5 Sequential Hypothesis Testing Introduction Sequential sampling is a fast efficient tool for many sampling problems. Se- quential sampling may be used (1) to obtain precise estimate(s) of the parame- ter(s), or (2) to test hypotheses concerning the parameters. Sequential estimation is used when the purpose of sampling is to obtain precise parameter estimates. Several sequential estimation procedures are discussed in Chapter 4. The focus of this chapter is sequential hypothesis testing. This approach is appropriate when we are interested in determining whether the population density is above or below astated threshold. As in sequential estimation, sequential hypothesis testing re- quires taking observations sequentially until some stopping criterion is satisfied. The observations are taken at random over the sampling area. Generally , the accumulated total of the observations relative to the number of observations taken determines when sampling is stopped. The sequential hypothesis testing we con- sider requires some prior knowledge of the population distribution. This permits most computations to be completed in advance of sampling and to be stored in handheld ca1culators, laptop computers, or printed on cards or sheets. Wald's sequential probability ratio test was the earliest sequential test and is described first. Lorden's 2-SPRT is a more recent development that has some exciting pos- sibilities for tests of hypotheses conceming population density and is discussed in the latter parts of this chapter. Wald's Sequential Probability Ratio Test Wald (1947) introduced a sequential method of testing a simple null versus a simple alternative hypothesis with specified error probabilities. He called this sequentiaI process the sequentiaI probability ratio test (SPRT). The SPRT was used as a basis for inspecting wartime weapons and evaluating war research prob- 153
Transcript
5
Sequential Hypothesis Testing
Introduction
Sequential sampling is a fast efficient tool for many sampling problems. Sequential sampling may be used (1) to obtain precise estimate(s) of the parameter(s), or (2) to test hypotheses concerning the parameters. Sequential estimation is used when the purpose of sampling is to obtain precise parameter estimates. Several sequential estimation procedures are discussed in Chapter 4. The focus of this chapter is sequential hypothesis testing. This approach is appropriate when we are interested in determining whether the population density is above or below astated threshold. As in sequential estimation, sequential hypothesis testing requires taking observations sequentially until some stopping criterion is satisfied. The observations are taken at random over the sampling area. Generally , the accumulated total of the observations relative to the number of observations taken determines when sampling is stopped. The sequential hypothesis testing we consider requires some prior knowledge of the population distribution. This permits most computations to be completed in advance of sampling and to be stored in handheld ca1culators, laptop computers, or printed on cards or sheets. Wald's sequential probability ratio test was the earliest sequential test and is described first. Lorden's 2-SPRT is a more recent development that has some exciting possibilities for tests of hypotheses conceming population density and is discussed in the latter parts of this chapter.
Wald's Sequential Probability Ratio Test
Wald (1947) introduced a sequential method of testing a simple null versus a simple alternative hypothesis with specified error probabilities. He called this sequentiaI process the sequentiaI probability ratio test (SPRT). The SPRT was used as a basis for inspecting wartime weapons and evaluating war research prob-
153
154/ Statistical Ecology
lems. Oakland (1950) and Morgan et al. (1951) presented early applications of the SPRT in the biological sciences. The SPRT was used increasingly in biology after Waters (1955) presented details and examples of its use in forest insect studies. By the 1960s, the SPRT was widely applied in integrated pest management for a wide variety of pests and cultivars. Today, the SPRT is most often used to determine whether a specified region needs to be treated with pesticides (see Onsager, 1976; Pieters and Sterling, 1974; Oakland, 1950; Allen et al., 1972; Rudd 1980).
As in any sampling program, the first step is to identify clearly the sampling unit and the associated random variable of interest, such as the number of lady beetles (random variable) on a cotton plant (sampling unit) or the number of pigweeds (random variable) in a one-row meter (sampling unit) of a com field. The SPRT assurnes that the parametric form ofthe random variable's distribution is known. In presence-absence sampling, the binomial distribution is the parametric form of the distribution. The negative binomial and Poisson distributions are likely parametric forms when the number of organisms within a sampling unit is the random variable of interest. For the negative binomial, the parameter k is assumed known; if it is unknown, it must be estimated before the test of hypotheses can begin. The equations associated with the normal distribution with a known variance will be given because these are used in some biological applications. Because Iwao's Patchiness Regression and Taylor's Power Law are based on a relationship in mean and variance, and not on a parametric distribution, these approaches are not appropriate for use in Wald's SPRT.
Once the sampling unit and associated random variable are clearly defined, the hypotheses to be tested must be stated:
Ho: 8 = 80
H I : 8 = 81, (5.1)
Two types of error are associated with the tests of these hypotheses. A type I error results when Ho is true, but it is rejected in favor of HI • A type 11 error occurs when Ho is false, and it is not rejected. The probabilities of type I and type 11 errors are a and ß, respectively.
Usually we actually want to test two composite hypotheses:
Ho: 8::; 80
H I : 82:8 1, (5.2)
In (5.2), the type I and type 11 error rates are associated with 80 and 81, respectively, because the maximum probability of error occurs at these points. Consequently, the test of the composite hypotheses in (5.2) is the same as the one for the simple hypotheses in (5.1). In many biological applications, the hypotheses
Sequential Hypothesis Testing / 155
are being tested to determine whether the population density is above the economic threshold, implying treatment is needed, or below a safety level in which case no treatment is necessary. For these cases, (5.2) may be stated as
Ho: The population is below astated safety level
H I : The population is above the econornic threshold. (5.3)
In (5.3), we have implicitly assumed that 90 < 91• We use this convention throughout the remainder of our discussion for mathematical reasons. However, from an applied view, the order of the hypotheses makes no difference because the probabilities of type land 11 errors are controlled.
We briefty discuss the foundation of hypothesis testing for fixed-sample sizes and describe how these ideas are extended to sequential testing. The NeymanPearson lemma states that, with a fixed sampIe size n and type I error probability a, the test that minimizes ß is the likelihood ratio test. The likelihood ratio is
n
TIfx(x;;91)
A = ,",;=,-,1 __ _ n
TIfx(x;;!:lo) ;=1
where fx is the probability mass (density) function associated with the random variable X of interest. The critical region that leads to rejection of Ho is given by A > Cn. Cn is obtained using the predeterrnined values of a and n. Consideration of ß may impact the choice of n and/or a, but once a and n are set, the type II error rate ß is determined.
Wald presented an alternative approach by setting a and ß and then rninimizing the sampie size n. Thus, the final sampie size N becomes a random variable of interest. A sequence of random observations (XI' X2, X3, ••• ) is drawn from one of the two hypothesized distributions resulting in a sequence of likelihood ratios:
n
TIfx(x;;9 1)
A = :-i=..:..I __ _ n n (5.4)
TIfx(x i ;90)
;=1
Adecision is made and the test terminated after the nth observation if An 2:: A (accept H I ) or An ::; B (accept Ho), where A and Bare constants, depending on the predetermined a and ß. Exact computations for A and B are extremely difficult. Thus, the approximations
J 56/ Statistical Ecology
A=l-ß (5.5) a
and
ß B=--1 - a
(5.6)
are used. The approximations for A and B result in error rates a ' and ß' that differ from the desired error rates a and ß. However, it may be shown that the total of the actual error rates, a'and ß', is no larger than the sum of the specified error rates, a and ß; that is, a ' + ß' $ a + ß·
Sampling continues as long as B < An < A. In this form, computations are required after each observation in order to determine whether the stopping criterion is met. However, this relationship can be simplified. To do so, first, take the naturallogarithm to obtain
b = In(B) < In(An ) < In(A) = a. (5.7)
For the distributions we consider, equation (5.7) may be rewritten as bounds on the cumulative sum of the observations. Let Tn denote the sum of the first n observations. Then, the decision procedure may be stated as
1. Terminate the test and accept the alternative hypothesis if
2. Terminate the test and accept the null hypothesis if
3. Continue sampling if
h1 + Sn< Tn < h2 + Sn.
Notice that the decision boundaries are straight lines. They are parallel because both have slope S. The intercepts are h1 and h2 for the lower and upper boundaries, respectively (see Figure 5.1). S, hp and h2 depend on the distribution, the hypothesized parameters, and the specified error rates (see Table 5.1).
The extensive use of Wald's SPRT in pest management programs gives added meaning to the three zones in Figure 5.1 as may be seen in Figure 5.2. Figure 5.2 is based on the assumption that a pest species is being sampled to determine whether treatment is necessary to control the population. Three zones are pro-
Sequential Hypothesis Testing /157
Accept H,
h,+ Sn
Continue Sampling
Accept Ho
n
Figure 5.1. Decision lines for Wald's SPRT.
Table 5.1. 1ntercepts and Slopes Jor Wald's SPRT Jor the Binomial (B), Negative Binomial (NB), Normal (N), and Poisson (P) Distributions
Distribution h, h2 S
b a In[~] B In[P,qO] In~,qO]
In[P,qO] Poq, oq, Poq,
b a ln[~' + k]
NB In[~'(~o + k)] In[~'(~o + k)] ~o + k
k In[~'(~o + k)] JliJl, + k) Jlo(Jl, + k)
Jlo(~, + k)
N bcr2 acr2 Jl, + Jlo
Jl, - Jlo Jl, - ~o 2
b a A, - Ao P In[~] In[~] ln[~] where a = InC : ß) and b = InC ~ J
Reprinted from Young and Young, 1994, p. 756, with kind permission from Elsevier Seien eeNL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.
158 I Statistical Ecology
T.
Treatment Zone
00 Nol Treal Zone
n
Figure 5.2. Wald's SPRT applied to integrated pest management.
duced: the zone of decision to recommend control or treatment, the zone of decision to recommend no treatment, and the indecision zone where sampling continues. Wald's SPRT is used extensively in professional scouting programs, such as those for cotton, fruits, vegetables, and forests, and in disease control. Extensive field evaluation has shown that the use of Wald's SPRT saves in time spent sampling when compared to fixed-sample-size methods. The greatest time savings result from decisions being made quickly when the populations are either considerably above the economic threshold or considerably below the safety level. Sampling takes more time when the population mean density is near or between the economic threshold and the safety level. Thus, sequential hypothesis testing results in a greater allocation of resources when decision-making is most critical.
The following information is needed to design a Wald's sequential sampling plan from a pest management view point:
I. Economic threshold: This is the population density or damage level where economic loss will occur if control procedures are not initiated
2. Safety level: This is the population density that will ensure that economic damage will not occur
3. The distribution of the pest or damage being sampled: If the distribution is negative binomial, the common kc' or a precise estimate of k, is needed. The variance, or a precise estimate of it, is needed when hypotheses conceming the mean of a normal distribution are being tested
Sequential Hypothesis Testing / 159
4. The probabilities of type I and type II errors, er and ß. respectively: Based on the hypotheses in (5.3), a type I error occurs when the population density is dec1ared to be above the economic threshold when it is in fact below the safety level. A type 11 error is committed when the population is stated to be below the safety level when it is above the economic threshold.
The safety level is often the most difficult to determine. In many scouting programs, a region is monitored throughout the growing season. Periodically, tests are conducted to determine whether treatment is needed. If the decision is made that no treatment is necessary, the scout wants to be assured that the population is not likely to exceed the economie threshold for some period of time; that is, the population density should be enough below the economic threshold that the likelihood of economic damage before the region is sampled again is smalI. The largest population density that offers this assurance is the safety level. Time and labor constraints and the relationship between pest density and damage are two of the primary factors considered when establishing this safety level. As the difference in the economic threshold and safety level increases, the distance between the decision lines decreases and the number of observations needed to reach adecision decreases.
The probabilities of type I and type 11 errors, Q' and ß, respectively, also influence the distance between the lines. As the probability of error that the sampIer is willing to accept increases, the distance between the lines decreases and the required sampling effort decreases. Often these error rates are set at the same level. Historically, many scouts and other pest management personnel have believed that it is more serious to make a type 11 error than a type I error. This largely results from the fact that when a producer suffers economic damage from a pest due to lack of treatment, the error is often more obvious than the error of having a producer apply a treatment when it was not really needed. Further, many producers who are paying for pest management advice may believe they are getting a greater return on their money if they are told a particular action is needed instead of being told that no action is required. With the increase in environmental concerns and the development of resistance to pesticides, this approach is debatable and is being reevaluated. If the sampling plan is based on sound ecological studies, each risk factor is equally important.
The equations for Wald's SPRT can be developed using the equations in Table 5.1 after these factors are determined. These provide the decision boundaries as in Figures 5.1 and 5.2. The totals needed to stop are generally presented in tabular, instead of graphie, form when implementing the SPRT.
SPRT for the Negative BinomiaI Distribution
The negative binomial distribution has been used extensively in developing sequential tests of hypotheses because it adequately models the distribution of
160/ Statistical Ecology
many organisms. Therefore, sampiing plans for the negative binomial distribution are widely available (see Pieters ami Stt!rling, 1974; Allen et al., 1972; Oakland, 1950). For the negative binomial distribution, the hypotheses of interest are
Ho: I.l = I.lo
H 1: J.l = J.ll(>J.lO)· (5.8)
Knowledge, or a precise estimate, of k is required to determine the boundaries ofthe SPRT. The robustness ofthe SPRT to misspecification of k has been studied (Hubbard and Allen, 1991). Improperly specifying k does not yield error rates that are seriously different from those attained at the true value of k. If k is underestimated, the test is conservative; that is, the error rates are less than stated. If the estimate of k is larger than the true k, the error rates are larger than specified. As k becomes larger, the effect of misspecification is less pronounced. Another concem is that k mayaiso differ, depending on the mean. A reasonable approach is to consider the estimates of k for means near and between the two hypothesized values. Use a common k if one can be found within that region. If not, a conservative test can be developed by using an estimate of k at the lower end of the range of the estimates from historical data. Hefferman (1996) has suggested adjusting a and ß so that a test constructed based on a cOlumon k has the desired error rates if k varies with J.l in a known manner or if the test is truncated.
Example 5.1 Suppose we want to determine whether a pest has sufficient population density
to warrant treatment. Earlier work has determined that the economic threshold and safety levels are 10 and 7 pests/sampling unit, respectively. The distribution of this pest is known to be modeled weIl by the negative binomial distribution with k = 3. Further, assume that we have specified a = 0.1 and ß = 0.1. Then the hypotheses of interest are
S = k = 3 = 8.35. In[l.ll(l.lo + k)] In[10(7 + 3)]
I.lO(l.ll + k) 7(10 + 3)
The upper intercept is
h - ---=-----= 2 - In[lll{J.lO + k)]
1l0(1l1 + k)
and the lower intercept is
Sequential Hypothesis Testing / 161
In(1 - 0.1) 0.1
In[1O(7 + 3)] = 23.30.
7(10 + 3)
In( 0.1 ) 1 - 0.1
-23.30. n[lO(7 + 3)] 1 7(10 + 3)
The upper decision boundary U in Figures 5.1 and 5.2 is given by
U = h2 + Sn = 23.30 + 8.35n,
and the lower decision boundary L is
L = hl + Sn = - 23.30 + 8.35n.
For the first observation,
L = - 23.30 + (8.35)(1) = -15.0.
It is not possible to have minus pests, so a minimum sampIe size (MSS) is required before adecision can be made in favor of Ho, and this is
MSS = !!:t! = 23.30 = 2.8. S 8.35
This is always rounded up, giving 3 as the MSS for the lower decision boundary. In other words, the MSS is the point where the smallest observation number multiplied by the slope is greater than the absolute value of the line intercept. It is customary to note the L for observation numbers below the MSS with nd for no decision. The upper line is calculated in the same manner. For example, at sampIe 11, we have
U = 23.30 + (8.35)(11) = 115.2.
In order to attain the desired error rates, U, the upper limit, should always be rounded up (giving a value of 116 for this example) and L, the lower limit, should
162/ Statistical Ecology
always be rounded down. It is customary to program these systems into handheld calculators or laptop computers or to construct charts or graphs as in Table 5.2.
ECOSTAT can be used to construct an SPRT for the binomial, geometrie, negative binomial, and Poisson distributions. Begin by selecting SPRT from the Sequential Sampling menu. Select the appropriate distribution. Then enter the hypothesized values and the error rates for the test. By pressing Set Boundaries, the equations for the upper and lower decision boundaries are displayed. A data sheet as in Table 5.2 can be stored by pressing Data Sheet. The data sheet can be printed using a lO-point fixed font, such as Courier, with one-inch margins.
In practice, sequential sampling is simple. An observation is made and recorded. Another observation is made, added to the previous sampie total, and the total count is recorded. This procedure is repeated until a stopping rule is satisfied. The stopping rule has the following form:
Stop if the total number or organisms counted equals or exceeds the upper limit,
OR
Stop if the total number of organisms counted equals or is lower than the lower limit.
For instance, in Table 5.2, if the running total at observation 11 is 68 or less, stop sampling and recommend that no treatment be used. If the total number of organism is 116 or more, stop sampling and recommend treatment. If the total number of organisms is between 68 and 116, continue to sampie until the total number of organisms crosses adecision boundary and satisfies the stopping rule. Excessive sampling may be needed to meet the stopping criterion. This may be due to the choices of hypothesized values and error probabilities. How to evaluate the reasonableness of a proposed sampling program is discussed later in this chapter. Also, if the true mean population density is between the two hypothesized means, sampling may continue beyond reasonable limits. If this case is frequently encountered, a 2-SPRT test (see 2-SPRT in this chapter) could be used. Many pest management scouts simply return to the field at a shorter interval. Knowledge of the pest and cropping system that is being sampled aids in this decision.
ECOSTAT can be used to construct an SPRT for the binomial, geometrie, negative binomial, and Poisson distributions. Begin by selecting SPRTfrom the Sequential Sampling menu. Select the appropriate distribution. Then enter the hypothesized values and the error rates for the test. By pressing Set Boundaries, the equations for the upper and lower decision boundaries are displayed. A data sheet as in Table 5.2 can be stored by pressing Data Sheet. The data sheet can be printed using a lO-point fixed font, such as Courier, with one-inch margins.
SPRT for the Poisson Distribution
The Poisson distribution has one parameter, A. The mean is A, and the mean and variance are equal. Expressing (5.3) in terms of A, we have
Sequential Hypothesis Testing / 163
Table 5.2. Sequential Sampling Jor the Negative Binomial Distribution'
SampIe Lower Running Upper SampIe Lower Running Upper number limit total limit number limit total limit
After the type I and type 11 error rates for the test have been specified, the decision boundaries may be constructed using the formulae in Table 5.1
Example S.2 Suppose that if 30,000 spiders per acre are present in a field, treatment to
control a pest population within that field is not needed. Further, assume that this field has 40,000 plants per acre and that 20,000 spiders per acre leave the field vulnerable to pest attack. Experience has shown that the distribution of the number of spiders tends to be Poisson in most fields. This example differs from earlier ones in that since the spiders are predators, no treatment is required if the population is large enough. Also, the example is more realistic, in that the sampling unit, the hypotheses, and the error rates have not been specified. In this case, it seems reasonable to use a plant as the sampling unit. The hypotheses of interest are
20,000 ~ = Äo = 40,000 = 0.5
~ = Ä1 = 30,000 = 0.75. 40,000
Notice that the hypotheses are stated in terms of the mean density on the sampling unit basis. Assume that we specify a = ß = 0.1. Then the slope is
Sequential Hypothesis Testing / 165
Al - 1.0 0.75 - 0.5 S = (Al) = (0.75) = 0.617.
In - In-1.0 0.5
Tbe upper intercept is
and the lower intercept is
In(1 - 0.1) O.l
In 0.75 ( ) = 5.42,
0.50
In(-ß ) In( 0.1 ) 1 - IX 1 - 0.1
= = -5.42.
In(~:) In(~:~~) The upper decision line is
U = 5.42 + 0.617n,
and the lower decision line is
L = -5.42 + 0.6I7n.
The minimum sampie size is
MSS = ~ = 1-5.421 = 8.78 S 0.617 .
Tberefore, at least nine observations must be made before deciding in favor of Ho that the population density of the spiders is not sufficient to protect the field from the pest species.
SPRT for the Binomial Distribution
Binomial sequential sampling plans are sometimes used in pest management programs. This usually arises when each observation may be considered an out-
166/ Statistical Ecology
come of a Bernoulli trial; that is, the fruit is damaged or not damaged, the organism is male or female, the plant is infested or not infested, and so on. When counting organisms is time consuming, the binomial mayaiso be used as an alternative to the negative binomial and Poisson distributions. In this case, each sampling unit is judged to have no organisms or at least one organism. Then the probability of 0 for the distribution of the number of organisms per sampling unit is equated with q of the binomial distribution.
The hypotheses to be tested are
Ho: P = Po
HI : P = PI(>PO)· (5.10)
The decision boundaries may be constructed using the formulae in Table 5.1.
Example 5.3
Dry cowpeas in storage are to be tested for weevil damage. If 50% or more of the cowpeas are infested, they must be destroyed. If 30% or less are infested, the cowpeas can be saved by treating them with a fumigant. The sampling unit is one cowpea. Suppose that the type I and type 11 error rates are specified to be 0.15 and 0.20, respectively.
The hypotheses to be tested are
Ho: P = 0.3
H I : P = 0.5.
For the binomial distribution, q = - p. A value of q is associated with each hypothesized value of P; that is,
qo = 1 - 0.3 = 0.7
and
ql = 1 - 0.5 = 0.5.
The slope of the decision lines is then
In(0.7) 0.5
In 0.5(0.7) [ ] = 0.397.
0.3(0.5)
The upper intercept is
Sequential Hypothesis Testing / 167
InC : ß) InC - 0.2) 0.15
h2 = ln[ 0.5(0. 7)]
1.98, In(PlqO)
POql 0.3(0.5)
and the lower intercept is
ln(-ß ) 1 - a
1 ( 0.2 ) n 1 - 0.15
hl -1.71. In[PlqO] In[0.5(0.7) ]
POql 0.3(0.5)
Notice that in the example above, h l oF- - h2 • When a = ß, the intercept of the lower decision boundary is the negative of the intercept of the upper decision boundary. This relationship no longer holds if a oF- ß.
The minimum sampIe size for deciding in favor of Ho is
SS Ihll 1-1.711 M = - = = 4.31.
S 0.397
Therefore, at least five observations must be taken before one can conclude that the proportion of infested cowpeas is no more than 30%.
Operating Characteristic and Average Sam pie Number Functions
The properties of any proposed sampling system should be known as completely as possible before that program is implemented. The evaluation ofWald's SPRT is generally based on the Operating Characteristic (OC) and Average Sampie Number (ASN) functions.
The OC function, L(9), is the probability that the null hypothesis is accepted given any parameter 9 in the parameter space. In constructing the test, the user specifies L(9o) = 1 - a and L(9z} = ß. Wald (1947) proved that, for general 9,
Ah(9) - 1 L(9)=---A h(9) - Bh(9)
(5.11 )
where A and Bare given in equations (5.5) and (5.6), respectively, and h(9) is the solution of
depending on whether f(x;9) is discrete or continuous, respectively. With the exception of h(Oo) = land h(Ol) = - I, it is difficult to solve for h(9) given 0 in equations (5.12) and (5.13). Therefore, Wald (1947) suggested solving for 9 given h values, rather than vice versa. For h(9) ~ 0, the solution for 9 is given in Table 5.3 (see also Waters, 1955; Fowler and Lynch, 1987). Then equation (5.11) is used to compute L(e). This is also given in Table 5.3 for completeness. When h(8) = 0, e is the solution of
and
E [ln(f(x;e1»)] = 0 f(x;9o)
a L(e) =-
a - b
(5.14)
(5.15)
where a and b, as defined in equation (5.7), are In(A) and In(E), respectively. The 8 for h(e) = 0 and the corresponding L(e) are given in Table 5.4. By computing L(e) for h values ranging from - 2 to 2 in increments of 0.2, the oe function can usually be adequately described. These computations may be done using ECOSTAT. Simply press Wald's Approximations after the decision lines have been found as described in Example 5.1.
One of the disturbing features of the SPRT for many first-time users is that the indecision zone lies between two parallel lines. The fear of staying in the indecision zone forever is common. Wald (1945, 1947) did prove that the probability of making a terrninating decision was one, regardless of the true value of O. However, this does not guarantee that the sampie size needed to reach adecision will be within the time and labor constraints of the sampier. Therefore, another important measure of the feasibility of a proposed sampling plan is the average sampie size that is required to satisfy the stopping criterion. The ASN function [Ee(n)] is the average number of observations needed to make a terrninating decision for each 9 in the parameter space 0. Wald (1947) approximated the ASN function using
Ta
ble
5.3
. O
pe
rati
ng
Ch
ara
cte
rist
ic F
un
ctio
n,
L(e
); A
vera
ge
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pie
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Fu
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nd
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Bet
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n 8
an
d h
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hen
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0, f
or
the
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om
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Ne
ga
tive
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om
ial
(NB
), N
orm
al
(N),
an
d P
oiss
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P)
Dis
trib
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on
s
Dis
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n e
B
p
NB
Jl
N
Jl
P
A
(I
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whe
re
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l-
and
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1
Ah(p
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Ah(~)
-Bh(~)
Ah(~)
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n)
bL
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+ a
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p Inf
EI(
I -
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pI]
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l) +
a[I
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L(J
l)]
k In[
JlO +
k] +
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k)]
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k)
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l)]
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; Jlo
]
bL
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ll
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A . ln[~]
+ A
-A
Ao
0
I
e =
f[h
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1 _ [1
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1 -
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p =
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= ;J(P)
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k
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k]h
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k
[JlI(Jl
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JlO(Jl
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2
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)
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58.
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herl
ands
.
f ~ [ ~
"::j
~ So
~ 1:;' ~
'" ~.
' .....
0-
\0
Sequential Hypothesis Testing / 170
Table 5.4. Operating Characteristic Value, L(9); Average Sampie Number Value, Eo(n); and 9, When h(9) = 0 Jor the Binomial (B), Negative Binomial (NB), Normal (N), and Poisson (P) Distributions
Distribution e L(9) Eo(n) 9 when h (9) = 0
-ab ln[~] a 1 - p,
B P -- In[el] In[l = po] p= a - b In~,(l - po)] Po 1 p,
0(1 - p,)
-ab In[~' + k]
NB a
[)i()i + k)][ln()i,()io + k»)T ~o + k
)i -- Il = k a - b In[Il,(llo + k)] k llo(Il, + k) llo(Il, + k)
-ab a Il = Il, + Ilo N Il -- (Il, - IlO)2
a - b 2 (J2
-ab A= (A, - Ao) A a
A[ln(~)T In[~] P a - b
where a = Ine : ß) and b = Ine ~ J Reprinted from Young and Young, 1994, p. 759, with kind permission from Elsevier Science-NL,
Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.
for h(9) "e 0 and
bL(9) + a( I - L(9) Ea(n) = --'-E-;:-[1-n(:--fx-"-(X-;9--'1»)~] ~
for h(9) = O. The forms (5.16) and (5.17) take for the binomial, negative binomial, Poisson, and normal distribution are shown in Tables 5.3 and 5.4 (see Waters, 1955; Fowler and Lynch, 1987).
Wald's equations for the oe and ASN functions are based on the assumption that sampling stops on the boundary. This rarely occurs. Usually, there is some
Sequential Hypothesis Testing / 171
overshooting of the boundary with the size of the overshoot being the absolute difference in the final total TN and the boundary at that N. The approximations tend to overstate the error probabilities and understate the average sampie size. The difference in the true and approximate error probabilities is not of practical significance in most cases. However, Corneliussen and Ladd (1970) found that Wald's formula under-represented the ASN at the maximum by about 20% when sampling from binomial distributions. Similar results were obtained by Fowler and Lynch (1987) and Seebeck (1989) for the Poisson and negative binomial distributions. In addition, the distribution of the sampie size tends to be highly skewed. This means that sampIe sizes substantially larger than the average may be required to reach adecision. If all available resources are needed to gather a sampie of average size, the sampier will often have to cease sampling before meeting the stopping criterion.
A realistic assessment of the ASN function and the possible sampie size range is needed. Corneliussen and Ladd (1970) developed an algorithm for computing the exact OC and ASN functions for the binomial distribution. This work was extended by Y oung (1994) to cover discrete distributions that are members of the exponential family with special emphasis on the Poisson and negative binomial distributions. These algorithms, along with Wald's approximations, are in the accompanying software ECOSTAT. The exact computations take much longer than Wald's approximations. Therefore, we recommend investigating the properties of a proposed SPRT using Wald's approximations. Once a reasonable test seems to have been developed, examine the exact properties of the test. In addition to the OC and ASN functions for various parameter values, the 95th percentile of sampie size N g5 is given; 95% of the sampies collected will have sampie sizes SoNg5 , and 5% of the sampie sizes will be >Ng5 •
Example 5.4
Consider again the SPRT developed in Example 5.1. Computing Wald's approximations for the OC and ASN functions as h(P) ranges from - 2 to 2 in increments of 0.2, we obtain Table 5.5. To verify these numbers, suppose h(P) = 0.4. Then the parameter /l corresponding to h(P) = 0.4 is
[/l0 + k]h(~)
1- -/lI + k
[ 7 + 3 ]0.4 1- --
10 + 3 3 "'""[1-0-(7-+-3-=)]:--0.4-_-1 = 7.77.
7(10 + 3)
For Ci = 0.10 and ß = 0.10, we can calculate
172 / Statistical Ecology
Table 5.5. Wald's Approximations ofthe oe and ASN Functionsfor Various Means with k = 3.'
k In[1l0 + k + Il In[1l1(llo + k)] 111 + k 1l0(1l1 + k)
[ 7 + 3] [10(7 + 3)] 3 In 10 + 3 + 7.77 In 7(10 + 3)
16.8.
The exact properties of the test were computed using ECOSTAT and are given in Table 5.6. Wald's approximate and the exact OC and ASN functions are shown in Figures 5.3 and 5.4, respectively. In ECOSTAT, the exact values of the Oe, ASN, and N 95 functions may be found by pressing Exact Computations after the decision boundaries have been determined as described in Example 5.1. They can be stored as an ASCII file by pressing Save Properties.
The 2-SPRT
No other test of a simple versus a simple hypothesis with type I and type II error rates at most IX and ß, respectively, results in a smaller average sampIe size at the hypothesized values than Wald's sequential probability ratio test (Wald and
Table 5.6. Exact Values 0/ the oe and ASN Functions and N 95 /or Wald's SPRT with k = 3."
'The economic threshold and safety levels are 10 and 7 pests per sampling unit, respectively.
174 / Statistical Ecology
oe 1.00
0.75
0.50
0.25
o 2 4 6 8 10
Exact Approximate
12 14
Figure 5.3. The SPRT oe function based on Wald's approximation and exact computations based on the negative binomial distribution with k = 3. Economic threshold and safety level are 10 and 7, respectively. a. = ß = 0.1.
Wolfowitz, 1948; Lehmann, 1959). However, since e may not always assume one of the hypothesized values, the bebavior of the ASN over tbe full parameter space is often of interest. As sbown in Figure 5.4, the ASN function peaks at a point in tbe parameter space intermediate to tbe two hypotbesized values, and this peak may represent a substantially larger average sampie size than that at either bypothesized value. In addition, tbe sampie size distribution tends to be bigbly skewed, and the sampling effort may greatly exceed tbe average as can be seen by tbe line denoting the 95th percentile of sampie size, N 9S ' in Figure 5.4
Suppose that sampling is conducted according to the SPRT and that no decision bas been made by tbe time tbe maximum possible sampie size bas been attained. Tbe sampIer could assume tbat tbe true parameter value is intermediate to the two hypothesized values. If sampling is being conducted to determine whether a field should be treated, a value of the parameter between the economic threshold and safety level indicates that a pest population may be building in numbers, and the field should be c10sely monitored. The sampier would return to the field sooner than tbe normal scouting scbedule would dictate, as a delay could result in econornic damage. Alternatively, if no decision has been made after attaining tbe maximum feasible sampie size, a rule can be established for deciding between
N 65
60
55
50
45
40
35
30
25
20
15
10
5
Sequential Hypothesis Testing / 175
Exact E(N) Approximate E(N) Exact Nos
o ~~~~~~~~TMno~~Tn~~~ o 2 4 6 8 m ~ M ~ ffi 20
Mean
Figure 5.4. SPRT ASN function based on Wald's approximation and exact computations as weil as N 95 based on the negative binomial distribution with k = 3. Economic threshold and safety level are 10 and 7, respectively. a = ß = 0.1.
the two hypotheses. This approach generally decreases the expected sampie size at the hypothesized values of the parameter and increases the actual error probabilities.
Several different schemes have been proposed to modify or replace the SPRT with a closed sequential test, a test with a bounded zone of indecision resulting in adecision on or before a known maximum sampie size. Weiss (1953) introduced the generalized SPRT. For the generalized SPRT, the predetermined constants A and B change at each stage of sampling instead of remaining constant as they do in the SPRT. Armitage (1957) proposed restricted SPRT' s for testing the mean of the normal distribution. The 2-SPRT, proposed by Lorden (1976, 1980), is another closed test and is presented here.
An alternative approach to minimizing the expected sampie size at the hypothesized values as in the SPRT is to minimize the average sampie size at an intermediate parameter value 8*. The problem of finding such a test is known as the modified Kiefer-Weiss problem. Minimizing the ASN at the value of 8* for which the ASN is a maximum provides a solution to the Kiefer-Weiss problem.
176/ Statistical Ecology
An asymptotic solution to the modified Kiefer-Weiss problem was given by Lorden (1976, 1980). He developed the 2-SPRT test, which simultaneously performs two one-sided SPRTs. Again consider testing
Ho: 9 = 90
H I : 9 = 91(>90) (5.18)
Let 9* be a value intermediate to 90 and 91 for which the ASN is to be minimized. Define a third hypo thesis
(5.19)
A one-sided hypothesis of H2 against Ho is conducted for possible rejection of Ho. Simultaneously, another one-sided SPRT of H2 against HI is conducted for the possible rejection of H I • The decision boundaries for this test are two converging lines that produce a triangular continuation region (see Figure 5.5). To gain an understanding of the 2-SPRT, we first develop the 2-SPRT for a given 9*. Then we show how to choose 9* to minimize the ASN asymptotically. However, in practice, we use ECOSTAT to perform all the computations.
Let XI' X2 , X3 , ••• be a random sampie from a density ofthe Koopman-Darmois form; that is,
Contlnue Sampling
Accept H I
°1 I
Figure 5.5. Decision boundaries for the 2-SPRT.
I I
M n
Sequential Hypothesis Testing / 177
f(x;9) = exp{k(x) + 9x - b(9)}
where k(x) is a function of x alone and b(9) is a function of 9 alone. 9 and b(9) for the binomial, Poisson, and negative binomial distributions are given in Table 5.7. Note that in each case, the original set of hypotheses of interest must be restated in terms of computational hypotheses involving 9 to develop the testjust as we did with the negative binomial when developing the SPRT. It is desired to test the hypotheses in (5.18) with type I and type II error probabilities equal to IX
and ß, respectively. As before, let
n
Tn ~Xj' i=1
Sampling will continue until
Table 5.7. Quantities Used for Lorden 's 2-SPRT for the Binomial, Negative Binomial, and Poisson Distributions
Binomial Poisson Negative binomial
8 In(~) In(A) In(-~ ) ~ + k
b(8) -n In(l - p) A _kln(_k ) ~ + k
( B(8*) ) ( B(8*) ) I ( B(8*) )
ho In I - A(8*) In I - A(8*) n I - A(8*)
In(P,q*) In(~~) In(~'(~* + k») q,p* ~*(~, + k)
InC - B(8*» InC - B(8*») InC - B(8*»)
h, A(8*) A(8*) A(8*)
In(P*QO) In(~:) In(~*(~O + k») q*po ~o(~* + k)
In(~~) A, - A* k In(~' + k)
In(~~ ) ~* + k
So In(P,q*) In(~'(~* + k»)
q,p* ~*(~, + k)
In(:~) A* - Ao (~* + k) kin ---
In(~:) ~o + k
s, In(P*qO) In(~*(~o + k»)
Q*Po ~o(~* + k)
Reprinted from Young and Young, 1994, p. 762, with kind permission from Elsevier Science-NL, Sara Burgerhartstraat 25, 1055 KV Amsterdam, The Netherlands.
178/ Statistical Ecology
(1) T. ;z: hin + SI OR
(2) Tn::S; hrJI + So (5.20)
(accept Ho).
It remains to determine ho, h l , So, and SI' The values h l and SI are determined by the one-sided test of H2 versus Ho. Similarly, ho and So are obtained from a onesided test of HI and H2•
We begin by defining some quantities that are used repeatedly in determining 9* and the bounds of the 2-SPRT. The Kullback-Leibler information numbers are
/;(9) = (9 - f)Jb'(9) - [b(9) - b(9J] (5.2l)
for i = 0, I. Further define
(5.22)
for i = 0, I. Now recall that the sequentiallikelihood ratio for the SPRT is
• fIfx(xßI)
A = :..-;=..:...1 __ -n •
fIfxCxßo) ;=1
Since two SPRTs are to be conducted simultaneously, two likelihood ratios need to be formed. Let
Ao• = ",-;;~I __ - (5.23)
fI fxCx;;fl*) ;=1
and
AI. = ;....;;....:..1 __ - (5.24)
fIfx<xß*) ;=1
The 2-SPRT takes the form:
Sequential Hypothesis Testing / /79
(1) Terminate the test, reject Ho, and accept HI if AOn :5 A; (2) Terminate the test, reject HI , and accept Ho if Aln :5 B; (3) Otherwise, continue sampling. (5.25)
The quantities A and B are chosen so that the desired errors rates (X and ß are attained when both one-sided tests are conducted simultaneously. Ignoring the excess over the boundary, Huffman (1983) recommended for practical uses that
(5.26)
and
(5.27)
The slopes and intercepts of the decision boundaries in (5.20) may now be obtained. The form of the distribution permits the quantities in the decision rule in (5.25) to be computed as folIows:
In[ 1 - B(9*)] A(9*)
h l 9* - 90
(5.28)
[ B(8*) ] In I - A(9*)
ho = 91 - 9*
(5.29)
SI = b(9*) - b(90)
9* - 90
(5.30)
and
(5.31 )
This gives a solution to the modified Kiefer-Weiss problem. The maximum sampie size for the SPRT occurs at the point of intersection of the two decision boundaries:
180/ Statistical Ecology
(5.32)
Intuitively, one might think that the maximum of the ASN function occurs equidistant from 90 and 9 I. When the hypotheses coneeming the binomial parameter p are symmetrie about 0.5, this is indeed the ease. The midpoint between 90
and 91 is p = 0.5, yielding a symmetrie distribution. However, when the distribution is skewed at the midpoint, the maximum of the ASN funetion does not oecur at the midpoint between 90 and 91. Henee, we now foeus our attention on determining the 9* for which the ASN is a maximum. By constructing the 2-SPRT at that point, an asymptotic solution to the Kiefer-Weiss problem is obtained.
In order to obtain 9*, 0' is first determined such that
In[A{9')]-1 In[B{9')]-1
11(9') 10{9') (5.33)
Let n' be the common value of the two sides in equation (5.33). Denote a j{9') as a:. Find r' such that
, <l>{r') = a l
a; - a~ (5.34)
where <l> is the distribution function for the standard normal random variable. Also, for 9 = 9',
a' = JVare.()().
The point 9* for which the ASN is to be minimized may be expressed as
r' 9* = 9' + --. a'p
The adjusted error rates based on H2 : 9 = 9* are
and
B{9*) = a l {9*) - ao{9*) ß. a l {9*)
(5.35)
(5.36)
(5.37)
(5.38)
Sequential Hypothesis Testing / 181
Finally, the values needed to construct the decision boundaries for Huffman's extension of the 2-SPRT are as in equations (5.28) to (5.31), with the value of 9* found in equation (5.36) and the corresponding corrected error rates from equation (5.37) and (5.38).
Huffman's (1983) extension of Lorden's work deterrnines the value of 9*, that minimizes the maximum expected sampIe size to within o((log a -1)1/2) as a and ß tend to zero. This provides an asymptotic solution to the Kiefer-Weiss problem.
The express ions for 9, b(9), ho, hl , So, and SI for the binomial, Poisson, and negative binomial distributions are given in Table 5.7.
Example 5.5
Consider again the test of hypotheses discussed in Examples 5.1 and 5.4:
The distribution of pests in the field is known to be negative binomial with k = 3. a and ß are each specified to be 0.1. From Table 5.7, we have the following computational hypotheses:
9 = In(~) = In(_7_) = - 0.357 o Ilo+ k 7+3
91 = In(-1l-1 -) = In( 10 ) = - 0.262 (>90),
fll + k 10 + 3
Also from Table 5.7, we have
b(9) = -k In(_k_). fl + k
The derivative of b(9) is the mean of the distribution for members in the Koopman-Darmois family of distributions. We can demonstrate that property for this distribution by using the chain rule for taking derivatives to obtain
b'(9) = 11.
Later, we will need the value of b(9) at the hypothesized values so we find these now:
182/ Statistical Ecology
b(80) = -k ln(_k_) = -3In(_3_) = 3.612 J.1+k 7+3
and
The KulIback-Leibier information number for this distribution may be expressed in terms of J.1 and k (see Table 5.7) as
1;(8) = J.1In[J.1(J.1i + k)] + k In[J.1i + k] J.1i(J.1 + k) J.1 + k
for i = 0, 1. Further, we have
for i = 0, 1. ao(8) and a1(8) are used Iater to compute the adjustments to the error rates.
To find 8*, the value of 8 for which the ASN function is to be minirnized, we first determine er such that
In --[ 1] A(er)
Solving for er iteratively, we find er - 0.307, corresponding to J.1' = 8.36. This can be verified by showing that equality holds in the above equation. Several steps are required to accomplish this. We begin by computing the information numbers
To determine the decision boundaries, we need to evaluate b(9) at 9 = 9*:
b(9*) = - k In(_k -) = - 3 In( 3 ) = 3.984 J.l + k 8.32 + 3
Sequential Hypothesis Testing / J 85
Now we are ready to detennine the intercepts and siopes of the decision boundaries:
and
ln( 1 - B(8*») A(8*)
h l =-----8* - 80
( B(8*) ) In 1 - A(G*)
ho =-----81 - 8*
b(G*) - b(80)
8* - 80
In(1 - 0.187) 0.214
---0-.3-08---(---0-.3-5-7) = 34.29,
( 0.187 ) In 1 - 0.214
- 0.262 - (- 0.308) -33.86,
3.98 - 3.61 ------ = 7.63, - 0.308 - (- .357)
b(81) - b(8*) 4.40 - 3.98 So = = = 9.12.
81 - G* - 0.262 - (- 0.308)
Therefore, after n observations, we
1. Stop sampling and conclude the population density exceeds the economic threshold if the total number of observed insects exceeded 34.29 + 7.63n;
2. Stop sampling and conclude the population density is less than the safety level if the total number of observed insects is less than - 33.86 + 9.12n;
3. Continue sampling otherwise.
The maximum sampie size is
SI - So 7.63 - 9.12 M = ho _ h, = _ 33.86 _ 34.29 = 45.81.
Therefore, at most, 46 observations will be taken. The example above clearly demonstrates why the 2-SPRT has not been used
extensively in biological, and other, applications; the computations are complex. In addition, if the reader works through some of the above caIculations, it soon becomes evident that round-off error can cause severe problems. Fortunately, ECOSTAT may be used to construct these decision boundaries using doubleprecision to reduce the impact of rouod-off error. The process is the same as that
186/ Statistical Ecology
for the SPRT except 2-SPRT is chosen from the Sequential Sampling menu. In addition, ECOSTAT deterrnines the values of the operating characteristic function, the average sampie number, and the 0.95 quantile of sampie size. For Example 5.5, these are given in Table 5.8. Plots with the exact characteristics associated with the SPRT and the 2-SPRT are shown in Figures 5.6, 5.7, and 5.8, so that the properties of the two tests can be compared. Notice that the error rates are comparable (Figure 5.6). If the mean is equal to one of the hypothesized values, then the 2-SPRT requires a few more observations than the SPRT whether observing the ASN function (Figure 5.7) or the 95th percentile of sample size (Figure 5.8). However, for means intermediate to the hypothesized values, the 2-SPRT can result in significantly lower sampie sizes, especially if one considers the 95th percentile of sampie size.
Summary
Wald's (1945, 1947) SPRT has been used extensively to make management decisions in integrated pest management. For the SPRT, the type I and type II error rates are specified in advance of sampling, and the sampIe size is random.
Table 5.8. Exact Values ofthe oe and ASN Functions and N 9Jor Lorden's 2-SPRT With k = 3a
'The economic threshold and safety levels are 10 and 7 pests per sampling unit, respectively.
oe 1.00
0.75
0.50
0.25
o 2 4 6 8
Mean
Sequential Hypothesis Testing / 187
10
SPRT 2-SPRT
12 14
Figure 5.6. Exact oe functions for the SPRT and 2-SPRT based on the negative binomial distribution with k = 3. Economic threshold and safety level are 10 and 7, respectively. IX = ß = 0.1.
If an infestation is high or low, few observations are required to make adecision. If the population is at or near the economic threshold or safety level, more observations are required. Therefore, sampling effort is increased during the critical period of decision-making. The properties of a proposed sampling program can be evaluated using Wald's approximation or using ECOSTAT to determine exact properties. It is recommended that the approximations be used for initial screening of a proposed test. Then the exact properties should be investigated. Use of the 95th percentile of sampie size provides insight into whether sufficient resources exist so that sampling can probably continue until adecision is made.
Lorden (1976, 1980) and Huffman (1983) have developed the 2-SPRT, which is based on simultaneously conducting two one-sided SPRTs. Instead of rninimizing the sampie size at the hypothesized values as in the SPRT, an effort is made to rninirnize the maximum sampie size. This results in a triangular nodecision zone, ensuring that sampling stops with a known maximum sampie size. Exact properties of the test can be investigated using ECOSTAT. Compared to the SPRT, the 2-SPRT has comparable error rates, slightly higher sample sizes at the
188/ Statistical Ecology
E(N) 24
22
20
18
16
14
12
10
8
6
4
2
/. /
,/
! !
! /
.I /
/
SPAT 2-SPRT
o ~~~~~~~~~~~~~~~ o 2 4 6 · 8 m ~ M ffi m ro
Mean
Figure 5.7. Exact ASN functions for the SPRT and 2-SPRT based on the negative binomial distribution with k = 3. Economic threshold and safety level are \0 and 7, respectively. a = ß = 0.1.
hypothesized values, and substantially lower sampie sizes at values intennediate to the hypothesized values. Now that software is available for ready implementation, we recommend this approach be strongly considered for IPM use.
Exercises
For Problems 1 through 5 below,
A. Develop Wald's SPRT to test the specified hypotheses.
B. Estimate the OC and ASN functions for Wald's SPRT.
C. Based on the results in B, adjust the error rates ifthis is needed to obtain a reasonable plan in terms of sampie size.
D. Compute the exact OC and ASN functions.
E. Based on the results in D, adjust the error rates to attain the specified error rates using the plan.
F. Print out a sampling sheet for the final sampling plan.
N.., 65
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50
45
40
35
30
25
20
15
10
5
0
Sequential Hypothesis Testing / 189
SPAT 2-SPAT
....... -" ..
o 2 4 6 8 ·10 ~ M ffi ffi 20
Mean
Figure 5.8. N 9, for the SPRT and 2-SPRT based on the negative binomial distribution with k = 3. Economic threshold and safety level are 10 and 7, respectively. ot = ß = 0.1.
1. The economic threshold for cotton fleahoppers is 40 fleahoppers/100 terminals; 25 fleahoppers/100 terminals is considered safe. Experience has shown that the distribution of cotton fleahoppers tends to be negative binomial with a k value of 2. The purpose of sampling is to determine whether the population is above the economic threshold or below the safety level with type I and type 11 error rates of 0.10 and 0.15, respectively.
2. The distribution of greenbugs in seedling oats is consistently geometric. For an oat field, the farmer wants to know whether the mean population density is above the economic threshold of 251linear row foot or below the safety level of IOllinear row foot with type I and type 2 error rates of 0.1 0 and 0.05, respecti vel y.
3. The distribution of snails in streams is negative binomial with a k value of 0.5, when the sampIe unit is a O.I-meter dredge with a standardized net. Develop a test of the hypotheses that the mean population density is above the economic threshold of 2 or below the safety level of I snaiVO.I-meter dredge with type I and type 11 error probabilities ofO.1 and 0.1, respectively.
4. The distribution of grasshoppers in rangeland is often Poisson. Develop a test of the hypotheses that the mean population density is above the economic
190 / Statistical Ecology
threshold of 3.5 or below the safety level of 1.5 grasshoppers/square yard with type I and type 11 error probabilities of 0.15 and 0.10, respectively.
5. If the percentage of damaged oranges in an orchard exceeds 10%, the orchard will usually not be harvested. Develop a test of the hypotheses that the percent damaged fmit is above this threshold versus it is below the safety level of 5% with type I and type 11 error rates of 0.1 and 0.05, respectively.
Consider problems I to 5 again. However, this time
A. Specify the hypotheses.
B. Develop the 2-SPRT to test the specified hypotheses.
C. Compute the OC and ASN functions for the SPRT.
D. Based on the results in C, adjust the error rates if this is needed to obtain a reasonable plan in terms of sampie size or to have the test with error rates more nearly equal to the specified levels.
E. If the test was modified in D, compute the exact OC and ASN functions for the modified test.
