The resulit of mcintosh_e index calculated by qiime diversity alpha/–p-metric mcintosh_e is incorrect. It may be heip_e index.
How are you testing this? What equation do you consider to be the “correct” McIntosh E?
From perusing the literature it seems that there are a few different definitions of McIntosh E out there, and some are incorrect.
In any case, all alpha diversity metrics used in QIIME 2 are imported directly from scikit-bio (which, in turn, are mostly imported from scikit-learn); you can see the scikit-bio McIntosh E equation here. See also the note on that page — perhaps you are using an incorrect implementation of McIntosh E? Or perhaps scikit-bio is wrong about their assertion and Heip 1974 is using the wrong McIntosh E equation in their work? Let us know which definition of E you are using, and if it differs from scikit-bio’s definition perhaps also provide us with the reference or paper you are using for that equation.