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Sample Size under Inverse Negative Binomial Group Testing for Accuracy in Parameter Estimation

Identifieur interne : 000516 ( Pmc/Corpus ); précédent : 000515; suivant : 000517

Sample Size under Inverse Negative Binomial Group Testing for Accuracy in Parameter Estimation

Auteurs : Osval Antonio Montesinos-L Pez ; Abelardo Montesinos-L Pez ; José Crossa ; Kent Eskridge

Source :

RBID : PMC:3310835

Abstract

Background

The group testing method has been proposed for the detection and estimation of genetically modified plants (adventitious presence of unwanted transgenic plants, AP). For binary response variables (presence or absence), group testing is efficient when the prevalence is low, so that estimation, detection, and sample size methods have been developed under the binomial model. However, when the event is rare (low prevalence <0.1), and testing occurs sequentially, inverse (negative) binomial pooled sampling may be preferred.

Methodology/Principal Findings

This research proposes three sample size procedures (two computational and one analytic) for estimating prevalence using group testing under inverse (negative) binomial sampling. These methods provide the required number of positive pools (), given a pool size (k), for estimating the proportion of AP plants using the Dorfman model and inverse (negative) binomial sampling. We give real and simulated examples to show how to apply these methods and the proposed sample-size formula. The Monte Carlo method was used to study the coverage and level of assurance achieved by the proposed sample sizes. An R program to create other scenarios is given in Appendix S2.

Conclusions

The three methods ensure precision in the estimated proportion of AP because they guarantee that the width (W) of the confidence interval (CI) will be equal to, or narrower than, the desired width (), with a probability of . With the Monte Carlo study we found that the computational Wald procedure (method 2) produces the more precise sample size (with coverage and assurance levels very close to nominal values) and that the samples size based on the Clopper-Pearson CI (method 1) is conservative (overestimates the sample size); the analytic Wald sample size method we developed (method 3) sometimes underestimated the optimum number of pools.


Url:
DOI: 10.1371/journal.pone.0032250
PubMed: 22457714
PubMed Central: 3310835

Links to Exploration step

PMC:3310835

Le document en format XML

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<alt-title alt-title-type="running-head">Sample Size under Negative Binomial Group Testing</alt-title>
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<name>
<surname>Montesinos-López</surname>
<given-names>Osval Antonio</given-names>
</name>
<xref ref-type="aff" rid="aff1">
<sup>1</sup>
</xref>
<xref ref-type="corresp" rid="cor1">
<sup>*</sup>
</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Montesinos-López</surname>
<given-names>Abelardo</given-names>
</name>
<xref ref-type="aff" rid="aff2">
<sup>2</sup>
</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Crossa</surname>
<given-names>José</given-names>
</name>
<xref ref-type="aff" rid="aff3">
<sup>3</sup>
</xref>
<xref ref-type="corresp" rid="cor1">
<sup>*</sup>
</xref>
</contrib>
<contrib contrib-type="author">
<name>
<surname>Eskridge</surname>
<given-names>Kent</given-names>
</name>
<xref ref-type="aff" rid="aff4">
<sup>4</sup>
</xref>
</contrib>
</contrib-group>
<aff id="aff1">
<label>1</label>
<addr-line>Facultad de Telemática, Universidad de Colima, Colima, Colima, México</addr-line>
</aff>
<aff id="aff2">
<label>2</label>
<addr-line>Departamento de Estadística, Centro de Investigación en Matemáticas (CIMAT), Guanajuato, Guanajuato, México</addr-line>
</aff>
<aff id="aff3">
<label>3</label>
<addr-line>Biometrics and Statistics Unit, Maize and Wheat Improvement Center (CIMMYT), Mexico D.F., Mexico</addr-line>
</aff>
<aff id="aff4">
<label>4</label>
<addr-line>Department of Statistics, University of Nebraska, Lincoln, Nebraska, United States of America</addr-line>
</aff>
<contrib-group>
<contrib contrib-type="editor">
<name>
<surname>Duffy</surname>
<given-names>Ken R.</given-names>
</name>
<role>Editor</role>
<xref ref-type="aff" rid="edit1"></xref>
</contrib>
</contrib-group>
<aff id="edit1">National University of Ireland Maynooth, Ireland</aff>
<author-notes>
<corresp id="cor1">* E-mail:
<email>oamontes1@ucol.mx</email>
(OM);
<email>j.crossa@cgiar.org</email>
(JC)</corresp>
<fn fn-type="con">
<p>Conceived and designed the experiments: OM AM JC KE. Performed the experiments: OM. Analyzed the data: OM AM. Contributed reagents/materials/analysis tools: OM AM JC KE. Wrote the paper: OM JC.</p>
</fn>
</author-notes>
<pub-date pub-type="collection">
<year>2012</year>
</pub-date>
<pub-date pub-type="epub">
<day>22</day>
<month>3</month>
<year>2012</year>
</pub-date>
<volume>7</volume>
<issue>3</issue>
<elocation-id>e32250</elocation-id>
<history>
<date date-type="received">
<day>16</day>
<month>8</month>
<year>2011</year>
</date>
<date date-type="accepted">
<day>25</day>
<month>1</month>
<year>2012</year>
</date>
</history>
<permissions>
<copyright-statement>Montesinos-López et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.</copyright-statement>
<copyright-year>2012</copyright-year>
</permissions>
<abstract>
<sec>
<title>Background</title>
<p>The group testing method has been proposed for the detection and estimation of genetically modified plants (adventitious presence of unwanted transgenic plants, AP). For binary response variables (presence or absence), group testing is efficient when the prevalence is low, so that estimation, detection, and sample size methods have been developed under the binomial model. However, when the event is rare (low prevalence <0.1), and testing occurs sequentially, inverse (negative) binomial pooled sampling may be preferred.</p>
</sec>
<sec>
<title>Methodology/Principal Findings</title>
<p>This research proposes three sample size procedures (two computational and one analytic) for estimating prevalence using group testing under inverse (negative) binomial sampling. These methods provide the required number of positive pools (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e001.jpg" mimetype="image"></inline-graphic>
</inline-formula>
), given a pool size (
<italic>k</italic>
), for estimating the proportion of AP plants using the Dorfman model and inverse (negative) binomial sampling. We give real and simulated examples to show how to apply these methods and the proposed sample-size formula. The Monte Carlo method was used to study the coverage and level of assurance achieved by the proposed sample sizes. An R program to create other scenarios is given in
<xref ref-type="supplementary-material" rid="pone.0032250.s002">Appendix S2</xref>
.</p>
</sec>
<sec>
<title>Conclusions</title>
<p>The three methods ensure precision in the estimated proportion of AP because they guarantee that the width (
<italic>W</italic>
) of the confidence interval (CI) will be equal to, or narrower than, the desired width (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e002.jpg" mimetype="image"></inline-graphic>
</inline-formula>
), with a probability of
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e003.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. With the Monte Carlo study we found that the computational Wald procedure (method 2) produces the more precise sample size (with coverage and assurance levels very close to nominal values) and that the samples size based on the Clopper-Pearson CI (method 1) is conservative (overestimates the sample size); the analytic Wald sample size method we developed (method 3) sometimes underestimated the optimum number of pools.</p>
</sec>
</abstract>
<counts>
<page-count count="11"></page-count>
</counts>
</article-meta>
</front>
<body>
<sec id="s1">
<title>Introduction</title>
<p>To detect the presence of a rare event, thousands of individuals need to be tested, and the cost of such testing usually exceeds the available budget and staff. The pooling methodology (Dorfman method) was first proposed to save a significant amount of money when detecting soldiers with syphilis
<xref ref-type="bibr" rid="pone.0032250-Dorfman1">[1]</xref>
. Significant cost savings were achieved by first testing a sample created by mixing blood from several people. If the sample tested positive, the blood from each individual in that pool would be retested; if the sample tested negative, all individuals in that pool were declared free of the disease
<xref ref-type="bibr" rid="pone.0032250-Dorfman1">[1]</xref>
. Currently the Dorfman method is used for detecting and estimating the proportion of positive individuals in fields such as medicine
<xref ref-type="bibr" rid="pone.0032250-Westreich1">[2]</xref>
,
<xref ref-type="bibr" rid="pone.0032250-Dodd1">[3]</xref>
,
<xref ref-type="bibr" rid="pone.0032250-Remlinger1">[4]</xref>
,
<xref ref-type="bibr" rid="pone.0032250-Verstraeten1">[5]</xref>
, agriculture
<xref ref-type="bibr" rid="pone.0032250-Tebbs1">[6]</xref>
, telecommunications
<xref ref-type="bibr" rid="pone.0032250-Wolf1">[7]</xref>
, and science fiction
<xref ref-type="bibr" rid="pone.0032250-Bilder1">[8]</xref>
. Most applications for detecting and estimating a proportion are developed using binomial sampling; however, Pritchard and Tebbs
<xref ref-type="bibr" rid="pone.0032250-Pritchard1">[9]</xref>
have suggested that inverse (negative) binomial pooled sampling may be preferred when prevalence
<italic>p</italic>
is known to be small, when sampling and testing occur sequentially, or when positive pool results require immediate analysis—for example, in the case of many rare diseases. Unlike binomial sampling, in this model the number of positive pools to be observed is fixed
<italic>a priori</italic>
, and testing is complete when the rth positive pool is reached
<xref ref-type="bibr" rid="pone.0032250-Pritchard2">[10]</xref>
.</p>
<p>George and Elston
<xref ref-type="bibr" rid="pone.0032250-George1">[11]</xref>
recommended using geometric sampling when the probability of an event is small; they gave confidence intervals for the prevalence based on individual testing. Also, according to Haldane
<xref ref-type="bibr" rid="pone.0032250-Haldane1">[12]</xref>
, using a binomial distribution may not provide an unbiased and precise estimate of
<italic>p</italic>
when
<italic>p</italic>
is small (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e004.jpg" mimetype="image"></inline-graphic>
</inline-formula>
). Lui
<xref ref-type="bibr" rid="pone.0032250-Lui1">[13]</xref>
extended George and Elston's work
<xref ref-type="bibr" rid="pone.0032250-George1">[11]</xref>
on the confidence interval (CI) by considering negative binomial sampling and showed that as the required number of successes increased, the width of the CI decreased. However, this extension was also under individual testing. Using negative binomial group testing sampling, Katholi
<xref ref-type="bibr" rid="pone.0032250-Katholi1">[14]</xref>
derived point and interval estimators of
<italic>p</italic>
, obtained by both classical and Bayesian methods, and investigated their statistical properties.</p>
<p>Recently Pritchard and Tebbs
<xref ref-type="bibr" rid="pone.0032250-Pritchard1">[9]</xref>
used maximum likelihood as a basis for developing three point and interval estimators for
<italic>p</italic>
under inverse pooled sampling; they compared its performance with Katholi's
<xref ref-type="bibr" rid="pone.0032250-Katholi1">[14]</xref>
proposed point and interval estimators. Pritchard and Tebbs
<xref ref-type="bibr" rid="pone.0032250-Pritchard2">[10]</xref>
extended their work to Bayesian point and interval estimation of the prevalence under negative binomial group testing. They used different distributions to incorporate prior knowledge of disease incidence and different loss functions, and derived closed-form expressions for posterior distributions and point and credible interval estimators
<xref ref-type="bibr" rid="pone.0032250-Pritchard2">[10]</xref>
. However, until now sample size procedures under inverse (negative) binomial sampling for group testing have not been proposed.</p>
<p>In practice, pooling is a simple process; for example, if 40,000 plants are collected from the field, they could be tested one at a time for detecting unwanted transgenic plants (AP). If each test takes 15 minutes and costs US$12, then this project will take 10,000 hours and cost US$480,000. A shorter approach would be to smash 10 plants together and test this pooled sample
<xref ref-type="bibr" rid="pone.0032250-Ebert1">[15]</xref>
. This approach would take 1000 hours and cost US$48,000. Even greater savings are achieved with larger pool sizes. However, because the maximum likelihood estimator (MLE) of
<italic>p</italic>
under binomial
<xref ref-type="bibr" rid="pone.0032250-Swallow1">[16]</xref>
and negative binomial
<xref ref-type="bibr" rid="pone.0032250-Pritchard1">[9]</xref>
,
<xref ref-type="bibr" rid="pone.0032250-Pritchard2">[10]</xref>
group testing is biased to the right, then, on average, the MLE of
<italic>p</italic>
overestimates the true prevalence for any pool size (assuming a perfect diagnostic test); however, this bias is usually small when
<italic>p</italic>
is small (
<italic>p</italic>
<0.1)
<xref ref-type="bibr" rid="pone.0032250-Katholi2">[17]</xref>
. In addition, if the diagnostic test is imperfect, a high rate of false positives is very likely. Thus, there are benefits and risks attached to the use of pooling methodology
<xref ref-type="bibr" rid="pone.0032250-Ebert1">[15]</xref>
. For this reason, it is important to choose the pool size with care in order to guarantee precision in the estimation process.</p>
<p>Under binomial group testing, some authors have proposed methods for determining the required sample size (number of required pools) to guarantee a certain level of power and/or precision
<xref ref-type="bibr" rid="pone.0032250-Yamamura1">[18]</xref>
,
<xref ref-type="bibr" rid="pone.0032250-HernndezSurez1">[19]</xref>
,
<xref ref-type="bibr" rid="pone.0032250-MontesinosLpez1">[20]</xref>
,
<xref ref-type="bibr" rid="pone.0032250-MontesinosLpez2">[21]</xref>
. Yamamura and Hino
<xref ref-type="bibr" rid="pone.0032250-Yamamura1">[18]</xref>
and Hernández-Suárez et al.
<xref ref-type="bibr" rid="pone.0032250-HernndezSurez1">[19]</xref>
developed sample size methods in terms of power considerations. This approach is consistent with the emphasis on hypothesis testing for inference, with results reported in terms of
<italic>p</italic>
-values. Montesinos-López et al.
<xref ref-type="bibr" rid="pone.0032250-MontesinosLpez1">[20]</xref>
,
<xref ref-type="bibr" rid="pone.0032250-MontesinosLpez2">[21]</xref>
developed sample size procedures under the
<italic>accuracy in parameter estimation</italic>
(AIPE) framework that guarantee narrow confidence intervals for estimating the parameter. The use of this approach is increasing, not only because the CIs ensure that the magnitude of the effect can be better assessed, but also because the effect in question can be readily identified by the reader. Furthermore, CIs also convey information about how precisely the magnitude of the effect can be ascertained from the data at hand
<xref ref-type="bibr" rid="pone.0032250-Beal1">[22]</xref>
. Another advantage of the AIPE approach is that it treats the estimates (from pilot studies or literature review) used to determine the required sample size as random to guarantee that the desired CI width for estimating the parameter of interest is achieved, as originally planned
<xref ref-type="bibr" rid="pone.0032250-Wang1">[23]</xref>
.</p>
<p>However, under binomial group testing sampling when the prevalence is low, the calculated sample size sometimes does not contain any pools with the trait of interest (i.e., failure to detect and estimate AP). For this reason, inverse (negative) binomial sampling is a good alternative because each sample will contain the desired number of rare units and also the sample size is not a fixed quantity
<xref ref-type="bibr" rid="pone.0032250-Haldane1">[12]</xref>
,
<xref ref-type="bibr" rid="pone.0032250-Pritchard1">[9]</xref>
,
<xref ref-type="bibr" rid="pone.0032250-Pritchard2">[10]</xref>
. In binomial group testing, the number of required pools is treated as a fixed quantity, whereas under inverse (negative) binomial group testing, the pools are drawn one by one until the sample contains exactly
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e005.jpg" mimetype="image"></inline-graphic>
</inline-formula>
positive pools (here the number of positive pools is fixed).</p>
<p>Based on the previous findings, the purpose of the present study is to develop methods for determining sample size (number of positive pools) under inverse (negative) binomial group testing with the objective of increasing accuracy in the estimation of the population proportion. This research proposes methods for determining the required number of positive pools, with the aim of estimating the proportion of AP (
<italic>p</italic>
) using inverse (negative) binomial group testing with a perfect test and fixed pool size (
<italic>k</italic>
) that will assure a narrow CI. Accuracy in the estimation of
<italic>p</italic>
is achieved because CI width is considered stochastic and thus treated as a random variable. The methods used for achieving the objectives of the present research are: point and interval estimation for the population proportion, delta method, and central limit theorem. We provide an R program that reproduces the results presented in this study and makes it easy for the researcher to create other scenarios.</p>
</sec>
<sec sec-type="materials|methods" id="s2">
<title>Materials and Methods</title>
<p>Suppose that
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e006.jpg" mimetype="image"></inline-graphic>
</inline-formula>
represents the number of pools tested until the first positive pool is detected and
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e007.jpg" mimetype="image"></inline-graphic>
</inline-formula>
are observed to obtain the
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e008.jpg" mimetype="image"></inline-graphic>
</inline-formula>
positive pool. Therefore,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e009.jpg" mimetype="image"></inline-graphic>
</inline-formula>
has a geometric distribution. Therefore, the overall number of pools that are tested to find
<italic>r</italic>
positive pools is equal to
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e010.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. In what follows, we shall denote the size of the pools collected as
<italic>k</italic>
and assume equal pool size; the prevalence of infection is denoted by
<italic>p</italic>
, the number of pools tested to find one positive pool is
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e011.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and the number of times this experiment is carried out is denoted by
<italic>r</italic>
. It is important to mention that in this paper we consider that: (i) the sample size is the value of
<italic>r</italic>
that represents the number of positive pools required to stop the sampling and testing process, and (ii) the overall number of pools tested is the value of
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e012.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. If the prevalence of infection is
<italic>p</italic>
, then the probability that a pool of size
<italic>k</italic>
tests positive is
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e013.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. Therefore, the sufficient statistics
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e014.jpg" mimetype="image"></inline-graphic>
</inline-formula>
follows a negative binomial distribution (nib) with waiting parameter
<italic>r</italic>
and success probability
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e015.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<xref ref-type="bibr" rid="pone.0032250-Pritchard1">[9]</xref>
,
<xref ref-type="bibr" rid="pone.0032250-Pritchard2">[10]</xref>
,
<xref ref-type="bibr" rid="pone.0032250-Katholi1">[14]</xref>
. According to Pritchard and Tebbs
<xref ref-type="bibr" rid="pone.0032250-Pritchard1">[9]</xref>
,
<xref ref-type="bibr" rid="pone.0032250-Pritchard2">[10]</xref>
and Katholi
<xref ref-type="bibr" rid="pone.0032250-Katholi1">[14]</xref>
, the maximum likelihood estimate (MLE) of
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e016.jpg" mimetype="image"></inline-graphic>
</inline-formula>
using inverse (negative) binomial group testing is
<disp-formula>
<graphic xlink:href="pone.0032250.e017"></graphic>
<label>(1)</label>
</disp-formula>
where
<italic>k</italic>
is the pool size and
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e018.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the fixed required number of positive pools. This MLE of
<italic>p</italic>
for inverse (negative) binomial group testing with groups of equal size assumes a perfect diagnostic test. On the other hand, the variance of
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e019.jpg" mimetype="image"></inline-graphic>
</inline-formula>
according to Pritchard and Tebbs
<xref ref-type="bibr" rid="pone.0032250-Pritchard1">[9]</xref>
,
<xref ref-type="bibr" rid="pone.0032250-Pritchard2">[10]</xref>
and Katholi
<xref ref-type="bibr" rid="pone.0032250-Katholi1">[14]</xref>
is given by
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e020.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. According to Pritchard and Tebbs
<xref ref-type="bibr" rid="pone.0032250-Pritchard1">[9]</xref>
, the corresponding Wald CI is as follows:
<disp-formula>
<graphic xlink:href="pone.0032250.e021"></graphic>
<label>(2)</label>
</disp-formula>
where
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e022.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e023.jpg" mimetype="image"></inline-graphic>
</inline-formula>
quantile of the standard normal distribution, and
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e024.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the MLE estimated from Eq. (1). This approximation of the CI is easy to calculate and allows deriving closed-form sample size formulas. However, when
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e025.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is small, the normal approximation for MLE is doubtful; in such cases, the Wald-type CI often produces negative endpoints. In addition, the coverage probability of the CIs constructed by Wald-type CIs is often smaller than
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e026.jpg" mimetype="image"></inline-graphic>
</inline-formula>
.</p>
<sec id="s2a">
<title>Derivation of the sample size formula for detecting transgenic plants</title>
<p>The quantity
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e027.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(added and subtracted from the observed proportion,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e028.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) in Eq. (2) is defined as
<italic>W/2</italic>
(where
<italic>W</italic>
is the full width of the CI;
<italic>W</italic>
or
<italic>W/2</italic>
can be set
<italic>a priori</italic>
by the researcher depending on the desired precision). The observed CI width for any realization of a confidence interval (from Eq. 2) can be expressed as:
<disp-formula>
<graphic xlink:href="pone.0032250.e029"></graphic>
<label>(3)</label>
</disp-formula>
Let
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e030.jpg" mimetype="image"></inline-graphic>
</inline-formula>
be the desired CI width; then the basic AIPE approach seeks to find the minimum sample size so that the expected CI width is sufficiently narrow
<xref ref-type="bibr" rid="pone.0032250-Kelley1">[24]</xref>
,
<xref ref-type="bibr" rid="pone.0032250-Kelley2">[25]</xref>
. In other words, the AIPE approach seeks the minimal sample size so that
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e031.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. The problem is that the expected CI width is an unknown quantity, although it can be approximated. As
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e032.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, where
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e033.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, the observed width,
<italic>W</italic>
, is a function of
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e034.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. Since the distribution of
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e035.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is unknown, it is not possible to obtain an analytic solution for
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e036.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. An alternative is to use the delta method to derive the asymptotic distribution of
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e037.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. From Result 1 in
<xref ref-type="supplementary-material" rid="pone.0032250.s001">Appendix S1</xref>
, we have that
<disp-formula>
<graphic xlink:href="pone.0032250.e038"></graphic>
</disp-formula>
where
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e039.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e040.jpg" mimetype="image"></inline-graphic>
</inline-formula>
for
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e041.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. Therefore, the expected value of
<italic>W</italic>
is
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e042.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. Now if we set the
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e043.jpg" mimetype="image"></inline-graphic>
</inline-formula>
to the desired width of the CI,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e044.jpg" mimetype="image"></inline-graphic>
</inline-formula>
:
<disp-formula>
<graphic xlink:href="pone.0032250.e045"></graphic>
<label>(4)</label>
</disp-formula>
Solving for
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e046.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, Eq. (4) yields the following formulation:
<disp-formula>
<graphic xlink:href="pone.0032250.e047"></graphic>
<label>(5)</label>
</disp-formula>
Note that if
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e048.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, Eq. (5) reduces to the formula derived by Lui
<xref ref-type="bibr" rid="pone.0032250-Lui1">[13]</xref>
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e049.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. However, Eq. (5) requires the population value of
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e050.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, which is unknown and in practice is replaced by an estimation of the true proportion. Eq. (5) finds the required sample size for achieving an expected CI width,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e051.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, that is sufficiently narrow for estimating the proportion of AP using pools; however, this does not guarantee that for any particular CI, the observed expected CI width,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e052.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, will be sufficiently narrow, because the expectation only approximates the mean CI width. Kelley and Rausch
<xref ref-type="bibr" rid="pone.0032250-Kelley2">[25]</xref>
state that this issue is similar to the case where a mean is estimated from a normal distribution; although the sample mean is an unbiased estimator of the population mean, the sample mean will almost certainly be smaller or larger than the population value. This is because the sample mean is a continuous random variable, as is the CI width, due to the fact that both are based on random data. Thus, approximately half of the time, the computed confidence interval will be wider than the desired (specified) width
<xref ref-type="bibr" rid="pone.0032250-Kelley2">[25]</xref>
.</p>
<p>Since Eq. (3) uses an estimate of
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e053.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<sup>,</sup>
the CI width (
<italic>W</italic>
) is a random variable that will fluctuate from sample to sample. This implies that, using
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e054.jpg" mimetype="image"></inline-graphic>
</inline-formula>
from Eq. (5), less than 50% of the sampling distribution of
<italic>W</italic>
will be smaller than
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e055.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(see the third column in
<xref ref-type="table" rid="pone-0032250-t001">Table 1</xref>
). To demonstrate this, we need to calculate the probability of obtaining a CI width that is smaller than the specified value (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e056.jpg" mimetype="image"></inline-graphic>
</inline-formula>
). This can be computed as:
<disp-formula>
<graphic xlink:href="pone.0032250.e057"></graphic>
</disp-formula>
where
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e058.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is an indicator function showing whether or not the actual CI width calculated using Eq. (3 ) is ≤
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e059.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e060.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the true population proportion and
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e061.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the sample size obtained using equation (5). To avoid possible computer limitations, the above probability can be approximated by the following:
<disp-formula>
<graphic xlink:href="pone.0032250.e062"></graphic>
<label>(6)</label>
</disp-formula>
where
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e063.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e064.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is considered a random variable because the exact value of
<italic>p</italic>
is not known and
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e065.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the value that satisfies
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e066.jpg" mimetype="image"></inline-graphic>
</inline-formula>
; we use this value of
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e067.jpg" mimetype="image"></inline-graphic>
</inline-formula>
because in the R package summing to infinity is not possible.</p>
<table-wrap id="pone-0032250-t001" position="float">
<object-id pub-id-type="doi">10.1371/journal.pone.0032250.t001</object-id>
<label>Table 1</label>
<caption>
<title>Underestimation of the sample size given by using Eq. (5) (
<xref ref-type="table" rid="pone-0032250-t001">Table 1A</xref>
).</title>
</caption>
<alternatives>
<graphic id="pone-0032250-t001-1" xlink:href="pone.0032250.t001"></graphic>
<table frame="hsides" rules="groups">
<colgroup span="1">
<col align="left" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
</colgroup>
<thead>
<tr>
<td align="left" rowspan="1" colspan="1">A</td>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e068.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e069.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e070.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e071.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e072.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e073.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e074.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e075.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e076.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
</tr>
</thead>
<tbody>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">0.005</td>
<td align="left" rowspan="1" colspan="1">8</td>
<td align="left" rowspan="1" colspan="1">0.4602923</td>
<td align="left" rowspan="1" colspan="1">18</td>
<td align="left" rowspan="1" colspan="1">0.9439192</td>
<td align="left" rowspan="1" colspan="1">28</td>
<td align="left" rowspan="1" colspan="1">0.9985824</td>
<td align="left" rowspan="1" colspan="1">48</td>
<td align="left" rowspan="1" colspan="1">0.9999997</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">0.0075</td>
<td align="left" rowspan="1" colspan="1">18</td>
<td align="left" rowspan="1" colspan="1">0.4937528</td>
<td align="left" rowspan="1" colspan="1">28</td>
<td align="left" rowspan="1" colspan="1">0.8677739</td>
<td align="left" rowspan="1" colspan="1">38</td>
<td align="left" rowspan="1" colspan="1">0.9860621</td>
<td align="left" rowspan="1" colspan="1">58</td>
<td align="left" rowspan="1" colspan="1">0.9999798</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">0.01</td>
<td align="left" rowspan="1" colspan="1">31</td>
<td align="left" rowspan="1" colspan="1">0.4764102</td>
<td align="left" rowspan="1" colspan="1">41</td>
<td align="left" rowspan="1" colspan="1">0.792025</td>
<td align="left" rowspan="1" colspan="1">51</td>
<td align="left" rowspan="1" colspan="1">0.9491423</td>
<td align="left" rowspan="1" colspan="1">71</td>
<td align="left" rowspan="1" colspan="1">0.9993324</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">0.0125</td>
<td align="left" rowspan="1" colspan="1">49</td>
<td align="left" rowspan="1" colspan="1">0.4825564</td>
<td align="left" rowspan="1" colspan="1">59</td>
<td align="left" rowspan="1" colspan="1">0.7531049</td>
<td align="left" rowspan="1" colspan="1">69</td>
<td align="left" rowspan="1" colspan="1">0.9091713</td>
<td align="left" rowspan="1" colspan="1">89</td>
<td align="left" rowspan="1" colspan="1">0.9962656</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">0.015</td>
<td align="left" rowspan="1" colspan="1">70</td>
<td align="left" rowspan="1" colspan="1">0.4831282</td>
<td align="left" rowspan="1" colspan="1">80</td>
<td align="left" rowspan="1" colspan="1">0.6966756</td>
<td align="left" rowspan="1" colspan="1">90</td>
<td align="left" rowspan="1" colspan="1">0.867216</td>
<td align="left" rowspan="1" colspan="1">110</td>
<td align="left" rowspan="1" colspan="1">0.9873122</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">0.0175</td>
<td align="left" rowspan="1" colspan="1">96</td>
<td align="left" rowspan="1" colspan="1">0.49556</td>
<td align="left" rowspan="1" colspan="1">106</td>
<td align="left" rowspan="1" colspan="1">0.6823066</td>
<td align="left" rowspan="1" colspan="1">116</td>
<td align="left" rowspan="1" colspan="1">0.83486</td>
<td align="left" rowspan="1" colspan="1">136</td>
<td align="left" rowspan="1" colspan="1">0.9736274</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">0.02</td>
<td align="left" rowspan="1" colspan="1">126</td>
<td align="left" rowspan="1" colspan="1">0.4923463</td>
<td align="left" rowspan="1" colspan="1">136</td>
<td align="left" rowspan="1" colspan="1">0.6682315</td>
<td align="left" rowspan="1" colspan="1">146</td>
<td align="left" rowspan="1" colspan="1">0.8073307</td>
<td align="left" rowspan="1" colspan="1">166</td>
<td align="left" rowspan="1" colspan="1">0.9575451</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">0.0225</td>
<td align="left" rowspan="1" colspan="1">159</td>
<td align="left" rowspan="1" colspan="1">0.4885201</td>
<td align="left" rowspan="1" colspan="1">169</td>
<td align="left" rowspan="1" colspan="1">0.6302238</td>
<td align="left" rowspan="1" colspan="1">179</td>
<td align="left" rowspan="1" colspan="1">0.7655083</td>
<td align="left" rowspan="1" colspan="1">199</td>
<td align="left" rowspan="1" colspan="1">0.9288043</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">0.025</td>
<td align="left" rowspan="1" colspan="1">198</td>
<td align="left" rowspan="1" colspan="1">0.5028085</td>
<td align="left" rowspan="1" colspan="1">208</td>
<td align="left" rowspan="1" colspan="1">0.631837</td>
<td align="left" rowspan="1" colspan="1">218</td>
<td align="left" rowspan="1" colspan="1">0.7583371</td>
<td align="left" rowspan="1" colspan="1">238</td>
<td align="left" rowspan="1" colspan="1">0.9121938</td>
</tr>
</tbody>
</table>
<table frame="hsides" rules="groups">
<colgroup span="1">
<col align="left" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
</colgroup>
<thead>
<tr>
<td align="left" rowspan="1" colspan="1">B</td>
<td align="left" rowspan="1" colspan="1">
<italic>k</italic>
</td>
<td align="left" rowspan="1" colspan="1">
<italic>r</italic>
 = 5</td>
<td align="left" rowspan="1" colspan="1">
<italic>r</italic>
 = 15</td>
<td align="left" rowspan="1" colspan="1">
<italic>r</italic>
 = 25</td>
<td align="left" rowspan="1" colspan="1">
<italic>r</italic>
 = 35</td>
<td align="left" rowspan="1" colspan="1">
<italic>r</italic>
 = 45</td>
<td align="left" rowspan="1" colspan="1">
<italic>r</italic>
 = 55</td>
<td align="left" rowspan="1" colspan="1">
<italic>r</italic>
 = 65</td>
<td align="left" rowspan="1" colspan="1">
<italic>r</italic>
 = 75</td>
</tr>
</thead>
<tbody>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">5</td>
<td align="left" rowspan="1" colspan="1">0.562</td>
<td align="left" rowspan="1" colspan="1">0.535</td>
<td align="left" rowspan="1" colspan="1">0.528</td>
<td align="left" rowspan="1" colspan="1">0.523</td>
<td align="left" rowspan="1" colspan="1">0.520</td>
<td align="left" rowspan="1" colspan="1">0.519</td>
<td align="left" rowspan="1" colspan="1">0.518</td>
<td align="left" rowspan="1" colspan="1">0.515</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">15</td>
<td align="left" rowspan="1" colspan="1">0.566</td>
<td align="left" rowspan="1" colspan="1">0.539</td>
<td align="left" rowspan="1" colspan="1">0.529</td>
<td align="left" rowspan="1" colspan="1">0.524</td>
<td align="left" rowspan="1" colspan="1">0.522</td>
<td align="left" rowspan="1" colspan="1">0.518</td>
<td align="left" rowspan="1" colspan="1">0.517</td>
<td align="left" rowspan="1" colspan="1">0.516</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">25</td>
<td align="left" rowspan="1" colspan="1">0.561</td>
<td align="left" rowspan="1" colspan="1">0.536</td>
<td align="left" rowspan="1" colspan="1">0.529</td>
<td align="left" rowspan="1" colspan="1">0.524</td>
<td align="left" rowspan="1" colspan="1">0.523</td>
<td align="left" rowspan="1" colspan="1">0.513</td>
<td align="left" rowspan="1" colspan="1">0.512</td>
<td align="left" rowspan="1" colspan="1">0.513</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">35</td>
<td align="left" rowspan="1" colspan="1">0.573</td>
<td align="left" rowspan="1" colspan="1">0.539</td>
<td align="left" rowspan="1" colspan="1">0.528</td>
<td align="left" rowspan="1" colspan="1">0.520</td>
<td align="left" rowspan="1" colspan="1">0.518</td>
<td align="left" rowspan="1" colspan="1">0.515</td>
<td align="left" rowspan="1" colspan="1">0.514</td>
<td align="left" rowspan="1" colspan="1">0.516</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">45</td>
<td align="left" rowspan="1" colspan="1">0.550</td>
<td align="left" rowspan="1" colspan="1">0.537</td>
<td align="left" rowspan="1" colspan="1">0.522</td>
<td align="left" rowspan="1" colspan="1">0.524</td>
<td align="left" rowspan="1" colspan="1">0.516</td>
<td align="left" rowspan="1" colspan="1">0.521</td>
<td align="left" rowspan="1" colspan="1">0.512</td>
<td align="left" rowspan="1" colspan="1">0.517</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">55</td>
<td align="left" rowspan="1" colspan="1">0.549</td>
<td align="left" rowspan="1" colspan="1">0.543</td>
<td align="left" rowspan="1" colspan="1">0.522</td>
<td align="left" rowspan="1" colspan="1">0.527</td>
<td align="left" rowspan="1" colspan="1">0.516</td>
<td align="left" rowspan="1" colspan="1">0.521</td>
<td align="left" rowspan="1" colspan="1">0.514</td>
<td align="left" rowspan="1" colspan="1">0.517</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">65</td>
<td align="left" rowspan="1" colspan="1">0.533</td>
<td align="left" rowspan="1" colspan="1">0.520</td>
<td align="left" rowspan="1" colspan="1">0.515</td>
<td align="left" rowspan="1" colspan="1">0.514</td>
<td align="left" rowspan="1" colspan="1">0.512</td>
<td align="left" rowspan="1" colspan="1">0.512</td>
<td align="left" rowspan="1" colspan="1">0.514</td>
<td align="left" rowspan="1" colspan="1">0.513</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">75</td>
<td align="left" rowspan="1" colspan="1">0.530</td>
<td align="left" rowspan="1" colspan="1">0.520</td>
<td align="left" rowspan="1" colspan="1">0.518</td>
<td align="left" rowspan="1" colspan="1">0.521</td>
<td align="left" rowspan="1" colspan="1">0.517</td>
<td align="left" rowspan="1" colspan="1">0.518</td>
<td align="left" rowspan="1" colspan="1">0.517</td>
<td align="left" rowspan="1" colspan="1">0.520</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">85</td>
<td align="left" rowspan="1" colspan="1">0.568</td>
<td align="left" rowspan="1" colspan="1">0.548</td>
<td align="left" rowspan="1" colspan="1">0.542</td>
<td align="left" rowspan="1" colspan="1">0.514</td>
<td align="left" rowspan="1" colspan="1">0.514</td>
<td align="left" rowspan="1" colspan="1">0.517</td>
<td align="left" rowspan="1" colspan="1">0.519</td>
<td align="left" rowspan="1" colspan="1">0.522</td>
</tr>
</tbody>
</table>
<table frame="hsides" rules="groups">
<colgroup span="1">
<col align="left" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
</colgroup>
<thead>
<tr>
<td align="left" rowspan="1" colspan="1">C</td>
<td align="left" rowspan="1" colspan="1">
<italic>k</italic>
</td>
<td align="left" rowspan="1" colspan="1">
<italic>r</italic>
 = 5</td>
<td align="left" rowspan="1" colspan="1">
<italic>r</italic>
 = 15</td>
<td align="left" rowspan="1" colspan="1">
<italic>r</italic>
 = 25</td>
<td align="left" rowspan="1" colspan="1">
<italic>r</italic>
 = 35</td>
<td align="left" rowspan="1" colspan="1">
<italic>r</italic>
 = 45</td>
<td align="left" rowspan="1" colspan="1">
<italic>r</italic>
 = 55</td>
<td align="left" rowspan="1" colspan="1">
<italic>r</italic>
 = 65</td>
<td align="left" rowspan="1" colspan="1">
<italic>r</italic>
 = 75</td>
</tr>
</thead>
<tbody>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">5</td>
<td align="left" rowspan="1" colspan="1">1.4E-05</td>
<td align="left" rowspan="1" colspan="1">2.3E-06</td>
<td align="left" rowspan="1" colspan="1">1.2E-06</td>
<td align="left" rowspan="1" colspan="1">8.2E-07</td>
<td align="left" rowspan="1" colspan="1">6.2E-07</td>
<td align="left" rowspan="1" colspan="1">5.0E-07</td>
<td align="left" rowspan="1" colspan="1">4.1E-07</td>
<td align="left" rowspan="1" colspan="1">3.5E-07</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">15</td>
<td align="left" rowspan="1" colspan="1">2.4E-05</td>
<td align="left" rowspan="1" colspan="1">2.4E-06</td>
<td align="left" rowspan="1" colspan="1">1.2E-06</td>
<td align="left" rowspan="1" colspan="1">8.2E-07</td>
<td align="left" rowspan="1" colspan="1">6.3E-07</td>
<td align="left" rowspan="1" colspan="1">4.9E-07</td>
<td align="left" rowspan="1" colspan="1">4.1E-07</td>
<td align="left" rowspan="1" colspan="1">3.5E-07</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">25</td>
<td align="left" rowspan="1" colspan="1">3.4E-05</td>
<td align="left" rowspan="1" colspan="1">2.3E-06</td>
<td align="left" rowspan="1" colspan="1">1.2E-06</td>
<td align="left" rowspan="1" colspan="1">8.2E-07</td>
<td align="left" rowspan="1" colspan="1">6.2E-07</td>
<td align="left" rowspan="1" colspan="1">4.9E-07</td>
<td align="left" rowspan="1" colspan="1">4.1E-07</td>
<td align="left" rowspan="1" colspan="1">3.6E-07</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">35</td>
<td align="left" rowspan="1" colspan="1">1.4E-04</td>
<td align="left" rowspan="1" colspan="1">2.3E-06</td>
<td align="left" rowspan="1" colspan="1">1.2E-06</td>
<td align="left" rowspan="1" colspan="1">8.2E-07</td>
<td align="left" rowspan="1" colspan="1">6.1E-07</td>
<td align="left" rowspan="1" colspan="1">5.0E-07</td>
<td align="left" rowspan="1" colspan="1">4.2E-07</td>
<td align="left" rowspan="1" colspan="1">3.5E-07</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">45</td>
<td align="left" rowspan="1" colspan="1">3.1E-04</td>
<td align="left" rowspan="1" colspan="1">2.4E-06</td>
<td align="left" rowspan="1" colspan="1">1.2E-06</td>
<td align="left" rowspan="1" colspan="1">8.3E-07</td>
<td align="left" rowspan="1" colspan="1">6.2E-07</td>
<td align="left" rowspan="1" colspan="1">5.0E-07</td>
<td align="left" rowspan="1" colspan="1">4.1E-07</td>
<td align="left" rowspan="1" colspan="1">3.6E-07</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">55</td>
<td align="left" rowspan="1" colspan="1">8.9E-04</td>
<td align="left" rowspan="1" colspan="1">2.4E-06</td>
<td align="left" rowspan="1" colspan="1">1.2E-06</td>
<td align="left" rowspan="1" colspan="1">8.4E-07</td>
<td align="left" rowspan="1" colspan="1">6.2E-07</td>
<td align="left" rowspan="1" colspan="1">5.0E-07</td>
<td align="left" rowspan="1" colspan="1">4.2E-07</td>
<td align="left" rowspan="1" colspan="1">3.6E-07</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">65</td>
<td align="left" rowspan="1" colspan="1">1.6E-03</td>
<td align="left" rowspan="1" colspan="1">2.4E-06</td>
<td align="left" rowspan="1" colspan="1">1.2E-06</td>
<td align="left" rowspan="1" colspan="1">8.2E-07</td>
<td align="left" rowspan="1" colspan="1">6.2E-07</td>
<td align="left" rowspan="1" colspan="1">5.0E-07</td>
<td align="left" rowspan="1" colspan="1">4.2E-07</td>
<td align="left" rowspan="1" colspan="1">3.6E-07</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">75</td>
<td align="left" rowspan="1" colspan="1">2.7E-03</td>
<td align="left" rowspan="1" colspan="1">2.4E-06</td>
<td align="left" rowspan="1" colspan="1">1.3E-06</td>
<td align="left" rowspan="1" colspan="1">8.3E-07</td>
<td align="left" rowspan="1" colspan="1">6.3E-07</td>
<td align="left" rowspan="1" colspan="1">5.1E-07</td>
<td align="left" rowspan="1" colspan="1">4.2E-07</td>
<td align="left" rowspan="1" colspan="1">3.6E-07</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">85</td>
<td align="left" rowspan="1" colspan="1">5.1E-03</td>
<td align="left" rowspan="1" colspan="1">2.4E-06</td>
<td align="left" rowspan="1" colspan="1">1.3E-06</td>
<td align="left" rowspan="1" colspan="1">8.3E-07</td>
<td align="left" rowspan="1" colspan="1">6.4E-07</td>
<td align="left" rowspan="1" colspan="1">5.1E-07</td>
<td align="left" rowspan="1" colspan="1">4.2E-07</td>
<td align="left" rowspan="1" colspan="1">3.6E-07</td>
</tr>
</tbody>
</table>
</alternatives>
<table-wrap-foot>
<fn id="nt101">
<label></label>
<p>
<xref ref-type="table" rid="pone-0032250-t001">Table 1A</xref>
. Preliminary sample size (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e077.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, number of required positive pools) for estimating the population proportion, computed with Eq. (5) and three sample size increments (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e078.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e079.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e080.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) with their corresponding probability that the confidence interval width
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e081.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is smaller than the specified value (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e082.jpg" mimetype="image"></inline-graphic>
</inline-formula>
),
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e083.jpg" mimetype="image"></inline-graphic>
</inline-formula>
computed with Eq. (6). For a 95% CI and
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e084.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e085.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the desired CI width.
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e086.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the probability that (
<italic>W</italic>
) is smaller than the specified value (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e087.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) calculated using Eq. (6).
<xref ref-type="table" rid="pone-0032250-t001">Table 1B</xref>
. Proportion of times the MLE of
<italic>p</italic>
is greater than the population proportion
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e088.jpg" mimetype="image"></inline-graphic>
</inline-formula>
for different combinations of values of
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e089.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e090.jpg" mimetype="image"></inline-graphic>
</inline-formula>
that produce simulated 40, 000 samples.
<xref ref-type="table" rid="pone-0032250-t001">Table 1C</xref>
. Mean Square Error for 40, 000 simulated samples with
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e091.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and different values of
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e092.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e093.jpg" mimetype="image"></inline-graphic>
</inline-formula>
.</p>
</fn>
</table-wrap-foot>
</table-wrap>
</sec>
<sec id="s2b">
<title>Degree to which the sample size is underestimated using Eq. 5</title>
<p>To show the degree to which
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e094.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is underestimated using Eq. (5), we give an example (
<xref ref-type="table" rid="pone-0032250-t001">Table 1A</xref>
) in which Eq. (6) is used to calculate
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e095.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, that is, the probability that
<italic>W</italic>
will be smaller than, or equal to, the desired CI width (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e096.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) for a given value
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e097.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(number of positive pools) obtained using Eq. (5). The numerical example in
<xref ref-type="table" rid="pone-0032250-t001">Table 1</xref>
is given for several values of the population proportion (
<italic>p</italic>
) for a CI of 95%,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e098.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and for a desired width of
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e099.jpg" mimetype="image"></inline-graphic>
</inline-formula>
.
<xref ref-type="table" rid="pone-0032250-t001">Table 1A</xref>
presents the preliminary sample size
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e100.jpg" mimetype="image"></inline-graphic>
</inline-formula>
computed with Eq. (5), and three other increments computed as
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e101.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e102.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e103.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. For each sample size, the probability that
<italic>W</italic>
is smaller than the specified value (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e104.jpg" mimetype="image"></inline-graphic>
</inline-formula>
),
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e105.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, is calculated using Eq. (6). This is done to show that the required number of positive pools for the proportion (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e106.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, second column in
<xref ref-type="table" rid="pone-0032250-t001">Table 1A</xref>
) computed using Eq. (5) has a probability of around 0.50 that
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e107.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(third column in
<xref ref-type="table" rid="pone-0032250-t001">Table 1A</xref>
). For example, when
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e108.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, the preliminary sample size (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e109.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) is 49 and the probability of obtaining a
<italic>W</italic>
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e110.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e111.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is 0.4825564. With
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e112.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e113.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, we can only be 49.235% certain that
<italic>W</italic>
will be
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e114.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e115.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. When the number of pools increases by 10 (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e116.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, fourth column,
<xref ref-type="table" rid="pone-0032250-t001">Table 1A</xref>
) or by 20 (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e117.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, sixth column,
<xref ref-type="table" rid="pone-0032250-t001">Table 1A</xref>
), the probability
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e118.jpg" mimetype="image"></inline-graphic>
</inline-formula>
increases. For example, when
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e119.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, there are
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e120.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 = 69 units (pools) in the sample with
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e121.jpg" mimetype="image"></inline-graphic>
</inline-formula>
; for
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e122.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 = 89 pools in the sample, the
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e123.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. Thus, results of
<xref ref-type="table" rid="pone-0032250-t001">Table 1A</xref>
show that in order to ensure a high
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e124.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, a bigger sample size (number of positive pools) than the preliminary one (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e125.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) calculated using Eq. (5), is required. Also, we see in
<xref ref-type="table" rid="pone-0032250-t001">Table 1A</xref>
that 8 times out of 9 the preliminary sample size (number of positive pools) resulting from using Eq. (5) produces a
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e126.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, that is, 88.89% of the time
<italic>P</italic>
(
<italic>W</italic>
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e127.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e128.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) was lower than 50%.</p>
<p>For
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e129.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and a different combination of values of
<italic>k</italic>
and
<italic>r</italic>
that produces 40,000 samples,
<xref ref-type="table" rid="pone-0032250-t001">Table 1B</xref>
shows that for larger values of
<italic>r</italic>
, the percentage of times that the MLE of
<italic>p</italic>
is larger than the population proportion is lower. These results also show that the level of underestimation of the required number of pools (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e130.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) caused by the use of Eq. (5) is important and is mainly due to the fact that half of the time the population proportion
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e131.jpg" mimetype="image"></inline-graphic>
</inline-formula>
will be lower than the estimated proportion
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e132.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(
<xref ref-type="table" rid="pone-0032250-t001">Table 1B</xref>
)
<sup>;</sup>
thus the obtained CI width (
<italic>W</italic>
) will be larger than the specified
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e133.jpg" mimetype="image"></inline-graphic>
</inline-formula>
about more than half of the time. However, the expected value of the computed
<italic>W</italic>
is the value specified
<italic>a priori</italic>
(
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e134.jpg" mimetype="image"></inline-graphic>
</inline-formula>
), provided the correct value of the population variance is used. Therefore, the use of Eq. (5) will ensure that the desired width
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e135.jpg" mimetype="image"></inline-graphic>
</inline-formula>
for the CI will be obtained less than 50% of the time, that is,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e136.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. The values of the Mean Square Error (MSE) for
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e137.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and different combinations of
<italic>k</italic>
and
<italic>r</italic>
(
<xref ref-type="table" rid="pone-0032250-t001">Table 1C</xref>
) indicate MSE increases for lower values of
<italic>r</italic>
, however, no values of
<italic>k</italic>
seem to guarantee low bias.</p>
<p>Since Eq. (5) underestimates the required number of pools, in the following section, we propose three new methods to estimate the optimum sample size (two computational and one analytic).</p>
</sec>
<sec id="s2c">
<title>Computational optimum sample size estimation–methods 1 and 2</title>
<p>The optimal sample size is the smallest integer value (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e138.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) such that
<disp-formula>
<graphic xlink:href="pone.0032250.e139"></graphic>
<label>(7)</label>
</disp-formula>
where
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e140.jpg" mimetype="image"></inline-graphic>
</inline-formula>
will start with a minimal sample size, say
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e141.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e142.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is an indicator function showing whether or not the actual CI width (
<italic>W</italic>
) is ≤
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e143.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. The CI width will be calculated as
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e144.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. We determined that method 1 is when an exact
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e145.jpg" mimetype="image"></inline-graphic>
</inline-formula>
CI for
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e146.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is used, and method 2 is when the CI is computed using the Wald CI (Eq. 2) and Eq. (7), which we call the computational Wald procedure.</p>
<p>The CI used for the exact method (method 1) is the Clopper-Pearson CI, as explained in the following. When equal pool sizes
<italic>k</italic>
are used,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e147.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, where
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e148.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. Using the relationship between the negative binomial distribution and the incomplete beta function, Lui
<xref ref-type="bibr" rid="pone.0032250-Lui1">[13]</xref>
derived an exact interval for
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e149.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. The lower and upper confidence limits are
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e150.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e151.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, respectively, where
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e152.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e153.jpg" mimetype="image"></inline-graphic>
</inline-formula>
denotes the
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e154.jpg" mimetype="image"></inline-graphic>
</inline-formula>
quantile of the two-parameter beta distribution
<xref ref-type="bibr" rid="pone.0032250-Pritchard1">[9]</xref>
. Thus an exact
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e155.jpg" mimetype="image"></inline-graphic>
</inline-formula>
CI for
<italic>p</italic>
can be obtained by suitably transforming the endpoints of the
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e156.jpg" mimetype="image"></inline-graphic>
</inline-formula>
interval, i.e.,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e157.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e158.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<xref ref-type="bibr" rid="pone.0032250-Pritchard1">[9]</xref>
. Also, this interval for
<italic>p</italic>
can be formed using the relationship between the negative binomial and F distribution, in this case
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e159.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e160.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, where
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e161.jpg" mimetype="image"></inline-graphic>
</inline-formula>
denotes the upper
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e162.jpg" mimetype="image"></inline-graphic>
</inline-formula>
quantile of the two-parameter F distribution. Again, an exact
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e163.jpg" mimetype="image"></inline-graphic>
</inline-formula>
CI for
<italic>p</italic>
is
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e164.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e165.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<xref ref-type="bibr" rid="pone.0032250-Casella1">[26]</xref>
. This last version of the Clopper-Pearson CI has the advantage that the exact CI for
<italic>p</italic>
can be calculated by hand using standard F tables.</p>
<p>In methods 1 and 2, we start with a minimal sample size, say
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e166.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and increase the initial number of pools (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e167.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) by one unit, recalculating Eq. (7) each time, until the desired degree of certainty (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e168.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) is achieved; this will produce a modified number of pools (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e169.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) that assures, with a probability
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e170.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, that the
<italic>W</italic>
will be no wider than
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e171.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. In other words,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e172.jpg" mimetype="image"></inline-graphic>
</inline-formula>
ensures that the researcher will have approximately 100
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e173.jpg" mimetype="image"></inline-graphic>
</inline-formula>
percent certainty that the computed CI will have the desired width or smaller. For example, if the researcher requires 90% confidence that the obtained
<italic>W</italic>
will be no larger than the desired width (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e174.jpg" mimetype="image"></inline-graphic>
</inline-formula>
), (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e175.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) would be defined as 0.10, and there would be only a 10% chance that the CI width, around
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e176.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, would be larger than specified (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e177.jpg" mimetype="image"></inline-graphic>
</inline-formula>
)
<xref ref-type="bibr" rid="pone.0032250-Kelley1">[24]</xref>
,
<xref ref-type="bibr" rid="pone.0032250-Kelley3">[27]</xref>
.</p>
<p>Contrary to Eq. (5) above, the computational sample size proposed by Eq.(7) with methods 1 and 2 considers
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e178.jpg" mimetype="image"></inline-graphic>
</inline-formula>
as a random variable and gives a non-closed-form solution for computing a minimum sample size (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e179.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) that guarantees that
<italic>W</italic>
is smaller than, or equal to,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e180.jpg" mimetype="image"></inline-graphic>
</inline-formula>
with a probability of at least
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e181.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. In the following section, we propose a closed-form analytic method for determining the optimal sample size (number of positive pools required) that uses a single formula which assures the estimation of a narrow confidence interval.</p>
</sec>
<sec id="s2d">
<title>Analytic optimum sample size estimation–method 3</title>
<p>The CI width using the Wald interval for
<italic>p</italic>
is
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e182.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and
<italic>W</italic>
must be smaller than a specified value (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e183.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) with probability (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e184.jpg" mimetype="image"></inline-graphic>
</inline-formula>
). Therefore, the optimal sample size is defined as being the smallest integer value (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e185.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) such that
<disp-formula>
<graphic xlink:href="pone.0032250.e186"></graphic>
<label>(8)</label>
</disp-formula>
From Result 2 in
<xref ref-type="supplementary-material" rid="pone.0032250.s001">Appendix S1</xref>
, for fixed
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e187.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, the number of required positive pools with method 3 is given by
<disp-formula>
<graphic xlink:href="pone.0032250.e188"></graphic>
<label>(9)</label>
</disp-formula>
where
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e189.jpg" mimetype="image"></inline-graphic>
</inline-formula>
represents the desired degree of certainty (required probability) of achieving a CI width (
<italic>W</italic>
) for
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e190.jpg" mimetype="image"></inline-graphic>
</inline-formula>
that is no wider than the desired value (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e191.jpg" mimetype="image"></inline-graphic>
</inline-formula>
).
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e192.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e193.jpg" mimetype="image"></inline-graphic>
</inline-formula>
quantile of the standard normal distribution.
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e194.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the probability of a positive pool. Note that if
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e195.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(because the 50% quantile of a standard normal distribution is required), then Eq. (9) reduces to Eq. (5), that is, the formula determines the required number of pools assuming that the proportion of the population
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e196.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is known and fixed; this means, as already anticipated, that the required width
<italic>W</italic>
will be achieved only 50% of the time approximately. On the other hand, if
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e197.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, Eq. (9) reduces to
<disp-formula>
<graphic xlink:href="pone.0032250.e198"></graphic>
<label>(10)</label>
</disp-formula>
which is appropriate for determining the sample size without grouping (without making pools) (individual testing because
<italic>k</italic>
 = 1) and guarantees that
<italic>W</italic>
will be smaller than, or equal to,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e199.jpg" mimetype="image"></inline-graphic>
</inline-formula>
with a probability
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e200.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. In other words, only
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e201.jpg" mimetype="image"></inline-graphic>
</inline-formula>
of the time will
<italic>W</italic>
be larger than the desired CI width,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e202.jpg" mimetype="image"></inline-graphic>
</inline-formula>
.</p>
<p>Also note that when
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e203.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, Eq. (10) [individual inverse (negative) binomial sample size] reduces to the formula proposed by Lui
<xref ref-type="bibr" rid="pone.0032250-Lui1">[13]</xref>
under individual inverse (negative) binomial sampling,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e204.jpg" mimetype="image"></inline-graphic>
</inline-formula>
when the stochastic nature of the CI width is not considered. It is important to point out that Eq. (7) and the proposed formulas Eq. (9) and (10) determine a minimum sample size (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e205.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) that guarantees that
<italic>W</italic>
will be smaller than, or equal to,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e206.jpg" mimetype="image"></inline-graphic>
</inline-formula>
with a probability of at least
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e207.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. In contrast to Eq. (5), Eqs. (7), (9), and (10) account for the stochastic nature of the random variable
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e208.jpg" mimetype="image"></inline-graphic>
</inline-formula>
via the desired degree of certainty (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e209.jpg" mimetype="image"></inline-graphic>
</inline-formula>
). It should be pointed out that
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e210.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is what we call the sample size obtained from Eq. (5) or from Eq. (9) or (7) using
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e211.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e212.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the sample size obtained with Eq. (9) or (7) when
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e213.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. For this reason, the level of assurance would be
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e214.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. When using Equations (9) or (7), we suggest three ways of specifying the value of
<italic>p</italic>
: (1) perform a pilot study, (2) use the value of
<italic>p</italic>
reported in the literature of similar studies, and (3) use the upper bound for
<italic>p</italic>
that was reported. The upper bound should be chosen carefully to avoid estimators with high bias and high MSE; also, the upper bound needs to be used when the study was performed under group testing and when the value of
<italic>r</italic>
is not small
<xref ref-type="bibr" rid="pone.0032250-Pritchard1">[9]</xref>
. In addition, if the value of
<italic>p</italic>
reported in the literature was not obtained using group testing (but rather individual testing), then using an upper bound for sample size determination is not recommended. On the other hand, it is important to point out that the sample size from Equation (5) or from Equation (7) or (9) when using
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e215.jpg" mimetype="image"></inline-graphic>
</inline-formula>
will be called preliminary sample size in order to distinguish it from the sample size obtained from Equations (7) or (9) when level
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e216.jpg" mimetype="image"></inline-graphic>
</inline-formula>
.</p>
</sec>
</sec>
<sec id="s3">
<title>Results</title>
<p>Sample sizes are shown for
<italic>k</italic>
values of 40 (
<xref ref-type="table" rid="pone-0032250-t002">Table 2</xref>
),
<italic>p</italic>
values ranging from 0.005 to 0.025, and
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e217.jpg" mimetype="image"></inline-graphic>
</inline-formula>
values from 0.007 to 0.010 by 0.001 for each method. Within this table, we delineated three sub-tables with the modified number of pools
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e218.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e219.jpg" mimetype="image"></inline-graphic>
</inline-formula>
values of 0.50, 0.80, and 0.90, each for a CI coverage of 95%. Each condition is crossed with all other conditions in a factorial manner; thus there are a total of 108 different cases for planning an appropriate sample size for each proposed method. To examine the results shown in
<xref ref-type="table" rid="pone-0032250-t002">Table 2</xref>
, a simulation study was performed to examine the coverage and assurances of the samples as compared with the nominal coverage and assurances [
<xref ref-type="table" rid="pone-0032250-t003">Table 3</xref>
for the analytic procedure (method 3);
<xref ref-type="table" rid="pone-0032250-t004">Table 4</xref>
for the computational Wald procedure (method 2), and
<xref ref-type="table" rid="pone-0032250-t005">Table 5</xref>
for the exact Clopper-Pearson procedure (method 1)].</p>
<table-wrap id="pone-0032250-t002" position="float">
<object-id pub-id-type="doi">10.1371/journal.pone.0032250.t002</object-id>
<label>Table 2</label>
<caption>
<title>Sample size (required number of positive pools) for the three methods
<xref ref-type="table-fn" rid="nt102">b</xref>
.</title>
</caption>
<alternatives>
<graphic id="pone-0032250-t002-2" xlink:href="pone.0032250.t002"></graphic>
<table frame="hsides" rules="groups">
<colgroup span="1">
<col align="left" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
</colgroup>
<thead>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td colspan="4" align="left" rowspan="1">Analytic formula (method 3)</td>
<td colspan="4" align="left" rowspan="1">Clopper-Pearson (method 1)</td>
<td colspan="4" align="left" rowspan="1">Computational Wald (method 2)</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e220.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e221.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td colspan="4" align="left" rowspan="1">
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e222.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">
<italic>p</italic>
</td>
<td align="left" rowspan="1" colspan="1">0.007</td>
<td align="left" rowspan="1" colspan="1">0.008</td>
<td align="left" rowspan="1" colspan="1">0.009</td>
<td align="left" rowspan="1" colspan="1">0.010</td>
<td align="left" rowspan="1" colspan="1">0.007</td>
<td align="left" rowspan="1" colspan="1">0.008</td>
<td align="left" rowspan="1" colspan="1">0.009</td>
<td align="left" rowspan="1" colspan="1">0.010</td>
<td align="left" rowspan="1" colspan="1">0.007</td>
<td align="left" rowspan="1" colspan="1">0.008</td>
<td align="left" rowspan="1" colspan="1">0.009</td>
<td align="left" rowspan="1" colspan="1">0.010</td>
</tr>
</thead>
<tbody>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td colspan="4" align="left" rowspan="1">
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e223.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td colspan="4" align="left" rowspan="1">
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e224.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td colspan="4" align="left" rowspan="1">
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e225.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.005</td>
<td align="left" rowspan="1" colspan="1">8</td>
<td align="left" rowspan="1" colspan="1">6</td>
<td align="left" rowspan="1" colspan="1">5</td>
<td align="left" rowspan="1" colspan="1">4</td>
<td align="left" rowspan="1" colspan="1">9</td>
<td align="left" rowspan="1" colspan="1">7</td>
<td align="left" rowspan="1" colspan="1">6</td>
<td align="left" rowspan="1" colspan="1">5</td>
<td align="left" rowspan="1" colspan="1">9</td>
<td align="left" rowspan="1" colspan="1">7</td>
<td align="left" rowspan="1" colspan="1">6</td>
<td align="left" rowspan="1" colspan="1">5</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0075</td>
<td align="left" rowspan="1" colspan="1">18</td>
<td align="left" rowspan="1" colspan="1">14</td>
<td align="left" rowspan="1" colspan="1">11</td>
<td align="left" rowspan="1" colspan="1">9</td>
<td align="left" rowspan="1" colspan="1">19</td>
<td align="left" rowspan="1" colspan="1">15</td>
<td align="left" rowspan="1" colspan="1">12</td>
<td align="left" rowspan="1" colspan="1">10</td>
<td align="left" rowspan="1" colspan="1">19</td>
<td align="left" rowspan="1" colspan="1">15</td>
<td align="left" rowspan="1" colspan="1">12</td>
<td align="left" rowspan="1" colspan="1">10</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.01</td>
<td align="left" rowspan="1" colspan="1">31</td>
<td align="left" rowspan="1" colspan="1">24</td>
<td align="left" rowspan="1" colspan="1">19</td>
<td align="left" rowspan="1" colspan="1">15</td>
<td align="left" rowspan="1" colspan="1">34</td>
<td align="left" rowspan="1" colspan="1">26</td>
<td align="left" rowspan="1" colspan="1">21</td>
<td align="left" rowspan="1" colspan="1">17</td>
<td align="left" rowspan="1" colspan="1">33</td>
<td align="left" rowspan="1" colspan="1">25</td>
<td align="left" rowspan="1" colspan="1">20</td>
<td align="left" rowspan="1" colspan="1">17</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0125</td>
<td align="left" rowspan="1" colspan="1">49</td>
<td align="left" rowspan="1" colspan="1">38</td>
<td align="left" rowspan="1" colspan="1">30</td>
<td align="left" rowspan="1" colspan="1">24</td>
<td align="left" rowspan="1" colspan="1">52</td>
<td align="left" rowspan="1" colspan="1">41</td>
<td align="left" rowspan="1" colspan="1">33</td>
<td align="left" rowspan="1" colspan="1">27</td>
<td align="left" rowspan="1" colspan="1">50</td>
<td align="left" rowspan="1" colspan="1">39</td>
<td align="left" rowspan="1" colspan="1">31</td>
<td align="left" rowspan="1" colspan="1">25</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.015</td>
<td align="left" rowspan="1" colspan="1">72</td>
<td align="left" rowspan="1" colspan="1">55</td>
<td align="left" rowspan="1" colspan="1">43</td>
<td align="left" rowspan="1" colspan="1">35</td>
<td align="left" rowspan="1" colspan="1">75</td>
<td align="left" rowspan="1" colspan="1">59</td>
<td align="left" rowspan="1" colspan="1">46</td>
<td align="left" rowspan="1" colspan="1">38</td>
<td align="left" rowspan="1" colspan="1">73</td>
<td align="left" rowspan="1" colspan="1">56</td>
<td align="left" rowspan="1" colspan="1">45</td>
<td align="left" rowspan="1" colspan="1">36</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0175</td>
<td align="left" rowspan="1" colspan="1">98</td>
<td align="left" rowspan="1" colspan="1">75</td>
<td align="left" rowspan="1" colspan="1">59</td>
<td align="left" rowspan="1" colspan="1">48</td>
<td align="left" rowspan="1" colspan="1">103</td>
<td align="left" rowspan="1" colspan="1">80</td>
<td align="left" rowspan="1" colspan="1">63</td>
<td align="left" rowspan="1" colspan="1">52</td>
<td align="left" rowspan="1" colspan="1">100</td>
<td align="left" rowspan="1" colspan="1">76</td>
<td align="left" rowspan="1" colspan="1">61</td>
<td align="left" rowspan="1" colspan="1">50</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.02</td>
<td align="left" rowspan="1" colspan="1">130</td>
<td align="left" rowspan="1" colspan="1">99</td>
<td align="left" rowspan="1" colspan="1">78</td>
<td align="left" rowspan="1" colspan="1">64</td>
<td align="left" rowspan="1" colspan="1">136</td>
<td align="left" rowspan="1" colspan="1">105</td>
<td align="left" rowspan="1" colspan="1">84</td>
<td align="left" rowspan="1" colspan="1">68</td>
<td align="left" rowspan="1" colspan="1">131</td>
<td align="left" rowspan="1" colspan="1">101</td>
<td align="left" rowspan="1" colspan="1">80</td>
<td align="left" rowspan="1" colspan="1">65</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0225</td>
<td align="left" rowspan="1" colspan="1">166</td>
<td align="left" rowspan="1" colspan="1">127</td>
<td align="left" rowspan="1" colspan="1">101</td>
<td align="left" rowspan="1" colspan="1">81</td>
<td align="left" rowspan="1" colspan="1">174</td>
<td align="left" rowspan="1" colspan="1">134</td>
<td align="left" rowspan="1" colspan="1">106</td>
<td align="left" rowspan="1" colspan="1">86</td>
<td align="left" rowspan="1" colspan="1">168</td>
<td align="left" rowspan="1" colspan="1">128</td>
<td align="left" rowspan="1" colspan="1">101</td>
<td align="left" rowspan="1" colspan="1">82</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.025</td>
<td align="left" rowspan="1" colspan="1">208</td>
<td align="left" rowspan="1" colspan="1">159</td>
<td align="left" rowspan="1" colspan="1">126</td>
<td align="left" rowspan="1" colspan="1">102</td>
<td align="left" rowspan="1" colspan="1">218</td>
<td align="left" rowspan="1" colspan="1">167</td>
<td align="left" rowspan="1" colspan="1">133</td>
<td align="left" rowspan="1" colspan="1">109</td>
<td align="left" rowspan="1" colspan="1">209</td>
<td align="left" rowspan="1" colspan="1">160</td>
<td align="left" rowspan="1" colspan="1">126</td>
<td align="left" rowspan="1" colspan="1">104</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td colspan="4" align="left" rowspan="1">
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e226.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td colspan="4" align="left" rowspan="1">
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e227.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td colspan="4" align="left" rowspan="1">
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e228.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.005</td>
<td align="left" rowspan="1" colspan="1">12</td>
<td align="left" rowspan="1" colspan="1">10</td>
<td align="left" rowspan="1" colspan="1">8</td>
<td align="left" rowspan="1" colspan="1">7</td>
<td align="left" rowspan="1" colspan="1">14</td>
<td align="left" rowspan="1" colspan="1">12</td>
<td align="left" rowspan="1" colspan="1">10</td>
<td align="left" rowspan="1" colspan="1">9</td>
<td align="left" rowspan="1" colspan="1">14</td>
<td align="left" rowspan="1" colspan="1">12</td>
<td align="left" rowspan="1" colspan="1">10</td>
<td align="left" rowspan="1" colspan="1">8</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0075</td>
<td align="left" rowspan="1" colspan="1">24</td>
<td align="left" rowspan="1" colspan="1">19</td>
<td align="left" rowspan="1" colspan="1">16</td>
<td align="left" rowspan="1" colspan="1">13</td>
<td align="left" rowspan="1" colspan="1">26</td>
<td align="left" rowspan="1" colspan="1">22</td>
<td align="left" rowspan="1" colspan="1">18</td>
<td align="left" rowspan="1" colspan="1">15</td>
<td align="left" rowspan="1" colspan="1">26</td>
<td align="left" rowspan="1" colspan="1">21</td>
<td align="left" rowspan="1" colspan="1">17</td>
<td align="left" rowspan="1" colspan="1">15</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.01</td>
<td align="left" rowspan="1" colspan="1">40</td>
<td align="left" rowspan="1" colspan="1">32</td>
<td align="left" rowspan="1" colspan="1">26</td>
<td align="left" rowspan="1" colspan="1">22</td>
<td align="left" rowspan="1" colspan="1">44</td>
<td align="left" rowspan="1" colspan="1">35</td>
<td align="left" rowspan="1" colspan="1">29</td>
<td align="left" rowspan="1" colspan="1">24</td>
<td align="left" rowspan="1" colspan="1">43</td>
<td align="left" rowspan="1" colspan="1">33</td>
<td align="left" rowspan="1" colspan="1">28</td>
<td align="left" rowspan="1" colspan="1">24</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0125</td>
<td align="left" rowspan="1" colspan="1">61</td>
<td align="left" rowspan="1" colspan="1">48</td>
<td align="left" rowspan="1" colspan="1">39</td>
<td align="left" rowspan="1" colspan="1">32</td>
<td align="left" rowspan="1" colspan="1">65</td>
<td align="left" rowspan="1" colspan="1">52</td>
<td align="left" rowspan="1" colspan="1">43</td>
<td align="left" rowspan="1" colspan="1">35</td>
<td align="left" rowspan="1" colspan="1">63</td>
<td align="left" rowspan="1" colspan="1">50</td>
<td align="left" rowspan="1" colspan="1">40</td>
<td align="left" rowspan="1" colspan="1">34</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.015</td>
<td align="left" rowspan="1" colspan="1">86</td>
<td align="left" rowspan="1" colspan="1">67</td>
<td align="left" rowspan="1" colspan="1">54</td>
<td align="left" rowspan="1" colspan="1">45</td>
<td align="left" rowspan="1" colspan="1">91</td>
<td align="left" rowspan="1" colspan="1">71</td>
<td align="left" rowspan="1" colspan="1">59</td>
<td align="left" rowspan="1" colspan="1">49</td>
<td align="left" rowspan="1" colspan="1">88</td>
<td align="left" rowspan="1" colspan="1">69</td>
<td align="left" rowspan="1" colspan="1">56</td>
<td align="left" rowspan="1" colspan="1">47</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0175</td>
<td align="left" rowspan="1" colspan="1">115</td>
<td align="left" rowspan="1" colspan="1">90</td>
<td align="left" rowspan="1" colspan="1">72</td>
<td align="left" rowspan="1" colspan="1">60</td>
<td align="left" rowspan="1" colspan="1">121</td>
<td align="left" rowspan="1" colspan="1">96</td>
<td align="left" rowspan="1" colspan="1">77</td>
<td align="left" rowspan="1" colspan="1">65</td>
<td align="left" rowspan="1" colspan="1">118</td>
<td align="left" rowspan="1" colspan="1">93</td>
<td align="left" rowspan="1" colspan="1">75</td>
<td align="left" rowspan="1" colspan="1">62</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.02</td>
<td align="left" rowspan="1" colspan="1">149</td>
<td align="left" rowspan="1" colspan="1">116</td>
<td align="left" rowspan="1" colspan="1">94</td>
<td align="left" rowspan="1" colspan="1">77</td>
<td align="left" rowspan="1" colspan="1">156</td>
<td align="left" rowspan="1" colspan="1">123</td>
<td align="left" rowspan="1" colspan="1">100</td>
<td align="left" rowspan="1" colspan="1">82</td>
<td align="left" rowspan="1" colspan="1">151</td>
<td align="left" rowspan="1" colspan="1">118</td>
<td align="left" rowspan="1" colspan="1">96</td>
<td align="left" rowspan="1" colspan="1">80</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0225</td>
<td align="left" rowspan="1" colspan="1">189</td>
<td align="left" rowspan="1" colspan="1">147</td>
<td align="left" rowspan="1" colspan="1">118</td>
<td align="left" rowspan="1" colspan="1">97</td>
<td align="left" rowspan="1" colspan="1">198</td>
<td align="left" rowspan="1" colspan="1">154</td>
<td align="left" rowspan="1" colspan="1">126</td>
<td align="left" rowspan="1" colspan="1">104</td>
<td align="left" rowspan="1" colspan="1">190</td>
<td align="left" rowspan="1" colspan="1">150</td>
<td align="left" rowspan="1" colspan="1">120</td>
<td align="left" rowspan="1" colspan="1">99</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.025</td>
<td align="left" rowspan="1" colspan="1">234</td>
<td align="left" rowspan="1" colspan="1">182</td>
<td align="left" rowspan="1" colspan="1">146</td>
<td align="left" rowspan="1" colspan="1">120</td>
<td align="left" rowspan="1" colspan="1">244</td>
<td align="left" rowspan="1" colspan="1">191</td>
<td align="left" rowspan="1" colspan="1">154</td>
<td align="left" rowspan="1" colspan="1">128</td>
<td align="left" rowspan="1" colspan="1">237</td>
<td align="left" rowspan="1" colspan="1">185</td>
<td align="left" rowspan="1" colspan="1">148</td>
<td align="left" rowspan="1" colspan="1">122</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td colspan="4" align="left" rowspan="1">
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e229.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td colspan="4" align="left" rowspan="1">
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e230.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td colspan="4" align="left" rowspan="1">
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e231.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.005</td>
<td align="left" rowspan="1" colspan="1">14</td>
<td align="left" rowspan="1" colspan="1">11</td>
<td align="left" rowspan="1" colspan="1">9</td>
<td align="left" rowspan="1" colspan="1">8</td>
<td align="left" rowspan="1" colspan="1">17</td>
<td align="left" rowspan="1" colspan="1">14</td>
<td align="left" rowspan="1" colspan="1">12</td>
<td align="left" rowspan="1" colspan="1">11</td>
<td align="left" rowspan="1" colspan="1">17</td>
<td align="left" rowspan="1" colspan="1">14</td>
<td align="left" rowspan="1" colspan="1">12</td>
<td align="left" rowspan="1" colspan="1">11</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0075</td>
<td align="left" rowspan="1" colspan="1">27</td>
<td align="left" rowspan="1" colspan="1">22</td>
<td align="left" rowspan="1" colspan="1">18</td>
<td align="left" rowspan="1" colspan="1">15</td>
<td align="left" rowspan="1" colspan="1">31</td>
<td align="left" rowspan="1" colspan="1">25</td>
<td align="left" rowspan="1" colspan="1">21</td>
<td align="left" rowspan="1" colspan="1">18</td>
<td align="left" rowspan="1" colspan="1">30</td>
<td align="left" rowspan="1" colspan="1">25</td>
<td align="left" rowspan="1" colspan="1">21</td>
<td align="left" rowspan="1" colspan="1">18</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.01</td>
<td align="left" rowspan="1" colspan="1">45</td>
<td align="left" rowspan="1" colspan="1">36</td>
<td align="left" rowspan="1" colspan="1">29</td>
<td align="left" rowspan="1" colspan="1">25</td>
<td align="left" rowspan="1" colspan="1">49</td>
<td align="left" rowspan="1" colspan="1">39</td>
<td align="left" rowspan="1" colspan="1">33</td>
<td align="left" rowspan="1" colspan="1">29</td>
<td align="left" rowspan="1" colspan="1">48</td>
<td align="left" rowspan="1" colspan="1">38</td>
<td align="left" rowspan="1" colspan="1">32</td>
<td align="left" rowspan="1" colspan="1">27</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0125</td>
<td align="left" rowspan="1" colspan="1">67</td>
<td align="left" rowspan="1" colspan="1">53</td>
<td align="left" rowspan="1" colspan="1">43</td>
<td align="left" rowspan="1" colspan="1">36</td>
<td align="left" rowspan="1" colspan="1">71</td>
<td align="left" rowspan="1" colspan="1">57</td>
<td align="left" rowspan="1" colspan="1">48</td>
<td align="left" rowspan="1" colspan="1">40</td>
<td align="left" rowspan="1" colspan="1">70</td>
<td align="left" rowspan="1" colspan="1">56</td>
<td align="left" rowspan="1" colspan="1">46</td>
<td align="left" rowspan="1" colspan="1">39</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.015</td>
<td align="left" rowspan="1" colspan="1">93</td>
<td align="left" rowspan="1" colspan="1">73</td>
<td align="left" rowspan="1" colspan="1">59</td>
<td align="left" rowspan="1" colspan="1">50</td>
<td align="left" rowspan="1" colspan="1">98</td>
<td align="left" rowspan="1" colspan="1">79</td>
<td align="left" rowspan="1" colspan="1">65</td>
<td align="left" rowspan="1" colspan="1">55</td>
<td align="left" rowspan="1" colspan="1">96</td>
<td align="left" rowspan="1" colspan="1">76</td>
<td align="left" rowspan="1" colspan="1">62</td>
<td align="left" rowspan="1" colspan="1">53</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0175</td>
<td align="left" rowspan="1" colspan="1">123</td>
<td align="left" rowspan="1" colspan="1">97</td>
<td align="left" rowspan="1" colspan="1">79</td>
<td align="left" rowspan="1" colspan="1">65</td>
<td align="left" rowspan="1" colspan="1">130</td>
<td align="left" rowspan="1" colspan="1">104</td>
<td align="left" rowspan="1" colspan="1">85</td>
<td align="left" rowspan="1" colspan="1">71</td>
<td align="left" rowspan="1" colspan="1">127</td>
<td align="left" rowspan="1" colspan="1">101</td>
<td align="left" rowspan="1" colspan="1">82</td>
<td align="left" rowspan="1" colspan="1">69</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.02</td>
<td align="left" rowspan="1" colspan="1">159</td>
<td align="left" rowspan="1" colspan="1">125</td>
<td align="left" rowspan="1" colspan="1">101</td>
<td align="left" rowspan="1" colspan="1">84</td>
<td align="left" rowspan="1" colspan="1">167</td>
<td align="left" rowspan="1" colspan="1">134</td>
<td align="left" rowspan="1" colspan="1">109</td>
<td align="left" rowspan="1" colspan="1">91</td>
<td align="left" rowspan="1" colspan="1">163</td>
<td align="left" rowspan="1" colspan="1">128</td>
<td align="left" rowspan="1" colspan="1">105</td>
<td align="left" rowspan="1" colspan="1">87</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0225</td>
<td align="left" rowspan="1" colspan="1">200</td>
<td align="left" rowspan="1" colspan="1">157</td>
<td align="left" rowspan="1" colspan="1">127</td>
<td align="left" rowspan="1" colspan="1">105</td>
<td align="left" rowspan="1" colspan="1">211</td>
<td align="left" rowspan="1" colspan="1">166</td>
<td align="left" rowspan="1" colspan="1">136</td>
<td align="left" rowspan="1" colspan="1">113</td>
<td align="left" rowspan="1" colspan="1">203</td>
<td align="left" rowspan="1" colspan="1">160</td>
<td align="left" rowspan="1" colspan="1">131</td>
<td align="left" rowspan="1" colspan="1">110</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.025</td>
<td align="left" rowspan="1" colspan="1">247</td>
<td align="left" rowspan="1" colspan="1">193</td>
<td align="left" rowspan="1" colspan="1">156</td>
<td align="left" rowspan="1" colspan="1">129</td>
<td align="left" rowspan="1" colspan="1">260</td>
<td align="left" rowspan="1" colspan="1">205</td>
<td align="left" rowspan="1" colspan="1">167</td>
<td align="left" rowspan="1" colspan="1">138</td>
<td align="left" rowspan="1" colspan="1">250</td>
<td align="left" rowspan="1" colspan="1">197</td>
<td align="left" rowspan="1" colspan="1">159</td>
<td align="left" rowspan="1" colspan="1">132</td>
</tr>
</tbody>
</table>
</alternatives>
<table-wrap-foot>
<fn id="nt102">
<label>b</label>
<p>For a CI of 95%,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e232.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, four desired widths (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e233.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) and three values of
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e234.jpg" mimetype="image"></inline-graphic>
</inline-formula>
(0.5, 0.8, and 0.90). The value of
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e235.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the population proportion,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e236.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the preliminary number of required positive pools,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e237.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the modified required number of positive pools, and
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e238.jpg" mimetype="image"></inline-graphic>
</inline-formula>
is the assurance for the desired degree of certainty of achieving a CI for
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e239.jpg" mimetype="image"></inline-graphic>
</inline-formula>
that is no wider than the desired CI width (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e240.jpg" mimetype="image"></inline-graphic>
</inline-formula>
).</p>
</fn>
</table-wrap-foot>
</table-wrap>
<table-wrap id="pone-0032250-t003" position="float">
<object-id pub-id-type="doi">10.1371/journal.pone.0032250.t003</object-id>
<label>Table 3</label>
<caption>
<title>Simulation study of the coverage and assurance for method 3 (analytic formula)
<xref ref-type="table-fn" rid="nt103">c</xref>
.</title>
</caption>
<alternatives>
<graphic id="pone-0032250-t003-3" xlink:href="pone.0032250.t003"></graphic>
<table frame="hsides" rules="groups">
<colgroup span="1">
<col align="left" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
</colgroup>
<thead>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e241.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e242.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">
<italic>P</italic>
</td>
<td align="left" rowspan="1" colspan="1">0.007</td>
<td align="left" rowspan="1" colspan="1">0.008</td>
<td align="left" rowspan="1" colspan="1">0.009</td>
<td align="left" rowspan="1" colspan="1">0.010</td>
<td align="left" rowspan="1" colspan="1">0.007</td>
<td align="left" rowspan="1" colspan="1">0.008</td>
<td align="left" rowspan="1" colspan="1">0.009</td>
<td align="left" rowspan="1" colspan="1">0.010</td>
</tr>
</thead>
<tbody>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td colspan="4" align="center" rowspan="1">-------Coverage
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e243.jpg" mimetype="image"></inline-graphic>
</inline-formula>
--------</td>
<td colspan="4" align="center" rowspan="1">----------Assurance
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e244.jpg" mimetype="image"></inline-graphic>
</inline-formula>
-----------</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0050</td>
<td align="left" rowspan="1" colspan="1">0.9550</td>
<td align="left" rowspan="1" colspan="1">0.9553</td>
<td align="left" rowspan="1" colspan="1">0.9590</td>
<td align="left" rowspan="1" colspan="1">0.9534</td>
<td align="left" rowspan="1" colspan="1">0.4670</td>
<td align="left" rowspan="1" colspan="1">0.4323</td>
<td align="left" rowspan="1" colspan="1">0.4764</td>
<td align="left" rowspan="1" colspan="1">0.4543</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0075</td>
<td align="left" rowspan="1" colspan="1">0.9530</td>
<td align="left" rowspan="1" colspan="1">0.9585</td>
<td align="left" rowspan="1" colspan="1">0.9534</td>
<td align="left" rowspan="1" colspan="1">0.9573</td>
<td align="left" rowspan="1" colspan="1">0.4917</td>
<td align="left" rowspan="1" colspan="1">0.4782</td>
<td align="left" rowspan="1" colspan="1">0.4863</td>
<td align="left" rowspan="1" colspan="1">0.4613</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0100</td>
<td align="left" rowspan="1" colspan="1">0.9512</td>
<td align="left" rowspan="1" colspan="1">0.9546</td>
<td align="left" rowspan="1" colspan="1">0.9508</td>
<td align="left" rowspan="1" colspan="1">0.9555</td>
<td align="left" rowspan="1" colspan="1">0.4573</td>
<td align="left" rowspan="1" colspan="1">0.4713</td>
<td align="left" rowspan="1" colspan="1">0.4669</td>
<td align="left" rowspan="1" colspan="1">0.4546</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0125</td>
<td align="left" rowspan="1" colspan="1">0.9508</td>
<td align="left" rowspan="1" colspan="1">0.9518</td>
<td align="left" rowspan="1" colspan="1">0.9551</td>
<td align="left" rowspan="1" colspan="1">0.9522</td>
<td align="left" rowspan="1" colspan="1">0.4601</td>
<td align="left" rowspan="1" colspan="1">0.4973</td>
<td align="left" rowspan="1" colspan="1">0.4920</td>
<td align="left" rowspan="1" colspan="1">0.4787</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0150</td>
<td align="left" rowspan="1" colspan="1">0.9497</td>
<td align="left" rowspan="1" colspan="1">0.9475</td>
<td align="left" rowspan="1" colspan="1">0.9527</td>
<td align="left" rowspan="1" colspan="1">0.9541</td>
<td align="left" rowspan="1" colspan="1">0.4886</td>
<td align="left" rowspan="1" colspan="1">0.4731</td>
<td align="left" rowspan="1" colspan="1">0.4485</td>
<td align="left" rowspan="1" colspan="1">0.4614</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0175</td>
<td align="left" rowspan="1" colspan="1">0.9513</td>
<td align="left" rowspan="1" colspan="1">0.9506</td>
<td align="left" rowspan="1" colspan="1">0.9533</td>
<td align="left" rowspan="1" colspan="1">0.9533</td>
<td align="left" rowspan="1" colspan="1">0.4821</td>
<td align="left" rowspan="1" colspan="1">0.4696</td>
<td align="left" rowspan="1" colspan="1">0.4826</td>
<td align="left" rowspan="1" colspan="1">0.4895</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0200</td>
<td align="left" rowspan="1" colspan="1">0.9525</td>
<td align="left" rowspan="1" colspan="1">0.9516</td>
<td align="left" rowspan="1" colspan="1">0.9539</td>
<td align="left" rowspan="1" colspan="1">0.9523</td>
<td align="left" rowspan="1" colspan="1">0.4867</td>
<td align="left" rowspan="1" colspan="1">0.4835</td>
<td align="left" rowspan="1" colspan="1">0.4893</td>
<td align="left" rowspan="1" colspan="1">0.4826</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0225</td>
<td align="left" rowspan="1" colspan="1">0.9483</td>
<td align="left" rowspan="1" colspan="1">0.9527</td>
<td align="left" rowspan="1" colspan="1">0.9458</td>
<td align="left" rowspan="1" colspan="1">0.9539</td>
<td align="left" rowspan="1" colspan="1">0.4949</td>
<td align="left" rowspan="1" colspan="1">0.4878</td>
<td align="left" rowspan="1" colspan="1">0.5046</td>
<td align="left" rowspan="1" colspan="1">0.4850</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0250</td>
<td align="left" rowspan="1" colspan="1">0.9527</td>
<td align="left" rowspan="1" colspan="1">0.9514</td>
<td align="left" rowspan="1" colspan="1">0.9481</td>
<td align="left" rowspan="1" colspan="1">0.9472</td>
<td align="left" rowspan="1" colspan="1">0.5019</td>
<td align="left" rowspan="1" colspan="1">0.4907</td>
<td align="left" rowspan="1" colspan="1">0.4992</td>
<td align="left" rowspan="1" colspan="1">0.4725</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td colspan="4" align="center" rowspan="1">------Coverage
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e245.jpg" mimetype="image"></inline-graphic>
</inline-formula>
-------</td>
<td colspan="4" align="center" rowspan="1">---------Assurance
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e246.jpg" mimetype="image"></inline-graphic>
</inline-formula>
-----------</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0050</td>
<td align="left" rowspan="1" colspan="1">0.9521</td>
<td align="left" rowspan="1" colspan="1">0.9542</td>
<td align="left" rowspan="1" colspan="1">0.9546</td>
<td align="left" rowspan="1" colspan="1">0.9581</td>
<td align="left" rowspan="1" colspan="1">0.7314</td>
<td align="left" rowspan="1" colspan="1">0.7523</td>
<td align="left" rowspan="1" colspan="1">0.7352</td>
<td align="left" rowspan="1" colspan="1">0.7334</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0075</td>
<td align="left" rowspan="1" colspan="1">0.9546</td>
<td align="left" rowspan="1" colspan="1">0.9571</td>
<td align="left" rowspan="1" colspan="1">0.9549</td>
<td align="left" rowspan="1" colspan="1">0.9542</td>
<td align="left" rowspan="1" colspan="1">0.7367</td>
<td align="left" rowspan="1" colspan="1">0.7324</td>
<td align="left" rowspan="1" colspan="1">0.7626</td>
<td align="left" rowspan="1" colspan="1">0.7191</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0100</td>
<td align="left" rowspan="1" colspan="1">0.9509</td>
<td align="left" rowspan="1" colspan="1">0.9515</td>
<td align="left" rowspan="1" colspan="1">0.9534</td>
<td align="left" rowspan="1" colspan="1">0.9548</td>
<td align="left" rowspan="1" colspan="1">0.7603</td>
<td align="left" rowspan="1" colspan="1">0.7573</td>
<td align="left" rowspan="1" colspan="1">0.7538</td>
<td align="left" rowspan="1" colspan="1">0.7743</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0125</td>
<td align="left" rowspan="1" colspan="1">0.9489</td>
<td align="left" rowspan="1" colspan="1">0.9557</td>
<td align="left" rowspan="1" colspan="1">0.9495</td>
<td align="left" rowspan="1" colspan="1">0.9494</td>
<td align="left" rowspan="1" colspan="1">0.7725</td>
<td align="left" rowspan="1" colspan="1">0.7594</td>
<td align="left" rowspan="1" colspan="1">0.7622</td>
<td align="left" rowspan="1" colspan="1">0.7653</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0150</td>
<td align="left" rowspan="1" colspan="1">0.9511</td>
<td align="left" rowspan="1" colspan="1">0.9488</td>
<td align="left" rowspan="1" colspan="1">0.9525</td>
<td align="left" rowspan="1" colspan="1">0.9520</td>
<td align="left" rowspan="1" colspan="1">0.7819</td>
<td align="left" rowspan="1" colspan="1">0.7704</td>
<td align="left" rowspan="1" colspan="1">0.7536</td>
<td align="left" rowspan="1" colspan="1">0.7839</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0175</td>
<td align="left" rowspan="1" colspan="1">0.9521</td>
<td align="left" rowspan="1" colspan="1">0.9538</td>
<td align="left" rowspan="1" colspan="1">0.9499</td>
<td align="left" rowspan="1" colspan="1">0.9511</td>
<td align="left" rowspan="1" colspan="1">0.7760</td>
<td align="left" rowspan="1" colspan="1">0.7781</td>
<td align="left" rowspan="1" colspan="1">0.7630</td>
<td align="left" rowspan="1" colspan="1">0.7678</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0200</td>
<td align="left" rowspan="1" colspan="1">0.9484</td>
<td align="left" rowspan="1" colspan="1">0.9495</td>
<td align="left" rowspan="1" colspan="1">0.9507</td>
<td align="left" rowspan="1" colspan="1">0.9493</td>
<td align="left" rowspan="1" colspan="1">0.7780</td>
<td align="left" rowspan="1" colspan="1">0.7692</td>
<td align="left" rowspan="1" colspan="1">0.7740</td>
<td align="left" rowspan="1" colspan="1">0.7522</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0225</td>
<td align="left" rowspan="1" colspan="1">0.9491</td>
<td align="left" rowspan="1" colspan="1">0.9514</td>
<td align="left" rowspan="1" colspan="1">0.9541</td>
<td align="left" rowspan="1" colspan="1">0.9495</td>
<td align="left" rowspan="1" colspan="1">0.7848</td>
<td align="left" rowspan="1" colspan="1">0.7636</td>
<td align="left" rowspan="1" colspan="1">0.7766</td>
<td align="left" rowspan="1" colspan="1">0.7656</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td colspan="4" align="center" rowspan="1">-------Coverage
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e247.jpg" mimetype="image"></inline-graphic>
</inline-formula>
-------</td>
<td colspan="4" align="center" rowspan="1">---------Assurance
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e248.jpg" mimetype="image"></inline-graphic>
</inline-formula>
-----------</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0050</td>
<td align="left" rowspan="1" colspan="1">0.9535</td>
<td align="left" rowspan="1" colspan="1">0.9524</td>
<td align="left" rowspan="1" colspan="1">0.9551</td>
<td align="left" rowspan="1" colspan="1">0.9546</td>
<td align="left" rowspan="1" colspan="1">0.8300</td>
<td align="left" rowspan="1" colspan="1">0.8007</td>
<td align="left" rowspan="1" colspan="1">0.7798</td>
<td align="left" rowspan="1" colspan="1">0.8127</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0075</td>
<td align="left" rowspan="1" colspan="1">0.9504</td>
<td align="left" rowspan="1" colspan="1">0.9534</td>
<td align="left" rowspan="1" colspan="1">0.9527</td>
<td align="left" rowspan="1" colspan="1">0.9532</td>
<td align="left" rowspan="1" colspan="1">0.8434</td>
<td align="left" rowspan="1" colspan="1">0.8537</td>
<td align="left" rowspan="1" colspan="1">0.8385</td>
<td align="left" rowspan="1" colspan="1">0.8301</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0100</td>
<td align="left" rowspan="1" colspan="1">0.9502</td>
<td align="left" rowspan="1" colspan="1">0.9503</td>
<td align="left" rowspan="1" colspan="1">0.9521</td>
<td align="left" rowspan="1" colspan="1">0.9508</td>
<td align="left" rowspan="1" colspan="1">0.8741</td>
<td align="left" rowspan="1" colspan="1">0.8686</td>
<td align="left" rowspan="1" colspan="1">0.8384</td>
<td align="left" rowspan="1" colspan="1">0.8583</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0125</td>
<td align="left" rowspan="1" colspan="1">0.9534</td>
<td align="left" rowspan="1" colspan="1">0.9495</td>
<td align="left" rowspan="1" colspan="1">0.9539</td>
<td align="left" rowspan="1" colspan="1">0.9552</td>
<td align="left" rowspan="1" colspan="1">0.8689</td>
<td align="left" rowspan="1" colspan="1">0.8672</td>
<td align="left" rowspan="1" colspan="1">0.8483</td>
<td align="left" rowspan="1" colspan="1">0.8580</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0150</td>
<td align="left" rowspan="1" colspan="1">0.9476</td>
<td align="left" rowspan="1" colspan="1">0.9545</td>
<td align="left" rowspan="1" colspan="1">0.9510</td>
<td align="left" rowspan="1" colspan="1">0.9501</td>
<td align="left" rowspan="1" colspan="1">0.8670</td>
<td align="left" rowspan="1" colspan="1">0.8722</td>
<td align="left" rowspan="1" colspan="1">0.8646</td>
<td align="left" rowspan="1" colspan="1">0.8677</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0175</td>
<td align="left" rowspan="1" colspan="1">0.9515</td>
<td align="left" rowspan="1" colspan="1">0.9538</td>
<td align="left" rowspan="1" colspan="1">0.9543</td>
<td align="left" rowspan="1" colspan="1">0.9521</td>
<td align="left" rowspan="1" colspan="1">0.8757</td>
<td align="left" rowspan="1" colspan="1">0.8682</td>
<td align="left" rowspan="1" colspan="1">0.8633</td>
<td align="left" rowspan="1" colspan="1">0.8570</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0200</td>
<td align="left" rowspan="1" colspan="1">0.9490</td>
<td align="left" rowspan="1" colspan="1">0.9484</td>
<td align="left" rowspan="1" colspan="1">0.9487</td>
<td align="left" rowspan="1" colspan="1">0.9549</td>
<td align="left" rowspan="1" colspan="1">0.8781</td>
<td align="left" rowspan="1" colspan="1">0.8723</td>
<td align="left" rowspan="1" colspan="1">0.8764</td>
<td align="left" rowspan="1" colspan="1">0.8644</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0225</td>
<td align="left" rowspan="1" colspan="1">0.9490</td>
<td align="left" rowspan="1" colspan="1">0.9500</td>
<td align="left" rowspan="1" colspan="1">0.9520</td>
<td align="left" rowspan="1" colspan="1">0.9544</td>
<td align="left" rowspan="1" colspan="1">0.8766</td>
<td align="left" rowspan="1" colspan="1">0.8767</td>
<td align="left" rowspan="1" colspan="1">0.8850</td>
<td align="left" rowspan="1" colspan="1">0.8744</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0250</td>
<td align="left" rowspan="1" colspan="1">0.9522</td>
<td align="left" rowspan="1" colspan="1">0.9488</td>
<td align="left" rowspan="1" colspan="1">0.9543</td>
<td align="left" rowspan="1" colspan="1">0.9492</td>
<td align="left" rowspan="1" colspan="1">0.8803</td>
<td align="left" rowspan="1" colspan="1">0.8671</td>
<td align="left" rowspan="1" colspan="1">0.8784</td>
<td align="left" rowspan="1" colspan="1">0.8698</td>
</tr>
</tbody>
</table>
</alternatives>
<table-wrap-foot>
<fn id="nt103">
<label>c</label>
<p>These coverages and these levels of assurance are for sample sizes obtained with the analytic formula (method 3) presented in
<xref ref-type="table" rid="pone-0032250-t002">Table 2</xref>
, for a CI of 95%,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e249.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, four desired widths (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e250.jpg" mimetype="image"></inline-graphic>
</inline-formula>
), and three values of assurance
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e251.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</p>
</fn>
</table-wrap-foot>
</table-wrap>
<table-wrap id="pone-0032250-t004" position="float">
<object-id pub-id-type="doi">10.1371/journal.pone.0032250.t004</object-id>
<label>Table 4</label>
<caption>
<title>Simulation study of coverage and assurance for method 2
<xref ref-type="table-fn" rid="nt104">d</xref>
.</title>
</caption>
<alternatives>
<graphic id="pone-0032250-t004-4" xlink:href="pone.0032250.t004"></graphic>
<table frame="hsides" rules="groups">
<colgroup span="1">
<col align="left" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
</colgroup>
<thead>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e252.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e253.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">
<italic>p</italic>
</td>
<td align="left" rowspan="1" colspan="1">0.007</td>
<td align="left" rowspan="1" colspan="1">0.008</td>
<td align="left" rowspan="1" colspan="1">0.009</td>
<td align="left" rowspan="1" colspan="1">0.010</td>
<td align="left" rowspan="1" colspan="1">0.007</td>
<td align="left" rowspan="1" colspan="1">0.008</td>
<td align="left" rowspan="1" colspan="1">0.009</td>
<td align="left" rowspan="1" colspan="1">0.010</td>
</tr>
</thead>
<tbody>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td colspan="4" align="center" rowspan="1">-------Coverage
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e254.jpg" mimetype="image"></inline-graphic>
</inline-formula>
--------</td>
<td colspan="4" align="center" rowspan="1">----------Assurance
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e255.jpg" mimetype="image"></inline-graphic>
</inline-formula>
-----------</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0050</td>
<td align="left" rowspan="1" colspan="1">0.9544</td>
<td align="left" rowspan="1" colspan="1">0.9580</td>
<td align="left" rowspan="1" colspan="1">0.9538</td>
<td align="left" rowspan="1" colspan="1">0.9581</td>
<td align="left" rowspan="1" colspan="1">0.5393</td>
<td align="left" rowspan="1" colspan="1">0.5431</td>
<td align="left" rowspan="1" colspan="1">0.5653</td>
<td align="left" rowspan="1" colspan="1">0.5498</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0075</td>
<td align="left" rowspan="1" colspan="1">0.9548</td>
<td align="left" rowspan="1" colspan="1">0.9523</td>
<td align="left" rowspan="1" colspan="1">0.9533</td>
<td align="left" rowspan="1" colspan="1">0.9595</td>
<td align="left" rowspan="1" colspan="1">0.5388</td>
<td align="left" rowspan="1" colspan="1">0.5329</td>
<td align="left" rowspan="1" colspan="1">0.5576</td>
<td align="left" rowspan="1" colspan="1">0.5337</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0100</td>
<td align="left" rowspan="1" colspan="1">0.9524</td>
<td align="left" rowspan="1" colspan="1">0.9502</td>
<td align="left" rowspan="1" colspan="1">0.9574</td>
<td align="left" rowspan="1" colspan="1">0.9536</td>
<td align="left" rowspan="1" colspan="1">0.5397</td>
<td align="left" rowspan="1" colspan="1">0.5012</td>
<td align="left" rowspan="1" colspan="1">0.5028</td>
<td align="left" rowspan="1" colspan="1">0.5383</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0125</td>
<td align="left" rowspan="1" colspan="1">0.9499</td>
<td align="left" rowspan="1" colspan="1">0.9508</td>
<td align="left" rowspan="1" colspan="1">0.9518</td>
<td align="left" rowspan="1" colspan="1">0.9557</td>
<td align="left" rowspan="1" colspan="1">0.5040</td>
<td align="left" rowspan="1" colspan="1">0.5134</td>
<td align="left" rowspan="1" colspan="1">0.5079</td>
<td align="left" rowspan="1" colspan="1">0.5015</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0150</td>
<td align="left" rowspan="1" colspan="1">0.9505</td>
<td align="left" rowspan="1" colspan="1">0.9522</td>
<td align="left" rowspan="1" colspan="1">0.9520</td>
<td align="left" rowspan="1" colspan="1">0.9507</td>
<td align="left" rowspan="1" colspan="1">0.5216</td>
<td align="left" rowspan="1" colspan="1">0.5116</td>
<td align="left" rowspan="1" colspan="1">0.5384</td>
<td align="left" rowspan="1" colspan="1">0.5107</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0175</td>
<td align="left" rowspan="1" colspan="1">0.9489</td>
<td align="left" rowspan="1" colspan="1">0.9497</td>
<td align="left" rowspan="1" colspan="1">0.9489</td>
<td align="left" rowspan="1" colspan="1">0.9479</td>
<td align="left" rowspan="1" colspan="1">0.5149</td>
<td align="left" rowspan="1" colspan="1">0.5069</td>
<td align="left" rowspan="1" colspan="1">0.5165</td>
<td align="left" rowspan="1" colspan="1">0.5317</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0200</td>
<td align="left" rowspan="1" colspan="1">0.9522</td>
<td align="left" rowspan="1" colspan="1">0.9485</td>
<td align="left" rowspan="1" colspan="1">0.9494</td>
<td align="left" rowspan="1" colspan="1">0.9509</td>
<td align="left" rowspan="1" colspan="1">0.5133</td>
<td align="left" rowspan="1" colspan="1">0.5112</td>
<td align="left" rowspan="1" colspan="1">0.5139</td>
<td align="left" rowspan="1" colspan="1">0.5113</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0225</td>
<td align="left" rowspan="1" colspan="1">0.9514</td>
<td align="left" rowspan="1" colspan="1">0.9519</td>
<td align="left" rowspan="1" colspan="1">0.9457</td>
<td align="left" rowspan="1" colspan="1">0.9548</td>
<td align="left" rowspan="1" colspan="1">0.5072</td>
<td align="left" rowspan="1" colspan="1">0.5076</td>
<td align="left" rowspan="1" colspan="1">0.5048</td>
<td align="left" rowspan="1" colspan="1">0.5151</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0250</td>
<td align="left" rowspan="1" colspan="1">0.9520</td>
<td align="left" rowspan="1" colspan="1">0.9512</td>
<td align="left" rowspan="1" colspan="1">0.9465</td>
<td align="left" rowspan="1" colspan="1">0.9516</td>
<td align="left" rowspan="1" colspan="1">0.5086</td>
<td align="left" rowspan="1" colspan="1">0.5115</td>
<td align="left" rowspan="1" colspan="1">0.5051</td>
<td align="left" rowspan="1" colspan="1">0.5179</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td colspan="4" align="center" rowspan="1">------Coverage
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e256.jpg" mimetype="image"></inline-graphic>
</inline-formula>
--------</td>
<td colspan="4" align="center" rowspan="1">---------Assurance
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e257.jpg" mimetype="image"></inline-graphic>
</inline-formula>
-----------</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0050</td>
<td align="left" rowspan="1" colspan="1">0.9543</td>
<td align="left" rowspan="1" colspan="1">0.9528</td>
<td align="left" rowspan="1" colspan="1">0.9532</td>
<td align="left" rowspan="1" colspan="1">0.9566</td>
<td align="left" rowspan="1" colspan="1">0.8286</td>
<td align="left" rowspan="1" colspan="1">0.8531</td>
<td align="left" rowspan="1" colspan="1">0.8413</td>
<td align="left" rowspan="1" colspan="1">0.8109</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0075</td>
<td align="left" rowspan="1" colspan="1">0.9554</td>
<td align="left" rowspan="1" colspan="1">0.9523</td>
<td align="left" rowspan="1" colspan="1">0.9551</td>
<td align="left" rowspan="1" colspan="1">0.9516</td>
<td align="left" rowspan="1" colspan="1">0.8206</td>
<td align="left" rowspan="1" colspan="1">0.8051</td>
<td align="left" rowspan="1" colspan="1">0.8029</td>
<td align="left" rowspan="1" colspan="1">0.8293</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0100</td>
<td align="left" rowspan="1" colspan="1">0.9516</td>
<td align="left" rowspan="1" colspan="1">0.9524</td>
<td align="left" rowspan="1" colspan="1">0.9560</td>
<td align="left" rowspan="1" colspan="1">0.9545</td>
<td align="left" rowspan="1" colspan="1">0.8296</td>
<td align="left" rowspan="1" colspan="1">0.8019</td>
<td align="left" rowspan="1" colspan="1">0.8206</td>
<td align="left" rowspan="1" colspan="1">0.8415</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0125</td>
<td align="left" rowspan="1" colspan="1">0.9476</td>
<td align="left" rowspan="1" colspan="1">0.9473</td>
<td align="left" rowspan="1" colspan="1">0.9508</td>
<td align="left" rowspan="1" colspan="1">0.9529</td>
<td align="left" rowspan="1" colspan="1">0.8092</td>
<td align="left" rowspan="1" colspan="1">0.8226</td>
<td align="left" rowspan="1" colspan="1">0.8016</td>
<td align="left" rowspan="1" colspan="1">0.8167</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0150</td>
<td align="left" rowspan="1" colspan="1">0.9477</td>
<td align="left" rowspan="1" colspan="1">0.9517</td>
<td align="left" rowspan="1" colspan="1">0.9511</td>
<td align="left" rowspan="1" colspan="1">0.9526</td>
<td align="left" rowspan="1" colspan="1">0.8077</td>
<td align="left" rowspan="1" colspan="1">0.8028</td>
<td align="left" rowspan="1" colspan="1">0.8161</td>
<td align="left" rowspan="1" colspan="1">0.8128</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0175</td>
<td align="left" rowspan="1" colspan="1">0.9504</td>
<td align="left" rowspan="1" colspan="1">0.9503</td>
<td align="left" rowspan="1" colspan="1">0.9502</td>
<td align="left" rowspan="1" colspan="1">0.9466</td>
<td align="left" rowspan="1" colspan="1">0.8108</td>
<td align="left" rowspan="1" colspan="1">0.8170</td>
<td align="left" rowspan="1" colspan="1">0.8063</td>
<td align="left" rowspan="1" colspan="1">0.8180</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0200</td>
<td align="left" rowspan="1" colspan="1">0.9508</td>
<td align="left" rowspan="1" colspan="1">0.9514</td>
<td align="left" rowspan="1" colspan="1">0.9504</td>
<td align="left" rowspan="1" colspan="1">0.9504</td>
<td align="left" rowspan="1" colspan="1">0.8089</td>
<td align="left" rowspan="1" colspan="1">0.8050</td>
<td align="left" rowspan="1" colspan="1">0.8180</td>
<td align="left" rowspan="1" colspan="1">0.8146</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0225</td>
<td align="left" rowspan="1" colspan="1">0.9498</td>
<td align="left" rowspan="1" colspan="1">0.9500</td>
<td align="left" rowspan="1" colspan="1">0.9460</td>
<td align="left" rowspan="1" colspan="1">0.9527</td>
<td align="left" rowspan="1" colspan="1">0.7995</td>
<td align="left" rowspan="1" colspan="1">0.8131</td>
<td align="left" rowspan="1" colspan="1">0.8092</td>
<td align="left" rowspan="1" colspan="1">0.8034</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td colspan="4" align="center" rowspan="1">-------Coverage
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e258.jpg" mimetype="image"></inline-graphic>
</inline-formula>
--------</td>
<td colspan="4" align="center" rowspan="1">---------Assurance
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e259.jpg" mimetype="image"></inline-graphic>
</inline-formula>
-----------</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0050</td>
<td align="left" rowspan="1" colspan="1">0.9492</td>
<td align="left" rowspan="1" colspan="1">0.9525</td>
<td align="left" rowspan="1" colspan="1">0.9527</td>
<td align="left" rowspan="1" colspan="1">0.9537</td>
<td align="left" rowspan="1" colspan="1">0.9223</td>
<td align="left" rowspan="1" colspan="1">0.9104</td>
<td align="left" rowspan="1" colspan="1">0.9223</td>
<td align="left" rowspan="1" colspan="1">0.9294</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0075</td>
<td align="left" rowspan="1" colspan="1">0.9504</td>
<td align="left" rowspan="1" colspan="1">0.9529</td>
<td align="left" rowspan="1" colspan="1">0.9526</td>
<td align="left" rowspan="1" colspan="1">0.9548</td>
<td align="left" rowspan="1" colspan="1">0.9050</td>
<td align="left" rowspan="1" colspan="1">0.9165</td>
<td align="left" rowspan="1" colspan="1">0.9242</td>
<td align="left" rowspan="1" colspan="1">0.9103</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0100</td>
<td align="left" rowspan="1" colspan="1">0.9505</td>
<td align="left" rowspan="1" colspan="1">0.9520</td>
<td align="left" rowspan="1" colspan="1">0.9518</td>
<td align="left" rowspan="1" colspan="1">0.9493</td>
<td align="left" rowspan="1" colspan="1">0.9130</td>
<td align="left" rowspan="1" colspan="1">0.9054</td>
<td align="left" rowspan="1" colspan="1">0.9106</td>
<td align="left" rowspan="1" colspan="1">0.9056</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0125</td>
<td align="left" rowspan="1" colspan="1">0.9524</td>
<td align="left" rowspan="1" colspan="1">0.9533</td>
<td align="left" rowspan="1" colspan="1">0.9512</td>
<td align="left" rowspan="1" colspan="1">0.9513</td>
<td align="left" rowspan="1" colspan="1">0.9113</td>
<td align="left" rowspan="1" colspan="1">0.9093</td>
<td align="left" rowspan="1" colspan="1">0.9039</td>
<td align="left" rowspan="1" colspan="1">0.9158</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0150</td>
<td align="left" rowspan="1" colspan="1">0.9484</td>
<td align="left" rowspan="1" colspan="1">0.9498</td>
<td align="left" rowspan="1" colspan="1">0.9492</td>
<td align="left" rowspan="1" colspan="1">0.9551</td>
<td align="left" rowspan="1" colspan="1">0.8985</td>
<td align="left" rowspan="1" colspan="1">0.8999</td>
<td align="left" rowspan="1" colspan="1">0.9016</td>
<td align="left" rowspan="1" colspan="1">0.9088</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0175</td>
<td align="left" rowspan="1" colspan="1">0.9486</td>
<td align="left" rowspan="1" colspan="1">0.9486</td>
<td align="left" rowspan="1" colspan="1">0.9510</td>
<td align="left" rowspan="1" colspan="1">0.9478</td>
<td align="left" rowspan="1" colspan="1">0.9070</td>
<td align="left" rowspan="1" colspan="1">0.9023</td>
<td align="left" rowspan="1" colspan="1">0.9090</td>
<td align="left" rowspan="1" colspan="1">0.9061</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0200</td>
<td align="left" rowspan="1" colspan="1">0.9518</td>
<td align="left" rowspan="1" colspan="1">0.9482</td>
<td align="left" rowspan="1" colspan="1">0.9495</td>
<td align="left" rowspan="1" colspan="1">0.9567</td>
<td align="left" rowspan="1" colspan="1">0.9019</td>
<td align="left" rowspan="1" colspan="1">0.9011</td>
<td align="left" rowspan="1" colspan="1">0.9074</td>
<td align="left" rowspan="1" colspan="1">0.9067</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0225</td>
<td align="left" rowspan="1" colspan="1">0.9494</td>
<td align="left" rowspan="1" colspan="1">0.9534</td>
<td align="left" rowspan="1" colspan="1">0.9509</td>
<td align="left" rowspan="1" colspan="1">0.9472</td>
<td align="left" rowspan="1" colspan="1">0.8969</td>
<td align="left" rowspan="1" colspan="1">0.9041</td>
<td align="left" rowspan="1" colspan="1">0.9064</td>
<td align="left" rowspan="1" colspan="1">0.9089</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0250</td>
<td align="left" rowspan="1" colspan="1">0.9492</td>
<td align="left" rowspan="1" colspan="1">0.9511</td>
<td align="left" rowspan="1" colspan="1">0.9533</td>
<td align="left" rowspan="1" colspan="1">0.9530</td>
<td align="left" rowspan="1" colspan="1">0.9056</td>
<td align="left" rowspan="1" colspan="1">0.8986</td>
<td align="left" rowspan="1" colspan="1">0.9036</td>
<td align="left" rowspan="1" colspan="1">0.9019</td>
</tr>
</tbody>
</table>
</alternatives>
<table-wrap-foot>
<fn id="nt104">
<label>d</label>
<p>These coverages and these levels of assurance are for sample sizes obtained with the computational Wald procedure (method 2) presented in
<xref ref-type="table" rid="pone-0032250-t002">Table 2</xref>
, for a CI of 95%,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e260.jpg" mimetype="image"></inline-graphic>
</inline-formula>
four desired widths (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e261.jpg" mimetype="image"></inline-graphic>
</inline-formula>
), and three values of assurance
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e262.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</p>
</fn>
</table-wrap-foot>
</table-wrap>
<table-wrap id="pone-0032250-t005" position="float">
<object-id pub-id-type="doi">10.1371/journal.pone.0032250.t005</object-id>
<label>Table 5</label>
<caption>
<title>Simulation study of coverage and assurance for method 1
<xref ref-type="table-fn" rid="nt105">e</xref>
.</title>
</caption>
<alternatives>
<graphic id="pone-0032250-t005-5" xlink:href="pone.0032250.t005"></graphic>
<table frame="hsides" rules="groups">
<colgroup span="1">
<col align="left" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
<col align="center" span="1"></col>
</colgroup>
<thead>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e263.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
<td align="left" rowspan="1" colspan="1">
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e264.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">
<italic>p</italic>
</td>
<td align="left" rowspan="1" colspan="1">0.007</td>
<td align="left" rowspan="1" colspan="1">0.008</td>
<td align="left" rowspan="1" colspan="1">0.009</td>
<td align="left" rowspan="1" colspan="1">0.010</td>
<td align="left" rowspan="1" colspan="1">0.007</td>
<td align="left" rowspan="1" colspan="1">0.008</td>
<td align="left" rowspan="1" colspan="1">0.009</td>
<td align="left" rowspan="1" colspan="1">0.010</td>
</tr>
</thead>
<tbody>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td colspan="4" align="center" rowspan="1">-------Coverage
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e265.jpg" mimetype="image"></inline-graphic>
</inline-formula>
--------</td>
<td colspan="4" align="center" rowspan="1">----------Assurance
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e266.jpg" mimetype="image"></inline-graphic>
</inline-formula>
-----------</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0050</td>
<td align="left" rowspan="1" colspan="1">0.9537</td>
<td align="left" rowspan="1" colspan="1">0.9564</td>
<td align="left" rowspan="1" colspan="1">0.9532</td>
<td align="left" rowspan="1" colspan="1">0.9566</td>
<td align="left" rowspan="1" colspan="1">0.5383</td>
<td align="left" rowspan="1" colspan="1">0.5426</td>
<td align="left" rowspan="1" colspan="1">0.5696</td>
<td align="left" rowspan="1" colspan="1">0.5513</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0075</td>
<td align="left" rowspan="1" colspan="1">0.9555</td>
<td align="left" rowspan="1" colspan="1">0.9535</td>
<td align="left" rowspan="1" colspan="1">0.9543</td>
<td align="left" rowspan="1" colspan="1">0.9593</td>
<td align="left" rowspan="1" colspan="1">0.5404</td>
<td align="left" rowspan="1" colspan="1">0.5303</td>
<td align="left" rowspan="1" colspan="1">0.5564</td>
<td align="left" rowspan="1" colspan="1">0.5375</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0100</td>
<td align="left" rowspan="1" colspan="1">0.9499</td>
<td align="left" rowspan="1" colspan="1">0.9547</td>
<td align="left" rowspan="1" colspan="1">0.9527</td>
<td align="left" rowspan="1" colspan="1">0.9537</td>
<td align="left" rowspan="1" colspan="1">0.5673</td>
<td align="left" rowspan="1" colspan="1">0.5402</td>
<td align="left" rowspan="1" colspan="1">0.5721</td>
<td align="left" rowspan="1" colspan="1">0.5426</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0125</td>
<td align="left" rowspan="1" colspan="1">0.9540</td>
<td align="left" rowspan="1" colspan="1">0.9517</td>
<td align="left" rowspan="1" colspan="1">0.9513</td>
<td align="left" rowspan="1" colspan="1">0.9527</td>
<td align="left" rowspan="1" colspan="1">0.5607</td>
<td align="left" rowspan="1" colspan="1">0.5776</td>
<td align="left" rowspan="1" colspan="1">0.5795</td>
<td align="left" rowspan="1" colspan="1">0.5863</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0150</td>
<td align="left" rowspan="1" colspan="1">0.9556</td>
<td align="left" rowspan="1" colspan="1">0.9493</td>
<td align="left" rowspan="1" colspan="1">0.9550</td>
<td align="left" rowspan="1" colspan="1">0.9500</td>
<td align="left" rowspan="1" colspan="1">0.5650</td>
<td align="left" rowspan="1" colspan="1">0.5945</td>
<td align="left" rowspan="1" colspan="1">0.5486</td>
<td align="left" rowspan="1" colspan="1">0.5651</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0175</td>
<td align="left" rowspan="1" colspan="1">0.9529</td>
<td align="left" rowspan="1" colspan="1">0.9525</td>
<td align="left" rowspan="1" colspan="1">0.9554</td>
<td align="left" rowspan="1" colspan="1">0.9509</td>
<td align="left" rowspan="1" colspan="1">0.5660</td>
<td align="left" rowspan="1" colspan="1">0.5968</td>
<td align="left" rowspan="1" colspan="1">0.5521</td>
<td align="left" rowspan="1" colspan="1">0.5635</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0200</td>
<td align="left" rowspan="1" colspan="1">0.9517</td>
<td align="left" rowspan="1" colspan="1">0.9506</td>
<td align="left" rowspan="1" colspan="1">0.9552</td>
<td align="left" rowspan="1" colspan="1">0.9505</td>
<td align="left" rowspan="1" colspan="1">0.5953</td>
<td align="left" rowspan="1" colspan="1">0.5836</td>
<td align="left" rowspan="1" colspan="1">0.5930</td>
<td align="left" rowspan="1" colspan="1">0.5740</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0225</td>
<td align="left" rowspan="1" colspan="1">0.9527</td>
<td align="left" rowspan="1" colspan="1">0.9488</td>
<td align="left" rowspan="1" colspan="1">0.9516</td>
<td align="left" rowspan="1" colspan="1">0.9545</td>
<td align="left" rowspan="1" colspan="1">0.5940</td>
<td align="left" rowspan="1" colspan="1">0.6096</td>
<td align="left" rowspan="1" colspan="1">0.5919</td>
<td align="left" rowspan="1" colspan="1">0.5859</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0250</td>
<td align="left" rowspan="1" colspan="1">0.9507</td>
<td align="left" rowspan="1" colspan="1">0.9491</td>
<td align="left" rowspan="1" colspan="1">0.9487</td>
<td align="left" rowspan="1" colspan="1">0.9523</td>
<td align="left" rowspan="1" colspan="1">0.6014</td>
<td align="left" rowspan="1" colspan="1">0.6093</td>
<td align="left" rowspan="1" colspan="1">0.6103</td>
<td align="left" rowspan="1" colspan="1">0.5903</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td colspan="4" align="center" rowspan="1">------Coverage
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e267.jpg" mimetype="image"></inline-graphic>
</inline-formula>
--------</td>
<td colspan="4" align="center" rowspan="1">---------Assurance
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e268.jpg" mimetype="image"></inline-graphic>
</inline-formula>
-----------</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0050</td>
<td align="left" rowspan="1" colspan="1">0.9549</td>
<td align="left" rowspan="1" colspan="1">0.9518</td>
<td align="left" rowspan="1" colspan="1">0.9526</td>
<td align="left" rowspan="1" colspan="1">0.9551</td>
<td align="left" rowspan="1" colspan="1">0.8299</td>
<td align="left" rowspan="1" colspan="1">0.8509</td>
<td align="left" rowspan="1" colspan="1">0.8453</td>
<td align="left" rowspan="1" colspan="1">0.8478</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0075</td>
<td align="left" rowspan="1" colspan="1">0.9538</td>
<td align="left" rowspan="1" colspan="1">0.9549</td>
<td align="left" rowspan="1" colspan="1">0.9529</td>
<td align="left" rowspan="1" colspan="1">0.9538</td>
<td align="left" rowspan="1" colspan="1">0.8182</td>
<td align="left" rowspan="1" colspan="1">0.8563</td>
<td align="left" rowspan="1" colspan="1">0.8384</td>
<td align="left" rowspan="1" colspan="1">0.8296</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0100</td>
<td align="left" rowspan="1" colspan="1">0.9511</td>
<td align="left" rowspan="1" colspan="1">0.9502</td>
<td align="left" rowspan="1" colspan="1">0.9505</td>
<td align="left" rowspan="1" colspan="1">0.9551</td>
<td align="left" rowspan="1" colspan="1">0.8403</td>
<td align="left" rowspan="1" colspan="1">0.8336</td>
<td align="left" rowspan="1" colspan="1">0.8388</td>
<td align="left" rowspan="1" colspan="1">0.8369</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0125</td>
<td align="left" rowspan="1" colspan="1">0.9511</td>
<td align="left" rowspan="1" colspan="1">0.9526</td>
<td align="left" rowspan="1" colspan="1">0.9547</td>
<td align="left" rowspan="1" colspan="1">0.9541</td>
<td align="left" rowspan="1" colspan="1">0.8422</td>
<td align="left" rowspan="1" colspan="1">0.8602</td>
<td align="left" rowspan="1" colspan="1">0.8517</td>
<td align="left" rowspan="1" colspan="1">0.8324</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0150</td>
<td align="left" rowspan="1" colspan="1">0.9523</td>
<td align="left" rowspan="1" colspan="1">0.9517</td>
<td align="left" rowspan="1" colspan="1">0.9537</td>
<td align="left" rowspan="1" colspan="1">0.9521</td>
<td align="left" rowspan="1" colspan="1">0.8493</td>
<td align="left" rowspan="1" colspan="1">0.8456</td>
<td align="left" rowspan="1" colspan="1">0.8631</td>
<td align="left" rowspan="1" colspan="1">0.8429</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0175</td>
<td align="left" rowspan="1" colspan="1">0.9489</td>
<td align="left" rowspan="1" colspan="1">0.9537</td>
<td align="left" rowspan="1" colspan="1">0.9471</td>
<td align="left" rowspan="1" colspan="1">0.9517</td>
<td align="left" rowspan="1" colspan="1">0.8444</td>
<td align="left" rowspan="1" colspan="1">0.8534</td>
<td align="left" rowspan="1" colspan="1">0.8478</td>
<td align="left" rowspan="1" colspan="1">0.8544</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0200</td>
<td align="left" rowspan="1" colspan="1">0.9513</td>
<td align="left" rowspan="1" colspan="1">0.9537</td>
<td align="left" rowspan="1" colspan="1">0.9537</td>
<td align="left" rowspan="1" colspan="1">0.9510</td>
<td align="left" rowspan="1" colspan="1">0.8567</td>
<td align="left" rowspan="1" colspan="1">0.8593</td>
<td align="left" rowspan="1" colspan="1">0.8587</td>
<td align="left" rowspan="1" colspan="1">0.8530</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0225</td>
<td align="left" rowspan="1" colspan="1">0.9494</td>
<td align="left" rowspan="1" colspan="1">0.9512</td>
<td align="left" rowspan="1" colspan="1">0.9512</td>
<td align="left" rowspan="1" colspan="1">0.9531</td>
<td align="left" rowspan="1" colspan="1">0.8525</td>
<td align="left" rowspan="1" colspan="1">0.8427</td>
<td align="left" rowspan="1" colspan="1">0.8692</td>
<td align="left" rowspan="1" colspan="1">0.8606</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1"></td>
<td colspan="4" align="center" rowspan="1">-------Coverage
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e269.jpg" mimetype="image"></inline-graphic>
</inline-formula>
--------</td>
<td colspan="4" align="center" rowspan="1">---------Assurance
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e270.jpg" mimetype="image"></inline-graphic>
</inline-formula>
-----------</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0050</td>
<td align="left" rowspan="1" colspan="1">0.9529</td>
<td align="left" rowspan="1" colspan="1">0.9543</td>
<td align="left" rowspan="1" colspan="1">0.9522</td>
<td align="left" rowspan="1" colspan="1">0.9509</td>
<td align="left" rowspan="1" colspan="1">0.9234</td>
<td align="left" rowspan="1" colspan="1">0.9112</td>
<td align="left" rowspan="1" colspan="1">0.9235</td>
<td align="left" rowspan="1" colspan="1">0.9280</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0075</td>
<td align="left" rowspan="1" colspan="1">0.9521</td>
<td align="left" rowspan="1" colspan="1">0.9536</td>
<td align="left" rowspan="1" colspan="1">0.9507</td>
<td align="left" rowspan="1" colspan="1">0.9534</td>
<td align="left" rowspan="1" colspan="1">0.9235</td>
<td align="left" rowspan="1" colspan="1">0.9140</td>
<td align="left" rowspan="1" colspan="1">0.9237</td>
<td align="left" rowspan="1" colspan="1">0.9086</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0100</td>
<td align="left" rowspan="1" colspan="1">0.9500</td>
<td align="left" rowspan="1" colspan="1">0.9516</td>
<td align="left" rowspan="1" colspan="1">0.9527</td>
<td align="left" rowspan="1" colspan="1">0.9522</td>
<td align="left" rowspan="1" colspan="1">0.9217</td>
<td align="left" rowspan="1" colspan="1">0.9107</td>
<td align="left" rowspan="1" colspan="1">0.9188</td>
<td align="left" rowspan="1" colspan="1">0.9350</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0125</td>
<td align="left" rowspan="1" colspan="1">0.9492</td>
<td align="left" rowspan="1" colspan="1">0.9501</td>
<td align="left" rowspan="1" colspan="1">0.9547</td>
<td align="left" rowspan="1" colspan="1">0.9529</td>
<td align="left" rowspan="1" colspan="1">0.9165</td>
<td align="left" rowspan="1" colspan="1">0.9263</td>
<td align="left" rowspan="1" colspan="1">0.9269</td>
<td align="left" rowspan="1" colspan="1">0.9185</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0150</td>
<td align="left" rowspan="1" colspan="1">0.9493</td>
<td align="left" rowspan="1" colspan="1">0.9533</td>
<td align="left" rowspan="1" colspan="1">0.9535</td>
<td align="left" rowspan="1" colspan="1">0.9494</td>
<td align="left" rowspan="1" colspan="1">0.9232</td>
<td align="left" rowspan="1" colspan="1">0.9284</td>
<td align="left" rowspan="1" colspan="1">0.9323</td>
<td align="left" rowspan="1" colspan="1">0.9385</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0175</td>
<td align="left" rowspan="1" colspan="1">0.9492</td>
<td align="left" rowspan="1" colspan="1">0.9531</td>
<td align="left" rowspan="1" colspan="1">0.9505</td>
<td align="left" rowspan="1" colspan="1">0.9518</td>
<td align="left" rowspan="1" colspan="1">0.9249</td>
<td align="left" rowspan="1" colspan="1">0.9355</td>
<td align="left" rowspan="1" colspan="1">0.9321</td>
<td align="left" rowspan="1" colspan="1">0.9307</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0200</td>
<td align="left" rowspan="1" colspan="1">0.9477</td>
<td align="left" rowspan="1" colspan="1">0.9512</td>
<td align="left" rowspan="1" colspan="1">0.9486</td>
<td align="left" rowspan="1" colspan="1">0.9520</td>
<td align="left" rowspan="1" colspan="1">0.9238</td>
<td align="left" rowspan="1" colspan="1">0.9402</td>
<td align="left" rowspan="1" colspan="1">0.9299</td>
<td align="left" rowspan="1" colspan="1">0.9336</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0225</td>
<td align="left" rowspan="1" colspan="1">0.9530</td>
<td align="left" rowspan="1" colspan="1">0.9471</td>
<td align="left" rowspan="1" colspan="1">0.9478</td>
<td align="left" rowspan="1" colspan="1">0.9539</td>
<td align="left" rowspan="1" colspan="1">0.9346</td>
<td align="left" rowspan="1" colspan="1">0.9380</td>
<td align="left" rowspan="1" colspan="1">0.9340</td>
<td align="left" rowspan="1" colspan="1">0.9347</td>
</tr>
<tr>
<td align="left" rowspan="1" colspan="1">0.0250</td>
<td align="left" rowspan="1" colspan="1">0.9511</td>
<td align="left" rowspan="1" colspan="1">0.9492</td>
<td align="left" rowspan="1" colspan="1">0.9504</td>
<td align="left" rowspan="1" colspan="1">0.9516</td>
<td align="left" rowspan="1" colspan="1">0.9381</td>
<td align="left" rowspan="1" colspan="1">0.9371</td>
<td align="left" rowspan="1" colspan="1">0.9416</td>
<td align="left" rowspan="1" colspan="1">0.9316</td>
</tr>
</tbody>
</table>
</alternatives>
<table-wrap-foot>
<fn id="nt105">
<label>e</label>
<p>These coverages and levels of assurance are for sample sizes obtained with the exact Clopper-Pearson (method 1) presented in
<xref ref-type="table" rid="pone-0032250-t002">Table 2</xref>
, for a CI of 95%,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e271.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, four desired widths (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e272.jpg" mimetype="image"></inline-graphic>
</inline-formula>
), and three values of assurance
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e273.jpg" mimetype="image"></inline-graphic>
</inline-formula>
</p>
</fn>
</table-wrap-foot>
</table-wrap>
<sec id="s3a">
<title>Comparing the proposed analytic formula with two exact computational procedures using group size
<italic>k</italic>
 = 40</title>
<p>Although the Clopper-Pearson CI is conservative, it is regarded as the gold standard reference method. First the sample size of methods 2 (computational Wald procedure) and 3 (analytic formula Eq. 9) are compared with the sample size resulting from using the exact Clopper-Pearson CI (method 1). For example, when
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e274.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and 0.8, the analytic method (method 3; Eq. 9) underestimates the sample size from 1 to 10 pools (
<xref ref-type="table" rid="pone-0032250-t002">Table 2</xref>
), while the computational Wald procedure (method 2) underestimates the sample size from 1 to 9 pools with regard to the Clopper-Pearson (method 1) sample size. When
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e275.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, the underestimation is from 3 to 13 pools using the analytic method (method 3; Eq. 9) and from 1 to 10 pools using the computational Wald procedure (method 2). It is important to point out that the level of underestimation increases for bigger values of the proportion (
<italic>p</italic>
); when the proportion is less than 0.01, the underestimation can be considered negligible because it is less than 5 pools and decreases for smaller values of
<italic>p</italic>
.</p>
<p>On the other hand, comparing the analytic method (method 3; Eq. 9) with the computational Wald procedure (method 2), the analytic method (method 3; Eq. 9) produces at most 5 pools less than the exact Wald procedure (
<xref ref-type="table" rid="pone-0032250-t002">Table 2</xref>
), which shows that the difference between these two methods is not important. For the analytic method (method 3; Eq. 9), the level of underestimation can be considered irrelevant when
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e276.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and of little relevance when
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e277.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, given that the Clopper-Pearson method (method 1) produces a considerable overestimation due to the use of a conservative CI procedure.</p>
<p>Suppose a researcher is interested in estimating
<italic>p</italic>
for AP maize in the region of Oaxaca, Mexico, where AP maize was reported to be found. With this information and after doing a literature review, it is considered that
<italic>p</italic>
 = 0.01, with a CI of 95%, and
<italic>k = 40</italic>
, and it is assumed that the final CIW is
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e278.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. The application of the proposed methods leads to the required number of preliminary pools of
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e279.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, each of size
<italic>k</italic>
 = 40, using the analytic (method 3; Eq. 9), Clopper-Pearson (method 1; Eq.7), and computational Wald methods (method 2; Eq.7), respectively. These sample sizes are contained in the first sub-table of
<xref ref-type="table" rid="pone-0032250-t002">Table 2</xref>
(
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e280.jpg" mimetype="image"></inline-graphic>
</inline-formula>
with
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e281.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, where
<italic>k</italic>
 = 40,
<italic>p</italic>
 = 0.01, and
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e282.jpg" mimetype="image"></inline-graphic>
</inline-formula>
).</p>
<p>Realizing that
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e283.jpg" mimetype="image"></inline-graphic>
</inline-formula>
will lead to a sufficiently narrow CI only about 50% of the time, the researcher incorporates an assurance of
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e284.jpg" mimetype="image"></inline-graphic>
</inline-formula>
 = 0.90, which implies that the width of the 95% CI will be larger than the required width (i.e., 0.008) no more than 10% of the time. From the third sub-table of
<xref ref-type="table" rid="pone-0032250-t002">Table 2</xref>
(
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e285.jpg" mimetype="image"></inline-graphic>
</inline-formula>
with
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e286.jpg" mimetype="image"></inline-graphic>
</inline-formula>
), it can be seen that the modified sample size procedure yields the necessary number of pools
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e287.jpg" mimetype="image"></inline-graphic>
</inline-formula>
for the analytic method (method 3), Clopper-Pearson method (method 1), and computational Wald procedure (method 2), respectively. Using these sample sizes (36, 39, and 38) will provide 90% assurance that the CI obtained for
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e288.jpg" mimetype="image"></inline-graphic>
</inline-formula>
will be no wider than 0.008 units. This sample size is contained in the third sub-table of
<xref ref-type="table" rid="pone-0032250-t002">Table 2</xref>
(
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e289.jpg" mimetype="image"></inline-graphic>
</inline-formula>
with
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e290.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, where
<italic>k</italic>
 = 40,
<italic>p</italic>
 = 0.01, and
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e291.jpg" mimetype="image"></inline-graphic>
</inline-formula>
).</p>
</sec>
<sec id="s3b">
<title>An example using the proposed formula (method 3)</title>
<p>In this subsection, we will illustrate the use of the developed formula (Eq. 9) called method 3. Assume that a researcher is interested in estimating
<italic>p</italic>
and she/he hypothesizes that
<italic>p</italic>
 = 0.02, and wants a CI of 95%, pool size
<italic>k = 40</italic>
, and a desired error equal to
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e292.jpg" mimetype="image"></inline-graphic>
</inline-formula>
with an assurance level of 99% (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e293.jpg" mimetype="image"></inline-graphic>
</inline-formula>
). First, it is necessary to calculate
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e294.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e295.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e296.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e297.jpg" mimetype="image"></inline-graphic>
</inline-formula>
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e298.jpg" mimetype="image"></inline-graphic>
</inline-formula>
because the CI is 95%,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e299.jpg" mimetype="image"></inline-graphic>
</inline-formula>
because it is assumed that the assurance level is
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e300.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e301.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e302.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. Therefore,
<disp-formula>
<graphic xlink:href="pone.0032250.e303"></graphic>
</disp-formula>
With Eq. (9), the optimum number of positive pools is calculated with a 99% probability that the CI width will be smaller than 0.008, the desired error. Note that for calculating
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e304.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, the double precision format was used; otherwise, a slight overestimation would have occurred. It should be pointed out that if
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e305.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, the value of
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e306.jpg" mimetype="image"></inline-graphic>
</inline-formula>
and the required number of pools reduces to Eq. (5), that is, 99 pools.</p>
<p>
<xref ref-type="supplementary-material" rid="pone.0032250.s002">Appendix S2</xref>
provides information for implementing the proposed methods and for obtaining sufficiently narrow CIs for any combination of
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e307.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e308.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e309.jpg" mimetype="image"></inline-graphic>
</inline-formula>
,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e310.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, and
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e311.jpg" mimetype="image"></inline-graphic>
</inline-formula>
using the R package
<xref ref-type="bibr" rid="pone.0032250-R1">[28]</xref>
. The R package computes the sample size using the proposed formula, Eq. (9), and the two proposed computational sample size methods.</p>
</sec>
<sec id="s3c">
<title>Coverage and assurance levels–simulation study</title>
<p>In this subsection we will examine whether the three sample size procedures [analytic (method 3), computational Wald (method 2) and exact Clopper-Pearson (method 1)] achieve: (1) the coverage probabilities of the nominal (1-α)100% CI used to calculate the CIs, and (2) the nominal levels of assurance, because this sample size formula (Eq. 9) and the two computational methods were derived under the AIPE approach.</p>
<p>For each sample size (number of positive pools, (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e312.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) from each combination of
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e313.jpg" mimetype="image"></inline-graphic>
</inline-formula>
reported in
<xref ref-type="table" rid="pone-0032250-t002">Table 2</xref>
and obtained from Equations (7) or (9), we took 40,000 random samples of size
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e314.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, where
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e315.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, to examine the coverage and assurance levels for each sample size (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e316.jpg" mimetype="image"></inline-graphic>
</inline-formula>
). First we obtained the corresponding CI from the 40,000 random samples, and then we counted the proportion of CI that contains the true value of
<italic>p</italic>
, and the proportion of CI that has a CI width narrower than the desired CI width (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e317.jpg" mimetype="image"></inline-graphic>
</inline-formula>
). In
<xref ref-type="table" rid="pone-0032250-t003">Table 3</xref>
, we can see that the coverage of the confidence intervals corresponding to the sample sizes for the analytic method (method 3) obtained from
<xref ref-type="table" rid="pone-0032250-t002">Table 2</xref>
is very similar to the nominal level (95%) and in most cases is slightly greater than 95%. These results are not in agreement with other studies that showed that the coverage of small sample sizes using the Wald CI is poor. The Wald CI performed very well here perhaps due to the relatively large sample sizes and also because the parameter
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e318.jpg" mimetype="image"></inline-graphic>
</inline-formula>
in the cases studied here is around 0.5, which causes less skewing in the distribution of
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e319.jpg" mimetype="image"></inline-graphic>
</inline-formula>
; consequently, the normal approximation is better. Also, the coverage of the sample sizes in
<xref ref-type="table" rid="pone-0032250-t004">Table 4</xref>
[for the computational Wald (method 2)] and in
<xref ref-type="table" rid="pone-0032250-t005">Table 5</xref>
[exact Clopper-Pearson (method 1)] is in most cases slightly greater than the nominal level (95%).</p>
<p>Concerning the level of assurance, we can see in
<xref ref-type="table" rid="pone-0032250-t003">Table 3</xref>
[for the analytic procedure (method 3)] that for the three levels studied (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e320.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) the obtained assurances are smaller than the specified nominal values. The results for
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e321.jpg" mimetype="image"></inline-graphic>
</inline-formula>
are consistent with the results in
<xref ref-type="table" rid="pone-0032250-t001">Table 1</xref>
, which indicates that sample sizes with no assurance (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e322.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) guarantee a desired CI width around 50% of the time and, in most cases, less than 50%. Also, when the assurance is 80% or 90%, the achieved levels of assurance are smaller than the nominal levels. For the computational Wald procedure (
<xref ref-type="table" rid="pone-0032250-t004">Table 4</xref>
), we can see that the assurance levels in most cases are slightly greater than the specified nominal level (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e323.jpg" mimetype="image"></inline-graphic>
</inline-formula>
). Finally, for the exact Clopper-Pearson procedure (
<xref ref-type="table" rid="pone-0032250-t005">Table 5</xref>
), the levels of assurance reached are larger than the nominal values in all cases, and we can say that there is an evident overestimation of the specified nominal values (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e324.jpg" mimetype="image"></inline-graphic>
</inline-formula>
).</p>
</sec>
</sec>
<sec id="s4">
<title>Discussion</title>
<p>This paper presented three methods for determining the optimal sample size for estimating the proportion of transgenic plants in a population, assuming perfect sensitivity and specificity, which must be taken into account when designing a study. The proposed methods guarantee that the desired CI width (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e325.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) will be achieved with a probability
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e326.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, because they take into account the stochastic nature of the confidence interval width. Of the three methods presented, two are computational and one is analytic. According to the Monte Carlo study, the computational Wald procedure (method 2) is the best option because its corresponding coverage and assurance levels are very close to the nominal specified values. On the other hand, the exact Clopper-Pearson procedure (method 1) is conservative (overestimates the required sample size) because the coverage (in most cases) and assurance levels (in all cases) are larger than the nominal values; the analytic procedure (method 3) slightly underestimates the required sample sizes because in most cases the observed levels of assurance are smaller than the nominal values, even though in most cases the coverage reached is slightly greater than the nominal level (95%).</p>
<p>The main advantage of the analytic procedure (method 3) is that a simple formula (Eq. 9) was derived which, within a certain range of
<italic>k</italic>
,
<italic>p</italic>
, and
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e327.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, is very precise and produces similar results to the two computational methods proposed. However, the proposed formula underestimated the optimum number of positive pools, mainly for
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e328.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, for
<italic>k</italic>
>75 at
<italic>p</italic>
>0.01. However, if the number of pools given by the formula (Eq. 9) of the analytic method increases to 6, the resulting sample size will be very close to the computational Wald CI, which produces, on average, 5 pools more than the analytic procedure (method 3).</p>
<p>The three proposed methods are good approximations for determining the optimal sample size under negative binomial group testing, because they were derived using two types of confidence intervals (Wald and Clopper-Pearson). Although the Clopper-Pearson CI is considered the gold standard, its corresponding sample size (method 1) is conservative (overestimates the sample size) and it is not possible to compute it analytically. For this reason, we recommend using the sample size resulting from the computational Wald procedure (method 2). A disadvantage of method 2 is that it does not have an analytic solution.</p>
<p>These methods using group testing are an excellent option under the assumption that AP concentration is low,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e329.jpg" mimetype="image"></inline-graphic>
</inline-formula>
. Pool size can be an important consideration, since from an economic perspective, it is always better to have a large pool size and a smaller number of pools than vice versa. However, pool size should be chosen carefully to avoid a high rate of false negatives. On the other hand, an important point to take into account when using the negative binomial group testing sampling method is that the sample size (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e330.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) given by Equations (7) and (9) represents the number of positive pools required to stop the sampling and testing process. The sampling and testing process is performed pool by pool using simple random sampling until we find the required number of positive pools (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e331.jpg" mimetype="image"></inline-graphic>
</inline-formula>
). That is, sampling and testing will stop when the number of positive pools,
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e332.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, is reached and we need to record the observed data
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e333.jpg" mimetype="image"></inline-graphic>
</inline-formula>
, to get the overall number of pools tested
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e334.jpg" mimetype="image"></inline-graphic>
</inline-formula>
.</p>
<p>Note that the sample size formula developed by Montesinos-López et al.
<xref ref-type="bibr" rid="pone.0032250-MontesinosLpez2">[21]</xref>
under binomial group testing looks similar to those developed in this study; however, here we derived the three procedures under inverse negative binomial group testing sampling, that is, using negative binomial distribution. In the method of Montesinos-López et al.
<xref ref-type="bibr" rid="pone.0032250-MontesinosLpez2">[21]</xref>
, the required sample size is a fixed quantity (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e335.jpg" mimetype="image"></inline-graphic>
</inline-formula>
: number of pools to study, which represents the number of laboratory tests to be performed); under negative binomial group testing, the number of positive pools (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e336.jpg" mimetype="image"></inline-graphic>
</inline-formula>
) is the quantity that is fixed in advance, whereas the overall number of pools tested is a random variable, because the sampling and testing process stops when the
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e337.jpg" mimetype="image"></inline-graphic>
</inline-formula>
positive pool is found. The methods proposed here give the value of the required number of positive pools (
<inline-formula>
<inline-graphic xlink:href="pone.0032250.e338.jpg" mimetype="image"></inline-graphic>
</inline-formula>
).</p>
<p>The R program (see
<xref ref-type="supplementary-material" rid="pone.0032250.s002">Appendix S2</xref>
) developed using the R package
<xref ref-type="bibr" rid="pone.0032250-R1">[28]</xref>
allows the user to quickly and simply plan the sample size according to her/his requirements or needs using the three proposed methods [the analytic (method 3), exact Clopper-Pearson (method 1) and computational Wald methods (method 2)]. However, if the researcher does not have access to the R program, the best practical solution is the analytic procedure using Eq. (9).</p>
</sec>
<sec sec-type="supplementary-material" id="s5">
<title>Supporting Information</title>
<supplementary-material content-type="local-data" id="pone.0032250.s001">
<label>Appendix S1</label>
<caption>
<p>(DOC)</p>
</caption>
<media xlink:href="pone.0032250.s001.doc" mimetype="application" mime-subtype="msword">
<caption>
<p>Click here for additional data file.</p>
</caption>
</media>
</supplementary-material>
<supplementary-material content-type="local-data" id="pone.0032250.s002">
<label>Appendix S2</label>
<caption>
<p>(DOC)</p>
</caption>
<media xlink:href="pone.0032250.s002.doc" mimetype="application" mime-subtype="msword">
<caption>
<p>Click here for additional data file.</p>
</caption>
</media>
</supplementary-material>
</sec>
</body>
<back>
<fn-group>
<fn fn-type="conflict">
<p>
<bold>Competing Interests: </bold>
The authors have declared that no competing interests exist.</p>
</fn>
<fn fn-type="financial-disclosure">
<p>
<bold>Funding: </bold>
These authors have no support or funding to report.</p>
</fn>
</fn-group>
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