Does high correlation imply proportionality? - Cross Validated most recent 30 from stats.stackexchange.com 2019-08-23T00:36:26Z https://stats.stackexchange.com/feeds/question/222143 http://www.creativecommons.org/licenses/by-sa/3.0/rdf https://stats.stackexchange.com/q/222143 2 Does high correlation imply proportionality? user1172468 https://stats.stackexchange.com/users/14163 2016-07-04T21:55:13Z 2016-07-05T00:42:42Z <p>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.</p> <p>Say I have:</p> <pre><code>m &lt;- matrix(rnorm(100*100),nrow=100) d1&lt;-as.matrix(dist(m, diag=T, upper = T)) d2&lt;-1.-cor(t(m)) plot(as.vector(d1),as.vector(d2)) cor(as.vector(d1),as.vector(d2)) </code></pre> <p>which results in </p> <p>correlation of 0.8716288 between d1 and d2</p> <p>and plot:</p> <p><a href="https://i.stack.imgur.com/W3bql.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/W3bql.png" alt="enter image description here"></a></p> <p>NOTE:</p> <ol> <li>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</li> <li>Would appreciate definitions of proportionality as part of the answer</li> </ol> https://stats.stackexchange.com/questions/222143/-/222145#222145 1 Answer by Tim for Does high correlation imply proportionality? Tim https://stats.stackexchange.com/users/35989 2016-07-04T22:16:20Z 2016-07-04T23:20:20Z <p>Since you do not provide definition of proportionality, it can be assumed that you mean the <a href="http://mathworld.wolfram.com/DirectlyProportional.html" rel="nofollow noreferrer">regular definition</a> that if \$ y = cx \$ then we can say \$ y \propto x \$.</p> <p>If this is what you mean then if \$Y\$ is \$X\$ times constant, then they are <a href="https://stats.stackexchange.com/questions/31270/what-is-the-difference-between-linearly-dependent-and-linearly-correlated">linearly dependent</a> and correlation between them is equal to \$-1\$ or \$1\$. That is the only relation between the two terms.</p> <p>As <em>@whuber</em> correctly pointed out, the relation between two terms is not symmetric. Correlation measures <a href="https://stats.stackexchange.com/questions/29713/what-is-covariance-in-plain-language/29715#29715">linear relationship</a>, 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\$.</p> https://stats.stackexchange.com/questions/222143/-/222146#222146 1 Answer by Kostia for Does high correlation imply proportionality? Kostia https://stats.stackexchange.com/users/102683 2016-07-04T22:48:30Z 2016-07-04T22:48:30Z <p>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 <a href="https://en.wikipedia.org/wiki/Correlation_and_dependence" rel="nofollow noreferrer">Wikipedia page</a>:</p> <p><a href="https://i.stack.imgur.com/qlvQH.png" rel="nofollow noreferrer"><img src="https://i.stack.imgur.com/qlvQH.png" alt="enter image description here"></a></p> <p>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.</p> <p>Moral: never use the correlation coefficient blindly, check the scatter plot! :)</p>