The Arduino SD library should not be used with more than one device on a SPI bus, you should use the SdFat library to get multiple devices to work on a SPI bus.
...unless the other device is the w5100 ethernet IC, then the standard SD library will work fine.
@LeahM: Try disabling the SD SPI in setup. If D4 is not the slave select, change that.
// then do your AD stuff
You must disable it or initialize it. Otherwise, it will cause garbage on the SPI bus.
If you initialize it, insure you disable your AD device SPI first. Otherwise, it could cause garbage on the SPI and mess up the SD startup.