International Journal Of Scientific And Statistical Computing (IJSSC) Volume (1) Issue (1)

---

INTERNATIONAL JOURNAL OF
SCIENTIFIC AND STATISTICAL
COMPUTING (IJSSC)





Volume 1, Issue 1, 2010










Edited By
Computer Science Journals
www.cscjournals.org

---

INTERNATIONAL JOURNAL OF SCIENTIFIC AND
STATISTICAL COMPUTING (IJSSC)
Book: 2010 Volume 1, Issue 1
Publishing Date: 20-12-2010
Proceedings
ISSN (Online): 2180 -1339

This work is subjected to copyright. All rights are reserved whether the whole or
part of the material is concerned, specifically the rights of translation, reprinting,
re-use of illusions, recitation, broadcasting, reproduction on microfilms or in any
other way, and storage in data banks. Duplication of this publication of parts
thereof is permitted only under the provision of the copyright law 1965, in its
current version, and permission of use must always be obtained from CSC
Publishers. Violations are liable to prosecution under the copyright law.

IJSSC Journal is a part of CSC Publishers
http://www.cscjournals.org

© IJSSC Journal
Published in Malaysia

Typesetting: Camera-ready by author, data conversation by CSC Publishing
Services – CSC Journals, Malaysia


CSC Publishers

---

EDITORIAL PREFACE

The International Journal of Scientific and Statistical Computing
(IJSSC) is an effective medium for interchange of high quality
theoretical and applied research in Scientific and Statistical Computing
from theoretical research to application development. This is the first
issue of volume first of IJSSC. International Journal of Scientific and
Statistical Computing (IJSSC) aims to publish research articles on
numerical methods and techniques for scientific and statistical
computation. IJSSC publish original and high-quality articles that
recognize statistical modeling as the general framework for the
application of statistical ideas.

Submissions reflect the important developments, extensions, and
applications
in
statistical
modeling.
IJSSC
goal
is
to
be
multidisciplinary in nature, promoting the cross-fertilization of ideas
between scientific computation and statistical computation. IJSSC is
refereed journal and invites researchers, practitioners to submit their
research work that reflect new methodology on new computational and
statistical modeling ideas, practical applications on interesting
problems which are addressed using an existing or a novel adaptation
of an computational and statistical modeling techniques and tutorials &
reviews with papers on recent and cutting edge topics in
computational and statistical concepts.

IJSSC editors understand that how much it is important for authors
and researchers to have their work published with a minimum delay
after submission of their papers. They also strongly believe that the
direct communication between the editors and authors are important
for the welfare, quality and wellbeing of the Journal and its readers.
Therefore, all activities from paper submission to paper publication are
controlled through electronic systems that include electronic
submission, editorial panel and review system that ensures rapid
decision with least delays in the publication processes.

To build international reputation of IJSSC, we are disseminating the
publication information through Google Books, Google Scholar,
Directory of Open Access Journals (DOAJ), Open J Gate,
ScientificCommons, Docstoc, Scribd, CiteSeerX and many more. Our
International Editors are working on establishing ISI listing and a good
impact factor for IJSSC. I would like to remind you that the success of
the journal depends directly on the number of quality articles
submitted for review. Accordingly, I would like to request your
participation by submitting quality manuscripts for review and
encouraging your colleagues to submit quality manuscripts for review.

---

One of the great benefits that IJSSC editors provide to the
prospective authors is the mentoring nature of the review process.
IJSSC provides authors with high quality, helpful reviews that are
shaped to assist authors in improving their manuscripts.


Editorial Board Member
International Journal of Scientific and Statistical Computing (IJSSC)

---

Editorial Board Members (EBMs)

Dr. De Ting Wu
Morehouse College (United States of America)

---

Table of Content




Volume 1, Issue 1, December 2010



Pages
1-6
Group Testing with Test Errors Made Easier

Nyongesa L. Kennedy, Paul J. Syaywa
















International Journal Of Scientific And Statistical Computing (IJSSC) Volume (1) Issue (1)

---

Nyongesa L.Kennedy & Syaywa J.Paul
Group Testing With Test Errors Made Easier

Nyongesa L. Kennedy



knyongesa@hotmail.com
Department of Mathematics,








Masinde Muliro University of Science and Technology,
190 kakamega, Kenya

Paul J. Syaywa





syaywa@yahoo.com
Department of Mathematics

Masinde Muliro University of Science and Technology,
190 kakamega, Kenya
Research Partially Supported by MMUST-URF

Abstract

Group testing is a cost-effective procedure for identifying defective items in a
large population. It also improves the efficiency of the testing procedure when
imperfect tests are employed. This study develops computational group-testing
strategy based on [5] testing strategy. Statistical moments based on this applied
design have been generated. With advent of digital computers in 1980‘s, group-
testing strategy under discussion is handled in the context of computational
statistics.

Keywords: False-negative; False-positive; Group; Imperfect-tests; Pool.


1. INTRODUCTION
Sequential testing of a population in the form of grouped sample started the way back in the
second world war by [2] as a cost-effective method for screening syphilis in US soldiers returning
from abroad. The [2] idea entails putting together individuals to form a group, and then testing the
group rather than testing each individual for evidence or absence of the characteristic of interest.
Epidemiological studies that use group testing have one of the two objectives. The first objective
is to screen a large population with a view to identifying those individuals with a trait (cf. [2]). The
second objective is to estimate the rate of the trait (cf. [10] and [9]). For either objective, group
testing is more cost effective than individual testing especially when the rate of the trait is low
because if a group tests negative it implies
that none of the individuals that constitute the
group have the trait, and thus it is not necessary to test each individual in the group.

In recent years, there has been renewed interest in group testing strategies of biological
specimens because of the application in HIV/Aids epidemiology (cf. [5]). The procedure has
potential in the application of HIV/Aids testing because disease prevalence is estimated without
necessarily identifying the subject (cf. [3]). [12] studied the cost-effectiveness of pooling algorithm
International Journal of Scientific and Statistical Computing (IJSSC), Volume (1): Issue (1) 1

---

Nyongesa L.Kennedy & Syaywa J.Paul
for the first objective of identifying individuals with the trait. In their procedure, each individual
group that test positive is divided into two equal groups, which are then tested. Groups that tested
positive were further sub-divided and tested and so on. [13] extended this work by considering
pooling algorithms when there are errors and showed that some of these algorithms can reduce
the error rates of the screening procedures (the false positives and false negatives) compared to
individual testing. [7] examined group testing with re-testing and observed that re-test improves
the sensitivity and specificity of the group-testing algorithm.

Recent studies have focused on the second objective of estimating the rate of the trait using
group-testing strategy. [11] discussed the procedure as a potential method for use by
pharmaceutical companies in discovering drugs in the early stages. [8] has proposed an
estimator in pool testing strategy that benefit from re-testing the pools. He observed that re-
testing improves the efficiency of the estimator.

In this study, we discuss the computation of statistical measures in pool testing strategy with
imperfect test via computer package MATLAB based on [5] design of pool testing strategy. To the
authors knowledge no article has appeared in the literature of group-testing as championed by [2]
that has discussed the procedure in computational aspect. The rest of the paper is arranged as
follows: Group testing strategy with imperfect test or in the presence of test errors is introduced in
Section 2. Various statistical moments are generated in Section 3. Misclassification in the
proposed algorithm as a result of test errors is discussed in Section 4.Section 5 provides the
conclusion to the present study.

2. THE TESTING STRATEGY
In this study, we generalize the group testing strategy by introducing the error component in the
testing scheme so that the earlier proposed strategies become special cases as proposed by [5].
The strategy proposed in this study is as follows. Initially, group the population under investigation
into a single group of size n and carry out a test on the group. If the test result is negative, further
testing is discontinued. If the test result is positive, the group is divided into groups of equal sizes
(nk), and each group is subjected to group-testing. If the group tests positive, individual testing is
carried out. Diagrammatic description of the procedure has been presented in Figure 1.

1
2
…….………………
i
……………………
nb
1
2
i





_

+



Negative result on the test
Positive result on the test


FIGURE 1: Block Testing Strategy

International Journal of Scientific and Statistical Computing (IJSSC), Volume (1): Issue (1) 2

---

Nyongesa L.Kennedy & Syaywa J.Paul
3. MOMENTS IN THE GROUP TESTING STRATEGY
Generation of random numbers from distributions form a basis for generation of moments in this
section. Notice that, in order to use a computer to initiate a simulation study, we must be able to
generate the values of a uniform (0,1) random variables; such variates are called random
numbers, most computers have in-built subroutines, called a random number generator. For
further discussion on this subject see [6].With the above in mind, we are in a position to generate
moment measures in our proposed group testing strategy. In our testing strategy, we shall
assume that tests under use are imperfect so that when tests are assumed to be perfect would be
a special case. Before the generation of moments, we shall require the composite probability of
classifying a group as positive, denoted by π and given by

(1)
where k is the group size, p is the probably of incidence,
and
are the sensitivity and
specificity of the test in use, respectively. Equation (1) is easily derived by the law of total
probability. In our study, (1) is the probability of success. Therefore, we shall generate random
numbers from a binomial distribution with probability of success
. Now, with (1) at hand, to
convert the data set {xi} generated from U (0, 1) into zeros and ones, we use the indicator
function


But from (1), it is clear that
since
(0, 1) and
and
.
Also, notice that in situations where the test kits are perfect,
. Then (1)
reduces to

(2)

and if the group size is one, i.e., k = 1, (1) reduces to
(3)

Let X denote the number of defective groups (groups that test positive on the test), then X ~
binomial(n, π ). Hence, various statistical measures; mean, standard deviation, Kurtosis and
skewness have been computed by the aid of statistical packages. In addition, the total numbers
of tests, cost, and relative savings have been computed. We utilize Equations (1) and (3) to
generate these moments as presented in Tables 1a (i), 1a(ii), 1a (iii), though 2(b). Graphical
presentations are provided in Figures2(a) through 2(c) in the Appendix.
The simulation at population size 100 with groups of size 10 when the sensitivity and specificity of
the tests in use are 99% is provided in Table 1(a)(i). It can be observed from the simulated results
that:
International Journal of Scientific and Statistical Computing (IJSSC), Volume (1): Issue (1) 3

---