Does Lin's concordance correlation coefficient assume that the 2 data sets have linear or monotonic tendencies? Or, can I measure the concordance between 2 data sets that have a sinusoidal tendency?
The concordance correlation can be thought of as a measure of agreement. The question is: do two variables $x$ and $y$ (say) have identical values? If so, the concordance correlation will be 1. The question makes no sense unless the variables have the same units of measurement or more generally are recorded in the same way.
You can calculate a concordance correlation for any variables you like, but the answer will be of no use unless your question is about agreement. You could have a deterministic relation $y = sin x$, but concordance between $y$ and $x$ will be a meaningless number, if only because concordance correlation does not adjust for different units.
For an informal introduction to this area, see
Cox, N.J. 2006. Assessing agreement of measurements and predictions in geomorphology. Geomorphology 76: 332-346. http://www.sciencedirect.com/science/article/pii/S0169555X05003740
Here "in geomorphology" indicates the field of the examples, not a restriction of statistical scope.
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