ChristianKl comments on Towards cause priotisation estimates for child abuse - Less Wrong Discussion
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Given that the abstract doesn't say so, that's very likely to be a misreading. My guess is that they report correlations but haven't ruled out genetic effects. The same is true for the other studies you link for harm, they also seem to talk about correlations.
You don't talk about the amount of government money that's invested into the cause.
No, there no reason that a 10% compliance rate would be equally distributed among different risk groups. The parents who make a deliberate conscious choice to use corporal punishment are less likely to go into stronger abuse than parents who react because they can't manage their emotions well. The parents who act because of deliberate conscious decisions are more likely to comply.