(I asked this question in the Stack Exchange http://stats.stackexchange.com/questions/13014/how-to-combine-the-results-of-several-binary-tests" [Broken], but didn't get anything that was helpful to me.)

I'm a programmer (former EE) working on a medical application. My last stats course was in engineering school 38 years ago, so I'm not exactly current on my stats lingo.

I have the results of essentially 18 different binary (in programmer-speak -- ie, yes/no results, with no adjustable threshold) medical tests, each one of which is a (presumably independent) "proxy" measurement for the disorder being tested for. For each of the tests I have statistics (though in a few cases "artistically derived") for # of true positive, false positive, true negative, false negative when compared to the "gold standard", so I can compute specificity, sensitivity, PPV, NPV, etc. (Typical specificity/sensitivity values are, in %, 50/71, 24/85, 29/84, 72/52.) I

What I want to do is, given the individual statistics, derive a formula that, for a given set of inputs (list of test results for a single patient), will decide "probably positive", "probably negative", or "ambiguous" (and, to keep the FDA and the medical bean counters happy, it would be nice if the formula had some small degree of rigor). (The whole idea here is to avoid the expensive and uncomfortable "gold standard" test where possible, especially on the negative side.)

The best scheme I've come up with so far is to combine specificities using the formula

combine the selectivities the same way, and then take the ratio

Using >> 1 for POSITIVE, << 1 for NEGATIVE, and near 1 for "ambiguous".

This works fairly well, though it seems to behave strangely for some inputs, and it clearly lacks any real mathematical/statistical rigor.

So I'm wondering how one should go about deriving the "correct" formula, given the available statistics.

I'm a programmer (former EE) working on a medical application. My last stats course was in engineering school 38 years ago, so I'm not exactly current on my stats lingo.

I have the results of essentially 18 different binary (in programmer-speak -- ie, yes/no results, with no adjustable threshold) medical tests, each one of which is a (presumably independent) "proxy" measurement for the disorder being tested for. For each of the tests I have statistics (though in a few cases "artistically derived") for # of true positive, false positive, true negative, false negative when compared to the "gold standard", so I can compute specificity, sensitivity, PPV, NPV, etc. (Typical specificity/sensitivity values are, in %, 50/71, 24/85, 29/84, 72/52.) I

**do not**have a collection of results for the entire suite of tests which would show which combination was true for a given specific patient, and I have no real prospect of making such measurements (or any other new measurements), at least not before an initial formula is produced.What I want to do is, given the individual statistics, derive a formula that, for a given set of inputs (list of test results for a single patient), will decide "probably positive", "probably negative", or "ambiguous" (and, to keep the FDA and the medical bean counters happy, it would be nice if the formula had some small degree of rigor). (The whole idea here is to avoid the expensive and uncomfortable "gold standard" test where possible, especially on the negative side.)

The best scheme I've come up with so far is to combine specificities using the formula

Code:

`spec_combined = 1 - (1 - spec_1) * (1 - spec_2) * ... (1 - spec_N)`

Code:

`(1 - sens_combined) / (1 - spec_combined)`

This works fairly well, though it seems to behave strangely for some inputs, and it clearly lacks any real mathematical/statistical rigor.

So I'm wondering how one should go about deriving the "correct" formula, given the available statistics.

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