As an application analyst, how can I test integration through the interface?

Great question!  First, let me state that the interface engineer that you are working with should test the logic of their code.  It should be their responsibility to ensure that the code is working as designed.  Second, as an application analyst there are two things that you should test.  The first is data validation within the receiving application.  For example, ADT interfaces you will want to make sure that the demographic information is being placed in the correct fields within the receiving application i.e. first name, last name, DOB, gender, and etc.  The second is what I call business logic.  Test to make sure that you get the desired patient population.  If you only want inpatients, test to make sure that outpatients do not come through.  If data is being mapped, make sure that the new values are coming through.  Next, verify that HL7 transaction are working accordingly in the receiving application.  For example if a patient is discharged, verify that the application shows they are discharge.
The best testing document is your interface specifications.  However, don't hesitation to test normal daily activities either.