Asymptotic Seed Bias in Respondent-driven Sampling

TL;DR

This paper investigates how initial seed selection biases affect the limiting distribution of estimators in respondent-driven sampling, revealing that some estimators are robust while others are significantly influenced by seed bias.

Contribution

It provides a theoretical analysis of seed bias effects on RDS estimators using branching process tools, highlighting conditions under which bias impacts the estimators' distributions.

Findings

GLS estimator is unaffected by seed bias under certain conditions.

VH estimator converges to a mixture distribution influenced by seed node.

Numerical experiments support theoretical results beyond Markov assumptions.

Abstract

Respondent-driven sampling (RDS) collects a sample of individuals in a networked population by incentivizing the sampled individuals to refer their contacts into the sample. This iterative process is initialized from some seed node(s). Sometimes, this selection creates a large amount of seed bias. Other times, the seed bias is small. This paper gains a deeper understanding of this bias by characterizing its effect on the limiting distribution of various RDS estimators. Using classical tools and results from multi-type branching processes (Kesten and Stigum, 1966), we show that the seed bias is negligible for the Generalized Least Squares (GLS) estimator and non-negligible for both the inverse probability weighted and Volz-Heckathorn (VH) estimators. In particular, we show that (i) above a critical threshold, VH converge to a non-trivial mixture distribution, where the mixture component depends on the seed node, and the mixture distribution is possibly multi-modal. Moreover, (ii) GLS converges to a Gaussian distribution independent of the seed node, under a certain condition on the Markov process. Numerical experiments with both simulated data and empirical social networks suggest that these results appear to hold beyond the Markov conditions of the theorems.