Causation and Lurking Variables

Association does not imply causation! • Example 1: • During the months of March and April of a certain year, the weekly weight increases of a puppy in New York were collected. For the same time frame, the retail price increases of snowshoes in Alaska were collected. • The data was examined and was found to have a very strong linear correlation. Causation and Lurking Variables 2 The weight of a growing puppy in New York (in pounds) The retail price of snowshoes in Alaska (in dollars) 8 32.45 8.5 32.95 9 33.45 9.6 34.00 10.1 34.50 10.7 35.10 11.5 35.63 The moral of this example is: So, this must mean that the weight increase of a puppy in New York is causing snowshoe prices in Alaska to increase, or the price increases of snowshoes are causing the puppy's weight to increase. Of course this is not true! • Be careful what you infer from your statistical analyses. Unfortunately, usually the situation is not as obvious as this one. • Be sure your relationship makes sense. • Also keep in mind that other factors may be involved in a potential cause and effect relationship. 3 4 The thymus example (shocking) Association does not imply causation! • Example 2: • In the early 1930s the relationship between the human population (response variable) of Oldenburg, Germany, and number of storks nesting in the town (explanatory variable) was investigated. The correlation coefficient turned out to be 0.97. • Does this mean that storks bring babies? • Can you give a possible explanation for this strong association? The thymus, a gland in your neck, unlike other organs of the body, doesn’t get larger as you grow—it actually gets smaller. Imagine the situation: many infants are dying of what seem to be respiratory obstructions, so doctors begin to do autopsies on infants who die with respiratory symptoms. They have done many autopsies in the past on adults who died of various causes, so they decide to rely on those autopsy results for comparison. 5 6 1 • What stands out most when they did autopsies on the infants is that they all have thymus glands that look too big in comparison to their body size. So they concluded that the respiratory problems are caused by an enlarged thymus. • It became quite common in the early 1900s for surgeons to treat respiratory problems in children by removing the thymus. In particular, in 1912, Dr. Charles Mayo published an article recommending removal of the thymus. He made this recommendation even though a third of the children who were operated on died. • What’s the lurking variable in this shocking example? What could be a lurking variable in these examples? • There is a strong positive correlation between the foot length of K-12 students and reading scores. • Students who use tutors have lower test scores than students who don’t. • A survey shows a strong positive correlation between the percentage of a country's inhabitants that use cell phones and the life expectancy in that country. 7 Important: Association does not imply causation! 8 Simpson’s Paradox • One of the most common mistakes people make is when they observe a high correlation between two variables and conclude that one must be causing the other. • Scatterplots and correlation do NOT demonstrate causation. • It’s hard to establish the nature and direction of causation, and there is always the risk of overlooking lurking variables. • A relationship between two variables that holds for each individual value of a third variable can be changed or even reversed when the data for all values of the third variable are combined. This is Simpson’s paradox. • Simpson’s paradox is an example of the effect of lurking variables on an observed association. 9 10 Simpson’s paradox Simpson’s Paradox • Simpson’s paradox is a severe form of confounding in which there is a reversal in the direction of an association caused by a lurking variable. Example: Is acceptance into a college (response variable) predicted by gender (explanatory variable)? Consider these data: Male Female Overall direction of association: positive But when we color different habitats in different colors, the data is separated by a lurking variable (different habitats) into a series of negative linear associations. 11 Success 198 88 Failure 162 112 Total 360 200 Proportions accepted by gender: Male success rate = 198 / 360 = 0.55 Female success rate = 88 / 200 = 0.44 Conclude: males were accepted at a higher rate than females. 12 2 Another example of Simpson’s paradox: kidney stones Broken down according to the lurking variable "major…" Male Female Male Female Business Success Failure 18 102 24 96 Success 198 88 Total 120 120 Failure 162 112 Male Female Total 360 200 Art Success Failure 180 60 64 16 • These data are from a medical study assessing different treatments for kidney stones (Treatment A includes all open procedures, and Treatment B is percutaneous nephrolithotomy): Total 240 80 Male proportion = 18 / 120 = 0.15 Female proportion = 24 / 120 = 0.20 Male proportion = 180 / 240 = 0.75 Female proportion = 64 / 80 = 0.80 Therefore: males were accepted at a lower rate than females. Therefore: males were accepted at a lower rate than females. Treatment A Treatment B Success 273 289 Failure 77 61 Total 350 350 • Treatment A success rate: 78% (273/350) • Treatment B success rate: 83% (289/350) • Conclude: Treatment B is better to treat kidney stones. 14 13 Broken down according to the lurking variable “size of kidney stone” Treatment A Treatment B Small kidney stones Success Failure 81 6 Tr. A Tr. B 234 36 Success 273 289 Total 87 270 Tr.A success rate = 81/87 = 0.93 Tr.B success rate = 234/270 = 0.87 Therefore: Treatment A is better to treat small kidney stones. Failure 77 61 Summary Total 350 350 Large kidney stones Success Failure 192 71 Tr.A Tr.B 55 25 Total 263 80 Tr.A success rate= 192/263 = 0.73 Tr.B success rate = 55/80 = 0.69 Therefore: Treatment A is better to treat large kidney stones. 15 • • • • Association does not imply causation! Association does not imply causation! Association does not imply causation! The issue of lurking variables and Simpson's paradox occur equally in both quantitative and categorical situations. So, in either case, be careful with your conclusion, and remember: Association does not imply causation! 16 3