An Application of Cluster Analysis to HIV/AIDS Prevalence in Nigeria

An Application of Cluster Analysis to HIV/AIDS Prevalence in Nigeria Data clustering is a vital tool when it comes to understanding data items with similar characteristics in a data set for the sake of grouping. Clustering may be for understanding or utility. Clustering for understanding, which is the focus of this work deals with grouping items with common characteristics in order to better understand a dataset and to identify possible or pre-interest sub-groups that could be formed from such data. The HIV prevalence statistics in Nigeria is measured bi-annually across 36 states and FCT which were zoned under 6 geo-political zones happens to be a suitable data to implement this subject matter. Cluster Analysis was implemented through the general methods of Hierarchical (agglomerative nesting) and Partitioning methods (K-Means). These techniques where implemented on the platform of R (Statistical Computing Language) to cluster HIV prevalence rate in Nigeria so as to find out states that could be considered same category and to investigate the concentration of the disease in respect to geo-political zones. Relative type of validation was used for cluster validation (a mechanism for evaluating the correctness of clustering).

2012

Raheem Kabir Kola Department Of Statistics, University Of Abuja, Nigeria

2012

1

AN APPLICATION OF CLUSTER ANALYSIS TO HIV/AIDS

PREVALENCE IN NIGERIA

RAHEEM KABIR KOLA

DEPARTMENT OF STATISTICS

FACULTY OF SCIENCE, UNIVERSITY OF ABUJA,

NIGERIA

IN PARTIAL FULFILMENT FOR

THE AWARD OF BACHELOR OF SCIENCE (B.Sc.) DEGREE IN

STATISTICS

2012

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CHAPTER ONE

INTRODUCTION

1.0 Introduction

Cluster analysis divides data into groups called clusters such that items in the

same cluster are similar while items in different cluster are distinct, the essence of

such grouping may be for understanding or for utility. The first case, clustering for

understanding has to do with understanding the behaviour of data or in some

respect knowing the characteristics that data items may have in common, this is

the focal point of this work. Meanwhile, clustering for utility has to do with

classification of data for further usage or analysis.

Over the years, the world had suffered several deaths due to a virus known as HIV,

several institutes, governments and international organisations had invested a lot

in providing measures to curb the spread of this disease, but till date HIV still

posed a significant challenge among other diseases particularly in Africa. Nigeria

as a case study has second highest number of people living with HIV (CIA World

Factbook, 2012) with an estimate of 3.3 million of its populace which amounts to

about 2% of the entire population. The country has 36 states zoned under 6 geo-

political zones. The HIV prevalence rate is measured periodically across these

states by means of a sentinel survey.

This research work applied the technique of cluster analysis on the platform of R

(statistical computing) to classify the spread of this virus among the populace of

Nigeria. Classification of this information would help to take better decisions as to

what part of the country needs certain attention.

1.1 Objective

This study intends to implement Cluster Analysis using R (Statistical Computing)

and to understand the nature of the spread of Human Immunodeficiency Virus

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state-wise by classifying the prevalence rate among states and to investigate if

states in a particular geographical location have similar prevalence. These goals

are itemised as follows:

1. Categorize states according to level of prevalence

2. Determine the areas where we have high concentration or dominance

3. Implement Cluster Analysis using R (Statistical Computing) and

4. Make further recommendation on the prevalence of most affected areas.

1.2 Scope

This study is concerned with identifying the possible categories for which Nigerian

states could be categorized by the rate of HIV prevalence among states by means

of cluster analysis. It does not in any way investigate or probe into factors that

could be responsible for concentration of the disease in certain geographical areas

with high concentration or dominance.

The implemented Cluster analysis was based on specific algorithms of

Agglomerative (hierarchical method) and K-means (partitioning method), in which

case all computations were made using the Cluster Library available in R, thus

there is less attention for detail theoretical computations usually involved in

Clustering techniques. Graphs (Dendogram and banner plot) were used in

appropriate places. Advanced concepts, such as model based clustering and

numerical clustering validation is beyond the scope of this work.

The relative type of cluster validation was used in attempt to validate the result of

our clustering.

1.3 Limitation

Research works that depends on data acquired over long periodic circles involving

a large population like Nigeria are likely be challenged by one or two drawbacks.

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A major restriction to this study is the inability to access sufficient and consistent

data such as quarterly prevalence rate which is as a problem attributable to

inefficient medical system. The HIV Sentinel survey usually commence bi-annually

which forms a sequence of 2 years lag as from 1991, but in 2007, no survey was

conducted until 2008 which resulted in a 3 years lag and altered the sequence.

Also, until 1999 HIV sentinel survey did not cover the entire states in Nigeria. For

this reasons the use of average prevalence rate was employed (average

prevalence rate from 1999 to 2003 and 2005 to 2010).

Unwillingness to disclose HIV status by infected persons due to Stigmatization is

another factor of concern as this restricted the source of data acquisition to

antenatal patients. Therefore the available data may not be a true representation.

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1.4 Definition of terms

AIDS: Acquired Immune Deficiency Syndrome

Cluster: A cluster is an ordered list of objects, which have some common

characteristics.

Cluster Membership: Items in a particular cluster have same cluster membership.

Cluster Size: number of items or members in a cluster.

Console: command line (not graphical)

Class: A class in an object oriented programming is a construct that serves as a

template for creating, or instantiating, specific objects within a program (plural

classes).

Dendogram: a tree like structure used to display the result of hierarchical

clustering.

Function: A chunk of code written to achieve a particular goal in programming,

usually gives a result in return.

GUI: Graphical User Interface

HIV: Human Immunodeficiency Virus, a retrovirus that causes AIDS.

HSS: HIV Sentinel Survey

IDE: Integrated Development Environment (an interface that enhances

programming and code writing)

Object: is an instance of a class (see class above).

Seroprevalence: is the number of persons in a population who test positive for a

specific disease based on serology (blood serum) specimens

Sentinel survey: a survey system in which a designated group of reporting sources

(hospitals, agencies e.t.c.) from certain geographical location agrees to report all

cases of one or more notifiable conditions for the purpose of acquiring high

quality data.

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CHAPTER TWO

LITERATURE REVIEW

2.0 HIV prevalence in Nigeria

The HIV and AIDS pandemic have constituted the greatest health challenge in

Nigeria over time. In 2012 Nigeria was adjudged to have the second highest

burden of HIV in the world after South Africa.

Since the first reported case of HIV and AIDS in Nigeria in 1986, the epidemic has

continued to unleash a huge blow on the commonwealth with about 3.3 million

Nigerians currently infected. To respond to this epidemic, the Federal Government

of Nigeria put in place several programmes aimed at controlling and mitigating its

effect. The sole purpose of these intervention programmes is usually the

continuous monitoring of the HIV epidemic via a biennial sentinel survey among

pregnant women at finishing antenatal clinics in Nigeria.

In the African region, active HIV sero-surveillance using pregnant women

attending ante-natal clinics as the survey population is employed in line with the

World Health Organization (WHO) plus the Joint United international locations

Programme on HIV and AIDS (UNAIDS). The HIV sentinel sero-surveillance survey

has been conducted biennially in Nigeria since 1991.

The HIV prevalence in Nigeria had been on a consistent increase from 1.8% in

1991 to 5.8% in 2001 before a decline to 5% in 2003 and 4% in 2005. The report of

the 2005 survey further reaffirms that no state or community is spared this

epidemic. There are wide variations in HIV prevalence among states and between

urban and rural places across the commonwealth. Data resulting from sentinel

survey could be inconclusive to get ready direct comparisons between aggregate

figures acquired inside a several surveys due to differences in location and counts

of survey sites.

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2.1 Research works on HIV /AIDS

Over the years, there are agencies responsible for monitoring and control of the

HIV/AIDs. Example of such agencies include NACA (National Agency for the

Control of Aids), UNFPA (United Nations Fund for Population Activities), These

agencies organises periodic survey to estimate data needed to investigate the

nature of HIV prevalence and also conduct research works based on specifications

of international organisation such as WHO (World Health Organisation), UNAIDS

(Joint United Nations Programme on HIV/AIDS) and so on.

There is little or no literature on the study HIV Prevalence using Cluster Analysis,

though certain clustering techniques were employed in available works majorly

from Federal Ministry of Health and some HIV/AIDS agencies in the country.

The Federal Ministry of Health, recent research on HIV prevalence presented a

clustering by state of HIV spread in a map view as found in the technical report of

National HIV Sero-prevalence Sentinel Survey 2010 (figure 2.0 below).

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Fig 2.0: Geographical distribution of HIV Prevalence by States (HSS 2010)

Source: Federal Ministry of Health, 2010 National HIV Sero-prevalence

Sentinel Survey - Technical Report

2.2 Cluster Analysis

Cluster analysis is a study of dividing data elements into groups, such that the

elements in each group are homogenous as much as possible.

The most common use of cluster analysis is classification. That is, subjects are

separated into groups such that each subject is more similar to other subjects in

its group than to subjects outside the group. Cluster analysis is thus concerned

ultimately with classification and represents a set of techniques that are part of

the field of numerical taxonomy (Frank and Green, [1968]; Punj and Stewart

[1983]; Aldenderfer and Blashfield [1984]).

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Though, Clustering is often confused or used interchangeably with classification,

but the two can be distinguish. In classification the objects are assigned to pre

defined classes or groups, whereas in clustering the classes or groups are also to

be found by the algorithm used. Thus classification is a supervised approach while

clustering is unsupervised.

The term cluster analysis does not identify a particular statistical method or

model, as do discriminant analysis, factor analysis, and regression. We do not have

to make any assumptions about the underlying distribution of the data. Using

cluster analysis, groups of related variables can be formed, similar to the case of

factor analysis. There are numerous ways to sort cases into groups. The choice of

a method depends on the size of the data and other factors. Methods commonly

used for small data sets are impractical for data files with thousands of cases.

CLARA (clustering for large sample) was suggested by Kaufman and Rousseeuw

(1990).

There are generally two major approaches used for clustering, these are

Hierarchical method and Partitioning method.

2.2.1 Hierarchical Clustering

The hierarchical procedures are characterized by the construction of a hierarchy

or tree-like structure. In some methods every single point starts as a unit (single-

point) cluster. At the next level the two closest points are placed in a cluster. At

the next level a third point joins the first two or else a second two-point cluster is

formed, based on different criterion rules for assignment. A popular technique for

hierarchical clustering is the Agglomerative Nesting which is discussed in

subsequent section. The results of hierarchical clustering are usually presented in

a dendogram, this clustering technique relies on dissimilarity (distance) matrix.

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2.2.1.1 Dissimilarity Matrix:

Dissimilarity matrix (also called distance matrix) describes pairwise distinction

between 𝑝 objects. It is a square symmetrical 𝑝 𝑋 𝑝 matrix with the (ab)th

element equal to the value of a chosen measure of distinction between the (a)th

and the (b)th object. The diagonal elements are either not considered or are

usually equal to zero - i.e. the distinction between an object and itself is

postulated as zero.

If we denote distance by 𝒅, then 𝒅(𝒂,𝒃) is the distance between one object and

the other (the value of distance depends on the chosen metric, may be Euclidean

or Manhattan as explained in further discussion). The pairwise distance matrix is

given as:

In order to decide which clusters should be combined (for agglomerative), or

where a cluster should be split (for divisive), a measure of dissimilarity between

sets of observations is required. In most methods of hierarchical clustering, this is

achieved by use of an appropriate metric and a linkage criterion which specifies

the dissimilarity of sets as a function of the pairwise distances of observations in

the sets. Metric is a measure of distance between pairs of observations. Examples

of some commonly used metric are:

Manhattan distance:

D = d(a, b) =

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Euclidean Distance:

Squared Euclidean Distance:

Maximum Distance:

2.2.1.2 Linkage Criteria

Linkage criterion determines how clusters should be formed; it computes the

distance between clusters.

A linkage criterion specifies the dissimilarity of sets as a function of the pairwise

distances of observations in the sets. Metric is a measure of distance between

pairs of observations. Examples of some commonly used linkage are single linkage,

complete linkage and the ward’s method. Some of these methods are discussed

below.

Single Linkage Rule

The single linkage or minimum distance rule starts out by finding the two points

with the minimum distance. These are placed in the first cluster. At the next stage

a third point joins the already-formed cluster of two if the minimum distance to

any of the members of the cluster is smaller than the distance between the two

closest unclustered points. Otherwise, the two closest unclustered points are

placed in a cluster. The process continues until all points end up in one cluster.

The distance between two clusters is defined as the shortest distance from a point

in the first cluster that is closest to a point

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Complete Linkage Rule

The complete linkage option starts out in just the same way by clustering the two

points with the minimum distance. However, the criterion for joining points to

clusters or clusters to clusters involves maximum (rather than minimum) distance.

That is, a third point joins the already formed cluster of two if the maximum

distance to any of the members of the cluster is smaller than the distance

between the two closest un-clustered points. In other words, the distance

between two clusters is the longest distance from a point in the first cluster to a

point in the second cluster.

Average Linkage

The average linkage option starts out in the same way as the other two. However,

in this case the distance between two clusters is the average distance from points

in the first cluster to points in the second cluster.

2.2.1.3 Dendogram

A dendogram (from Greek dendron "tree", -gramma "drawing") is a tree diagram

frequently used to illustrate the arrangement of the clusters produced by

hierarchical clustering. Dendograms are often used in computational biology to

illustrate the clustering of genes.

The x-axis of a dendogram usually consists of clusters while the y-axis (height)

denotes the distance between one cluster and the other.

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Fig 2.0: Sample Dendogram

2.2.2 Partitioning Method

The partitioning methods generally result in a set of M clusters, each object

belonging to one cluster. Each cluster may be represented by a centroid or a

cluster representative; this is some sort of summary description of all the objects

contained in a cluster. The precise form of this description will depend on the type

of the object which is being clustered. In case where real-valued data is available,

the arithmetic mean of the attribute vectors for all objects within a cluster

provides an appropriate representative; alternative types of centroid may be

required in other cases, e.g., a cluster of documents can be represented by a list of

those keywords that occur in some minimum number of documents within a

cluster. If the number of the clusters is large, the centroids can be further

clustered to produces hierarchy within a dataset. K-Means is the most popularly

known partitioning technique. The technique of K-means is discussed below.

2.2.2.1 K Means:

K-means (MacQueen, 1967) is one of the simplest unsupervised learning

algorithms that solve the well-known clustering problem. The procedure follows a

simple and easy way to classify a given data set through a certain number of

Distance or Height

Objects

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clusters (denoted by k) fixed a priori. The main idea is to define k centroids, one

for each cluster. These centroids are then placed far away from each other as

much as possible. The next step is to take each point belonging to a given data set

and associate it to the nearest centroid. When no point is pending, the first step is

completed and an early groupage is done. At this point we need to re-calculate k

new centroids as barycenters of the clusters resulting from the previous step.

After we have these k new centroids, a new binding has to be done between the

same data set points and the nearest new centroid. A loop has been generated. As

a result of this loop we may notice that the k centroids change their location step

by step until no more changes are done. In other words centroids do not move

any more.

Finally, this algorithm aims at minimizing an objective function, in this case a

squared error function.

A. Algorithm for K-means

1. Place K points into the space represented by the objects that are being

clustered. These points represent initial group centroids.

2. Assign each object to the group that has the closest centroid.

3. When all objects have been assigned, recalculate the positions of the K

centroids.

4. Repeat Steps 2 and 3 until the centroids no longer move. This produces a

separation of the objects into groups from which the metric to be

minimized can be calculated.

B. Benefits of using K-means

1. With a large number of variables, K-Means may be computationally faster

than hierarchical clustering (if K is small).

2. K-Means may produce tighter clusters than hierarchical clustering,

especially if the clusters are globular.

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C. Drawback of using K-means

1. Difficulty in comparing quality of the clusters produced (e.g. for different

initial partitions or values of K affect outcome).

2. Fixed number of clusters can make it difficult to predict what K should be.

3. Does not work well with non-globular clusters.

4. Different initial partitions can result in different final clusters. It is helpful to

rerun the program using the same as well as different K values, to compare

the results achieved.

2.3 Cluster Validation

Since there are diverse ways of conducting cluster analysis, and each method has

combination of possible alternatives whereby choice of selection depends on

users discretion, there is need to do a check on the resulting output. Cluster

validation has to do with the evaluation of the result obtained in a particular

clustering technique.

Conducting a Cluster Analysis is not much of an easy task but cluster validation is

proven to be more difficult.

“The validation of clustering structures is the most difficult and frustrating part of

cluster analysis. Without a strong effort in this direction, cluster analysis will

remain a black art accessible only to those true believers who have experience

and great courage.” - Algorithms for Clustering Data, Jain and Dubes.

Techniques for validating the result of cluster analysis are generally categorized as

Intrinsic, Extrinsic and Relative. Intrinsic technique uses certain properties()

present in the result of the conducted analysis for evaluation while the Extrinsic

uses parameters outside the result for evaluation. The Relative Technique

evaluates a clustering structure by comparing it to other clustering schemes. The

Relative type of cluster validation was used in attempt to validating the result of

our clustering.

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CHAPTER THREE

METHODOLOGY

Cluster analysis is a study of dividing data items into groups such that the

elements of each group are homogenous as much as possible. There are basically

two major approaches used for grouping in cluster analysis, these are Hierarchical

and Partitioning method.

For the sake of this work, Agglomerative nesting and K-Means approach which are

methods of Hierarchical and Partitioning clustering would be our focus

3.0 Agglomerative Nesting (Hierarchical)

Agglomerative nesting often called AGNES uses an algorithm in a bottom-up

manner till a single rooted tree like diagram (dendogram) is formed. The algorithm

is as follows:

1. Initially, put each article in its own cluster.

2. Among all current clusters, pick the two clusters with the smallest distance.

3. Replace these two clusters with a new cluster, formed by merging the two

original ones.

4. Repeat the above two steps until there is only one remaining cluster in the

pool.

3.0.1 Distance Measure

In order to compute clustering, a measure of dissimilarity between sets of

observations is required. This is sometime refers to as proximity matrix (a matrix

of distance measure which could either be similarity measure or dissimilarity

measure as in the case of hierarchical clustering). The proximity matrix is achieved

by use of an appropriate metric such as Euclidean distance and Manhattan

distance measures.

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Euclidean distance: is an inter point distance, it takes the magnitude of the

expression data into account and therefore, preserves more information about

the data. Euclidean distance is the length of the shortest path between two points

and required standardization when used. Mathematically given as:

Manhattan Distance:

The Manhattan distance also known as the City-Block distance is the sum of

distances along each dimension. This distance measure corresponds to the

distance of travel between two points. By formula:

The cluster analysis computation of this work uses the Manhattan distance.

3.0.2 Ward’s Linkage Criterion

Linkage criterion determines how clusters should be formed; it computes the

distance between clusters.

Ward's method is a criterion applied in hierarchical cluster analysis. Ward's

minimum variance method is a special case of the objective function approach

originally presented by Joe H. Ward, Jr. (1963). Ward suggested a general

agglomerative hierarchical clustering procedure, where the criterion for choosing

the pair of clusters to merge at each step is based on the optimal value of an

objective function

Ward's minimum variance criterion minimizes the total within-cluster variance. At

each step the pair of clusters with minimum cluster distance are merged

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This method is distinct from other methods because it uses an analysis of variance

approach to evaluate the distances between clusters. In general, this method is

considered very efficient.

3.0.3 Agglomerative Coefficient

Agnes computes a coefficient, called Agglomerative Coefficient (AC), which

measures the clustering structure of the data set.

Agglomerative Coefficient is a dimensionless quantity, varying between 0 and 1.

When AC is close to 1 there is an indication that a very clear structuring has been

found. Otherwise when AC is close to 0 it indicates that the algorithm has not

found a natural structure. In other words, the data consists of only one big cluster.

3.1 K-Means (Partitioning)

This is a method of cluster analysis which aims to partition n observations into k

clusters in which each observation belongs to the cluster with the nearest mean.

K-means is a prototyped based approach that requires that the number of cluster

be specified before the analysis is conducted.

Determining number of cluster to specify is a major challenge of using the k-

means. There are several theories on how to compute and obtain the appropriate

number; these would not be discussed since it is beyond the scope of this work.

There is a Rule of thumb for convenience sake which is an estimate given as:

𝑘 ≈ 𝑛/2

Where: 𝑘 – Number of clusters to specify

𝑛 – Sample size

3.2 R-Console

R-console is a platform independent open source package built for statistical

computing. It based on object oriented programming (works with objects and

classes). The power of R lies in the flexibility to write customized functions and

classes that makes up an R package. This feature of R makes it possible to have

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thousands of packages by several developers. These packages are known as

LIBRARY. These Libraries can be found on CRAN (Comprehensive R Archive

Network).

The cluster library in R consists of all required function for cluster analysis

computation. The cluster library is usually invoked with the function

library(cluster)

3.2.1 R-Studio

R Studio Is an IDE (Integrated Development Interface) for R. It is a powerful and

productive interface that GUI for R that provides room for convenience coding.

Since R-Console is more of based only on Console, R-Studio makes a perfect

alternative for computation. Below is the interface of R-Studio

Fig 3.0 Interface of R-Studio 0.96.330

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CHAPTER FOUR

DATA PRESENTATION AND ANALYSIS

This section discussed the detail examination and explanation of the findings

made at the process of this work. Graphical Illustrations were employed to depict

the result of findings.

Table 4.0 Average HIV Prevalence rate in Nigerian States (1999-2003 and 2005-2010)

S/N State Average (1999-2003)

Average (2005-2010)

1 Abia 3.33 5.43 2 Adamawa 5.7 4.93 3 Akwa Ibom 10.13 9.53 4 Anambra 5.43 6.17 5 Bauchi 4.87 2.83 6 Bayelsa 5.17 6.7 7 Benue 13.2 11.1 8 Borno 4.07 3.73 9 Cross River 8.6 7.07

10 Delta 5 3.83 11 Ebonyi 6.67 3.53 12 Edo 5.3 5.03 13 Ekiti 2.47 1.33 14 Enugu 4.93 5.8 15 Gombe 6.57 4.37 16 Imo 5.07 3.83 17 Jigawa 1.83 1.63 18 Kaduna 7.73 5.9 19 Kano 4.07 3 20 Katsina 2.87 2.43 21 Kebbi 3.4 2.63 22 Kogi 5.53 5.47 23 Kwara 3.4 2.27 24 Lagos 4.97 4.5 25 Nasarawa 8.47 8.07 26 Niger 6.07 5.17 27 Ogun 2.5 2.8 28 Ondo 3.97 2.63 29 Osun 3.07 1.97 30 Oyo 3.87 2.33 31 Plateau 6.97 5.07 32 Rivers 5.87 6.23 33 Sokoto 3.33 4.17 34 Taraba 5.9 5.7 35 Yobe 3.07 2.83 36 Zamfara 3.17 2.4 37 FCT (capital) 8.6 8.27

Source: Federal Ministry of Health, 2010 National HIV Sero-prevalence

Sentinel Survey - Technical Report

21

The table 4.0 above shows the average HIV prevalence rate across the 36 states of

the Federal Republic of Nigeria, adopted from the technical report by Federal

Ministry of health, 2010 National HIV Sero-prevalence Sentinel Survey. The states

would be taken for objects while the periods (i.e. 1999-2003 and 2005-2010)

represent the variables.

4.0 Agglomerative Nesting (Before Standardization):

The following R code computes the Agglomerative Hierarchical Clustering using

the default (distance measure = average and variables are not standardized)

parameters as contained in the cluster library in R. At this point the data was not

standardized.

Code listing 4.0 A: Code to compute AGNES

Explanation of Code Listing 4.0 A:

The first line of code is used to invoke the cluster library (package that contains

code for computing cluster analysis).

library(cluster)

The next line reads the data shown above (table 4.0) from text file named HIV.txt

into an array variable called data. Header=True means that the columns should be

named and row.names=1 implies that the rows should also have title (state

names).

data <- read.table(file="HIV.txt", header=TRUE, row.names=1, sep="\t")

> library(cluster) > data <- read.table(file="HIV.txt", header=TRUE, row.names=1, sep="\t") > data.agnes <- agnes(data) > data.agnes

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The line of code below creates and object called data.agnes (an instance of agnes

class). Where agnes implies Agglomerative nesting

data.agnes <- agnes(data)

The last line (data.agnes) executes the agnes computation.

Below is the output (result) of the execution of the code listing 4.0. The execution

produced a summary table as shown below.

R Output 4.0 A

Interpretation of R Output 4.0 A:

a. Agglomerative Coefficient (AC):

The output above summarises the result of the computed analysis. The

Agglomerative coefficient measures the clarity of the structure of clusters formed

as shown in R Output 4.0 B. Agglomerative coefficient of the computed agnes was

0.8885348 (i.e. about 89%) indicates a clear structure or a structure close to a

natural structure since the value is close to unity (see section 3.03 above). The AC

of 89% therefore means that the clusters generation formed a very good and clear

structure.

b. Order of Objects:

This shows the order in which the objects (States) were selected to form the

clusters, that is the order in which the agglomerative algorithm selected the States

at the process of forming the cluster. It is noticed that States with a reasonably

Call: agnes(x = data) Agglomerative coefficient: 0.8885348 Order of objects: [1] Abia Borno Sokoto Delta Imo [6] Lagos Bauchi Kano Ondo Oyo [11] Katsina Yobe Kebbi Kwara Zamfara [16] Osun Ogun Ekiti Jigawa Adamawa [21] Edo Kogi Taraba Niger Anambra [26] Rivers Bayelsa Enugu Ebonyi Gombe [31] Plateau Kaduna Akwa Ibom Cross River Nasarawa [36] FCT Benue Height (summary): Min. 1st Qu. Median Mean 3rd Qu. Max. 0.0700 0.4742 0.8066 1.2940 1.4820 7.2070

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low or average prevalence rate were found at the top of the ordering while States

with high prevalence were towards the end. Though this seem to look as if the

States were arranged in ascending order of prevalence rate, but it does not exactly

appear that way. Further result shed more light to the ordering. This ordering

appears as nodes of the dendogram in R Output 4.0 B.

c. Height Summary:

The height measures the distance between clusters formed, this forms the y-axis

of the hierarchical structure below. The minimum distance within the clusters

formed is as low as 0.0700 and a maximum as high as 7.2070. This implies that the

nearness or farthest of any two cluster is within the range of the stated values.

Objects (States) in the same clusters are naturally expected to have minimal

height difference. Clusters with a low distance would likely have more members

with close level of prevalence than Clusters with a distance far apart.

The average (mean) 1.294 is the average distance between clusters

In order to see the graphs of the analysis, the data.agnes object is passed into a

function called plot( ) as in Code listing 4.0 B. The execution produces a dendogram

as shown in R Output 4.0 B.

Code listing 4.0 B

4.0.1 Dendogram of agnes

Dendogram was used to depict the result of hierarchical clustering by grouping

similar items in the same cluster in a tree-like manner. As we move upward the

dendogram the level of Homogeneity within cluster members increases and

decreases by downward movement. Items in the same cluster were grouped due

to common characteristics in their prevalence rate.

> plot(data.agnes)

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R Output 4.0 B: Dendogram of AGNES

Interpretation of R Output 4.0 B:

The x-axis consists of the names of States according to the order in which they

were selected to form clusters. Choosing the number of clusters is subjective since

number of clusters depends on a given height (x-axis). The number of clusters

varies as we move along the height upwardly or downwardly.

The height value labelled 0 to about 8 on the y-axis, recall that we have a

minimum distance of 0.0700 and a maximum of 7.2070 as stated in summary

table above. The optimal number of clusters would be determined based on a

criteria to maximise distance between clusters and yet identify distinct group.

Interpreting the clusters could be sometimes difficult. A better result was

obtained when the data was standardized (See R Output 4.1 below) for further

interpretation.

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4.1 Agglomerative Nesting (after Standardization)

After computing the agglomerative nesting (agnes) with the un-standardised data,

the analysis was re-computed with the standardised data this is to ensure that one

variable do not dominate the analysis.

The code listing 4.1 was used to execute the agnes specifying the distance

measure (Manhattan) and enforcing that the data be standardized with the

Ward’s method of linkage as in the next line.

metric=”manhattan”, methord=”ward”, stand=TRUE

Code listing 4.1 A

R Output 4.1 A

Interpretation of R Output 4.0 A:

The summary table (R Output 4.1 A) above shows the summary of agnes after

standardizing the variables. After standardizing the data, the Agglomerative

Coefficient (AC) increased from 89% to 96% meaning that we have a better

structure than the first one. The height range in the earlier result was 0.0700 to

7.2070, but we now have a range of about 0.04004 to 16.7100 which implies a

> data.agnes.ALT <- agnes(data, metric="manhattan", method="ward", stand=TRUE) > data.agnes.ALT

Call: agnes(x = data, metric = "manhattan", stand = TRUE, method = "ward") Agglomerative coefficient: 0.9645148 Order of objects: [1] Abia Sokoto Bauchi Borno Kano [6] Ondo Oyo Ekiti Jigawa Katsina [11] Zamfara Osun Kebbi Kwara Ogun [16] Yobe Adamawa Edo Kogi Taraba [21] Niger Anambra Rivers Bayelsa Enugu [26] Kaduna Plateau Delta Imo Lagos [31] Ebonyi Gombe Akwa Ibom Cross River Nasarawa [36] FCT Benue Height (summary): Min. 1st Qu. Median Mean 3rd Qu. Max. 0.04004 0.36470 0.67400 1.75600 1.78000 16.71000

26

wider difference between clusters. This enhances the aim of maximizing the inter-

cluster distance. Several changes were also noticed in the order of objects.

R Output 4.1 C: Two Cluster Solution

Interpretation of R Output 4.1 C:

The height on the y-axis indicates the distance between one cluster and the other.

Looking at this structure from the top, we would see that we generally have two

major groups from the dendogram. This is evident at the height of 12, if we move

from a height of 12 on the y-axis across the x-axis we would cross two lines, which

indicate the two major groups that we have in the result. We call this a two cluster

(2-cluster) solution. This implies that Akwa-Ibom, Cross-River, Nasarawa, F.C.T and

Benue are in the same category against the other 33 States as marked with the

27

red rectangles. A reflection of this is showed on the average prevalence rate;

these states have the highest prevalence rates.

As we move down the dendogram and read the line in the same manner used to

obtain a 2-Cluster solution above, we would encounter more clusters. The

dendogram was further analysed below.

R Output 4.1 D: Five Cluster Solution

Interpretation of R Output 4.1 D:

Furthermore, by taken a closer look at the two-cluster solutions, the cluster on the

left is further divided into two sub-groups at about a height of 10 as we moved

downward from 12, at this level we would say we have three-cluster solutions.

Further look into the dendogram reveals that we have a five cluster solution as

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shown in the figure R Output 4.1 D above. At this point, the States are distributes in 5

clusters of sizes 7, 9, 11, 5, and 5 taking each group from the left.

4.1.0 Banner Plot

The banner plot is an alternative to dendogram. The white area on the left of the

banner plot (R Output 4.0 b) represents the un-clustered data while the white lines

that stick into the red shows the heights at which the clusters were formed. We

considered the dendogram preferable though interpreting the banner plot would

lead to same conclusion.

The choice using the banner plot is determined by the analyst, we found the

dendogram to be better for this reason we stick to it.

R Output 4.1 B: Banner Plot

Abia Sokoto Bauchi Borno Kano Ondo Oyo Ekiti Jigawa Katsina Zamfara Osun Kebbi Kwara Ogun Yobe Adamawa Edo Kogi Taraba Niger Anambra Rivers Bayelsa Enugu Kaduna Plateau Delta Imo Lagos Ebonyi Gombe Akwa Ibom Cross River Nasarawa FCT Benue

29

4.2 K-Means (partitioning)

K-means is a prototype based technique, the number of clusters to be formed

needs to be specified before the computation as discussed earlier. By applying the

Rule of thumb, with sample size of 37, our k (number of clusters) is around 4.30.

Since choosing a number of clusters in k-means could be decided as willed, k=5

was used. The following code listing was used to scale the average HIV prevalence

data, the output of the scaled data appears in R output 4.2 A

Code listing 4.2 A

K MEANS

The output below shows the scaled data after been scaled. Scaling the variables

ensures that the variables are on the same metric.

R Output 4.2 A

> data.scale <- scale(data) > data.kmeans <- kmeans(data.scale, 5) > data.kmeans

K-means clustering with 5 clusters of sizes 13, 5, 7, 10, 2 Cluster means: X1999_2003 X2005_2010 1 0.3009821 0.3436609 2 1.9106795 1.8553624 3 -0.3790514 -0.2508853 4 -0.8173938 -0.9216925 5 -1.3194336 -1.3856407 Within cluster sum of squares by cluster: [1] 3.04525944 4.79277415 1.41354797 0.58009386 0.04531497 (between_SS / total_SS = 86.3 %) Clustering vector: Abia Adamawa Akwa Ibom Anambra Bauchi Bayelsa 3 1 2 1 3 1 Benue Borno Cross River Delta Ebonyi Edo 2 3 2 3 1 1 Ekiti Enugu Gombe Imo Jigawa Kaduna 5 1 1 3 5 1 Kano Katsina Kebbi Kogi Kwara Lagos 4 4 4 1 4 3 Nasarawa Niger Ogun Ondo Osun Oyo 2 1 4 4 4 4 Plateau Rivers Sokoto Taraba Yobe Zamfara 1 1 3 1 4 4 FCT 2 Within cluster sum of squares by cluster: [1] 3.04525944 4.79277415 1.41354797 0.58009386 0.04531497 (between_SS / total_SS = 86.3 %)

30

Interpretation of R Output 4.2 A:

The implementation of K-means generated 5 clusters of sizes 13, 5, 7, 10, and 2.

The clustering vector, a number under each state denotes which cluster a State

belongs to. For better understanding, the members of each of these clusters are

listed in the table below:

Table 4.0: Cluster Membership for K-means Clusters

Cluster 1 Cluster 2 Cluster 3 Cluster 4 Cluster 5

Adamawa Akwa Ibom Abia Kano Ekiti

Anambra Benue Bauchi Katsina Jigawa

Bayelsa Cross River Borno Kebb Ebonyi Nasarawa Delta Kwara Edo FCT Imo Ogun Enugu

Lagos Ondo

Gombe

Sokoto Osun Kaduna

Oyo

Kogi

Yobe Niger

Zamfara

Rivers Taraba

The output in the summary table above indicates 86.3% total variation, this

implies that about 86% of the variation present in our data was explained by the

clustering result. This shows that we have a reasonably good clustering as the

clusters formed were able to account for 86% of the differences in the HIV

prevalence rate.

Cluster means indicates the centres of the clusters formed. It is the mean of the

specific variables combined to form a particular cluster. K-means attempt to

minimise variation (difference in prevalence rate) within members of a cluster and

maximise it between clusters so that members in the same cluster are

homogeneous as much as possible and heterogeneous to other clusters.

For clarity of what the centres shows we present the table below:

31

Table 4.1: Cluster Means (centres) for K-means

Cluster Size X1999_2003 X2005_2010 Final Mean

13 0.3009821 0.3436609 0.644643

5 1.9106795 1.8553624 3.7660419*

7 -0.3790514 -0.2508853 -0.6299367

10 -0.8173938 -0.9216925 -1.7390863

2 -1.3194336 -1.3856407 -2.7050743

The mean of the second cluster (3.766) in the above table is the most distinct

among others, we observed that members (states) of this cluster have extreme

prevalence rate as evident in our original data. It is also noticed that these states

are exactly the same as found in the last cluster of our 5-Cluster solution, far right

of the dendogram in R Output 4.1 D above.

The result of the K-means to a large extent agreed with the agglomerative nesting

at k=5 no of cluster. Clusters formed by both analysis (Agglomerative nesting and

K-means) have much similarities, this is clear by comparing the Clustering Vector

(Table 4.0) and the five cluster solution in the Dendogram (R Output 4.1 D).

4.3 Clusplot Graph Function

The clusplot( ) is a function that display Cluster on a bi-component plot. It uses the

PCA (Principal Component Analysis) approach on the given data.

Principal component analysis (PCA) is a dimension reduction technique; it

summarizes the information of all variables into a couple of new variables called

Components. Each component is responsible for explaining certain percentage of

the total variability. The clusplot uses the first two components which usually

explain the maximum variation in the given data to display a principal plane.

The bi-plot obtained from clusplot is used to identify the effectiveness of

clustering. Clearly separated cluster in the principal plane indicates successful

clustering. On the other hand, a merged cluster implies unsuccessful clustering

32

The clusplot function takes the parameter of a k-means object and specification of

colours to differentiate groupings or cluster membership. The code listing below

was used to implement clusplot.

Code listing 4.2 C

The resulting output of the clusplot is shown in R Output 4.2 C.

R Output 4.2 C: Clustering Plot

Interpretation of R Output 4.2 A:

It is obvious from the plot that the objects (States) formed a kind of clusters on

the principal plane, though this clusters differs from that of k-means, it is not a

> fit <- kmeans(data.scale, 5) > clusplot(data.scale, fit$cluster, color = TRUE, shade = TRUE, label=2, lines=0)

33

problem since the clustering plot aims to show the spread of the objects on the

plane, we are interested in knowing if these objects worth having a substantial

grouping as claimed by the K-means by looking at their spread on the plane.

The different colour shadings indicates different groups and objects that are likely

to have same cluster membership, it is evident from this figure that the objects

have sufficient groupings. The two components explained 100% of the total

variability, meanwhile the first component explains much of the variation, this is

customary to principal component analysis. The values on the axis are not

necessarily important.

An important observation is that states with high prevalence rate falls to the right

of the red line at the centre of the clusplot. In fact, the farther the state to the

middle line the higher the prevalence rate. Thus, the plot clearly shows States that

policies makers should give more attention.

4.4 Summary of Results

The hierarchical clustering analysis (agglomerative nesting) yielded five clusters

solution, and revealed states that could be considered to be in the same category.

From observation, it cannot be said absolutely that states with highest prevalence

rate are from a particular geographical area (geopolitical zones). Though places

with high prevalence rate are more dominated in North-Central (Benue, Nasarawa

and F.C.T) and South-South (Akwa-Ibom and Cross-River).

The states that could be considered to have low prevalence rate includes Ekiti,

Jigawa, Katsina, Zamfara, Osun, Kebbi, Kwara, Ogun, and Yobe, most of these

states belong to North-West zone.

From the clustering result it was clear that different level of HIV prevalence (low,

average and high) sparse across the six zones, except where it was noted that

North Central and South-South States were the dominants of High level

34

prevalence rate (see Appendix II: Geopolitical Zones on Nigeria). Despite this, the

clustering map for the agglomerative nesting reveals something different as

shown in figure 4.4 below

Fig 4.4: Clustering Map for the Agglomerative Nesting

Though, going by geopolitical zones various categories of prevalence sparse across

the country, but the clustering map reveals a direct association between states

with high level of prevalence, for example states with the highest HIV prevalence

(cluster 1) appeared to have direct connections and shared boundaries with each

other. The same was also noticed for states with second highest prevalence

(cluster 2).

This may mean that states that have close bilateral relationship are likely to have

socio-cultural similarities that resulted in similar prevalence.

35

4.5 Cluster Validation

The result of our clustering was subjected to validation by means of Relative

approach as discussed earlier (Section 2.3)

K-means produced clusters of sizes 13, 5, 7, 10, and 2 while the Hierarchical

produced clusters of sizes 7, 9, 11, 5, and 5. If we compare members of a given

cluster in k-means to the members of a similar cluster in hierarchical we would see

that there are lots of similarity between the clusters produced by the two

methods. For clarity, we assign a group number to each cluster in both methods as

follows:

K-means Clusters:

Group No. Group 1 Group 2 Group 3 Group 4 Group 5

Cluster Size 13 5 7 10 2

Agglomerative Nesting Clusters:

Group No. Group 1 Group 2 Group 3 Group 4 Group 5

Cluster Size 11 5 7 5 5

The table below summarizes the relationship between the clusters generated by

K-means and the Agglomerative Nesting.

Table 4.2: Comparison Table for K-Means and Hierarchical Clusters Membership

Clusters

K Means (Cluster size)

Hierarchical (Cluster size)

Difference in Cluster Size

No. of Agreed Members

No. of Different Members

Group 1 13 11 2 10 3

Group 2 5 5 0 5 (All) 0

Group 3 7 7 0 4 3

Group 4 10 9 1 6 4

Group 5 2 5 3 0 7

Starting with Group 1, K-Means and Agnes have 13 and 11 cluster sizes

respectively, whereas 10 of the members are equal in both clusters (that is 10 of

the States in Group 1 cluster of K-Means is found in Group 1 cluster of Agnes).

These include Adamawa, Anambra, Bayelsa, Edo, Enugu, Kaduna, Kogi, Niger,

Rivers, and Taraba States. But Ebonyi and Gombe were found in Group 1 of K-

36

Means and Plateau in Group 1 of Agnes which implies a difference of 3 members,

2 from K-Means and 1 from Agnes.

There is a high level of similarity between the cluster sizes produced by the two

methods as shown in the column “Difference in Cluster Size”. In fact, they both

have equal cluster sizes on two occasions (Group 2 and 3). Group 2 and 3 have

equal number of members in both K-Means and Agglomerative methods, though

there is member difference in Group 3. A special case is the Group 2, they do not

only have equal number of members, but they also have the same States (Akwa-

Ibom, Cross-River, Nasarawa, FCT, and Benue) as cluster members.

Group 5 appears to be the only case where there no similar member in both cases.

Therefore, both clusters have 7 distinct members in which 2 belong to K-Means

and 5 to Agnes. Exact match is not expected as both methods uses different

algorithm to make their clustering.

It is reasonable to say that the difference in cluster sizes in both K-means and

Agnes is low and there exist a high level of similarity in clustering and cluster

membership yielded by both methods, this indicates goodness in the clustering

results obtained.

37

CHAPTER FIVE

SUMMARY, RECOMMENDATION AND CONCLUSION

5.0 Summary

The HIV prevalence rate in Nigeria is measured bi-annually, pregnant women

under antenatal care in survey sites are usually the major source of measuring this

statistics. The HIV prevalence data used for this work covers the periods of 1999

to 2010, though this period was split into two variables, the first variable being the

period between 1999 – 2003 and the second variable is between 2005 – 2010,

each variable consists 3 years data point since the measurement is done bi-

annually. The split was necessary to correct the possible anomalies that could be

posed by inconsistent lag in the second variable (2005 – 2010), which also led to

the use of average prevalence rate in our analysis (see Appendix I).

We intend to find possible classification for which we can determine States with

common situation of the HIV spread. We conducted a Cluster analysis using the

clustering schemes from two clustering methods, Hierarchical and the Partitioning

methods.

Agglomerative nesting an approach of Hierarchical clustering, divides data items

into homogeneous group by means of a proximity matrix and a linkage criterion,

this agglomerative techniques was first used on the prevalence data. This resulted

in a five cluster solution after detail study of the output and a hierarchical

structure (dendogram) as shown in R Output 4.1 D. The 36 States were grouped into

clusters of sizes 7, 9, 11, 5, and 5. This result was accepted to a large extent since

it generated the number of clusters that coincides with the recent literatures by

(FMOH 2010) as recommended by WHO.

In the case of Partitioning method, we used the K-means, a prototype based

approach in which a priori information about number of cluster must be specified.

38

In conducting the K-means clustering, we specified k (number of cluster) to be

formed as five (k=5), the same number of clusters generated in the agglomerative

approach which is also the approximated value for using the rule of thumb for

determining the appropriate number cluster for K-means clustering. This was

done to see how well we can rely on the grouping done by the Hierarchical

procedure, another reason for this is for cluster validation. The result of the K-

means agreed with the agglomerative to a large extent, since they both have more

similarities than differences.

The Relative type of Cluster validation (a check mechanism) was used to critically

compare the clustering produced by the two clustering shames to ascertain that

we did not cluster in noise (that is clustering when it is not necessary). A

comparison table Fig 4.0 was established in order to ease the task of validating the

clustering result.

5.1 Recommendation

The quality of data plays a vital role in any statistical analysis. It is highly important

that the sentinel survey for acquiring the HIV prevalence rate be made consistent.

There should be consideration for having more sources of acquiring the

prevalence rate rather than patients under antenatal care only.

Conducting a cluster analysis is not an easy task, the fact that there are multiple

ways of implementing cluster analysis in which combination of steps and possible

alternatives depends on the choice of analyst makes it subjective. Meanwhile,

each combination of steps and alternatives has its own weakness. It is important

to use different clustering schemes on the data of interest in order to see the

direction of the obtained result and to avoid misleading clustering.

A tabular mechanism for comparing result of two clustering scheme used in Table

4.2 above provides an easy alternative for comparing clustering results from

39

different clustering scheme. Further research works could develop numerical

computations to enhance or optimize this approach and make the process faster.

A major aim of Governments and Agencies concerned with HIV issues is to

minimize the spread of this disease as much as possible. The plot above (R Output

4.2 C) could be an easy way to clearly identify States that require more attention.

More of the programmes such as awareness and education on HIV should be

channelled to such States. Another point of interest that government and agencies

in control of HIV in Nigeria should look into is the connection between bilateral

relationships and prevalence rates; the clustering map (figure 4.4) gives an insight

on this.

5.2 Conclusion

From the results obtained it can be said that we have a successful clustering that

classified the spread of HIV among Nigerian states where North-Central and the

South-South geopolitical zones are important areas that require more attention.

Clustering Plot (R Output 4.2 C), a graph of K-means clustering reveals an

important measure for identifying areas (states) under risk; this graph requires a

closer look.

Despite the fact that K-means and Agglomerative clustering are different

clustering techniques and choice of using any may be based on the type of data;

the final results of these techniques for the same data shows lots of similarities.

40

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