I have the test results of a blood test administered to 2500 people four times at six-month intervals. The results primarily consist of two measures of immune response - one in the presence of certain tuberculosis antigens, one in the absence. Currently, each test evaluates to either positive or negative based on the difference between the antigen response and the nil response (with the idea being that if your immune system responds to TB antigens, you've likely been exposed to the bacterium itself at some point). In essence, the test supposes that a non-exposed individual's distributions of nil and TB responses should be basically identical, whereas a person with TB exposure will have TB responses drawn from a different distribution (of higher values). Caveat: the responses are very, very non-normal, and values clump at both the natural floor and the instrument-truncated ceiling.
However, it's seems pretty clear in this longitudinal setting that we're getting "false positives" (no actual gold standard for latent tuberculosis, I fear) that are caused by (typically small) fluctuations in the antigen and nil responses. While this might be hard to avoid in some situations (you may only get one chance to test someone), there are many situations in which people are routinely tested for TB every year or so - in the US, this is common for healthcare workers, the military, homeless people staying at shelters, and so on. It seems a shame to ignore prior test results because the extant criteria happen to be cross-sectional.
I think that what I'd like to do is what I crudely conceive of as longitudinal mixture analysis. Much like the cross-sectional criteria, I'd like to be able to estimate the probability that an individual's TB and nil responses are drawn from the same distribution - but have that estimate incorporate prior test results, as well as information from the sample as a whole (e.g., can I use the sample-wide distribution of within-individual variabilities to improve my estimates of a specific individual's distribution of nil or TB?). The estimated probability would need to be able to change over time, of course, to account for the possibility of new infection.
I've gotten myself totally twisted around trying to think about this in unusual ways, but I feel like this conceptualization is as good as any I'm going to come up with. If something doesn't make sense, please feel free to ask for clarification. If my understanding of the situation seems wrong, please feel free to tell me. Thank you so much for your help.
In response to Srikant: It's a case of latent classification (TB-infected or not) using the two continuous (but non-normal and truncated) test results. Right now, that classification is done using a cutoff (in its simplified form, TB - nil > .35 -> positive). With test results presented as (nil, TB, result), the basic archetypes* are:
Probable Negative: (0.06, 0.15, -) (0.24, 0.23, -) (0.09, 0.11, -) (0.16, 0.15, -)
Probable Positive: (0.05, 3.75, +) (0.05, 1.56, +) (0.06, 5.02, +) (0.08, 4.43, +)
Wobbler: (0.05, 0.29, -) (0.09, 0.68, +) (0.08, 0.31, -) (0.07, 0.28, -)
The positive on the second test for the Wobbler is pretty clearly an aberration, but how would you model that? While one line of my thinking is to estimate the "true difference" between TB and nil at each time point using a repeated-measures multilevel model, it occurred to me that what I really want to know is if the person's nil response and TB response are drawn from the same distribution, or if their immune system recognizes the TB antigens and activates, producing an increased response.
As for what could cause a positive test other than infection: I'm not sure. I suspect it's typically just within-person variation in results, but there's certainly a possibility of other factors. We do have questionnaires from each time point, but I haven't looked into those too much yet.
*Fabricated but illustrative data