sets and fish, respectively, and are included as finite population corrections which express the degree to which the whole population is sampled. AS!I

~.nd!2 increase, the overall variance decreases because we gain

progressively more information about the entire population. In the

menhaden sampling surveY'!2 is always infinitely small (and

generally unknown) because each set contains large numbers of

fish, and!1 is generally < 0.10. For these n:asons the variance formula can be simplified to:

vG)= Among Set Mean Square.

nm (6)

The estimated variance is, therefore, dependent only on the unit

(set) means. 1f!1 is not small, this simplification gives a conservatively high estimate of the variance. The standard error is, of

course, Ii vG), and confidence regions may be estimated as y ±

2(SE), if the sample size is large.

To illustrate the calculation of yand its variance, consider the

sample data for Port Monmouth for the week ending 6/16/79

(Table 5). For this case, y was estimated at 252.19 g/fish landed.

Here, the true population statistics, o~ and o?v, are best estimated

by the among set component of variance (s~) and the within set

component of variance (s?v) calculated from the analysis of variOptimal allocation of sampling resources.-The data in

Table 2 may be used to examine the efficient allocation of sampling effort within a two-stage sampling design. To demonstrate

the effect of varying the number of sets sampled (n) and the

number of fish sampled per set (m), we first consider the

theoretical variance of the population mean:

n

L (Y. - y)l 2 I

= SI j=l v(y) g;; - =-'------''----

n n(n-l) (5)

0 2 0 2

- a w v(y)=-+-· n nm (7)

Table 5.-Survey data collected during the week ending 6/16/79 at Port Monmouth, N.J. Values

reported are weight (g) for eacb fish in the sample. Values In parentheses are fish ages estimated

from scales. The analysis or variance table, estimate of mean weight per fish <Y), variance and

standard error of y, and approximate 95% confidence intervals are also given.

Sampling dates

Fish

6/12 6/12 6/12 6/13 6/13 6/14 6/14 6/15 6/15 no.

1 329(2) 274(2) 247(2) 278(2) 160(2) 237(2) 184(2) 227(2) 215(2)

2 254(3) 244(3) 216(2) 305(3) 260(3) 257(2) 223(3) 271(2) 258(3)

3 274(2) 407(3) 178(2) 176(2) 208(2) 204(2) 401(3) 240(2) 209(2)

4 314(3) 401(3) 318(3) 185(2) 175(2) 246(3) 250(2) 252(3) 268(2)

5 312(3) 338(3) 263(3) 161(2) 269(3) 219(2) 157(2) 371(3) 305(3)

6 288(2) 332(3) 276(3) 355(2) 272(3) 207(2) 147(2) 230(3) 231(2)

7 339(3) 385(3) 261(2) 309(3) 181(2) 174(2) 197(2) 219(2) 154(2)

8 259(2) 194(2) 200(3) 218(2) 278(2) 263(i) 205(2) 213(3) 213(2)

9 176(2) 293(3) 271(3) 266(2) 237(2) i63(3) 294(2) 218(2) 241(2)

10 296(2) 350(3) 327(3) 221(2) 203(2) 278(3) 147(2) 246(2) 230(2)

Yi 284.1 321.8 255.7 247.4 224.3 234.8 220.5 248.7 232.4

s 47.36 70.07 47.71 64.97 44.41 32.91 78.75 46.51 40.34

ANOVA table

Source df SS MS

Total 89 324,155.7889

Among sets 8 83,874.0889 10,484.2611

Within sets 81 240,281.7000 2,966.4407

y n Yi :1: - = 252.19

;=1 n

n

:1:

;=16;-y)2",

n(n-I) -

Among Set MS ;:" I 16.49

nm

SE Y= 10.79

95% Confidence Intervals about y;:" 252.19 ±2(10.79);:" 230.60 - 273.78

7
ance of the existing sample (Snedecor and Cochran 1967):

S2 = S2 _ =Among Set MS - Error MS = 10,484 - 2,96~

a t m m 10

=751.8 (8)

minimum cost required to halve the variance, first replace m with

mopt to solve for n in the variance equation and then solve the cost

equation:

ZJ~7~ + 2,9~~.44 = 58.25,

sJ, = s~ =Error MS = 2,966.44. (9) n = 23 sets, and

If we were to conduct a new survey of the catch for that particular

port-week, the predicted variance of the mean weight would be: 45 (23) + 7.5 (23) (5) = 1,898 min.

where e 1 is the cost of sampling a set and c 2 is the cost of processing a fish. To estimate the optimal number of fish per set, the

product of the cost and variance functions are minimized

(Snedecor and Cochran 1967):

,-tf

,,'"DECREASING VARIANCE,

," INCREASING COST

"

"

,

" ,

""

~COSTCURVE m

Conversely, given a time resource limit of X minutes, it is a simple

matter to choose II and m such that variance is minimized. For example, given mopt = 5, what is the minimum possible variance for

our original cost of 1,080 min? Substitute mopt into Equation (11)

and solve for n (answer: n = 13 sets; minimum variance =

103.47).

The relationship between cost and variance may be represented

graphically (Fig. 6). As nand m increase, the variance decreases

and the cost ofsampling increases. To solve for minimum cost at a

stated variance, the cost curve is moved, and the point of tangency

with the variance isoline defines optimal nand m. To solve for

minimum variance, the cost curve is fixed and the variance curve

is moved to a point of tangency.

n

(12)

(11)

116.49. (10)

(6) (2,966.44) =5 fish.

(1) (751.78)

cost =e In + e2nm,

vG) = sJ + s~ = 751.78 + 2,966.44

n nm 9 (9) (10)

These data can be used to help address two related matters. First,

we can now predict the variance we would obtain with different

values of nand m. Second, given s~ and s?v we can determine what

combinations of nand m will yield a required variance. For example, by sampling 5 fish per set from 30 sets we would expect the

variance to be lowered to 44.84. These conclusions are valid only

for the given port-week. More general conclusions rely on similar

analyses for several port-week combinations. Clearly, however,

sampling requirements in the menhaden fishery vary according to

location and time of year.

For a given level of variance, several different values of nand m

are possible. The "best" combination depends on the relative cost

of sampling primary and secondary units. For the menhaden

survey it has been estimated (Huntsman2 ) that a port sampler

needs 45 min to collect a subsample and 7.5 min to age. weigh,

and measure the length of each fish. The relative cost of subsampie selection versus individual fish analysis is approximately 6:1.

If we ignore fixed costs, the expense of the sample can be

estimated by:

Assuming that ctie 2 remains constant, the optimal subsample size

(m) is directly related to Vs~/s~. As the variance among sets

increases relative to the variance within sets, this fraction

becomes smaller. and optimal allocation requires greater numbers

of sets to be sampled with fewer fish per set. Quick calculations

for Port Monmouth. Reedville, and Beaufort reveal optimal subsampling sizes of about 4 fish with an upper 95% confidence

bound of about 10 fish.

Knowing the optimal value for m, one can use the cost and

variance equations [Equations (10) and (11)] to either 1)

minimize the sampling cost subject to a variance constraint, or 2)

minimize the variance subject to a cost constraint. In the Table 5

example, with /I = 9, m = 10, and variance = 116.49, the cost of

sampling is (45) (9) + (7.5) (9) (10) = 1,080 min. To predict the

'0. R. Huntsman. Fishery Biologist, Southeast Fisheries Center Beaufort Laboratory, National Marine Fisheries Service, NOAA, Beaufort, NC, 28516-9722, pen.

commun. April 1981.

8

Figure 6.-Theoretlcal relationship between n (number of sets sampled) and m

(number of nsh sampled per set) and the variance In average weight and cost of

the sampllDg program.

Estimating the Total Number of Fish Landed

Number of fish landed by port-week.-The total number of

fish landed per port-week (Cpw) is presently estimated by

dividing the total weight of the catch for each plant (Wt) by the

mean weight per fish landed cY):

(13)

I assume that Wt is measured without error and that yis a random

variable subject to the uncertainty described previously. For the

sample data shown in Table 5, the weekly catch total was

2,356.87 t. Dividing this figure by 252.19 glfish, an estimated
9.346 X 106 fish were harvested at Port Monmouth during the

week ending 6/16/79.

The variance of the estimate of total fish caught is derived by

considering the basic calculation to be of the form uX, where a is

the constant Wt

, and xis the inverse of the mean weight per fish.

11 y. The variance of this product may be expressed as:

v(ax) = a2 v(x), and by analogy,

Tabulated in Table 7 are the weekly and season totals for Port

Monmouth and Reedville; only the fall fishery totals are given for

Beaufort. Confidence intervals, expressed as a percentage of the

individual estimates, lead to the conclusion that the northern summer fishery is less uncertain with respect to catch estimates than is

the North Carolina fall fishery. As indicated earlier, it is the

greater variance associated with estimates of average fish weight

that supplies the explanation.

v(Wt / y) =Wi v<y- 1). (14)

Table 7.-Estlmate of the total number of nsh landed by port-week and for the

entire 1979 season. Approximate 95% conndence Intervals are Included and are

expressed as a percentage of the estimate.

5/12

5/19

5126 3.678x10' ± 7.5%

61 2 5.528x10' ± 8.1%

61 9 5.534xIO' ± 7.2%

6/16 9.346x IO' ± 8.6% 6.898x10' ±16.0%

6/23 6.837xI0' ±20.4% 5.963xIO' ± 8.6%

6/30 6.833 x IO' ±13.1% 5.933xIO' ±16.8%

71 7 1.043x lO' ±23.7% 6.271xIO' ±30.0%

7/14 5.984x10' ±25.2%

7/21 1.368xIO' ± 9.6% 4.226x10' ± 11.8%

7128 6.236xIO' ±17.1% 7.168xI07 ±16.0%

81 4 9.652"10' ±33.3% 4.740xIO' ±12.9%

8/11 8.316xI0' ±15.8% 3.917'<10' ±27.6%

8/18 8.011'<10' ± 4.9% 6.718x10' ±22.7%

8/25 3.017'<10' ±15.2%

91 I 2.347"10' ± 4.4% 4.474xIO' ±30.9%

91 8 3.951'<10' ±21.9% 4.871'<10' ± 5.7%

9/15 4.144'<10' ±J6.6% 5.120'<10' ±10.7%

9/22 5.843xIO' ±22.0% 5.799x10' ±31.2%

9/29 2.044'<10' ± 6.5% 4.801'<10' ±16.3%

10/ 6 2.165xI0' ±13.5% 4.812x10' ±23.0%

10/13 5.906x10' ±24.4%

10/20 1.478xIO· ±15.3%

10127 2.235xl0' ± 3.3% 5.140xIO' ±36.4%

I II 3 4.514'<107 ±29.4% 4.743'<10' ±164.2%

11/10 3.245xI07 ±45.4% 3.858'<10' ±395.0%

11/17

11/24 8.157xIO' ± 5.7% 1.428'<10' ± 15.8%

121 1 6.825'<IU' ± 33.9%

12/ 8 6.544'<107 ± 73.0%

12/15 9.171'<10' ±I19.8%

11/22 4.088xIO· ± 18.3%

12/29

1/ 5.955XIO· ± 11.3%

1/12 1.750><10· ± 18.4%

Season

total 1.021><10· ± ~.l % 1.398x 109 ± 4.3% 1.367xI09 ± 11.8%

The variance of y'l is estimated as for y in the previous section,

but an inverse transformation is applied to the individual fish

weights initially. The ANDVA table derived from the data in

Table 5, and the calculation of variance and confidence intervals

are given in Table 6. For that example, the 95% confidence intervals extended to ± 8.6% of the total catch estimate.

Table 6.-Estimate of the total number of nsh landed at Port Monmouth. N.J.,

during the week ending 6/16/19 with approximate 95% conndence Intervals.

ANOVA table

Source df SS MS

Total 89 9.178x I0"

Among sets 8 2.085xlO-' 2.606IxIO·'

Within sets 81 7.093xIO·' 8.7569xIO-'

v<J") = Among Set MS = 2.900xlO-.

nm

WI = 2.35687xI09 g

2.35687xI09= 9,345.612 fish landed

252.19

v(WI I y) = w: v<J-') = (2.35687XI09)' 2.900"10-· = 1.611 x IO"

SE = 401,361 fish

95% Confidence Interval:; 9.346x I0' ± 2(401.361)

:; 8,708,694 - 9,982,550

:; estimate ± 8.6%

Number of fish landed over ports and weeks.-Similar calculations can be made for any port-week combination and we can

build on these individual catch estimates to obtain composite

measures of the total catch over any period of time or combinations of ports. For example, to get an estimate for the total number

of fish caught in the Beaufort fall fishery, simply add up the weekly totals. (I assume that samples are available for every week included in the estimate.) If k weeks are to be summed:

Week Port Monmouth, N.J. Reedville, Va. Beaufort, N.C.

(15) Estimating the Age Composition of the Catch

Assuming that weekly totals are independently estimated, the

variance of the grand total (CT) may be expressed as:

This equation ignores any possible covariance terms among weekly catch totals. These terms may exist, for example between CpW1

and CpW2, if seasonal trends recur. Such an effect would inflate

the variance and, hence, decrease the confidence about seasonal

totals.

9

Basic calculations and variance formulatlons.-The age

composition of the catch for any particular port-week is presently

estimated by averaging the relative proportions of each age class

over the n sampled sets. The age of each sampled fish, determined

from scale readings, is assumed to be known without error, Let p"

be the proportion of fish falling in the jth age category for the sUb~ sample from the ith sampled set. The proportion of all other age

categories within the subsample may then be expressed as qr =

I-Pi)' The mean proportion of the jth age category in the popKlation is then estimated to be:
This estimate i's subject to the same sort of biases as encountered

for the mean weight estimate, That is, we must assume that age

composition is relatively independent of overall set size (Mi

).

Assuming the Pij's are normally distributed within the population of N sets, an estimate of the sample variance may be derived.

The two parts to the complete expression, the among set (s~) and

the within set (s~) variances, are given by Cochran (1977) as:

n -

Pi = L Pi)! n,

i=l

(17)

(18)

(Tables 8-10).

It is an extremely practical matter for the management of the

fishery to be able to state an average age composition with approximate confidence intervals over time intervals and geographic

areas greater than 1 wk and one port. For example, the 1981

Menhaden Fishery Management Plan set a goal that at least 10%

of all fish landed be age 3 or older. Specific administrative actions

are keyed to this criterion. Obviously then, a knowledge of the

precision with which overall age proportions are estimated is

critical to the credibility of this part of the management plan. We

approached the problem by conceptualizing a stratified two-stage

sampling design whereby strata are individual weeks whose

relative population sizes are indexed by the estimates of total

catch. The underlying assumption remains that all sets within a

given port-week have the same number of fish, but now set size

may vary from week to week. For a given port over the entire

season:

(19)

(23)

Once more, assuming thatfl andf2 are small, the estimate simplifies to:

95% Confidence interval 0.6222 ± 2(0.06827)

0.4857 - 0.7588.

Estimates of mean age compositions and variances are presented

for each of the three ports by week for the 1979 fishing season

n

L Pi2

- i=l = 5'960 = 0.622222 P2 = -n-

(24)

52

v(PjkST) = L WW k2 v(Pjwk)'

w=l

Similarly, average age compositi<rn over the en-tire fisl\ery may be

estimated by considering each port as a stratum within the- Jotal

fishery. Overall estimates of the average age composition (for ages

0, 1, 2, 3+) and approximate 95% confidence intervals by port

and over all three ports for the 1979 fishing season are given in

Tables 8-11.

Optimal allocation of resources in age composition sampUng.-The solution of the optimal allocation problem for proportional age sampling is by no means simple. Where only two

age categories are present, the binomial distribution provides an

appropriate theoretical framework within which to proceed. For

more than two age classes, however, the multinomial distribution

is preferable, but the theoretical properties with respect to twostage sampling have not been adequately addressed. Cochran

(1977) advised that sample size be determined by considering

each age category separately versus the remaining categories as a

binomial distribution. In the case of simple random sampling, one

would presumably use the largest estimated sample size for the

survey. For a two-stage design the optimal number of primary and

secondary units is more difficult to determine. It is clear, however,

that what is really wanted is the simultaneous confidence interval

approach provided by the multinomial distribution. Tortora

(1978) has devised such a technique for simple random sampling

in a multinomial population, and has concluded that the binomial

approximation underestimates the required sample size. The

degree of understatement ranges by a factor of 1.7 to 2.4 as the

number of categories increases from 3 to 10.

Several authors have approached the related problem of optimal allocation of resources in two-stage sampling designs when

where PjkST is the overall average proportion of the jth age

category over all strata from the kth port, Nwk is the estimated

totbl catch for the wth week at the kth port, Ntk is the total catch

over all weeks at the kth port, and Pjwk is the average proportion

of age j fish in the wth week and the kth port. Wwk is, therefore, a

week specific weighting factor. The variance about this overall

proportion is:

(22)

(20)

(21)

n

L epij- p;)2

v(jjj) = = i=ln(n_-l)-'

_ 1-.f1 n _ f/l-f2) n vep·) = -- L ep ..-p -)2 + -- L p .. q .. J n(n-l). IJ J n 2(m-l). IJ IJ

/=1 /=1

= I-f1S2 +ft(l-f2)s2.

n I mn 2

v(jj ) = 0.33556 = 0.00466 2 (9) (8)

SE =VV(jj2) = 0.06827

As an example, consider the Port Monmouth data from Table 5.

The proportions of age 2 fish for those nine sampled sets are 0.6,

0.2, 0.4, 0.8, 0.7, 0.7, 0.8, 0.6, 0.8, respectively. The following

calculations may be performed:

The derivation of these formulae and their relationship to the

analysis of variance are not as readily apparent as in the case of

mean weight per fish. These variance components are derived in

the Appendix.

Given s~ and s~, the variance ofPj may now be estimated in a

fashion similar to that of the variance of mean weight:

10
Table 8.-Total catch per week (stratum), weighting factor per week, mean age proportion and

variance for ages I, 2, and 3+ by week, and the overall 1979 seasonal mean proportions (PST)'

variances, and approximate 95% confidence Intervals for Port Monmouth, N.J.

Total Age I Age 2 Age 3+

catch Weighting

Week (XIO·) ractor 1',(%) viP,) 1',<%) v{j>,) 1',.(%) v{j>,.l

6/16 9.35 0.09154 62.2 4.66xlO-' 37.8 4.66xIO-'

6/23 6.84 0.06697 46,0 6.60xlO-' 54.0 6.60xI0-'

6/30 6.84 0.06697 70.0 5.00xI0-' 30.0 5.00xI0-'

7/ 7 10.44 0.10221 95.0 5.00xlO-· 5.0 5.00xlO-·

7/14

7/21 1368 0.13393 79.0 4.77xlO-' 21.0 4.77xlO-'

7/28 6.24 0.06109 39.3 1.27xlO-' 60.7 1.27x10-'

8/ 4 9.66 0.09458 1.4 2.04xI0-· 27.1 1.61xIO-' 71.4 1.59x10-'

8/11 8.32 0.08146 30.0 1.27xIO-' 70.0 l.27xlO-'

8/18 8.01 0.07842 20.4 1.35x 10-' 79.6 1.35XIO-'

8/25

9/ I 2.35 0.02301 90.0 5.00x IO-' 10.0 5.00x lO-'

9/ 8 3.95 0.03867 2.5 6.25x IO-· 52.0 3.56xIO-' 45.5 4.25XI0-'

9/15 4.15 0.04063 2.5 6.25xIO-· 97.5 6.25XI0-·

9/22 5.85 0.05727 31.7 1.89x10-' 68.3 1.89x10-'

9129 2.05 0.02007 64.2 1.40xlO-' 35.8 1.40xlO-'

10/ 6 2.17 0.02125 35.0 1.75xIO-' 65.0 1.75x10-'

10/27 2.24 0.02193 22.8 2.07xI0-' 77.2 2.07xI0-'

PST v(PST) 95% Confidence Interval

Seasonal Age I 0.23% 2.76xIO-· ±0.33%

estimates Age 2 51.29% 5.99xI0-· ±4.89%

Age 3+ 48.47% 6.08xlO-· ±4.93%

Table 9.-Total catch per week (stratum), weighting factor per week, mean age proportion and variance for ages 0, 1,2,

and 3+ by week, and the overall 1979 seasonal mean proportions (PST), variances, and approxlmatc 95% confidence

Intervals for Reedville, Va.

Total Age 0 Age I Age 2 Age 3+

catch Weighting

Week (XIO·) ractor 1'.(%) v(p.) 1',(%) viP,) P,(%) V(P,) 1',.(%) v{j>,,)

5/26 36.80 0.02646 2.5 2.68xlO-· 96.3 6.92xIO-· 1.3 1.S6xIO-·

6/ 2 55.30 0.03976 98.0 l.78xlO-· 2.0 l.78xlO-·

6/ 9 55.36 0.03980 97.0 4.56x IO-· 3.0 4.56xI0-·

6/16 69.01 0.04961 3.0 456x lO-' 97.0 4.56x IO-·

6/23 59.66 0.04289 2.0 l.78xlO-· 97.0 2.33xlo-· 1.0 1.00xlO-·

6/30 59.36 0.04267 23.0 1.I6x10-' 77.0 1.I6x10-'

7/ 7 62.74 0.04511 22.9 7.00xlO-' 77.1 7.00xlO-'

7/14 59,87 0.04304 18.0 4.40xlO-' 82.0 4,40xlO-'

7/21 42.28 0.03040 18.2 5.77xlO-' 81.8 5.77xlO-'

7128 71.71 0.05155 8.0 1.07xlO-' 92.0 1.07x 10-'

8/ 4 47.42 0.03409 9.2 161xlO-' 90.8 1.61xlO-'

8/11 39.18 0.02817 11.2 3.0IxI0-' 79.4 3.92xI0-' 9.3 3.90xlO·'

8/18 67.21 0.04832 12.1 l.73xlO-' 86.9 2.23xlO-' 1.0 1.00xlO-·

8/25 30.19 0.02170 7.0 9.00x lO-· 88.0 1.07x10-' 5.0 5.00xlO-·

9/ I 44.76 0.03218 12,5 3,l2xlO-' 75.0 2.50xlO-' 12.5 3.84xI0-'

9/ 8 48.73 0.03503 47.2 6.IOxlO-' 52.8 6.IOxlO-'

9/15 51.22 0.03682 65.0 3.57x lO-' 35.0 3.57xI0-'

9122 58.02 0.04171 2.0 4.00xlO-· 32.1 1.03xlO-' 58.7 8.8IxI0-' 7.2 1.86x lO-'

9/29 48.03 0.03453 5.0 1. 17X 10-' 66.7 4.44xI0-' 28.3 4.94xIO-'

10/ 6 48.14 0.03461 38.4 8.2IxlO-' 60.6 8.12xI0-' 1.0 1.00xlO-·

10/13 59.09 0.04248 1.2 1.S6xI0-· 65.0 1.46x10-' 33.8 1.46x10-'

10/20 147.84 0.10628 78.7 6.84xIO-' 21.3 6.84xIO-'

10127 51.43 0.03697 68.6 1.7Ox 10-' 28.6 1.32xIO-' 2.7 7.72xI0··

III 3 45.16 0.03247 57.1 1.57x10-' 42.9 1.57xIO-'

11110 32.47 0.02334 25 2.68xlO-· 82.5 1.S3x10-' 11.2 7.66xI0-' ' 3.8 1.41xlO-'

PST V{j>ST) 95% Confidence Interval

Age 0 0.37% 2.5IXI0-· ±0.32%

Seasonal Age I 32.28% 2.62xI0-· ±3.24%

estimates Age 2 65.73% 2.54xI0-· ±3.19%

Age 3+ 1.62% 1.40xI0-' ±0.75%

11
Table 10.-Total catcb per week (stratum), welgbtlng factor per week, mean age proportIon and variance for ages 0, I,

2, and 3+ by week, and tbe overall 1979 seasonal mean proportions (PST), variances, and approximate 95% confidence

intervals for Beaufort, N.C.

Total Age 0 Age I Age 2 Age 3+

catch Weighting

Week (XIO') factor i'.(%) vlJ'o) 1',(%) v(P,) 1',(%) v(p,) P,.(%) vii',)

5/12 11.43 0.00737 96.7 I.IlxIO-' 3.3 I.Il x lO-'

5/19 14.44 0.00931 5.0 2.50xlO-' 95.0 2.50xlO-'

5/26 6.94 0.00447 3.3 1. 11 X10-' 96.7 1.1IxlO-'

61 2 12.43 0.00801 12.0 5.40xlO-' 86.0 5.60xlO-' 2.0 4.00xlO-·

61 9 11.16 0.00719 97.5 6.25x lo-· 2.5 6.25xI0-·

6/16 18.57 0.01197 8.3 2.28xI0-' 88.3 1.61xlO-' 3.3 4.44xIO-·

6/23 15.68 0.01010 12.5 3.96xlO-' 87.5 3.96xIO-'

6/30 18.16 0.01170 12.0 6.40xIO-' 86.0 9.60xlO-' 2.0 4.00xlO-·

71 7 22.34 0.01440 40.0 1.60xlO-t 60.0 1.60xlO-t

7/14 27.76 0.0"89 15.0 2.50xlO-' 85.0 2.50xlO-'

7/21 6.18 0.00399 7.5 2.29xIO-' 90.0 1.67xIO-' 2.5 6.25xIO-·

7/28 2.12 0.00137 10.4 l.37xlO-' 86.3 1.00xlO-' 3.3 I.IlxIO-'

81 4 3.05 0.00196 15.0 2.25xIO-' 65.0 2.50xlO-' 20.0 4.00xlO-'

8/11 0.95 0.00061 5.0 2.50xlO-' 95.0 2.50x lO-'

8/18

8/25

91 I 5.76 0.00371 50.0 2.50XIO-t 40.0 1.60xlO-' 10.0 1.00xlO-'

91 8

9/15

9/22

9/29

101 6 2.33 0.00150 76.3 7.30xIO-' 23.7 7.30x lO-'

10/13

10120 1.22 0.00078 37.5 2.06xIO-' 62.5 2.06xIO-'

10/27 3.83 0.00247 40.0 6.00xIC-' 8.0 6.40xlO-' 44.0 3.76xIO-' 8.0 6.40xlO-'

III 3 4.76 0.00307 55.0 3.18xlO-' 21.7 1.16xIO-' 23.3 2.58xlO-'

11/10 3.86 0.00249 42.9 4.08xlO-' 54.3 3.76x lO-' 2.8 8.16xIo-·

11117

11/24 14.82 0.00955 10.9 2.8IxlO-' 79.6 2.92xlO-' 9.5 1.63x IO-'

121 I 6.83 0.00440 17.2 9.53xIO- 1 75.6 8.29xIO-' 7.2 9.69xIO-·

121 8 65.48 0.04220 28.7 8.61xlO-' 41.4 8.15X!0-' 26.8 5.5I x I0-' 3.2 4.76xIO-·

12/15 91.76 0.05913 59.2 1.44xIO-' 7.4 1.09x lO-' 23.8 9.03xlO-' 9.6 3.9IxIO-'

12/22 409.01 0.26359 92.1 1.56xIO-' 7.9 1.56xIO'·'

12/29

11 5 595.74 0.38394 100.0 0.00

1/12 175.05 0.11282 100.0 0.00

PST vIJ'ST) 95% Confidence Interval

Age 0 79.22% 1.78x I0-' ±2.67%

Seasonal Age 1 6.12% 1.64XIO-· ±2.56%

estimates Age 2 13.59% 8.14xIO-' ±1.80%

Age 3+ 1.06% 1.53xIO-' ±0.78%

Table H.-Estimated age composition for the combined catcb of Port Mon·

moutb, N.J., Reedville, Va., and Beaufort, N.C., during tbe 1979 fisblng season.

Variances and approximate 95% confidence Intervals are given for tbe age pro·

portion estimates.

PST v(PST) 95% Confidence Intervals

-----"

Age 0 40.54% 4.69xl0-< ± 1.37%

Age I 17.87% 9.72xlO-' ± 1.97%

Age 2 18.67% 7.48x lO-' ± 1.73%

Age 3+ 2.91% 7.57xlO-' ± 0.55%

----_.------_._-------------

more than one variable is measured. Saila et al. (1976) pointed out

that the traditional approach requires solutions for the optimal

primary and secondary sample size for each variable separately,

followed by a compromise to reach a final allocation decision.

Saila et al. (1976) acknowledged that this solution is not

necessarily optimal and may incur excessive costs in large

surveys. but their final recommendation appears to be an ineffi12

cient compromise. Schweigert and Sibert (1983) determined the

optimal allocation of sampling resources for a multiage fishery,

but their method is similar to the one proposed by Saila et al.

(1976). The more correct approach would be a simultaneous solution for all age groups that minimizes cost subject to several

variance constraints. This approach relies on an iterative solution

(Kelley 1971) that may incur significant computer costs; a

graphical solution may be more practical.

The allocation of sampling effort when two age classes occur is

demonstrated for the Port Monmouth data (Table 5). The

theoretical variance of the population is:

_ oJ o~ v(p') = ._- +-.

} /I nm

where o~ and o~ may best be estimated by s~ and s~. These estimates are derived by analogy:
sJ = s~ = 0.33556 _ (10) (1.78) = 0.01997

m 8 (9) (9) (10)

exact representation of the variance proposed by Goodman

(1960):

For a new sample we would, therefore, predict the variance of the

average proportion of age 2 fish to be:

s~ = s~ (10) (1.78) = 0.219753.

(9) (9)

v(Cpw} = Cpw v(Pj ) + PJ v(Cpw)

_ L(CpW-Cpw) (Pij-Pj

) + 2 C p. _-"--"------"--"-------:l'------''-

pw } n(n-l)

sJ + s~ = 0.01997 + 0.219753 = 0.004661.

n nm 9 (9) (10)

Different values of nand m may be computed to satisfy this

relationship for any specified variance. Assuming the same

overall cost function as before (c l:C 2 = 6), optimal subsample size

may be found by: + L(CpW-CpW)2 (Pij-Pj)Z _ [L(Cpw-Cpw) (Pij-Pj )] 2. (27)

n(n-l) n(n-l)

(6) (0.219753) = 9.

(1) (0.01997)

where Cpwj is the number of agej fish landed per port week. An

estimate of the variance of Cpwj requires an expression for the

variance of the product of two random variables. A first approximation of this relationship that has been previously applied to

fishery data (Southward 1976) is:

CONCLUSIONS AND RECOMMENDATIONS

The method of solution for the Port Monmouth data is given in

Table 12. In this case, the 95% confidence region extended to ±

26.3% of the number of age 2 fish caught (5,815,054 ± 1,526,988

fish). As in the case of total catch, various age-specific catches

and their variances may be summed to obtain estimates and confidence intervals over time and space. We have accomplished this

by port over the entire 1979 fishing season and arrive at confidence intervals that are very similar to those obtained by

multiplying the upper and lower confidence bounds of the portseason age proportion estimate by total number of fish landed.

In this paper I have attempted to evaluate constructively the effectiveness of the present menhaden sampling scheme within the

framework of probability sampling theory. The derivation of the

basic population statistics and their associated variance functions

has been accomplished, within the constraints of stated assumptions. In so doing, I have examined the correspondence of those

who have addressed this problem previously, explored the

literature available on related fishery sampling problems, interviewed several people who have been involved with data collection and analysis, and analyzed a limited body of data to gain a

preliminary feel for the sampling variability.

In general, a theoretical approach is outlined which is consistent with present estimation techniques and can be applied to the

raw data. Although it appears, from the limited amount of data

considered, that the current number of fish sampled per set is adequate for estimating mean weight and total catch, it may be too

small for estimating age composition. It also appears, at least for

the 1979 data, that a greater number of sets should be sampled.

The specific number depends on the area and time concerned, but

generally no less than 10 sets/port-week should be sampled, since

fewer degrees of freedom will raise the Hable value used to

calculate confidence intervals.

Ultimately it is up to fishery planners to decide what level of

uncertainty is acceptable. This, of course, depends on the actual

use of the data and the sensitivity of management decisions. If information for every port-week combination is critical, and if 1979

data are representative, then we probably need to sample more intensively. Ifsummary information over the fishing season is most

important, then our level of sampling may be sufficient. For example, on a weekly basis, 95% confidence intervals about

numbers of fish landed average ± 14.7% (range 3.3-33.3%), ±

(25)

The optimal number of primary units would be found based on the

total budget available for sampling or from the predetermined

variance requirement. The calculations for mopt are only computed for the one example shown, but for that case the optimal subsample size is larger for age proportions than for average weight.

Again, this conclusion depends on the actual age structure of the

menhaden population at different times and places. Individual sets

at Port Monmouth appear to be nonuniform with respect to age

distribution, so larger subsamples are indicated. This result will

hold in all but cases of single (or almost single) age populations.

When more than two age groups are present, the optimal allocation of sampling effort is found by minimizing the cost of the

sampling program, provided that predetermined levels of variance

are satisfied for each proportion. As Tortora (1978), Saila et al.

(1976), and Kelley (1971) illustrated, this problem is difficult,

and I will be addressing it in a future report.

Number of fish at age.-To conclude this section, I estimate

the number of fish in each age group and generate an approximate

solution to the variance function. The present method requires

that the total number of fish landed, Cpw, be multiplied by the

proportion of fish in age group j, P/

This equation ignores any possible covariance between the

estimates of catch and age proportions. Clearly, however, the

estimate of catch is not independent of age structure, since, within

anyone set, average weight and age are related. That is, a greater

proportion of young fish generally implies a lower average weight,

which in turn leads to a higher estimate of total catch. The

covariance terms are included in the following modification of the

13
Table 12.-EsUmate of tbe number of age 2 fisb landed at Port Monmoutb, N.J., during tbe week ending

6/16179, wltb metbod to calculate approximate 95% confidence Intervals. Data taken from Table 5.

Total catch Proportion of

Sampled Mean weight based on Yi age 2 fish

set no. eY;) (Cpwi) (Pi,)

I 284.1 8.296xI0· 0.6

2 321.8 7.324xIO· 0.2

3 255.7 9.217xIO· 0.4

4 247.4 9.527xI0· 0.8

5 224.3 10.508xIO· 0.7

6 234.8 10.038xIO· 0.7

7 220.5 10.669x IO· 0.8

8 248.7 9.477xI0· 0.6

9 232.4 10.14IxIO· 0.8

P, = 0.62222 v(P,) = 0.00466 SE = 0.06827

Number of age 2 fish landed = CpwP, = (9.346XIO·) (0.62222) = 5.815xI0· fish

2 C p. ~(Cpw - Cpw) (Pij - Pj)

pw J n(n-I)

+

_. 2 -

~(Cpw - Cpw) (Pij - Pj)

n(n-I)

(9.346x I0·)' (0.00466) + (0.622)' (1.014XI0")

+ 2(9.346XI0') (0.622) (1.459XIO' ) + 2(9.346XIO') (-3.239XIO' )

(8) (9) (8) (9)

+ 2(0.622) (-1.532X~ ) + (8.936XI01I ) _ (1.459XIO' ),

(8) (9) (8) (9) (8) (9)

v(Cpwj) = 5.829xIO" SE = 763,494 fish

95% Confidence Interval 5,815,054 ± 2(763,494)

5,815,054 ± 26.3%

19.2% (5.7-45.4%), and ± 94.4% (11.3-395.0%) for Port Monmouth, Reedville, and Beaufort, respectively. Corresponding confidence intervals by season, however, are ± 5.1 %, ± 4.3%, and ±

11.8% (Table 7). Similarly, confidence bounds about mean age

proportions are much narrower for the entire season than for individual weeks (Tables 8-11).

Throughout this discussion, the assumptions basic to probability sampling theory have been emphasized, and potential breaches

have been identified. The most serious violations stem from 1) the

method of set selection, and 2) the use of sets as the primary

sampling unit.

With regard to set selection there are at least two areas of concern. First, it appears that when trips occur both nearshore and

offshore (e.g., inside and outside Chesapeake Bay) the last set

taken is more likely to occur close to shore. Consequently, these

nearshore sets are prone to oversampling since they are the ones

available to port samplers. Given the current sampling design, this

flaw is difficult to correct, but the introduced bias may be small,

since a minority of vessel trips (35% for Reedville) are involved.

It may also be possible to estimate the bias by stratifying the sample into nearshore and offshore sets and comparing estimates of

average age and size.

The second area of concern is that vessels themselves are not

being sampled at random. This nonrandom behavior can occur on

the level of plant or individual vessel. For example, at Reedville,

14

plant B appears to be preferentially sampled over plant A for the

1979 season. In addition, within plants, certain vessels are more

or less likely to be sampled (Table 3). This problem may be avoided by providing port samplers with explicit instructions dictating

the method to be used for vessel selection. As is true for nearshore

vs. offshore sets, the degree of bias introduced by nonrandom

selection depends on actual population differences between sets

selected for and sets selected against.

The second major problem involves the implicit assumption

that set size is uniform with respect to numbers of fish. When set

size does vary, current sampling theory requires a knowledge of

these sizes. Paradoxically, that is the very quantity we want to

estimate. As a consequence, the range of available procedures is

narrowed. It is essential that we explore the effect of unknown and

unequal set size on the parameter estimates. In most cases, for the

data studied, violating the assumption of equal set size does not

appear to appreciably bias the estimates, except perhaps for the

North Carolina fall fishery (Table 4). Even with unequal set sizes

it is still possible to estimate some population characters if accurate estimates of each set's approximate weight is recorded, but

the calculation can become rather circular. Use of captains' daily

fishing reports coupled with weight data from the sampled set can

provide the estimate of relative set size necessary to a weighted

calculation (ratio-to-size estimate). Appropriate variance formulations may be derived from Cochran (1977).
The advantages of a different sampling design might also be

considered. For example, fewer assumptions would be needed if,

instead of sampling sets at random, we could sample full boatloads. To accomplish this would require that fish be sampled at

various and randomly selected times throughout the unloading

process. This sort of sampling plan might offer at least one other

advantage. Instead of estimating numbers of specific age fish indirectly from total catch and age composition, we could calculate

the numbers directly. Tomlinson (1971) offered a similar plan for

schooling pelagic fish and applied it to the California anchovy

fishery. Basically, primary units are taken to be boatloads and

secondary units are now defined as clusters of fish, each cluster

weighing a known amount. By taking two or more random clusters

from each boatload sampled, we could obtain replicate counts of

the numbers of specific age fish. Since we know (approximately)

the weight of each boatload and the weight of a cluster, we can

estimate the number of secondary units in a boatload (assuming

nonoverlapping clusters). It is then possible to devise a sampling

scheme whereby boatloads are sampled with probabilities proportional to estimated catch size, a method which often yields the

smallest sampling variance.

Another avenue of research is the optimal allocation of sampling resources in the case of a multiage fishery. As indicated

earlier, we have been exploring this problem and feel that significant progress can be made, at least in generating a general solution. Initially we will examine the behavior of the age-specific

variance function for representative menhaden samples. After

specifying a cost function, our goal will be to derive graphical and

analytical procedures for estimating the optimal number of

primary and secondary sampling units.

ACKNOWLEDGMENTS

Robert Chapoton, Dean Ahrenholz, and William Nicholson

made valuable contributions to this project by acquainting me

with the realities and practicalities of the menhaden survey and by

acting as sounding boards for many of the ideas expressed in this

report. I would also like to acknowledge James Waters for the

many hours of discussion on sampling theory and optimal allocation. Many of his ideas are represented here and we hope to collaborate further on the project. Mayo Judy contributed to the

report by providing summaries from daily fishing reports used to

investigate random sampling assumptions. Finally, I thank

Charles Krouse for developing computer programs to estimate

sampling statistics from raw bioprofiles and catch information.

LITERATURE CITED

COCHRAN, W. G.

1977. Sampling techniques. 3d ed. Wiley, N.Y. 428 p.

GOODMAN, L. A.

1960. On the exact variance of products. J. Am. Stat. Assoc. 55:708-

713.

15

GULLAND, J. A.

1966. Manual of sampling and statistical methods for fisheries biology. Part

1. Sampling methods. FAO Man. Fish. Sci. 3, 87 p.

JUNE, F. C.

1972. Variations in size and length composition of Atlantic menhaden groupings. Fish. Bull., U.S. 70:699·713.

JUNE, F. c., and J. W. RE1NTJES.

1959. Age and size composition of the menhaden catch along the Atlantic

coast of the United States, 1952-55; with a brief review of the commercial

fishery. U.S. Fish Wild\. Serv., Spec. Sci. Rep. Fish. 317,65 p.

KELLEY, J. C.

1971 Multivariate oceanographic sampling. Math. Gen\. 3:43-50.

KENNEY, B. C.

1982. Beware of spurious self-correlations! Water Resour. Res. 18:1041-

1048.

NELSON, W. R., M. C. INGHAM, and W. E. SCHAAF.

1977. Larval transport and year-class strength of Atlantic menhaden,Brevoorria tyrannus. Fish. Bul\., U.S. 75:23-41.

NICHOLSON, W. R.

1'I7Ia. Coastal movements of Atlantic menhaden as inferred from changes in

agc and length distributions. Trans. Am. Fish. Soc. 100:708-716.

1971 b. Changes in catch and effort in the Atlantic menhaden purse-seine

fishery 1940-68. Fish. Bul\., U.S. 69:765-781.

1972. Population structure and movements of Atlantic menhaden, Brevovrtia

tyrannus, as inferred from back-calculated length frequencies. Chesapeake

Sci. 13:161-174.

POPE, J. A.

1956. An outline of sampling techniques. Rapp. P.-V. Reun. Cons. Int.

Explor. Mer 140: 11-20.

SAILA, S. B., R. A. PIKANOWSKI, and D. S. VAUGHAN.

1976. Optimum allocation strategies for sampling benthos in the New York

Bight. Estuarine Coastal Mar. Sci. 4:119-128.

SCHAAF, W. E.

1975. Status of the Gulf and Atlantic menhaden fisheries and implications

for resource management. Mar. Fish. Rev. 37(9):1·9.

1979. An analysis of the dynamic population response of Atlantic menhaden,

Brevoortia tyrannus, to an intensive fishery. Rapp. P.-V. Reun. Cons. Int.

Explor. Mer 177:243-251.

SCHAAF. W. E., and G. R. HUNTSMAN.

1972. Effects of fishing on the Atlantic menhaden stock: 1955-1969. Trans.

Am. Fish. Soc. 101 :290-297.

SCHAAF, W. E., J. E. SYKES, and R. B. CHAPOTON.

1975. Forecasts of Atlantic and Gulf menhaden catches based on the historical relation of catch and fishing effort. Mar. Fish. Rev. 37(10):5-9.

SCHWEIGERT, J. F., and J. R. SIBERT.

1983. Optimizing survey design for determining age structure of fish stocks:

An example from British Columbia Pacific herring (Clupea harengus pallasi).

Can. J. Fish. Aquat. Sci. 40:588-597.

SNEDECOR, G. W., and W. G. COCHRAN.

1967. Statistical methods. 6th ed. Iowa State Univ. Press, Ames, 593 p.

SOKAL, R. ·R., and F. J. ROHLF.

1981. Biometry. 2d ed. Freeman, San Franc., 859 p.

SOUTHWARD, G. M.

1976. Sampling landings of halibut for age composition. Int. Pac. Halibut

Comm. Sci. Rep. 58, 31 p.

SUKHATME, P. V., and B. V. SUKHATME.

1970. Sampling theory ofsurveys with applications. 2d ed. Iowa State Univ.

Press, Ames, 452 p.

TOMLINSON, P. K.

1971. Some sampling problems in fishery work. Biometrics 27:631-641.

TORTORA, R. D.

1978. A note on sample size estimation for multinomial populations. Am.

Stat. 32:100-102.

WILLIAMS, W. H.

1978. A sampler on sampling. Wiley, N.Y., 254 p.
APPENDIX

Derivation of Variance Formulae in Two-Stage Sampling for Age Proportions

According to Cochran (1977), the among set and within set

components of variance for the mean proponion of agej fish in a

two-stage design are:

S21

n

L (Pij - Pj)2

i=1

n-l

SET MS = S~ * m = (0.04195) (10) = 0.41945

S~ = S~ = ERROR MS = _,!!- L p .. q .. n(m-l) IJ IJ

10 (1.78) = 0.21975

(9) (9)

let Xij = 1 if in class j and 0 if not in class j

L L (X··-X? i j IJ

(m-l) (n)

(

U. m-u.) __, * , = L:r (X·· - X-)2 m m IJ I

S2 =_m_ LP.. q .. 2 n(m-l) IJ IJ

L L (X··-X·)2 m L p .. q .. = 1/ I

n(m-I) IJ IJ (m-l) (n)

this is true because when Xi = 1, Xi = 1

:. U i = ui·

(Ui)2 [U·J2 L [U·--- )= L (U~ __'_) I m I m

Proof that Cochran's (1977) formula for S~ is equal to the error

mean square (Sa,) of the ANOVA.

m L p .. q .. = L L (X .. - X·)2 = sum of each set's sum of squares IJ IJ IJ I

mL

The best way to appreciate these formulae is by utilizing the

device of coding fish in age classj as 1 and all other fish as 0 and

perform an ANOVA. For example in Table 3 for age 2 fish:

1 1 1 1 1 I 1 1 1

1 1 1 1 1 1 1 1 1

1 0 1 1 I 1 1 1 1

1 0 1 1 1 1 I 1 1

1 0 0 1 1 1 1 1 1

1 0 0 1 1 1 1 1 1

0 0 0 1 1 1 1 0 1

0 0 0 1 0 0 1 0 1

0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0

Pi2 .6 .2 .4 .8 .7 .7 .8 .6 .8

ANOVA table

Source df SS MS

Total 89 21.15556

Sets 8 3.35556 0.41944

Error 81 17.80000 0.21975

S2 L(PiCP;)2 0.33556 = 004195 I n-l 8 .

16