Network driven sampling; a critical threshold for design effects

Abstract

Web crawling, snowball sampling, and respondent-driven sampling (RDS) are three types of network sampling techniques used to contact individuals in hard-to-reach populations. This paper studies these procedures as a Markov process on the social network that is indexed by a tree. Each node in this tree corresponds to an observation and each edge in the tree corresponds to a referral. Indexing with a tree (instead of a chain) allows for the sampled units to refer multiple future units into the sample. In survey sampling, the design effect characterizes the additional variance induced by a novel sampling strategy. If the design effect is some value $DE$, then constructing an estimator from the novel design makes the variance of the estimator $DE$ times greater than it would be under a simple random sample with the same sample size $n$. Under certain assumptions on the referral tree, the design effect of network sampling has a critical threshold that is a function of the referral rate $m$ and the clustering structure in the social network, represented by the second eigenvalue of the Markov transition matrix, ${\lambda }_2$. If $m<1/{\lambda }_2^2$, then the design effect is finite (i.e. the standard estimator is $\sqrt{n}$-consistent). However, if $m>1/{\lambda }_2^2$, then the design effect grows with $n$ (i.e. the standard estimator is no longer $\sqrt{n}$-consistent). Past this critical threshold, the standard error of the estimator converges at the slower rate of $n^{{\log }_m{\lambda }_2}$. The Markov model allows for nodes to be resampled; computational results show that the findings hold in without-replacement sampling. To estimate confidence intervals that adapt to the correct level of uncertainty, a novel resampling procedure is proposed. Computational experiments compare this procedure to previous techniques.

References in corpus (2)

• Stochastic blockmodels and community structure in networks
• Modeling social networks from sampled data

Cited by in corpus (5)

• Generalizing the Network Scale-Up Method: A New Estimator for the Size of Hidden Populations
• Novel Sampling Design for Respondent-driven Sampling
• Generalized least squares can overcome the critical threshold in respondent-driven sampling
• Central limit theorems for network driven sampling
• Reducing Seed Bias in Respondent-Driven Sampling by Estimating Block Transition Probabilities