Central limit theorems for network driven sampling

Abstract

Respondent-Driven Sampling is a popular technique for sampling hidden populations. This paper models Respondent-Driven Sampling as a Markov process indexed by a tree. Our main results show that the Volz-Heckathorn estimator is asymptotically normal below a critical threshold. The key technical difficulties stem from (i) the dependence between samples and (ii) the tree structure which characterizes the dependence. The theorems allow the growth rate of the tree to exceed one and suggest that this growth rate should not be too large. To illustrate the usefulness of these results beyond their obvious use, an example shows that in certain cases the sample average is preferable to inverse probability weighting. We provide a test statistic to distinguish between these two cases.arXiv:1509.04704v2 [stat.ME] 28 Aug 2016 population that is HIV+). Extensive previous statistical research has proposed various estimators of µ which are approximately unbiased based upon various types of models for an RDS sample [Salganik and Heckathorn, 2004, Volz and Heckathorn, 2008, Gile, 2011. We note that in the papers cited above (except [Gile, 2011]), RDS is assumed to sample with replacement. Previous research has also explored the variance of these estimators Salganik, 2009, Rohe, 2015]. This paper studies the asymptotic distribution of statistics related to these estimators.Results on asymptotic distributions for RDS are useful for two obvious reasons. First, they allow us to construct asymptotic confidence intervals for µ. Second, they provide essential tools to test various statistical hypotheses. The only central limit theorem associated considered in the RDS literature studied the case when the tree indexed process reduces to a Markov chain [Goel and Salganik, 2009]; this presumes that each individual refers exactly one person. Previous research suggests that the number of referrals from each individual is fundamental in determining the variance of common estimators [Rohe, 2015]. This paper establishes two central limit theorems in settings which allow for multiple referrals.The main results apply to both the sample average and the Volz-Heckathorn estimator, which is an approximation of the inverse probability weighted estimator (cf Remark 1). Because the inverse probability weighted (IPW) estimator and its extensions are asymptotically unbiased, these estimators are often preferred to the sample average. However, sometimes survey weights are not needed and they only introduce additional variance to the estimator [Bollen et al., 2016]. This issue is particularly salient when sampling weights are highly heterogeneous, as is often the case in RDS. Proposition 3 shows that if the outcomes y i are uncorrelated with the sampling weights, then the sample average is unbiased. Theorem 3 extends this result to RDS to show that the IPW estimator can have a larger variance than the sample average. Taken together, these results imply that the sample average can have a lower mean squared error (MSE) than the IPW estimator. Section 4 introduce...

Asymptotic seed bias in respondent-driven sampling

“…m < λ −2 2 High variance, i.e. m > λ −2 2 Variance IPW O(n −1 ) (Rohe, forthcoming) O(n 2 log m λ 2 ) (Rohe, forthcoming) GLS O(n −1 ) (Roch and Rohe, forthcoming) Distribution IPW&VH Asymptotically normal (Li and Rohe, 2017) Non-trivial mixture [Current paper] GLS Asymptotically normal [Current paper] adjustment. For technical reasons, our analysis of the GLS estimator is restricted to a special case of the Markov model that was first used to study RDS in Goel and Salganik (2009).…”

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.…”

A critical threshold for design effects in network sampling

“…In the simulations, the studentized intervals from the atree-bootstrap often fail to be contained in [0, 1], despite the fact that y i ∈ {0, 1} for all nodes i. Perhaps one reason for these strange results is that the accuracy of the studentized intervals depends onμ V H being asymptotically normal, while results in Li and Rohe [2015] suggest that it is not. In the limited simulations that were performed, the percentile interval was often (i) narrower and (ii) more likely to cover µ true than the studentized interval.…”

“…If the design effect is growing, then it should not be used for sample size or power cal- culations for two reasons. First, there might not be a central limit theorem to justify this approach [Li and Rohe, 2015]. Second, if DE changes with n, then many of the standard formulas are not well defined (or they are incorrect).…”