  Integrating datasets from different disciplines is hard because the data are
often qualitatively different in meaning, scale, and reliability. When two
datasets describe the same entities, many scientific questions can be phrased
around whether the (dis)similarities between entities are conserved across such
different data. Our method, CLARITY, quantifies consistency across datasets,
identifies where inconsistencies arise, and aids in their interpretation. We
illustrate this using three diverse comparisons: gene methylation vs
expression, evolution of language sounds vs word use, and country-level
economic metrics vs cultural beliefs. The non-parametric approach is robust to
noise and differences in scaling, and makes only weak assumptions about how the
data were generated. It operates by decomposing similarities into two
components: a `structural' component analogous to a clustering, and an
underlying `relationship' between those structures. This allows a `structural
comparison' between two similarity matrices using their predictability from
`structure'. Significance is assessed with the help of re-sampling appropriate
for each dataset. The software, CLARITY, is available as an R package from
https://github.com/danjlawson/CLARITY.
