Correlation is actually a remarkably simple concept, which makes it all the more frustrating <\/p>\n Let’s look at the pieces of that, to really nail down the meaning of the concept precisely.<\/p>\n What’s a random variable<\/em>? <\/p>\n A random variable is a rather dreadful misnomer – because it’s not actually a variable. A random variable is a function<\/em> which maps the outcome of an experiment to a numeric value. That can be as simple as mapping, say, “3 milliliters” (a measured outcome) to the number 3. It can also A random variable is typically described in terms of a probability distribution (to be more precise, a probably distribution or<\/em> a probability density function.). The details So given two independent random variables, x and y, what does it mean to say that X and Y are correlated<\/em>? It means that given a change in x, you can with high probability predict As usual, an example helps. Suppose I’m measuring the shoe sizes and heights of a group of men. In general, (not always, but in general), measurements have shown that the size of your feet is On the other hand, if I were to measure the size of the vocabulary that people use in the course If you’re reading carefully, you’ll notice that I went from a definition that said when two The way that we compute the correlation coefficient of a set of data is as follows. Given a set of data with two random variables X and Y, where there is a set of measurements {x1<\/sub>…xn<\/sub>} of X with mean x<\/span> and standard deviation σx<\/sub>, and a series of measurements {y1<\/sub>,…,yn<\/sub>} of Y with mean y<\/span> and standard deviation σy<\/sub>, the correlation coefficient is:<\/p>\n The closer the correlation coefficient is to either 1 or -1, the stronger the linear relationship between the two random variables. If it’s close to +1, then it’s called a strong positive correlation<\/em>; if it’s close to -1, it’s called a strong negative correlation<\/em>.<\/p>\n One thing you’ll constantly hear in discussions is “correlation does not imply causation”. Causation isn’t really a mathematical notion – and that’s the root of that confusion. Correlation means that as one value changes, another variable changes in the same way. Causation means that when one value changes, it causes<\/em> the other to change. There is a very big difference between The catch – and it’s a big one – is that correlation does strongly suggest<\/em> a causal relationship. (There’s a Yale professor of statistics who’s famous for saying something close to “Correlation is not the same thing as causation – but it’s a darn good hint!”. ) It may not be the case that X causes Y or Y causes X – but if there’s a strong correlation between them, you should<\/em> suspect that there’s a causal relationship. It may be that both X and Y are dependent on some third factor (is in the vaccine case). But too often, the mantra “correlation does not imply causation” is a hand-waving way of dismissing data that the speaker doesn’t feel like dealing with.<\/p>\n On the other hand, you constantly see people waving around statistics that show correlations, and One more really interesting example that I read about when my daughter was little. There was a study published a few years ago showing a pretty strong correlation between leaving a night-light on in a child’s room, and that child developing nearsightedness. This caused a big stir at the time; it got written up in newspapers, reported on by various television programs, etc. But other studies were unable to reproduce that correlation. Finally a group from Ohio State found that while they could not consistently reproduce a direct linkage between night-lights and nearsightedness, they could<\/em> find a correlation between nearsighted parents and nearsighted children, and that there was also<\/em> a correlation between nearsighted parents and the likelihood of having a night-light in their child’s room. In other words, children whose parents are nearsighted are more likely to have a night-light in their childrens’ rooms, and children are also likely to inherit nearsightedness from their parents. Correlation, but not causation: both the night-lights and the nearsightedness are caused by the parents’ nearsightedness; there’s no causal connection between the night-lights and the nearsightedness.<\/p>\n","protected":false},"excerpt":{"rendered":" Correlation and Causation Yet another of the most abused mathematical concepts is the concept of correlation, along with the related (but different) concept of causation. Correlation is actually a remarkably simple concept, which makes it all the more frustrating to see the nonsense constantly spewed in talking about it. Correlation is a linear relationship between […]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"jetpack_post_was_ever_published":false,"_jetpack_newsletter_access":"","_jetpack_dont_email_post_to_subs":false,"_jetpack_newsletter_tier_id":0,"_jetpack_memberships_contains_paywalled_content":false,"_jetpack_memberships_contains_paid_content":false,"footnotes":"","jetpack_publicize_message":"","jetpack_publicize_feature_enabled":true,"jetpack_social_post_already_shared":false,"jetpack_social_options":{"image_generator_settings":{"template":"highway","default_image_id":0,"font":"","enabled":false},"version":2}},"categories":[74],"tags":[],"class_list":["post-288","post","type-post","status-publish","format-standard","hentry","category-basics"],"jetpack_publicize_connections":[],"jetpack_featured_media_url":"","jetpack_shortlink":"https:\/\/wp.me\/p4lzZS-4E","jetpack_sharing_enabled":true,"jetpack_likes_enabled":true,"_links":{"self":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/288","targetHints":{"allow":["GET"]}}],"collection":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/comments?post=288"}],"version-history":[{"count":0,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/posts\/288\/revisions"}],"wp:attachment":[{"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/media?parent=288"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/categories?post=288"},{"taxonomy":"post_tag","embeddable":true,"href":"http:\/\/www.goodmath.org\/blog\/wp-json\/wp\/v2\/tags?post=288"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nto see the nonsense constantly spewed in talking about it. Correlation is a linear relationship<\/em> between two random variables.<\/p>\n
\nbe much more complicated – for example, the methods used in assigning a numeric value to the
\nquality of a chessboard position. But the thing to remember is that it’s a mapping from the outcome of an experiment to a numeric value.<\/p>\n
\naren’t appropriate for this post, but the basic idea is just that to understand a measured quantity
\nfrom an experiment, you need to understand how that quantity varies. The things you can meaningfully
\nconclude from measurements of a random variable with a normal distribution are quite different from things you can conclude from measurements of a random variable with a logarithmic distribution.<\/p>\n
\nan equivalent change in Y using a linear calculation. In informal use, we frequently drop the linear part: if when X increases, Y also increases; when X decreases, Y also decreases; and when X stays the same, then Y stays the same; then we say X and Y are correlated.<\/p>\n
\nproportional to your maximum adult height. So if I did a mapping between the size of feet and height in adult men, I would find a strong correlation between height and foot size.<\/p>\n
\nof a normal day, and compare it to the length of their ring fingers, I would find that there’s no
\ncorrelation to speak of. (Yes, I have actually seen a study that measured this. The hypothesis was
\nthat the length of ring fingers (at least, I think it was the ring finger) is related to the quantity
\nof certain hormones that a fetus is exposed to in the womb, and that that hormone level also affects
\nthe development of the speech centers of the brain.)<\/p>\n
\nthings are correlated, but then in the example, I said “strong” correlation. The reason is that in
\npractice, things are rarely quite as clear as they are in theory. Even things that really do
\ncorrelate perfectly, if we’re measuring them in a series of experiments, experimental error is going
\nto create enough variation that in the measured data, the correlation won’t be perfect. So we have a
\nmeasurement, called the correlation coefficient, of a relation between two random variables X and Y (usually abbreviated cx,y<\/sub><\/em> or rx,y<\/sub><\/em>), which measures the
\nstrength<\/em> of a correlation between X and Y. cx,y<\/sub><\/em> varies from -1 to +1. cx,y<\/sub>=0 indicates absolutely no correlation at all between X and Y; cx,y<\/sub>=+1 means that X and Y are perfectly correlated; cx,y<\/sub>=-1 means that X and -Y are perfectly correlated.<\/p>\n
\n![\"corr.mmf.jpg\"](\"https:\/\/i0.wp.com\/scientopia.org\/img-archive\/goodmath\/img_138.jpg?resize=144%2C32\")
<\/p>\n
\ncausation and correlation. To give a rather commonly abused example: the majority of children with autism are diagnosed between the ages of 18 months and three years old. That’s also the same<\/em> period of time when children receive a large number of immunizations. So people see the correlation<\/em> between receiving immunizations and the diagnosis of autism, and assume that
\nthat means that the immunizations cause<\/em> autism. But in fact, there is no causal linkage. The causal factor in both cases is age: there is a particular age when a child’s intellectual development reaches a stage when autism becomes obvious; and there is a particular age when certain vaccinations are traditionally given. It just happens that they’re roughly<\/em> the same age. <\/p>\n
\nwho insist that the correlation implies causation. (Again, look at that vaccine example!) To show causation, you need to show a mechanism<\/em> for the cause, and demonstrate that mechanism
\nexperimentally. So when someone shows you a correlation, what you should<\/em> do is look for a plausible causal mechanism, and see if there’s any experimental data to support it. Without
\na demonstrable causal mechanism, you can’t be sure that there’s a causal relationship – it’s just
\na correlation.<\/p>\n