  Network analysis is often focused on characterizing the dependencies between
network relations and node-level attributes. Potential relationships are
typically explored by modeling the network as a function of the nodal
attributes or by modeling the attributes as a function of the network. These
methods require specification of the exact nature of the association between
the network and attributes, reduce the network data to a small number of
summary statistics, and are unable provide predictions simultaneously for
missing attribute and network information. Existing methods that model the
attributes and network jointly also assume the data are fully observed. In this
article we introduce a unified approach to analysis that addresses these
shortcomings. We use a latent variable model to obtain a low dimensional
representation of the network in terms of node-specific network factors and use
a test of dependence between the network factors and attributes as a surrogate
for a test of dependence between the network and attributes. We propose a
formal testing procedure to determine if dependencies exists between the
network factors and attributes. We also introduce a joint model for the network
and attributes, for use if the test rejects, that can capture a variety of
dependence patterns and be used to make inference and predictions for missing
observations.
