Onsager1976_SeqSamplg

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THE RATIONALE OF SEQUENTIAL SAMPLING, WITH EMPHASIS ON ITS USE

IN PEST MANAGEMENT

o Technical Bulletin No. 152^

fV3

I CO >

CO N>

Agricultural Research Service UNITED STATES DEPARTMENT OF AGRICULTURE

in cooperation with

South Carolina Agricultural Experiment Station



Washington, D.C. ssued March 1976



CONTENTS Page

Abstract 1 Introduction 1 Mechanics of sequential sampling 2 Sampling from a binomial distribution 8

Formulas for sequential plans 8 Binomial example 9

Sampling from a Poisson distribution 0

Formulas for sequential plans 1

Poisson example 2 Sampling from a negative binomial distribution 3 Formulas for sequential plans 3 Negative binomial example 4

General considerations 5

Literature cited 7 Appendix.—Definition of symbols 8

Illustrations Fig.

1. equential sampling chart 3 2 . perating characteristics curve for sequential sampUng

plan 6

3. Average sample number curve for sequential sampling plan 7

Table 1 . artial probability distribution of expected results

of sampling when p'=0.5, p^=0.2, p2=0.8, and a=ß=0M882S



THE RATIONALE OF SEQUENTIAL SAMPLING, WITH EMPHASIS ON ITS USE

IN PEST MANAGEMENT By J E RO ME A. ONSAGER, entomologist, Agricultural Research SeTuice, U.S. Department ofAgriculture, Charleston, S.C.^

ABSTRACT The rationale of sequential sampling is described in nonmathematical

terms. Formulas are given for dichotomous sampling plans based on the

binomial, Poisson, and negative binomial distributions. Solved examples are given fo r each type of distribution, and some applications to sampling for insect pests are discussed.

INTRODUCTION Knipling (3)^ has discussed pest management programs designed to

maximize the role of natural biological agents and minimize the use of insecticides in maintaining pest populations below economic threshold

levels. The concept relies on critical monitoring of both destructive and beneficial insect populations and presupposes that the presence or absence of critical densities can be accurately detected. Since emphasis is essentially on proper classification of populations, rather than on estima- tion of population parameters, sequential sampling is eminently appropriate to present-day pest management programs (10),

According to Wald (9), the concept of sequential sampUng was brought

to fruition in 943 under wartime military contract for use in quality

control. The savings realized from sampling sequentially were so significant (often exceeding 5 0 % ) and the potential for other uses was so obvious that the information was released to the pubUc in 1945. A decade later. Waters (10) observed that the application of sequential sampling to biological problems had been somewhat limited and gave an excel- lent discussion of the advantages, applications, basic requirements, and essential features of sequential sampling plans in relation to entomologi- ca l problems. While the use of sequential sampling has subsequently

proliferated, its maximum potential will not be realized until all those

who could use it are in full command of it .

^Present address: Rangeland Insect Laboratory, Montana State University, Bozeman 59715.

^Italic numbers in parentheses refer to items in Literature Cited, p. 7.



2 ECHNICAL BULLETIN 526, U.S. D E P T . O F AGRICULTURE

Ordinary sampling methods usually require a fixed number of samples that will estimate a key density with some desired level of precision. However, the specifiednumber of samples will invariably be inadequate

for constant precision at low densities and excessive fo r high densities. Sequential sampling is designed simply to choose between tw o alterna-

tives (i.e., is the density higher or lower than a critical level?). Sampling continues only until results dictate that one alternative is more likely true han he other at some constant acceptable evel of probability.

When densities are either very high or very low, the most likely alterna-

tive quickly becomes apparent. Thus, while the tw o methods m ay have

similar sampling requirements within moderate population densities,

the sequential method is much more efficient over the entire range of possible densities.

I believe hat he apparent mathematical complexity of sequential

sampling has unfortunately nhibited ts use n entomology. he

rationale behind sequential plans is actually quite simple, but the expla- nations in literature are mathematical and therefore seem complicated to he reader who s not a mathematician. urthermore, here s a tendency fo r each textbook on the subject to contain a unique approach, a unique set of symbols, and a unique variation of common formulas,

omitting at the same time helpful information that has been exploited elsewhere. While this lends an air of originality to each discourse and contributes to conciseness, it annoys and confuses beginning students of sequential sampling.

The intent of this bulletin is to bring together some of the information now scattered in previous treatments of the subject and present it in the

simplest possible mathematical erms. omprehension will equire

some facility in the use of common logarithms (log to the base 10) and familiarity with the point-slope and slope-intercept forms of the general

equation of a straight line. The mechanics rather than the mathematics

of sequential sampling will be emphasized, however, and methods for developing dichotomous sampHng plans will be given fo r three distribu-

tions commonly encountered in sampling fo r insects: the binomial, the

Poisson, and he negative binomial. Unfortunately, his requires yet

another unique approach, unique set of symbols,^ and a unique variation

of common formulas. It is hoped that such means can be used to make sequential sampling easier to understand.

MECHANICS OF SEQUENTIAL SAMPLING All dichotomous sequential plans begin with the establishment of tw o

alternative hypotheses, here designated H i and Hg. Consecutive samples are examined and evaluated until cumulative results dictate that

^ A U symbols used in this bulletin are defined in the appendix.



RATIONALE O F SEQUENTIAL SA MPLING one hypothesis s more ikely o be correct han he other at ome preestablished degree of reliabihty.

With the help of figure 1, let us develop a trivial example based on the

binomial distribution. Suppose we are examining ears of sweet corn for the presence of corn earworm larvae. W e expect to find no larvae in noninfested ears, and exactly one larva per infested ear, because the

larvae are cannibahstic. Let us assume that (1 ) lots having 2 0 percent or fewer infested ears are good lots, and it is very important that they be so classified; (2 ) lots having 8 0 percent or more infested ears are badlots, and it is very important that they be so classified; and (3) lots having

more than 2 0 percent but less than 8 0 percent infestation can be classified as either good or bad without serious consequence.^ Then, for any given lo t to be sampled, we state, H i is that the true proportion (p') of infested ears is equal to or less than 0.2 ; or, more concisely, H j: p'^0.2. (H i always contains the lowest level of whatever we are sampling for.) Likewise, Hg: p'^0.8.

Intuition tells us that whenp' is extremely lo w or extremely high, our samples will yield a preponderance of noninfested or nfested ears,

respectively, and will quickly lead to acceptance of H i or Hg, respec-

tively. However, f we are examining a lo t for which p' s exactly 0 .5

(halñvay between he pecified imits H j and H2), t s possible or

'^The reader should not feel disturbed about the wide range between H , and H2 in this simple example. It will later become apparent that H , and H2 can be set with any practical difference between heir pecified imits. The given evels are imply onvenient or subsequent discussion.

X-RttULTt THAT OtCIATt AOOITIONAL SAMPLtt CXAMPLCt THAT OICTATI ACCEPTANCI or .

-, 2 3 5 6

I 8 9 10 il 1 2 13 1 4 1 9 16

TOTAL UMBER F AMPLES XAMINED (X ) FIGURE 1.—Sequential sampling chart.




several consecutive pairs of samples to each yield one infested and one noninfested ear. This could theoretically continue indefinitely (fig. 1), so the only way our results could approach H i or H2 would be through

obtaining by chance an overabundance of noninfested or infested ears, respectively.5 Let us therefore agree to sample by examining pairs of ears chosen at random from a lo t and to pause after each pair to evaluate

results. Let us further agree that (1 ) if neither ear is infested, we shall accept H i (that is , we will stop samipKng and classify the lo t as good ); (2 ) if both ears are infested, we shall accept Hg (that is , we will stop

sampling and classify the lo t as bad ); and (3) if only one of the ears is infested, we shall withhold judgment and examine another pair of ears

chosen at random.

Figure 1 shows four of the possible sample points that dictate acceptance of a hypothesis. Given these points, we can calculate the lope- intercept equation, Y=a-^bX, for tw o parallel lines which will intersect

all possible sample results that will yield a decision to stop sampling. The point-slope form of the equation defines slope (b ) as the change in Y per

unit change inZ. For H i, 6-(4-3)/(6-4)=0.5. Since the lines are paral-

lel, b for H2 is also 0.5. Given b and any point on a line, the F-intercept (a) is calculated as a=r-6Z. For H i, ai=-l; fo r H2 , a2=+l. With the solid

lines in figure 1 fo r guidance, it is no longer essential that we examine a pair of ears before we pause to assess results. However, we m ay note

that it is impossible to reach a decision on an odd-numbered ear in this

example, o nothing would be lost by continuing to examine tw o at a time.

Clearly, our sampling plan is not devoid of error. It is entirely possible to draw a pair of infested ears and accept H2 when H i is true. This is often called a type I error, and the maximum allowable risk of its occurrence is referred to as alpha (a). Likewise, we could accept H i and H2 is true.

This is often called a type II error, and its maximum allowable risk is referred to as beta (ß).

The ignificance of a and ß can be llustrated by calculating these

values or our example. To make uch calculations, om e additional symbols will be helpful. Let us designate the specified Hmits of H i and H

as pi and P2 , respectively. The quantities (1-pi) and (l-pg) are designated Q i and q 2 y respectively. Thus, n our example, Pi==0.2, qi=0.8 P 2 = 0 . 8 , and (72=0.2.

2

^The perceptive reader will here declare that we are relying on sampling errors to steer

us to on e of tw o classifications, neither of which is technically correct. This observation is absolutely true. W e plan to capitalize on the fact that sampling errors are governed by the

laws of probability. W e are completely agreeable to making incorrect classifications in borderline cases where there is no serious consequence. Such error is inevitable in any decisionmaking process, an d the concession protects us ro m making serious errors n classification.



RATIONALE O F SEQUENTIAL SAMPLING Wh en the rue proportion of infestation is exactly as the pecified

limits of H i or H2 (that is , when p'=pi orp'=p2), the associated p is the

binomial probability that any given ingle ear drawn at random will

contain a larva; the associated q is the probability that the same ear will not contain a larva.

If p'=Pi, he probability of finding a larva in each of tw o random

ears—which would lead to incorrect acceptance of H2—ispi^ or 0.04 (the

logic is akin to calculating that the chances of observing consecutive heads in tw o flips of a coin is {V^y^^V^). Likewise, the probability of no

larva in tw o ears is Ç i or 0.64, which would lead to correct acceptance of H j. Since a m ay be estimated as the ratio of incorrect classifications to total classifications, a=0.04/(0.04+0.64)=0.058823. Obviously, asp' becomes progressively less han pi, he probability of accepting H2 becomes progressively less than a; the converse is also true.

If p'=P2, he probability of a larva in each of two ears and correct

classification is p2^=0.64. The probability of incorrect classification is ^^2=0.04, o 8 = 0 . 0 4 / ( 0 . 0 4 + 0 . 6 4 ) = 0 . 0 5 8 8 2 3 . It s not essential hat

a = j 8 ; however, a-\-ß must be less than 1 .) It can also be demonstrated that a and ß are constant, regardless of

the number of samples required to yield acceptance of H j or H2. For

example, if p =P2, the probability of finding three larvae in four ears and correctly accepting H2 is P2^q2=(P2Q2)(P2^)=(^ 16)(0.64). The probability

of one larva in four ears and incorrect classification isp2Q'2^ = P 2 Q 2 ) Q 2 ^ ) = (0.16)(0.04). The quantity P 2 g 2 , or any equivalent value derived f i ^ - o m a different number of samples, ill always ancel out n he division process, leaving a ratio of 0.04/0.68 as before.

Figure 1 has now become an empirical sequential sampling plan with

H i: Pi^O.2, H2 : P2^0.8, and a = ) 8 = 0 . 0 5 8 8 2 3 . At this point we are ready

to examine the plan to assure ourselves that it is feasible. This requires

calculation of an operating characteristic OC) curve and an average sample number (ASN) curve.

The typical OC curve (solid line in fig. 2 ) gives the level of probabiUty (LP) of accepting H i fo r any givenp'. Four of the points on the OC curve

are always known. 1 ) Wh en p'=0, P=1; hat s, n our example, accepting H i s 00 percent certain when ll ears n a lo t are ree of earworms. Likewise, (2 ) whenp' = l, LP=0. (3) Whenp'=p2, LPhasby definition been set at ß. Likewise, (4) whenp'=pi, (1-LP) has been set

at a o LP=(l-a). Completion of the OC curve essentially involves proper interpolation between these known values. The inverse of the

typical OC curve (the broken Une in fig. 2) gives the LP of accepting H2 for any given p'. This inverse curve is seldom plotted but always under-

stood to exist. The ASN curve (fig. 3) gives the number of samples that will, on the

average, dictate acceptance of either H i or H2 for any givenp'. W e must



6 TECHNICAL BULLETIN 526, U.S. D E P T . OF AGRICULTURE

now develop an A SN curve to determine whether or not our plan is feasible. If the A SN dictates examination of several bushels of ears per

decision, we m ay be forced to either relax our standards or abandon the

idea of sampling altogether. Conversely, f the sampling of only a fe w ears per decision is required, we m ay either accept the plan in its present

form or decide that we can afford gi-eater accuracy. The A SN is reduced

at the expense of accuracy by increasing a, / 3 , or the interval between H j and H2 . Conversely, accuracy is increased at the expense of additional sampling requirements by reducing a, / 3 , or the interval between H j and H ,.

Two of the points on our A SN curve m ay be determined by inspection of figure . W h e n p'=0, he only possible number of samples s . Likewise, when p' = l, he A SN miust also be 2 . Other points are not

found so easily, but intuition tells us that the maximum A SN will occur whenp'=0.5, because the example has to this point been symmetrical in all aspects. Table 1 gives the expected distribution of results possible when p =0.5. Notice that the probability of accepting either H j or H2 after examining any pair of ears equals the probability that we will have to examine at least one additional pair of ears. Thus, 50 percent of the

time we can stop after tw o samples, 2 5 percent of the time after four

T 0 J 0 2 0 3 0 4 0 5 0 6 0 7 08 0 9 1

n P t TRUE R OP OR TION tp 1 O F NFESTATION

FIGURE 2.—Operating characteristics curve fo r sequential sampling plan.



RATIONALE OF SEQUENTIAL SAMPLING

TRUE ROPORT ION pi OF NFESTATION FIGURE 3.—Average sample number curve or sequential sampling plan.

samples, and so forth. Obviously, as we approach an infinite number of samples, he probability that additional amples will be required approaches zero. If we attempt to average all possible sample numbers

weighted by their probabilities, i.e.,

(2)(0.50)+(4)(0.25). + (^)(0) 0 . 5 0 + 0 . 2 5 . +0

we will find that the numerator approaches 4 while the denominator

approaches 1 as the number of samples approaches infinity. Thus, we say that the limit of the above equation is 4, which is hen the ASN or p'=0.5. Stated otherwise, f we were o ample ears rom a million different lots, each of which was exactly 50 percent infested, we would expect o examine a total of 4 million ears n the process. It can be confirmed by similar calculations fo r other values of p' that 4 is indeed a maximum on the A SN curve for our example.

TABLE 1 .— Partial probability distribution of expected results of sampling when p'=0.5, Pi=0.2, P 2 = 0 . 8 , and a=jÖ =0.05 8 8 2 3

Expected result Number of samples examined

0 1 0

Accept H, Keep sampling Accept H2 0 25

50 25

0.125

. 2 5 0

. 1 2 5

0.0625

. 1 2 5 0

. 0 6 2 5

0.03125

. 0 6 2 5

. 0 3 1 2 5

0.015625

. 0 3 1 2 5

. 0 1 5 6 2 5



8 ECHNICAL BULLETIN 1526, U.S. D E P T . OF AGRICULTURE

Now, to relate our example to real life. n practice, the steps are as follows: 1 ) The parameters by which populations of an insect can be estimated are determined and (2 ) tolerable and intolerable population

levels that can be set as limits of H j and H2 are determined. These are logical tasks of the research scientist. Next, (3) a and ß are set arbitrar-

ily at acceptable levels of risk for committing a type I and type II error,

respectively. deally, this is not the responsibiUty of the scientist, but

should be done by parties that are affected by the decision; e.g., he

farmer, he manager of a couting program, he company hat has contracted to buy the crop, etc. Then (4) appropriate general formulas (which are given ater or binomial. oisson, and negative binomial distributions) are used to calculate a^, a^, and 6; OC and A SN values are

calculated and evaluated; and, if necessary or feasible, H j, Hg, a, or ^ 0 are adjusted to yield an adequate compromise between accuracy and practi-

caUty. Finally, 5) a sampling chart like figure or a table containing equivalent information is prepared by using the formulas y^=a^^bX and y2=a2+bX to determine the points of acceptance fo r H i and H2, respec-

tively. Now, subject to certain considerations mentioned later, we are

ready to sample in the field.

SAMPLING FROM A BINOMIAL DISTRIBUTION The binomial distribution applies when random samples from an infi-

nite population (o r at least a very large one) fall into one of tw o categories

(e.g., units are either ound or square, ale or emale, nfested or

noninfested, etc.). The parameter of interest s p, he proportion of samples that bear the attribute of interest. The proportion that lacks

that attribute is q; obviously then, q = l-p.

Formulas for Sequential Plans Given a binomial distribution, H j p'=$pi, H2 ' ^2, and acceptable

values for a and ß, necessary calculations follow the general formulas below, all of which are variations of or derivations from Waters' formulas (10),

n t e r c e p t s

log a^=-^ ß

lo g Pñi \P1q2



RATIONALE OF SEQUENTIAL SAMPLING

a.r

Slope:

- hlog JM±

log

log MLL PlQ2

Binomial

OC curve.—It

has already

been

given

hat

when p'=0,

LP=1; whenp'=Pi, LP=l-a; whenp'==p2, P=iS; and whenp' = l, LP=0. Also, when p =b, LP^as/Cag-aj). hese are he only values

essential or crude evaluations of tentative sampling plans, since he

rest of the O C curve can be estimated by nterpolation. or any pbetween 0 and 1, the exact LP and the associated ASN can be calculated

by an implicit method through use of a '*dummy variable (10) . This is not a simple task unless it is done by a programmable calculator or a

computer, however.

Binomial ASN curve.—Wald (9 ) reported that the weighted-average method we used earlier to calculate ASN's is usable only under certain

rigid conditions. To circumvent these Hmitations, he developed an alternate method that yields what he termed arbitrarily fine approxima-

tions of values that cannot be calculated exactly. The general formula is

ASN= LP(ai-a2)+a2

p -b By substitution, we observe that whenp' = 0 and LP=l,ASN=ai/-6.

Likewise, whenp' = l and LP=0, ASN^a.Jil-b). We also observe that

the general formula is not valid when p =b because we cannotdivide by

zero. However, Wald 9 ) has shown hat he ASN when p'=6 s he

product of the ASN's at p =0 and p = l; hat is, when p =b, ASN= (^1X0^2)

-b a-b

The ASN atp b will be maximum ASN when a=^; it will also suffice as

an estimate of maximum ASN when a and ß are similar.

Binomial Example While all constants in the trivial example could be verified here, let us

rather develop a more significant hypothetical example. Assume that an

infestation of aphids on alfalfa has been classified as ''threatening and

that wewant to assess the potential of a population of chalcid parasites.



10 TECHNICAL BULLETIN 1526, U.S. D E P T . O F AGRICULTURE

Previous experience has established that (1 ) aphid populations are apt

to increase, and that some remedial action is required if parasitism is 8 percent or ess, 2 ) aphid populations are ikely o be reduced f 3

percent or more of the aphids exhibit symptoms of parasitism, and (3 ) the balance is precarious when parasitism is between 8 and 1 3 percent.

Thus, Hi p'^0.08 and H2 p'^0.13. If Hi is true we definitely want to

be alerted to that fact, so let us set a at 0.05. If Hi is not true, it will not

be an extremely serious mistake to incorrectly assume it is true because

we retain the option of correcting the error in subsequent monitoring.

Let us therefore set ß at 0.15. Calculations may proceed as follows:

P , = 0.13)(0.92) 0.1196 1 7 1 8 3 9 1

P1Ç2 (0.08)(0.87) 0.0696

log

a, =

/0.95\

\0.15/ =. 0.801632 3-^^.409

log 1 . 7 1 8 3 9 1 0.235122

ao

lo g /M5^\

lo g 1.71839r 0. 2351 22

lo g /0.92\

\0.87J = 0 . log 1 . 7 1 8 3 9 1

0242686^=0.1032 0.235122

PLP

Formula Value

ASN

Formula Value

0 PI

b V

1

Definition 1-a

(tta-ai)

ß Definition

1 .9 5

. 6 0 5 5

.1 5 0

aj-b General

(ai)(a2)

(-0X1-6) General

aja-b)

33.0 1 2 8 . 2

192.8

147.0

5.8

A true maximum ASN of 1 9 3 . 7 was determined by several iterations

of the implicit method. The above value forp'=6 is obviously an excel-

lent approximation of the maximum ASN.

SAMPLING FROM A POISSON DISTRIBUTION

The Poisson s a distribution n ts wn right, ut s most easily

visualized as an approximation to the binomial when p is very small and



R A T I O N A L E O F SEQUENTIAL SA MPLING the number of samples is large. W e therefore assume a random distribu-

tion of insects and expect a preponderance of samples to be devoid of insects. Unlike the binomial, however, the Poisson distribution allows the presence of tw o or more insects per sample. The required parameter o f the Poisson is the mean, which is usually designated as lambda (À).

Formulas for Sequential Plans The mechanics of the Poisson sequential sampling plan is similar to

that of the binomial plan. However, the mathematics differs, because Poisson probabilities are more complex. Given a Poisson distribution, H j : X'=^Xi, H2 y^X2, and acceptable values fo r a and / 3 , necessary calcula-

tions follow the general formulas below, which are variations of formulas by Waters (10), Intercepts:

lo g a, =- (Y ) tto

lo g 4r)

Slope:«

f , = 0.4343(\2-\i)

lo g

Poisson C curve,—Obviously, hen K =0, P==1; hen X ' = X i , LP=l-a; and when X = X 2 , LP==^. Wh en k =b, LP^ag/Cag-aj), as for the binomial distribution. These four LP values will suffice for crude evaluation of sampling plans. Exact LP's and ASN's for any X again may be calculated by an implicit method (10).

Poisson ASN curve.—The general formula is

ASN- LP(ai-a2 )+^2 X'-6

By substitution, when X=0 and LP=1, ASN=a,/-6. W h e n k =b, the

^The constant 0.4343 s /ln 0, onversion hat llows use of the more amiliar logarithms to th e base 10, rather than the natural logarithms thatWaters (10) specified.




general formula again is not valid, but the A SN can be calculated as (ai)(a2)/-6. Thus, four points on the Poisson A SN curve m ay be easily calculated.

The maximum A SN will be very close to one of the three ASN's for X ' = 0 , X ' = X i, orX'=è. Exactly which of the three ASN's will be greatest

varies with the levels of a and ) 8 , and with the magnitude of X 2 relative to X i. Experience has shown that when a and ß are 0.05 or greater, and when 2 does not exceed 0 X i, he ighest ASN calculated by he

formulas above will be at least 9 4 percent of the true maximum A S N .

Poisson Example Consider asequential plan for detecting the presence or absence of an economic infestation of Pacific Coast wireworms in soil to be planted to

potatoes. The distribution of larvae in soil cores having a surface area of one-fourth square foot and a depth of 9 inches is Poisson, an average of 0. 022 larva per core will cause no significant damage, and band treat-

ments with chemical insecticide become economical when larvae average

0.030 per core ( Ö , 7 ). Let us arbitrarily set a=0.4 and ) 8 = 0 . 1 . Then, H i X^Xi=0.022, and H2 X ' ^ X 2 = 0 . 0 3 0 .

ai

a y=

V 0.022/

lo g 6 _ 0 . 7 7 8 1 5 1 _5 ^g lo g 1.3636 0.134698

^ lo g 2 . 2 5 ^ 0. 3521 82 ^g 6 1 0.134698 0.134698 0.134698

¿ 0.43 43 )(0.03 -0.022 ) 0 2 5 8 0.134698

\LP SN

Formula Value ormula alue 2240

637.9 584.7 4 2 1 .7

The exact maximum A SN was determined by several iterations of the

implicit method to be 639. This dictates examination of up to nearly 4.5 cubic yards of soil per decision. Clearly, this plan is not feasible, and no reasonable manipulation of H,, H j, a, or / 3 will make it so .

0 Definition

X , 1-a h dilißi—d^ X, ß

aj-b .6 0 General

.3 1 1 1 {a^)(a<^l-b

.1 0 General



RATIONALE OF SEQUENTIAL SAMPLING 3 SAMPLING FROM A NEGATIVE BINOMIAL

DISTRIBUTION The negative binomial has often been used to describe the distribution

of insects that tend to occur in aggregates or clumps. Sampling from this distribution typically yields more zero counts and more very high counts than one expects ro m a andom distribution ike he Poisson. he required parameters are the mean (X) and an exponent, called / c , which is a function of the degree of aggregation.

Formulas for Sequential Plans Given a negative binomial^ distribution, H i Z'^Zi, H2 Z'^Zg, a value oik that is valid whenZi Z'=^X2, and acceptable values for a and

ß, the subsequent procedures are described by Oakland (5). Two new parameters, P and Q , are calculated for each hypothesis as

follows:

Parameter H , H,

k Q=l+P

k Q,=l+P,

k The new parameters are used in all subsequent calculations.

Intercepts:

lo g a, = m

og

log ao

fP2QA \P.Q2J

Slope:

log

lo g

b =k

\P.Q2J

log /P2QÄ \P1Q2J

O C curve.—Again, when Z ' = = 0 , LP^l; whenZ'=Zi, LP=l-û:;when

X =X2, LP=j 8; and whenZ'=6, LP^ag/Cag-aj). These four LP values




will uffice or crude evaluation of sampling plans. akland 5) and Waters {10) give formulas whereby other LFs and associated ASN^s m ay be calculated implicitly.

ASN cî^ry^.— Oakland's (5 ) formula is essentially as follows:

A SN = 2-i-(ai-a2)(LP)

X -b Wh en X =Q and LP=1, he ormula again educes o ASN=ai/-6.

Wh en Z' = 6 , the general formula again is not valid, but theA SN can be calculated as {a,){a^)l-{bm+h), ^ Once again, our points on the A SN

curve m ay be calculated easily.

The maximum A SN will be very close to the ASN forZ'=6 when a^ß. As a becomes progressively greater than ß, the reliability of this approximation apidly diminishes and use of the mplicit method becomes mandatory fo r a realistic estimate of the maximum A S N .

Negative Binomial Example Let us consider a sequential plan reported by Sylvester and C ox (8)

for sampling sugar beetplants to detect aphid infestationswairanting

treatment. Given H j Z'^Zi=0.9 aphids per plant, Hg X X^ IA, ¿=0.81, and a and 3=0.1, then

Parameter Hi H o

P Q

i0 .9 = 1 .1 1 1 1 1 1

0.81 Q i = l+Pi=2.111111

P.= -1 = 1. 3 5 80 25 0. 81

( 3 2 = 1 + ^ 2 = 2 . 3 5 8 0 2 5

Intercepts:

a, = - lo g f lo g

0.954242 0 . 0 3 9113

( 1 . 3 5 8 0 2 5 X 2 . 1 1 1 1 1 1 ) ( 1 . 1 1 1 1 1 1 X 2 . 3 5 8 0 2 5 )

= 24.40

lo g 9 log 1.094241

ao= log

m og 1.094241 0.94242

0.039113=24 40

I am not aware of this formula having been pubhshed previously, but it has worked fo r every type of example that I have been able to contrive.



RATIONALE OF SEQUENTIAL SA MPLING 5 Slope:

0.81 log ^ 2 . 3 5 8 0 2 5 \

b

=

2 - 1 1 1 1 1 1 / = (0 . 81X0 . 0 4 80 3 7) = o . 9 9 4 8

lo g 1.094241 . 0 3 9113

XLP SN

Formula alue ormula alue 0 _ efinition j-b 4.5 Z, -a 9 0 eneral 05.8

b 2/(a2-a,)

5 0 -M

6 8. 5

X^ 1 0 eneral 85.5

The above values differ only by rounding errors from reported values. The exact maximum A SN w as determined by several iterations of the

implicit method to be 2 6 8 . 9 5 .

GENERAL CONSIDERATIONS Morris i U ) reported that persons engaged in sampKng preferred to evaluate results with the aid of tables rather than graphs like figure 1 .

Situations will arise occasionally in which a rather large number of samples have failed to yield a decision. Several authors have offered arbitrary suggestions for resolving this predicament, but none of these

solutions appears to have universal merit. The necessary considerations

are based largely on economics and available options, which are apt to be unique for each sampling plan. n sampling for foUage-feeding insects,

for example, if the maximum A SN has been examined without a decision, we could assume that the need for corrective action is not immi- nent. However, n the case of soil insects that must be controlled by

insecticide broadcast before the crop is planted, the decision cannot be postponed ndefinitely. n uch a situation, s n he case of disease

vectors, we might decide to apply control measures whenever sampKng fails o dictate hat controls are not required. n general, arbitrary

decisions of this sort force us to make a guess, and thereby defeat the

basic reason for sampling. Granted that sampling should have increased

the probability that the guess will be a good one, the guess nevertheless remains a guess.

O ne possible solution to indecisiveness lies in careful study of tenta-

tive sampling plans. The nature of the problem often dictates that most sampling will be of populations n he vicinity of H i and H2 . This s especially true for scouting programs, because (1 ) natural trends are to

populations in excess of H i or the insect would not be an economic pest in




the first place, (2 ) infestations greater than Hg are constantly reduced by control measures, nd 3) einfestation o he evel of Hg does not proceed in a predictable manner. (If it did, we could justifiably adapt a

rigid treatment schedule and dispense with sampling.) W e should therefore be prepared to cope with maximum rather than minimum ASN's, as well as he occasional examination of an unusually arge number of samples. O ne hopes that H i, Hg, a, or ß can be adjusted to preclude a high incidence of indecisiveness.

An alternate olution to indecisiveness hat appears easible when units are sampled several times in succession is to consider the estimated

mean f i : - o m each examination in scheduling the subsequent examination. The rules of maximum likelihood tell us that the population is probably

lower in field I where H i was accepted after examining n samples than in field I where H i was accepted after 2n amples. Likewise, ield I

probably has ewer nsects han ield III, where a decision was not reached after examining an arbitrary imit of 3 n amples. ield III

obviously needs to be resampled sooner than the others, and we can efficiently give it the attention it needs if we ignore field I for a while.

A word of caution concerning the selection of a value fork in negative

binomial sequential plans appears to be in order. For some insects, such

as potato beetles and leatherjackets, k is essentially constant (i). That is , /c is apparently independent of density, so a common value (referred

to as ommon k or k^) s an adequate parameter of aggregation or clumping or any ypical population density. ommon k poses no problem to sequential sampling. However, with other insects, such as certain wireworms (1) and aphids ( < ^ ) , aggregation (and therefore k also) varies with density. It is therefore essential that the k used in a sequential ampling plan be ndicative of aggregation between he critical densities that are specified as limits of H , and Hg.

Sylvester and C ox < ^ ) and Ingram and Green (2) report that when the negative binomial distribution s rue and critical evels are reached

before 00 percent of the sample units become infested, t m ay under

some conditions be more expedient to determine only whether samples are infested or not infested, as opposed to the tedious chore of counting each insect in each infested sample. Given a negative binomial distribu-

tion, Z, and Ä : , the probability of a given sample being noninfested is

\k+xj and is calculated as the antilog of A : times

lo g / „ ^ \ \k+Xj

This probability is equivalent toq of the binomial distribution. Since the



RATIONALE O F SEQUENTIAL SAMPLING 7 binomial p = l-q,Xi and X2 of the negative binomial can be converted to Pi and P 2 of the binomial. A binomial sequential plan based on he

converted parameters will theoretically yield the same decisions as the

negative binomial plan with identical values of a and ß. The binomial plan will require the higher ASN's, but compensation lies in the fact that

only noninfested samples are examined in total.

A o mmo n amentation hese ays s hat ccurate conomic thresholds are not available for several important pests that, like the

proverbial mountain, have often been assaulted imply because hey

were there. However, sequential samphng techniques can help prevent

this lack of information from inhibiting the inception of a pest manage-

ment

rogram

hat

therwise

would proceed.

n

he

irst place,

economic thresholds that can be accurately defined as a point probably do not exist, and I would question the wisdom of efforts to refine them to that degree. Given that most thresholds vary within a range from season to season and field to field, he limits of that range can be set as the

specified limits of H i and H2 of the sequential sampling plan. Conse- quently, as infestations increase from the upper hmit of H i to the lower limit of H2, it becomes increasingly probable that they will indeed cause economic injury, and corrective measures accordingly will be dictated

with increasing probability. Secondly, in the absence of precise data, a minimum of research combined with experience and astute judgment

can often produce an estimated range that will subtend most of the true

range of given economic hresholds. erious eservations about he

accuracy of initial estimates for H i and H2 can be compensated for by setting a and ß at low levels (about 0.05 or lower). The OC curve will then be relatively flat in the neighborhood of the specified limits of H i and H2 (see fig. 2 ), ndicating that relatively large errors in economic thresholds will have relatively Httle effect on the proportion of erroneous

classifications rendered by use of the plan. Wood (11) has aptly discussed the justification for proceeding on the basis of the best available information in response to urgency.

LITERATURE CITED (1 ) liss, C . ., and Owen, A. R . G. 1 9 5 8 . Negative binomial distributions with acommon

k. Biometrika 45 : 37-58.

(2 ) ngram, . ., nd reen, . . 9 7 2 . equential ampling or bollworms n raingrown cotton in Botswana. Cotton Grow. Rev. 49 : 265-275.

(3 ) Knipling, . . 9 7 2 . ntomology and he management of man's environment. J.

Austr. Entomol. Soc. 1 : 153-167.

(4) Morris, R. F. 94 5. A sequential sampling technique or spruce bud worm egg sur-

veys. Can. J. Zool. 32 : 302- 13.

(5 ) akland, . B . 9 5 0 . An application of sequential analysis o whitefish sampling.

Biometrics 6 : 59-67.

(6 ) nsager, J. A. 9 6 9 . Sampling to detect economic infestations of Limonius spp. J.

Econ. Entomol. 2 : 183-189.




(7 ) and Foiles, L. L. 970. Relative efficiency of three methods or chemical control of Pacific Coast wireworms in potatoes. Am. Potato J. 47 : 379^85.

(8 ) Sylvester, E. S., an d Cox, E. L. 1961. Sequential plans for sampling aphids on sugar

beets in Kern County, California. J. Econ. Entomol. 4: 0 8 0 - 1 0 8 5 . (9 ) Wald, A. 945. equential ests of statistical hypotheses. nn. ath. tat. 6 :

1 1 7 - 1 8 6 . (10) Waters, W. E. 1955. Sequential sampling in forest insect surveys. For. Sei. 1 : 6 8 - 7 9 . (11) Wood, B. J. 971. Development of integrated control programs fo r pests of tropical

perennial rops n Malaysia. n C. . uffaker ed.), iological Control, p. 422-455. Plenum Press, N ew York.

APPENDIX.—DEFINITION OF SYMBOLS General Discussion

H i = the hypothesis hat an attribute of interest n an unknown population equals or is less than a predescribed level which is less than a level predescribed for Hg.

Hg = the hypothesis hat an attribute of interest n an unknown population equals or is greater than a predescribed level which is greater than a level predescribed for H j.

a = (alpha) the maximum allowable probability of committing a type

I error; i.e., the probability of accepting Hg when H i is true. ß = (beta) the maximum allowable probability of committing a type

II error; i.e., the probability of accepting H i when H2 is true.

a = the F-intercept in an equation of a straight line. b = the measure of slope in an equation of a straight line; i.e., the

rate of change in Y per unit change in X. LP = level of probability.

OC = operating characteristic.

A SN

= average sample number.

Binomial Distribution p = a general abbreviation for the proportion of samples in an unknown

population that bear the attribute of interest.

q = l-p=a general abbreviation for the proportion of samples in an unknown population that lack the attribute of interest.

P i = the proportion specified as the upper hmit of H i.

P 2 the proportion specified as the lower Hmit of Hg. Ç2 =1 2p any proportion that is known or assumed to be true.

Poisson Distribution X = (lambda) a general symbol for a Poisson mean.



RATIONALE OF SEQUENTIAL SAMPLING 9 X i the mean that is specified as the upper limit of Hi.

X 2 the mean that is specified as he ower limit of H2 . \ = any mean that is known or assumed to be true.

Negative Binomial Distribution X =8L general symbol for the mean.

Zi = the mean that is specified as the upper limit of Hi.

Xj = the mean that is specified as the lower Hmit of Hg. X = any mean that is known or assumed to be true.

k = an exponent which is a parameter of thedegree of aggregation.

P = a contrived parameter calculated as P=Xlk. Q = a contrived parameter calculated as Q = l4-P. Pi =Zi/fc.

Qi 1_+Pi.

X 2 ^^ - 2

Q 2 = 1+^2-

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