10 ms·

Berkson's Paradox

39 points by xtacy 12y ago

sp332 12y ago
In this notation: P(A|B,C) what does the comma mean?
- tredontho 12y ago
  I think it's the intersection, so, "and". P(A|B,C) is "Probability of A, given both B and C".
  - spacehome 12y ago
    That's true, may be a slightly confusing way to interpret it, given that C = A∪B. In context I think it's better to read it as P(A|B), in the case that C occurs.
    - theophrastus 12y ago
      wouldn't that be P((A|B)|C) ? or even more precedence dubious: P(A|B|C) ?
      - DrewAllyn 12y ago
        The "given" symbol should only appear once in any probability statement.
        spacehome 12y ago
        That's true from an objective, mathematical standpoint. But the paradox is really saying something about how humans perceive the statement, so different (mathematically equivalent) ways of framing the same thing can make a difference. It's a minor point.
- deleted 12y ago
  [deleted]
- stepstep 12y ago
  Yeah it's tricky because in most programming languages, '|' has precedence over ',', but in math sometimes it's the opposite. That tripped me up too.
- Chinjut 12y ago
  In this context, the comma is as good as &.
btilly 12y ago
Implicit population selection biases can lead to all sorts of fun. One that I saw a paper get wrong a while ago was one which claimed to find an inverse correlation between two different traits that individually improved intelligence. The catch? The sample population was chosen from university students at no name school X. But people who did well on both traits would have done well enough to go to a better school and so were underrepresented in the population sample! (I forget the paper, but pay attention and you'll find lots of other examples...)
- gwern 12y ago
  Speaking of intelligence and Berkson's paradox, it looks like Berkson's paradox may be responsible in a very similar way for why the personality trait Conscientiousness seems to correlate negatively in some studies with intelligence - the studies were using selected samples while more representative population samples show the expected independence: "How are conscientiousness and cognitive ability related to one another? A re-examination of the intelligence compensation hypothesis", Murray et al 2014 (https://pdf.yt/d/Dfl1N6pbR-4vYaKk https://pdf.yt/d/Dfl1N6pbR-4vYaKk ; excerpts: https://plus.google.com/103530621949492999968/posts/aQ51UnLCLee https://plus.google.com/103530621949492999968/posts/aQ51UnLC... )
Tarrosion 12y ago
This reminds me of a thought experiment I read a while ago, I believe on the Atlantic. I'll paraphrase with a bit more math: Suppose that acting ability and attractiveness are independently normally distributed with mean 0 and standard deviation 1. Further suppose that to be a successful movie star, an individual must have an acting ability plus attractiveness of 6. Then among people with the necessary attributes to be movie stars, attractiveness and acting ability will be negatively correlated. In this example, we might expect to see movie stars with (intelligence, attractiveness) around (3,3) or (2,4) or (4,2), but it's much more unlikely that we see many people around (4,4).
- btilly 12y ago
  For a more thought-provoking variation, suppose that intelligence and test taking ability are independent, and IQ is the sum of the two. What, then, does your IQ score say about your intelligence? See http://bentilly.blogspot.com/2010/02/what-is-intelligence.html http://bentilly.blogspot.com/2010/02/what-is-intelligence.ht... for some of my thoughts on that from a few years back.
  - Retric 12y ago
    That's an unlikely assumption. A better case might be for students at big name university's where GPA (proxy work etc) and intelligence (proxy SAT score etc) need to be over some threshold for admittance. If that's the case bankers and others that want both a big name school and high GPA are actually negatively selecting for intellect. Which may account for a lot of fairly dumb behavior at banking institutions etc. As they might have a lot of people that can talk about statistics without actually understanding it.
    - btilly 12y ago
      That's an unlikely assumption. On what basis do you believe it is unlikely? As an example I submit that test preparation courses like Kaplan do nothing but improve how well you'll do on a certain type of test without improving your general intelligence. As another example I am quite aware of how much of an advantage I gain on tests from my ability to relax in a situation where other people tense up. I've described this advantage before with, "Comparing me to a normal person based on the resulting test score is like starting with two runners, taking one out back and beating on him for a while, then expecting them to run a fair race."

Berkson's Paradox

39 points by xtacy 12y ago

sp332 12y ago

In this notation: P(A|B,C) what does the comma mean?

tredontho 12y ago
I think it's the intersection, so, "and". P(A|B,C) is "Probability of A, given both B and C".
- spacehome 12y ago
  That's true, may be a slightly confusing way to interpret it, given that C = A∪B. In context I think it's better to read it as P(A|B), in the case that C occurs.
  - theophrastus 12y ago
    wouldn't that be P((A|B)|C) ? or even more precedence dubious: P(A|B|C) ?
    - DrewAllyn 12y ago
      The "given" symbol should only appear once in any probability statement.
      - spacehome 12y ago
        That's true from an objective, mathematical standpoint. But the paradox is really saying something about how humans perceive the statement, so different (mathematically equivalent) ways of framing the same thing can make a difference. It's a minor point.
deleted 12y ago
[deleted]
stepstep 12y ago
Yeah it's tricky because in most programming languages, '|' has precedence over ',', but in math sometimes it's the opposite. That tripped me up too.
Chinjut 12y ago
In this context, the comma is as good as &.

btilly 12y ago

Implicit population selection biases can lead to all sorts of fun. One that I saw a paper get wrong a while ago was one which claimed to find an inverse correlation between two different traits that individually improved intelligence. The catch? The sample population was chosen from university students at no name school X. But people who did well on both traits would have done well enough to go to a better school and so were underrepresented in the population sample! (I forget the paper, but pay attention and you'll find lots of other examples...)

gwern 12y ago
Speaking of intelligence and Berkson's paradox, it looks like Berkson's paradox may be responsible in a very similar way for why the personality trait Conscientiousness seems to correlate negatively in some studies with intelligence - the studies were using selected samples while more representative population samples show the expected independence: "How are conscientiousness and cognitive ability related to one another? A re-examination of the intelligence compensation hypothesis", Murray et al 2014 (https://pdf.yt/d/Dfl1N6pbR-4vYaKk https://pdf.yt/d/Dfl1N6pbR-4vYaKk ; excerpts: https://plus.google.com/103530621949492999968/posts/aQ51UnLCLee https://plus.google.com/103530621949492999968/posts/aQ51UnLC... )

Tarrosion 12y ago

This reminds me of a thought experiment I read a while ago, I believe on the Atlantic. I'll paraphrase with a bit more math: Suppose that acting ability and attractiveness are independently normally distributed with mean 0 and standard deviation 1. Further suppose that to be a successful movie star, an individual must have an acting ability plus attractiveness of 6. Then among people with the necessary attributes to be movie stars, attractiveness and acting ability will be negatively correlated. In this example, we might expect to see movie stars with (intelligence, attractiveness) around (3,3) or (2,4) or (4,2), but it's much more unlikely that we see many people around (4,4).

btilly 12y ago
For a more thought-provoking variation, suppose that intelligence and test taking ability are independent, and IQ is the sum of the two. What, then, does your IQ score say about your intelligence? See http://bentilly.blogspot.com/2010/02/what-is-intelligence.html http://bentilly.blogspot.com/2010/02/what-is-intelligence.ht... for some of my thoughts on that from a few years back.
- Retric 12y ago
  That's an unlikely assumption. A better case might be for students at big name university's where GPA (proxy work etc) and intelligence (proxy SAT score etc) need to be over some threshold for admittance. If that's the case bankers and others that want both a big name school and high GPA are actually negatively selecting for intellect. Which may account for a lot of fairly dumb behavior at banking institutions etc. As they might have a lot of people that can talk about statistics without actually understanding it.
  - btilly 12y ago
    That's an unlikely assumption. On what basis do you believe it is unlikely? As an example I submit that test preparation courses like Kaplan do nothing but improve how well you'll do on a certain type of test without improving your general intelligence. As another example I am quite aware of how much of an advantage I gain on tests from my ability to relax in a situation where other people tense up. I've described this advantage before with, "Comparing me to a normal person based on the resulting test score is like starting with two runners, taking one out back and beating on him for a while, then expecting them to run a fair race."