Basically the decoder will have multiple inputs (three i think in this case). If the inputs change state (high / low) but the none of the outputs change (high / low) then the decoder has failed.
There is two types of aux lamp boards with slightly different logic, but basically works the same way (and i dont remember off the top of my head what they are). The other logic chips can fail as well. With these you would probably need a logic probe and know how to test that particular kind of logic gate.
Do the connectors first tho. If an enable pin, 5v pin, ground, has a cracked solder joint/bad female connector none of the aux lamps will work.