Introduction to Two-DimensionalitySo far, we have seen how to organize and summarize information related to a single variable, also called univariate analysis, but we are often interested in analyzing the joint behavior of two or more random variables.The analysis of the behavior of two variables together (how one behaves if we increase or decrease the other) is called bivariate analysis or two-dimensional analysis.What might we want to study?- temperature in city A x temperature in city B- Married people's salary x Single people's salary- Education x Salary- Number of police officers on the street x Number of crimes- etc...This analysis makes it possible to establish relationships between variables, that is, to determine whether the differences between the distribution of two variables are statistically significant, with the aim of investigating causalities or correlations.What Correlation MeansIn probability and statistics, correlation, dependence, or association is any statistical relationship (causal or non-causal) between two variables, and correlation is any relationship within a broad class of statistical relationships involving dependence between two variables.The word correlation can be replaced by synonyms such as: relationship, equivalence, link, correspondence, analogy, and connection.A positive correlation means that the two items studied increase or decrease together. A negative correlation means that when one item increases, the other decreases.Examples> There is a positive correlation between opening an umbrella and it starting to rain;> There is a negative correlation between education and unemployment;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 LeonIn direct terms, A is the cause of B when A is a necessary requirement, but not necessarily a sufficient requirement, for B to occur. To characterize a causal relationship, distinguishing it from a simple correlation, one must establish not only the significant correlation between the events in question but also the physical mechanisms of action that lead A to determine B, or rather, the absence of A to inhibit B.Causal study becomes a more qualitative study; we try to find where an event B came from. This study is already old and goes back to Aristotle, but it gained a lot of strength in determinism: it is the philosophical theory that every event (including mental ones) is explained by determination, that is, by causal relationships.Thus, modern studies seek causality among various elements: what causes a case of dengue?- Current temperature- Past rainfall- Past cases- Type of vegetation- etc...If we stop to analyze it, MANY things influence and CAUSE dengue cases in Rio de Janeiro. Therefore, our model only needs to include the largest and most common effects. A reductionist process of the data.If we omit a large number of small and independent effects, we can treat them as "noise." We stop talking about things as being completely determined by the considered causes. Instead, we talk about the cause as amplifying the chances of its effects. You move from intuitions like “higher temperature causes dengue cases” to intuitions like “higher temperature increases the chance of dengue cases occurring.”Thus, the bunch of “what if” questions arising from the unlikely exceptions we had not considered is resolved. They are all situations incorporated as noise.The truth is that causality can hardly be determined with absolute certainty. Hence, in science it is already understood that there are no absolute truths and that all theories are open to revision in light of new evidence.Why Correlation Is Not Causality> Correlation, that is, the link between two events, does not necessarily imply a causal relationship, meaning that one of the events caused the occurrence of the other. Correlation may, however, indicate possible causes or areas for further study, or in other words, correlation may be a clue.Just because (A) happens together with (B) does not mean that (A) causes (B). Determining whether there is in fact a causal relationship requires additional investigation, because five situations may occur:- it really causes (B);- (A): rain; (B): opening an umbrella;- it may be the cause of (A);- (A): opening an umbrella; (B): rain;- A third factor (C) may be the cause of both (A) and (B);- (A): opening an umbrella; (B): umbrella sales; (C): rain;- It may be a combination of the three previous situations. For example, (A) causes (B) and at the same time (B) also causes (A);- (A): rain; (B): thunder;- The correlation may be just a coincidence, that is, the two events have no relationship other than the fact that they occur at the same time. (If we are talking about a scientific study, using a large sample helps reduce the probability of coincidence).We are all going to dieGeneral Examples> “The bigger a child's feet are, the greater their ability to solve math problems. Therefore, having big feet leads to better grades in math.” Although there is a correlation, this line of thought commits a fallacy by immediately assuming that foot size is the primary cause, because there are other factors involved that are not taken into account. Foot size has a positive correlation with many other developmental changes. As children grow, so do their feet, as well as their reasoning ability, amount of knowledge acquired, and many other characteristics. Age is the true common cause of both foot size and the ability to solve math problems.> “I had a cold, so I ate chicken soup; two weeks later the cold disappeared. Therefore, chicken soup cures colds.”> “I entered the classroom with my right foot first; that day the math test went really well. Therefore, entering the classroom with your right foot first before tests brings luck.”> “A Viking observes a solar eclipse. Believing that the Sun is being devoured by a giant wolf, he starts shouting to scare it away. Moments later, the Sun returns to normal.” He therefore 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 causal agent because it occurred before the effect (B). Both the cold and the eclipse would have disappeared regardless of the action taken. As for the test, it may have gone well because the student studied the material, was not nervous, or the test was relatively easy.Real examples> Several studies initially indicated that menopausal women who received hormone replacement therapy (HRT) also had a lower risk of coronary heart disease, which led to the idea that HRT provided protection against coronary heart disease. However, controlled and randomized studies (more rigorous ones), carried out later, showed that HRT actually caused a small but significant increase in the risk of coronary heart disease. A reanalysis of the studies revealed that women who received HRT were also more likely to belong to a higher socioeconomic class, with better diet and exercise habits. 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 high socioeconomic status. [4]> In 1998, a British doctor, Andrew Wakefield, published a study revealing a correlation between autism and the measles, mumps, and rubella (MMR) vaccine. A wave of panic followed, leading many parents to stop vaccinating their children and, as a result, measles outbreaks began to reappear around the world. The fact that autism is usually diagnosed after the child has received the MMR vaccine led many parents to become convinced of the causal link. Several other researchers also tried to confirm the connection, but none succeeded. Wakefield’s study eventually turned out to be nothing more than a fraud, and was retracted by the journal in which it was published. Wakefield had received money to prove the link between autism and MMR and was also preparing a competing vaccine that he would only be able to sell if confidence in MMR were shaken. Despite this discovery, the damage had already been done. Because of fraud and irrational paranoia, many lives were, and continue to be, put at risk. Wakefield moved to the USA, where he is supported by celebrities and seen as a martyr of the anti-vaccination movement.Relationship between COVID and scented candlesObvious correlations, but ones we should be careful withIn one of comedian Don McMillan’s videos, he 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? Of course it is true! But is it worth dying early in order not to have Alzheimer’s?What alternatives can we think of?Let’s start from a recent study. One of the findings of the paper "Impact of Tobacco Smoking on the Risk of COVID‑19: A Large Scale Retrospective Cohort Study" was that, among the crew members of an aircraft carrier, current smokers had a statistically lower risk of contracting COVID‑19 compared with former smokers + non-smokers.Why did smokers have a lower chance of COVID-19 in this study? We can think of several alternatives.Explanatory alternativesDirect effect of smoking or nicotinePerhaps something in cigarettes (or in nicotine) alters the viral receptor, or modifies the immune response, so that the risk of infection is reduced. If that were the case, we could think that there is causality: smoking → lower risk of infection. But be careful: smoking is also associated with many adverse effects, so “lower risk of infection” does not mean “smoking is good.”Difference in behavior or lifestyleSmokers may have different patterns of socialization, due to setting, smoking breaks, specific areas, or less participation in certain activities that non-smokers do — which may reduce exposure to the virus. Here, the explanation would not be “smoking causes lower risk,” but rather “a factor linked to smokers (C) reduces risk.”Demographic or health differences between groupsThere may be differences in age, sex, comorbidities, workload, or other factors between smokers and non-smokers. In the study, the population was relatively young (median ~28 years) and healthy. If, for example, smokers had less work pressure or slept under different conditions, this could explain part of the difference.Reverse causality or another feedback effectIt may not be that “smoking protects” — perhaps those who already had symptoms or suspected infection avoided smoking, or smokers with symptoms were hospitalized or left out of the sample. Or also: infection may lead to reduced smoking, reversing the direction of the relationship.Hidden factors (confounders)A third factor (C) may affect both “smoking” and “COVID risk” — for example: type of barracks, proximity of contact, rest schedule, ventilation of sleeping quarters, workplace. Therefore, the relationship between “smoking” and “lower risk” could reflect this hidden variable.Absurd correlations [3]We have to be very careful with correlations. They can often explain something to us, but at other times they may be nothing more than pure coincidence. On the following site, there are some examples of absurd correlations found in real life: 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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