Abstract
Link-tracing designs are often used to estimate the size of hidden populations by utilizing the relational links between their members. A major problem in studies of hidden populations is the lack of a convenient sampling frame. The most frequently applied design in studies of hidden populations is respondent-driven sampling in which no sampling frame is used. However, in some studies multiple but incomplete sampling frames are available. In this article, we introduce the B-graph design that can be used in such situations. In this design, all available incomplete sampling frames are joined and turned into one sampling frame, from which a random sample is drawn and selected respondents are asked to mention their contacts. By considering the population as a bipartite graph of a two-mode network (those from the sampling frame and those who are not on the frame), the number of respondents who are directly linked to the sampling frame members can be estimated using Chao’s and Zelterman’s estimators for sparse data. The B-graph sampling design is illustrated using the data of a social network study from Utrecht, the Netherlands.
References
1.
Bishop, Y.M.M., S.E. Fienberg, and P.W. Holland. 1988. Discrete Multivariate Analysis: Theory and Practice . Cambridge, MA, and London: The MIT Press.
2.
Bogaerts, S. and A. Daalder. 2011. “Measuring Childhood Abuse and Neglect in a Group of Female Indoor Sex Workers in the Netherlands. A Confirmatory Factor Analysis of the Dutch Version of the Childhood Trauma Questionnaire-Short Form.” Psychological Reports 108: 856–860. Doi: https://dx-doi-org.web.bisu.edu.cn/10.2466/02.10.13.16.PR0.108.3.856-860.
3.
Böhning, D. 2010. “Some General Comparative Points on Chao’s and Zelterman’s Estimators of the Population Size.” Scandinavian Journal of Statistics 37: 221–236. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1111/j.1467-9469.2009.00676.x.
4.
Böhning, D. and P. van der Heijden. 2009. “Recent Developments in Life and Social Science Applications of Capture-Recapture Methods.” AStA Advances in Statistical Analysis 93: 1–3. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1007/s10182-008-0097-7.
5.
Brittain, S. and D. Böhning. 2009. “Estimators in Capture-Recapture Studies With Two Sources.” AStA Advances in Statistical Analysis 93: 23–47. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1007/s10182-008-0085-y.
6.
Brugal, M.T., A. Domingo-Salvany, A. Maguire, J.A. Cayla, J.R. Villalbi, and R. Hartnoll. 1999. “A Small Area Analysis Estimating the Prevalence of Addiction to Opioids in Barcelona.” Journal of Epidemiology and Community Health 53: 488–494. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1136/jech.53.8.488.
7.
Chao, A. 1987. “Estimating the Population Size for Capture-Recapture Data with Unequal Catchability.” Biometrics 43: 783–791. Doi: https://dx-doi-org.web.bisu.edu.cn/10.2307/2531532.
8.
Chao, A. 1988. “Estimating Animal Abundance with Capture Frequency Data.” Journal of Wildlife Management 52: 295–300. Doi: https://dx-doi-org.web.bisu.edu.cn/10.2307/3801237.
9.
Chao, A. 1989. “Estimating Population Size for Sparse Data in Capture-Recapture Experiments.” Biometrics 45: 427–438. Doi: https://dx-doi-org.web.bisu.edu.cn/10.2307/2531487.
10.
Chao, A. 2001. “An Overview of Closed Capture-Recapture Models.” Journal of Agricultural, Biological, and Environmental Statistics 6: 158–175. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1198/108571101750524670.
11.
Chao, A., H.Y. Pan, and S.C. Chiang. 2008. “The Petersen-Lincoln Estimator and its Extension to Estimate the Size of Shared Population.” Biometrical Journal 50: 957–970. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1002/bimj.200810482.
12.
Cormack, R.M. 1989. “Log-Linear Models for Capture-Recapture.” Biometrics 45: 395–413. Doi: https://dx-doi-org.web.bisu.edu.cn/10.2307/2531485.
13.
Cormack, R.M. 1992. “Interval Estimation for Mark-Recapture Studies of Closed Populations.” Biometrics 48: 567–576. Doi: https://dx-doi-org.web.bisu.edu.cn/10.2307/2532310.
14.
Ten Den, C., B. Bieleman, E. De Bie, and J. Snippe. 1995. Pijn in het hart . Groningen and Rotterdam: Intraval.
15.
Fienberg, S.E. 1972. “The Multiple Recapture Census for Closed Populations and Incomplete 2k Contingency Tables.” Biometrika 59: 591–603. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1093/biomet/59.3.591.
16.
Frank, O. 1979. “Estimation of Population Totals by Use of Snowball Samples.” In Perspectives on Social Network Research , edited by P.W. Holland and S. Leinhardt, 319–348. New York: Academic Press.
17.
Frank, O. and T.A.B. Snijders. 1994. “Estimating the Size of Hidden Populations Using Snowball Sampling.” Journal of Official Statistics 10: 53–67.
18.
Goodman, L.A. 1961. “Snowball Sampling.” Annals of Mathematical Statistics 32: 148–170.
19.
Heckathorn, D.D. 1997. “Respondent-Driven Sampling: A New Approach to the Study of Hidden Populations.” Social Problems 44: 174–199. Doi: https://dx-doi-org.web.bisu.edu.cn/10.2307/3096941.
20.
Holland, R., R. Vivancos, V. Maskrey, J. Sadler, D. Rumball, I. Harvey, and L. Swift. 2006. “The Prevalence of Problem Drug Misuse in a Rural County of England.” Journal of Public Health 28: 88–95. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1093/pubmed/fdl009.
21.
Jansson, I. and M. Spreen. 1998. “The Use of Local Networks in a Study of Heroin Users: Assessing Average Local Networks.” Bulletin de Méthodologie Sociologique 59: 49–61. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1177/075910639805900105.
22.
Klovdahl, A.S. 1989. “Urban Social Networks: Some Methodological Problems and Possibilities.” In The Small World , edited by M. Kochen, 176–210. Norwood, NJ: Ablex.
23.
Kunst, M.J.J., F.W. Winkel, and S. Bogaerts. 2010. “Prevalence and Predictors of Posttraumatic Stress Disorder Among Victims of Violence Applying for State Compensation.” Journal of Interpersonal Violence 2010: 1631–1654. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1177/0886260509354591.
24.
McCullough, D.R. and D.H. Hirth. 1998. “Evaluation of the Petersen-Lincoln Estimator for a White-tailed Deer Population.” Journal of Wildlife Management 52: 534–544. Doi: https://dx-doi-org.web.bisu.edu.cn/10.2307/3801606.
25.
Palusci, V.J., S.J. Wirtz, and T.M. Covington. 2010. “Using Capture-Recapture Methods to Better Ascertain the Incidence of Fatal Child Maltreatment.” Child Abuse & Neglect 34: 396–402. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1016/j.chiabu.2009.11.002.
26.
Salganik, M.J. and D.D. Heckathorn. 2004. “Sampling and Estimation in Hidden Populations Using Respondent-Driven Sampling.” Sociological Methodology 34: 193–239. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1111/j.0081-1750.2004.00152.x.
27.
Särndal, B.E., B. Swensson, and J.H. Wretman. 1992. Model Assisted Survey Sampling . New York: Springer-Verlag.
28.
Seber, G.A.F. 1986. “A Review of Estimating Animal Abundance.” Biometrics 42: 267–292. Doi: https://dx-doi-org.web.bisu.edu.cn/10.2307/2531049.
29.
Smit, F., W. Brunenberg, and P. van der Heijden. 1996. “Het Schatten van Populatiegroottes.” Tijdschrift voor Sociale Gezondheidszorg 74: 171–176.
30.
Smit, F., J. Toet, and P. van der Heijden. 1997. “Estimating the Number of Opiate Users in Rotterdam Using Statistical Models for Incomplete Count Data.” In European Monitoring Centre for Drugs and Drug Addiction (EMCDDA) Methodological Pilot Study of Local Prevalence Estimates , Lisbon: EMCDDA.
31.
Spreen, M. 1992. “Populations, Hidden Populations, and Link-Tracing Designs: What and Why?” Bulletin de Méthodologie Sociologique 36: 34–58. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1177/075910639203600103.
32.
Surjadi, B., J. van Horn, S. Bogaerts, and R. Bullens. 2010. “Internet Offending: Sexual and Non-Sexual Functions within a Dutch Sample.” Journal of Sexual Aggression 16: 47–58. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1080/13552600903470054.
33.
Thompson, S.K. and O. Frank. 2000. “Model-Based Estimation with Link-Tracing Sampling Designs.” Survey Methodology 26: 87–98.
34.
Volz, E. and D.D. Heckathorn. 2008. “Probability Based Estimation Theory for Respondent Driven Sampling.” Journal of Official Statistics 24: 79–97.
35.
Watters, J.K. and P. Biernacki. 1989. “Targeted Sampling: Options for the Study of Hidden Populations.” Social Problems 36: 416–430. Doi: https://dx-doi-org.web.bisu.edu.cn/10.2307/800824.
36.
Wilson, R.M. and M.F. Collins. 1992. “Capture-Recapture Estimation with Samples of Size One Using Frequency Data.” Biometrika 79: 543–553. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1093/biomet/79.3.543.
37.
Zelterman, D. 1988. “Robust Estimation in Truncated Discrete Distributions with Application to Capture-Recapture Experiments.” Journal of Statistical Planning and Inference 18: 225–237. Doi: https://dx-doi-org.web.bisu.edu.cn/10.1016/0378-3758(88)90007-9.