Part 6 RDS/RDS+ Methods Summary

  • Use the coupons, the timing of surveys and the structure of recruitment network to approximate the size of the population

  • Relies on network connectedness, i.e. if the seeds are within a “small world” the method will likely depart significantly from the truth

  • Do not rely on proportional sampling and likely are cheaper

Crawford, Wu, and Heimer (2018)

  • Use data from RDS \(\mathbf{Y} = (G_{R}, \mathbf{d}, \mathbf{t}, \mathbf{C})\), where:

    • \(G_{R}\) – Directly observed recrutiment graph with \(n\) (out of \(N\)) vertices (not the same as \(G_{S}\) – full subgraph of graph of hidden population \(G\) induced by recruitment)

    • \(\mathbf{d}\) – vector of number of connections of each recruited respondents in hidden population \(G\) from survey

    • \(\mathbf{t}\) – directly observed vector of recruitment timings

    • \(\mathbf{C}\) – coupon matrix in which \(C_{ij} = 1\) iff subject \(i\) has at least one coupon just before the \(j\)-th subject recruitment event, and zero otherwise

  • Using this data we are trying to estimate \(N\) using the likelihood of \(N | G_{S}\) marginalizing over all \(G_{S}\) consistent with \(G_{R}\)

  • Assumptions:

    1. \(G\) is finite graph with no self-loops
    2. Use Erdos-Renyi Model assumptions for hidden population graph \(G\): assume \(d_{i} \sim \mathrm{B} (N-1, p)\) (similar to (Killworth et al. 1998) and thus to (Maltiel et al. 2015))
    3. The edges connecting the newly recruited respondent \(i\) to other unrecruited respondents do not affect \(i\)’s probability of being recruited [this is very strong assumption]
    4. Vertices become recruiters immediately upon entering the study and receiving one or more coupons. They remain recruiters until their coupons or susceptible neighbors are depleted, whichever happens first [this is very strong assumption]
    5. When a susceptible neighbor \(j\) of a recruiter \(i\) is recruited by any recruiter, the edge connecting \(i\) and \(j\) is immediately no longer susceptible [this is very strong assumption]
    6. The time to recruitment along an edge connecting a recruiter to a susceptible neighbor has exponential distribution with rate \(\lambda\), independent of the identity of the recruiter, neighbor, and all other waiting times

  • Assumptions 2 and 3 imply that the “unrecruited degree” of subject recruited in \(i\)-th order can be represented as \(d^{u}_{i} \sim \mathrm{B} (N - i, p)\)

  • This in turns allows us to write \(\mathcal{L} (N, p \,|\, G_{S}, \mathbf{Y}) = \prod_{i}^{n} {N - i \choose d^{u}_{i}} p^{d^{u}_{i}} (1-p)^{N-i-d^{u}_{i}}\)

  • Assumptions 4-6 imply \(\mathcal{L} (G_{S}, \lambda \,|\, \mathbf{Y}) = \left( \prod_{j \notin M} \lambda \mathbf{s}_{j} \right) \exp (-\lambda \mathbf{s}' \mathbf{w})\), where \(s = \mathrm{lowerTri} (\mathbf{AC})'\mathbb{1} + \mathbf{C}'u\) – vector of the number of susceptible edges just before each recruitment event

  • Appendix also has very useful ideas on network structure simulation using block approach similar to (Feehan and Salganik 2016)

Handcock, Gile, and Mar (2014)

  • Uses the same RDS data (no need for supplementary data from general population)

  • Models degrees of un-sampled vertices as being drawn independently from a pre-specified parametric distribution

  • Does not follow strict topological rules on \(G_{S}\) as the ones imposed by (Crawford, Wu, and Heimer 2018)

  • This lack of graphical constraints in the SS-size model suggests a view of RDS recruitment that is not network-based: subjects’ reported degrees might be regarded as surrogate measures of “visibility” in the population which in turn refers to (Maltiel et al. 2015) and standard HT estimator

  • The SS-size model also assumes that the degrees of recruited subjects should decrease over time as the sample accrues (Johnston et al. 2017)

  • We can basically estimate this approach and the one in (Crawford, Wu, and Heimer 2018) at the same time: While the models are very different the data used is very similar. It seems that the NSUM methods and specifically (Maltiel et al. 2015) model has a better “compatibility” with (Handcock, Gile, and Mar 2014)

Berchenko, Rosenblatt, and Frost (2017)

  • Current RDS estimators resort to modeling recruitment as a homogeneous random walk, which culminates in the assumption that the sampling probability is proportional to degree, that is, \(\pi_{k} \propto k\) which in turns allows for use of HT estimator

  • Uses the same RDS data (no need for supplementary data from general population) in the context of epidemiological model setup in (Andersson and Britton 2000)

  • Assumptions:

    1. Sampling is done without replacement, with \(n_{k,t}\) being the (right-continuous) counting process representing the number of people with degree \(k\) recruited by time \(t\).

    2. Between times \(t\) and \(t + \Delta t\), an individual with degree \(k\) is sampled with probability \[\lambda_{k,t} = \frac{\beta_{k}}{N} I_{t} (N_{k} - n_{k,t}) \Delta t + \mathcal{o} (\Delta t)\]

      where \(I_{t}\) is the number of people actively trying to recruit new individuals, and the constant \(\beta_{k}\) is a degree-dependent recruitment rate

    3. The multivariate counting process \(n_{t} \equiv (n_{1,t}, \dots, n_{k_{\max}, t})\) has intensity \[\lambda_{t} = N^{-1} \left( \beta_{1} I^{-}_{t} (N_{1} - n^{-}_{1,t} ), \dots, \beta_{k_{\max}} I^{-}_{t} (N_{k_{\max}} - n^{-}_{k_{\max},t} ) \right)\]

      such that \(m_{t} \equiv n_{t} - \int_{0}^{t} \lambda_s \mathrm{d} s\) is a multivariate martingale.

  • We are interested in the estimator \(\hat{P}_{CP} = \sum_{k \geq 1} \hat{f}_{k} \hat{p}_{k}\) where \(\hat{f}_{k}\) is the estimate of proportion of degree \(k\) in the population and \(\hat{p}_{k}\) is the estimate of prevalence in the degree \(k\) group

Service Multiplier

Andersson, Håkan, and Tom Britton. 2000. “Stochastic Epidemics in Dynamic Populations: Quasi-Stationarity and Extinction.” Journal of Mathematical Biology 41 (6): 559–80. https://doi.org/10.1007/s002850000060.

Berchenko, Yakir, Jonathan D. Rosenblatt, and Simon D. W. Frost. 2017. “Modeling and Analyzing Respondent-Driven Sampling as a Counting Process.” Biometrics 73 (4): 1189–98. https://doi.org/10.1111/biom.12678.

Crawford, Forrest W., Jiacheng Wu, and Robert Heimer. 2018. “Hidden Population Size Estimation from Respondent-Driven Sampling: A Network Approach.” Journal of the American Statistical Association 113 (522): 755–66. https://doi.org/10.1080/01621459.2017.1285775.

Feehan, Dennis M., and Matthew J. Salganik. 2016. “Generalizing the Network Scale-up Method: A New Estimator for the Size of Hidden Populations.” Sociological Methodology 46 (1): 153–86. https://doi.org/10.1177/0081175016665425.

Handcock, Mark S., Krista J. Gile, and Corinne M. Mar. 2014. “Estimating Hidden Population Size Using Respondent-Driven Sampling Data.” Electronic Journal of Statistics 8 (1): 1491–1521. https://doi.org/10.1214/14-EJS923.

Johnston, Lisa G., Katherine R. McLaughlin, Shada A. Rouhani, and Susan A. Bartels. 2017. “Measuring a Hidden Population: A Novel Technique to Estimate the Population Size of Women with Sexual Violence-Related Pregnancies in South Kivu Province, Democratic Republic of Congo.” Journal of Epidemiology and Global Health 7 (1): 45–53. https://doi.org/10.1016/j.jegh.2016.08.003.

Killworth, Peter D., Eugene C. Johnsen, Christopher McCarty, Gene Ann Shelley, and H. Russell Bernard. 1998. “A Social Network Approach to Estimating Seroprevalence in the United States.” Social Networks 20 (1): 23–50. https://doi.org/10.1016/S0378-8733(96)00305-X.

Maltiel, Rachael, Adrian E. Raftery, Tyler H. McCormick, and Aaron J. Baraff. 2015. “Estimating Population Size Using the Network Scale-up Method.” The Annals of Applied Statistics 9 (3): 1247–77. https://www.jstor.org/stable/43826420.