  An important aspect of modeling spatially-referenced data is appropriately
specifying the covariance function of the random field. A practitioner working
with spatial data is presented a number of choices regarding the structure of
the dependence between observations. One of these choices is determining
whether or not an isotropic covariance function is appropriate. Isotropy
implies that spatial dependence does not depend on the direction of the spatial
separation between sampling locations. Misspecification of isotropy properties
(directional dependence) can lead to misleading inferences, e.g., inaccurate
predictions and parameter estimates. A researcher may use graphical
diagnostics, such as directional sample variograms, to decide whether the
assumption of isotropy is reasonable. These graphical techniques can be
difficult to assess, open to subjective interpretations, and misleading.
Hypothesis tests of the assumption of isotropy may be more desirable. To this
end, a number of tests of directional dependence have been developed using both
the spatial and spectral representations of random fields. We provide an
overview of nonparametric methods available to test the hypotheses of isotropy
and symmetry in spatial data. We summarize test properties, discuss important
considerations and recommendations in choosing and implementing a test, compare
several of the methods via a simulation study, and propose a number of open
research questions. Several of the reviewed methods can be implemented in R
using our package spTest, available on CRAN.
