Methods in Epidemiology: Population Size Estimation (Advances in Experimental Medicine and Biology, 1333) 3030754634, 9783030754631

Table of contents :
Preface
Contents
1 Review of Size Estimation Methods
1.1 The Importance of Population Size Estimation for Key Groups in the Context of HIV
1.2 Direct Population Size Estimation Methods
1.2.1 Population-Based Surveys
1.2.2 Census/Enumeration Method
1.2.3 Nomination Method
1.2.4 Capture-Recapture Method
1.2.5 Multiplier Method
1.3 Indirect Population Size Estimation Methods
1.3.1 Cross-Wise Method
1.3.2 Network Scale-up Method
1.3.3 Proxy Respondent Method
1.4 Advantages of Network Scale-up Over Other Size Estimation Methods
1.5 Requirements of Network Scale-up
1.5.1 Network Size
1.5.2 Preliminary Size Estimation
1.5.3 Correction Factors
References
2 Methods to Estimate the Average Social Network Size
2.1 Introduction
2.2 Direct Estimation of C
2.2.1 Global Method
2.2.2 Summation Method
2.3 Indirect Estimation of C: Reference Groups Approach
2.3.1 Incorporating Multiple Reference Groups
2.3.1.2 Means of Sums Estimator
2.4 Considerations in Selecting Appropriate Reference Groups
2.4.1 Visibility
2.4.2 Barrier Effect
2.4.3 Cognitive Error
2.4.4 Attractiveness Bias
2.4.5 Racal Bias
2.4.6 Conclusion
2.5 Calibration of Calculated Average Social Network Size
2.6 Which Factors Affects Average Network Size?
2.6.1 Summation versus Traditional Reference Group Method
2.6.2 Missing Data
2.6.3 Joint Effect of Estimator, Definition of Ratio, and Its Tolerable Range
2.6.4 Joint Effect of Estimator, Missing Data, and Calibration
2.6.5 Digit Preference
2.6.6 Effect of Type of Network Size and Geographical Zone
2.6.7 Conclusion
2.7 Network Size Variation at the Subnational Level
2.8 Aggregation of Different Estimates
2.9 Effect of Demographic Characteristics on Network Size
2.9.1 Conclusion
2.10 Age-Sex Distribution of the Network Size
2.11 Confidence Interval Calculation for Average Network Size
References
3 Estimating the Size of Hidden Groups
3.1 Crude Estimations of the Size of Hidden Groups
3.2 Correction Factors
3.3 Visibility Factor
3.3.1 Methods to Estimate the Visibility Factor
3.3.2 Popularity Ratio
3.4 Influence of Missing Data on Size Estimations of Hidden Groups
3.4.1 Types of Missing Data
3.4.2 Ad Hoc Approaches to Adjust for Missing Data
3.4.3 Likelihood-Based Approaches to Impute Missing Data
3.5 Which Factors Influence Replies to Sensitive Questions
3.5.1 Background Characteristics
3.5.2 Definition, Frequency of Acts, and Question Order Effect
3.5.3 Data Collection Methods
3.5.4 Matching Between Interviewers and Interviewees
3.5.4.1 Conclusions
3.6 Methodological Considerations in the Size Estimation of Gender and Age-Specific Stigmatized Behaviors: Abortion
3.6.1 Conclusions
References
4 Data Smoothing, Extrapolation, and Triangulation
4.1 Smoothing of District Level Estimates
4.1.1 Spatial Smoothing by Weighted Head-Banging Algorithm
4.1.2 Spatial Smoothing Based on Kernel Functions
4.1.3 Empirical Bayesian Smoothing of Prevalence by Beta-Binomial Model
4.1.4 Empirical Bayes Smoothing of Standard Morbidity Ratio (SMR) by Poisson-Gamma Model
4.1.5 Practical Challenges in Bayesian Smoothing of NSU Studies
4.2 National Estimates by Extrapolation from Subnational-Level Estimates
4.2.1 How to Select Provinces?
4.2.2 Proportion-Based Extrapolation
4.2.3 Extrapolation by Linear Regression
4.2.3.1 Pitfalls in Linear Regression Extrapolation
4.2.3.2 Extrapolation by Spatial Regression
4.3 Triangulating Results
4.4 Conclusion
References

Citation preview

Advances in Experimental Medicine and Biology 1333

George Rutherford   Editor

Methods in Epidemiology Population Size Estimation

Advances in Experimental Medicine and Biology Volume 1333

Series Editors Wim E. Crusio, Institut de Neurosciences Cognitives et Intégratives d’Aquitaine, CNRS and University of Bordeaux, Pessac Cedex, France Haidong Dong, Departments of Urology and Immunology, Mayo Clinic, Rochester, MN, USA Heinfried H. Radeke, Institute of Pharmacology & Toxicology, Clinic of the Goethe University Frankfurt Main, Frankfurt am Main, Hessen, Germany Nima Rezaei, Research Center for Immunodeficiencies, Children’s Medical Center, Tehran University of Medical Sciences, Tehran, Iran Ortrud Steinlein, Institute of Human Genetics, LMU University Hospital, Munich, Germany Junjie Xiao, Cardiac Regeneration and Ageing Lab, Institute of Cardiovascular Science, School of Life Science, Shanghai University, Shanghai, China

Advances in Experimental Medicine and Biology provides a platform for scientific contributions in the main disciplines of the biomedicine and the life sciences. This series publishes thematic volumes on contemporary research in the areas of microbiology, immunology, neurosciences, biochemistry, biomedical engineering, genetics, physiology, and cancer research. Covering emerging topics and techniques in basic and clinical science, it brings together clinicians and researchers from various fields. Advances in Experimental Medicine and Biology has been publishing exceptional works in the field for over 40 years, and is indexed in SCOPUS, Medline (PubMed), Journal Citation Reports/Science Edition, Science Citation Index Expanded (SciSearch, Web of Science), EMBASE, BIOSIS, Reaxys, EMBiology, the Chemical Abstracts Service (CAS), and Pathway Studio. 2019 Impact Factor: 2.450 5 Year Impact Factor: 2.324

More information about this series at http://www.springer.com/series/5584

George Rutherford Editor

Methods in Epidemiology Population Size Estimation

123

Editor George Rutherford University of California San Francisco, CA, USA

ISSN 0065-2598 ISSN 2214-8019 (electronic) Advances in Experimental Medicine and Biology ISBN 978-3-030-75463-1 ISBN 978-3-030-75464-8 (eBook) https://doi.org/10.1007/978-3-030-75464-8 © Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The Iran Ministry of Health and Medical Education started an initiative in 2015 to advance health science by a decentralization program. As part of this national program, Kerman University of Medical Sciences (KUMS) was assigned to advance knowledge on health data modeling and forecasting. This book has been written and published in order to fulfill this mission with the support from KUMS and the Institute for Futures Studies in Health (IFSH) based at KUMS. San Francisco, USA

George Rutherford

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Contents

1 Review of Size Estimation Methods . . . . . . . . . . . . . . . . . . . . . . . Mohammad Reza Baneshi, Azam Rastegari, and Ali Akbar Haghdoost

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2 Methods to Estimate the Average Social Network Size . . . . . . . . 17 Mohammad Reza Baneshi, Saiedeh Haji-Maghsoudi, Azam Rastegari, and Ali Mirzazadeh 3 Estimating the Size of Hidden Groups . . . . . . . . . . . . . . . . . . . . . 39 Mohammad Reza Baneshi, Farzaneh Zolala, Saiedeh Haji-Maghsoudi, Maryam Zamanian, Ali Akbar Haghdoost, and Ali Mirzazadeh 4 Data Smoothing, Extrapolation, and Triangulation . . . . . . . . . . 61 Ali Mirzazadeh and Mohammad Reza Baneshi

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Review of Size Estimation Methods Mohammad Reza Baneshi, Azam Rastegari, and Ali Akbar Haghdoost

1.1

The Importance of Population Size Estimation for Key Groups in the Context of HIV

Understanding and measuring the burden of HIV faces several remaining challenges around the globe. HIV prevention, care, and treatment efforts, including advocacy for populations at risk of HIV, the design and implementation of national guidelines, monitoring the coverage of HIV programs and evaluating their impact, are not feasible without relatively precise estimates of the impact and magnitude of HIV among different subpopulations. In particular, estimating the population size of the key population (KP) plays a vital role in informing HIV policy makers and epidemiologists about the prevalence

M. R. Baneshi
A. Rastegari Modeling in Health Research Center, Institute for Futures Studies in Health, Kerman University of Medical Sciences, Kerman, Iran M. R. Baneshi Faculty of Medicine, Center for Longitudinal and Life Course Research, School of Public Health, The University of Queensland, Herston, Queensland 4006, Australia A. A. Haghdoost (&) HIV/STI Surveillance Research Center, and WHO Collaborating Center for HIV Surveillance, Institute for Futures Studies in Health, Kerman University of Medical Sciences, Kerman, Iran e-mail: [email protected]

and incidence of HIV and estimating the future course of the epidemic within countries. While most countries have developed HIV surveillance programs to track the epidemic and HIV-related risk behaviors, most lack the capacity to estimate the size of KPs, including people who inject drugs (PWID), female sex workers (FSW), men who have sex with men (MSM), transgenders and clients of FSWs. Given the importance of providing public health policy makers reliable information about the number of people at risk of HIV, it is crucial to quantify the population size of these populations. However, populations at increased risk or most-at-risk of HIV are sometimes composed of hidden or hard-to-reach individuals that engage in behaviors deemed as “criminalized,” “illegal” or “stigmatized.” Therefore, these populations may be reluctant to participate in public health programs that might lead to their identity being revealed (e.g., HIV surveillance programs and HIV prevention, care, and treatment programs). Recognizing the limitations of routine surveillance activities in capturing KPs, guidelines for developing methods to estimate the size of populations at risk of HIV were first developed in 2003 through a collaborative effort of Family Health International, the Impact Project, the United States Agency for International Development (USAID), the Joint United Nations Programme on HIV/AIDS (UNAIDS), the World Health Organisation (WHO) and the UN Drug Control Programme (UNDCP).

© Springer Nature Switzerland AG 2021 G. Rutherford (ed.), Methods in Epidemiology, Advances in Experimental Medicine and Biology 1333, https://doi.org/10.1007/978-3-030-75464-8_1

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M. R. Baneshi et al.

Population size estimation (PSE) of KPs provides valuable information for improving our understanding of the epidemiology of HIV given the importance of KPs in both generalized and concentrated epidemic and advocating for public health action and developing appropriate contextspecific interventions targeted to those at the highest risk of HIV. Indeed, developing compelling arguments to convince policy makers and funders about the magnitude of the epidemic can be quite challenging when data on the size of KPs are lacking and when the estimates are often baseless and inconsistent. Moreover, PSE is a vital component of planning and implementing HIV prevention, care, and treatment programs. For example, considering the sensitivities around HIV, investing in services aimed at improving the health of KPs can be quite challenging from a political perspective. PSEs can help inform resource allocation for better planning, management and monitoring provision of adequate services to these populations, tasks that are difficult without a picture of the number of people at risk of or living with HIV. Using a direct estimation technique of population size has various challenges and limitations. Surveys for direct PSE (owing to their relatively small population size) require a large sample size, which is often not feasible. The challenges are even more notable in countries, where due to religion and culture, sexual behaviors and the various sexual subpopulations are taboo, and so people may underreport their risk behaviors or sexual orientation. In the next section, we briefly review the main direct and indirect size estimation methods.

1.2

Direct Population Size Estimation Methods

PSE methods can be divided into two broad categories: direct and indirect. There is a need to contact members of groups whose size is being estimated in the direct method. These methods include population-based survey, census, enumeration, capture-recapture and multiplier methods.

1.2.1 Population-Based Surveys Conducting surveys among the general population or subgroups of the general population is a commonly used method for size estimation in many countries. Household surveys are the most often used surveys, which can vary based on the sampling frame deployed from telephone surveys in industrialized settings to face-to-face interviews in low- and middle-income countries. Assuming the sample is representative of the total population, the prevalence of risk behaviors among respondents yields an estimate of the total proportion of individuals engaging in the behavior (World Health Organization 2003, 2010). The total number of people with such behavior will be estimated by multiplying the proportion by the total number of people from census data. Population-based surveys are complicated and expensive, and there is a general hesitancy to increase the sample size. Nonetheless, given sufficiently high participation rates, the results are usually generalizable to the general population. In fact, they should be considered as a reliable method to estimate a common behavior in the general population. However, the real challenge arises when targeting a hidden or hardto-reach population. First of all, where behavior like injecting drugs or selling sex is rare, those engaging in the behavior may not be included in the sample even with random selection and large sample size. Second, many members of key population like PWIDs or FSWs may not have a stable or fixed place of residence. They may be living on the streets, brothels, jails or places that routine household surveys may not reach. So typical household surveys may miss a significant proportion of these people. Moreover, the stigmatized and illegal nature of many HIV risk behaviors poses a more serious problem when utilizing survey-based methods. Since people are less likely to provide honest answers to these types of questions or deny engaging in risk behaviors, the prevalence of such behaviors is often underestimated among

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Review of Size Estimation Methods

targeted populations (Brown 2003; Rehle et al. 2004; Purcell et al. 2012). Behavioral surveys have become a valuable approach to track risk behaviors and estimate hard-to-reach populations’ size in many countries. These surveys are conducted in targeted subpopulations with readily available instruments and within short time periods. However, they are also subject to some challenges. The implementation of these surveys usually needs an enriched review of existing data and rapid assessments. Otherwise, valuable resources will be wasted, and the survey may not provide essential information on target populations (Brown 2003).

1.2.2 Census/Enumeration Method Census and enumeration (mapping methods) have been often cited as the simplest methods of population size estimation. Census methods entail fielding teams that go to each site where KP members gather and count each person they find at each site. This process should ideally happen simultaneously since movements of population members between sites may lead to double counting (World Health Organization 2010; UNAIDS 2010). Enumeration is a modification of census mapping which utilizes the same method except that a sampling frame with a pre-determined size and structure is applied. A sample of units is selected from a sampling frame, and only the individuals within those chosen units are counted. Multiplying the average number of individuals per unit by the total number of units can yield the size of the key population in the sample frame (World Health Organization 2010). The main advantage of both of these methods is their comprehensibility. However, they are inefficient strategies (UNAIDS 2009). Where a list of places is available, the implementation of census/enumeration is straightforward. While the census and enumeration methods in theory provide very specific information about the distributions and types of KPs in different spots, they are often subject to underestimation or

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overestimation biases (World Health Organization 2010). As a practical matter, census methods do not operate strictly correctly when targeting hidden populations, in particular when those populations do not usually gather at fixed places or when they tend to move from their locations because of law enforcement or stigma they may encounter. On the contrary, when a KP is not properly defined and those who are not really members of the population are counted, or in cases where estimation is conducted over a long period of time and individuals are counted twice or more, overestimation can occur. Geographical diversity and population dispersion can influence the magnitude of overestimation or underestimation and are the other two serious challenges we must confront when estimating the size of a hard-to-reach population by mapping methods (World Health Organization 2010; UNAIDS 2010; Abdul-Quader et al. 2014; Mutagoma et al. 2014). Example To estimate the population size of street children in six cities in Iran, multiple methods including mapping and enumeration were used (Vameghi et al. 2018). Using key informant interviews and group discussions, we created a list of all potential venues where street children could be found and identified the days and time periods when the maximum number of street children were present. Key informants included diverse persons with knowledge of street children at the city level (e.g., municipal social welfare personnel, public and non-governmental service providers, academicians) and the street or venue level (e.g., street children themselves). The qualitative mapping phase was followed by a quantitative phase conducted at the venues in two visits. Our team interviewed 114 key informants with knowledge of street children at the city level, 223 key informants with local or neighborhood knowledge, and 933 street children with street and venue level

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knowledge. Key informants and group discussants identified 370 venues (range 23 to 113 per city across the six cities) where street children could purportedly be found. Interviews with street children in the field identified an additional 94 venues (range 4 to 32 across cities). Of the 464 total venues, our team visited 226 (48.7%) at randomly selected venue-day-times for the one-hour counting periods (visit 1), and 200 venues (43.1%) for four-hour periods. During the one-hour observation periods at the venues in the six cities, we counted 53 to 709 street children per city, or 3.4 to 10.6 children per venue per city on average. Additionally, we counted 320 to 2,364 street children in the sub-sample of venues during four-hour periods to generate incremental numbers of children, ranging from 7.1 to 16.3 per venue per city. Applying these averages, we estimated the total number of street children at the venues in a 14-hour turn-over period for all six cities at 9657. These direct count estimates ranged from 486 in Zahedan to 4,785 in Tehran.

1.2.3 Nomination Method The nomination method starts with the accessible part of the hidden population, for instance, PWID receiving services from non-governmental organizations (NGO). These persons are asked to provide contact information of others who share their risk behaviors. However, the accessible group will likely not provide the names or locating information of others they know due to the stigmatized nature of their behavior and concerns about law enforcement (Rees Davis et al. 2003). Despite this limitation, the nomination method may partially work for groups that NGOs or similar organizations support. However, as explained earlier, in countries with strict

religious or cultural restrictions against certain risk behaviors, the majority of KPs are hidden. This limits the practicality of the nomination method in such situations.

1.2.4 Capture-Recapture Method The capture-recapture method (CRM) relies on the utilization of (at least) two independent samples or (at least) two non-interrelated existing data sources (direct or indirect CRM approaches). The direct CRM approach entails counting KPs members, like FSWs interviewed in a survey, and then having a second survey revisit the population ideally using different methods. By counting the overlap, that is KP members that appeared in both the first and second surveys, the total population size can be estimated mathematically. Distributing a card or some memorable gift at the first sampling and asking participants whether they have received the object or completed the survey in the second sampling is a way to enhance individuals’ memories of having participated in the first survey. In the indirect CRM approach, we use two existing lists of FSWs. Records from, for instance, a sexually transmitted infection (STI) clinic or data from a social insurance system could serve as the first source. A police database or a brothel registry could be the second source. We then determine those individuals captured on both lists (World Health Organization 2010; Mutagoma et al. 2014; Xu et al. 2014). The number captured in the first sample (M) is multiplied by the number captured in the second sample (C), and then divided by the number captured in both rounds (R): N ¼ ðM  CÞ=R

ð1:1Þ

Where census or enumeration is not feasible or when the quality of data is subject to fluctuations, CRM seems a better option. This method, however, also suffers from potential biases (World Health Organization 2010; Kimani et al.

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Review of Size Estimation Methods

2013). There are a number of critical assumptions that need to be addressed properly when using CRM. The most important one is probably the population being closed with no one migrating in or out between the two rounds. This criterion is often violated when working with hard-to-reach individuals who typically move in and out of specific geographic areas. PWIDs, for example, frequently tend to leave the population. Some may stop using drugs, some may be incarcerated, some may die from overdoses or violence, and some may move away. Moreover, new drug users may enter the target population between the two survey rounds, which again undermines the credibility of our estimates. The randomness and independence of samples/sources are other important criteria. Each individual in the population must have an equal (or known) chance of being selected in both of the surveys; otherwise, the results will be biased. For instance, returning to the same facilities or brothels to capture FSWs would lead to recruiting a high percent of the same individuals, but they would not be a random sample of all members of the FSW population in an area, leaving some individuals less likely to be recruited. Another important source of bias in CRM appears when matching individuals from two sample sources. A firm and clear definition is often needed to properly match individuals; this is sometimes difficult to achieve. Members of hard-to-reach populations usually do not provide similar names at different visits. It has been argued that “in practice, estimation of drugmisusing populations using simple, two-sample capture-recapture methods is neither easy nor reliable” (Abdul-Quader et al. 2014; Mutagoma et al. 2014). It is possible to generalize the setting and to identify individuals reliably in more than two rounds. This allows the application of modeling techniques, such as log-linear models, which can provide more accurate estimates. In this case, the assumption of independence can be relaxed, and it is possible to take into account the interaction between sources. However, the analyses of data are more complicated.

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Example 1 Fars Province, located in southern Iran, has a population of 4,336,878. In a study to estimate the completeness of cancer mortality in Fars province, three data sources were selected to use CRM and estimate the true number of cancer deaths. The sources were the Fars provincial death registry, cancer mortality data created from followup of pathology-based cancer registry data and follow up of the hospital cancer diagnoses. Results of log-linear capture–recapture modeling suggested that the current death registry underestimates the true cancer mortality rate by about 40%. After correction for underestimation, cancer mortality rate increased from 44.8 per 100,000 to 76.3 per 100,000 (Marzban et al. 2015). Example 2 In another study to estimate the number of HIV patients in Fars, information from three sources was gathered: hospitals, voluntary counseling and testing centers and prisons. A total of 5,167 HIV-infected patients were identified. The most recorded cases (3,347) were from the voluntary counseling and testing centers. The fewest recorded cases were the cases recorded at both prisons and hospitals. The revised estimate of HIV-infected patients in Fars province was about 18,914 (Joulaee et al. 2013).

1.2.5 Multiplier Method The most commonly used type of survey-based method developed to overcome the limitations of mapping methods, particularly the need for a comprehensive list of sites, is the multiplier method (World Health Organization 2010). This method depends on comparing the derived information from two distinct and independent data sources—a list followed by a survey—that overlap in a known way (Paz-Bailey et al. 2011; Johnston et al. 2011; Khalid et al. 2013).

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A list or count of individuals in the target population who have either accessed some type of services or received a “unique object” that has been distributed as a part of the method is the first source of data (World Health Organization 2010; Abdul-Quader et al. 2014). For example, the number of FSWs who attended an STI clinic in the last month is a possible first list that can be used in the multiplier method (service multiplier). Furthermore, drug treatment and needle exchange programs, social services, prisons or laboratories where individuals are tested for HIV, hepatitis C virus (HCV) infection or hepatitis B virus (HBV) infection could be potential data sources for the list, that is the first element, for estimating the number of PWID. A unique object multiplier can also be applied by randomly distributing an object, which has been preferably designed to be uniquely memorable. Participants would be asked then in the survey if they had received the object or not. Examples of this include a key ring or another small item that does not have sufficient value to warrant sale. The second source of data is a representative survey of the target population defined in the same way in the same area. As a practical matter, the difference between definitions of KPs may yield vastly different results in data sources. Multiplying the number of those who received the service or the unique object by the inverse of the proportion reporting receiving the service or object would yield the population size (Khalid et al. 2013; Johnston et al. 2014). To provide greatest accuracy, the population should be randomly captured in the survey so that everyone who had a chance to be on the list can be captured in the survey. The independence of the two data sources is the key challenge in the multiplier method. The multiplier method is preferable to census/enumeration when the population is hard to reach or when it is difficult to develop a standard sampling frame; however, it comes with certain, similar biases. Duplication can be a source of overestimation in the service multiplier method. Consider as an example, in an STI clinic, and a service provider may have only the

M. R. Baneshi et al.

number of visits during a timeframe, not the number of individuals that attended that clinic. The unique object multiplier has been able to overcome this problem to some extent since such problems could be controlled by the survey team (World Health Organization 2010). The magnitude of biases in the multiplier heavily depends on a list followed by a survey on the data quality on the list. The existing data sources may not provide precise and valid data for estimations. On the other hand, in the survey phase conventional sampling methods are not well-suited for accessing all group members in most of the hard-to-reach populations; hence probability sampling methods such as timelocation cluster sampling and respondent-driven sampling should be considered as two options for obtaining key populations samples. However, these methods also suffer from a potentially unacceptable level of bias (World Health Organization 2010; Paz-Bailey et al. 2011; Purcell et al. 2012; Abdul-Quader et al. 2014). Another type of multiplier that has been widely recommended in the literature is using information from two separate population-based survey samples that intersect in important ways like female sex workers and their clients. This requires the size of one of the groups to be relatively well known. Potential sources of information, in this case, could be investigated to be utilized to arrive at estimates of subpopulations at risk for HIV. In all types of multiplier methods, the need for precise and timely demographic and geographic information to link the data sources limits the usefulness of the approach (Abdul-Quader et al. 2014). Example We used three PSE methods including the multiplier method to calculate the number of female sex workers (FSW) in 13 cities in Iran (Sharifi et al. 2017). For the multiplier method, we required two sources of data. For the first data source, we used administrative data from social welfare organizations or facilities supported by the

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Ministry of Health. These data included unduplicated FSW client counts of services for HIV testing, free condoms, drug treatment and clinical care. We also used participation in a 2010 survey of FSW as a source for the first count. The second data source was a 2015 FSW survey, which determined the proportion of the FSW who received a unique objects (a pocket-size mirror distributed during one month before the survey) or any of services at the specific facility during the specified period. Averaging the various multipliers in each city, the overall population size of FSW was estimated to be 19,800 (95% uncertainty interval: 10,900–38,100).

1.3

Indirect Population Size Estimation Methods

Indirect PSE methods are those in which there is no direct contact with hidden groups. The main approaches in this category are Cross-Wise (CW), Network Scale-Up (NSU), and proxy respondent (PR) methods (Catania et al. 1990; SA et al. 2014).

1.3.1 Cross-Wise Method The basis of this model is the matching of one sensitive question with one non-sensitive question in a survey (Jann et al. 2010; Veen 2014). The non-sensitive question must be independent of the sensitive one. We can ask a random sample of respondents, selected from the population to which we wish to generalize the results, to answer both questions simultaneously. The respondent is asked to choose the option “A” if his or her answers to both questions are the same (both yes or both no) and choose “B” if his or her answers are different (one of them yes, another is no). According to the frequency of option “A,” we estimate the prevalence of sensitive trait. Suppose that the probability of non-sensitive trait

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is p, and n/N is the proportion that said yes to both questions (i.e., selection of option A divided by the total sample size, A + B). The prevalence of sensitive trait, p, can be estimated applying the following formula: p ¼

n N

þ p1 2p  1

ð1:2Þ

The CW method is an efficient way to estimate the size of several hidden groups in one single survey. In addition, it guarantees confidentiality and increases the chance of getting reliable answers. However, our experiences so far suggest that this method works well in educated populations such as students (Kazemzadeh et al. 2016), but not in general populations [unpublish data]. We postulate that, for members of the general population who are a mix of well and less-educated people, it might be difficult to utilize the CW method as participants might not trust the research team. Example We conducted a cross-sectional study on students of Kerman University of Medical Science (KMU), one of the main universities in southeastern Iran, in 20122013 (Kazemzadeh et al. 2016). This university consists of seven faculties with approximately 5200 students (60% women). A proportional to size stratified sample was used to recruit 563 students. The prevalence of opium use, alcohol consumption, amphetamine use, taking tramadol without a medical indication, having relationships with the opposite sex (RWOS), and extra/ pre-marital sex (EPMS) in the previous year were estimated (even one episode). Data were collected using an anonymous researcher-made questionnaire designed based on the objectives of the study. The reliability and validity of the questionnaire were first evaluated in a pilot study on 28 subjects. The questionnaire consisted of two parts: the first part consisted of six pairs questions; that is, one

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sensitive question (e.g., have you used opium in the last year?) and a non-sensitive question (e.g., were you born in winter?). We asked respondents whether their reply to both questions is yes or not. Prevalence of measured risky behaviors were as follows: opium 2.2%, alcohol 16.8%, methamphetamine 7.2%, tramadol without a medical indication 14.8%, friendship and close relationship with a member of the opposite sex 42.8%, and extra/ pre-marital sexual contact 12.4%. In another study, to estimate the frequency of STI-related symptoms, we conducted a cross-sectional study in Kerman in 2013. A total sample of 128 individuals from the general population, aged 18 to 60, was recruited using multi-stage random sampling. The frequency of genital ulcer and urethral or vaginal discharge were estimated at about 43.7% (male: 53.1%, female: 34.9%) and 79.3% (male: 72.3%, female: 86.3%), respectively. Estimated prevalence was much higher that the results of other studies across the country and were thought to be inaccurate by experts. The prevalence obtained by faceto-face interviewing of the general population of Kerman in the same year was much lower than the estimates obtained by the CW method (unpublished data). We surmise that one issue that might partially explain the findings is that some of the respondents may not have properly understood the method or may have been confused and randomly answered the questions, hence leading to marked overestimates of the prevalence.

1.3.2 Network Scale-up Method In the NSU method, a random sample of the general population describes the presence of individuals belonging to hidden groups in their social networks (C). Based on the prevalence of

hidden groups in the social network of the selected sample, the population size of these groups in a community can be estimated. Let us assume a population T with size t and a subpopulation E of size e. All members of population E also belong to population T but not vice versa. If, on average, members of T know m subjects in E, then the following formula seems reasonable (Eq. 1.3). Here C corresponds to the average number of people known by members of T: m=C ¼ e=t

ð1:3Þ

This is the basic definition. It is possible to generalize it by integrating information from several reference groups (see Chap. 2). Here a clear definition of “know” is required. It is usual practice for “know” to be defined as the mutual recognition of each other by sight or name, the known person can be contacted, and in the past one year, the respondent has had contact with the known person either in person, face-to-face, phone or by email. This definition fits a social network. It is also possible to limit the definition. For example, in Rwanda, a “meal definition” was applied. Participants were asked to describe the frequency of risky behaviors among members of their network with at least one meal in the past 12 months (Rwanda Biomedical Center 2012). In another study, to estimate the number of deaths due to the Bam earthquake in Iran in 2003, we selected one member from each randomly selected household. We then asked them to tell us the number of their neighbors who died in the earthquake. Three houses on the right and three houses on the left of the randomly selected home were defined as the neighborhood (Daneshi et al. 2014). Several studies have been carried out in the United States to estimate hidden subgroups, including a number of HIV-infected groups (Killworth 1998); heroin or crack users (Kadushin et al. 2006); women who have been raped (Killworth 1998) or were victims of robbery (Killworth 1998), assault or burglary (Killworth 1998); alcohol consumers (Nikfarjam et al. 2017); and marijuana users (Salganik et al.

1

Review of Size Estimation Methods

2011a). Other studies were also conducted in Moldova (Stroup 2010), Ukraine (Paniotto et al. 2009), Thailand (Kanato 2015), Rwanda and Kazakhstan to estimate the population size of PWID, FSWs and MSM (Rwanda Biomedical Center 2012; Hansson et al. 2008); in Brazil to estimate the population size of heavy drug users (Salganik et al. 2011a, b); in Japan to estimate MSM (Ezoe et al. 2012); and in Iran to estimate groups at high risk of HIV/AIDS (Nikfarjam et al. 2016). It has been noted that NSU methods can be used as a part of surveys of the general population with the minimum level of resources (Paniotto et al. 2009). One advantage is that NSU methods require no direct contact with members of hidden groups. Another is that the NSU method allows the estimation of the size of many groups in one study. One of the assumptions of this method is that respondents are aware of the sensitive behaviors of those from their network (Salganik et al. 2011a, b). However, sensitive characteristics are not visible. Therefore, the application of appropriate correction factors to correct for the problem of visibility is necessary. These issues are discussed later. Example 1 We designed a cross sectional study in the city of Kerman (the capital of Kerman province), located in southeastern Iran, with a population of around 500,000. Our target population was males between 18 and 45 years old who had lived in Kerman in the past five years (t = 132,651). We selected a random sample of 500 from this group to explore their social networks. We asked respondents how many people they knew in any of hidden groups, with at least one episode of a specific act in the past last year. The biggest populations were alcohol users followed by the opium users. The

9

size of these two groups were 13.7% (95% CI: 11.3%, 16.1) and 13.1% (95% CI: 10.9%, 15.3%), respectively. The smallest population estimate was PWID at 1.2% (95% CI: 1%, 1.4%). In the study by Kazemzadeh and colleagues, the aim was to estimate the prevalence of risky behaviors among university students, applying NSU and CW methods to cross validate the findings (Kazemzadeh et al. 2016). To calculate the size of the active social network of the students, participants were asked about a number of their close friends. A close friend was defined as “a person whom the student knows by face and name, the close friend knows the study participant as well, they are in touch/contact several times a week and they spend at least 2 h a week, continuously or discontinuously together”. Respondents were then asked to introduce how many of their close friends practice a series of risky behaviors. Estimates derived from the NSU method were lower than those derived from the CW approach. For example, alcohol consumption and extra/pre-marital sexual contact were estimated at 8.1% and 7.1%, respectively. Corresponding estimates using the CW method were 16.8 and 12.4%. We should emphasize that results of NSU studies should be adjusted for visibility. In this study, no correction factor was applied, and this may partially explain the differences. We believe that ratio of NSU to CW methods provides a measure of visibility. This is because in CW studies respondents reply on behalf of themselves while in NSU studies they on behalf of their network. In our work, visibility of alcohol use and pre/extra marital sexual contact was estimated at around 50%.

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M. R. Baneshi et al.

Example 2 As another example, in 2014, we implemented an NSU study to estimate the number of deaths that occurred in the Bam earthquake of 2003 and to compare it with other organizations’ estimates (Daneshi et al. 2014). The Bam earthquake killed thousands of people. After the earthquake, different statistics about the number who had died in the earthquake were published. We randomly selected 80 households from different parts of the city. One person was selected per household. We asked the respondent how many of his or her neighbors had died due to earthquake. The neighborhood was defined as three houses on the right and three on the left sides of the participant’s home (6 in total). We estimated a number of deaths to be approximately 54,000. Official statistics published by the Kerman council and governor estimated the number of death to be 45,000.

1.3.3 Proxy Respondent Method The PR method is a modified version of the NSU method in which there is no need to estimate the size of the social network. In this method, a random sample of respondents (proxy respondents) is asked about whether they know someone with a specific name (termed an “alter”) and whether that person practices the risky behavior of interest. The main assumption of PR method is that the random sample of the selected respondents and alters forms a representative sample of the community (Rwanda Biomedical Center 2012; McCarty et al. 2001). The most common and popular names in both sexes with a frequency of between 0.1% and 4% are selected, without belonging to a specific religion, race or ethnic group. Two sets of cards (for male and female names) are prepared that include a few names of the same sex. We ask the respondent to randomly select a card and a name.

If the proxy respondent knows more than one person with the name he or she selects, the interviewer will ask the respondent to choose someone who knows better (except people in the same home) (Rwanda Biomedical Center 2012). We then ask respondents to report whether that select person practices the behavior of interest or not. The PR method does not require an estimation of network size. The cost, however, is a much larger sample size. This is because here, each respondent replies only on behalf of one person. Also, as the proxy respondents know that we wish to ask them to describe the risky behavior of one of their close friends, they might prefer to select subjects who are not engaged in such behaviors (i.e., prestige bias). Example We recruited a random sample of 500 students from Kerman University of Medical Sciences (KMU) (Sheikhzadeh et al. 2016) to measure the lifetime prevalence of stigmatized behavior including alcohol use, drugs use (opium, cannabis, heroin and drug injection) and extramarital sex by two PSE methods: NSU and the PR method. For NSU, we first asked participants to report the total number of same-sex students they knew and how many of them had any of the above risk behaviors. For the PR method, we choose 30 female names and 30 male names with unique spellings and a frequency of at least 0.1% and at most 4% of the general population of Iran. Names were distributed on six cards for males and six cards for females. Male respondents were asked to select one of the cards with male names, and similarly, females were asked to select a card with female names. Respondents were then asked to identify the closest person (i.e., an alter) to them with one the listed names in the card. Then, the respondents were asked to report if the alter had any of the above risky behaviors. The frequency of risky behaviors among alters was used as an

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Review of Size Estimation Methods

11

1.4 estimate for the frequency of those behaviors among college students. We found in both PSE methods, higher proportion of male students had risk behaviors than female students (Table 1.1). Estimates from the PR method were consistently higher than NSU estimates with a PRM to NSU ratio ranging from 1.79 for extra-marital sex (in males) to 16.57 for opium use (in females). These findings suggest that the PR method is less prone to information bias because respondents were asked to recall the behaviors of only one known alter whose name was shown on the selected card without disclosing that alter’s identity. In contrast, in the NSU method, they had to look for, recall and count the total number of individuals with risky behaviors within their social networks. Overall, NSU estimates comes closer to estimates using the PR method for more common behaviors like alcohol use and extra-marital sex. These behaviors might have more social visibility and so less affected by recall bias and transmission error (see next chapter for details). So, it is crucial to adjust NSU estimations for such biases.

Table 1.1 Prevalence of risky behaviors among university students: comparison of NSU and PR methods

Risk behaviors

Sex

Alcohol use

Female Male

Opium use

Female Male

Cannabis use

Female Male

Heroin use

Female

Advantages of Network Scaleup Over Other Size Estimation Methods

The main advantages and limitations of PSE methods are summarized in Table 1.2. An important issue in selecting PSE methods is the availability and ease of access to the inputs required. For example, in the capture-recapture method, at least two sources of independent data that record information on high-risk groups is required. Other methods described, need also immediate contact with a random sample of hidden groups. However, as some groups at high risk of HIV have low social respect, respondents will most likely not reveal such personal information. Our experience in Iran shows that access to a random sample of hidden groups is extremely difficult. This is mainly because no NGO supports the majority of the hidden groups, particularly those with risky sexual behaviors. In the case of other groups, such as PWID, those PWID who register at NGOs or governmental organizations are fundamentally different than those who remain hidden. This violates the assumption of the randomness of respondents and leads to biased estimates due to selection bias. In contrast to other methods, three main issues make the NSU methods more attractive and

PR method (%)

NSU (%)

2.32

0.44

PRM to NSU ratio 5.27

18.12

8.68

2.09

1.16

0.07

16.57

9.39

3.02

3.11

0.77

0.00

–

2.01

0.17

11.82

0.38

0.00

–

Male

2.01

0.17

11.82

Drug injection

Female

0.39

0.00

–

Male

0.67

0.23

2.91

Extra-marital sex

Female

3.47

0.95

3.65

13.42

7.48

1.79

Male

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M. R. Baneshi et al.

Table 1.2 Summary of main population size estimation methods with their advantages and limitations Method

How it works

Requirement

Advantages

Disadvantages

Other issues

Census

Counts all members of hidden group

Direct contact with the whole population of hidden group

Straightforward

Expensive Under-estimates the size

When hidden population is geographically dispersed, it does not perform well

Enumeration

Counts a fraction of hidden group

Direct contact with a part of population of hidden group

Straightforward

Under-estimates the size

Fraction studies should be a random sample of the hidden population

Capturerecapture

Number of captures of two rounds

Two independent captures of hidden group

Easy to use and to apply the formula

Assumptions are hard to meet Sources might be inaccurate

Nomination

Contact with visible part and ask them to provide information of other members

Limited access to hidden group

Useful for presurveillance activities

Visible part might not report other members, and differ with others Selection bias

Multiplier

Compares two independent sources of data for hidden groups

Two independent sources of data about a hidden group

Straightforward if data are available

Needs independent sources of data with a similar definition of the population Sources might be inaccurate

Highly dependent on the quality of the data

Cross-wise

Matching answers of one sensitive question with one nonsensitive question

Accurate probability of answers to the nonsensitive question

Can estimate the size of several hidden groups in one survey, guarantees confidentiality

Individual with risk behaviors cannot be identified for further research or services

Not easily understood by participants

Proxy respondent

Participants report the risk behaviors of their alters, not themselves

Random sample of participants and a process to choose the alters

Straightforward

Prone to the visibility of behaviors and recall of alters’ behaviors

Transmission rate should be estimated

Network scale-up

Frequency of risky behaviors in network size of the respondent

Network size should be known or estimated

Possible to estimate size of many groups in one study

Prone to visibility of behaviors and recall of social network size and recall of behaviors

Transmission rate and popularity ratio should be estimated

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Review of Size Estimation Methods

applicable. First, direct contact with hidden groups is not required. Second, participants are not questioned about their behavior but the behaviors of members of their networks. Third, it is possible to estimate the size of several hidden groups in a single NSU study. Therefore, the remainder of this chapter focuses on practical issues in conducting NSU studies.

1.5

Requirements of Network Scale-up

The four main steps of an NSU study are an estimation of network size (C), an examination of the estimation’s reliability, a preliminary estimation of the size of hidden groups, and estimation of correction factors required to correct the preliminary estimations (Zhang et al. 2007; Hickman et al. 2006). These issues are addressed in great detail in Chaps. 2 and 3.

1.5.1 Network Size The NSU method depends on the active network size of the population (C). The idea behind the NSU method is that the proportion of individuals belonging to a subpopulation in the network of a representative sample has a direct association with the real size of that subpopulation in the general population. Therefore, the estimation of C is paramount.

13

behaviors in their active network (Russell Bernard et al. 1991; McCarty et al. 2001; Jackson et al. 2005; Stroup 2010). However, the justification of this assumption is difficult because of transmission error, a concept implying that sensitive information is not always transmitted. This error results in an underestimation of the size of target populations (Killworth et al. 2003; Salganik et al. 2011a, b). In addition, we assume the network size of the members of the hidden and general populations are the same. However, this may not always be the case, and the network size of populations may differ, introducing a concept called relative network size (Salganik et al. 2010). For instance, the network size of HIV patients might be onethird of that of the general population (Paniotto et al. 2009). Member of KPs might have smaller networks, possibly due to the stigmatized nature of their behaviors and less mixing with the general population. Therefore, transmission error and relative network size should be taken into account for adjusting NSU crude estimates. So far, only a few studies conducted in Ukraine, Brazil, Rwanda and Japan have measured and reported such correction factors (Paniotto et al. 2009; Salganik et al. 2011a, b; Rwanda Biomedical Center 2012; Ezoe et al. 2012). However, these statistics could be generalized to highstigmatizing settings like Iran or other neighboring countries in the Eastern Mediterranean region.

1.5.2 Preliminary Size Estimation

References An average number of hidden subgroups reported by every respondent constitutes a fraction of their network. Therefore, we can scale-up this fraction to a fraction of the total population and thus to obtain and estimate the number of the hard-toreach groups of interest (Kadushin et al. 2006).

1.5.3 Correction Factors One assumption in the NSU method is that respondents are aware of people’s sensitive

Abdul-Quader Abu S, Baughman Andrew L, Hladik Wolfgang (2014) Estimating the size of key populations: current status and future possibilities. Current Opinion in HIV and AIDS 9(2):107–114 Brown Tim (2003) Behavioral surveillance: current perspectives, and its role in catalyzing action. JAIDS J Acq Immune Def Synd 32:S12–S17 Catania J et al (1990) Methodological problems in AIDS behavioral research: influences on measurement error and participation bias in studies of sexual behavior. Psychol Bull 108(3):339–362 Daneshi S et al (2014) The estimated frequency of spinal cord injury, amputation (Hands and Feet) and death in

14 the bam earthquake using the network scale up method. Iran J Epidemiol 10(3):9–14 World Health Organization. Estimating the size of populations at risk for HIV: issues and methods (2002) Updated 2003 Ezoe S et al (2012) Population size estimation of men who have sex with men through the network scale-up method in Japan. PLoS One 7(1): Hansson M et al (2008) HIV/AIDS awareness and risk behavior among students in Semey, Kazakhstan: a cross-sectional survey. BMC Int Health Human Rights 8(1):14 Hickman M et al (2006) Estimating prevalence of injecting drug use: a comparison of multiplier and capture-recapture methods in cities in England and Russia. Drug Alcohol Rev 25(2):131–140 Jackson D et al (2005) Social network analysis and estimating the size of hard-to-count subpopulations. Connections 26(2):49–60 Jann B, Jerke J, Krumpal I (2010) Asking sensitive questions using the crosswise model: some experimental results. University of Leipzig, Institute of Sociology Johnston L, Saumtallyb A, Corcealb S et al (2011) High HIV and hepatitis C prevalence amongst injecting drug users in Mauritius: findings from a population size estimation and respondent driven sampling survey. Int J Drug Policy 22:252–258 Joulaee H et al (2013) Estimated number of patients with HIV in fars province using capture-recapture method, 1990-2010. Hakim 16(2):128–136 Kadushin C et al (2006) Scale-up methods as applied to estimates of heroin use. J Drug Iss 36(2):417 Kanato M (2015) Size estimation of injecting drug users through the network scale-up method in Thailand. J Med Assoc Thai 98(Suppl 6):S17–S24 Kazemzadeh Y et al (2016) The frequency of high-risk behaviors among Iranian college students using indirect methods: network scale-up and crosswise model. Int J High Risk Behav Add 5(3): Khalid FJ, Hamad FM, Othman AA et al (2013) Estimating the number of people who inject drugs, female sex workers, and men who have sex with men, Unguja Island, Zanzibar: results and synthesis of multiple methods. AIDS Behav. https://doi.org/10. 1007/s10461-013-0517-x Killworth PD et al (1998) Estimation of seroprevalence, rape, and homelessness in the United States using a social network approach. Eval Rev 22(2):289–308 Killworth PD et al (2003) Attempting to quantify transmission and barrier errors in scale-up methods Kimani J, McKinnon LR, Charles Wachihi C et al (2013) Enumeration of sex workers in the central business district of Nairobi, Kenya. PLoS One 1:1–5 Marzban M, Haghdoost AA, Dortaj E, Bahrampour A, Zendehdel K (2015, March) Completeness and underestimation of cancer mortality rate in Iran: a report from Fars Province in southern Iran. Arch Iran Med 18 (3):160–166. doi:0151803/AIM.005. PubMed PMID: 25773689

M. R. Baneshi et al. McCarty C et al (2001) Comparing two methods for estimating network size. Human Organ 60(1):28–39 Mutagoma M, Kayitesi C, Gwiza A, Ruton H, Koleros A, Gupta N, Balisanga H, Riedel DJ, Nsanzimana S (2014) Estimation of the size of the female sex worker population in Rwanda using three different methods. Int J STD & AIDS 0956462414555931 Nikfarjam A et al (2016) National population size estimation of illicit drug users through the network scale-up method in 2013 in Iran. Int J Drug Policy 31:147–152 Nikfarjam A et al (2017) The frequency of alcohol use in Iranian urban population: the results of a national network scale up survey. Int J Health Policy Manag 6 (2):97 Paniotto V et al (2009) Estimating the size of populations with high risk for HIV using the network scale-up method. Kiev Int Inst Sociol, Ukraine, pp 1–47 Paz-Bailey G, Jacobson JO, Guardado ME et al (2011) How many men who have sex with men and female sex workers live in El Salvador? Using respondentdriven sampling and capture recapture to estimate population sizes. Sex Transm Infect 87:279–282. https://doi.org/10.1136/sti.2010.0456332013 Purcell DW, Johnson CH, Lansky A, et al (2012) Estimating the population size of men who have sex with men in the United States to obtain HIV and syphilis rates. The Open AIDS J 6(Suppl 1: M6):98–107 Rees Davis W et al (2003) An enumeration method of determining the prevalence of users and operatives of cocaine and heroin in Central Harlem. Drug Alcohol Depend 72(1):45–58 Rehle T, Lazzari S, Dallabetta G, Asamoah-Odei E (2004) Second-generation HIV surveillance: better data for decision-making. Bull World Health Organ 82 (2):121–127 Russell Bernard H et al (1991) Estimating the size of an average personal network and of an event subpopulation: some empirical results. Soc Sci Res 20(2):109– 121 Rwanda Biomedical Center (2012) I.o.H.A., Disease Prevention and Control Department, Estimating the Size of Populations through a Household Survey SA et al (2014) Estimating the size of key populations: current status and future possibilities. Epidemiol Concent Epid 9(0):1–8 Salganik MJ et al (2010) The game of contacts: estimating the social visibility of groups. Social Networks Salganik MJ et al (2011a) Assessing network scale-up estimates for groups most at risk of HIV/AIDS: Evidence from a multiple-method study of heavy drug users in Curitiba, Brazil. Am J Epidemiol 174 (10):1190–1196 Salganik MJ et al. (2011b) Web Appendix Assessing network scale-up estimates for groups most at risk for HIV/AIDS: evidence from a multiple method study of heavy drug users in Curitiba, Brazil Sharifi H et al (2017) Population size estimation of female sex workers in Iran: synthesis of methods and results. PLoS One 12(8):

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Sheikhzadeh K et al (2016) Comparing direct, network scale-up, and proxy respondent methods in estimating risky behaviors among collegians. J Subs Use 21 (1):9–13 Stroup DF (2010) Size estimations for high risk groups Moldova. Unpublished, Chisinau UNAIDS (2010) Training manual on methods for size estimation of key at-risk populations in the AsiaPacific region UNAIDS (2009) Guidelines on estimating the size of populations most at risk to HIV Vameghi M et al (2018) Population size estimates of street children in Iran: synthesis of multiple methods. The Journal of Adolescent Health (under review)

15 Veen D (2014) Multivariate analysis for using the Crosswise- and Triangular method. Bachelor Thesis World Health Organization (2010) Guidelines on estimating the size of populations most at risk to HIV Xu Yuan, Murray Fyfe, Liz Walker, Laura LE Cowen (2014) Estimating the number of injection drug users in greater Victoria, Canada using capture-recapture methods. Harm Red J 11(1):1 Zhang D et al (2007) Advantages and challenges of using census and multiplier methods to estimate the number of female sex workers in a Chinese city. AIDS Care 19 (1):17–19

2

Methods to Estimate the Average Social Network Size Mohammad Reza Baneshi, Saiedeh Haji-Maghsoudi, Azam Rastegari, and Ali Mirzazadeh

2.1

Introduction

One of the requirements to calculate the size of a KP by the NSU method is to know the average social network size of the general population accurately (shown by C). There is no universal network size that can be used for different settings and populations. The figure varies across countries and in different populations due to differences in cultural and social structure factors. As shown in Fig. 2.1, the average social network size has a

wide range from 175 in Ukraine [64] to 364 in Japan (Tourangeau and TWJPoq 1996). These values are not fully comparable as the methodologies applied are not the same. This chapter discusses direct and indirect methods (Snidero et al. 2004) to estimate the average social network size. The indirect method involves several steps that need to be carefully designed and implemented. We provide enough details, including real-world examples, for each method that will allow readers to design and conduct their own studies.

2.2 Electronic Supplementary Material The online version of this chapter (https://doi.org/10.1007/978-3030-75464-8_2) contains supplementary material, which is available to authorized users. M. R. Baneshi (&)
S. Haji-Maghsoudi
A. Rastegari Modeling in Health Research Center, Institute for Futures Studies in Health, Kerman University of Medical Sciences, Kerman, Iran e-mail: [email protected]; [email protected] M. R. Baneshi Faculty of Medicine, Center for Longitudinal and Life Course Research, School of Public Health, The University of Queensland, Herston, Queensland 4006, Australia A. Mirzazadeh Department of Epidemiology and Biostatistics, Institute for Global Health Sciences, University of California San Francisco, San Francisco, USA

Direct Estimation of C

2.2.1 Global Method In the direct approach, we simply ask respondents about the number of people they know. The question that is usually being asked from the respondents is “over the last two years, how many people you have met in-person or contacted by phone, etc., whom you recognize by face and they also can recognize you by face and can contact you if they wanted to.” This is the standard definition of “know” used in the literature. While this may appear to be a simple and clear question, providing an accurate answer to it is challenging. Try to answer to this question you yourself, and you will find it to be a difficult question to answer. This is because you would not be able to accurately recall all members of your social network. Most of the time, respondents

© Springer Nature Switzerland AG 2021 G. Rutherford (ed.), Methods in Epidemiology, Advances in Experimental Medicine and Biology 1333, https://doi.org/10.1007/978-3-030-75464-8_2

17

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M. R. Baneshi et al.

400

364

355

350 308 286

300

236

250 200

251

175

150 100 50

social network into mutually exclusive categories, that is, friends, colleagues, family members, etc., and ask every respondent about the number of people he or she knows in each category. Then, for each respondent, we add them up to calculate the overall social network size. This method is known as the summation method. Breaking down the social network into subcategories can help people recall the number they know better and thus produce more reliable estimates (Snidero et al. 2004). However, there is some evidence that this method suffers from underestimation. In addition to that, it is not possible to check the reliability of responses.

0

Fig. 2.1 Average network size of selected countries

provide a rough estimate of their social network size like 50, 100, 200, 500, or even sometime unbelievably high numbers such as 1,000. It is hard to believe that a respondent can recall and count 1,000 individuals in his or her social network; in fact, he or she is unable to count them precisely and thus gives a number, like 1,000, which indicates that he or she had too many people to count or that this represented an upper limit. In some cases, when you are estimating smaller networks, the direct method can work. For example, the network size of students in a class, those in the same prison, or those in a neighborhood can be determined by asking a global question. For example, in a school, you can ask each student about the overall number of classmates they know by name and in-person [65]. Obviously, by limiting the geographical location and asking only about a specific small portion of an individual’s network, we expect to receive more reliable responses.

2.2.2 Summation Method Instead of asking one question for the overall social network size, we can break down the

Example Brewer and Webster (1999) asked 217 residents of a students hall to recall their friends in the hall. Then a list of all residents at the hall, from which they were asked to recognize their friends, was given to them. It was found that, on average, 20% of friends were forgotten. Failure to recall was independent of characteristics of informants, and no difference was found between those recalled and those recognized from the list. This indicates that the summation method suffers from recall bias (Sulaberidze et al. 2016).

2.3

Indirect Estimation of C: Reference Groups Approach

To determine C, there are indirect methods in which we ask each respondent about the number of people in his or her network who belong to specific subpopulations, for example, teachers, policemen, university students, etc. These subpopulations are known as reference groups, and their real sizes are known. People with larger social networks theoretically know more people from these subpopulations. Conversely, the more people you know from these subpopulations, the bigger your social network size is. The indirect method assumes that the number of people a respondent knows in reference groups in society,

2

Methods to Estimate the Average Social Network Size

and his or her social network size is more or less the same. Numerical Example Let us assume in the city with 500,000 residents, the number of teachers is 10,000. This means that the proportion of teachers is 2%. If you know only 5 teachers among people in your social network, your expected social network size is 500;000 10;000  5 ¼ 250..

Suppose we are looking for the average social network size of all people in your city. In that case, we need to ask the above questions from a representative sample and make an average of all estimated social network sizes. Now, let us put this concept into a generalizable formula. In a population with size t, where there are e members of a specific subpopulation, the frequency of this subpopulation is e/t. If person ‘i’ knows mi members of this subpopulation, given the overall frequency of e/t, his social network size (Ci) is calculated by Eq. 2.1. The average of Ci values is used as the average network size of the general population. Ci ¼

mi e t

¼t

mi e

ð2:1Þ

19

[66]. Now, the question is how best to integrate replies. Two main methods to do so are known as the traditional and Means of Sum (MoS) estimators.

2.3.1.1 Traditional Estimator Equation 2.1 can simply be extended by putting, for each person, the total number known in all reference groups in the nominator, and the summation of real sizes as the denominator. Here j and i stands for reference groups and respondent, and L shows the number of reference groups [43, 67]. The mean of the values is considered to be the average network size. PL j¼1

Ci ¼ t PL

mij

j¼1 ej

ð2:2Þ

Numerical Example Suppose number of teachers and diabetic patients who live in a city (with total population of 500,000) is 10,000 and 5,000 respectively. If you know 5 teachers, then your network size is 250. If you know 2 diabetic patients then your network size is 200. To get one figure, traditional approach gives network size at 500; 000 

5þ2 ¼ 233 10; 000 þ 5; 000

2.3.1 Incorporating Multiple Reference Groups To minimize the influence of a given reference group, we ask respondents about the frequency of their acquaintances (also known as alters) in several reference groups. Remember that overlap between reference groups is not of concern. People you know in one reference group might fall in another. For example, you might ask respondents about the number of teachers and also the number of diabetic patients they know. In this case, the overlap between groups is likely to happen. However, it has been shown that overlapping does not lead to biased estimations

2.3.1.2 Means of Sums Estimator Habecker et al. argued that the traditional method masks the performance of individual reference groups (Sheikhzadeh et al. 2016). In other words, when the difference between the size of reference groups is high, the significance of knowing individuals in small reference groups makes little difference in estimating network size. He suggested another approach that works based on the means of proportions of reference groups (i.e., m/e) that we know (Sheikhzadeh et al. 2016):

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Ci ¼ t

mi1 e1

þ

mi2 e2

þ...þ j

mij ej

ð2:3Þ

Numerical Example In our previous example, proportion of teachers and diabetic patients you know 5 2 was 0.0005 (10;000 ) and 0.0004 (5;000 ) respectively, resulting in a mean of 0.00045. Multiplying it by 500,000, we estimate the network size to be 225. Numerical Example Habecker et al. (Sheikhzadeh et al. 2016) provided the following example. Suppose three reference groups each with size of 1,000 are selected and a respondent knows 1 person in each. Based on the traditional method, the network size of this person becomes 0.1% of the total population 1þ1 (1;000 þ1 þ1;000 þ 1;000). However, if the size of a reference group is much lower (say 100) and the other is much higher (say 10 000), then the network size of this respondent is 0.027% of the total population þ1þ1 (100 þ 11;000 ). þ 10;000

2.4

Considerations in Selecting Appropriate Reference Groups

To be able to estimate the average social network size accurately, we need to know the actual size of reference groups. Examples of reference groups include patients with diabetes, students at primary schools, newborn babies in the last year and people with specific first names. The good thing about the names is that their real prevalence in the population can be obtained from the subnational or national birth registries. Sometimes two organizations have two different sizes for the same reference group. For example, the number of registered diabetic cases at the Iranian Ministry of Health and Medical

Education is different from the number reported by the Iranian Society of Diabetics. Careful assessment of the completeness and the validity of data is needed to decide which number should be used for this reference group. If it is not possible to find the correct real size, consider excluding this reference group. Moreover, data on the size of some reference groups in some countries may be considered sensitive data and must be kept confidential. An example of such kind of data might be the number of individuals who have been in prison last year. Access to this kind of data requires close collaboration with representatives of these organizations. There are several other issues that affect our final estimate, including visibility of reference groups, barrier effects, cognitive error, attractiveness and recall bias. These sources of errors are explained below.

2.4.1 Visibility Suppose an interviewer wishing to estimate C selects two reference groups and asks the following two questions: (1) how many people do you know named “Hamed”? (2) How many people do you know in prison? For you, which one is easier to answer? Clearly, the first question is a visible characteristic with no sensitivity. However, the second one has a stigma and may not be visible. Indeed although the actual number of people in prison is available, this is not an appropriate reference group as respondents cannot provide an accurate count. This is because respondents are not aware of the low visible characteristics of their acquaintances. Using reference groups with low visibility leads to a small ‘m’, and therefore, the average network size will be underestimated. Example Kadushin et al. recruited individuals aged 16 to 44 to estimate the number of heroin users (Paniotto et al. 2009b). They estimated the average network size was 55. The reference groups used to estimate

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Methods to Estimate the Average Social Network Size

average network size included the number of those who were attacked, hit or burglarized in the last year. The sensitivity of questions used in the estimation of C might partially explain why a low value was obtained.

2.4.2 Barrier Effect In NSU, it is assumed that respondents have equal chance to know members of reference groups. There are some reference groups that are more known by respondents in a specific gender age cohort. This is because the popularity of some names changes over time. ‘Abolfazl’ was a popular name four decades ago but not these days. This means that older respondents are more likely to know those with this given name. This error is known as the barrier effect. To avoid it, we recommend selecting names with even distribution across time. Example McCormick et al. (Brewer 2000). argued that uneven distribution of reference groups in the society causes serious problems in estimation of network size. They proposed a latent non-random mixing model to take into account the issue of unevenness. As an easier approach, they proposed an approach based on the distribution of the reference group in the general population. For example, if one chooses a male name with 10% prevalence among those aged 20-30 as reference group, then in the recruited sample, the proportion of male in the same age group should be the same.

21

of these replies are incorrect. Remember the definition of know. Do heads of state know you in person? Can you contact them by email, phone, or in-person? By cognitive error, we mean participants’ ability to understand the meaning of ‘know’ and reply to our questions regarding the definition. Example McCarty et al. (2001) conducted focus groups and realized that respondents tended to guess the number of people they know in large populations rather than trying to count them (McCarty et al. 2001). This is not a classic example of cognitive error but shows that sometimes respondents do not pay close attention to the definition of ‘know’.

2.4.4 Attractiveness Bias Sometimes over-reporting of a particular answer can make a respondent appear more socially acceptable. For example, people may underreport the number of families whose children live with only one parent. This is known as attractiveness bias [68]. Such groups are not appropriate for being selected as a reference.

2.4.5 Racal Bias The assumption behind the NSU method is that the association between the prevalence of reference groups in a society (i.e., e/t) and the average number of people known by respondents in each  is linear. However, reference group (i.e., m) respondents usually under-report the number of people they know from prevalent reference groups (Rastegari et al. 2013) (Fig. 2.2).

2.4.3 Cognitive Error How many heads of state do you know? Possible answers would be 1, 5, 10, or even 50. Indeed all

Example Killworth et al. showed that mean number of individuals known by respondents was

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M. R. Baneshi et al.

Fig. 2.2 Association between the prevalence of reference groups in the population and mean number known by respondents

2.4.6 Conclusion associated with square root of the prevalence of reference groups in the general population (Rastegari et al. 2013). This means that respondents usually underreport the size of more prevalent groups. Example: Iranian National Data Recruiting nearly 7,000 respondents from 15 provinces, we designed a national study to calculate the average network size of Iranians (Killworth et al. 1998). Sample size at the province level varied from 400 to 1,000, based on the provinces’ populations. Combining information from 23 reference groups, we estimated the average network size of the Iranian population at 239. We then plotted prevalence of reference groups in the community versus mean number known by respondents. Lowes smoother showed that number of prevalent groups is underreported.

Due to the four biases that may affect the results of C estimation by the indirect method, we recommend using multiple reference groups; a minimum of six and ideally 20 reference groups can produce a robust average social network size. The frequency of the reference groups should be between 0.1 and 4%. Choose the reference groups that have the maximum transparency. If you select names as reference groups, their popularity overtime should be constant. We recommend that investigators exclude combined names like Ali-Akbar or Mary Jane to avoid confusion. All reference groups’ prevalence should be about 0.02% of the total population (Brewer 2000). When possible, apply both summation and reference groups methods. A short version of the summation method to engage respondents in the thinking and recall process, followed by the

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Methods to Estimate the Average Social Network Size

known reference groups method, has been shown to improve the accuracy of responses and C estimation results (Sheikhzadeh et al. 2016).

2.5

Calibration of Calculated Average Social Network Size

One of the advantages of the reference group method is that you are able to explore the reliability of C. That is, you are able to back calculate the size of reference groups based on your observed C. This allows you to address the internal validity of the estimated average network size [67]. To back calculate the size of reference group j, substitute the summation known by respondents as the numerator and that of network size as the denominator. Then multiply this proportion by the total population (Eq. 2.4). PN

ej ¼ t Pi¼1 N i¼1

mij Ci

ð2:4Þ

Equation 2.4 allows you to back calculate the size of all reference groups. You might see that size of some of them are remarkably over- or underestimated. This suggests that you should refine your analysis by excluding reference groups that have substantially underestimated sizes on back calculation. To refine the average social network size, we recommend the following steps. Each of these steps involves methodological considerations, which are discussed later. 1. Estimate C using candidate reference groups 2. Back calculate the size of reference groups 3. Calculate the ratio of real to the predicted size 4. If all ratios are within a predefined bound (say 0.5 to 2), stop, otherwise exclude the reference group whose ratio is the farthest from 1 and return to step 1 In the next section, we discuss whether the definition of ratio and its tolerable bound affects the final C. In some applications, all out-of-

23

bounds reference groups are removed at one step [20, 69]. However, as NSU methods depend on how reference groups work together, we recommend deleting unsuitable reference groups in an iterative fashion. Example 1 Ukraine (McCarty et al. 2001) In Ukraine, nearly 1,000 respondents’ age above 14 were recruited. Combining information from 22 reference groups, the average network size was estimated at 202. The trimmed mean was 193. The investigators used the trimmed mean to back calculate the size of reference groups. Calculating the ratio of back calculated size to actual size, they found that the sizes of nine reference groups underestimated the size on back calculation poorly. All of the nine groups were deleted simultaneously rather than in an iterative fashion. The mean average network size based on the remained 13 reference groups was 175. Example 2 Japan (Tourangeau and TWJPoq 1996) In Japan, an internet survey was conducted among 1,500 respondents to estimate the size of the MSM population. Using 10 reference groups, a pilot study was conducted among 225 respondents. Based on the pilot study results, in one step, eight reference groups were deleted as the discrepancy between actual and back calculated size was large. In addition, two reference groups were deleted as their visibility was not high (the number of who had cancer and the number with motorcycle licenses). In the main study only three reference groups were used. However, their information was not aggregated. Average network size based on each name was computed (418, 338, and 335) and average of three values (i.e., 364) was considered as the most likely estimate.

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M. R. Baneshi et al.

Example 3 China (Zamanian et al. 2016) An NSU study was undertaken to estimate the size of the MSM population in Shanghai. About 4,000 respondents were recruited. Known population methods were applied to calculate the network size. Based on 22 reference groups, the value of C was calculated as 241. To calibrate this statistic, a combination of regression and ratio approaches was applied. The investigators plotted the mean number known in each reference group as dependent variables versus the relative sizes of the reference groups as independent variables. This was followed by a graphical analysis of residuals to detect abnormal populations. In a parallel analysis, reference groups for which the ratio of estimated to actual size was outside the range of 0.5 and 2 were detected. Finally unsuitable reference groups were excluded by combining both regression and ratio approaches. Only six reference groups were selected as appropriate by two approaches. This gave an adjusted C of 236. Example 4 Rwanda (Ezoe et al. 2012) In Rwanda, investigators have applied the standard definition of “know” but restricted its boundaries to one year. Combining replies to 22 reference groups, the estimated network size was 251. Applying the summation method, where relationships were broken into 28 categories, corresponding figure was 168. Example 5 Georgia (Habecker et al. 2015) In Tbilisi, Georgia, a household survey was used to estimate C. The invstigators used the known population method with 24 reference groups to calculate C. They created a ratio of the back calculated value to the actual size. Groups with ratios outside 0.5 and 1.5 were excluded. Combining information from 20 reference groups, C was calculated at 355.

Example 6 Iran (Killworth et al. 1998) We used known size reference group method in Iran to estimate C. To exclude unreliable reference groups, we used two approaches, ratio and regression. In the ratio approach, we defined the ratio as actual population size to the back calculated size estimate and set the plausible range from 0.5 to 2. Excluding 12 reference groups, we estimated the average C to be 308. In regression approach, we used predicted and actual sizes of all 23 reference groups as dependent and independent variables and calculated standardized DFBETA residuals (SDFBETA) for all reference groups. SDFBETA measures changes in regression coefficient per deletion of each reference group. The reference group with the highest SDFBETA was excluded to justify linearity. The whole process was continued in an iterative fashion to remove all reference groups with SDFBETA higher than 3/√n (where n is the number of reference groups retain in the analysis). In total we excluded six reference groups, one at a time. Based on information from the 17 reference groups that were retained, the average C was estimated to be 380. To address the internal validity of these values, we back calculated size of the reference groups that contributed to our estimation of C. In the regression and ratio approaches, intraclass correlation coefficient for agreement between real size of eligible reference groups and back calculated sizes was 0.93 and 0.96, respectively. Residual mean square error values were 69,402 and 61,758. We also fit a linear regression line where we considered back calculated and real sizes as dependent and independent variables. In the case of perfect agreement, we expected a slope of one. Both slopes were above 0.60 but were significantly different from one.

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Methods to Estimate the Average Social Network Size

2.6

Which Factors Affects Average Network Size?

We discussed summation and known reference group approaches (traditional and MoS) earlier. In addition, we explained the use of the ratiobased method to calibrate network size. Clearly, the method of estimation and the acceptable range for the ratio of estimation to real size affect the final estimate. In addition, missing data and digit preference may also change the network size estimated. Such effects on network size are explained below.

2.6.1 Summation versus Traditional Reference Group Method McCarty et al. argued that the reference groups method relies on accurate replies to a large number of reference groups (Snidero et al. 2004). He noted that, in developing countries, it might be difficult to obtain accurate sizes of reference groups. Tourangeau et al. [70] mentioned that the summation method poses a recall process and helps participants to remember more people. On the other hand, Habecker et al. (Sheikhzadeh et al. 2016) discussed that the summation method provides us with estimates of network size but not an enumeration. Although the results of McCarty et al. [65] support the usefulness of the summation method, there is no concrete evidence to support it. Also, in the summation method, it is not possible to check the accuracy of respondents’ replies, but in the known reference group method, as real sizes are known, one can check the accuracy of the C estimation. Example 1 (Snidero et al. 2004) McCarty et al. recruited 796 (in Survey 1) and 574 (in Survey 2) respondents to compare summation and known reference groups approaches. In the known population approach, respondents were asked

25

about the number of people they know in each of 29 reference groups. In the summation method study, group network was broken into 16 relations. The mean (SD) network sizes were almost the same (290.8±264.4 versus 290.7±258.8). Corresponding values in Survey 2 were 291.2 ±259.3 versus 281.2±255.4. Example 2 (Sheikhzadeh et al. 2016) Habecker et al. estimated network size of people living in Nebraska using summation and known reference group methods. They conducted 19 cognitive interviews. Interviewees reported that summation method was easier and that their replies were more accurate. Replies to known reference groups involved several replies ending in zeroes and ones. To tackle this problem, authors recommended use of more prevalent reference groups.

2.6.2 Missing Data

Example (Snidero et al. 2004) McCarty et al. compared summation and traditional reference group methods in terms of their flexibility to deal with missing data. First, for subjects with missing data, neither ‘m’ (in numerator) nor ‘e’ (in the denominator) contributed to calculations. Secondly, subjects had no contribution in the numerator but in the denominator. The resulting change was 0.33% for any respondent’s C. In the summation method, out of 1370 respondents in the combined sample, authors reported a missing rate of 35% (25% for at most two relations, and 10% for more than two). Missing data in each category were replaced by the mean of available replies. This led to underestimation of 24 or 8 percent of the average C.

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M. R. Baneshi et al.

In McCarty et al.’s experience (Snidero et al. 2004), the performance of the ability of these two methods to handle missing data was not compared in a head-to-head fashion. This is because questions that led to missing data were not the same, and approaches applied to address the problem were different as well. In the coming pages, we will explore the impact of missing data on the estimation of the size of hidden groups.

2.6.3 Joint Effect of Estimator, Definition of Ratio, and Its Tolerable Range

or overestimation in prediction. Corresponding figures in the case of real to back calculated sizes are 33% 100%. Some other studies used 0.5 and 2 as lower and upper limits (Sheikhzadeh et al. 2016). In this case, the way we define ratio is not a matter of concern. However, none of these approaches provide symmetrical ratios around 1, and, therefore, groups that are overestimated are more likely to be excluded than those underestimated. Habecker et al. used log base 2 of the ratios, which is the same as ratios between 0.5 and 2 in the original scale but guarantees values symmetric around 1 (Sheikhzadeh et al. 2016). Example Kerman Female Data We recruited 1275 adult women in Kerman and asked them to provide us with number of people they knew in 25 reference groups (Kadushin et al. 2006). Changing the definition of “know”, its plausible range, and method of estimation were compared in 12 scenarios (2*3*2). In each scenario, the value of C derived was used to back calculate the sizes of all 25 reference groups. To assess internal validity, we estimated the performance statistics comparing actual and back calculated sizes of eligible reference groups (i.e., those contributed in calculation of C). To assess external validity, statistics were compared for reference groups that did not contribute to the calculation of C. Traditional Method Analyzing the data by traditional approach, we have seen that C is fairly robust with respect to definition of ratio and its tolerable range. Minimum and maximum values estimated were 174 and 186 respectively. The best performance

To see how the definition of ratio can affect the results, we provide a hypothetical example. Suppose that we have two reference groups with real sizes 20,000 and 35,000. Based on these two reference groups C is calculated and applied to back calculate the size of these two reference groups. Assume that back calculated sizes are 45,000 and 20,000. We used two definitions for the ratio. In scenario 1, we defined the ratio as the back calculated estimate of population size to the actual size of the population. In scenario 2, its inverse is used. We then defined the absolute difference between the ratio and one as a measure of bias. In scenario 1, group two is poorer. However, in scenario 2 the opposite is true. This example shows the way we define the ratio directly affects the composition of final reference groups that contribute to the estimation of network size. Numerical Example See Table 2.1 Regarding the plausible bound, some studies used 0.5 to 1.5 (Killworth et al. 1998). Remember that defining the ratio as back calculated to real size corresponds to acceptance of 50% under

Table 2.1 Influence of definition of ratio on exclusion of reference groups Group number

Size Real

Scenario 1

Scenario 2

Back calculated

Real/ Back calculated

Bias

Back calculated/ Real

Bias

1

20,000

45,000

0.44

0.56

2.25

1.25

2

35,000

20,000

1.75

0.75

0.57

0.43

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Methods to Estimate the Average Social Network Size

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Table 2.2 Effect of estimator, definition of ratio and its range on Internal Validity of C Method

Ratio

Range

C

Traditional

Actual/Estimate

0.50 to 1.5

Estimate/Actual

MoS

Actual/Estimate

Estimate/Actual

#eligible groups

ICC

Intercept (P-value)

Slope (Pvalue)

RMSE

186

20

0.92

702 0.002

0.65 0.001

427

0.50 to 2

186

20

0.92

702 0.002

0.65 0.001

427

Abs(log (ratio))