A critical threshold for design effects in network sampling

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 $\mathrm{DE}$, then constructing an estimator from the novel design makes the variance of the estimator $\mathrm{DE}$ 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 $$ and the clustering structure in the social network, represented by the second eigenvalue of the Markov transition matrix, $$. If $$, then the design effect is finite (i.e., the standard estimator is $$-consistent). However, if $$, then the design effect grows with $$ (i.e., the standard estimator is no longer $$-consistent). Past this critical threshold, the standard error of the estimator converges at the slower rate of $$. 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.