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

To sample marginalized and/or hard-to-reach populations, respondent-driven sampling (RDS) and similar techniques reach their participants via peer referral. Under a Markov model for RDS, previous research has shown that if the typical participant refers too many contacts, then the variance of common estimators does not decay like [Formula: see text], where n is the sample size. This implies that confidence intervals will be far wider than under a typical sampling design. Here we show that generalized least squares (GLS) can effectively reduce the variance of RDS estimates. In particular, a theoretical analysis indicates that the variance of the GLS estimator is [Formula: see text] We then derive two classes of feasible GLS estimators. The first class is based upon a Degree Corrected Stochastic Blockmodel for the underlying social network. The second class is based upon a rank-two model. It might be of independent interest that in both model classes, the theoretical results show that it is possible to estimate the spectral properties of the population network from a random walk sample of the nodes. These theoretical results point the way to entirely different classes of estimators that account for the network structure beyond node degree. Diagnostic plots help to identify situations where feasible GLS estimators are more appropriate. The computational experiments show the potential benefits and also indicate that there is room to further develop these estimators in practical settings.