[Epistemic status: just looked into this briefly out of curiosity, not an official EAS analysis]
When I looked at this briefly in a generalized mixed model, I didn’t find a significant interaction effect for gender * engagement * the specific factor people were evaluating (e.g. EAG or group etc.) which comports with your observation that there doesn’t seem to be anything particularly interesting going on in the contrast between male and non-male interaction with low/high engagement. (In contrast, there were significant fixed effects for the interaction between engagement and the specific factor and gender and the specific factor.) Looking at the specific ‘getting involved factors’ in the interaction effect, it was clear where the only one where there was much of a hint of any interaction with gender * engagement was personal contact, which was “borderline significant” (though I am loathe to read much into that).
Probably the simplest way to illustrate the specific thing you mentioned is with the following two plots: looking at both male and non-male respondents, we can see that highly engaged respondents are more likely to select EAG than less engaged respondents, but the pattern is similar for both male and non-male respondents.
[Epistemic status: just looked into this briefly out of curiosity, not an official EAS analysis]
When I looked at this briefly in a generalized mixed model, I didn’t find a significant interaction effect for gender * engagement * the specific factor people were evaluating (e.g. EAG or group etc.) which comports with your observation that there doesn’t seem to be anything particularly interesting going on in the contrast between male and non-male interaction with low/high engagement. (In contrast, there were significant fixed effects for the interaction between engagement and the specific factor and gender and the specific factor.) Looking at the specific ‘getting involved factors’ in the interaction effect, it was clear where the only one where there was much of a hint of any interaction with gender * engagement was personal contact, which was “borderline significant” (though I am loathe to read much into that).
Probably the simplest way to illustrate the specific thing you mentioned is with the following two plots: looking at both male and non-male respondents, we can see that highly engaged respondents are more likely to select EAG than less engaged respondents, but the pattern is similar for both male and non-male respondents.