Sampling and Estimation in Hidden Populations.pdf

5
SAMPLING AND ESTIMATION IN
HIDDEN POPULATIONS USING
RESPONDENT-DRIVEN SAMPLING
Matthew J. Salganik*
Douglas D. Heckathorny
Standard statistical methods often provide no way to make
accurate estimates about the characteristics of hidden popula-
tions such as injection drug users, the homeless, and artists. In
this paper, we further develop a sampling and estimation techni-
que called respondent-driven sampling, which allows research-
ers to make asymptotically unbiased estimates about these
hidden populations. The sample is selected with a snowball-type
This research was made possible by grants from the Centers for Disease
Control and Prevention (U62/CCU114816-01), the National Institute on Drug Abuse
(RO1 DA08014), and the National Endowment for the Arts. For MJS this material is
based on work supported under a National Science Foundation Graduate Research
Fellowship, Cornell’s IGERT program in nonlinear systems funded by NSF Grant
DGE-9870681, and a Fulbright Fellowship with support from the Netherland-
America Foundation, which allowed him to spend the year at the ICS/Sociology
department at the University of Groningen, where most of this paper was written. We
would like to thank Marijtje van Duijn, Henk Flap, Andrew Gelman, Steve Morgan,
Michael Schweinberger, Tom Snijders, and Frans Stokman for valuable comments on
earlier versions of this paper. Peter Dodds and Steve Thompson provided helpful
conversation. Some parts of this paper were presented at the 2003 Sunbelt
International Social Network Conference, 2003 American Sociological Association
Annual Meeting, and the 2003 Joint Statistical Meetings.
Direct correspondence to Matthew J. Salganik at <mjs2105@columbia.
edu> and Douglas D. Heckathorn at <douglas.heckathorn@cornell.edu>.
*Columbia University
yCornell University
193

design that can be done more cheaply, quickly, and easily than
other methods currently in use. Further, we can show that under
certain specified (and quite general) conditions, our estimates
for the percentage of the population with a specific trait are
asymptotically unbiased. We further show that these estimates
are asymptotically unbiased no matter how the seeds are
selected. We conclude with a comparison of respondent-driven
samples of jazz musicians in New York and San Francisco, with
corresponding institutional samples of jazz musicians from these
cities. The results show that some standard methods for studying
hidden populations can produce misleading results.
1. INTRODUCTION
The problem of collecting accurate information about the behavior
and composition of social groups arises in many areas of research. In
most cases, standard sampling and estimation techniques, developed
over the past 70 years, provide a means for collecting such informa-
tion. However, there are a number of important groups for which
these techniques are not applicable.
For example, injection drug users and men who have sex with men
are populations of great interest to researchers because the behavior of
these groups affects the spread of HIV/AIDS and other diseases.
Unfortunately, standard sampling and estimation techniques cannot be
used on these populations. A recent report by the World Health
Organization cited inability to monitor the behavior and HIV seropre-
valence of at-risk subpopulations, like injection drug users and men who
have sex with men, as a major weakness in current HIV prevention
efforts (World Health Organization, 2000). This limitation of current
statistical methodology has received attention not just from public health
researchers, but also from statisticians and sociologists.
Standard sampling and estimation techniques require the
researcher to select sample members with a known probability of
selection. In most cases this requirement means that researchers must
have a sampling frame, a list of all members in the population. However,
for many populations of interest such a list does not exist.
A researcher wishing to study a population without a sampling
frame could attempt to construct such a frame. However, for a
number of populations this frame construction is made impractical
or impossible by, first, the small size of the target population and,
194 SALGANIK AND HECKATHORN

second, the difficulty of locating members of the target population.
This difficulty could be caused by the sensitive nature of the behaviors
in the population (for example, injection drug users) or simply
because members of the target population are difficult to distinguish
from members of the general population (for example, jazz musi-
cians). These special populations that cannot be studied using
standard sampling and estimation techniques are called hidden popula-
tions. A short list of examples includes injection drug users, men who
have sex with men, artists, commercial sex workers, illegal immigrants,
participants in some social movements, draft resisters, and the homeless.
Inability to collect information about hidden populations has
complicated existing studies and forced researchers to focus on other
problems. Imagine a researcher who wishes to begin a study of, for
example, injection drug users. This researcher is immediately faced
with a huge problem—how to get a sample from the hidden popula-
tion from which one can generalize.
A first thought might be to attempt to construct a sampling
frame and then, once the frame is complete, select people with known
probability of selection from the frame. This approach brings the
problem back under the purview of standard sampling and estima-
tion. However, for reasons discussed earlier, frame construction is
extremely expensive and probably impossible—imagine trying to
make a list of all the injection drug users in a large city.
Another approach would be to reach a large number of people
via random digit dialing and then screen them for membership in the
hidden population. Again, this approach is extremely costly and
potentially very inaccurate. For example, if 1 percent of a city is a
member of the hidden population it would take approximately 50,000
screening interviews to yield a sample of size 500. Further, it is unlikely
that the resulting sample would be a simple random sample from the
hidden population because many members are not reachable by phone,
or would not reveal their behavior to an unknown interviewer over the
phone. The nature and the magnitude of these biases would be
unknown, and thus there would be no way to generalize from this
type of sample to the entire hidden population.
A more efficient way to collect information about the behavior
and composition of hidden populations is to take a sample of target
population members in an institutional setting—for example, injection
drug users in a drug rehabilitation program. This can provide researchers
RESPONDENT-DRIVEN SAMPLING 195

with valuable information, but since the members of the population who
enter institutional settings are a nonrandom sample from the hidden
population, it is impossible to use samples from institutional settings to
make accurate estimates about the entire hidden population. For
example, Watters and Cheng (1987) found that in San Francisco,
injection drug users outside of drug treatment programs were twice as
likely to be infected with HIV as injectors in treatment programs.
Because of the problems with these three approaches, they are
rarely used. The two most common approaches, targeted sampling
and time-space sampling, will be discussed more thoroughly in Sec-
tion 2. These two methods, as well as those previously discussed, often
fail to provide researchers with a way of making accurate estimates
about a hidden population based on a sample. These methods also
have another common feature: they implicitly treat members of the
population as discrete, atomized units. By doing so, these methods fail to
make use of an important feature of many hidden populations—they are
made of real people connected in a network of relationships.
Switching to this network perspective gives us a fresh and novel
approach to the study of hidden populations. Using the extra infor-
mation that is available in the social network allows one to design a
sampling and estimation scheme that, in many cases, is both cheaper
and more accurate than existing methods commonly in use.
This new sampling and estimation method, called respondent-
driven sampling, is a variation of chain-referral sampling methods that
were first introduced by Coleman (1958) under the name snowball
sampling.1
The basic idea behind these methods is that respondents are
selected not from a sampling frame but from the friendship network of
existing members of the sample. The sampling process begins when the
researchers select a small number of seeds who are the first
people to participate in the study. These seeds then recruit others to
participateinthestudy.Thisprocessofexistingsamplemembersrecruiting
future sample members continues until the desired sample size is reached.
Experience with chain-referral methods has shown them to be
effective at penetrating hidden populations. However, researchers also
correctly realized that the promise of chain-referral methods was tempered
by the difficulty of making statistical inferences from this type of sample.
1
Other variations of snowball sampling have also appeared under the
names link-tracing sampling and random-walk sampling.

This difficulty occurs because chain-referral methods produce
samples that are not even close to simple random samples. For example,
in a simple random sample all people have the same probability of
selection. However, in chain-referral samples this is far from the
case. Since people recruit their friends, those with many friends are
more likely to be included in the sample than social isolates.
Another main concern among researchers centers around the
choice of seeds (the first people to be included in the sample). Since all
people in the sample are indirectly recruited by the seeds, researchers
believed that any small bias in selecting the seeds would be com-
pounded in unknown ways as the sampling process continued.
Both of these specific problems occur because chain-referral
designs fall outside of the realm of traditional probability sampling,
where units are selected from a sampling frame with known prob-
ability of selection. Instead, because of a lack of sampling frame and
unknown probability of selection, chain-referral samples have been
considered to be nonprobability or convenience samples ‘‘which can
only be assessed by subjective evaluation’’ (Kalton 1983).
Because of the nature of the chain-referral samples, numerous
researchers have questioned the estimates that can be drawn from them
(Welch 1975; Erickson 1979; Kalton 1983; Kalton and Anderson 1986;
Berg 1988; Spreen 1992; Friedman 1995; Eland-Goosensen et al.,
1997). This research is fairly well summarized by Berg (1988) when
he writes, ‘‘as a rule, a snowball sample will be strongly biased toward
inclusion of those who have many interrelationships with or are
coupled to, a large number of individuals. In the absence of knowl-
edge of individual inclusion probabilities in different waves of the
snowball sample, unbiased estimation is not possible.’’ It would be
fair to say that the conventional wisdom among sociologists, public
health researchers, and statisticians is that chain-referral sampling
holds great promise for a number of problems, especially the study
of hidden populations, but that it is so hopelessly biased that it cannot
be used to make reliable estimates.
We believe that previous researchers have been overly pessi-
mistic about chain-referral samples. In this paper we will show that it
is possible to make unbiased estimates about hidden populations from
these types of samples. Further, we will show that these estimates are
asymptotically unbiased no matter how the seeds are selected.

2. EXISTING METHODS FOR STUDYING HIDDEN
POPULATIONS
Most studies about the characteristics of hidden populations use
either targeted sampling or time-space sampling.2
These approaches
differ in their strengths and weaknesses, but one problem that they
both share is that, for many hidden populations, they allow no
systematic or principled way to use the information collected to
make inferences about the population from which they are drawn.
In targeted sampling (Watters and Biernacki 1989) researchers
use a number of different outreach techniques to attract a sample of
people in the hidden population. This technique is also sometimes
called ‘‘street outreach’’ because it generally involves sending field-
workers into to the streets to find, and recruit, members of the hidden
population. Some of these studies also make use of some type of
chain-referral technique to recruit additional people into the sample.
Targeted sampling does succeed in giving researchers access to
a large sample of noninstitutional members of the population. How-
ever, the targeted sample is clearly not a random sample where all
people have the same probability of selection. For example, in studies
of injection drug users, safety concerns often require that recruitment
occur only during the day when most drug scenes are less active. Also,
injection drug users who do not congregate in public are almost
certain to be missed in the sampling process. There is no way to
know the magnitude of these selection biases, so it is not possible to
generalize from the sample to the target population.
A more refined alternative to targeted sampling is time-
space sampling (Muhir et al. 2001). Under this procedure,
ethnographic fieldwork is done to construct a sampling frame
identifying times when members of the target population gather
at specific locations—for example, Tuesday afternoon from 2 PM
2
There have also been a number of approaches to estimating the size of
hidden populations. This research has shown that estimating the size of a
population can be different from estimating its behavior and characteristics. Some
examples are based on snowball sampling (Frank 1979; Frank and Snijders 1994;
Dávid and Snijders 2002), network scale-up designs (Killworth et al. 1998a, b),
multiplicity sampling (Sirken 1970), and capture-recapture (Sudman, Sirken, and
Cowan 1988; Hogan 1993; Heckathorn et al. 2002; Heckathorn and Jeffri 2003).

to 6 PM at a specific park. These specific venue-day-time segments
are the primary sampling units. These units are randomly selected,
in some cases with probabilities based on the expected sample
yield at the location, and members of the target population enter-
ing the venue are intercepted and interviewed.
Because the venue-day-time segments are sampled with a
known probability, it is possible to use the sample to make statistical
inferences about the population that attends the identified venues.
Unfortunately, all venues are not accessible. For example, some
studies do not sample from venues with low expected sample yields
because it is prohibitively costly (Stueve et al. 2001). Additionally,
private venues are generally not accessible to the researchers and in
some studies safety concerns further limit the choice of venues (see
Kanouse et al. [1999] for an example).
Time-space sampling produces probability samples of the
population that attend venue-day-time segments, which are accessible
to researchers. However, in some situations this venue-attending
population differs in unknown ways from the true population of
interest. For example, drug injectors who appear in public places
accessible to researchers probably are not representative of all drug
injectors.3
These coverage problems introduce unknown bias into the
estimates, a bias that could be substantial in some situations and
minimal in others depending on the population under study.
In summary, most current studies of hidden populations
require researchers to expend a lot of effort to collect a sample from
which they can not generalize to the population of interest. Instead,
they present summary statistics about the sample and then leave
interpretation to the reader. As we will see in Section 9, this approach
can lead people to make extremely misleading conclusions. Clearly,
we would prefer a way to collect a sample from which it is possible to
accurately generalize to the population.
3
One potential solution to this coverage problem is to change the
definition of the population of interest. For example, we could study brothel-
based sex workers if all brothels are accessible for researchers. However, if there
are differences between the brothel-based sex workers and nonbrothel-based sex
workers, it must be made clear that the estimates refer only to brothel-based sex
workers and not sex workers in general.

3. A NEW APPROACH TO THE STUDY OF HIDDEN
POPULATIONS
In this paper we develop a new approach to studying hidden popula-
tions that is based on a technique called respondent-driven sampling
(Heckathorn 1997, 2002). In respondent-driven sampling, a sample is
collected using a chain-referral procedure. That is, respondents are
selected not from a sampling frame but from the social network of
existing members of the sample.
Unlike a conventional probability sampling design, where each
unit has a known and constant probability of selection, respondent-
driven sampling is based on an adaptive sampling design where the
selection procedure is affected by the realized network in the popula-
tion (Thompson and Seber 1996; Thompson and Frank 2000).
Therefore, special estimation procedures must be used.
Rather than attempting to directly estimate from the sample to
the population, as in traditional sampling and estimation, as shown in
Figure 1(a), respondent-driven sampling uses an indirect method. First,
the sample is used to make estimates about the social network connect-
ing the population. Then, this information about the social network is
Sample
Collection Estimation
Population
(a)
Social Network
Sample
Population
Estimation
Estimation
Collection
(b)
Schematic of traditional
sampling and estimation
Schematic of respondent-driven
sampling
FIGURE 1. Schematic representations showing the differences between
traditional sampling and estimation and respondent-driven
sampling. By not attempting to estimate directly to from the
sample to the population, respondent-driven sampling avoids many
of the well-known problems with estimation from a chain-referral
sample.

used to derive the proportion of the population in different groups (for
example, HIVþ or HIV). This process is illustrated in Figure 1(b).
To explain how respondent-driven sampling works, in Section
4 we present the method for using information about the social
network to make estimates about the characteristics of the popula-
tion. In Section 5, we then describe how to collect a sample that can
be used to estimate the necessary information about the social
network. Section 6 presents the estimation techniques and conditions
under which they are unbiased. These analytic results are supported
by simulation results presented in Section 7. In Section 8 we present
results showing that the estimates are asymptotically unbiased no
matter how the seeds are selected. We conclude in Section 9 with
some real data from a study of jazz musicians that shows that institu-
tional samples and naive snowball samples can produce misleading
results.
4. USING THE SOCIAL NETWORK TO MAKE ESTIMATES
ABOUT POPULATION PROPORTIONS
The key idea behind the estimation procedure is that the estimates do
not come from the sample proportions. Rather, the sample is used to
make estimates about the network connecting the population. Then,
using information about this network, we can derive the population
proportion in different groups. By not attempting to estimate directly
from the sample to the population, we avoid many of the well-known
problems with chain-referral samples. To estimate from the social
network to the population, we will use the reciprocity model that
was introduced in Heckathorn (2002).
Consider a hypothetical population such as that shown in
Figure 2(a). The population is made up of two groups of people (for
example, HIVþ and HIV), and we would like to estimate the
proportion of the population in each of these groups. By redrawing
the same population in a different way, as shown in Figure 2(b), we
are able to notice that the number of ties from someone in group A to
someone in group B, which in this example is 6, is the same as the
number of ties from someone in B to someone in group A. This
statement may seem quite trivial, but it turns out to be very useful

because there are two different ways to calculate this number of cross-
group ties. First, we must begin with some notation.
We can store all the information about the social network in an
adjacency matrix X. That is, xij ¼ 1 if there is a directed edge between
person i and j, otherwise xij ¼ 0. In this paper we will consider only
reciprocal relations, so if it is the case that xij ¼ 1, then it is also the
case that xji ¼ 1. For convenience, we will call these relationships
friendships, although other relationships are possible.
An important property of a person is the number of friends
that he or she has. We define the degree of a person i, di, to be the
number of friendships that involve person i, di ¼ j xij. The total
number of friendships radiating from people in group A, RA, is the
sum of the degree of all people in group A and is defined as
RA ¼
X
i2A
di ¼ NA DA; ð1Þ
(a) (b)
A visual representation of a
hypothetical population
An alternative representation
of the same population
FIGURE 2. Two different representations of the same hypothetical population
made up of two types of people—circles ( ) and diamonds ( ). In
(b) the population is drawn to emphasize the difference between
within-group friendships (represented by dotted lines) and between-
group friendships (represented by solid lines). Using just
information about the network structure and equations (8) and
(9), we are able to correctly estimate that 60 percent of the
population are circles and 40 percent are diamonds.

where NA is the number of people in group A and DA is the average
degree of people in group A.
Now consider, for a given network X, the probability that if we
follow a randomly chosen friendship beginning with a person in group
A that we cross groups and end up at someone in group B. We can
define this probability, CA,B, as
CA;B ¼
TAB
RA
; ð2Þ
where TAB is the number of ties that contain a person in group A and
a person in group B.
Since we are considering only reciprocal ties, we know that the
number of relationships from group A to group B, in this example 6, is
the same as the number of relationships from group B to group A. We
can calculate that number two different ways: (1) the number of friend-
ships radiating from group A, RA, times the probability that one of those
relationships will lead to someone in group B, CA,B or (2) the number of
friendships radiating from group B, RB, times the probability that one of
those relationships will involve someone from group A, CB,A. That is,
RA CA;B ¼ TAB ð3Þ
RB CB;A ¼ TAB: ð4Þ
Setting equations (3) and (4) equal to each other and using the
definition of RA and RB, we can write
NA DA CA;B ¼ NB DB CB;A: ð5Þ
Note that equation (5) brings together both information about
the characteristics of the nodes and characteristics of the network.
Thus, we can begin to see how we can infer properties of the nodes
from information about the network.
However, even if we had complete information about the social
network—that is, if we knew DA, DB, CA,B, and CB,A—we still have
one equation with two unknowns—NA and NB, the sizes of the
population in group A and group B. If we divide both sides of
equation (5) by N, the total size of the population, then we can rewrite

equation (5) in terms of population proportions, PPA and PPB. This
allows us to add a second constraint—that the sum of the population
proportions must be 1. So now we have
PPA DA CA;B ¼ PPB DB CB;A ð6Þ
PPA þ PPB ¼ 1; ð7Þ
where PPA is the proportion of the population in group A and PPB is
the proportion of the population in group B.
Now we have a system with two equations and two unknowns.
Using ordinary algebra, we can derive that
PPA ¼
DB CB;A
DA CA;B þ DB CB;A
ð8Þ
PPB ¼
DA CA;B
DA CA;B þ DB CB;A
: ð9Þ
Examination of equations (8) and (9) reveals that we can
recover the population proportions, PPA and PPB, with knowledge
only of the network structure connecting the population.
Again we can return to the hypothetical population in Figure 2.
Using the values D ¼ 4, D ¼ 3, C , ¼ 0.25, and C , ¼ 0.5 as well
as equations (8) and (9), we can correctly estimate that 60 percent of
the population are circles and 40 percent are diamonds. It is import-
ant to note that these equations are true for any network structure
that contains only reciprocal relations.
We can see now that it is possible to estimate the proportion of
the population in group A and group B, but only if we know some
other information about the network connecting people in these
groups. Next we will discuss methods for collecting a sample that
can be used to estimate this network information.
5. COLLECTING THE SAMPLE
A respondent-driven sample is collected via a chain-referral design.
These designs were originally introduced by Coleman (1958), and

later elaborated by Goodman (1961) in order to study characteristics
of social networks. Work by Snijders (1992) has further reinforced the
point that chain-referral designs are more appropriate for inference
about the structure of the network than the characteristics of the
people in the network. However, much confusion has arisen because
people have ignored this advice and attempted to use chain-referral
samples to make estimates directly about the population, and not the
network connecting the populations.
To conduct a respondent-driven sample, one begins by select-
ing a set of s initial seeds that are chosen based on preexisting contact
with the study population. These seeds are paid to be interviewed and
form wave 0 of the sample.
Interviews for these studies can be organized in a number of ways
depending on the target population. A study of injection drug users
conducted interviews in a storefront office in a location accessible to the
target population (Heckathorn 2002). However, in a study of jazz musi-
cians, interviewers traveled to locations convenient to the respondents
(Heckathorn and Jeffri 2001). In some cases telephone interviews may be
preferred, as in a study of Vietnam War draft resisters (Hagan 2001).
No matter how the interview is conducted, each seed in wave 0
is supplied with c unique recruitment coupons similar to the one in
Figure 3. Subjects are told to give these coupons to other people they
know in the target population. Because each coupon is unique, it can
be used to trace the recruitment patterns in the population. When a
new member of the target population participates in the study, the
recruiter of that person is paid an additional bonus. Thus, subjects are
paid to participate and to recruit others.
FIGURE 3. An example of a recruitment coupon given to subjects.

The recruits of people in wave 0 form wave 1. Upon com-
pletion of the screening, a person from wave 1 is provided with c
recruitment coupons and the process continues. The sampling con-
tinues in this way until the desired sample size is reached.
The choice of the number of recruitment coupons given to each
subject, c, depends on the expected recruitment behavior of the people
in the sample. For the sampling process to continue, each participant
must recruit on average at least one new participant. So, the choice of
c should be large enough so that the recruitment continues even if
some subjects choose not to recruit. However, it is also desirable to
have a small c because it means that the sample can go through many
waves before the desired sample size is reached.
If the seeds are not drawn as desired (as explained in Section
6.1), many sampling waves are preferable because they allow the
sampling process to explore parts of the network that may have had
a zero probability of being included as a seed. These long chains are
important because all members of the hidden population must have a
nonzero chance of being included in the estimation procedure. Work
on the ‘‘small-world’’ problem (Watts and Strogatz 1998; Watts 1999;
Dodds et al., 2003) seems to indicate that in most networks the
average path length between any two people is often quite short.
Thus, we can be fairly confident that with reasonably long recruit-
ment chains all members of the population have a nonzero probability
of being reached even if the seeds occupy an unusual place in the
network structure. Also, in the case where the seeds are not selected as
desired, many recruitment waves are preferable because they allow the
sampling process to converge to an equilibrium (as we will see in
Section 8).
In other chain-referral type sampling techniques—for example,
the random-walk design by Klovdahl (1989)—respondents are asked
to list the people they know in the population under study, and then
the researcher uses a randomization device to select the people to be
included in the next wave of the sample. While appropriate for
Klovdahl’s study of the social network of a large city, there are
problems with this approach to sampling in hidden populations.
First, since the populations under study are often subject to
social stigma and/or criminal prosecution, respondents are often
hesitant to give researchers information about their colleagues in the
population. This bias, sometimes called masking (Erickson 1979),

can cause respondents to provide inaccurate information to research-
ers leading to unknown biases in the sample selection process.
Additionally, once the next member of the sample is selected via the
randomization device, it is often difficult to locate the person so that
they can be interviewed (McGrady et al. 1995; Eland-Goosensen et al.,
1997). Finally, this procedure is sometimes forbidden in the Uni-
ted States because, by requiring respondents to divulge sensitive
information about their peers, it violates federal guidelines for the
protection of human subjects (Heckathorn et al., 2002).
Respondent-driven sampling solves these problems by allowing
the subjects to do the recruitment. Thus, participants are not required
to divulge any sensitive information to the researcher, and the
researcher does not have to spend time looking for the named recruitee.
One concern with having the subjects do the recruitment is that it
could introduce unknown bias into the sampling process if the recruit-
ments are not random. However, there is some evidence that is con-
sistent with the idea that respondents recruit randomly from their
friends (Heckathorn et al. 2002).
In addition to the substantive information a researcher desires
to collect, two additional pieces of information must be collected from
each participant. These additional pieces of information are needed to
implement the estimation procedure.
First, the researcher must collect the degree of each person in
the sample.4
Second, the researcher must use the coupons to record
the recruitment patterns so that each sample member can be linked to
the person who recruited them. These two pieces of information, the
self-reported degree and the observed recruitment patterns, will be
critical for the estimation procedures that will be described more fully
in Section 6.
There are two additional important steps in the sample selec-
tion process: nonduplication and population membership verification.
It is a common problem in studies of drug injectors that some respond-
ents attempt to participate in the study multiple times in order to earn
additional money (Biernacki and Waldorf 1981). This duplication
can affect the quality of the data and thus accuracy of the estimates.
4
It is known that there are a number of difficulties in collecting the
degree of a person accurately (McCarty et al. 2001). This source of nonsampling
error is not considered in this paper.

It is an important, but difficult, problem to ensure that each person
can participate only once. Because the respondents in many hidden
populations may be unwilling or unable to provide photographic
identification, some other methods must be used to prevent
duplication. Previous researchers have had success using biometric meas-
urements that are not threatening to the respondents (Heckathorn,
Broadhead, and Sergeyev 2001).
Another important step in ensuring the quality of the sample
data is to verify that sample members are indeed members of the target
population. Again, there is a possibility that people who are not in the
population under study may attempt to participate in the study for
financial reasons. Methods of population verification must be tailored
to the specific population under study. For example, urinalysis can be
used to verify recent use of certain drugs (Kral et al. 1998).
This sampling methodology was designed to be implementable
in the real world and does not require an unrealistic amount of
funding, time, or cooperation by subjects. Because the design is
relatively simple and robust, it has already been used successfully in
a number of studies. Also, a recent review article by Semaan, Lauby,
and Liebman (2002) found that respondent-driven sampling is
cheaper, quicker, and easier to implement than other methods that
have been used for sampling hidden populations in evaluations of
HIV risk-reduction interventions.
At the end of the sampling procedure, the researcher has a
sample that consists of people who are seeds, people who are not
seeds, and a set of recruitments. In the next section, we will discuss
how to use the information in the sample to make estimates about the
social network. This information about the social network will be
used to make estimates about the population.
6. USING THE SAMPLE TO MAKE ESTIMATES ABOUT THE
SOCIAL NETWORK
Once we have selected a sample, we must then have a procedure to use
information from that sample to make estimates about the social net-
work from which it was drawn. Previous work on estimating propri-
eties of networks have often required a random sample of nodes (Frank
1981). However, it is not possible to collect such a sample from a

hidden population. This impossibility forces us to make estimates from
a chain-referral sample, which is notoriously difficult.
6.1. Assumptions
In order to make estimates from a respondent-driven sample, we must
first assume some things about the population under study and the
way that the recruitment occurs. By making these assumptions expli-
cit, we allow for them to be tested and for research to be done about
the nature of the bias introduced by their violation.
It is helpful to think of the sample selection as a process that
alternates between the selection of nodes and the selection of edges.
That is, nodes are first drawn to form wave 0 of the sample. Then
these nodes choose edges that define recruitment period 1. The edges
in recruitment period 1 determine the nodes drawn in wave 1. The
process continues in this way with nodes selecting edges that in turn
select nodes until the desired sample size is reached.
To make our assumptions more clear, we must first develop
some notation. We begin by setting up two indicator functions: one
for the nodes, NI(j)w¼x, which tells us whether a given node, j, is
selected in wave x and one for the directed edges, EI(ej!k)r¼x, which
tells us whether a given directed edge ej!k is selected in recruitment
period x. These indicator functions are defined as follows:
NIðjÞw¼x ¼
1 if node j is selected in wave x
0 otherwise
n
ð10Þ
EIðej!kÞr¼x ¼
1 if edge ej!k is selected in recruitment x
0 otherwise
n
ð11Þ
First, throughout this paper we will consider only the case of
sampling with replacement, even though, in practice, the actual sam-
pling is sometimes without replacement. This assumption simplifies
the calculations because it eliminates changes in the population as the
sampling progresses.5
5
Preliminary simulations for the case of sampling without replacement
show that it has a very small effect on the estimates when the sample size is small
relative to the population. However, this question deserves further research.

Next, we will assume that the network of the hidden popula-
tion forms one connected component. That is, we assume that there is
a path between every person and every other person. This may seem
like a restrictive assumption, but a number of results from random
graph theory predict that even for very sparse graphs, almost all
nodes belong to one giant component (Newman, 2003b). For many
hidden populations, like jazz musicians or injection drug users, the
friendship network used in the recruitment is dense enough that this
assumption is reasonable.6
It is important to note that the assumption
that the network forms only one component is the only assumption
that we need to make about the structure of the network. This lack
of potentially untestable network assumptions makes respondent-
driven sampling applicable to many different types of hidden popula-
tions.
Additionally, we assume that all respondents receive and use
one coupon,7
and that when respondents recruit others, they recruit
randomly from all edges that involve them. Some readers may doubt
this random recruitment assumption, but there is some empirical
evidence that is consistent with this assumption (Heckathorn et al.
2002). To specify this recruitment assumption more clearly, we can
write
Pr½EIðej!kÞr¼xþ1 ¼ 1jNIðjÞw¼x ¼ 1 ¼
1
dj
: ð12Þ
Finally, we will assume, initially, that the seeds are drawn with
probability proportional to their degree. That is, a person with 10
6
However, given this assumption, it may not be wise to do the
recruitment over the sexual or drug-sharing networks, which are less dense and
potentially consist of multiple components. Also, this network assumption means
that respondent-driven sampling is not appropriate for groups like tax evaders,
who do not have frequent contact with each other.
7
These results easily extend to the case of multiple coupons as long as all
coupons are used. However, this extension requires messier algebra, and so for
ease of presentation we presented the one coupon case. However, once multiple
coupons are allowed, new possibilities emerge for heterogeneous recruitment
behavior within a group and heterogeneous recruitment behavior between groups.
These situations clearly deserve further research.

friends is twice as likely to be a seed as a person with 5 friends. This
assumption can be expressed mathematically as8
Pr½NIðjÞw¼0 ¼ 1 ¼
dj
P
i2N
di
: ð13Þ
We chose to make this assumption because in studies of hidden
populations, the people drawn as seeds are often those who are
known to the researchers. These more well-known people tend to
have more friends than average. Thus, it seems reasonable to assume
that a person’s chance of being drawn as a seed increases with his or
her degree. In order to increase the generality of respondent-driven
sampling, this assumption about the seeds being drawn with prob-
ability proportional to degree will be relaxed in Section 8.
6.2. Consequences of the Assumptions
Since we have assumed that the seeds are drawn with probability
proportional to degree, we can make some conclusions about the
the probability that certain edges will be drawn in recruitment
period 1 and then the probability that certain nodes will be
drawn in wave 1.
A relationship ej!k can be selected in recruitment period 1 only
if the node from which it points, node j, is selected in wave 0. Using
the indicator function notation and the rules of conditional probabil-
ity, we can calculate the probability that a given relationship, ej!k,
will be drawn in recruitment period 1 as follows:
Pr½EIðej!kÞr¼1 ¼ 1 ¼
Pr½NIðjÞw¼0 ¼ 1 Pr½EIðej!kÞr¼1 ¼ 1jNIðjÞw¼0 ¼ 1:
ð14Þ
8
For equation (13) the probabilities should not be considered the
overall probability of being selected as a seed but the draw probability in each one
of the s draws that select the seeds. To see this further, compare equation (13)
with the overall probability that node j will be selected as a seed, which
is 1
ð
P
i2N
diÞdj
P
i2N
di
s
.

Since we have assumed that the seeds were drawn with prob-
ability proportional to their degree and that people choose randomly
from their relationships when making a recruitment, we can rewrite
equation (14) as
dj
P
i2N
di

1
dj
; ð15Þ
which can be simplified to,
1
P
i2N
di
: ð16Þ
Equation (16) tells us that if the nodes in wave 0 are drawn with
probability proportional to degree, then each relationship has the
same probability of being drawn in recruitment period 1.
This fact also gives us some information about the draw prob-
abilities of the nodes in wave 1. The probability that a node j will be
drawn in wave 1 is equal to the sum of the probabilities that the dj
relationships leading to it will be drawn in recruitment period 1. That
is,
Pr½ðNIðjÞw¼1 ¼ 1 ¼
X
dj
1
P
i2N
di
¼
dj
P
i2N
di
: ð17Þ
Equation (17) shows us that if the seeds are drawn with
probability proportional to degree, then the nodes in wave 1 will
also be drawn with probability proportional to degree. Repeating
this argument iteratively shows that, if the nodes in wave 0 are
drawn with probability proportional to degree, then the nodes in all
successive waves also will be drawn with probability proportional to
degree.
This argument can also be repeated iteratively to show that, if
the seeds are drawn with probability proportional to degree, then the

probability that a specific edge, ej!k, will be drawn in recruitment
period x is constant and equal for all edges:9
Pr½EIðiÞr¼x ¼ 1 ¼
1
P
i2N
di
: ð18Þ
It is important to note that these arguments apply for any
network structure with reciprocal ties and thus are quite general.
The consequences of the assumptions will prove useful when consid-
ering our estimators.
6.3. Making the Estimates
Using the results derived in the previous section, we can now derive
estimators for specific network properties and show that these esti-
mates are asymptotically unbiased. First, we show how to use the
observed recruitment behavior to estimate the probability of cross-
group connections, CA,B and CB,A. Second, we will estimate the
average degree for people in the groups, DA and DB, using the self-
reported network-size information. Third, we will combine these esti-
mates to estimate the proportion of the population falling into one of
two distinct groups, PPA and PPB.
6.3.1. Estimating CA,B and CB,A
Recall that we wanted to use the information from the sample to
estimate some properties of the network connecting the population.
The first thing that we would like to estimate is the probability that if
we follow a random friendship beginning in group A that we cross
groups and end up in group B. One way to estimate this probability,
CA,B, would be to ask respondents what percentage of their friends
fall into certain groups. However, this is not possible because for
9
As was pointed out by a reviewer, equal draw probabilities for
all edges does not mean that these probabilities are independent. The
edge draw probabilities are correlated. That is, Pr[EI(ei!j)r¼t ¼ 1,
EI(ej!k)r¼tþ1 ¼ 1] 6¼ Pr[EI(ei!j)r¼t ¼ 1] Pr[EI(ej!k)r¼tþ1 ¼ 1]. This dependence in
the edge selection process could introduce difficulties into the estimation of
certain network properties. However, for the properties of interest in this paper it
does not present a problem.

many items of interest—for example HIV status—respondents may
not have enough information about their peers to make accurate
assessments. The accuracy of this self-reported data is also doubtful.
Instead of basing the cross-group friendship probabilities on
self-reported data, we base them on actual behavior. When one
respondent recruits another, this behavioral link represents a network
link that can be verified by asking the recruitee to characterize the
relationship to the recruiter. Verification requires that the recruitee
identify the recruiter as an acquaintance, friend, or closer than friend,
rather than as a stranger. Only verified links should be used for
estimation.
To be able to construct an estimate for CA,B and CB,A, we must
know something about how the recruitments we observed are selected
from the set of possible recruitments. In Section 6.2, we showed that
each edge, ej!k is equally likely to be selected in each recruitment
period. That is, we showed that the recruitments we observe are a
random sample of all possible recruitments.
Recall that for each sample we observe a set of recruitments.
These recruitments can be divided into four groups—recruitments
from a person in group A to another person in group A, rAA, recruit-
ments from a person in group A to a person in group B, rAB, etc.
Since the observed recruitments are a random sample from all
edges, unbiased estimates for CA,B and CB,A are
d
CA;B
CA;B ¼
rAB
rAA þ rAB
ð19Þ
d
CB;A
CB;A ¼
rBA
rBB þ rBA
: ð20Þ
We now turn our attention to deriving estimators for DA and
DB, the average degree of nodes in group A and group B.
6.3.2. Estimating DA and DB
Recall that during the sampling process we collected the degree of
each member of our sample. If we tried to estimate the mean degree of
people in group A by taking the mean degree of the people in the
sample, our estimate would be too high, in some cases much too high
because chain-referral methods overrepresent people with high degree

(Erickson 1979; Kalton and Anderson 1986; Eland-Goosensen et al.
1997).
Because the simple mean is not a good estimator, a different
estimation procedure is required. That is, we must take the sample
data and somehow adjust it so that it yields accurate information
about the population.10
Two distinct approaches can be used to
motivate the exact form of this adjustment, and, as we will see, both
of these approaches lead us to the same estimator. We can then see
that this estimator is asymptotically unbiased.
One approach for constructing an estimator for the average
degree, which we call the degree distribution approach, is motivated
by the degree distributions of the sample and the population. We
begin by recalling that in Section 6.2 we showed that if our assump-
tions are met, then the nodes are drawn with probability proportional
to their degree in all waves. We can use this fact, together with the
observed sample degree distribution, qA(d), to estimate population
degree distribution pA(d). This population degree distribution can be
used to estimate the average degree of people in group A, DA.
Previous work on unrelated problems (Feld 1991; Newman,
Strogatz, and Watts, 2001; Newman 2003a) has found that if nodes
are drawn with probability proportional to their degree, the sample
degree distribution, qA(d), is given by
qAðdÞ ¼
d pAðdÞ
P
maxðdÞ
d¼1
d pAðdÞ
; ð21Þ
10
When attempting to estimate the average degree of people in group A,
DA, we recommend that researchers do not include the degree of the seeds. The
seeds are selected with a different and potentially unknown mechanism, and they
should really be treated differently from the data that are collected via the chain
referrals. In practice, since the number of seeds is small (about 5) relative to the
sample size (about 500), only a small amount of data are not used. For the
simulations presented in this paper, all estimates of average degree exclude the
degrees of the seeds. It is somewhat unusual not to use all of the information
collected when making the estimates from a sample, but this is not without
precedent when estimating after adaptive sampling procedures (for example, see
Thompson [1990]).

where pA(d) is the population degree distribution and
PmaxðdÞ
d¼1 d pAðdÞ is a normalizing constant to ensure that qA(d) sums
to 1.
Equation (21) allows us to predict the sample degree distribu-
tion given the population degree distribution. However, we have the
opposite problem. That is, we know the sample degree distribution
and want to predict the population degree distribution. Since, as
described in equation (21), d pA(d) is proportional to qA(d), it is
also the case that pA(d) is proportional to 1
d qAðdÞ. So, if a sample
has a degree distribution, qA(d), then the population degree distribution,
pA(d), can be estimated as
d
pAðdÞ
pAðdÞ ¼
1
d qAðdÞ
P
maxðdÞ
d¼1
1
d qAðdÞ
; ð22Þ
where
PmaxðdÞ
d¼1
1
d qAðdÞ is a normalizing constant.
From this predicted population distribution, d
pAðdÞ
pAðdÞ, we can
estimate the average degree in the population, DA, by recalling that
the average of a discrete probability density function f(x) is
P1
x¼0 x f ðxÞ. Since we are using a population distribution approach,
we will annotate our estimator, c
DA
DA, with the superscript dist:
d
Ddist
A
Ddist
A ¼
X
maxðdÞ
d¼1
d d
pAðdÞ
pAðdÞ: ð23Þ
In equation (23) the summation in the estimator is indexed by
degree and not by sample element. If we rewrite equation (23)
indexed by sample element (see Appendix B for the derivation), we
get
d
Ddist
A
Ddist
A ¼
nA
P
nA
i¼1
1
di
: ð24Þ
The estimator presented in equation (24) at first appears unlike
an estimator for a population mean because the sample size, nA, is in
the numerator and not the denominator.

However, as we will show, this estimator is equivalent to some-
thing that is more familiar—the ratio of two Hanson-Hurwitz estima-
tors,11
one that estimates RA, the total number of stubs radiating from
people in group A, and one that estimates NA, the number of people
in group A.
We write this Hansen-Hurwitz based estimator for c
DA
DA with the
superscript hh,
d
Dhh
A
Dhh
A ¼
c
RA
RA
c
NA
NA
¼
1
nA
P
nA
i¼1
1
pi
di
1
nA
P
nA
i¼1
1
pi
; ð25Þ
where pi is the probability that person i will be selected on a specific
draw.
At first it seems we have no hope for working with equation
(25) because the draw probabilities, pi, are unknown. However,
because people are drawn with probability proportional to degree,
we do know that the relative draw probabilities for two nodes, i and k,
will be
pk
pi
¼
dk
di
8 i; k: ð26Þ
So for each person we can rewrite the draw probability in terms
of a chosen reference person k. Thus, using equation (26) to rewrite
equation (25), we get
d
Dhh
A
Dhh
A ¼
1
nA
P
nA
i¼1
dk
dipk
di
1
nA
P
nA
i¼1
dk
dipk
: ð27Þ
11
Hansen-Hurwitz estimation is a standard procedure when estimating
from data where the sampling is with replacement and the units have unequal
draw probabilities (Hansen and Hurwitz 1943; Cochran 1977; Brewer and Hanif,
1983). The basic idea, similar to the Horvitz-Thompson estimator, is that the
procedure weights each sample element by the inverse of its draw probability.
That is, units with a small chance of being selected are counted more.

Since dk
pk
is constant, we can remove it from the numerator and
denominator. Also, canceling the 1
nA
terms, we are left with
d
Dhh
A
Dhh
A ¼
nA
P
nA
i¼1
1
di
: ð28Þ
Note that equation (28) does not require the unknown draw
probabilities, pi, for each node. It requires only information that is
collected in the sample.
Also, surprisingly, equations (24) and (28) are identical. That
is, we can see that these two very different approaches—one based on
degree distributions and one based on Hansen-Hurwitz estimators—
have led us to the same estimator. Since these estimators are equal, we
will drop the superscripts hh and dist and refer to the estimator as
simply c
DA
DA.
The numerator and denominator of c
DA
DA are both Hansen-
Hurwitz estimators that are known to be unbiased (Brewer and
Hanif 1983). It is also the case that the ratio of two unbiased estimators
is asymptotically unbiased with bias on the order of n1
, where n is the
sample size (Cochran 1977). So as nA, the sample size in group A, gets
large, E½ c
DA
DA ! DA. Generally, this bias is considered negligible in
samples of moderate size (Cochran 1977).
We have now shown that two distinct approaches to estimate
the average degree for group A are equivalent and that this estimator,
c
DA
DA, is asymptotically unbiased.
6.3.3. Putting It All Together to Estimate PPA and PPB
Now that we have estimated certain information about the social
network that connects the population, we can use that information
to estimate PPA and PPB. Plugging equations (28) and (19) into the
equations that express the population proportions in terms of net-
work information (equations [8] and [9]), we can now estimate the
population proportion as
d
PPA
PPA ¼
c
DB
DB d
CB;A
CB;A
c
DA
DA d
CA;B
CA;B þ c
DB
DB d
CB;A
CB;A
: ð29Þ

Again we have a ratio of asymptotically unbiased estimates that
is also asymptotically unbiased. Equation (29) is the central result of
this paper. With it, we are claiming that we can make an asymptotically
unbiased estimate of the proportion of the population with a
specific trait based only on data collected during respondent-driven
sampling.
7. COMPUTER SIMULATIONS
In the previous section, we have presented a number of analytic
results arguing that our proposed population proportion estimator
is asymptotically unbiased. These analytic arguments can be further
supported with numerical simulation. Also, simulation will allow us
to consider questions that are not analytically tractable, like those
considered in Section 8.
In these simulations, we generate a population and then
repeatedly take samples (with replacement) from that population. The
properties of our proposed estimators can be evaluated by averaging
the estimates that come from these many samples. For example, if the
average of these estimates matches the true population value, then we
have further support for our argument that the estimation procedures
we have developed are unbiased. The algorithms used to perform
these simulations are described in Appendix A.
In Figure 4 we can see the results of these numerical
simulations for a wide range of population proportion values. The
average value of the estimates, represented by the circles, is the same
as the true value, represented by the dotted line. It seems that
averaging these estimates over many repetitions does in fact yield
the true population proportion in group A, thus providing strong
support for our argument that these estimates are asymptotically
unbiased.
Because the estimate is only asymptotically unbiased, and not
strictly unbiased, we might be interested in knowing the rate of
convergence and the magnitude of the resulting bias. Since our esti-
mate for population proportions is a ratio of two asymptotically
unbiased estimates, we know that the bias is on the order of n1
.
Thus, as the sample size gets big, the bias gets small very quickly.

Numerical simulations from a population with parameters
described in Appendix A are presented in Figure 5. We see that, in
this case, the rate of convergence is rapid and the magnitude of the
bias is small. With a sample size of 500, the bias is less than one-tenth
of 1 percent (0.001) and therefore not a serious cause for concern.
However, more general results about the convergence properties, and
the resulting bias, would be desirable.
8. A NOTE ON THE CHOICE OF SEEDS
So far in our derivations we have assumed that the seeds were drawn
with probability proportional to degree. This assumption deserves
further consideration. Previously, numerous researchers have doubted
estimates from chain-referral samples because it is not possible to
select the seeds randomly from the target population (Erickson
1979; Spreen 1992; Snijders 1992; Friedman 1995). In a review article
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Actual PPA
Estimated
PP
A
True value
RDS estimate
90% confidence interval
FIGURE 4. Numerical simulations of equation (29) supporting our analytic
argument that the respondent-driven sampling population
proportion estimates are asymptotically unbiased. With a sample
size of only 500, the bias is negligible. The vertical lines show 90
percent confidence intervals around the estimate.

on chain-referral methods, Spreen (1992) summarizes this and other
literature: ‘‘[T]he central question in the methodological discussion
about sampling and analyzing hidden populations is basically: How
to draw a random (initial) sample?’’ Given that the choice of seeds has
long been considered a critical weakness of chain-referral methods, it
is natural to devote further attention to this topic.
Most previous chain-referral methods have assumed that the
seeds are a simple random sample from the population. However,
when actually implementing chain-referral sampling, researchers often
select people known to them to serve as seeds. These known
individuals are far from representative, and usually have much higher
degrees than other members of the hidden population. Unlike
other chain-referral methods, in respondent-driven sampling the
estimates are unbiased if the seeds are drawn with probability
proportional to degree—much closer to what actually happens in
the field.
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
0.2995
0.3
0.3005
0.301
0.3015
0.302
Sample size
Estimated
PP
A
RDS estimate
True population value
FIGURE 5. Numerical simulations supporting our argument that the rate of
convergence to the true estimate is quite rapid. The circles represent
the average estimate from 100,000 samples. In this case, the
magnitude of the bias is also small. For a sample size of 500 the
respondent-driven sampling estimate is within one-tenth of 1 percent
(0.001) of the true population value.

However, even though our assumption is closer to what actually
happens, it is still the case that it is unlikely that the seeds will be
drawn exactly with probability proportional to degree. Fortunately, it
turns out that using a Markov chain argument we can show that our
estimates are asymptotically unbiased no matter how the seeds are
chosen.
8.1. Convergence of Estimates for Arbitrary Choice of Seeds: Analytic
Results
Previously in this paper, we have shown that our estimates are
asymptotically unbiased if the seeds are drawn with probability
proportional to degree. This assumption about the seeds selection
mechanism was important because it allowed us to derive two import-
ant conclusions: (1) that nodes are drawn with probability propor-
tional to their degree in all waves and (2) that edges are drawn with
uniform probability. These conclusions were critical in showing that
our estimates were asymptotically unbiased.
However, it turns out that even if the seeds are not drawn with
probability proportional to degree, the sampling process converges to
one in which people are drawn with probability proportional to
degree. So, after a number of waves the draw probability for the
nodes will converge to the situation that we needed to make our
estimates unbiased.
We can see this by setting up a Markov chain on the nodes12
as
in Thompson (2003). A Markov chain is a natural way of modeling
this problem because the draw probabilities for nodes in wave w are
only dependent on the draw probabilities of nodes in wave w 1.
To begin, let’s construct a 1 N vector, ~

ðtÞ
, which stores the
draw probability for each person j in wave t. That is,
~

ðtÞ
j ¼ Pr½NIðjÞw¼t ¼ 1: ð30Þ
12
In a previous work on respondent-driven sampling, Markov chains
have been used on the population groups (Heckathorn 1997, 2002). These
previous Markov chain arguments showed that the sample proportions converged
independent of the seeds. This is different than the claim here, which is that the
respondent-driven sampling estimates converge to the true population value.

Now let’s construct an N N transition probability matrix, P,
which stores the probability that if we are at node i at time w then we
will be at node j at time w þ 1. That is,
pij ¼ 1=di if xij= 1
0 otherwise.

ð31Þ
Since we assumed that the network is connected and that every
person is reachable from every other person, then this transition
probability matrix P describes a regular Markov chain.13
This type
of Markov chain has a number of nice properties. In this case, the
most useful is that for all possible initial probability vectors (for any
way of selecting the seeds), the draw probability in later waves con-
verges to a unique probability vector, ~

ðÞ
, which satisfies the follow-
ing self-consistency equation (Häggström 2002):
~

ðÞ
P ¼ ~

ðÞ
: ð32Þ
The only probability vector ~

ðÞ
where equation (32) holds (see
Appendix C for proof) is
~

ðÞ
j ¼
dj
P
i2N
di
ð33Þ
So we can now see that even if the seeds are not drawn with
probability proportional to degree, the selection of the nodes in the
later waves more and more closely approximates probability propor-
tional to degree. As ~

ðtÞ
! ~

ðÞ
, the selection process for the sample
becomes closer and closer to desired. As the sample size gets larger, the
13
In order for the Markov chain to be regular, it must be irreducible and
aperiodic. Because the network consists of one connect component, we know that
the chain is irreducible. Further, if the network contains one triangle, then there
are paths of both odd and even length from each state back to itself. This
condition ensures aperiodicity. Since the networks under study are large, and
since social networks have a strong tendency toward transitivity, the existence of
at least one triangle has probability approaching 1. So, since the chain is
irreducible and aperiodic, it is regular (Häggström 2002).

people selected as desired make up a larger and larger part of the
sample and so our estimate converges to the true population value.
8.2. Convergence of Estimates for Arbitrary Choice of Seeds:
Simulation Results
Asymptotic convergence results are reassuring, but we will never be
able to select an infinitely large sample of people. Therefore, it is
important to consider the rate of convergence. Also, it is important
to note that this convergence, due to the seeds not being drawn as
desired, is different, and potentially more serious than, the previous
problems of convergence that were caused by the use of ratio estima-
tion.
Previously, some work has been done on the rate of conver-
gence to ~

ðÞ
in the case of random walks on networks (Lovász 1993)
and in Markov chain Monte Carlo methods14
(Cowles and Carlin
1996). However, we are interested in the rates of convergence of our
estimates, not the vector ~

ðÞ
. Simulation using the algorithm in
Appendix A will be used to explore rates of convergence.
To begin to explore convergence, we can consider a family of
different ways to draw the seeds:
Pr½NIðjÞw¼0 ¼ 1 ¼
ðdjÞ
P
i2N
ðdiÞ: ð34Þ
When ¼ 1, the seeds are drawn with probability proportional to
degree. However, it may be the case that the degree of the nodes has a
smaller or larger effect on the chance that a node is drawn as a seed.
For example, the nodes could be drawn with probability proportional
to degree squared ( ¼ 2) or the nodes could be drawn independent of
degree—that is, via simple random sampling ( ¼ 0).
A natural first question to ask is, for a typical network struc-
ture such as the one described in Appendix A, and a reasonable
sample size (say, 500 people): How close will our estimate be to the
14
Readers familiar with Markov chain Monte Carlo (MCMC) methods
will notice a similarity between our convergence argument and the method of
selecting random draws from a difficult distribution. However, unlike in MCMC,
we do not discard the data collected during the convergence process.

true population value even if the seeds are not drawn as assumed? For
a wide range of population proportion values, we have presented
simulation results when ¼ 0 and ¼ 2 in Figure 6. We see that for
a sample size of only 500, the estimates are all within one-half of one
percentage point (0.005) of the true population values. That is, for this
network structure, our estimates are essentially unbiased when we
selected a reasonable sample size, even though in both cases we violate
the assumption that the seeds are drawn with probability proportional
to degree.
Another natural question to ask is, how fast does our estimate
converge? For this question we can offer only preliminary results. In
Figure 7 we present simulation results for a population with 30
percent of its members in group A, and where the seeds are drawn
with probability proportional to degree squared, ¼ 2 (the other
population parameters appear in Appendix A). Note that even for
small sample sizes of fewer than 200 people, the average estimate has
a small bias—a little more than two-tenths of one percentage point
(0.002). As the sample size gets larger, the bias gets smaller. In this
case, for a sample size of 500 the bias is about one-tenth of one
percentage point (0.001), or well within rounding error and not a
serious cause for concern.
The figures are provided as illustrations—the actual rates of
convergence will depend on the characteristics of the population and
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Actual PPA
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Actual PPA
Estimated
PP
A
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
Estimated
PP
A
True value
RDS estimate
True value
RDS estimate
Seeds selected randomly from the entire
population
(b)
Seeds selected with probability proportional to degree
squared
(a)
FIGURE 6. Simulation results for a wide range of population proportion values.
In (a) the seeds are drawn with probability proportional to degree
squared ( ¼ 2). In (b) the seeds are drawn independent of degree
( ¼ 0). In both cases, the selection of the seeds violates our
assumptions but has almost no effect on the estimates. The vertical
lines show 90 percent confidence intervals.

the method of selecting the seeds. Further research in this area is
called for.
9. A WARNING ABOUT IGNORING SAMPLE DESIGN
Up until now, in almost all research on hidden populations, research-
ers have collected a sample with some nonrandom sample design and
then presented summary statistics about the sample they collected—
for example, the proportion of the sample that is HIV positive. There
is never an explicit claim that this sample can be generalized to the
population, but that claim is often implicit. This procedure can lead
researchers to draw very misleading conclusions (Thompson and
Frank 2000; Thompson and Collins 2002).
0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000
0.299
0.2995
0.3
0.3005
0.301
0.3015
0.302
0.3025
0.303
Sample size
Estimated
PP
A
RDS estimate
True population value
FIGURE 7. Numerical simulations showing that the respondent-driven sampling
estimate converges to the true value even if the seeds are not drawn
as desired. (In this case the seeds are drawn with probability
proportional to degree squared.) The circles represent the average
estimate from 100,000 samples. Note that, in this case, for all sample
sizes greater than 200, the bias is less than three-tenths of 1 percent
(0.003) and not a serious cause for concern.

We can see this problem more clearly if we look at some real
data collected from jazz musicians in San Francisco and New York. A
major problem for arts organizations is that it is often difficult to get
accurate information about the communities they seek to represent
and support (Throsby and Mills 1989). In the case of jazz musicians,
there is no sampling frame from which one can draw a sample.
To resolve this situation, researchers took a sample of jazz
musicians via respondent-driven sampling in several American cities
(Heckathorn and Jeffri 2001). A random sample from the American
Federation of Musicians was also taken and screened for jazz musi-
cians. With this data it is possible to compare the estimates from two
methods currently in use—institutional sampling and chain-referral
sampling ignoring the sample design—with the estimates from
respondent-driven sampling.
Some of the data from this study are presented in Table 1. It is
important to note the differences between the estimates depending on
the data source and the method of estimation. These differences are
not merely of academic concern. Results from studies of hidden
populations are often used to set public policy. If these policies are
set based on incorrect information, they could waste valuable
resources or even have effects counter to the goal of the policy.
In San Francisco, using the union sample we would conclude
that 81.6 percent of the city’s jazz musicians have had their work
played on the radio. However, the respondent-driven sampling
estimate, which, based on the arguments presented in this paper we
believe to be accurate, is that only 31.4 percent have had their work
played on the radio. This same pattern, overrepresentation of
successful musicians in the union sample, is also observed in New
York, but to a smaller extent. This overrepresentation of successful
musicians in the union could have been predicted ahead of time, and
in some sense serves as a reality check for the respondent-driven
sampling procedure. However, even though the direction of the bias
could have been predicted ahead of time, it would have been hard to
guess its magnitude.
In other situations, the relationship between the estimate from
the union sample and respondent-driven sampling estimate is quite
difficult to guess ahead of time. For example, in San Francisco the
union sample overestimates the percentage of females by about 10
percentage points. But in New York we observe the opposite; females

are underestimated in the union sample by almost 7 percentage
points. The magnitude—or even direction—of this bias could not
have been guessed at ahead of time. Because of these types of prob-
lems, the sample proportion from institutional samples should be
viewed as a bad estimate of the true population proportion.
Another technique currently used by researchers of hidden
populations is to select a sample via some nonrandom sample design
and then to ignore the sample design and treat the sample as if it were
a simple random sample. In these cases, the proportion of the sample
with a given characteristic (for example, HIV positive) is used as an
estimate of the proportion of the population with that characteristic.
By comparing this naive estimate, which ignores the sample design,
with the respondent-driven sampling estimate, which correctly
accounts for the sample design, we can see that ignoring the sample
design can produce misleading estimates.
TABLE 1
Three Different Estimates of the Characteristics of Jazz Musician Populations
JAZZ MUSICIANS IN NEW YORK
Characteristic
Union Sample
(n ¼ 415)
Chain-Referral Sample
(n ¼ 251)
RDS Estimate
(n ¼ 251)
Union member 100 39.6 25.4
Female 16.1 26.8 23.7
Solo only 19.9 8.8 11.6
Received airplay 79.6 81.6 74.9
JAZZ MUSICIANS IN SAN FRANCISCO
Characteristic
Union Sample
(n ¼ 237)
Chain-Referral Sample
(n ¼ 221)
RDS Estimate
(n ¼ 221)
Union member 100 11.2 4.0
Female 21.9 14.5 11.1
Solo only 15.0 9.2 21.0
Received airplay 81.6 48.7 31.4
Note: The data reflect different conclusions researchers would make depending on
the source of data used. The estimates from the union sample and chain-referral sample are
sample means. The respondent-driven sampling (RDS) estimates use the data from the
chain-referral sample and the estimation techniques presented in this paper.

As one might expect, the sample proportion from a chain-
referral sample overestimates those with many contacts in the popula-
tion—for example, union members. In New York, the naive estimate
from the chain-referral sample estimates that 39.6 percent of jazz
musicians are in the union. However, when we properly account for
the method used to collect the sample, we estimate that only 25.4
percent are union members. We observe a similar pattern, where
the naive chain-referral estimate (11.2 percent) is larger than the
respondent-driven sampling estimate (4.0 percent), in San Francisco.
The sample proportion from the chain-referral sample also
underestimates those with fewer relationships in the population—for
example, musicians who perform solo. In San Francisco, the chain-
referral estimate is that only 9.2 percent of jazz musicians perform
solo. The respondent-driven sampling estimate indicates that this was
an underestimate, and estimates the actual value to be 21.0 percent. A
similar bias, but of different magnitude, is observed in the New York
data.
We can draw a number of conclusions from this jazz musician
data. Most importantly, failure to successfully consider the sampling
design can lead researchers to make misleading conclusions. The
magnitudes, and sometimes even the directions, of these biases are
often difficult to predict ahead of time.
10. CONCLUSIONS
For many years researchers have known that chain-referral samples
are an extremely useful way to collect samples from hidden popula-
tions. However, they have also considered chain-referral samples mere
convenience samples, so hopelessly full of bias as to be useful only for
exploratory purposes. However, we have shown here that when
handled properly, chain-referral samples can produce estimates that
are asymptotically unbiased.
We have also shown that our estimation method is robust to
the actual selection mechanism for the seeds, even though this is
traditionally considered to be one of the largest problems with
chain-referral methods.
There are several other nice properties of respondent-driven
sampling. First, the sample gives us information about not just the

people in the population but also the network connecting them.
Currently, researchers are beginning to explore ways of extracting
useful social network information from the sample (Heckathorn and
Jeffri 2001). Given the role that networks play in the transmission of
disease, this information could prove extremely useful in public health
studies.
Another desirable property of respondent-driven sampling is
that sample data can be combined with institutional data to estimate
the size of a hidden population (Heckathorn and Jeffri 2001). Pre-
vious methods for estimating the sizes of hidden populations did not
also allow for unbiased estimates of population composition.
Respondent-driven sampling is also cheaper, quicker, and
easier to implement than other methods commonly used to study
hidden populations (Semaan, Lauby, and Liebman 2002). This is a
significant advantage because it means that for a given amount of
resources, respondent-driven sampling allows researchers to have
more study sites or larger sample sizes than other methods.
There are also still many open questions. In order to produce
analytic results about the properties of the respondent-driven sam-
pling estimators we had to make assumptions that in some cases may
be violated. For example, nonrandom recruitment of friends could
influence the estimates in unknown ways. Additionally, differential
recruitment success by different types of people could bias the sample
of edges that we observe. Nonsampling error related to estimating
subject degree could also introduce bias into the estimates. Empir-
ically checking the reasonableness of the assumptions and further
research related to the robustness of the estimation procedure are
both problems worthy of further study.
We hope this work opens up new possibilities for researchers
interested in carrying out substantive research on hidden populations.
We also hope that this will spur further work not just about
respondent-driven sampling but chain-referral methods in general.
These methods have great promise for answering a number of
important questions.

APPENDIX A: ALGORITHMS USED FROM COMPUTER
SIMULATION
It is possible to simulate the respondent-driven sampling procedure in
order to provide support for the arguments offered in this paper. The
simulation model can also be used to explore questions that are not
analytically tractable.
There are five different population parameters that we vary in
the simulations. We can vary the number of people in the population,
N, and the proportion of the population in group A, PPA. We can
also vary the super-population degree distribution, G(d), from which
the node degrees are drawn. That is, for a given degree d, GA(d) is the
probability that a node in group A will be assigned degree d. When the
number of nodes is large, the difference between the super-population
distribution, G(d), and the population degree distribution, p(d),
becomes small. There is some evidence that degree distributions of
friendship networks are right skewed (Killworth et al. 1998a,b;
McCarty et al. 2001), so unless otherwise stated we will draw from
an exponentially distributed super-population. However, the mean for
group A and group B can be different.
Finally, we can vary the interconnectedness of the two groups.
In some situations the friendship structure may be highly homo-
philous (McPherson, Smith-Lovin, and Cook 2001)—that is, people
in group A are more likely than chance to be connected to people in
group A. We formalize this with a parameter I, which represents the
interconnectedness of the two groups. I is defined as the ratio of the
actual number of cross-group ties to the possible number of
cross-group ties, where the number of possible cross-group ties is
limited by the min(RA,RB). That is,
I ¼
TAB
minðRA; RBÞ
: ðA:1Þ
To actually construct the network, we modify the algorithm pre-
sented in Malloy and Reed (1995) and Newman, Strogatz, and Watts
(2001). First, the nodes are created and assigned to each group. Then for
each node i, we draw a random number di from GA(d) if i 2 A or from

GB(d) if i 2 B. We then assign di ‘‘stubs’’ to node i.15
The stubs represent
the friendships to be connected with other nodes. Then each di is
increased by 1 to avoid any members of the population having degree 0.
Once the stubs are set, the required number of cross-group ties
is calculated, TA,B ¼ I min(RA,RB). Then, a randomly chosen stub
from group A is paired with a randomly chosen stub from group B
until the desired number of cross-group edges has been added. The
remaining stubs are meant for within-group edges, so they are ran-
domly connected to other stubs in their group. There is a check before
each edge is added that prevents self-loops and multiple edges
between the same two nodes.
Occasionally, the graph generation procedure can get ‘‘stuck’’.
That is, there are stubs remaining but no way to connect them without
adding self-loops or multiple edges between the same nodes. In that
case, the algorithm removes all the edges and begins adding them
again from the beginning. The algorithm described here ensures that
we are randomly sampling from the ensemble of all graphs with the
specified parameters.
Once the network is complete, we can simulate the sampling
process. First, a set of s seeds is chosen to make up wave 0. Then we
randomly sample from the s c coupons that are eligible to recruit
people for wave 1.
Once a coupon is selected, we choose a friend of the person
who holds that coupon. This new recruitee is now a member of wave 1
of the sample. We continue choosing randomly from the set of cou-
pons of people in wave 0 until all the people from wave 0 have used all
their c coupons. At this point there are s c c coupons to be used to
recruit wave 2. We then begin randomly sampling from the set of
coupons to be used to recruit wave 2. This process continues until the
desired sample size is reached.
In Section 6.1, we assumed that people were selected with replace-
ment. So in the simulation, each time a person is selected they are
considered a new member of the sample. In this way the composition
of the population does not change as the sampling procedure progresses.
Unless otherwise shown, all simulations are conducted with the
following population and sampling parameters:
15
All random numbers were generated using routines from Numerical
Recipes (Press et al. 1992).

* Population size (N): 10000
* Proportion of the population in group A (PPA): 0.3
* Super-population degree distribution for group A (GA(d)): expo-
nential with mean 20
* Super-population degree distribution for group B (GB(d)): expo-
nential with mean 10
*
Interconnectedness (I): 0.6
* Sample size (n): 500
* Replicate samples generated for each population (r): 1000
* Seeds (s): 5
* Coupons (c): 2
APPENDIX B: DERIVATION OF d
Ddist
A
Ddist
A
In the text we made the claim that the estimator d
Ddist
A
Ddist
A takes on a more
convenient form when it is rewritten indexed by sample element. We
will show that derivation here.
First, we begin with the equation
d
Ddist
A
Ddist
A ¼
X
maxðdÞ
d¼1
d d
pAðdÞ
pAðdÞ: ðB:1Þ
By plugging in the definition of d
pAðdÞ
pAðdÞ we get,
d
Ddist
A
Ddist
A ¼
X
maxðdÞ
d¼1
qAðdÞ
P
maxðdÞ
d¼1
1
d qAðdÞ
ðB:2Þ
Since the denominator is constant, we can pull it outside of the
summation and write
d
Ddist
A
Ddist
A ¼
1
P
maxðdÞ
d¼1
1
d qAðdÞ

X
maxðdÞ
d¼1
qAðdÞ: ðB:3Þ
Because qA(d) is a probability distribution, we know that
PmaxðdÞ
d¼1 qAðdÞ ¼ 1. Therefore we can rewrite

d
Ddist
A
Ddist
A ¼
1
P
maxðdÞ
d¼1
1
d qAðdÞ
: ðB:4Þ
Now let’s consider the frequency distribution, fA(d), which
records the number of people in group A of degree d observed in
the sample. It is the case that fAðdÞ
nA
¼ qAðdÞ. We can now rewrite in
terms of fA(d):
d
Ddist
A
Ddist
A ¼
1
P
maxðdÞ
d¼1
1
d fAðdÞ
nA
: ðB:5Þ
Since it is the case that 1
d fAðdÞ ¼
P
i2M
1
d where M ¼ {ijdi ¼ d},
we can substitute and rewrite the sum indexed by sample element i.
Moving nA to the numerator we get
d
Ddist
A
Ddist
A ¼
nA
P
nA
i¼1
1
d
: ðB:6Þ
This completes the derivation.
APPENDIX C: PROOF OF CONVERGENCE OF MARKOV
CHAIN
In the text we argued that the vector ~

ðtÞ
converges to a unique vector
~

ðÞ
such that,
~

ðÞ
P ¼ ~

ðÞ
ðC:1Þ
We then argued that the only such vector, ~

ðÞ
that satisfies
equation (C.1) is
~

ðÞ
j ¼
dj
P
i2N
di
: ðC:2Þ

We now rewrite equation (C.1) in expanded form

ðÞ
1
ðÞ
2
ðÞ
N
h i
p11 p12 p1N
p21 p22 p2N
.
.
. .
.
. ..
. .
.
.
pN1 pN2 pNN
2
6
6
6
4
3
7
7
7
5
¼
ðÞ
1
ðÞ
2
ðÞ
N
h i
: ðC:3Þ
We can now check to see if it is the case that
X
N
i¼1
~

ðÞ
i pij ¼ ~

ðÞ
j : ðC:4Þ
To verify equation (C.4), we can begin by ignoring all terms in
the sum where pij ¼ 0. These are cases where nodes i and j are not
friends, so there is no possibility of person i directly recruiting person
j. We can then rewrite both sides of equation (C.4) using the defin-
itions of the transition probability matrix P and the probability vector
~

ðÞ
j ,
X
fijxij¼1g
di
P
i2N
di

1
di
¼
dj
P
i2N
di
; ðC:5Þ
which simplifies to
X
fijxij¼1g
1
P
i2N
di
¼
dj
P
i2N
di
: ðC:6Þ
By recalling that the sum on the left side of equation (C.6) is
over dj terms, we can see that the equation does indeed hold.
So now we have shown that for any selection of the seeds, ~

ð0Þ
,
the probability of a node being selected converges to a value which is
proportional to its degree.

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