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In The Idiot Brain: A Neuroscientist Explains What Your Head is Really Up To, Dean Burnett wrote

The correlation between height and intelligence is usually cited as being about $0.2$, meaning height and intelligence seem to be associated in only $1$ in $5$ people.

To me, this sound wrong: I understand the correlation more like the (lack of) error we get when we try to predict one measure (here intelligence) if the only thing we know about that person is the other measure (here height). If the correlation is $1$ or $-1$, then we don't make any error in our prediction, if the correlation is $0.8$, then there is more error. Thus the correlation would apply to anyone one, not just $1$ in $5$ people.

I have looked at this question but I am not good enough in maths to understand the answer. This answer which talks about the strength of the linear relationship seems in line which my understanding but I am not sure.

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    $\begingroup$ @JamesPhillips, what you are referring to is $r^2$, not $r$ itself. If $r=0.2$ then $r^2=0.04$ so 4%. $\endgroup$
    – Richard Hardy
    Commented Feb 14, 2018 at 20:28
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    $\begingroup$ 4 percent makes much more sense than 20 percent, thank you kindly for the correction, I agree with you. $\endgroup$
    – James Phillips
    Commented Feb 14, 2018 at 20:39
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    $\begingroup$ This 0.01% sample of the book makes me wonder what nonsense is to be found in the rest... $\endgroup$
    – Nick Cox
    Commented Feb 15, 2018 at 13:19
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    $\begingroup$ I have favorited this post because it's precisely the kind of extremely simple question that, when asked of a stats 001 student (or any other neophyte, or a job applicant), will instantly and unmistakably determine whether they understand what correlation means. $\endgroup$
    – whuber ♦
    Commented Feb 15, 2018 at 16:25
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    $\begingroup$ I'd stop reading that book, right about now $\endgroup$
    – PhD
    Commented Feb 16, 2018 at 20:36

4 Answers 4

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The quoted passage is indeed incorrect. A correlation coefficient quantifies the degree of association throughout an entire population (or sample, in the case of the sample correlation coefficient). It does not divide the population into parts with one part showing an association and the other part not. It could be the case that the population actually consists of two subpopulations with different degrees of association, but a correlation coefficient alone doesn't imply this.

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    $\begingroup$ What's more, even in a population where 20% of people showed perfect correlation between height and intelligence and 80% showed zero correlation, the population-wide correlation isn't necessarily 0.2. The statement is wrong in several ways! $\endgroup$
    – Nuclear Hoagie
    Commented Feb 15, 2018 at 13:54
  • $\begingroup$ Weird things happen with threads that get into the Hot Network list. This answer is obviously correct and fine... but 57 upvotes?! :-) $\endgroup$
    – amoeba
    Commented Feb 17, 2018 at 22:27
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    $\begingroup$ @amoeba If you think that's wild, check out my highest-scoring answer. $\endgroup$
    – Kodiologist
    Commented Feb 17, 2018 at 23:27
  • $\begingroup$ Haha, you are the master! $\endgroup$
    – amoeba
    Commented Feb 17, 2018 at 23:30
  • $\begingroup$ Does it count as trolling to +1 something just to further the strangeness that is SE? $\endgroup$
    – Nat
    Commented Feb 18, 2018 at 1:50
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No, 0.2 doesn't mean 1 in 5 people show correlation. I don't know how he could write this nonsense.

Here's the source of 0.2 number: "On the sources of the height–intelligence correlation: New insights from a bivariate ACE model with assortative mating", https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3044837/ Apparently, the correlation is robust.

I already knew about it: my IQ rose considerably with my height as I grew taller. Now I know why am I not getting smarter anymore: my height is stable.

This was a joke, of course, but it points out the issue with that "Idiot" book's author's argument: nobody's measuring within subject correlation of height and IQ, at least as far as I know. I'm not sure how would you do it cleanly, there'd be so much confounding.

Having said that the researchers are using tricks such as looking at within twins and within family correlations of height and IQ, this helps them address confounding issues. Presumably, twins are growing up in similar environment and have the same DNA, so in observation studies it helps to address endogeneity and other issues. However if you set this all aside, the bottom line is that "0.2 correlation" gives no basis to saying nonsense like in some people there is correlation and in others there is none. It's just a ridiculous interpretation of correlation study results.

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    $\begingroup$ -1: while I understand the spirit of the last paragraph of this answer, I believe that it increases the confusion as it unnecessarily introduces the concept of causality (the why here is not relevant). $\endgroup$
    – Jorge Leitao
    Commented Feb 15, 2018 at 7:39
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    $\begingroup$ You must be the one in five for whom there is a correlation. $\endgroup$
    – Carsten S
    Commented Feb 15, 2018 at 12:24
  • $\begingroup$ @JorgeLeitão Of course not, no causation is implied, they grew together, it's correlational! :) $\endgroup$
    – Firebug
    Commented Feb 15, 2018 at 13:23
  • $\begingroup$ @JorgeLeitão, if anything NN research shows that size matters. Both a bigger brain and bigger sample. So we when we grow up our brains increase and we run more stuff through them, hence, we should become smarter. Also, men are taller than women in average, so they must be smarter too, in average. $\endgroup$
    – Aksakal
    Commented Feb 15, 2018 at 15:55
  • $\begingroup$ Hah, what a load of nonsense. $\endgroup$
    – Lightness Races in Orbit
    Commented Feb 16, 2018 at 13:24
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The irony in the statement is almost too thick to parse. Given the title of the text, I'm assuming some tongue-in-cheek was intended. However, your "gut" saying that this is wrong is probably on the right track, if intuition counts for anything. Unfortunately, a lot of scientific reporting eludes intuition when dealing with concepts we haven't encountered.

It is possible that, when measuring an association between $X$ and $Y$, the correlation between $X$ and $Y$ is 1.0 in 20% of the population, and 0 in the remaining 80%. The net effect is that overall the correlation of $X$ and $Y$ is 0.2. We see this all the time in pharmacoepidemiology: an experimental drug is deemed "effective" if on average there is a positive benefit; many drugs in common circulation, some of which you could be taking, may harm you because of interactions with your behavior or genetics, but nobody actually knows this.

The above is but one possible interpretation of a correlation of 0.2; it is extremely far-fetched because so few things in life have a correlation of either 1 or 0, and fewer things still have effect modification strong enough to produce such discrepant correlations.

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    $\begingroup$ "It is possible that, when measuring an association between X and Y, the correlation between X and Y is 1.0 in 20% of the population, and 0 in the remaining 80%." - in the study I posted they looked at within family and between twins correlations, they're different than overall population. However, I'm sure that's not what the book author meant, it's how he interpretes what correlation is the problem $\endgroup$
    – Aksakal
    Commented Feb 14, 2018 at 21:56
  • $\begingroup$ @Aksakal interestingly these variance-components methods are intended to estimate the same population level correlation as would be measured in a population, they just claim to use heritability to "shave off" phenotypic variance attributable to current environmental influence (the E component of the ACE model): an important source of confounding in the hypothesis considered. $\endgroup$
    – AdamO
    Commented Feb 14, 2018 at 22:19
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    $\begingroup$ If correlation is 1 in 20% of the population and 0 in 80%, it does not follow that overall it's 0.2. Depends on the relative variance in each sub-population. $\endgroup$
    – amoeba
    Commented Feb 14, 2018 at 22:24
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    $\begingroup$ @amoeba yes, good point, further underscores what an idiopathic scenario would justify such a claim. $\endgroup$
    – AdamO
    Commented Feb 14, 2018 at 22:29
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    $\begingroup$ "I'm assuming some tongue-in-cheek was intended." You are more charitable than I would be. It is simply wrong. I don't see any reason to think that the author was intentionally wrong in a way that was supposed to be clever. The most charitable interpretation is that the author wanted to underscore the probabilistic nature of a correlation and unthinkingly chose a bad way to illustrate it, a way that the author himself would likely agree doesn't really make sense if it were pointed out to him. $\endgroup$
    – John Coleman
    Commented Feb 15, 2018 at 14:09
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It would be difficult to come up with an interpretation of this that is meaningful, let alone correct. Association is not a property of individual data points. If you had just the height and intelligence of one person, how could you possibly say whether height and intelligence are associated? I suppose if we had the mean of height and intelligence, we could say that everyone above the mean in both, or below the mean in both, is showing an "association". But if you had completely random data (no correlation), you should expect that half of the people to show "association" in this sense. I generated a random data set with correlation around .2 (actually .22), and found that 55 showed "association" in this sense.

It's possible for Y to be an increasing function of X, and the correlation between them be only .5; it would be silly to say that only half the people show an association if every person has a higher intelligence than every shorter person and a lower intelligence than everyone taller. Moreover, it's theoretically possible to have one outlier creating all of the correlation, and the correlation of the set without that point be zero. It's even possible to have 20% of the population have a negative correlation, and the other 80% to also have a negative correlation, and the total correlation be .2.

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