Advances in Applied Sociology
2012. Vol.2, No.4, 245-252
Published Online December 2012 in SciRes (http://www.SciRP.org/journal/aasoci) http://dx.doi.org/10.4236/aasoci.2012.24032
Estimating the Size of the Methamphetamine-Using Population in
New York City Using Network Sampling Techniques
Kirk Dombrowski1*, Bilal Khan1, Travis Wendel1, Katherine McLean2, Evan Misshula2,
Ric Curtis1
1Social Networks Research Group, John Jay College, CUNY, New York, USA
2CUNY Graduate Center, New York, USA
Received July 21st, 2012; revised August 24th, 2012; accepted September 10th, 2012
As part of a recent study of the dynamics of the retail market for methamphetamine use in New York City,
we used network sampling methods to estimate the size of the total networked population. This process
involved sampling from respondents’ list of co-use contacts, which in turn became the basis for cap-
ture-recapture estimation. Recapture sampling was based on links to other respondents derived from
demographic and “telefunken” matching procedures–the latter being an anonymized version of telephone
number matching. This paper describes the matching process used to discover the links between the solic-
ited contacts and project respondents, the capture-recapture calculation, the estimation of “false matches”,
and the development of confidence intervals for the final population estimates. A final population of
12,229 was estimated, with a range of 8235 - 23,750. The techniques described here have the special vir-
tue of deriving an estimate for a hidden population while retaining respondent anonymity and the ano-
nymity of network alters, but likely require larger sample size than the 132 persons interviewed to attain
acceptable confidence levels for the estimate.
Keywords: Population Estimation; Network Methods; Methamphetamine; Anonymous Sampling
Introduction
Statistics such as the size of hard-to-enumerate populations
are both important and difficult challenges for social science:
important in that they represent one area where sociological
results impact the allocation of public funds for both law en-
forcement and public health resources (Aceijas et al., 2006,
Dengenhardt & Hall, 2012), yet difficult because they often
require estimation procedures that pit ideal methods against the
difficulties of research implementation. Such questions lie at
the heart of applied sociology. In particular, estimates of the
size of hidden populations often hinge on data drawn from a
single source, such as arrests or hospital admissions, whose
relationship to overall population levels remains largely un-
known, leaving both policy makers and researchers unsatisfied
with results. Recent modeling work not withstanding (Simeone
et al., 2003; Zhao, 2011; see Berchenko & Frost, 2011 for dis-
cussion) this represents a less than ideal situation, a point aptly
summed up in the titled of a recent article: “The numbers game:
Let’s all guess the size of the illegal drug industry!” (Thoumi,
2005). As noted by Thoumi, such problems are particularly true
for drug using populations, where limited data from disparate
sources often indicates countervailing trends, yet population
estimates and overall community dynamics continue to occupy
important policy decisions. In these situations, research con-
fronts hidden populations whose illegal behaviors invoke the
need for anonymous sampling, further exacerbating an already
difficult research scenario.
New York City methamphetamine users represent such a
population. Indeed, meth-users in NYC have received little
attention until recently when concern about growing levels of
methamphetamine use were associated HIV risk behaviors in
the MSM (men who have sex with men)/gay community
(Hirshfield et al., 2004; Morin et al., 2005). Methamphetamine
has actually been available in New York City for decades (Drug
Enforcement Administration (DEA) 2004, 2006, National Drug
Intelligence Center (NDIC) 2008). Yet New York’s metham-
phetamine markets have remained mostly inaccessible to re-
searchers, and the small body of literature that is currently
available on methamphetamine use in New York City focuses
mainly on use among MSM while offering little information
about market size, numbers of users, or distribution in general;
nor about use outside of MSM communities, and what effect
this has on the total number of users in the area. Local data such
as these are important. While DAWN (2009: pp. 18-19) reports
that the national estimate of methamphetamine-related emer-
gency room visits in the US dropped from 132,576 in 2004 to
66,308 in 2008, and ADAM II (2009) data show significant
declines in those testing positive for methamphetamine upon
arrest, the NDIC (2008) notes that “the number of ampheta-
mine-related (including methamphetamine-related) admissions
to publicly funded treatment facilities in the New York/New
Jersey Region increased 15 percent overall from 2002 (685) to
2006 (787)”.
Network-Based Population Estimates
Estimation techniques for hidden population sizes using so-
cial network techniques have grown as sociological exposure to
social network analysis has exploded over the last two decades.
Among the most popular of these techniques is Respondent
*Corresponding author.

K. DOMBROWSKI ET AL.
Driven Sampling developed by sociologist Douglas Heckathorn
(1997, 2002, 2007; see recent review of 128 RDS studies by
Johnston et al., 2008). However, RDS does not present overall
population sizes (rather, only population prevalences) and has
recently received some criticism for its base estimation proce-
dures (see Gile et al., 2012 for a summary of those criticisms).
Handcock and Giles’ proposed replacement estimator (the “se-
quential sampling” estimator, see Gile & Handcock, 2010)
relies, however, on an estimate of the total size of the hidden
population—and thus reintroduces a variable that the original
RDS estimators had sought to escape. Given the adoption of
RDS estimation by the World Health Organization (for esti-
mating national rates of HIV and AIDS) and UNAIDS, and
growing interest in using network techniques in determining
overall size estimations of hidden populations, in this paper we
propose a method of network-based capture-recapture popula-
tion estimation that involves only a single sampling round
(rather than the two rounds implied by standard capture-re-
capture techniques) and which can be used to supplement
RDS data collection or more conventional venue based ap-
proaches.
The method proposed below is capable of producing total
population estimates which can be used with the Gile and
Handcock estimator, or as a means for supplementing the
original RDS estimator with a total population estimate for the
group in question. And perhaps most importantly, it does so
while maintaining respondent anonymity, a crucial considera-
tion when dealing with drug using and other illegal or highly
stigmatized behaviors. This factor, taken together with the fact
that the recapture phase takes place simultaneous with the
original capture phase of the sampling, and the easy fit of the
technique with ordinary RDS methods, creates what we feel to
be an important new tool for applied sociological research. To
show an application of the technique in concrete terms, we
demonstrate the development of an estimate for the population
of methamphetamine users in New York City.
This method contrasts with two other network-related at-
tempts to estimate total population size: 1) network scale-up
methods and 2) other capture-recapture methods using multiple
RDS samples. In the words of a recent summary, network
scale-up methods (or NSUM) “rests on the assumption that
people’s social networks—the set of people whom you ‘know’
—are, on average, representative of the general population in
which you live and move” (Bernard et al., 2010: p. ii12). In this
procedure, individual estimates of sub-populations are “scaled”
to aggregate levels, and the estimates of many individuals are
combined. For example, if a respondent answers that he/she
knows two pregnant women out of a total of 100 contacts, we
could estimate the number of pregnant women in his/her county
of 10,000 people (via consistent proportion) to be 200. By
combining this estimate with the estimates drawn from many
others, more accurate figures can be obtained. NSUM advo-
cates see this as a means for estimating the size of sub-popula-
tions that may be known but difficult to enumerate. Still, sig-
nificant problems arise for NSUM methods when trying to
estimate rates of participation in activities that individuals
might try to keep secret even (or especially) from close associ-
ates (see Salganik et al., 2011). Such a situation, obviously,
could occur with any illegal or highly stigmatized activity, such
as illegal drug use.
A second popular method of estimation depends less on in-
formation known to individuals and more on researchers ability
to reach hidden populations repeatedly (by means, for example,
such as successive waves of Respondent Driven Sampling).
According to the logic of capture-recapture studies, successive
samples that discover a proportion of identical individuals can
be used to estimate the total population size by the well-known
Lincoln-Peterson formula (discussed below). Multiple resam-
pling increases the accuracy of these predictions. Where RDS
has proven capable of reaching large samples of hidden popula-
tions, it would appear ideally suited to such tasks. Problems
arise, however, where initial sampling paths can be seen to
affect subsequent referral paths, thus skewing the “recapture”
process to those in the original sample (and resulting in an in-
accurate recapture number, see Berchenko & Frost, 2011).
Given these issues, what seems needed is a process that is less
susceptible to discovery bias around stigmatized behaviors (a
problem for NSUM) and not dependent on resampling proce-
dures that may be biased by initial sampling (as is the issue for
RDS-based capture-recapture methods), and finally, one that is
capable of retaining respondent anonymity throughout the re-
search process. Below we propose such a method.
Estimating the Size of the NYC
Methamphetamine Using Population
In an attempt to estimate the size of the New York City
methamphetamine using population, we have developed a net-
work-based variant of standard capture-recapture methods that
is capable of estimating the total size of a hidden, networked
population from a network sample of current users, even while
maintaining respondent anonymity. The proposed method re-
quires sampling from each respondent’s network connections,
and matching these connections against both the other respon-
dents in the sample and the list of their respective contacts.
Such methods are not particularly complex, and make use of
capture/recapture methods with a long history in both social
and biological sciences. In current circumstances, however,
considerable modifications are required, as network sampling in
the context of illicit and often socially stigmatized activity re-
quires retaining anonymity of both research subjects and their
network connections. These concerns necessarily complicate
the matching of contacts assumed by the capture-recapture
methods. For this reason, a naïve matching strategy of simply
matching the names of respondents and contacts across inter-
views is not possible. We address this challenge by a novel
means of establishing network connections while maintaining
the anonymity of participants and their contacts which we refer
to as the “telefunken method”.
This process requires the recruitment of a sample pool of
network participants and the elicitation of a number of contacts
from each. In addition to personal descriptives later used in the
matching process, each participant was asked for his/her own
“telefunken code”, derived from the last three digits of their
own mobile phone number. To arrive at the code, each of the
three digits is encoded as being either even or odd, and low or
high (with 4.5 being the threshold). Together with height, ap-
proximate weight, hair color, eye color, gender, and race/eth-
nicity, this produced a six bit code for each respondent that
served in matching the respondent to contacts reported by other
246

K. DOMBROWSKI ET AL.
study respondents1. Importantly, the telefunken encoding en-
sures (and assures) that actual telephone numbers of respon-
dents remain unknown to researchers throughout the study. As
will be seen below, a critical question raised by this method is
the estimation of error scores (in the event of false matches
between individuals who by coincidence have the same code)
and error estimation of the resulting population estimate. We
note that these questions would be greatly simplified by attain-
ing a code for more phone number digits. In our case, however,
pre-testing found that asking for more than 3 digits raised sus-
picion among our research subjects and equally importantly,
questions about the assurance of anonymity by our Institutional
Review Board. Given these concerns, a method capable of
producing and bounding an estimate within a range of confi-
dence estimates seems particularly important.
In the current study, respondents were recruited using Re-
spondent Driven Sampling (RDS), an established research
method for anonymously recruiting hard-to-reach populations
(Heckathorn, 1997, 2002, 2007) such as the New York City
methamphetamine user network. This process resulted in the
recruitment of 132 eligible participants, starting from (n = 37)
RDS “seeds” reached using a Craigslist advertisement. Addi-
tional (n = 95) respondents were obtained by referrals via the
standard RDS protocol. Respondent interviews included a
number of use-related questions, and the appearance-based and
demographic information. Further, in addition to their own
personal information and telefunken code, each respondent was
asked to select up to five methamphetamine-using contacts
whose phone number they currently had in their mobile phone’s
directory. This selection was carried out by choosing initial
letters of last names from a randomized list of alphabet letters2.
The respondent was then questioned about the randomly se-
lected contacts, in order to obtain data on the contacts’ personal
characteristics (approximate height, approximate weight, hair
color, eye color, gender, and race/ethnicity) and telefunken code.
For purposes of the population estimate, project respondents
were treated as the “capture” population, while each of the
contacts provided during the interviews (“reports”) was consid-
ered a “recapture assay”. By finding the number of original
respondents discovered via recapture assays (as a proportion of
the total number of assays), researchers had a basis for esti-
mateing the overall size of the population under consideration.
Again, among the main contribution of the proposed method is
that anonymity can be maintained throughout the process, with
personal descriptions and telefunken codes together forming the
sole means of identification and matching.
Capture-recapture methods have been used extensively in es-
timating population levels in biology and epidemiology, and
more recently, employed in conjunction with methods designed
to sample hidden populations of people (Bouchard, 2007; Hope
et al., 2005; Paz-Bailey et al., 2011). At issue in these ap-
proaches is not normally the validity of the standard Lin-
coln-Peterson methodology or its appropriateness to the prob-
lem, but rather the question of whether the original “capture” or
subsequent “recapture” techniques are, in fact, sufficiently ran-
dom (see Berchenko & Frost, 2011 for review and discussion).
This issue is taken up in the discussion, below, but we note here
that one difference between past studies and the method de-
scribed here is that this method does not depend on data from
outside the study (such as arrest numbers or hospital admissions)
to determine either the capture or recapture statistic. Both are
determined simultaneously during the sampling/recruitment
process. Whether this results in an advantage or disadvantage
over capture-recapture methods dependent on external data
sources likely depends on context. Regardless, in this sense the
proposed strategy represents a significant departure from other
uses of capture-recapture in drug use and other research.
The remainder of this paper details the steps involved in two
separate attempts to estimate the methamphetamine using
population in New York City3. As will be seen below, an esti-
mate from the joint population was required due to the small
sample size of the research population. Even with this second
step, the range of estimates is still quite wide. One may con-
clude from this fact that the current method leaves much to be
desired. The “cup half full” interpretation, however, is that the
current method is able to produce a statistically sound method
for population estimation of a hidden population from a rela-
tively small sample, and to do so while maintaining anonymity.
It is this fact that, we feel, makes this method an important new
tool in research on illegal activities where questions of ano-
nymity and the protection of human subjects are paramount.
Baseline Estimate
The population estimate (P) entails a capture/recapture form
of estimation using the respondents (n = 132) to define the
capture population, and matches between the reports (s = 466)
and the respondents to define the recaptured subset. Matches
are defined by considering seven categorical variables: tele-
funken code, gender, race, height, weight, hair color, and eye
color. A respondent from the original sample was said to
“match” a report if the two agreed on all seven of these vari-
ables. With this definition, we found there were 11 matches
between the 466 reports and the 132 respondents4. These 11
matches were used to define the recapture number (t = 11).
Naïve extrapolation from this capture/recapture paradigm using
the Lincoln-Peterson method yields:
ns
Pt

 (1)
where P is the total estimated population, n is the size of the
capture population, t is the recapture number, and s is the num-
ber of recapture assays. Using 11 matches between 466 reports,
and an initial sample of 132 respondents, yields a population
estimate P = 5592. The sections that follow provide successive
refinements to this figure.
3The choice of NYC was not arbitrary. We received a grant to do a popula-
tion estimate of methamphetamine users in New York City (among other
things) from the US National Institute of Justice, and so the necessary data
was collected there. No similar data is available for a similar population in
another city for comparative purposes, nor are other formal estimates for
the size of the NYC meth using population available via other methods.
This significantly limits the comparability of the results and the opportuni-
ties for their verification by other means, though we hope this will be reme-
died in the future.
4The details of the matching procedure, which utilized approximate match-
ing of height, weight, and other continuous variables, is described in the
appendix of the original project report (Wendel et al., 2011).
1For example, the telefunken code for any phone numbers which end in 123
(or 343, or 301) is odd-even-odd-low-low-low, while for phone numbers
ending in 701 (or 523) the code is odd-even-odd-high-low-low. The name
“telefunken” is borrowed from a Frank Zappa song (from the album
J
oe’s
Garage). It is intended to imply “funky telephone” code, as we felt like this
was a good description of the coding method used here.
2Those respondents with five or fewer use-contacts in their mobile phone
directory simply selected all of them without using the randomized alphabet
page.

K. DOMBROWSKI ET AL.
False Matches
The matching technique maintains anonymity of both re-
spondents and reports by considering general characteristics
that are shared by entire segments of the ambient population of
methamphetamine users, but the technique also introduces the
possibility of “false matches” during the matching process. In
particular, a false match occurs whenever a report “matches” a
respondent based on agreement across all seven criteria, but
when the report actually refers to someone outside of our sam-
ple. Indeed, because false matches are possible, we have possi-
bly over-estimated the recapture number (t = 11), and hence the
P = 5592 estimate should be taken as a conservative lower es-
timate of population size.
To further refine the population estimate, it is necessary to
consider the probability distribution governing the number of
matches (amongst the 11 telefunken matches observed) that are
likely to be “false”.
Initial Estimation via Marginals
To estimate the expected number of false matches E[F], we
shall need to refer to the marginal sample distributions of each
categorical variable involved in the matching process (see Ta-
ble 1). We assume that the sample size is large enough so that
its marginals approximate the population marginals. In addition,
in this first attempt at refining the population estimate, we as-
sume that the six categorical variables are independent. We
begin by way of illustrative example. Consider a categorical vari-
able V, say Gender. The possible values assumed by V are known:


12 3
Male, Female, Transgenderxx x 
and associated probabilities are computable from the marginals
in Table 1:
Prob(V = Male) = 119/132
Prob(V = Female) = 11/132
Prob(V = Transgender) = 2/132.
Suppose we choose two individuals at random from an infi-
nite population satisfying the above marginal distribution for
the Gender variable. Since 119 of the 132 respondents were
male, the probability that both individuals in this pair will be
male is (119/132)2, or 0.81 (i.e. about 81% of the time). Simi-
larly, the probability of the two individuals both being female is
(11/132)2 = 0.007, or about 0.7% of the time. Finally, the
probability of the individuals both being transgender is (2/132)2
= 0.0002, a mere 0.02% of the time. The total probability of a
match across the Gender variable is then given by:

222
1191321113221320.82.
Repeating this same calculation we can determine the prob-
ability of agreement between the two individuals for each of the
other variables (race, gender, hair color, eye color, height and
weight). The results are shown in Table 2. Now, assuming
independent sequential assignment of categorical variables, the
probability that two randomly chosen individuals will match on
all six descriptive categorical variables is the product of the
individual probabilities listed in Table 2:
3
0
.3805 0.8198 0.32570.70720.2320 0.22363.7210


Since each telefunken code is 6 bits, there are 26 = 64 distinct
codes, and thus, the probability that two individuals will match
by sheer chance, is given by:
Table 1.
Sample distributions by attribute values.
Attribute
(k) Category (n=)
Race (5) Black/African American (71) Hispanic (26) White (30)
Asian (3) Other (2)
Gender (3)Male (119) Female (11) Trans (2)
Hair (5) Black (65) Brown (21) Blonde (3) Grey/Salt and Pepper
(10) Other (30)
Eye (2) Brown/Dark (109) Blue/Green/Light (21)
Height (5) Below 5’4” (8) 5’4”-5’8” (36) 5’7”-5’11” (45)
5’10”-6’2” (24) Over 6’1” (9)
Weight (5)Below 125-145 (15) 135-165 (41) 155-185 (36) 175-205
(23) Over 195 (13)
Table 2.
Probability of agreement between randomly selected sample members
by attributes.
AttributeSum of the Squares of the Marginals Probability of
Agreement
Race 0.2893 + 0.0388 + 0.0517 + 0.0005 + 0.0002 0.3805
Gender 0.8127 + 0.0069 + 0.0002 0.8198
Hair Color0.2425 + 0.0253 + 0.0005 + 0.0057 + 0.0517 0.3257
Eye Color0.6819 + 0.0253 0.7072
Height 0.0037 + 0.0744 + 0.1162 + 0.0331 + 0.0046 0.2320
Weight 0.0129 + 0.0964 + 0.0743 + 0.0303 + 0.0097 0.2236
35
3.72 101645.8110


 .
For any specific respondent then, the expected the number of
reports (drawn from a population represented accurately by the
sample itself) that would telefunken match by sheer chance is:


52
4665.81102.71 10

 
.
The expected total number of false matches over all (n = 132)
respondents can now be estimated using linearity of expecta-
tion:


2
'1322.71 103.58.F 
The number F' = 3.58 provides an initial estimate of E[F] ≈
F' which takes into account the marginal distributions of the
population from which the sample is drawn (to the extent that
the marginals of the population conform to those of the sample).
Adjusting the recapture number t' = t − F' to incorporate these
findings yields t' = 11 − 3.58 = 7.42 and the revised population
estimate P' = 8290.
Better Estimate via the Joint
The previous estimate of false matches provided a first at-
tempt at correcting for the fact that the number of matches gen-
erally exceeds the true recapture set. Nonetheless, there are
some shortcomings to the false match estimation procedure
described above. In particular, the procedure outlined above
248

K. DOMBROWSKI ET AL.
assumed independent assignment of categorical variables,
where in actuality our sample did not always reflect this as-
sumption, since several variables were clearly not independent
(e.g. height and weight). In more formal terms, the joint prob-
ability of randomly finding someone of African American eth-
nicity with blond hair, for example, was not well-estimated by
the product of probabilities specified in the marginal distribu-
tions of ethnicity and hair color. Indeed, the only property that
one could safely assume to be independent of all others is the
telefunken code.
One approach to the problem of non-independence would be
to establish the relationships among the six attributes used in
the matching process. However, quantifying the dependencies
between the six variables would be daunting. Instead, we chose
to consider all six variables simultaneously using a single joint
distribution across all possible combinations of their values.
Such an approach presented its own difficulties, however. To
describe these issues, it is helpful to define the notion of a class
to be a six-tuple of attribute values (one value for each of the
six variables). Let C denote the set of distinct classes that might
be manifested by study respondents. Examining the categories
listed in Table 1, we see that:
5352553750 classes.C
Although 3750 classes were potentially possible, only 128
classes were actually manifested by the (n = 132) sampled re-
spondents. Thus, the sample provided very little information
about the relative likelihoods of classes under the joint distribu-
tion, since the sample distribution over C was either 0 or 1/132
across almost all classes. The source of this difficulty was due
to having too small a sample to effectively model the joint dis-
tribution, and was this addressed by the best-case-available
remedy of adding the (s = 466) reports to the (n = 132) sample
to obtain a larger “extended sample” of 598 individuals. When
the joint distribution was estimated using this extended sample,
it was found to manifest non-zero probabilities for 290 distinct
classes in C with broad variations in probability mass. For ex-
ample, two classes exhibiting non-zero probability were:
Hispanic, male, black hair, brown eyes, 5’4”-5’8”,
135-165lbs (One of the (n = 132) respondents exhibited
these characteristics)
and
Black, female, black hair, brown eyes, 5’7”-5’11”,
155-185lbs (One of the (s = 466) reports exhibited these
characteristics).
Restated more formally, the joint distribution is defined over
the set of classes ci in C, and the joint probability of an indi-
vidual belonging to class ci, denoted p(ci), can be estimated
using the proportion of individuals in the extended sample that
were found to belong to class ci. To the extent that the distribu-
tion



ii
pc cC
reflects the characteristics of the ambient population, the prob-
ability that two individuals a and b, randomly chosen from an
infinite population, would be found to belong to a particular
class ci is:
 
2.
ii i
pc pcpc
Since class membership is mutually exclusive, the probabil-
ity that a and b would belong to the same class (irrespective of
which particular class), is given by:
 



2
Prob classclassi
cC i
ab p


c (2)
In the specific case of our data on New York City’s
methamphetamine-using population, the expression in Equation
(2) evaluates to 6.21 × 10−3. Multiplying this number by the
probability that a and b will share the same telefunken code
(1/64), yields the probability that two randomly chosen indi-
viduals will match by sheer chance:


3
1 646.21109.710


5
(3)
Applying linearity of expectation, each specific participant
expects:


52
4669.7 104.5210


 
reports (from among the 466) to match him/her by sheer chance.
Linearity of expectation applied once more yields the total
number of matches between the (n = 132) respondents and the
(s = 466) reports that are attributable to sheer chance:


2
1324.52105.97.F  
The number F′′ provides a more refined estimate of E[F] ≈
F′′, since it takes into account the joint distribution of the am-
bient population from which the sample was drawn (to the ex-
tent that the distribution of attributes in the population con-
forms to that of the extended sample). Adjusting the recapture
number t′′ = t − F′′ to incorporate this more refined analysis of
the expected false matches, yields t′′ = 11 − 5.97 = 5.03, from
which we derive the revised population estimate of P′′ =
12,229.
Range of Estimates
Developing a range of plausible population estimates re-
quires moving beyond the study of expected values (i.e. E[F] ),
to acquire a deeper understanding of the probability distribution
governing the number of false matches F. We begin by noting
that F represents the number of successes in a Bernoulli se-
quence of 132 × 466 = 61,512 trials—or 466 throws at 132
possible hits per throw—where the probability of success in
any given trial is 9.7 × 10−5 (see Equation (3)). The standard
deviation of F is thus given by a well-known fact concerning
Bernoulli distributions:





55
61,5129.7 1019.7 102.44std F 
This standard deviation can be used as a measure of the
variability of F.
Population estimates based on the expected number of false
matches should be seen as the midpoint of a range of estimates.
Our estimate F′′ can be better adjusted to incorporate this vari-
ability




5.97 2.44.EF FstdF
The population estimate corresponding to 5.97 + 2.44 = 8.41
false matches is:


=132 466118.4123,750P.
while considering 5.97 − 2.44 = 3.53 false matches yields:

K. DOMBROWSKI ET AL.


=132466 113.538235P.
By considering one standard deviation of the random vari-
able F around its estimated mean, we obtain a range of popula-
tion estimates [8235, 23,750].
Confidence Intervals
To obtain confidence intervals for population estimates we
use the Chernoff bound for the upper and lower tail of the dis-
tribution:
  



δ
1δ
Pr1 δ
1δ
.
E
F
e
FEF 


 



  



δ
1δ
Pr1 δ
1δ
E
F
e
FEF




 



.
Using the previous F′′ estimate of E[F], the upper and lower
bounds corresponding to these two equations are listed in the
Table 3. As is evident from the table, one needs to expand to
fairly wide estimates around F′′ in order for the upper and
lower bound confidence values to equalize, e.g., by considering
the number of false matches F to lie between 3 (60%) and 9
(49%). This analysis indicates considerable sensitivity to false
match frequencies, a result that is perhaps not surprising given
the value of std(F).
As such, the P′′ = 12,229 estimate based on F′′ = 5.97 should
be taken as a central value with a fairly wide range, with the
understanding that the actual population size could be as high
as 30,756 (if there were 9 false matches among the 11), or as
low as 7689 (if there were only 3 false matches among the 11).
Discussion
Perhaps more interesting than the actual methampheta-
mine-using population estimates themselves, however, is the
Table 3.
Population estimates by confidence intervals.
Upper bound
on false
matches (k)
Upper bound
on Prob.
(F < k)
Lower
bound on #
of true
matches
Lower
bound on
population
size
Bound
confiden ce
1 0.04 10 6151 0.96
2 0.17 9 6835 0.83
3 0.40 8 7689 0.60
4 0.69 7 8787 0.31
5 0.92 6 10,252 0.08
Lower
bound on
false
matches (k)
Upper bound
on Prob.
(F > k)
Upper bound
on # of true
matches
Upper bound
on popula-
tion size
Bound
confiden ce
7 0.92 4 15,378 0.08
8 0.73 3 20,504 0.27
9 0.51 2 30,756 0.49
10 0.32 1 61,512 0.86
estimation method. Capture-recapture techniques have retained
an important place in socio-medical studies (e.g. Chao et al.,
2001; Kruse et al., 2003; Hall et al., 2006; Vuylsteke, 2010),
despite acknowledgment of long standing limitations (Hook &
Regal, 1995). Few of these methods have involved social net-
work data, however, with recent network attention focused on
scale-up methods, as discussed by Kadushin et al., (2006),
McCormick et al., (2010) and Bernard et al., (2010). The
method discussed in this paper is not a substitute for large scale
estimation of the sort addressed by scale-up methods, but it
does take steps toward alleviating the largest problems associ-
ated with traditional capture-recapture techniques: the need for
two distinct samplings of the population (see Laska & Meisner,
1993 for discussion), and the need for subject anonymity
throughout the matching process when dealing with illegal or
highly stigmatized behaviors (see Hook & Regal, 1995). Be-
cause our method depends on data captured during a single
survey and involves (what we feel to be) a reliable way to rec-
ognize matches while maintaining anonymity, as well as means
for estimating the number of false matches, it addresses tradi-
tional problems associated with capture-recapture techniques
for population estimates of illegal drug users.
We note, however, that the method described here assumes
that the researcher has access to the hidden population, though
not complete access, and that this access is capable of produc-
ing a representative sample5. The latter is perhaps the most
problematic of these assumptions, and we recognize the diffi-
culty of establishing, rather than simply assuming representa-
tiveness. Nevertheless, where population estimates of specific
local subpopulations are sought, the method described here
avoids complex issues such as determination of degree distribu-
tions of the population from which contact information is gath-
ered, so-called “transmission errors”, barrier effects, and recall
error (as discussed by McCormick et al., 2010).
Obvious limitations contextualize these results. The most
obvious of these is the representativeness of the sample to the
larger population from which it is drawn, which is a fundamen-
tal assumption for both estimates, and one that rests on shaky
ground. This was a small sample by RDS standards, and as
such it is very likely that sample equilibrium had not been
reached, and that sample skewing as a result of seed selection,
volunteerism, and other peer-driven pitfalls affected the repre-
sentativeness of the 132 recruits, and perhaps the 466 reports as
well (the latter is equally important because the reports were
used to estimate the space of variability of the ambient popula-
tion in the second estimate as well). RDS recruitment methods
also generally tend to enroll higher-than-representative numbers
of well-connected individuals, simply by virtue of the fact that
they have more chances to be recruited, which could skew the
results should the ego-networks of these well-connected indi-
viduals differ from those of the remainder of the population in
significant ways, i.e. ways that affect the demographics of the
sample connections (see Berchenko & Frost, 2011). And fi-
5Ideally, one would like to begin the matching procedure from a random
sample of the population of interest. As has been clear from the beginning
of the paper, however, the method proposed here is intended for situations
where this is not possible. Inevitably, this means that we begin with some-
thing that is less than a random sample, but something more than a simple
convenience sample (as the RDS method does provide some semblance of a
random walk in the referral process, and means to estimate the limits of that
randomness). As no current alternative exists for this situation, this remains
an explicit and acknowledged limit of the method here, but one for which
we currently do not have any alternative.
250

K. DOMBROWSKI ET AL.
nally, the use of peer referrals forces us to wonder whether the
number matches discovered here (t = 11) were a result of the
fact that recruits were drawn from a closely connected seg-
ments of the larger population, leading to a greater likelihood
that individuals knew one another by virtue of being part of the
same social clique (and thus lowering the estimated population
figure). Given that both estimates assumed that the respondents
had been chosen randomly from the population, such consid-
erations cast doubt on the validity of the final estimate, which is
likely larger than the figures given here6.
Nevertheless, the methods described here are in no way de-
pendent on RDS as a method of recruitment, and may in fact be
better suited to other methods (venue-based sampling or other
techniques used to recruit hard-to-reach populations). In such
cases, the likelihood that matches are the result of over-re-
cruitment among a quasi-clique of well-connected respondents
remains an open question as well. Still, with the growing popu-
larity of mobile phones all over the world, the possibility of
telefunken encoding as a means of anonymously matching
network alters is rapidly expanding. In that case, the ano-
nymized identification method of encoding phone numbers
(even/odd, 0-4/5-9) as unique identifiers can potentially remedy
one of the more difficult questions about how to expand
ego-network data to larger chains of sociometric connection. As
such, there may be potential for the extension of this method to
other hard-to-reach populations, or to any population where
network connections are a concern but where the solicitation of
connection via name is not possible. Perhaps as importantly,
this technique has the special virtue of deriving an estimate
while retaining respondent anonymity and the anonymity of
network alters, a frequent requirement of human subject protec-
tion and a common difficulty in attempting to link ego-data
information gained in individual interviews into a larger net-
work whole.
Acknowledgements
This project was supported by Award No. 2007-NIJ-CX-
0110 from the National Institute of Justice, Office of Justice
Programs, U.S. Department of Justice. The opinions, findings,
and conclusions or recommendations expressed in this publica-
tion are those of the authors and do not necessarily reflect those
of the U.S. Department of Justice. See Wendel et al., 2011 for
an expanded discussion of the research project from which the
data for this analysis were taken.
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