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predicting yelds and regulating a fishery. Without measures of statis-
tical variability for the component parts, there can be no estimates of
precision for the yield predicted by the whole model, and this may be
of considerable economic importance.

This sampling plan was designed to estimate the relative abundance
of ocean shrimp from trawl surveys in the Klamath River-Eedding
Rock bed off the northern California coast. The methods should be
applicable to other types of aquatic populations as well. Relative abun-
dance estimates will be used to compute mortality rates and also to
evaluate the validity of similar determinations from commercial fishing
data.



'Accepted for publication February 1968.

(257)
3—77541



258 CATiTFORXIA FISTT AXD r,A:\rE

We first give a narrative account of tlie sampling and estimating
procedures, then results from a survey, derivations of the estiinaling
formulas, and finally a discussion of sampling efficiency.

SAMPLING AND ESTIMATING PROCEDURES

A region which contains the population ut! interest must be delin-
eated. Since estimated relative abundance is a function of tlie size of
area sampled, this area will ordinarily remain constant from survey
to survey. Otherwise, special weighting procedures are required. We
assume that the sampled region will be divided into areal strata based
on some prior knowdedge of the population or for adnnnistrative con-
venience. However, simple random sampling over the entire region
can be carried out if there is no basis for stratitication. Stratum boun-
daries, unlike those of the region, can be changed from survey to
survey without aft'ecting the overall estimate of relative abundance.

Tlu' surve}' is accomplished by selecting random starting j)()ints for
the constantdength hauls (first-stage sampling unitsj within each
stratum. Haul directions are arbitrary. AVhen a haul is completed, the
catch is placed in containers divided into serially numbered compart-
ments. Tills procedure is carried out so that consecutively numbered
compartments (second-stage sampling units) are filled. A table of ran-
dom numbers is then used to select the subsample for complete enumera-
tion with respect to the characteristics of interest. Jn ordiu- to obtain
an estimate of the standard error, at least two first-stage units must be
selected from each stratum and one second-stage unit drawn from each
first-stage unit. If separate estimates of the variance components from
the first and second stages are desired, at least two second-stage units
are required from each first-stage unit.

The estimated catch of a single haul is the product of the re-
ciprocal of the second-stage sampling fraction and the sample mean per
compartment. There may, of course, be more than one sample mean
for each ha\d. dt'pending upon the number of characteristics being
estimated, ]^'lativ(^ abnndance (estimates for each stratum are the means
of the estimated haul catches; relative abundance estimates over the
entire survey area are weighted means of the stratum estimates using
stratum areas as the weights. Stratum and population variance esti-
mates are obtained from standard formulas for uidnased estimates of
the variance of a mean and the variance of a weighted mean. The
sample of haul starting points is assumed to be from an infinite pop-
ulation. Explicit formulations for the mean and vai"iance estimates
are given later.

While this procedure is sinular to two-stage sampling plans described
in standard sampling texts such as Sukhatme's (1954), it differs in th(>
treatmeid of the tii-st stage since we assume an infinite population of
sampling units.

RESULTS FROM THE FALL, 1965 SURVEY

The survey ai-ea of 269.9 square nautical miles was divided into 14
strata ranging in size from 11.2 to 32.4 square miles. Stratum boun-
daries were determined by examining commercial fishing logbooks and



ESTIMATION OF OCEAN SHRIMP ABUNDANCE 259

research vessel survey data with respect to the homogeneity of catches
in terms of pounds of shrimp per unit haul time. Four strata, consti-
tuting about 36% of the total area, are regions of historically small
catches and it was arbitrarily decided to apportion only one-fourth of
the hauls to them. These hauls were allocated among the four strata in
proportion to their areas and the remaining three-fourths of the hauls
were also proportionally allocated to the other strata.

Random haul starting points in units of seconds of latitude and longi-
tude were independently selected for each stratum. Since the use of a
random number table for the selection of points in the irregularly
bounded strata would have been a very lengthy procedure, a computer
program was written to produce the random haul coordinates. For
each stratum, the program generates pairs of random numbers, checks
them against the stratum boundaries, and prints a list containing the
desired number of haul coordinates.

Hauls were one-half mile in length with a 41-ft. semiballoon otter
trawl. The catches were placed in boxes divided into 80 compartments.
Second-stage samples consisted of two compartments except that when
only one compartment was filled it constituted the entire sample. Com-
puter-prepared random number tables of various ranges were used so
the subsample selections could be made rapidly.

One hundred and seventeen hauls were completed during the Fall,
1965 Survey. The proportional allocation described previously was not
achieved completely, since vessel time was a limiting factor and some
of the scheduled hauls were not made. Nevertheless, the unbiased prop-
erty of the estimates is not dependent on any particular allocation.

Mean-per-haul estimates of the number in age groups I through III
and of their combined numbers and weights were calculated, as were
the area-weighted means or relative abundance estimates over the en-
tire survey area (Table 1). The standard errors associated with the
relative abundance estimates are larger than one would like, consid-
ering the number of hauls involved. However, an inherent high vari-
ability seems characteristic of marine populations and usually there
is no remedy except larger samples. A discussion of the efficiencies
which could have been obtained under various allocation schemes occurs

FORMULATION OF THE ESTIMATORS

The notations and definitions which follow are in terms of the num-
ber of shrimp of all ages. Since the same procedure is used for esti-
mating any population characteristic of interest, the formulas apply
as well to the estimation of numbers or weights in individual age groups
or any combination of them. The object of our estimation, the relative
number of shrimp contained in a prescribed area, is Y ... = q¥ ... ,
where Y ... is the total number of shrimp in the area and g is a con-
stant. If we let q ■= ap/A, with a the mean area swept by the net dur-
ing a standard haul, p the mean proportion of shrimp caught from
the water column over the s\vept area of a standard haul, and A
the entire survey area, then Y ... is the relative abundance pertinent
to the survey procedure. It clearly must be assumed that a, p and A
are constants in order for the relative abundance to be useful. We also



260



CALIFORNIA FISH AXD GAME





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1-1



ESTIMATION OP OCEAN SHRIMP ABUNDANCE 261

must adopt the premise that the quantities being estimated remain con-
stant during the survey.
From the definitions,

Ah Area of stratum /;; h = 1, 2, . . . , L,

Yh.. Total number of shrimp in stratum h,
and

Wh = A,/A ,

Yh.. = apYh.. /Ah ,

L

A = ^Ah ,
h=i

L

Y... = SF,..,
we obtain

L

F... = 2 WhYn.. .

h=i



(1)



It is further assumed that the catch of the ith. haul in the hih stratum
is an unbiased estimate of Yh.. , or E{J\IhiYhi.) = Yh.. ,

where

Mhi Number of cells filled on the ith haul; i = 1, 2, . . . , Hh,

Yhi. Mean number of shrimp per filled cell on the ith haul,
Uh Number of hauls in the hth stratum.

This assumption will be clearer if we consider MhiYhi. as the product of
three mutually independent random variables, Y'hi , a, and p, with
expectations Yh../Ah, a, and p. Then E{MhiYhi.) = E{apY'hi) =
apYh../Ah = Yh..
The following definitions referring to the hih. stratum are also needed:

yhij Number of shrimp in the j'lh cell from the /th haul,
rrihi Number of cells in the sample from the ith haul,

_ . mhi

ijhi. = S ijhij ,

1 ^'' _

Vh.. = 2 MhiYhi. ,

Uh ,.^1

as is the result, £'(?/Ai. | Mhi) = Yh. .



262



CALIFORNIA FISH AND C.XSIF.



A logical choice for an estimator of Yh.. is the mean of the estimated
haul totals given b}'



y\



1



nh



rih



S MhilJhi.



1=1



Thai y'h is unbiased can be seen with the use of standai'd theorems on
conditional expectation (Parzen, 1902). Thus,



E{y\)



1



ni,



^h ,=1



2: E{MHiiiki. )



n,,



tih



S E[M,,E(yni. \ M^^]



1=1



= j: h.. .

(2)
From (1) and (2) it follows that //'. = S TF/,?/'/, is an unbiased estimator
of n.. .
The variance of the estimated relative abundance is given by



V{y'.) = -^ W>:-V{y'n)



3)



A=l



In developing the expression for \'{y'h) we assume that \' {yiuj \ -V/,,) =
Sihi"^ and V{MhiYhi.) = Sbh' • That is to say, cell observations have a
constant variance within each haul and haul totals have a constant
varian(;e within each stratum. Thc^ variance of the stratum estimate

y'h is

V{y\) = 7?(?/. - Yn.r^ ;

and this is separated into within- and between-haul compoiuMits by sub-
tracting and adding ///,.. before squaring to obtain

v(]j\) = eQ', - 7j,„y- + eQ,., - n.y-

+ 2E[(7j', - 7,,„){y,„ - }\„)] .

Using theorems on conditional expectation and conditional variance it
can be shown that



Eiy'n- y,..)' = ^ E

nh



AIhi{Mhi — nihi)



nihi



Sihi^



E(yn.. - Y,.y = -— S,,r ,



ESTIMATION OF OCEAN SHRIMP ABUNDANCE



263



and

E[{y'u-']jn.){yZ. - Yn..)] = .
TIkmi,



vG'.) = -^ E



MhijMhi — mh,



S



Ihf



+ -^ S,n' , (4)



nh



where the within- and between-hauls components are the first and second
terms on the right-hand side respectively.

To estimate Viy'h) we use



Sth



2 — y



i=l



{MuiVhi. — y'h]



The unbiasedness of Sa^ can be seen by expanding and taking the ex-
pectation of its nmiierator to obtain,

nh _

E[{n, - l)sa;-] = S E[Mui'E{y,^.' \ M,,)] - nnE{y\^)



= TinE



Mhi{Mki — nihi) i, , . ^i,T ^ N,9
^ihr + (.iiJAi-t hi.) I



nihi



- n,[V{y'n) + Y,..'] .
Substituting from (4) and simphfying gives the desired result,

E (^) = V{y',) .
\ nil /

It follows that an unbiased estimator for V(y' .) is

A L



Viy'^ =






h=l



nh



(5)



An unbiased estimator for the within-hauls component of Viy'h) is



Sivh-



rih Uh'



■^ Sihi ,



i=l



nihi



where



Sihi" —



mhi

2 (yiiij — yiii.)'
1=1

nihi — 1



264 CALIFORNIA FISH AND GAME

The uiibiaiscd cliaiactcr of s,,/." can l)e vcrilicd \)y applying ihc prm'iuusly
mentioned theorems on conditional expectation, and an unbiased es-
timator for the between-hauls component is clearly

O O

Sbh" _ Sih-^ ^ Suh'

Uh iih nh

ALLOCATION OF THE SAMPLE

The initial surveys were carried out with modified proportional al-
location at the first stap'o and a constant sample size of two at the
second stage, as mentioned previously, since no prior information on
first- and second-stage variance components was available.

Here we w'ill examine several methods of allocating sampling be-
tween strata and botwoen the two stages: proportional sam])lin<:' at both
stages, proportional sampling at the first stage with a constant sample
size at the second stage, and unrestricted first-stage sampling Avith a
constant within-stratum sample size at the second stage.

For any given type of allocation it is necessary to determine the
distribution of sampling effort between the stages so that the variance
will be a minimum for a fixed cost, or so that the cost will be mini-
mized for a fixed variance. Once these determinations are made, the
efficiencies of the allocations may be compared.

A simple linear cost fuiiclioii suitable^ for 1lu> Irawl surveys is

L L

C = Ci S rih + fo S nniuik. - I) , (6)

where

C total cost of the survej^,

Ci cost of sampling one first-stage unit (haul),

C2 cost of sampling one second-stage unit (cell),

???/,. mean second-stage sample size in stratum //.

The second term on tlieriglil contains {mh. — 1^ I'alher than ////,, l)e('ause
one second-stage unit can be processed at \irlually no additional cost
while the next haul is in progress. It also should be noted that C does
not include costs invoK'ed in vessel travel to and from llie surx'ey area
nor costs incurred when I he vessel caimol operate du(> to rough \v(>athei';
such fixed costs have no eifcct on ilie allocations between stages.

Because in this case it is much easiei- to calculate costs for a fixed
variance than the converse, we have set up the following function from
(3), (4), and ((>) with a constraint on tlu^ variance. Sanipl(> sizes are
obtained by minimizing

L

F = ^ [Uhic, - Co) -f n^h.c:} (7)



ESTIMATION OP OCEAN SHRIMP ABUNDANCE



265



+ A



2 W,



( nh L



MhiiMhi - nihi)



nihi



Q 2



+ Sbh'



V



where X is a Lagrange multiplier and V is the fixed variance.

To determine the allocation with the first-stage sample proportional
to the stratum size and with the second-stage sample size a constant, we
let A^i = Uh/Wh and fcz = nihi in (7). Then solve dF/dki = 0,dF/dk2 = 0,
(3), and (4) for ki and k^. This procedure yields



k. =



N







L








(c


1


c-^ 2


W,E{Ah


■'S,k


^')




L










C2


2


Wn[S,n'


- E{Mn


iSllr,


')]



(8)



and



A-i = y I 2 W,[S,h' - EiMHiSvn') + E{I\h^Sui')/k,] } . (9)

The allocation with proportional sampling at both stages is found by
letting /ci = Hh/WhSindko = 7H/,//il//,, in (7) and (4). Solving 3F/3A-1 = 0,
dF/dk. = 0, and (3) gives



k2 =



A



(ci - c^) 2 WnEiMmS^H^)
h=i



L L

c, 2 WhMh. 2 W,[S,>r- - E{M,,Svu') ]



(10)



A=l



h=l



and



'^1 — "p"



1 - ko



2 W,E(MniS,Hr) + 2 W.Stk'



h=l



h=l



(11)



When no restrictions are placed on first-stage sampling and the second-
stage sample size is constant within strata, let A'o;, = mi,i in (7) and solve
dF/drig = 0, dF/dk^g = 0, and (3) to obtain sample sizes for the gih.
stratum. The solutions are



^"20 —



(Cl - C,)E{M„,'S^gi')
C2[Sf>g' - E{M„^S,g^)]



(12)



266
and



CALIFORNIA FISH AND GAME



rio



= El / C2H, 2^ hnWnHn



V \/ ci — fo + ^2/1



(13)



■0 h=i



Vl^Ahi'Sy,^)



whoro






- A\J/;,„S,„r)



By ciilculat in;^' what the Fall, 1I»()5 Survey wuukl liax'e cost undrr
each of these three types of allocation for the precision actually ob-
tained, we may compare the relative efSciencies of the samplinf?
schemes. Altliough five quantities were estimated from the survey and
each may be associated witli a different total cost for the attaiiietl ])re-
eision, a cost calculation for tlie number estimates in age groups I
and II should suffice for comparative purposes. Of course, in actual
practice one would try to provide sufficient sampling funds so that the
standard error of tlie estimate with the lowest precision was less than
a predetermined maximum.

The cost formula was evaluated w^ith Ci^ $125.00 and Co = $0.58.
Vessel operation costs and the wages of scientific pei-sonnel on board
tlie ship are included in Ci, while c-2 is the cost of processing one sec-
ond-stage sample ashore. Applying formulas (6) througli (13) to data
from the Fall, 1!)()5 Survey gave total costs and values yielded by A'l
and k2 (Table 2) and by n^ and A'or; (Table 3). In the cost calculations,
the actual estimates of ki and k^ were used for the allocations whieh
employ proportional sampling at the first stage. Costs of tlie allocations
with unrestricted first-stage sampling were eoiii]iuted with //_. and /. - u
rounded upward to integers.

TABLE 2

Values of /cj, k2 and C From the Fall, 1965 Survey for the Two Plans With
Proportional First-stage Sampling to Yield the Precision Actually Attained



Proportional
second stage


Constant
second stage


Ace erouD _ _ _ _


I


II


I


II






ki


132.0


105.2


132.7


107.5






ki


0.018


0.053


2.3


8.6






C - - - -


$16,530


$13,338


316,689


§13,913







For the Fall, 1965 Survey, C was $14,683. With proportional allo-
cation at both stages, the total cost would have been $16,530 or $13,338
in order to estimate the relative numbers in age groups I or II, re-
spectivelj', at the same levels of precision actually obtained. Also, for
the same precision, the allocation with proportional sampling at the
first stage and a constant second-stage sample size yielded C's of $16,-
689 and $13,913, while the plan using unrestriced first-stage sam-



ESTIMATION OF OCEAN SHRIMP ABUNDANCE



267



TABLE 3

Values of nn, k^t? and C From the Fall, 1965 Survey for the Plan With Unrestricted

First-stage Sampling and a Constant Second-stage Sample Size

to Yield the Precision Actually Attained





Age Group I


Age Group II


Stratum


Jig k^a


Tig kzg


1... .


2 9

3 4
11 1
15 3

6 2
10 1

4 4

1 4

7 3

2 3
2 4
2 4
6 6
2 1


3 7


2


2 12


3 .


5 3


4 ...


7 8


5


3 10


6


13 10


7 ...


2 12


8


1 8


9_ -


4 7


10

11


1 13

2 5


12

13 .


2 12

4 6


14 .


3 12






Totals


72


52


C


.?9,078


S6,727



pling and constant second-stage sampling within strata could have been
carried out for the relatively low costs of $9,078 and $6,727.

It is notable that the cost formula, when applied to plans which use
proportional allocation at the first stage, gives costs somewhat higlier
than tlie actual expense of the Fall, 1965 Survey for age group I and
costs lower than those of the survey for age group II. However, because
the sample sizes involved in estimating both the first- and second-stage
variance components were rather small, the differences between the cost
of the actual survey and the calculated costs of the two plans using
first-stage proportional sampling probably are not significant. The plan
with unrestricted first-stage sampling is clearly more efficient than
either of the projDortional allocation schemes, as evidenced by its sub-
stantially lower cost figures.

Examination of equations (8) through (13) reveals that all of these
allocation schemes require an advance knowledge of the first- and sec-
ond-stage A'ariance components. The sample sizes of the plans that
allocate proportionally at the first stage depend mainly on relative
magnitudes of the first- and second-stage variance components aver-
aged over the strata. The more efficient plan with unrestricted first-
stage allocations, however, requires for each stratum a knowledge of
the first-stage variance components, as M'ell as of the mean of the sec-
ond-stage components. The practicality of tlie unrestricted or optimum
allocation tlien depends on the ability to predict these witliin-stratum
variance components.

Unfortunately, prediction of the variance components needed for
optimum allocation seems virtually impossible in the case of the ocean
shrimp surveys. If we let fi/,, denote the fraction of C which would have
been expended on the 7th stage of the h\\\ stratum during the Fall,



268



CAI.IFORXIA FISH AND GAME



1965 Survey and lei f-jhi repre.sent llie currespoiuliii^f quantity from the
Spring, 3 966 Survey, then {fsju — fnn)/fm estimates tlic rehitive chanofp
in sampling costs between the two surveys for stage i of stratum A in
the plan with first-stage optimum allocation. A perfect prediction
would yield a value of zero for the relative change. Kelative changes
in sampling costs for age group I between the Fall. 1965 Survey and
the Spring, 1966 Survey range from —10.675 to 0.863 (Table 4). The
jiuNins of the absolute values of the relative changes are 1.65 and 2.14
for the first and second stages, respectively ; the change in the first-
stage allocation is, of course, the more serious in terms of cost and
precision. An examination of data from the surveys indicated that the
variance components from individual strata continued to be quite
variable and not useful for predicting optimum sampling allocations.



TABLE 4

Relative Changes in Sampling Costs Between the Fall, 1965 and the Spring, 1966
Surveys for the Plan With Unrestricted First-stage Allocation



Stratum


1


2


3


4


5


6


7


First stage

Second stage


+0.314
-0.447


-3.579

-3.579


+0.751

+0.257


+0.725
+0.451


-1.199
-5.531


+0.863
+0.594


+0.657
+0.833


Stratum


8


9


10


11


12


13


14






First stage

Second stage


-0.372
-0.040


+0.804
+0.612


+0.314
+0.553


-1.748
-6.510


-10.675
-4.804


+0.771
+0.694


+0.314


1 2 4 6 7 8 9 10

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