Like many games of the era with discrete ICs such as RAM chips, there are a few tricks you can use to try to diagnose a failed chip, if the self-test mode of the game isn't being too illuminating.

• Look at the symptom. If you're missing colors in the test mode, background colors change randomly, the screen appears to wash out, etc. -- the manual suggests you may suspect one of the screen RAMs (2101). There's also some 2114 RAM as well as 27LS00 RAM. Other symptoms I've encountered are moving objects draw correctly, but text folds over at a point, which would indicate a stuck address line.
• Chips that fail sometimes become much hotter than their counterparts. This isn't a sure-fire indicator -- some chips just run hotter than others, and it may not always get hot when it fails. But, it may provide a quick way to narrow things down.
• If you have spare chips stocked, you can "piggyback" as a means of testing. Place the good chip over the suspect chip, and see if the issue goes away or changes. If it does, try replacing the chip you just "covered. If the issue goes away, you're done!