Simpson's Paradox
This is my first attempt at an elementary statistics post, which I hope is suitable for Less Wrong. I am going to present a discussion of a statistical phenomenon known as Simpson's Paradox. This isn't a paradox, and it wasn't actually discovered by Simpson, but that's the name everybody uses for it, so it's the name I'm going to stick with. Along the way, we'll get some very basic practice at calculating conditional probabilities. A worked example The example I've chosen is an exercise from a university statistics course that I have taught on for the past few years. It is by far the most interesting exercise in the entire course, and it goes as follows: You are a doctor in charge of a large hospital, and you have to decide which treatment should be used for a particular disease. You have the following data from last month: there were 390 patients with the disease. Treatment A was given to 160 patients of whom 100 were men and 60 were women; 20 of the men and 40 of the women recovered. Treatment B was given to 230 patients of whom 210 were men and 20 were women; 50 of the men and 15 of the women recovered. Which treatment would you recommend we use for people with the disease in future? The simplest way to represent these sort of data is to draw a table, we can then pick the relevant numbers out of the table to calculate the required conditional probabilities. Overall A B lived 60 65 died 100 165 The probability that a randomly chosen person survived if they were given treatment A is 60/160 = 0.375 The probability that a randomly chosen person survived if they were given treatment B is 65/230 = 0.283 So a randomly chosen person given treatment A was more likely to surive than a randomly chosen person given treatment B. Looks like we'd better give people treatment A. However, since were given a breakdown of the data by gender, let's look and see if treatment A is better for both genders, or if it gets all of its advantage from one or the other. Wo
My first thought on reading this was that given that people tend to be overconfident in just about every other area of their lives, I would find it exceedingly surprising if it were in fact the case that people's estimates of their own attractiveness was systematically lower than the estimates of others. I notice that there isn't actually a citation for this claim anywhere in the article.
Indeed, having looked for some evidence, this was the first study I could find that attempted to investigate the claim directly: Mirror, mirror on the wall…: self-perception of facial beauty versus judgement by others.. To quote the abstract:
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