A Bayesian Approach to Estimation with Link-tracing Sampling Designs

For inference from link-tracing designs, Frank and Thompson (1998) derived the likelihood function for the graph model. In addition, they provided the likelihood functions considered under the symmetric model and also an asymmetric model. In that paper, they used maximum likelihood estimators to est imate the graph model parameters. Here, we propose a Bayesian approach for the estimation problem. For problems with sampling designs that follow social links from one person to another, it is quite often tha t prior information is available on the proportions tha t one wants to estimate. Thus, using these information effectively via a Bayesian approach should yield bet ter estimators. Also, under the Bayesian setup, obtaining interval estimates and assessing the accuracy (posterior variances) of the estimators can be done without much added difficulties whereas such tasks would be very difficult to perform using the classical approach. In general, a Bayesian analysis yields one distribution (the posterior distribution) for the unknown parameters, and from this a large number of questions can be answered simultaneously. 1 I n t r o d u c t i o n Social network data include measurements on the relationships between social entities. Collecting network data on entire networks requires a great deal of t ime and effort, especially when networks are large. It is thus important to be able to estimate network properties from samples. In link-tracing sampling designs, social links are followed from one respondent to another to obtain the sample. For hidden and hard-to-access human populations, such sampiing designs are considered the most practical way to obtain a sample large enough to study. For example, in a study of injection drug use in relation to the spread of the HIV infection, initial respondents may be asked to identify drug-injection partners who are then added to the sample. Social entities with social s tructure are often modeled as graphs, with the nodes of the graph representing social entities and the arcs of the graph representing social links, relationships, or transactions. The population graph itself can be viewed either as a fixed structure or as a realization of a stochastic graph model. Samples are taken to obtain information about the population graph. Usually, the sampling method will take advantage of the arcs or links from one entity to another. There is a large literature on network sampling, both applied and theoretical. Frank (1977a, 1977b, 1977c, 1978, 1979a, 1979b, 1980, 1997) has many important results in sampling for social networks. His classic work (Frank 1971) presents a basic solution for estimating graph quantities from the sample data. Snijders and Nowicki (1997) proposes various statistical approaches, including a Bayesian approach, to estimation and prediction for stochastic block models for graphs with latent block structure. Snowball sampling is one type of link-tracing sampling design in which individuals in an initial sample were asked to identify a fixed number of acquaintances, who in turn were asked to identify the same number of acquaintances and so on for a fixed number of stages or waves. This very clever network sampling idea originated from Goodman(1961). Erickson (1978) and Frank (1979b) review snowball sampling design with the goal of understanding how other "chain methods" (methods designed to trace ties through a network from a source to an end) can be used in practice. Snijders (1992) used the same term "snowball sampling" to include designs in which only a subsample of links from each node is traced. Frank and Snijders (1994) consider modeland design-based estimation of a hidden population