7.6% (17/224) of people in the general population have both bone disease and respiratory disease. Similarly, 7.2% (184/2560) of people have bone disease without having respiratory disease in the general population. However, once put into a hospitalization setting and only taking data from people that are hospitalized rather than the general setting, this number of people with both bone disease and respiratory disease jumps up to 25% (5/20). This shows the specific environment the data is collected from can show the correlation between these two factors rather than by the given general population extraction

Berkson's paradox example

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. 

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

When a single node is caused by two other nodes we encounter something called "collider-bias" where one condition explains-away another. 

Since A and C both effect the outcome of B, by conditioning on B, A and C become **dependent**.

$A \rightarrow B \leftarrow C$: The Collider

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

Berkson's Paradox

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)

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. 

Gambler's Fallacy

Cole, S. R., Platt, R. W., Schisterman, E. F., Chu, H., Westreich, D., Richardson, D., & Poole, C. (2010). Illustrating bias due to conditioning on a collider. $\mathit{International \ Journal \ of \ Epidemiology, 39}$, 417-420. [doi.org/10.1093/ije/dyp334](https://doi.org/10.1093/ije/dyp334)
[ncbi.nlm.nih.gov/pmc/articles/PMC2846442/](https://www.ncbi.nlm.nih.gov/pmc/articles/PMC2846442/) 

Illustrating bias due to conditioning on a collider

In a path that would impact both variables but also contains a collider, M-bias occurs when the variable in the center of the collider is adjusted for. 

The general structure for M-bias is A <- B -> C <- D -> E where adjusting for C would create bias. 

M-bias

The statistical procedure of controlling for everthing that can be measured is misguided. 

For instance, in a collider $A\rightarrow B \leftarrow C$ where the path between $A$ and $C$ is initially independent, controlling for B would make them dependnet due to the explain-away effect. This means that a back-door is opened.

Also, controlling for descendents of the varibles we are interested in should also be avoided. This is like "partially" controlling for the interested variable itself. 

Controlling for everything is misguided

Proposed by Hans Reichenbach, the common cause principal argues against correlation doesn’t imply causation. It states that if there are two variables I and J and they are correlated, then I causes J or J causes I or there is another variable X that precedes and causes both I and J. 

This theory is not true because it doesn't account for how the data is selected and appeals to our human nature to look for casual explanations whenever there is a pattern. This is an example of collider bias.

Common Cause Principal 

A proxy is used as a substitute measure when a variable that is not directly measurable is needed for analysis.  For a variable to be a proxy for another variable, the proxy variable needs to be highly correlated with the variable of interest.  The proxy can be used for any variables such as a mediator, confounder, collider.  

Proxy

In the Book of Why, some of the classic examples of collider bias are discussed:

1. Celebrity's talent and beauty
2. Birth-weight paradox (Simpson's paradox)
3. Admission-rate bias (Berkson's paradox)

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