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Let's say we are using logistic regression model for binary classification with the goal of classifying subjects as healthy or diseased. The output is the probability of the subject being diseased.

Does the output probability have anything to do with the level of disease? For example can we say if the output probability is higher this means that the disease is more sever for this subject?

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Logistic regression works with binary outcome variables; for disease either a "yes" or a "no" (usually a 1 or a 0, numerically). The fitted values of a logistic regression will be the probability that a particular observation is a "yes" (1 or 'has disease'). This will not say anything about the severity of the disease, since you are not giving the model that information in the outcome variable (the outcome is binary and says nothing about severity: has disease or not).

If you have information about disease severity, say BMI (body mass Index), or something, then you could perhaps work with a linear regression, later sorting which cases are likely to have BMI over / under whatever threshold is used to classify the presence of the BMI-related disease.

There is a nice chapter on this stuff here:

Diez, D. M., Barr, C. D., & Cetinkaya-Rundel, M. (2012). OpenIntro statistics (Vol. 12): CreateSpace.

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