It seems to me that the authors of the study used the phrase 'I'm proud of you' as a catch-all measurement, to reflect a parent's emotional connection with, and support, for, their child - and that it was this emotional involvement that provided protection against suicidal ideation.
How I long for somebody to say that to me.
I care about your life and your happiness.
I just felt important.
Posted articles do not constitute advice or recommendations.
I believe the effort has been worth it — for my children, and our relationship.