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I have a dataset that contains ~7,500 blood tests from ~2,500 individuals. I'm trying to find out if variability in the blood tests increases or decreases with the time between two tests. For example - I draw your blood for the baseline test, then immediately draw a second sample. Six months later, I draw another sample. One might expect the difference between the baseline and the immediate repeat tests to be smaller than the difference between the baseline and the six-month test.

Each point on the plot below reflects the difference between two tests. X is the number of days between two tests; Y is the size of the difference between the two tests. As you can see, tests aren't evenly distributed along X - the study wasn't designed to address this question, really. Because the points are so heavily stacked at the mean, I've included 95% (blue) and 99% (red) quantile lines, based on 28-day windows. These are obviously pulled around by the more extreme points, but you get the idea.

alt text http://a.imageshack.us/img175/6595/diffsbydays.png

It looks to me like the variability is fairly stable. If anything, it's higher when the test is repeated within a short period - that's terribly counterintuitive. How can I address this in a systematic way, accounting for varying n at each time point (and some periods with no tests at all)? Your ideas are greatly appreciated.

Just for reference, this is the distribution of the number of days between test and retest:

alt text http://a.imageshack.us/img697/6572/testsateachtimepoint.png

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  • $\begingroup$ +1 for interesting real case ! can you help us to understand what is blood test ? (how do you calculate it) $\endgroup$ – robin girard Aug 7 '10 at 9:35
  • $\begingroup$ There was an article on LFT (liver function tests) and how it seemed to vary in a cyclic basis (in months). I think it was in Annals of Internal Medicine. $\endgroup$ – Ming K Aug 7 '10 at 15:24
  • $\begingroup$ Sure - briefly, it's a test for latent tuberculosis infection. Blood is drawn from the patient into three tubes: one that has no antigens, one that has selected TB antigens, and one that has mitogen. We then compare amount of immunologic response in the tube with no antigens to the one with TB - if they have a strong boost of response in the TB tube, they may have latent TB. (The mitogen tube serves as a check to make sure that the person is capable of producing an immunologic response at all - most people have very strong reactions to it) $\endgroup$ – Matt Parker Aug 7 '10 at 17:25
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From your description I can't see any reason to distinguish the "baseline test" from the immediately drawn "second sample". They are simply 2 baseline measurements and the variance (at baseline) can be calculated on that basis. It would be better plotting the mean of the two baseline measurements versus the third "six month" sample.

The problem is with the 6 month sample. As only one sample is taken at this point there is no way of estimating the "variability" at this point, or rather separating sampling variation from longitudinal (real) change in TB reading.

If we consider this a longitudinal data analysis problem we would probably choose the a random intercept (baseline TB) and a random slope (to fit the 6 month TB). The sampling variability would be estimated from the two baseline measurements and the slope from the third 6 month measure. We can't estimate the variability at 6 months without strong distributional assumptions on the chnage over those six months, such as assuming no change.

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