If you're nervous about one of your stats but have really high other stats you can keep spamming your bad stat to force one of the judges out. I rarely do this, but it saved me a couple times in my first few playthroughs of boxm@s.
Let's see if I can bullet point the process.
3 phases of 9 appeals each.
The idol with the most stars wins. Stars are only awarded at the end of each phase.
Each phase operates on a bidding system. You may choose to put points in each of three categories, nine times per phase.
The effectiveness of your bids is influenced by your stat and your idol's tension (higher = more efficient bids).
Top 3 bidders per phase get stars, but different categories give more stars than others.
Last place gets a negative star.
When any idol (of the six available) appeals to a judge, he gives points, but loses interest.
When a judge's interest reaches 0, he walks. All stars he has given (including negative ones), are lost.
Succeed a memory appeal to gain points in all stats (large amount), and recover all judge's interest.
Fail a memory appeal to lost points in all stats (small amount), and damage all judge's interest.
Notes:
The judge that awards the most stars is the most likely to be appealed to. Thus, he is the most prone to walking. Over-appeal, and you'll come out with nothing.
Some people attempt to rank in the two worst categories, then kill off the most popular judge (sometimes with failed memory appeals).