Department of Mathematics, University of Wisconsin, Madison, WI 53706.
Department of Statistics, University of Wisconsin, Madison, WI 53706
Proc Natl Acad Sci U S A. 2018 Oct 9;115(41):10299-10304. doi: 10.1073/pnas.1706699115. Epub 2018 Sep 25.
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.
为了对边缘化和/或难以接触的人群进行抽样，响应驱动抽样（RDS）和类似技术通过同伴推荐来联系参与者。在 RDS 的马尔可夫模型下，先前的研究表明，如果典型参与者推荐了太多的联系人，那么常见估计量的方差不会像 [公式：见正文] 那样衰减，其中 n 是样本量。这意味着置信区间将远远宽于典型抽样设计。在这里，我们表明广义最小二乘法（GLS）可以有效地降低 RDS 估计量的方差。特别是，理论分析表明 GLS 估计量的方差为 [公式：见正文]。然后我们推导出两类可行的 GLS 估计量。第一类基于底层社交网络的校正度数随机块模型。第二类基于二阶模型。在这两个模型类别中，理论结果表明，有可能从节点的随机游走样本中估计出总体网络的谱性质，这可能具有独立的意义。这些理论结果为那些考虑到节点度数之外的网络结构的完全不同的估计类别指明了方向。诊断图有助于识别可行 GLS 估计量更适用的情况。计算实验表明了潜在的好处，并表明在实际环境中还有进一步开发这些估计量的空间。
