CHOOSING THE GROUP SIZE FOR GROUP TESTING TO ESTIMATE A
PROPORTION
Jacqueline IV1. HUGHES-OLIVER and \iVilliam H. SvVALLOvV
Institute of Statistics Mimeograph Series No. 2209
Janua.ry, 1992
NORTH CAROLINA STATE UNIVERSITY
Raleigh, North Carolina
t
,.
Mimeo
Jacqueline M. HughesSeries
Oliver and William H.
2209
Swallow
Choosing the Group Size for
Group Testing to Estimate a
Proportion
Name
Date
r
Choosing the Group Size for Group Testing to Estimate a
Proportion
Jacqueline M. HUGHES-OLIVER and William H. SWALLOW·
Department of Statistics, North Carolina State University,
Raleigh, North Carolina 27695-8209, U.S.A.
January 1992
Abstract
Group testing, which tests individuals in groups or batches rather that one-at-a-time, can
be very effective in estimating the proportion (P) of individuals that are infected or defective in
some sense, especially when P is small. However, choosing an appropriate group size (Ie) to use is
critical if the potential benefits of group testing are to be realized. Three criteria for selecting Ie
are: (1) choose Ie to make the probability that a group contains one or more defective individuals
equal to 1/2. This is equivalent to equating the influence of a defective group (one with one
or more defective individuals) and a nondefective group on the calculation of the maximum
likelihood estimator Pof Pi (2) choose Ie to minimize the asymptotic variance of Pi (3) choose Ie
to minimize the exact mean squared error of p. For implementation, all three approaches require,
the user to specify an initial value Po for P, and the value of Ie determined will depend in part
on Po. This paper compares the large- and small-sample properties of these three approaches in
detaili none is uniformly best, but (3) is recommended overall. Having determined Ie in advance,
the typical group-testing procedure .fixes Ie for the entire experiment, even though there might
-The authors thank Bibhuti B. Bhattacharyya. and Richard A. Johnson for their many helpful comments and
suggestions.
1
be some later indication that this k was a poor choice. A new multistage method is therefore
proposed; this procedure uses the data accumulated through stage i to re-evaluate the choice of
k to be used in the next stage of data collection. For this method, we show that the resulting
group size is asymptotically independent of Po.
KEY WORDS: asymptotic variance; influence function; maximum likelihood; mean squared
error; multistage procedure; recursive procedure.
1
Introduction
Group-testing has been in use for several decades, though not always under this label. The basic
concept involves collecting data on a group ofindividuals simultaneously, rather than one-at-a-time.
The data might be the result of a test, the measurement of some characteristic, or the recording of
status of some kind, to name a few possibilities.
Although group-testing procedures have received much attention over the years, there has been
a degree of discomfort associated with them because of the sensitivity of the procedures to the
size used for the groups in obtaining the data. Whereas great gains can be achieved by testing
in groups (rather than one-at-a-time) when near-optimal group sizes are used, the losses can be
overwhelming when highly inappropriate group sizes are used. The possibility of such occurrences
limits the appeal of group testing for potential users who prefer a procedure for which one is assured
a minimal level of performance (e.g., testing individuals one-at-a-time) to one that could behave
badly. For this reason, there has been much discussion and many recommendations given in the
literature about how to choose the group size to be used in experimentation. (See, for example,
::. '. ·Gibbs and Gower (1960), Chiang and Reeves (1962), Thompson (1962), Griffiths (1972), Sobel and
Elashoff (1975), Chen (1987), Swallow (1987).)
This paper examines the following three methods for determining the group size when estimating
a proportion: (1) choose the group size that makes the probability that a group shows the trait
2
equal to the probability that a group does not show the trait (Chiang and Reeves, 1962); (2) choose
the group size that minimizes the asymptotic variance of the estimate of the proportion (Thompson,
1962); (3) choose the group size that minimizes the exact mean squared error of the estimate of
the proportion (Thompson, 1962). The resulting group sizes are compared and their effect on the
estimate of the proportion is assessed, both when a large number of groups (large sample) is used
for the estimation procedure, and when a smaJl number of groups (smaJl sample) is used.
In application, each of the methods mentioned above requires the user to specify an initial value
for the proportion, based on whatever prior information or intuition the user may possess. The
group size obtained via each of the methods then depends on this inital value and remains the same
throughout the experiment--even in the event that, as the experiment proceeds, the data collected
begin to indicate that the prior information was "bad," and a different group size should have been
...
used.
A new recursive method for obtaining group sizes through a multistage procedure is therefore
introduced here, and its asymptotic properties determined. This method utilizes the accumulating
information as the data are collected to improve on the initial value specified for the proportion,
and thereby adjusts the group size to be used as the data collection continues.
Section 2 contains a brief introduction to the group-testing estimator for proportions, and
the three nonrecursive methods referenced above for choosing the group size for the estimation
procedure. Section 3 discusses the asymptotic (large-sample) behavior of the group sizes obtained
via the methods of Section 2. Section 4 contains (smaJI-sample and large-sample) comparisons of
the group sizes and the resulting estimates obtained from the methods of Section 2. Section 5
::. '. 'describes and discusses the new recursive method for obtaining the group sizes in a multistage
procedure.
3
2 . Group-testing for estimating a proportion-A review
2.1
The estimator
Suppose one is interested in estimating the proportion p, 0 < p < 1, of individuals possessing a
certain trait in an infinite population. Assume that individuals with this trait are independently
distributed tHf>ughout the population; this excludes situations where, for example, individuals with
"
the trait are clustered. Assume also that this trait is accurately revealed by some "test system,"
that is, there will be no false negatives and no false positives.
Under the usual group-testing scheme for estimation, a total of M individuals are divided into
N groups of size Ie each (M = N
X
Ie). Each of these N groups is then tested and labeled as
including one or more individuals with the trait of interest, or not. Under the assumptions made
in the preceding paragraph, the probability that a group includes at least one member with the
trait is 1 - (1- p)k. Furthermore, if the random variable X is used to represent the number of
the N groups that possess the trait, then X is distributed as binomial with parameters N and
l-(I-p)k.
If no retesting is done (Chen and Swallow (1990) show that retesting, even when possible,
provides little benefit), then the usual group-testing estimator for p is the maximum likelihood
estimator under the binomial model. That is,
(1)
This estimator has positive bias, and mean squared error
where Ok
= 1- (1- p)k.
Furthermore,
p is
asymptotically unbiased, normally distributed and
4
efficient. That is, for fixed k and N
-+ 00,
(3)
2.2
The design parameters N and k
In the usual group-testing scheme, it is assumed that N, the number of groups (or tests), and k,
the size of each group, are determined before data collection begins. In choosing N, one usually
considers constraints related to costs (the budget) and feasibility of implementation (ava.ilable
facilities). The choice of group size, k, may also be constrained. For instance, there might be
practical limitations on the size of the group to, say, no more than 50 individuals. This limitation
might reflect, for example, concern about a dilution effect (false negative) if more than 50 individuals
were to be tested simultaneously, or a violation of the independence assumption when, say, the group
is a number of insect vectors (transmitters) of a disease and their behavior could be affected by
crowding.
If M is fixed, then one-at-a-time or individual testing (k = 1) provides the most information
(Chiang and Reeves, 1962; Thompson, 1962), but at high testing cost. In other words, if the total
number of individuals is fixed, group testing does not offer any advantage over individual testing
in terms of information, but may in terms of cost. On the other hand, if constraints are such that
N is fixed and k is unlimited (the case where individuals are considered very inexpensive, relative
to the tests), then it is not immediately obvious what values of k should be used. Indeed, one
must use some criterion (beyond an upper bound) on which to base the choice of k. The next
'.:. '.. three subsections discuss three such criteria. [In using group testing for estimation, it is, in fact,
commonly the case that N is more or less fixed, and k (and thereby M) can be any value desired.]
Application of these methods is discussed in Section 2.2.4, and some additional guidance in using
r
these methods is given in Section 2.2.5.
5
..
2.2.1
Choosing k: Method 1
Chiang and Reeves (1962) consider a criterion that equaJizes the probability of obtaining a group
that shows the trait and the probability of obtaining a group that does not show the trait. That
is, they choose k such that 1 - {1- p)k = (I - p)k =
i.
In other words, they take k = ki =
lnO.5/ln{l- pl. [This is also the strategy that equates the absolute value of the influence function
for a group that shows the trait and one that doesn't (Chen and Swallow, 1990). Hence, the i in ki
stands for influence.] With this choice of k, one would then obtain the estimate p using (I). The
mean squared error is as given in (2), using k, for k.
2.2.2
Choosing k: Method 2
A method discussed by several authors (Peto, 1953; Thompson, 1962; Kerr, 1971; Griffiths, 1972;
Sobel and Elashoff, 1975) is based on choosing the k that minimizes the asymptotic variance of
p. Recall that this asymptotic variance is N'iig:~t'J' and let ka. be the k that minimizes this
quantity, for fixed p. That is, for fixed p,
Approximations for ka. include ka. == {1.5936/p)-1 (Thompson, 1962) and ka. == -1.5936/ln{l- p)
(Sobel and Elashoff, 1975). By choosing the group size to be ka.' one maximizes the asymptotic
efficiency of the estimator p. Analogous to what was done in Method 1, one would then obtain the
estimate p as described in (I). The mean squared error is as given in (2), using ka. for k.
2.2.3
Choosing k: Method 3
Yet another criterion, similar to that used in Method 2, is based on choosing the k that minimizes
the exact mean squared error of p (Gibbs and Gower, 1960; Thompson, 1962; Griffiths, 1972; Loyer,
6
1983; Swallow, 1985). That is, for fixed p, choose k
ke
=ke' where
=ke(N) =argminl;{MSE(fr, k,p)} =argminl;{N x MSE(fr, k,p)},
and MSE(fr, k,p) is given by (2). By choosing the group size to be ke , one maximizes the (smallsample) efficiency of the estimator p. The estimate is obtained as in (1) and the mean squared
error is as in (2), except that ke replaces k in both formulas.
2.2.4
Choosing k: Practical Use
Whether the group size is obtained by Method 1, 2, or 3, some initial value for the proportion
is needed, since all of these methods require specifying a value for p, one way or another, in the
process of choosing k. Let us represent this initial value for p as Po. For example, Po might be an
assumed upper bound on p, or an estimate of p obtained from historical data. The methods of the
three previous sections would then be implemented with Po in place of p. In particular,
k
iO
lnO.5
=In(l- Po)'
(4)
(5)
(approximations include
kilO
== (1.5936/Po) - 1 and
kilO
== -1.5936/ln(1- Po) ), and
(6)
Of course, if Po is very far from p, these group sizes can be very different from the group sizes
obtained with the true p.
7
2.2.5
Additional comments on practical use
Regardless of the method used for choosing a group size, whether it be one of the methods discussed
above or a different method, there are certain pitfalls to avoid and recommendations for avoiding
these pitfalls. Although the recommendations contained in this section cannot be called methods for
obtaining k, they give more insight into the problems associated with determining an appropriate
k.
The first recommendation is due to Gibbs and Gower (1960). They argue that the possible
values of the estimator
p should not be such that the true value of the proportion is between the
two largest possible values of p. The largest possible value of p is the value one, and this occurs
(*) 11k, and occurs when X = N -1 in (1). Thus,they
/k
~ p ~ 1 because this will result
argue that k should not be chosen in such a way that 1- (*
when X = N. The next highest value is 1-
r
in a very biased estimate. Equivalently, they argue that k should be such that N(I- p)k > 1.
The second recommendation is related to Method 2, given in Section 2.2.2. The approximation
ka
== -1.5936/ln(1 - p) (given in Section 2.2.2) is obtained by solving (1 - p)k == 0.2032 for k.
The solution to this equation is the k that minimizes the asymptotic variance of p. Following Peto
(1953), Figure 1 plots the asymptotic variance of p as a percentage of its minimum value versus
different values of (1 - p)k. We see that this function is indeed minimized at (1 - p)k
= 0.2032,
but we also see that there is a range of values of (1- p)k that are close to the minimum value. For
example, if (1- p)k E [0.1641,0.2477], then there is no more than a 1% increase in the asymptotic
variance, so that any k such that (1 - p)k is in this range might be acceptable. For example, if
p
= 0.05, then this region is 27 ~ k ~ 35, instead of the single value ka = 31.
Figure 1 also shows
that outside of this interval, the rate of increase of the asymptotic variance is greater for values
of k such that (1 - p)k < 0.2032 than for values of k such that (1 - p)k
> 0.2032. Since larger
values of k will result in smaller values of (1- p)k, the message is that the percent increase in the
asymptotic variance is greater when k is larger-than-optimal than when k is smaller-than-optimal.
8
~
as
..
0
CO
"C
~
'0
..
E
::s
E
..
.2
I
'E
~
0
N
E
~
0
..
0
0
0.1
0.2
0.4
0.3
0.5
0.6
(1-p)AJ<
Figure 1: The asymptotic variance of p as a percentage of its minimum value for different values
of (1- p)k.
Hence, if one cannot use the optimal value of k or a value such that (1 - p)k E [0.1641,0.2477],
then it is better to use a smaller-than-optimal value rather than a larger-than-optimal value. This,
in fact, has been the recommendation contained in virtually all of the literature on group testing
(see, e.g., Thompson (1962), and Swallow (1985».
3
Asymptotic behavior of the group sizes obtained from the
three methods
Both Methods 1 and 2, namely equalizing the absolute influence and minimizing the asymptotic
variance, respectively, yield group sizes that are independent of the value of N, that is, constant for
all N. Hence the asymptotic distributional result given in (3) is unchanged, except that ki replaces
9
k for Method 1 and kG replaces k for Method 2.
In contrast, Method 3, minimizing the exact mean squared error, generaJly yields a different
group size for each different value of N. The behavior of ke , particularly as N tends to infinity, is
clearly of interest. Under reasonable assumptions, the following theorem provides a simple limiting
property of ke •
Result 1 If the first derivative of N x MSE(fr,k,p) with respect to k equals zero at most once,
kG < 00, and ke(N)
~
1 for all N
> N'lJl
then limN_ooke(N) = kG, where kG is defined in
Section 2.2.2.
(The proof is obtained as a special case of Result 2 in Section 5.) This result says that under mild
conditions, as N tends to infinity, the group sizes obtained from minimizing the mean squared error
converge to the group size obtained from minimizing the asymptotic variance. Indeed, this is an
expected, but not immediate, result that has the following desirable implication: Itlthough for each
different value of N, ke(N) has the potential for being different, the asymptotic distribution of the
estimator p has the same form as in Method 2. That is, (3) holds with kG in place of k.
As a practical note, as N tends to infinity, kea(N) does not tend to ka. Rather, kea(N) tends
to k ao • That is, the effect of the initial value Po persists even in the limit.
In order to illustrate the mildness of the first assumption of Theorem 1, Figure 2 shows the
typical shape of N x M S E(fr, k, p) as a function of k. Figure 3 shows this same function plotted
against log k, for 1 ~ k
~
120. In both figures, p
= 0.01 and N = 20. The function is not convex,
but it is convex in a region containing the minimum. Outside of this region the function is strictly
. .:. '..increasing. Thus,it is quite reasonable to assume that the function has a derivative that vanishes
at most once, ensuring that Method 3 will generaJly be applicable since it will provide a unique
choice for the value of k.
r
10
....
It)
Q:
~
<g,
w
en
z
0
....
:E
)(
It)
o
o
100
200
300
400
500
600
k
Figure 2: The normaJized mean squared error function N
,,.
11
X
M S E(Pi k, p) for N = 20 and p = 0.01.
....
0
.It)
d
0
c: ....
0
¥.
ew
d
UJ
::e
)(
Z
It)
0
0
d
o
d
0.0
0.5
1.0
1.5
2.0
Iogk
Figure 3: The normaJized mean squared error function N x MSE(y, k,p) for N = 20 and p.= 0.01,
plotted against log k.
12
4
Comparing the three methods
Methods 1 and 2 (equations (4) and (5» share the questionable property that the recommended
group size remains the same; regardless of the number of tests, N, being performed. Method
3 (equation (6» recommends different group sizes for different valueS of N.
The criterion on which Method 1 is based, namely equalizing the absolute influence of a group
showing the trait and the influence of a group not showing the trait, has some appeal, but seems
to be of less practical interest than the criteria behind Methods 2 and 3, namely minimizing the
asymptotic variance or the exact mean squared error, respectively. In practice, these are the
quantities by which estimators are usually evaluated. Comparing Methods 2 and 3, although the
asymptotic variance might be an adequate approximation to the mean squared error when N is
large, it may be very misleading when N is small. Overall, based on these arguments alone, one
might prefer Method 3 for determining the group size to be used for data collection.
Asymptotically, Methods 2 and 3 are equivalent, since limN_co ke(N) = kG' To make an
asymptotic comparison of Methods 1 and 2, one needs to compare the values of the asymptotic
variance of p evaluated when k
= ki and when k = kG'
The asymptotic variance of p when k
= ki
is 2.0814(1- p)2[ln(1 - p)]2/N. The value when k = kG is 1.5441(1- p)2[ln(1- p)]2/N. Hence,
a procedure using kG to obtain the group size is asymptotically 2.0814/1.5441
= 1.35 times more
efficient than one using ki. Moreover, since kG and ke are asymptotically equivalent, a procedure
that uses ke to obtain the group size is also asymptotically 1.35 times more efficient than one using
ki. Therefore, Methods 2 and 3 are preferable to Method 1 in an asymptotic sense.
Tables 1-8 present further comparisons of these methods for choosing the group size for several
...
choices of p, Po, and N. For each value of p, the values of Po shown are Po = 0.5p, p, 1.5p, and
2p. The tables show the values of kiO, kGo and keD for these choices, as well as the true mean
squared error of p obtained with these different group sizes. For example, consider p
.~
Po
=0.01. When
= 0.01 =p, then kiO = 69, kGo = 159 and kea(20) = 65.2. These group sizes would be truncated
13
to 69, 159 and 65, respectively. The (true) mean squared errors (MSE's) of p when N = 20 for
these group sizes are 1.29 x 10-5 , 1.04 X 10-2 and 1.28 x 10-5 , respectively. In this example, the
MSE's with kiO and kea are very similar. When Po = 0.015 = 1.5p, then kiO = 45.9, kao = 105 and
kea (20) = 45.7. The (true) mean squared errors of p when N = 20, are 1.52 x 10-5, 2.08
X
10-4
and 1.52 X 10-5 • Once again, the MSE's with kiO and kea are very similar. Now suppose Po = 0.015
and N = 200. Then kiO = 45.9, kao = 105 and ken = 99.8. The corresponding mean squared errors
are 1.38 x 10-6 , 8.49 X 10-7 and 8.67 x 10-7 • In this case, kao and kea give similar MSE's.
There are several observations that can be made from Tables 1-8. In general, recommended
group sizes decrease as p increases. Also, for fixed N and p, the recommended group size (kiO, kao ,
or ken) decreases as Po increases.
In support of Result 1, it is seen that as N gets larger, ken indeed approaches kao (from below).
It is also seen that kiO is always exceeded by kao , and that kea starts below kiO for small N, then
increases towards kao as N increases.
The dependence of kiO, kao , and ken on the initial value Po, can be summarized as three different
cases. When Po = p, that is, when the initial value equals the true p, then kea provides the smallest
mean squared error. Indeed, this is no surprise since ken was derived to minimize the mean squared
error. However, kiO is a close second for small N and kao is a close second for large N.
Now consider Po
< p.
In this case, all methods choose larger group sizes than they did when
Po = p. When N is such that kiO < kea < kao (usually when N is large), the group size kiO provides
the smallest mean squared error, with ken yielding the next smallest, and kao the largest. H N is
such that k ea < kiO
< kao (usually when N is small), then ken provides the smallest mean squared
"":.. "" "error, with kiO giving the next smallest, and again kao the largest.
Finally, consider Po
> p.
In this case, all methods recommend a smaller group size than they
did when Po = p. When N is such that kiO < ken < kao (usually when N is large), the group
sizes kao and ken yield mean squared errors that are approximately equal, and smaller than that
14
obtained with kiO. If N is such. that keD
< kiO < k 40 (usually when N is small), either keD or kiO
(usually keD) provides the smallest mean squared error, with the other providing the next smallest,
and k 40 the largest.
Thus, there are cases wherein each. of the three methods performs best, sometimes much. better
than the other two, in recommending a group size to reduce the mean squared error of p. However,
the group size keD of Method 3 consistently gave the smallest or second-smallest mean squared
error. It seemingly balances the advantages and disadvantages of the other methods. Of the three
methods considered in this section, we recommend Method 3 overall.
5
An iterative method for choosing the group size through a
multistage procedure
Section 2.2 discussed different methods for obtaining the group size when only one group size is to
be used throughout the experiment. However, in a multistage procedure, one could alter the group
size to be used in the next stage of experimentation according to the data collected as of the end
of the previous stage. For example, at the beginning of an experiment one might have an initial
value, Po, for the proportion, and use that value to obtain a group size by any of the methods of
Section 2.2, or any other method. A number, say Nt, of groups of this size are then tested and an
estimate of the proportion,
PN I , is obtained. It is then decided that data collection will continue,
but that the group size to be used will now be redetermined using
PN I
in place of Po. The group
size for the second stage is then a function of Po and of the data collected in the first stage. The
'.:.....data for the second stage are then collected and a new estimate PN l+ N 2 is obtained, where N 2 is
the number of groups in the second stage. One could repeat the process of obtaining a new group
size based on the information accumulated to that point as often as desired. The question arises
"How do these multistage group sizes behave as the total number of tests increases?" This section
15
addresses that question.
Let N represent the total number of tests to be performed over all stages; AiN is the number
of tests performed in the i'h stage, where i ~ 1 and Al
+... + Ai =
~i ~ 1; PiiN is the estimate
of the proportion obtained at the end of the ith stage; and k;~+lN is the group size for stage i + 1,
rAiN
obtained using the estimate P>'iN' for i ~ 0 and PioN
=Po.
Method 3 of Section 2.2 will be used to determine the group size. That is,
where
and 8h
=1-
(1 - p)h. Result 2 describes the limiting behavior of these group sizes, when the
intermediate estimates P>'iN' i
~
0, are strongly consistent for p.
Result 2 Suppose the estimator P>'iN is such that with probability one, 0 < P>'iN < 1 for all
N
> Ni and PiiN ~1 pas N -
00.
Then, provided that with probability one the first derivative of
Q>'i+ 1N (k;Pi.N) with respect to k equals zero at most once, and ka,
all i
~
where i
< 00, and k;~:~N(N) ~
•
1 for
0 and N > Ni,
~
0 and ka, is as defined in Section 2.2.2.
The proof is given in the Appendix. [Result 1 is obtained as a special case by letting i = 0, Al = 1
and Po = p.] Result 2 says that, even if the proportion used to obtain the group size varies with
N in a random fashion (rather than the deterministic Po), we still have convergence of the group
sizes as N tends to infinity, provided the proportions used to obtain the group sizes converge with
16
probability one to the true proportion, as N tends to infinity. Moreover, the limit is independent
of the initial value Po. In other words, if the intermediate estimates are strongly consistent, then as
the total number of tests increases, the group size for the next stage approaches the optimal group
size with probability one, whatever the value of Po.
6
Discussion and conclusions
Three criteria for determining the size to be used for the groups in the usual group-testing procedure
for the estimation of proportions have been presented and compared. Of these three, the one that
has the greatest practical appeal is the criterion of minimizing the exact mean squared error of
the estimator p. Under this method, the group size selected depends on the number of tests to be
performed, as well as on the initial value Po one uses in place of the true value of the proportion
p. Moreover, as the number of tests increases, this method converges to that based on minimizlng
the asymptotic variance. As a result, when the number of tests is large, one could approximate the
group size obtained from the method based on minimizing the mean squared error by using the
r
group size obtained from minimizing the asymptotic variance. The group size obtained by equating
the absolute value of the influence of a group showing the trait with one not showing the trait
provides an estimate with a smaller mean squared error when the number of tests is small, but is
not, in general, the method of choice.
A new recursive method for obtaining group sizes through a multistage procedure has also
been presented. Although this method is more involved than the single-group-size methods, the
asymptotic properties are very similar. Moreover, this procedure has the property that, under
certain reasonable conditions, as the total number of tests increases, the group size approaches
the optimal group size, even when the value of Po was a poor choice. This is in contrast to the
single-group-size procedures, wherein, as the number of tests increases, the group size approaches
the group size that would be optimal if the value of Po exactly equaled the true p. That is, the
17
single-group-size procedures lack the ability to overcome the misinformation provided by Po, since
they do not alter the group size throughout the experiment as does the recursive method in a
multistage procedure. As the quality of Po is often unknown in practice, the multistage procedure
offers an appealing way to avoid an estimate with potentially large bias and mean squared error,
resulting from a poor choice of Po. The possible advantages of the multistage procedure will be
addressed in detail in a subsequent paper.
References
[1] Chen, C. L. (1987). Estimation problems in group testing. Ph.D. dissertation, North Carolina
State University, Raleigh, N.C.
[2] Chen, C. L. and Swallow, W. H. (1990). Using group testing to estimate a proportion, and to
test the binomial model. Biometrics 46, 1035-1046.
[3] Chiang, C. L. and Reeves, W. C. (1962). Statistical estimation of virus infection rates in
mosquito vector populations. American Joumal of Hygiene 75, 377-391.
[4] Gibbs, A. J. and Gower, J. C. (1960). The use of a multiple-transfer method in plant virus
transmission studies-some statistical points arising in the analysis of results. Annals of Applied
Biology 48(1), 75-83.
[5] Griffiths, D. A. (1972). A further note on the probability of disease transmission. Biometrics
28, 1133-1139.
-::.
".
[6] Kerr, J. D. (1971). The probability of disease transmission. Biometrics 27, 219-222.
[7] Loyer, M. (1983). Bad probability, good statistics, and group testing for binomial estimation.
The American Statistician 37(1),57-59.
18
[8] Peto, S. (1953). A dose response equation for the invasion of micro-organisms. Biometrics 9,
320-335.
[9] Sobel, M. and Elashoff, R. M. (1975). Group testing with a new goal, estimation. Biometrika
62, 181-193.
[10] Swallow, W. H. (1985). Group testing for estimating infection rates and probabilities of disease
transmission. Phytopathology 75, 882-889.
[11] Swallow, W. H. (1987). Relative mean squared error and cost considerations in choosing group
size for group testing to estimate infection rates and probabilities of disease transmission. Phy-
topathology 77, 1376-1381.
[12] Thompson, K. H. (1962). Estimation of the proportion of vectors in a natural population of
insects. Biometrics 18, 568-578.
APPENDIX:
PROOF OF RESULT 2
Define the function Q(kjp) =
J(l~;~~'J. The proof consists of two main parts. First, it will be
shown that
Then it will be shown that
--:.. '- -Part 1: Show that for all i
~
0
19
Proof: It is convenient to write the function QlIi+1 N (kjp>'iN ) as a conditional expectation:
where
and
Let i>'iN= !(1 - 6iiN ) for N ~ 1. Then 0 < i>'iN < 1, 6>'iN + i>'iN < 1 for all N ~ 1, and
6>'iN ~1 !(1- 6) as N _
00,
where 6 = 1- (1- p)h. Then
Case 1: YAi+1N ~ (6iiN + i>'iN)Ai+lN
Then
which implies that
'".:. "...
20
This implies that
Moreover,
So that
is uniformly integrable. But (1-oiiN-iiiN)2
and !(YAi+lN)
~
~1 l(1-0)2, I (YAi+lN < (OiiN +iiiN)Ai+lN) ~ 1
[P(t-6)] X~. Hence
which implies
21
Therefore,
Part !: Show that
Proof: Since Q(klliP) < Q(kiP) if k :f:
kll,
then for all 0 > 0, there exists € > 0 (€ depends on 0)
such that
Q(kll - OiP) - Q(klliP) > €
and
Q(kll + 0iP) - Q(klliP) >
€.
Furthermore, by Part 1 above, with probability one there exists Ne such that for all N
i'
B
IQ'\;+1N(kB + iiPi;N) - Q(k + OiP)1 < i·
IQ'\;+IN(kll - DiPi;N) - Q(k
Hence, with probability one, for all N
> oNe,
22
ll
-
DiP)1 <
> Nt.,
which implies
Similarly,
Since Q>'i+lN(k;p~iN) is continuously differentiable in k, then its derivative with respect to k equals
zero for some k E (kCl - 6, k Cl +6), N
is, limN_co k:~+lN(N) E (k Cl
"iN
-
> N u k:~+1N(N) E (k
"iN
Cl
-
6, k Cl +6) with probability one. That
6,kCl + 6) with probability one. But 6 is arbitrary, so
23
Table 1
Group sizes, and their associated mean squared errors, arising from three different
methods, for p
...
=0.001.
P
Po
N
k;o
0.001
0.0005
0.001
0.0010
0.001
0.0015
0.001
0.0020
5
10
15
20
25
30
40
50
100
150
200
5
10
15
20
"25
30
40
50
100
·150
200
5
10
15
20
25
30
40
50
100
150
200
5
10
15
20
25
30
40
50
100
150
200
1386 2.37 x 10- 1
5.63 x 10- 2
1.34x10- 2
"
3.17x10- 3
7.53x10- 4
1.79 x 10- 4
1.01 X 10- 5
6.04 X 10- 7
1.67 X 10- 8
1.09 x 10- 8
8.05 x 10- 9
693 3.12xl0- 2
9.75 X 10- 4
3.06 x 10- 5
1.07 x 10- 6
1.24 x 10- 7
7.75xl0- 8
5.59 x 10- 8
4.40 X 10- 8
2.14 x 10- 8
1.41 x 10- 8
1.05 x 10- 8
462 6.92 x 10- 3
4.83 x 10 - 5
5.46 x 10- 7
1.56 x 10- 7
1.20 x 10- 7
9.84 x 10- 8
7.24xlO- 8
5.73 x 10- 8
2.81 x 10- 8
1.86 x 10 - 8
1.39 x 10- 8
346 2.15x 10- 3
5.04 x 10- 6
2.70 x 10 - i
1.89 x J.O - 7
1.48 x 10- 7
1.22 x 10- 7
9.01 x 10- 8
7.14xlO- 8
3.51xlO- 8
2.32 x 10- 8
1.74 x 10-8.
MSE{Pik;oIP)
kao
MS~PikaoIP)
3186 8.08 x 10- 1
6.55 x 10- 1
5.31 x 10- 1
"
4.30 x 10- 1
3.48 x 10- 1
2.82 x 10- 1
"
1.85 X 10- 1
1.21 x 10- 1
1.48 x 10- 2
1.80 x 10- 3
2.19 x 10 - 4
159:' 3.21 x 10- 1
1.03x10- 1
3.31 x 10- 2
1.06xl0- 2
3.41 x 10 - 3
1.l0xl0- 3
1.13 x 10- 4
1.17xlO- 5
1.69 x 10- 8
1.09 x 10- 8
8.03 x 10- 9
1062 1.20 x 10- 1
1.43 x 10 - 2
1.72 x 10- 3
2.06 x 10- 4
2.48 x 10 - 5
3.03 x 10- 6
8.98 x 10- 8
3.74 x 10- 8
1.75 x 10 - 8
1.15xl0- 8
8.57 x 10- 9
796 4.98 x 10 - 2
2,48 x 10- 3
1.24 x 10 - ..
6.30 x 10- 6
3.97 x 10- 7
8.71 x 10- 8
5.22 x 10- 8
4.IOx 10- 8
1.98xl0- 8
1.31 x 10 - 8
9.74xl0- 9
keo
!JsEt Pi keol p)
98.6
364
646
905
1137
1345
1700
1993
2840
2953
3015
62.4
209
358
492
610
716
895
1041
1427
1481
1508
47.8
151
253
344
424
495
615
712
953
987
1004
39.6
120
198
267
328
381
471
543
715
740
753
1.01 x 10- 5
7.43 x 10- 6
1.48 x 10- 5
3.18 x 10- 5
6.35 x 10- 5
1.17xl0- 4
3.14xl0- 4
6.64 x 10- 4
2.44 x 10- 3
3.26 x 10- 4
4.33 x 10- 5
5.05 x 10- 6
6.78·x 10- 7
2.69 x 10- 7
1.53 x 10- 7
1.04 x 10- 7
7.71xlO- 8
5.04 x 10- 8
3.73 x 10- 8
1.66 x 10- 8
1.08x10- 8
8.02 x 10- 9
5.66 x 10- 6
8.22 x 10- 7
3.33 x 10- 7
1.90 x 10- 7
1.27 x 10- 7
9.37 x 10- 8
5.99 x 10- 8
4.34 x 10- 8
1.82 x 10- 8
1.18 x 10- 8
8.74xl0- 9
6.59xl0- 6
1.01xlO- 6
4.09x 10- 7
2.32 x 10- 7
1.55 x 10- 7
1.13 x 10- 7
7.14xl0- 8
5.12 x 1.0- 8
2.10xl0- 8
1.36 x 10- 8
1.00xI0~8
Table 2
Group sues, and their associated mean squared errors, arising from three different
methods, for p = 0.005.
P
Po
N
kio
MSE{fi;kiO'p)
k ao
AIS ~ p; k ao' p)
0.005
0.0025
.277
0.0050
0.005
0.0075
0.005
0.0100
2.36xI0- 1
5.61 x 10- 2
1.33 x 10- 2
3.18 x 10- 3
7.58xI0- 4
1.82 x 10- 4
1.14 x 10- 5
1.47 x 10- 6
4.15 x 10- 7
2.70 x 10- 7
2.00 x 10- 7
3.09 x 10- 2
9.74 x 10- 4
3.45 x 10- 5
3.97 x 10- 6
2.36 x 10- 6
1.90 x 10- 6
1.39 x 10- 6
1.10xl0- 6
5.32xl0- 7
3.52xl0- 7
2.62xl0- 7
6.85 x 10- 3
5.59 x 10 - 5
5.63 x 10- 6
3.82 x 10- 6
2.98 x 10- 6
2.45 x 10- 6
1.80 x 10- 6
1.43xl0- 6
6.99 x 10- 7
4.63 x 10- 7
3.46 x 10- 7
2.14 x 10 - 3
1.50 x 10 - 5
6.49 x 10- 6
4.70x 10- 6
3.69 x 10- 6
3.04 x 10- 6
2.24 x 10- 6
1.78 x 10 - 6
8.74xl0- 7
5.80 x 10 - 7
4.33x 10- 7
637
0.005
5
10
15
20
25
30
40
50
100
150
200
5
10
15
20
25
30
40
50
100
150
200
5
10
15
20
25
30
40
50
100
150
200
5
10
15
20
25
30
40
50
100
150
200
8.03 x 10- 1
6.51xl0- 1
5.27 X 10- 1
4.27 X 10- 1
3.47 x 10- 1
2.81xl0- 1
1.85 X 10- 1
1.21 X 10- 1
1.49 X 10- 2
1.82 x 10- 3
2.23 x 10- 4
3.18xI0- 1
1.02xl0- 1
3.28xl0- 2
1.05 x 10 - 2_
3.39 x 10- 3
1.09 x 10 - 3
1.13xlO- 4
1.25 x 10- 5
4.18x 10- 7
2.70 x 10- 7
2.00 x 10- 7
1.18xl0- 1
1.42 x 10 - 2
1.70 x 10 - 3
2.05 x 10- 4
2.62xl0- 5
4.53xl0- 6
1.22 x 10- 6
9.16xlO- 7
4.36x 10- 7
2.87 x 10- 7
2.13 X 10- 7
4.91xl0- 2
2.44 x 10- 3
1.25 x 10- 4
8.84 x 10- 6
2.49x 10- 6
1.80 x 10 - 6
1.30 x 10 - 6
1.02 x 10 - 6
4.93 X 10- 7
3.25 x 10- 7
2,43 x 10- 7
138
92.1
69.0
318
212
159
keo
34.2
101
164
220
269
311
383
441
572
592
602
21.7
·58.0
91.2
120
144
166
202
230
286
296
301
16.7
42.0
64.7
83.9
100
115
138
157
190
197
200
13.8
33.5
50.7
65.2
77.6
88.3
106
119
143
148
150
MS~p;keo,P)
1.39 x 10- 4
1.03x10- 4
1.73 x 10- 4
3.11 x 10- 4
5.30 x 10- 4
8.41 x 10- 4
1.75xl0- 3
2.95 x 10- 3
2.85 x 10- 3
3.60 x 10- 4
4.43 x 10- 5
7.46 x 10 - 5
1.29 x 10 - 5
5.63 x 10- 6
3.37 x 10- 6
2.35 x 10- 6
1.79 x 10- 6
1.20 x 10- 6
9.06 x 10- 7
4.14xl0- 7
2.69 x 10- 7
2.00 x 10- 7
8:29 x 10- 5
1.54 x 10 - 5
6.81 x 10- 6
4.07 x 10- 6
2.81 x 10- 6
2.12xI0- 6
1.39 x 10 - 6
1.03 x 10- 6
4.54 x 10- 7
2.94 x 10- 7
2.18xI0- 7
9.56 x 10- 5
1.86 x 10 - 5
8.26 x 10- 6
4.91xlO- 6
3.37 x 10- 6
2.52xl0- 6
1.64 x 10- 6
1.20 x 10- 6
5.23 x 10- 7
3.39 x 10- 7
2.50 x 10- 7
Table 3
Group sues, and their associated mean squared errors, arising from three different
methods, for P = 0.010.
.'.:.
-.
P
Po
N
k io
MSFi... p;kiOIP)
kao
MS~p;kaoIP)
keo
MS~p;keoIP)
0.010
0.0050
138
0.0100
0.010
0.0150
0.010
0.0200
2.34 x 10- 1
5.59 x 10- 2
1.33 x 10- 2
3.19 x 10- 3
7.68xl0- 4
1.88 x 10- 4
1.50 x 10- 5
4.14 x 10- 6
1.65 x 10- 6
1.07 x 10- 6
7.97xI0- 7
3.07 x 10- 2
9.85 x 10- 4
4.69 x 10- 5
1.29x 10- 5
9.31 x 10- 6
7.57 x 10- 6
5.53 X 10- 6
4.36 x 10- 6
2.12 x 10- 6
1.40 x 10 - 6
1.04 x 10- 6
6.81 x 10- 3
8.07 x 10 - 5
2.14x10- 5
1.52 x 10 - 5
1.19xlO- 5
9.74 x 10- 6
7.18 x 10- 6
5.68 x 10- 6
2.78 x 10- 6
1.84 x 10 - 6
1.38 x 10- 6
2.16 x 10- 3
4.59 x 10 - 5
2.58 x 10 - 5
1.87 x 10 - 5
1.47 x 10 - 5
1.21 x 10 - 5
8.94 x 10- 6
7.09 x 10- 6
3.49 x 10- 6
2.31 x 10- 6
1.73 x 10 - 6.
318
0.010
5
10
15
20
25
30
40
50
100
150
200
5
10
15
20
25
30
40
50
100
150
200
5
10
15
20
25
30
40
50
100
150
200
5
10
15
20
25
30
40
50
100
150
200
7.95xIO- 1
6.45 x 10- 1
5.23 x 10- 1
4.25 x 10- 1
3.45 x 10- 1
2.80 x 10- 1
1.84 x 10- 1
1.21 x 10- 1
1.50x10- 2
1.85 x 10- 3
2.30 x 10-"
3.15xIO- 1
1.01 X 10- 1
3.25 x 10 - 2
1.04 x 10 - 2
3.36 x 10- 3
1.08 x 10- 3
1.16x10- 4
1.51x10- 5
1.66 x 10 - 6
1.08x10- 6
7.96 x 10- 7
1.17xlO- 1
1.39 x 10 - 2
1.67 x 10 - 3
2.08 x 10- 4
3.17 x 10 - 5
9.32 x 10- 6
4.71 x 10- 6
3.64 x 10- 6
1.74 x 10- 6
1.14xlO- 6
8,49xl0- 7
4.82x 10- 2
2.39 x 10- 3
1.33 x 10 - 4
l.71x10- 5
9.02 x 10- 6
7.12xlO- 6
5.17 x 10- 6
4.06x 10- 6
1.97 x 10- 6
1.30 x 10 - 6
9.67 x 10- 7
21.7
58.0
91.2
120
144
166
202
230
286
296
301
13.8
33.5
50.7
65.2
77.6
88.3
106
119
143
148
150
10.6
24.3
35.9
45.7
54.0
61.0
72.5
81.0
94.8
98.1
99.8
8.8
19.3
28.1
35.5
41.7
46.9
55.3
61.5
71.0
73.4
74.7
4.23 x 10- 4
3.07 x 10- 4
4.76xI0- 4
7.88x·l0- 4
1.25 x 10- 3
1.85 x 10- 3
3.42 x 10- 3
5.21 x 10- 3
2.89 x 10- 3
3.69 x 10- 4
4.73xl0- 5
2.37 x 10- 4
4.58 x 10 - 5
2.08 x 10 - 5
1.28 x 10 - 5
9.01 x 10- 6
6.92 x 10- 6
4.71 x 10- 6
3.58xl0- 6
1.65 x 10- 6
1.07 x 10- 6
7.94 x 10- 7
2.62 x 10- 4
5.40 x 10- 5
2.49 x 10 - 5
1.52xl0- 5
1.07 x 10 - 5
8.10xl0- 6
5.39xl0- 6
4.01 x 10- 6
1.80 x 10- 6
1.17xl0- 6
8.67 x 10- 7
3.00 x 10- 4
6.49 x 10 - 5
3.00 x 10 - 5
1.82 x 10- 5
1.27 x 10 - 5
9.59 x 10- 6
6.31 x 10- 6
4.66 x 10- 6
2.08 x 10- 6
1.35 x 10- 6
9.97 x 10- 7
69.0
45.9
34.3
159
105
i8.9
•
Table ..
Group sIZes, and their associated mean squared errors, arising from three different
methods, for
."
p = 0.020.
P
Po
N
k io
MSE{pjkio,p)
k ao
MS!lpjkao,p)
keo
MS!lpjkeo,p)
0.020
0.0100
69.0
0.0200
0.020
0.0300
0.020
0.0400
2.31 x 10- 1
5.54x10- 2
1.33 x 10- 2
3.23 x 10- 3
7.99 x 10- 4
2.10xl0- 4
2.89 x 10- 5
1.47 x 10 - 5
6.52xl0- 6
4.25 x 10- 6
3.16 x 10- 6
3.02 x 10- 2
1.05 x 10- 3
9.63 x 10- 5
4.83 x 10 - 5
3.67 x 10- 5
2.99 x 10- 5
2.19 x 10 - 5
1.72 x 10- 5
8.38xl0- 6
5.54 x 10- 6
4.13xl0- 6
6.8IxI0- 3
1.80 x 10 - 4
8.34 x 10 - 5
6.00 x 10 - 5
4.69xI0- 5
3.86 x 10 - 5
2.84 x 10 - 5
2.25xI0- 5
I.IOxlO- 5
7.31 x 10 - 6
5.46 x 10- 6
2.36x 10- 3
1.68 x 10 - 4
1.02 x 10 - 4
7.4IxlO- 5
5.83 x 10 - 5
4.80xI0- 5
3.55 x 10 - 5
2.82 x 10 - 5
1.39 x 10 - 5
9.18x 10- 6
6.87 x 10- 6
159
0.020
5
10
15
20
25
30
40
50
100
150
200
5
10
15
20
25
30
40
50
100
150
200
5
10
15
20
25
30
40
50
100
150
200
5
10
15
20
25
30
40
50
100
150
200
7.81x10- 1
6.34 x 10- 1
5.16xl0- 1
4.19 x 10- 1
3.41xl0- 1
2.77xl0- 1
1.83 x 10- 1
1.21 x 10- 1
1.52 x 10 - 2
1.92 X 10- 3
2.46 x 10- 4
3.09 x 10- 1
9.9IxI0- 2
3.19 x 10 - 2
1.03 x 10- 2
3.32 x 10- 3
1.08 x 10- 3
1.28x10- 4
2.57 x 10- 5
6.57 x 10- 6
4.26 x 10- 6
3.15xI0- 6
1.14x10- 1
1.35 x 10- 2
1.65 x 10- 3
2.30 x 10- 4
5,43 x 10 - 5
2.84 x 10 - 5
1.85 x 10 - 5
1.44 x 10 - 5
6.88 x 10- 6
4.52x 10- 6
3.36 x 10- 6
4.65x 10- 2
2.35 x 10- 3
1.72 x 10 - 4
5.02 x 10 - 5
3,48 x 10- 5
2.8IxlO- 5
2.05 x 10 - 5
1.61 X 10- 5
7.80x 10- 6
5.15 x 10 - 6
3.84 X 10- 6
13.8
33.5
50.7
65.2
77.6
88.3
106
119
143
148
150
8.8
19.3
28.1
35.5
41.7
46.9
55.3
61.5
71.0
73.4
74.7
6.8
14.0
19.9
24.8
28.9
32.3
37.7
41.5
47.1
48.7
49.6
5.6
ILl
15.6
19.2
22.2
24.8
28.7
31.3
35.2
36.4
37.0
1.27 x 10- 3
8.98 x 10- 4
1.27 x 10- 3
1.93 x 10- 3
2.82 x 10- 3
3.90 x 10- 3
6.39 x 10- 3
8.68 x 10- 3
3.00 x 10- 3
3.91 x 10- 4
5.43 x 10- 5
7.43 x 10- 4
1.62 x 10- 4
7.69xl0- 5
4.81xlO- 5
3.45xl0- 5
2.67 x 10- 5
1.84 x 10 - 5
1.41 x 10- 5
6.52 x 10- 6
4.24 x 10- 6
3.14 x 10- 6
8.15 x 10- 4
1.89 x 10- 4
9.08 x 10- 5
5.67 x 10- 5
4.02 x 10- 5
3.09xl0- 5
2.08 x 10- 5
1.57x10- 5
7.15 x 10- 6
4.64 x 10- 6
3.43 x 10- 6
9.27 x 10- 4
2.25 x 10- 4
1.09 x 10- 4
6.75xl0- 5
4.77xl0- 5
3.64 x 10 - 5
2.43 x 10 - 5
1.82 x 10 - 5
8.26 x 10- 6
5.35 x 10- 6
3.96 x 10- 6
34.3
22.8
17.0
78.9
52.3
39.0
Table 5
Group sizes, and their associated mean squared errors, arising from three different
methods, for p 0.050.
=
P
Po
N
kio
MSE{pj kio' p)
k ao
AfSE(pj kao' p)
keo
MSE(pjkeo,p)
0.050
0.0250
27.4
0.0500
0.050
0.0750
2.21 x 10- 1
5.42 x 10- 2
1.35 x 10- 2
3.47 X 10- 3
9.82 x 10- 4
3.49 x 10- 4
1.22 x 10- 4
8.54xl0- 5
3.94 x 10- 5
2.57 x 10- 5
1.91xl0- 5
2.92 x 10 - 2
1.53 x 10 - 3
4.26 x 10 - 4
2.84 x 10 - 4
2.20 x 10- 4
1.80 x 10 - 4
1.32 x 10 - 4
1.04 x 10- 4
5.07 x 10 - 5
3.35 x 10- 5
2.50 x 10 - 5
7.61 x 10- 3
8.44xl0- 4
4.99 x 10- 4
3.61 X 10- 4
2.83 x 10- 4
2.33 x 10- 4
1.72 x 10- 4
1.37 x 10 - 4
6.71 X 10- 5
4.45 x 10 - 5
3.33 x 10 - 5
3.92 x 10- 3
9.80 x 10- 4
6.15xl0- 4
4.50x 10- 4
3.54 x 10- 4
2.92 x 10 - 4
2.17 x 10 - 4
1.72 x 10 - <I
8.48xl0- 5
5.63 x 10 - 5
4.21 X 10- 5
62.9
0..050
5
10
15
20
25
30
40
50
100
150
200
5
10
15
20
25
30
40
50
100
150
200
5
10
15
20
25
30
40
50
100
150
200
5
10
15
20
25
30
40
50
100
150
200
7.37xl0- 1
6.03 x 10- 1
4.92 x 10- 1
4.02 X 10- 1
3.29 X 10- 1
2.68 x 10- 1
1.79 x 10- 1
1.20 x 10- 1
I.S9x 10- 2
2.15 x 10- 3
3.14xl0- 4
2.90 x 10- 1
9.33 x 10 - 2
3.02 x 10 - 2
9.84 x 10- 3
3.28 x 10- 3
1.15 x 10- 3
2.17 x 10 - 4
9.78 x 10 - 5
3.97xl0- 5
2.57 x 10- 5
1.90xl0- 5
1.05 x 10- 1
1.25 x 10 - 2
1.75 x 10- 3
4.14xl0- 4
2.11 x 10- 4
1.57 x 10- 4
l.11xlO- 4
8.71 x 10- 5
4.17xl0- 5
2.74xl0- 5
2.04 x 10 - 5
4.21 x 10- 2
2.50 x 10- 3
4.70x 10- 4
2.75 x 10- 4
2.09 x 10- 4
1.70 x 10 - 4
1.24 x 10-<1
9.79x 10- 5
4.76xlO- 5
3.14 X 10- 5
2.34xl0- 5
7.6
16.2
23.3
29.2
34.1
38.2
44.8
49.6
56.6
58.6
59.6
4.9
9.3
12.9
15.8
18.1
20.1
23.1
25.1
28.0
28.9
29.4
3.8
6.7
9.1
10.9
12.5
13.7
15.5
16.7
18.5
19.1
19.4
3.1
5.3
7.1
8.4
9.5
10.4
11.7
12.4
13.7
14.1
14.4
5.13 x 10- 3
3.52 x 10- 3
4.37 x 10- 3
5.87 x 10- 3
7.71xl0- 3
9.69 x 10- 3
1.33 x 10- 2
1.51 x 10 - 2
3.32 x 10- 3
4.86 x 10- 4
9.11 x 10- 5
3.24 x 10- 3
8.40xl0- 4
4.24 x 10- 4
2.74xl0- 4
2.00 x 10- 4
1.57 x 10- 4
1.l0xl0- 4
8.44 x 10 - 5
3.94 x 10- 5
2.57 x 10- 5
1.90 x 10 - 5
3.51 x 10- 3
9.64xl0- 4
4.92 x 10- 4
3.17xl0- 4
2.29 x 10- 4
1.78xl0- 4
1.23 x 10- 4
9.36 x 10 - 5
4.33 x 10- 5
2.81 x 10- 5
2.08 x 10 - 5
3.94)(10- 3
1.14xl0- 3
5.84 x 10- 4
3.75 x 10- 4
2.71 x 10- 4
2.10 x 10- 4
1.44 x 10 - 4
1.09 x 10 - 4
5.03 x 10 - 5
3.26 x 10- 5
2.41 x 10 - 5
•
0.050
...
0.1000
13.5
8.9
"
6.6
31.1
20.4
15.1
Table 6
Group sues, and their associated mean squared
error~,
arising from three different
methods, for p = 0.100.
0"
•
P
Po
N
kio
MSF{p;kiO'p)
k ao
MSElp;kao'p)
keo
0.100
0.0500
la·5
0.1000
0.100
0.1500
0.100
0.2000
2.05 x 10- 1
5.26 x 10- 2
1.41 x 10- 2
4.18x 10- 3
1.55 x 10- 3
7.92xl0- 4
4.26 x 10- 4
3.19 x 10- 4
1.48 x 10- 4
9.69 x 10 - 5
7.20xl0- 5
2.89 x 10 - 2
3.15xl0- 3
1.50 x 10 - 3
1.05xl0- 3
8.21 x 10- 4
6.73 X 10- 4
4.95xl0- 4
3.91 x 10- 4
1.91xlO- 4
1.27 x 10- 4
9.46 x 10 - 5
l.11xlO- 2
2.98 x 10- 3
1.86 x 10- 3
1.36 x 10 - 3
1.07 x 10 - 3
8.83 x 10 - 4
6.54xl0- 4
5.19x 10- 4
2.56 x 10 - 4
1.70 x 10 - 4
1.27 x 10 - 4
9.21xlO- 3
3.63 x 10- 3
2.33 x 10- 3
1.71xlO- 3
1.35 x 10 - 3
1.12xl0- 3
8.33 x 10- 4
6.63 x 10- 4
3.28 x 10- 4
2.18xl0- 4
1.63 x 10 - 4
31.1
0.100
5
10
15
20
25
30
40
50
100
150
200
5
10
15
20
25
30
40
50
100
150
200
5
10
15
20
25
30
40
50
100
150
200
5
10
15
20
25
30
40
50
100
150
200
6.68 x 10- 1
5.51 x 10- 1
4.54xl0- 1
3.74 x 10- 1
3.09 X 10- 1
2.54 X 10- 1
1.73 x 10- 1
1.18 x 10- 1
1.73 x 1O~ 2
2.66 x 10- 3
4.96 x 10- 4
2.61 x 10 - 1
8.44 x 10 - 2
2.78 x 10 - 2
9.48 x 10 - 3
3.49 x 10 - 3
1.49 x 10 - 3
5.19 x 10 - 4
3.36 x 10- 4
1.50 x 10- 4
9.71 X 10- 5
7.19xl0- 5
9.15xl0- 2
1.18 x 10- 2
2.41 x 10 - 3
1.06 x 10 - 3
;.:10 x 10 -·1
5.81 X 10-.1
4.19xlO-·1
3.28 x 10- 4
1.58 x 10 - 4
1.04xl0- 4
7.76xl0- 5
3.67xl0- 2
3.65xl0- 3
1.48 x 10 - 3
1.02 x 10 - 3
7.90xl0- 4
6.46 x 10 - 4
4.74x 10- 4
3.74xlO- 4
1.82 x 10 - 4
1.21xlO- 4
9.02 x 10 - 5
1.38 x 10- 2
4.9
9.32 x 10- 3
9.3
12.9 1.05 x 10- 2
15.8 1.28 x 10- 2
18.1 1.54 x 10- 2
20.1 1.79 x 10- 2
23.1 2.12xlO- 2
25.1 2.05 x 10- 2
28.0 4.01 x 10- 3
28.9 7.27xl0- 4
29.4 2.05 x 10- 4
9.21 x 10- 3
3.1
2.79 x 10- 3
5.3
1.48 x 10- 3
7.1
9.83 x 10- 4
8.4
7.29 x 10- 4
9.5,
10.4 5.78 x 10- 4
11.7 . 4.09 x 10- 4
12.4 3.17xl0- 4
13.7 1.48 x 10- 4
14.1 9.68 x 10- 5
14.4 7.18 x 10- 5
9.90 x 10- 3
2.4
3.16 x 10- 3
3.9
1.70 x 10- 3
4.9
1.13xl0- 3
5.8
8.29xl0- 4
6.4
6.53 x 10- 4
7.0
4.57
x 10- 4
7.7
3.52
x
10- 4
8.1
4
1.64
x
108.9
4
1.07
x
109.2
7.91 x 10- 5
9.3
1.10xl0- 2
2.0
3.69
X 10- 3
3.0
2.01
x
10- 3
3.8
1.33
x
10- 3
4.4
4
9.80
x
104.9
4
7.71xlO5.2
5.38 x 10- 4
5.7
4
4.14xl06.0
4
1.93
x
106.5
1.25 x 10- 4
6.7
9.27 x 10- 5
6.8
6.6
4.3
3.1
15.1
9.8
7.1
MSElp;keo'p)
Table 7
Group sIZes, and their associated mean squared errors, arising from three different
methods, for p = 0.200.
P
Po
N
k io
MSF{ Pi kiO' P)
l~ao
J.!SD.Pi kao' p)
keo
0.200
0.1000
6.6
0.2000
0.200
0.3000
0.200
0.4000
1.76xl0- 1
5.02 x 10- 2
1.62 x 10- 2
6.42x10- 3
3.37 x 10- 3
2.25 x 10- 3
1.45 x 10- 3
1.11 x 10- 3
5.21 x 10- 4
3.41xl0- 4
2.53 X 10- 4
3.21 x 10- 2
8.36 x 10- 3
4.97 x 10- 3
3.60x 10- 3
2.83 x 10- 3
2.33 x 10- 3
1.72 x 10- 3
1.37 x 10 - 3
6.73 x 10- 4
4.47xI0- 4
3.34 x 10- 4
2.32 x 10 - 2
9.97 x 10- 3
6.45 x 10- 3
4.78 x 10- 3
3.79 x 10- 3
3.14xI0- 3
2.34 x 10- 3
1.87 x 10- 3
9.27x 10- 4
6.16x 10- 4
4.62x 10- 4
2.65 x 10": 2
1.27 x 10 - 2
8.37 x 10- 3
6.24 x 10- 3
4.98 x 10- 3
4.14xI0- 3
3.IOx 10- 3
2.47 x 10- 3
1.23 x 10 - 3
8.21 x 10- 4
6.16 x 10 - ...
15.1
0.200
5
10
15
20
25
30
40
50
100
150
200
5
10
15
20
"25
30
40
50
100
150
200
5
10
15
20
25
30
40
50
100
150
200
5
10
15
20
25
30
40
50
100
150
200
5.39 x 10 - 1
4.53 x 10- 1
3.81 x 10- 1
3.20 x 10- 1
2.69 X 10- 1
2.26 x 10- 1
1.60 x 10- 1
1.13 x 10- 1
2.05 x 10- 2
4.13xlO- 3
I.lIxlO- 3
2.08 x 10- 1
6.90 X 10- 2
2.44 x 10- 2
9.7IxI0- 3
4.63 x 10- 3
2.74 x 10- 3
1.54 x 10- 3
1.14 X 10- 3
5.24 x 10- 4
:1.42 x 10- 4
2.54 x 10- 4
7.05 x 10- 2
1.26 x 10- 2
5.04 x 10 - 3
3.25 x 10- 3
2.47 x 10 - 3
2.01 X 10- 3
1.47x10- 3
1.16 x 10- 3
5.63 x 10- 4
3.72x 10- 4
2.78 x 10- 4
3.24 x 10 - 2
8.36 x 10- 3
4.96 x 10- 3
3.59 x 10- 3
2.82 x 10- 3
2.33 x 10 - 3
1.72 x 10 - 3
1.36xl0- 3
6.72x 10- 4
4A5x 10- 4
:1.:1:, x 10 - ..
3.1
5.3
7.1
8.4
9.5
10.4
11.7
12.4
13.7
14.1
14.4
2.0
3.0
3.8
4.4
4.9
5.2
5.7
6.0
6.5
6.7
6.8
1.6
2.2
2.6
2.9
3.2
3.4
3.6
3.8
4.1
4.2
4.3
1.3
1.7
2.0
2.2
2.3
2.5
2.6
2.7
2.9
3.0
3.0
"
-::.
".
3.1
1.9
1.4
7.1
4.5
3.1
.MSD.Pikeo,P)
3.25 x 10 - 2
2.24x10- 2
2.31 x 10- 2
2.57 x 10- 2
2.84x10- 2
3.04 x 10 - 2
3.09 x 10- 2
2.64 x 10- 2
5.91 x 10- 3
1.52 x 10- 3
6.17x10- 4
2.31 X 10- 2
8.35x 10- 3
4.74x10- 3
3.25 x 10- 3
.2.46x10- 3
1.97 x 10- 3
1.41 x 10- 3
I.lOx10- 3
5.21 x 10- 4
3.41 x 10- 4
2.53 x 10 - 4
2.46 x 10- 2
9.35 x 10- 3
5.39x10- 3
3.70x10- 3
2.79 x 10- 3
2.23 x 10- 3
1.59 x 10- 3
1.24 x 10- 3
5.83 x 10- 4
3.81 x 10- 4
2.83 x 10- 4
2.67 x 10 - 2
1.08 x 10 - 2
6.37 x 10- 3
4.41 x 10- 3
3.34 x 10- 3
2.68 x 10- 3
1.91 x 10 - 3
1.49 x 10- 3
7.03 x 10- 4
4.60 x 10- 4
3.41 x 10- 4
Table 8
Group sues, and their associated mean squared errors, arising from three different
methods, for P = 0.300.
P
Po
N
kio
MSE(PikiO'P)
kao
MSElPikao,P)
keo
MSElPikeo,P)
0.300
0.1500
.4.3
0.3000
0.300
0.4500
0.300
0.6000
1.48 x 10- 1
4.81 X 10- 2
1.86 x 10- 2
9.09 x 10- 3
5.61 x 10- 3
4.09 x 10- 3
2.77xl0- 3
2.13xl0- 3
1.01 X 10- 3
6.63 x 10-"
4.93 x 10-"
3.80 x 10 - 2
1.46x10- 2
9.26 x 10- 3
6.8Ixl0- 3
5.39 x 10- 3
4.46 X 10- 3
3.32 X 10- 3
2.64 x 10- 3
1.31 x 10 - 3
8.70 x 10- 4
6.5.2 x 10 - 4
3.88 x 10 - 2
1.90x10- 2
1.26x10- 2
9.41 X 10- 3
7.52 x 10- 3
6.26 x 10- 3
4.69x 10- 3
3.75 x 10- 3
1.87 x 10 - 3
1.25 x 10 - 3
9.34 x 10- 4
5.0IxlO- 2
2.58 x 10 - 2
1.74 x 10- 2
1.31 x 10 - 2
1.05 x 10 - 2
8.76 x 10- 3
6.59 x 10 - 3
5.28 x 10- 3
2.65 x 10- 3
1.76 x 10 - 3
1.32 x 10 - 3
9.8
0.300
5
10
15
20
25
30
40
50
100
150
200
5
10
15
20
"25
30
40
50
100
150
200
5
10
15
20
25
30
40
50
100
150
200
5
10
15
20
25
30
40
50
100
150
200
4.24 x 10- 1
3;63 x 10- 1
3.11xl0- 1
2.67 x 10- 1
2.29 x 10- 1
1.96xl0- 1
1.44 x 10- 1
1.06 x 10- 1
2.41xl0- 2
6.19xlO- 3
2.08 X 10- 3
1.63x10- 1
5.63 x 10- 2
2.23xl0- 2
1.06 x 10 - 2
6.24 x 10- 3
4.36 x 10- 3
2.83 X 10- 3
2.16 x 10- 3
1.01 x 10 - 3
6.64 x 10- 4
4.94 x 10- 4
5.66 x 10- 2
1.56 x 10 - 2
8.56 x 10- 3
6.06 x 10 - 3
4.73x 10- 3
3.89 X 10- 3
2.87 x 10- 3
2.27 x 10- 3
I.l1xlO- 3
7.38 x 10- 4
5.52 x 10- 4
3.61 x 10 - 2
1.52 x 10 - 2
9.77xlO- 3
7.23xlO- 3
5.74xlO- 3
4.76xlO- 3
3.54 X 10- 3
2.82 x 10- 3
1.40 x 10 - 3
9.33x 10- 4
6.99 x 10- 4
2.4
3.9
4.9
5.8
6.4
7.0
7.7
8.1
8.9
9.2
9.3
1.6
2.2
2.6
2.9
3.2
3.4
3.6
3.8
4.1
4.2
4.3
1.3
1.5
1.8
1.9
2.0
2.1
2.3
2.3
2.5
2.5
2.6
1.1
1.2
1.3
1.4
1.4
1.5
1.5
1.6
1.7
1.7
1.7
4.81 x 10- 2
3.43 x 10- 2
3.39 x 10- 2
3.60 x 10 - 2
3.80 x 10 - 2
3.90 x 10 - 2
3.70xl0- 2
3.08xl0- 2
8.45 x 10- 3
2.74xl0- 3
1.27 x 10- 3
3.56 x 10- 2
1.44 x 10- 2
8.55 x 10- 3
6.00 x 10- 3
4.60 x 10- 3
3.72xl0- 3
2.69 x 10- 3
2.11xl0- 3
1.01 x 10- 3
6.62 x 10- 4
4.93 x 10- 4
3.75xl0- 2
1.60 x 10- 2
9.71xI0- 3
6.85 x 10- 3
5.25 x 10- 3
4.25 x 10- 3
3.07 x 10- 3
2.40 x 10- 3
1.15 x 10 - 3
7.54 x 10- 4
5.61 x 10- 4
4.03 x 10- 2
1.85 x 10 - 2
1.16 x 10- 2
8.30 x 10- 3
6.44 x 10- 3
5.25 x 10- 3
3.82 x 10- 3
3.01 x 10- 3
1.45xl0- 3
9.54 x 10- 4
7.11xlO- 4
"
0"
1.9
1.2
0.8
4.5
2.7
1.7