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  1. I don't understand how the following [I reworded the original] instances Berkson's Paradox; and can be analogized to the University Admissions example, where $\color{limegreen}{\text{corr(SAT, GPA)}> 0}$, but criteria (SAT + GPA = constant) makes it appear $< 0 $.

  2. Beneath, nothing is stated about $\text{corr(pregnancy, AIDS)}$. But this may be relevant:

This can occur when the subset is not an unbiased sample of the whole population. It has been frequently cited in medical statistics. For example, if patients only present at a clinic with disease A, disease B or both, then even if the two diseases are independent, a negative association between them may be observed.

  1. Does the bolded italicized sentence assume mutual exclusivity of pregnancy and AIDS?
    If yes, isn't this wrong?

  2. Is this post relevant here? If yes, what'd the blue and green excluded data be here?

However, in the same way that universities accept students based on a combination of positive traits, hospitals accept patients based on a combination of symptoms. For example, if a study wants to know if pregnancy increases or decreases the time for an HIV-positive woman to develop full-blown AIDS, and the study is conducted at a antenatal clinic, the study may be biased. Women will come either to be seen about pregnancy or about AIDS risk [2]. So, a woman who comes to the clinic and is not pregnant is likelier to have AIDS than someone in the general population, as they likely came to the clinic for some reason barring pregnancy. There may seem to be a correlation even if there is not one.

[2] Westreich D. Berkson’s bias, selection bias, and missing data. Epidemiology (Cambridge, Mass). 2012;23(1):159-164. doi:10.1097/EDE.0b013e31823b6296.

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    $\begingroup$ Please do not make us click through three links to figure out what it is you're asking... I realize that may make the question longer, but you'll get more responses if it's self-contained. $\endgroup$
    – jbowman
    Dec 25 '17 at 22:40
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    $\begingroup$ @jbowman Sorry. You needn't click through 3 links though. The quotation is my rewrite of the first link; you needn't read the first link. If 2 links are still too much, then please LMK. $\endgroup$
    – NNOX Apps
    Dec 25 '17 at 22:43
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The answer to your first question is "No."

To see this, consider the following model of patient disease and behavior, which is generic, i.e., not specific to AIDS and pregnancy. We have four patient states: $N$ for no disease, $A$ for disease A, $B$ for disease B, and $AB$ for both diseases. Now, the state transitions are such that the patient can go from $N \to A$, $N \to B$, $A \to AB$, $B \to AB$, but very rarely will a patient go directly from $N \to AB$.

Now for patient behavior. When does a patient go to the hospital? Presumably shortly after becoming symptomatic with whichever disease they get, either $A$ or $B$. This does not give much time for transitions $A \to AB$ or $B \to AB$ to occur; consequently, patients who show up for treatment tend to have $A$ or $B$ but not both.

Now for the effect of hospital intervention. Assuming the diseases can be cured, the hospital intervention induces the transitions $A \to N$ or $B \to N$, again not leaving much time for transitions $A \to AB$ or $B \to AB$ to occur. Consequently, after treatment, the patient returns to state $N$, from which they may of course transition to $A$ or $B$ at a later date.

If, on the other hand, the patient doesn't seek treatment, the transitions $A \to AB$ or $B \to AB$ occur more frequently, because the treatment-induced transitions $A \to N$ or $B \to N$ don't occur as often (this presumes there is some value in the treatment.) Consequently, the proportion of people in the population with a disease who are in state $AB$ is greater than the proportion of people arriving at the hospital in state $AB$. This will give the hospital a biased view of $AB$ incidence relative to $A$ or $B$ incidence.

On to a rather extreme concrete example. Assume time-to-disease is distributed Exponentially with mean 1 year for both disease $A$ and $B$, and that the population is equally divided between people who will visit the hospital and people who won't. If we assume hospital visits occur instantaneously upon getting a disease, and the cure is also instantaneous, it should be clear that everyone visiting the hospital will have $A$ or $B$ but no-one will have both. However, among the population of non-hospital-visitors, after a few years, say five, almost everyone will have both $A$ and $B$. What the hospital sees, however, is that no-one has both $A$ and $B$, which is as strong a negative association as you can get,as well as being strongly non-representative of the population at large.

With respect to your second question - I think you should stop thinking in terms of blue and green data. You've made several posts which indicate, cumulatively, that that isn't a helpful way to think about this problem for you. Instead, draw state transition diagrams or Venn diagrams or work through examples using Bayes' rule or some other approach.

Let's assume there are only two reasons why someone would show up at a clinic: AIDS or pregnancy. If someone shows up at the clinic but isn't pregnant, there's a 100% chance they have AIDS, because otherwise they wouldn't have shown up at the clinic (in our two-disease world.) This does not mean that 100% of the non-pregnant population has AIDS, however. All it means is that people who are not pregnant and who don't have AIDS don't visit the clinic.

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