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"Type III error occurs when you correctly conclude that the two groups are statistically different, but you are wrong about the direction of the difference. Say that a treatment increases some variable. But in your experiment, random sampling leads the value of that variable to be lower (on average) in the treated group, and enough lower that the difference is statistically significant."

It sounds like a test which is better able to find significant differences (i.e., more powerful) should thus mechanically lead to more type III errors. Is there any literature on the topic?

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I'm not sure that this is entirely true. There is some literature on the topic that seems to indicate that in fact a lower power is what leads to a type III error. From your description a type III error is a type S error in the linked paper.

You might try some simulation to verify these results.

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