  We study the data-driven selection of causal graphical models using
constraint-based algorithms, which determine the existence or non-existence of
edges (causal connections) in a graph based on testing a series of conditional
independence hypotheses. In settings where the ultimate scientific goal is to
use the selected graph to inform estimation of some causal effect of interest
(e.g., by selecting a valid and sufficient set of adjustment variables), we
argue that a "cautious" approach to graph selection should control the
probability of falsely removing edges and prefer dense, rather than sparse,
graphs. We propose a simple inversion of the usual conditional independence
testing procedure: to remove an edge, test the null hypothesis of conditional
association greater than some user-specified threshold, rather than the null of
independence. This equivalence testing formulation to testing independence
constraints leads to a procedure with desriable statistical properties and
behaviors that better match the inferential goals of certain scientific
studies, for example observational epidemiological studies that aim to estimate
causal effects in the face of causal model uncertainty. We illustrate our
approach on a data example from environmental epidemiology.
