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 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, λ2. If m < 1/λ 2 2 , then the design effect is finite (i.e. the standard estimator is √ n-consistent). However, if m > 1/λ 2 2 , then the design effect grows with n (i.e. the standard estimator is no longer √ n-consistent). Past this critical threshold, the standard error of the estimator converges at the slower rate of n log m λ 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. * Thank you Zoe Russek and Emma Krauska for recording the referral trees used in Figure 2 and helpful comments on this draft. Thank you

Generalized least squares can overcome the critical threshold in respondent-driven sampling

“…Under the Markov model where the covariance between samples is known, Theorems 1 and 2 show that the variance of the GLS estimator decays like . To estimate the covariance between samples, we use the fact that the covariance between adjacent samples can be exactly specified in terms of the spectral properties of the Markov transition matrix ( 5 , 20 – 24 ). These essential spectral properties of the network can be estimated from the observed data under the DC-SBM and the rank-two model.…”

“…In all cases, the lines are less than 1, indicating that the GLS estimator has a smaller variance than the sample proportion. Under this simulation model, if , then the variance of the sample proportion decays slower than ( 5 , 8 ). As Theorems 1 and 2 below show, the variance of the GLS estimator converges to 0 like .…”

“…Let be the eigenvalues of corresponding to . For each node , decomposes as follows: Under the , the covariance is stationary with autocovariance function That is, the covariance matrix has the form , where is the graph distance (i.e., minimum path length) between and in ( 5 ).…”

“…Under a Markov model, ref. 5 showed how the dependence induced by RDS can drastically inflate the variance of traditional estimators, making it decay at a rate slower than . This implies that reducing the sampling error by 50% can require far more than 4 times as many samples.…”

“…Using the covariance function derived in ref. 5 , this paper studies the generalized least squares (GLS) estimator for RDS. Our theoretical analysis establishes that the variance of the GLS estimator is .…”