I'll chime in on this.

I'm very familiar with one of the four antibody tests currently being performed and approved under FDA emergency authorization. I am also familiar with the the other 3, but not as well.

Any test method has to be accurate, precise, robust, linear, sensative and specific.

accuracy... as it sounds, if your result should be 90, the closer to 90 the better.

Precision.. how close to the bullseye you get in multiple attempts. This used to be called reliability and is measured by r2.

Robustness... having reliable and accurate results on different days, different personnel, different laboratories/clinical settings, different instruments/equipment, reagents etc. This is used to reduce the chance of bias. "On sunny days good results, rainy days, bad results"

Linearity... evaluate the response of the test to ensure a little amount produces a little result and a large amount of analyte has a proportional large result.

Sensitivity... determine the methods detection limit and also the level the method can reliability produce an accurate, precise and linear result.

Specificity.. determine if the method selectivity only analyzes what you want. Is the method prone to interference from other analytes? This was once called selectivity.

Under the emergency authorization by the FDA some of these studies were very abbreviated and developed and validated by groups (universities) that commonly don't validate these tests.

Antibody tests are looking for immunoglobulins, specifically IgG and IgM. In order for the test to be selective for COVID19, a way to seperate the COVID19 antibodies from the gagillion of other antibodies and other proteins floating around in a patients blood. So a synthetic antigen is made with the hope of teasing the immunoglobulin out. This is the tricky part as the COVID19s spike protien is unique, it's not completely novel and people have similar antibodies, though not COVID19 antibodies that may or may not conjugate with the antigen. Also for some reason, genetics, compromised immune system, people's antibodies, although present just don't bind with the antigen. The tests build that fudge factor in, but that's a guess based on other immuno assays.

The degree of error of the tests haven't been determined yet. A guess of error has been made, up to 60% for one said test, but many of these leaked studies are actually the clinical trail and data aggregation of the test itself.

What I'm saying is don't get too caught up in the news and numbers yet. We still have a lot of science to do and science takes time as frustrating as it is.

I worked on a monoclonal antibody a way back. We thought it was going to change the world, and our data looked great and in a controlled clinical setting the therapy outperformed all expectations. Several months after approval, adverse events and deaths piled up. A recall was initiated and a year later we finally figured out what we missed. Science takes time.