Simpson's paradox

Vector interpretation of Simpson's paradox

Simpson's paradox can also be illustrated using a 2-dimensional vector space.[21] A success rate of ${\textstyle {\frac {p}{q}}}$ (i.e., successes/attempts) can be represented by a vector ${\displaystyle {\vec {A}}=(q,p)}$, with a slope of ${\textstyle {\frac {p}{q}}}$. A steeper vector then represents a greater success rate. If two rates ${\textstyle {\frac {p_{1}}{q_{1}}}}$ and ${\textstyle {\frac {p_{2}}{q_{2}}}}$ are combined, as in the examples given above, the result can be represented by the sum of the vectors ${\displaystyle (q_{1},p_{1})}$ and ${\displaystyle (q_{2},p_{2})}$, which according to the parallelogram rule is the vector ${\displaystyle (q_{1}+q_{2},p_{1}+p_{2})}$, with slope ${\textstyle {\frac {p_{1}+p_{2}}{q_{1}+q_{2}}}}$.

Simpson's paradox says that even if a vector ${\displaystyle {\vec {L}}_{1}}$ (in orange in figure) has a smaller slope than another vector ${\displaystyle {\vec {B}}_{1}}$ (in blue), and ${\displaystyle {\vec {L}}_{2}}$ has a smaller slope than ${\displaystyle {\vec {B}}_{2}}$, the sum of the two vectors ${\displaystyle {\vec {L}}_{1}+{\vec {L}}_{2}}$ can potentially still have a larger slope than the sum of the two vectors ${\displaystyle {\vec {B}}_{1}+{\vec {B}}_{2}}$, as shown in the example. For this to occur one of the orange vectors must have a greater slope than one of the blue vectors (here ${\displaystyle {\vec {L}}_{2}}$ and ${\displaystyle {\vec {B}}_{1}}$), and these will generally be longer than the alternatively subscripted vectors – thereby dominating the overall comparison.

Simpson's reversal can also arise in correlations, in which two variables appear to have (say) a positive correlation towards one another, when in fact they have a negative correlation, the reversal having been brought about by a "lurking" confounder. Berman et al.[22] give an example from economics, where a dataset suggests overall demand is positively correlated with price (that is, higher prices lead to more demand), in contradiction of expectation. Analysis reveals time to be the confounding variable: plotting both price and demand against time reveals the expected negative correlation over various periods, which then reverses to become positive if the influence of time is ignored by simply plotting demand against price.

Psychological interest in Simpson's paradox seeks to explain why people deem sign reversal to be impossible at first, offended by the idea that an action preferred both under one condition and under its negation should be rejected when the condition is unknown. The question is where people get this strong intuition from, and how it is encoded in the mind.

Simpson's paradox demonstrates that this intuition cannot be derived from either classical logic or probability calculus alone, and thus led philosophers to speculate that it is supported by an innate causal logic that guides people in reasoning about actions and their consequences.[4] Savage's sure-thing principle[12] is an example of what such logic may entail. A qualified version of Savage's sure thing principle can indeed be derived from Pearl's do-calculus[4] and reads: "An action A that increases the probability of an event B in each subpopulation C_i of C must also increase the probability of B in the population as a whole, provided that the action does not change the distribution of the subpopulations." This suggests that knowledge about actions and consequences is stored in a form resembling Causal Bayesian Networks.

A paper by Pavlides and Perlman presents a proof, due to Hadjicostas, that in a random 2 × 2 × 2 table with uniform distribution, Simpson's paradox will occur with a probability of exactly 1⁄60.[23] A study by Kock suggests that the probability that Simpson's paradox would occur at random in path models (i.e., models generated by path analysis) with two predictors and one criterion variable is approximately 12.8 percent; slightly higher than 1 occurrence per 8 path models.[24]

A second, less well-known paradox was also discussed in Simpson's 1951 paper. It can occur when the "sensible interpretation" is not necessarily found in the separated data, like in the Kidney Stone example, but can instead reside in the combined data. Whether the partitioned or combined form of the data should be used hinges on the process giving rise to the data, meaning the correct interpretation of the data cannot always be determined by simply observing the tables.[25]

Judea Pearl has shown that, in order for the partitioned data to represent the correct causal relationships between any two variables, ${\displaystyle X}$ and ${\displaystyle Y}$, the partitioning variables must satisfy a graphical condition called "back-door criterion":[26][27]

1. They must block all spurious paths between ${\displaystyle X}$ and ${\displaystyle Y}$
2. No variable can be affected by ${\displaystyle X}$

This criterion provides an algorithmic solution to Simpson's second paradox, and explains why the correct interpretation cannot be determined by data alone; two different graphs, both compatible with the data, may dictate two different back-door criteria.

When the back-door criterion is satisfied by a set Z of covariates, the adjustment formula (see Confounding) gives the correct causal effect of X on Y. If no such set exists, Pearl's do-calculus can be invoked to discover other ways of estimating the causal effect.[4][28] The completeness of do-calculus [29][28] can be viewed as offering a complete resolution of the Simpson's paradox.

One criticism is that the paradox is not really a paradox at all, but rather a failure to properly account for confounding variables or to consider causal relationships between variables.[30]

Another criticism of the apparent Simpson's paradox is that it may be a result of the specific way that data is stratified or grouped. The phenomenon may disappear or even reverse if the data is stratified differently or if different confounding variables are considered. Simpson's example actually highlighted a phenomenon called noncollapsibility,[31] which occurs when subgroups with high proportions do not make simple averages when combined. This suggests that the paradox may not be a universal phenomenon, but rather a specific instance of a more general statistical issue.

Critics of the apparent Simpson's paradox also argue that the focus on the paradox may distract from more important statistical issues, such as the need for careful consideration of confounding variables and causal relationships when interpreting data.[32]

Despite these criticisms, the apparent Simpson's paradox remains a popular and intriguing topic in statistics and data analysis. It continues to be studied and debated by researchers and practitioners in a wide range of fields, and it serves as a valuable reminder of the importance of careful statistical analysis and the potential pitfalls of simplistic interpretations of data.

• Aliasing – Signal processing effect
• Anscombe's quartet – Four data sets with the same descriptive statistics, yet very different distributions
• Berkson's paradox – Tendency to misinterpret statistical experiments involving conditional probabilities
• Cherry picking – Fallacy of incomplete evidence
• Condorcet paradox – Situation in social choice theory where collective preferences are cyclic
• Ecological fallacy – Logical fallacy that occurs when group characteristics are applied to individuals
• Gerrymandering – Form of political manipulation
• Low birth-weight paradox – Statistical quirk of babies' birth weights
• Modifiable areal unit problem – Source of statistical bias
• Prosecutor's fallacy – Error in thinking which involves under-valuing base rate informationPages displaying short descriptions of redirect targets
• Will Rogers phenomenon – phenomenon in which moving an element from one set to another set raises the average values of both setsPages displaying wikidata descriptions as a fallback
• Spurious correlation
• Omitted-variable bias

• Leila Schneps and Coralie Colmez, Math on trial. How numbers get used and abused in the courtroom, Basic Books, 2013. ISBN 978-0-465-03292-1. (Sixth chapter: "Math error number 6: Simpson's paradox. The Berkeley sex bias case: discrimination detection").

Wikimedia Commons has media related to Simpson's paradox.

• Simpson's Paradox at the Stanford Encyclopedia of Philosophy, by Jan Sprenger and Naftali Weinberger.
• How statistics can be misleading – Mark Liddell – TED-Ed video and lesson.
• Pearl, Judea, "Understanding Simpson’s Paradox" (PDF)
• Simpson's Paradox, a short article by Alexander Bogomolny on the vector interpretation of Simpson's paradox
• The Wall Street Journal column "The Numbers Guy" for December 2, 2009 dealt with recent instances of Simpson's paradox in the news. Notably a Simpson's paradox in the comparison of unemployment rates of the 2009 recession with the 1983 recession.
• At the Plate, a Statistical Puzzler: Understanding Simpson's Paradox by Arthur Smith, August 20, 2010
• Simpson's Paradox, a video by Henry Reich of MinutePhysics