Estimating the Size of an Injecting Drug User Population

doi:10.4236/wja.2011.13013

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World Journal of AIDS, 2011, 1, 88-93

doi:10.4236/wja.2011.13013 Published Online September 2011 (http://www.SciRP.org/journal/wja)

Estimating the Size of an Injecting Drug User

Population

Yang Zhao

Department of Mathematics and Statistics, University of Regina, Saskatchewan, Canada.

Email: zhaoyang@uregina.ca

Received April 6th, 2011; revised May 27th, 2011; accepted June 23rd, 2011.

ABSTRACT

This article describes a sampling and estimation scheme for estimating the size of an injecting drug user (IDU) popula-

tion by combining classical sampling and respondent-driven sampling procedures. It is designed to use the information

from prevention programs, especially, Needle Exchange Programs (NEPs). The approach involves using respon-

dent-driven sampling design to collect a sample of injecting drug users who appear at site of NEP in a certain period of

time and to obtain retrospective self-report data on the number of friends among the IDUs and number of needles ex-

changed for each sampled injecting drug user. A methodology is developed to estimate the size of injecting drug users

who have ever used the NEP during the fixed period of time, and which allows us to estima te the proportio n of injectin g

drug users in using NEP. The size of the IDU population is estimated by dividing the total number of IDUs who using

NEPs during the period of time by the estimated proportion of IDUs in the group. The technique holds promise for pro-

viding data needed to answer questions such as “What is the size of an IDU population in a city?” and “Is that size

changing?” and better understand the dynamics of the IDU population. The methodology described here can also be

used to estimate size of othe r hard-to-reach po p ul ati o n by usi n g i nf ormation from preve n t i on programs.

Keywords: Injecti ng Drug Users, Needle Exchange Programs, Respondent-Driven Sampling, Estimating the Size of an

IDU Population

1. Introduction

It is reported that HIV epidemics are primarily driven by

injecting drug use in Eastern Europe and Central Asia [1].

In Central and South America, injecting drug use and

unsafe sex have been the main route of HIV transmission

[1,2]. The negative health consequences of injecting drug

use are not limited to just HIV infection. Sharing injec-

tion equipment carries a high risk of transmission of

other blood-borne infectious diseases such as hepatitis B

and hepatitis C. Also, injection drug use contributes to

the epidemic's spread far beyond the circle of those who

inject. Injection drug users (IDUs), their partners, and

their children account for at least 36% of all AIDS cases

reported in the U.S. through 1999 [3]. In Canada, injec-

tion drug use is also a problematic activity. Although it is

difficult to obtain accurate data on the prevalence and

profile of IDUs in Canada, it is clear that there are large

numbers of IDUs across the country [4-6]. Policy makers

and researchers have realized that data on the size of

IDUs and pattern of drug injection need to be systemati-

cally collected for a comprehensive understanding of the

HIV epidemics among IDUs [7-9]. Understanding some-

thing about the dynamics of the injection drug users

makes it possible not only to assess the likely impact of

the spread of HIV/AIDS and other related diseases, but

also alert policy makers to a worsening situation, or al-

ternatively to provide evidence so that other initiatives

may be working. However, it is difficult to get reliable

estimates of the number of IDUs by systematic surveil-

lance. Policy makers and researchers have met the prob-

lem of collecting accurate information about IDUs, since

they are not easily captured in a general population based

survey (a typical general population based survey is to

survey individuals in a random sample of households in a

district, province or a country, depending on the scale of

the study). Injectors sometimes even hide their habit

from those with whom they live, including parents,

roommates, and sexual partners because injecting drug

could be viewed as socially unacceptable. Therefore, the

traditional sampling and estimation methods can not be

used to collection information on this population, so

Estimating the Size of an Injecting Drug User Population89

called as hard-to-reach population. To overcome the dif-

ficulties, a network sampling technique, respondent-

driven sampling (RDS), has been developed and widely

used to sample from the hard-to-reach population [10-14].

The methodology shown in this article is to estimate the

size of an IDU population and to understand the HIV/

AIDS epidemics among IDUs, based on information

from needle exchange programs (NEPs) by systematic

surveillance.

2. Estimating the Size of IDU Population

Who Use NEPs

Needle or syringe exchange programs, which sterile nee-

dles and syringes are free or at a minimal cost for IDUs,

are a convenient means of monitoring the prevalence of

blood borne viral infections among large numbers of

IDUs who are currently injecting drugs. There are well

over two hundred NEPs in Canada, with more under de-

velopment [15]. In addition, there are numerous pharma-

cies that provide needle exchange services [15]. For ex-

ample, within the province of Ontario, 34 NEPs operate

distributing over 3.2 million clean syringes annually to

an approximate 41,100 people who inject drugs and it is

estimated that 53 needles are distributed per injector per

year [16].

The purpose in this Section is to estimate the size of

IDU population who are using NEPs, based on informa-

tion of the number of needles distributed by NEP centres.

To obtain such information from NEP centres, a two

stage-samples procedure is provided as the following: a

sample of NEP centres is selected, then, in a survey week,

all individuals in the selected NEP centers will be inter-

viewed, which allows for the sampling of individuals at a

limited number of centres. Because of time and cost con-

straints, considering characteristics of IDU population, a

sampling strategy has to be developed to select NEP

centers. They can be chosen completely at random from

a list of the all centers of NEPs. However, to decrease the

variability of parameter estimators based on data from

the completed survey, sampling theory suggests that it is

better to sample large centers (covering large-IDU-po-

pulation) with higher probability than small centers

(covering small IDU-population). So, the sampling de-

sign for NEP centers is quite general and frequently used,

which is the Stratified Probability-Proportional-to-Size

sampling. Based on the number of needles which has

exchanged in each NEP centre in the last year, all centers

can be divided across the country into strata with

needle exchange centers in the th stratum. In ad-

dition, needles are legally available for purchase through

pharmacies [17], although the willingness of pharmacists

to sell syringes to IDUs is variable. This kind of ex-

change sites can be one of strata. From the th stratum,

NEP centers are sampled with inclusion probabilities

1, 2,, ,h

hk . The inclusion probabilities are de-

termined by the number of needles that were exchanged

in the last year of the survey week. The could be

determined in the following way: assuming that sampling

is without replacement, the inclusion probability of the

th NEP center in the list of NEP centers is

πh

πhπ

πhi



πhi

h h



, where is the total needles that

were exchanged at the th centre in th strata in the

last year of the survey week. It is necessary to require

π1



. This is the case for all , when . It could

be expected that there is no very large hj within each

stratum. So, the requirement can be satisfied. Otherwise,

it could be done by setting for the following

inequality being valid [18]

πhi

k

1

hhi j



hj

Q.

It is clear that the size of the IDU population which a

NEP center covers is correlated with the number of nee-

dles distributed. Therefore, Probability-Proportional-Size

sampling offers the possibility of decreased variability of

estimators of totals [19]. It is also offer some operational

advantages in multistage sampling in that it can equalize

the workload across geographic areas sampled at the first

stage of sampling. All IDUs attending the selected needle

exchange centres during the survey week will be asked to

complete a brief questionnaire, a question, such as “how

many needles did you use in the last three months?”

could be in the questionnaire.

Let hi be the number of needles that are exchanged

during the period of the three months in the th sampled

NEP centre of th stratum. Suppose there are hi

IDUs who were exchanged the needles in the last three

months of the survey week in this centres and

hi is

the number of IDUs who has used needles during the

three month and all those needles come from this centre.

The numbers of hi and

hi include not only IDUs

who attend the NEP centre but also those IDUs who use

the needles from the centre through possible second hand

exchange. A survey [20] conducted in American shows

that 90% of NEPs actively encouraged secondary ex-

change which is defined as “providing needles that you

know will be used by persons other than the exchanger”.

Therefore, the number of clients for a NEP centre does

not equal to the number of IDUs who exchange needles

from this centre.

Now we can get relation between the number of IDUs

who use needles from the centre and the total number of

needles that are exchanged in the centre for the period of

the three months,

Estimating the Size of an Injecting Drug User Population

hihi hihi

TjNNjR





the is the maximum number of needles that an IDU

could be used in the three month, the jj

hihihi

which can be estimated by the sample obtained in the

survey week. Notice that hi and

RNN

hi are unknown.

However, what we need to know is the proportion

Rand it can be estimated by the sample proportion

hihihi , where the hi is the number of IDUs who

attend the needle exchanges in the exchange centre and

complete the questionnaire in the survey week and among

of them there are

rnnn

hi IDUs who used needles in the

three months. Therefore, the can be estimated by

n j

hi hihi

NT jr



.

In fact, the denominator

hi hi

rjr



 is the mean of

numbers of needles that exchanged for IDUs who used the

NEP for their changes in the period of three months. The

estimator of the size of IDU population who used

NEPs at least once during the three months in all juris-

dictions of the country (or the area) is the weighted mean

hi hi





 .

Suppose the survey week is a typical one for the IDUs

who use the NEP centre. It is assumed that the sample of

IDUs who are observed in the survey week is obtained

by a simple random sampling without replacement in the

IDUs who use the NEP centre in the period of the three

months. From the surveys which were carried out for

studying the risk behavior of IDUs in Australian [21] and

in Canada [22], the IDUs observed from a survey week

are representative for the IDUs who use the NEP centers

in a period of three months. So, the above assumption

seems reasonable. Algorithms for Stratified Probability

Proportion-to-Size Sampling selection without replace-

ment could be realized, based on many methods, such as

Hanurav-Vijayan algorithm [23,24].

3. Estimating the Proportion by RDS

Under the assumption in the last Section, the IDU popu-

lation is made up of two groups of people based on their

status of participating NEPs. To estimate the size of IDU

population, we have to know not only the size of the

population who use NEPs but also the proportion of the

population who participate the NEPs during the three

months. The idea from respondent-driven sampling [14]

will be used to estimate the proportion.

The sampled IDUs in the survey week in the th

centre can be divided into two groups: one is for having

used NEPs at least one time in the period of the three

months and the other one is for having never used NEPs

in the period, the groups are denoted by

 and

, respectively. The size of the group hi

 usually

is small. To increase the size and coverage, we can use

each IDU in the group of hi

EP as an initial seed to

conduct a respondent-driven sample, that is, each seed

will recruit certain number of IDUs and provide informa-

tion on how many friends they have with NEPs users

group and non-users group respectively. The new sample

collected by members of the group



 will contain

NEP users and non-NEP users.

Now, to estimate the population proportion, we need

to know the networking structure, such as the average

degree of friendships, and probability that a non-NEP

user (or NEP user) have a friendship with a NEP user.

The total number of friendships radiating from the IDUs

in the group hi

 is denoted by hi

 which can be

written as

hihi hi

EPNEP NEP

n ,

where hi

hi

is the average friendships of an IDU in the

group and

EP is the number of IDUs in the group. Let n

EP NEP hi

 be the number of friendships that all IDUs in

group

 have with IDUs who don’t use NEPs. Then

the probability ,

EP NEP that an IDU who uses NEPs

has a friendship with an IDU who has never used the

NEPs in the period of the three months can be estimated

EP NEP

NEP NEPhi

NEP

P









 .

The average number of friendships of IDUs in the

group



can be estimated by pooling the samples

from each sampled centre together:



NEP

NEP Lk

dhi

NEP







 

 ,

the





NEP

d is the number of IDUs in the group hi



who have number friendships among IDUs. Simi-

larly, we can get

NEP

P and

.

Based on the status of an IDU using NEPs in the pe-

riod of the three months, we say that an IDU in state

EP if he or she has ever used NEPs in this period,

otherwise we say he or she is in state

NEP . Suppose we

have respondent-driven sampling design to collect a

sample, the initial IDU (a seed) is chosen in step 0. An-

other IDU could be chosen based on the degree of friend-

ships of the initial seed. We say this IDU is chosen in

step 1, and so on. Suppose the chance of recruiting an-

other IDU with state

S depends on this chosen IDU

Estimating the Size of an Injecting Drug User Population91

only through his or her degree of friendships among

IDUs. Suppose also that if the IDU chosen in step 0 is in

state

EP , then an IDU in state S

EP will be chosen

with probability

EP NEP

P; and if the IDU chosen in step

0 is not in state

EP , then an IDU in state S

EP is

chosen with probability

NEP

P in step 1. Letting n

denote the status of an IDU chosen in the th step, then

is a two-state Markov chain having a

following transition probability matrix:



,0Xn



,1,

n

,NEP N

NEP

EP NEP NEP

NEP NEP NEP







It is clear that it is an irreducible argotic Markov chain

[25]. In fact, we assume that IDUs within the group of

using NEPs have similar proportions of degree of

friendships with group of IDUs not using NEPs (or using

NEPs). Also, notice that ,

EP NEP

NEP NEP and

PP

NEP NEPNEP. In practice, it is much easier for a

sample is collected by 1 wave. So, we have to use the

different method to approach the proportions

PP

EP and

πNEP of a chain of IDUs in states

EP and SNE re-

spectively. Based on the Markov theory, we have P

NEP

EP NEP

NEP



NEP

and

NEP NEP

NEP

EP NEP

NEP





NEP

The π

EP and π

EP can be estimated by plug-in es-

timates ,

EP NEP and P,

NEP

P. Now, based on the re-

sult given by [14], the proportion



of IDUs who have

ever used NEPs in the period of the three months can be

estimated by

NEPNEP

NEP NEP

EP NEP









 .

Now we have estimated the size of IDU popula-

tion who use NEPs and the proportion



of IDUs in

using NEPs among all IDUs, then, this information can

be used to estimate the size of IDU population as N



4. Discussions and Conclusions

For many years researchers have tried to get accurate size

of IDU population. We have shown that this goal could

be fulfilled by the sample design combining prevention

programs—NEPs. Switching to this prevention programs

and using the information of number of needles ex-

changed in the NEP centres give us a fresh and novel

approach to the estimation of the size of IDU population.

Using NEPs allows us to design a sampling and estima-

tion scheme which could be both cheaper and more ac-

curate. The prevention programs, such as NEPs, provide

a comprehensive HIV and blood-born infections preven-

tion model to prevent the further spread of the diseases

among IDUs and they have been proved to be effective

for intervention of risk behaviour among IDUs. It is pos-

sible that the network of NEPs could embrace a well dis-

tributed, age and sex representative population of the

area where we are interested in. However, it may be dif-

ficult to choose all NEPs to participate the data collection

system. For the purpose of estimating the size of IDUs

using NEPs, the sample design we proposed follows the

basic principle of sampling theory which is that each

individual in the target population should have some

nonzero chance of being sampled in the survey [26]. No-

tice that the target population in our first stage of estima-

tion is all IDUs who have ever used the NEPs in the pe-

riod of the three months. Therefore, each IDU using

NEPs has some nonzero chance of being sampled in the

survey of estimating the size of IDUs of participating

NEPs. We use the idea of respondent-driven sampling to

estimate the proportion of IDU population in using NEPs.

However, in order to estimate the proportion, what we

need to know is transition probability that a non-NEP

user (or NEP user) has a friendship with a NEP user.

Then the results from Markov chain and the article [14]

were used for the estimates. We have provided estimators

corresponding to the sample design.

Our approach has concentrated on estimating the size

of IDU population. It may be necessary to combine this

sampling and estimation strategy with study of risk be-

haviour of IDUs. Further study for variances of the esti-

mators has to be carried out. We have presented a num-

ber of analytic results and these analytic arguments could

be further supported with numerical simulation. Also, the

possible bias exists when we estimate the transition

probabilities, because the sample size of non-NEP users

is usually small. However, it may be possible to predict

the magnitudes and direction of this bias ahead of time.

5. Acknowledgements

The authors thank referees for their valuable suggestions

on improving the manuscript. Some results of this paper

were presented at the Statistics Canada Symposium: In-

novative Methods for Surveying Difficult-to-reach Popu-

lations; Center for Diseases Control Symposium on Sta-

tistical Methods. This article only represents the opinion

of the authors and it does not present any official views

of an organization.

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Appendix

We discuss the variance of the estimator in this ap-

pendix. First, we look at the estimator hi . Let hi

denote the number of needles exchanged by an injecting

drug user in the centre for a period of the three months.

The probability mass function of is denoted by

Y



y. The mean of hi

Y is





Y

jj. The

variance of conditioned on is

Y N

 



hi hi

hi hi hi

Njf jjf j

N











The



j can be substituted by

r; the 1



can be replaced by 1 because hi is usually very large

in our case. Notice that hi is known and

Thi

Y is esti-

mated by hi , linearization methods [27,28] are used to

approximate the variance of the estimator , then, we

have



var var

hihi hi

Y,

where denote the expectation. Finally, the variance

can be estimated by







var

11hi

hi hi hi

NNn







 ,

where can be approximated by



var hi



hi hi

hi hi hi

Njr jr

N









.

Because the sampling is carried in each stratum inde-

pendently and within a stratum the Probability Propor-

tional-to-size sampling with fixed size without-replace-

ment design is used, the variance of the estimator

for the size of IDU who use the NEP in the period of the

three months can be written as the following [29]:



111

var,1, ,

1ππ π

hi h

LKKhi hl

hi hlhil

hil hi hl

NN iK





















where is the covariance between indicator variables

πhil

and hl

(hi

= 1 if the centre hi is included in the

sample, 0 otherwise). In the case of fixed size without

replacement design, the calculation of could be

complicated. For example, if the sample size in th

stratum is 2, the has the following expression under

the scheme [29]:

πhil h

πhil









 

j



πhi hlhhihl

hhjhhi hl

QQ QQQ

cQQ



QQ

hil Q,

where





and



 



hj hj

cQhhjhh

QQ QQ hj

Q.

For the first draw, the scheme gives the center the

probability

hi hi hj

j of being selected; with-

out replacing the first drawn element, (say ), it give

the other element the probability:



pc

,hi



|,h i,hihh i

pQQQ

hi .





N can be calculated by The variance va













var

var,1, ,

ar, 1,

NNi

ENN i







 v

Considering that all are estimated independently,

we could have











,1, ,

var

hi h

hhi

ENN ik







var

.

By the Taylor linearization technique, an approxima-

tion of the variance of the estimator for the size of

injecting drug users who use NEP in the period of the

three months is



varππ π1ππ

var

kk hl

hihlhil

hilhl

hhi

































Nhi

hi .