How to denote the contribution of a co-author who did not do any experiments or analyses?
This does not sound like a special case at all, but rather like a typical contribution of a supervisor.
When describing such a co-author’s contributions, I would let myself be inspired by other similar descriptions for supervisors.
The definite answer may depend on the particular style of this section and details of the contribution, but one suggestion would be:


  X designed the study, supervised the research, and wrote the manuscript.