I am studying a test X in a population A, and obtained a ROC curve A. After using another test Y, to rule out some patients, I obtain population B. Applying again test X in population B, I obtain a ROC curve B. Any methods and lit ref about comparing ROC curve in that context, instead of using LH would be great!
Very interesting. If I understand population B is a subset of population A. X and Y are two tests or models that produce a score for observations. What model are your tests based on? Perhaps some modeling methods will have a more straightforward way to assess quality of fit on testing data.
Otherwise, my suggestion would be to bootstrap confidence intervals on the AUROCs for X and Y. Resample the observations in A with replacement (so the total number of observations is the same as originally) and calculate the AUROC for X on the resampled distribution. Repeat this many times to get a distribution of AUROCs. Take the 2.5 and 97.5 percentiles to identify the 95% confidence interval. Then repeat with B and Y and check for confidence interval overlap. Or you could potentially test for equality between the two bootstrapped AUROC sets with a Mann–Whitney U test.
I think a cross-validation strategy may be preferable, but I will need to know the specifics of your model. Are you training your models or do your tests not require any parameter estimation?
Not completely sure I understand population A and B vs ROC A and ROC B. If tests X and Y are used in the same sample, we could compare using DeLong test. Bootstrapping would also work, as always. If we study incremental value of Y over X, we could fit a logistic regression model, and the likelihood ratio test would give us the p-value, see e.g. http://www.ncbi.nlm.nih.gov/pubmed/21276237
Your question should be answered in view of diagnostic pathways.
To give an example, your approach is the difference to apply a Tumor marker (test X) at a screening population on the one hand and a population with symptoms (symptoms=test B) on the other Hand, whereby the latter is a subpopulation of the screening population.
Formally, we can describe your setting in this way:
1. application of test X on population A
2. application of text Y on population A
then you have 4 populations:
a: X+Y+, b: X+Y-, c: X-Y+, d: X-Y-
Population B consists of 2 of these subgroups, suppose those with positive Y: X+Y+ and X-Y+.
I have concerns whether a larger AUC for test X in population B then for test X in population A is really a proof for the advantage of the diagnostic pathway you have in mind. In fact, it is easy to construct a situation where a lower AUC in A could be more advantegeous then a larger ROC for B.
The way to analyse these or similar situations is described in papers describing strategies about add-on-tests, triage test etc.
e.g. Bossuyt (2006) Comparative accuracy: assessing new tests against existing diagnostic pathways
