Taking drug D lowers the risk of heart attack in people, however, it raises the risk of heart attack in both men and women. When accounting for gender it will raise the risk of heart attack, but when gender is not separated from the general it actually lowered the risk of heart attack. 
We found this to be a numerical calculation that are biases simply do not account for. We found that:
3/40 > 1/20 or (A/B > a/b),
8/20 > 12/40 or (C/D > c/d),
but that does not mean (3 + 8)/(40 + 20) > (1 + 12)/(20 + 40) or (A + C)/(B + D) > (a + c)/(b + d).
We found that (3 + 8)/(40 + 20) < (1 + 12)/(20 + 40)


Simpson's Paradox Example

In this example illustrating Simpson's paradox, height is seen as a a beneficial factor (positive slope) in each age group, but harmful (downward slope) overall.

Simpson's Paradox Diagram

In 1973, Eugene Hammel, an associate dean at the University of California, noticed a worrisome trend in the university’s admission rates for men and women. According to his data, it is shown that 44 percent of the men who applied to graduate school at Berkeley had been accepted, compared to only 35 percent of the women.
After a careful investigation, Eugene found out that even though for each apartment, the admissions decisions were consistently more favorable to women than to men, the admission rate for women is lower than that of men.
The causal story Eugene discovered is that women were rejected in greater numbers because they applied to harder departments to get into, like history and literature, therefore having a lower acceptance rate.

Berkeley Admissions Paradox

If you were to graph a scatterplot of cholesterol levels and amount exercise for a number of people it would appear that as exercise increases, so does cholesterol (graph on the right). When you account for age in labeling the data points however, it becomes obvious that exercise helps to instead reduce cholesterol (graph on the left).

Simpson's Paradox: Cholesteral and Exercise example

When groups of data show one particular trend, but this trend is reversed when the groups are combined together. (A/B) > (a/b) & (C/D) > (c/d) but (A+C)/(B+D) < (a+c)/(b+d)

University of Michigan - Ann Arbor

A few of the common statistical paradoxes are:"

- Monty Hall's paradox 
- Berkson's paradox
- Simpson's paradox

Common statistical paradoxes

Pearl, J., & Mackenzie, D. (2018). $\mathit{The \ Book \ of \ Why: \ The \ New \ Science \ of \ Cause \ and \ Effect.}$ Basic Books.

The Book of Why

Monty Hall is a famous paradox from a reader's letter to "Ask Marilyn" column in Parade magazine in 1990.

The problem is defined as: suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

While the instinctive reaction is usually to think that the probabilities of either door are the same, the conditional probabilities that the car is behind door 1 and door 2 are 1/3 and 2/3 respectively. Thus the game show participant should always switch their choice, given the option. 


Monty Hall’s Paradox

This a subcategory of a collider bias (Collider is the selection of the data). When two factors individually are usually not severe enough to cause some effect but the combination of the two is. Usually both factors are not thought to be correlated with each other in real life settings but in a certain condition they are. This seems to occur because the resulting effect is being adjusted for by not considering the situations where both factors do not occur. This association is caused based on the conditional collider of the environment the data is selected. 

Berkson's Paradox

Simpson's Paradox

Low Birth weight babies have a significantly higher mortality rate than others. The smoking mothers are more likely to have low birth weight babies than non-smoking mothers. Thus, people inferred that the child mortality rate should be higher among children of smoking mothers. However, the observation runs contrary to our expectation. Low birth weight babies of smoking mothers have a lower child mortality than low birth weight babies of non-smokers.

Although smoking may contribute to low weight and be harmful for babies, other causes like genetic abnormalities (birth defect) are much more harmful than smoking. Those underweight babies would weight normal if their mother were non-smoker. But they still have a lower mortality rate than babies who have more severe medical reasons that made them born underweight.

We can explain this paradox with a causal diagram.  In this case, birth weight is the collider, a variable which is a common effect of two other variables, smoking and U(other causes like birth defects). Conditioning on the collider will open a back-door path that goes smoking -> birth weight <-U(birth defects) -> Infant death. The collider bias is so large that even makes smoking seems beneficial. 


Birth-weight paradox

The Gambler's fallacy is the false equivalence of prior occurrences with independent future probabilities. The classic example is a coin toss game, where participants can bet on the outcome of an established 50/50 coin. If the coin flips 10 heads in a row, the gambler may purportedly conclude that the next flip will be heads. This cannot be true, given that the coin has been verified as fair. 

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