Introduction to two-dimensional analysisSo far we have seen how to organize and summarize information about a single variable, the so-called univariate analysis. But we often want to study how two or more variables behave together.The analysis of two variables together (how one behaves when the other increases or decreases) is called bivariate analysis or two-dimensional analysis.What might we want to study?Temperature in city A × temperature in city BSalary of married people × salary of single peopleEducation × salaryNumber of police officers on the street × number of crimesetc.This analysis lets us check whether there is a relationship between the variables and measure how strong it is. It is the first step in investigating whether there is only a correlation or an actual cause-and-effect relationship.What correlation meansIn probability and statistics, dependence or association is any statistical relationship, causal or not, between two variables. In everyday language the word correlation is used in this broad sense, but in the technical sense it usually refers to a linear relationship: how closely the points of the two variables line up around a straight line (this is what Pearson's coefficient, which we will see later, measures).A positive correlation means that the two variables tend to increase or decrease together. A negative correlation means that when one increases, the other tends to decrease.ExamplesThere is a positive correlation between opening an umbrella and starting to rain.There is a negative correlation between education and unemployment.In a city, the higher the average temperature of the month, the lower the consumption of heating gas. How is this relationship described?Answer: Negative correlation, because when one increases the other tends to decrease — the sign of the correlation says whether the variables move in the same or in opposite directions. Different units do not prevent correlation, and a relationship that looks obvious is not proof of cause.What causality means“Causality is the relationship between an event A (the cause) and a second event B (the effect), provided that the second event is a consequence of the first”- Your teacher LeonPut simply, A is a cause of B when the occurrence of A produces B, or at least increases the chance that B happens. To characterize a causal relationship and distinguish it from a mere correlation, it is not enough to show that the events are correlated: we must also establish the mechanism by which A leads to B — or, put another way, show that the absence of A inhibits B.The study of causality is more qualitative: we try to find out where an event B came from. This discussion is old and goes back to Aristotle, but it gained a lot of strength with determinism, the philosophical theory that every event (including mental ones) is explained by causal relationships.Modern studies look for causality among many elements. What causes a case of dengue fever?Current temperaturePast rainfallPast casesType of vegetationetc.If we stop to think about it, MANY things influence and CAUSE dengue cases in Rio de Janeiro. So our model only needs to include the largest and most common effects — a reductionist process.If we leave out a large number of small, independent effects, we can treat them as "noise". We stop talking about things as completely determined by the causes we considered and start talking about a cause as something that amplifies the chances of its effect. We move from intuitions like "high temperature causes dengue cases" to intuitions like "high temperature increases the chance of dengue cases".This settles the flood of "what if..." questions that come from unlikely exceptions we had not considered: all of them are absorbed into the noise.The truth is that causality can hardly ever be determined with absolute certainty. That is why science takes it for granted that there are no absolute truths and that every theory is open to revision in the face of new evidence.Why correlation is not causalityCorrelation, that is, the link between two events, does not necessarily imply causation, that is, that one of the events brought about the other. Correlation can, however, point to possible causes or to areas for further study: it can be a clue.Just because (A) happens together with (B) does not mean that (A) causes (B). Determining whether there really is a causal relationship requires further investigation, because there are five possible situations:(A) really causes (B). Example: (A) rain; (B) opening an umbrella.(B) causes (A), the so-called reverse causality. Example: (A) opening an umbrella; (B) rain.A third factor (C) causes both (A) and (B). Example: (A) opening an umbrella; (B) umbrella sales; (C) rain.Both directions happen at once: (A) causes (B) and (B) also causes (A). Example: (A) poverty; (B) low education — poverty makes studying harder, and low education makes it harder to escape poverty.The correlation is just a coincidence: the two events have no relationship other than happening at the same time. In a scientific study, using a large sample helps reduce the chance of coincidence.We are all going to dieEveryday examples"The bigger a child's feet, the greater their ability to solve math problems. Therefore, having big feet leads to better math grades." Although the correlation exists, this reasoning commits a fallacy by assuming that foot size is the cause and ignoring other factors. As children grow, so do their feet, their reasoning ability, the amount of knowledge they have acquired and many other characteristics. Age is the real common cause of both foot size and math skill."I caught a cold, so I had chicken soup. Two weeks later the cold was gone. Therefore, chicken soup cures colds.""I walked into the classroom right foot first and, that day, the math test went really well. Therefore, entering right foot first before tests brings good luck.""A Viking watches a solar eclipse. Believing the Sun is being devoured by a giant wolf, he starts shouting to scare it away. Moments later the Sun returns to normal." He then concludes that shouting during an eclipse makes the wolf spit the Sun back out.In all these examples it is assumed, in a very simplistic way, that an action (A) is the cause just because it happened before the effect (B). Both the cold and the eclipse would have gone away regardless of what was done. As for the test, it may have gone well because the student studied, was not nervous or the test was easy.Real examplesSeveral studies indicated that menopausal women receiving hormone replacement therapy (HRT) had a lower risk of coronary heart disease, which led to the idea that HRT protected the heart. However, randomized controlled trials (more rigorous) carried out later showed that HRT actually caused a small but significant increase in the risk of coronary heart disease. A re-analysis of the studies revealed that women on HRT were also more likely to belong to a higher socioeconomic group, with a better diet and more exercise. The use of HRT and the low incidence of coronary heart disease were not cause and effect, but the result of a common cause: the benefits associated with a high socioeconomic status. [4]In 1998, the British doctor Andrew Wakefield published in The Lancet a study suggesting a link between autism and the MMR vaccine (measles, mumps and rubella). A wave of panic followed that led many parents to stop vaccinating their children and, as a result, measles outbreaks reappeared in several countries. Because autism is usually diagnosed around the same age at which children receive this vaccine, many parents became convinced that the relationship was causal. Several other researchers tried to confirm the link, and none succeeded. Wakefield's study turned out to be a fraud and was retracted by the journal in 2010; he was struck off the medical register in the United Kingdom. Wakefield had received money from lawyers suing the vaccine manufacturers and had applied for a patent on a rival vaccine that would only have a market if confidence in MMR were shaken. Even so, the damage was done: because of a fraud and irrational fears, many lives were and still are put at risk. Wakefield moved to the United States, where he is supported by celebrities and seen as a martyr by the anti-vaccine movement.Relationship between COVID and scented candlesA study shows that cities with more firefighters have more fires. What is the most likely explanation?Answer: A third factor, city size, increases both the number of firefighters and the number of fires — this is the common-cause case (C). Bigger cities have more buildings and more fires and therefore hire more firefighters. Calling it a coincidence would discard a simple explanation too soon.In the hormone replacement therapy story, what explained the lower rate of coronary heart disease among women on the treatment?Answer: The higher socioeconomic status, which led both to the treatment and to healthier habits — it was a common cause (a confounder). Randomized trials, which remove this effect, showed that HRT actually increased the risk slightly.Obvious correlations that call for cautionIn one of his videos, the comedian Don McMillan points out an "obvious fact":“The chances of having Alzheimer’s at age 85 are 10%. A smoker lives, on average, 66 years. Therefore, smoking decreases your chances of having Alzheimer’s.”- Don McMillanTrue? As written, yes! But is it worth dying early so as not to get Alzheimer's? Smoking does not protect against the disease: it simply makes people die before the age at which the disease usually appears.What alternative explanations can we think of?Let us start from a real study. One of the findings of "Impact of Tobacco Smoking on the Risk of COVID-19: A Large Scale Retrospective Cohort Study" (Paleiron and colleagues, 2021), carried out with the 1,688 crew members of the French aircraft carrier Charles de Gaulle during an outbreak in 2020, was that current smokers had a statistically lower risk of catching COVID-19 than former smokers and non-smokers combined.Why would smokers have had less COVID-19 in this study? We can think of several alternatives.Possible explanationsDirect effect of smoking or nicotinePerhaps something in cigarettes (or in nicotine) alters the receptor the virus uses or changes the immune response, reducing the risk of infection. If that were the case, there would be causality: smoking → lower risk of infection. But beware: smoking is associated with many serious harms, so "lower risk of infection" does not mean "smoking is good".Differences in behavior or lifestyleSmokers may socialize differently — smoking breaks in open areas, less participation in certain group activities — which could reduce their exposure to the virus. Here the explanation would not be "smoking reduces risk" but "a factor linked to smokers (C) reduces risk".Demographic or health differences between the groupsThere may be differences in age, sex, prior illnesses, workload or other factors between smokers and non-smokers. In the study, the population was young (median age 28) and healthy, and being over 50 was associated with higher risk. If, for example, smokers worked in areas with less contact or slept in different conditions, that could explain part of the difference.Reverse causalityPerhaps it is not that "smoking protects": people who already had symptoms or suspected infection may have stopped smoking and started declaring themselves non-smokers. In that case, it is the infection that leads to less smoking, reversing the direction of the relationship.Hidden factors (confounders)A third factor (C) may affect both smoking and the risk of COVID-19 — for example: job on board, closeness of contact, rest schedule, ventilation of the sleeping quarters or workplace. The relationship between smoking and lower risk could reflect this hidden variable. The authors themselves concluded that smoking cannot be considered effective protection and that the mechanism needs further research.In a study, people who sleep more than 9 hours a night die earlier. Which explanation is an example of reverse causality?Answer: Pre-existing illnesses make people sleep more, and it is those illnesses that shorten life — in reverse causality, the supposed effect (poor health) is what produces the supposed cause (sleeping a lot). The age option is a common cause, not reverse causality.Absurd correlations [3]We must be very careful with correlations. They often point to something real, but other times they are pure coincidence. The following site has many examples of absurd correlations found in real data: http://www.tylervigen.com/spurious-correlations[1] https://papodehomem.com.br/se-correlacao-nao-implica-causalidade-entao-o-que-implica[2] http://comcept.org/cepticismo/correlacao-nao-implica-necessariamente-causalidade/[3] http://www.tylervigen.com/spurious-correlations[4] https://academic.oup.com/ije/article/33/3/464/716652
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