Individual Measurement in Overlapping Distributions I have a situation where I have measurements taken from samples of healthy and pathological populations. The pathological populations tend to have much higher measurements, but the distributions are overlapping. It would be reasonable to model the distributions as Guassian.
Then I take a new individual's measurement and I want to say something about his health status. Something like "You have a X% chance of having this pathology, with specificity Y and sensitivity of Z."
I assume this is a very well-trodden problem, but unfortunately I am not sure where to get started here...
Some initial thoughts:

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*Its easy enough to tackle ths if there was one distribution and one measurement (p-value or such), but here there are two. How do you incorporate info from the 2nd distribution?

*Parametric vs non-parametric statistical approaches

*How would this relate to ROC/PR plots? The measurement would correspond to a specific point on the ROC plot w/ a given TPR & FPR. Can we say something like "Someone w/ this measurement or greater has the following TPR & FPR of being pathological"?

Thanks for the help!

 A: With this data, it looks like a logistic regression could work. It allows you to define $\mathbb{P}(\text{Pathological})$ as a function of your independent variable.
Alternatively, if you know the probability of being pathological in the global population, you could use bayesian statistics.
A: Indeed, this is well-trodden ground. You have most of the data necessary to estimate the probability of disease. In addition to the data you present, consider adding other variables that are associated with disease such as age or other disease-specific factors (eg, smoking).
The measurement depicted and other factors associated with the disease can be entered into a logistic regression model, which can be used to estimate the probability of disease in patients similar to those for whom you have data.
From the logistic model, most packages can estimate an AUC of ROC curve. True positives, false positives, etc. can also be estimated, but require you to create a dichotomy (high risk, low risk) from the spectrum of risk, which may be tempting to do but is rarely biologically plausible.
