R2 is not a good effect size measure, but none of these arguments should convince anyone. http://data.library.virginia.edu/is-r-squared-useless/ …
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H2 is a variance type metric, actually. You can and in some cases should convert to correlation-type metric.
Another criterion: whether the metric is comparable across models, datasets, range restriction etc. There's likely no one best metric.
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And whether it is susceptible to outliers etc. E.g. the squared-family metrics are strongly affected, but e.g. median absolute error is not.
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