Unless students were randomised to attend ALEKS or MOEMS sessions - and the results analysed on the basis of where students were randomised to, regardless of whether they actually attended - this comparison has no value. Were they?
( My intuition is that you're basically right, *but* that there will be significant numbers of students, including some with huge potential, not reached by any optional competition programme. My first step would be to make actually competing in the competitions optional, and advertise that it's perfectly fine to come to sessions to work on problems and never actually compete.)
I wouldn't say useless. It tells you that there is some combination of the relative strength (and potential to improve) of the students choosing MOEMS (relative to those who don't so choose), and the benefits of doing the MOEMS activities, that leads to improvements in other scores.
It's like if I tell you x+y=7.5, you can't determine x or y, but the equation certainly tells you something about (x,y).
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One thing I thought of when I asked the question a few posts up was, it may sometimes happen that someone that didn't think of themselves as super strong in math, may do really well in a competition, realize they've underestimated themselves, and improve a lot. (I.e. the competition could "discover" unrecognized/undeveloped talent, which is subsequently developed.)
