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suppose I produce two vectors d1 and d2 using the below scheme which results in a correlation of 0.8716288 between the two vectors. Can I make a precise statement on the proportionality of d1 and d2.

Say I have:

m <- matrix(rnorm(100*100),nrow=100) 
d1<-as.matrix(dist(m, diag=T, upper = T))
d2<-1.-cor(t(m))
plot(as.vector(d1),as.vector(d2))
cor(as.vector(d1),as.vector(d2))

which results in

correlation of 0.8716288 between d1 and d2

and plot:

enter image description here

NOTE:

  1. I'm leaving the definition of proportionality open -- since I personally do not know rigorous definitions of the concept beyond what is taught at grade school
  2. Would appreciate definitions of proportionality as part of the answer
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  • $\begingroup$ Actually that was part of the question -- thanks -- let me add that to the note. $\endgroup$ – user1172468 Jul 4 '16 at 22:00
  • $\begingroup$ @Tim, thanks updated the question to reflect your comment $\endgroup$ – user1172468 Jul 4 '16 at 22:02
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Since you do not provide definition of proportionality, it can be assumed that you mean the regular definition that if $ y = cx $ then we can say $ y \propto x $.

If this is what you mean then if $Y$ is $X$ times constant, then they are linearly dependent and correlation between them is equal to $-1$ or $1$. That is the only relation between the two terms.

As @whuber correctly pointed out, the relation between two terms is not symmetric. Correlation measures linear relationship, so both if $Y = cX$ and $Z = a + cX$, have correlation with $X$ equal to $-1$ or $1$, while only the $Y$ is proportional to $X$.

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  • $\begingroup$ many thanks -- I tried to find definitions that might imply a loose relationship, say one with random noise -- do you know of any? $\endgroup$ – user1172468 Jul 4 '16 at 22:19
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    $\begingroup$ @user1172468 Maybe... correlation? covariance? ;) $\endgroup$ – Tim♦ Jul 4 '16 at 22:20
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    $\begingroup$ The logic used in this answer is invalid. In fact, a correlation of $1$ is consistent with a near-complete lack of proportionality. You have implicitly changed your definition to mean $y=cx + b$ for some possibly nonzero constant $b$. $\endgroup$ – whuber♦ Jul 4 '16 at 22:27
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    $\begingroup$ @whuber wait... I don't get it... cor(X, cX) = 1, we agree on that? I guess you meant the "implies" sentence? Deleted it. $\endgroup$ – Tim♦ Jul 4 '16 at 22:41
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Even if the correlation is high, the two variables maybe far from being "proportional". Here I define "proportionality" as linear dependence ($y=ax+b$), because this is what the correlation coefficient is meant to measure. Look at the following four data sets (Anscombe's quartet) from the Wikipedia page:

enter image description here

All four sets have the same correlation 0.816. For the two sets in the left column, we can say that $x$ and $y$ are approximately linearly dependent (ignoring the outlier in the bottom set). But this is not the case for the two sets in the right column.

Moral: never use the correlation coefficient blindly, check the scatter plot! :)

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    $\begingroup$ But defining proportionality as linear dependence contradicts a longstanding definition across mathematics. That correlation measures linear dependence and nothing else is fair comment and pertinent, but the question can't be validly answered by this redefinition $\endgroup$ – Nick Cox Jul 5 '16 at 0:00
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    $\begingroup$ That is exactly why I put "proportionality" in quotes in my answer. The author of the question mentioned that the definition of proportionality is left open. Linear dependence is what I believe the author intuitively meant by proportionality. $\endgroup$ – Kostia Jul 5 '16 at 0:04
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    $\begingroup$ But $y = ax$ not $y = ax + b$ See mathworld.wolfram.com/Proportional.html etc. What the OP intended is for them to clarify; it's entirely possible that they are confused too. $\endgroup$ – Nick Cox Jul 5 '16 at 0:07
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    $\begingroup$ @NickCox Please don't teach me about the difference between $y=ax$ and $y=ax+b$ :) Instead, try to here what I am saying. The author does not know the property measured by correlation. He or she called this property "proportionality" and asked to suggest the right definition. The right definition is linear dependence. I simply used his or her terminology. $\endgroup$ – Kostia Jul 5 '16 at 0:11
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    $\begingroup$ I am sure that you know the difference between $y = ax + b$ and $y = ax$ and I am not implying otherwise. The point is what terms to use. The former is not proportionality as defined in mathematics unless $b = 0$. Proportionality is a special case of linear dependence and you throw away a useful word if you conflate the ideas. I've supplied a reference. I can't start to know what the OP is thinking beyond what they said. We don't help OPs or anyone else by redefining standard terms capriciously to try to match what we infer is their intuition. $\endgroup$ – Nick Cox Jul 5 '16 at 0:16

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