# "Are All Correlations Spurious But Some Correlations Are More Spurious?" - Not George Orwell

(Sorry for the George Orwell/Animal Farm reference)

I am an MBA student taking courses in Statistics. Over the Thanksgiving weekend, we have an informal assignment - we have to find publicly available data on two "concepts" and make a small presentation on the possible correlations between these two "concepts" and if the correlation might be "spurious" or not.

I chose to explore the relationship of countries between the "Number of Doctors Per 1000 People" and the "Life Expectancy" of people in that country. On the surface, this sounds like there might be some correlation - as a brutish example, imagine if there are few doctors (relative to the population) - assuming if not seeing a doctor can kill you, countries with fewer doctors would result in sitting in the waiting/emergency room for longer periods of times (assuming that the prevalence and severity of medical conditions are similar across different countries, as well as the quality of doctors) ... ultimately resulting in more people dying in general and lowering the life expectancy in countries with smaller doctor/people ratios. Although I have greatly and comically oversimplified this situation, I would hope that there is some grain of truth in this oversimplification and it would be reflected within the data.

I found data on the ratio of doctors/population in different countries over here (https://data.oecd.org/healthres/doctors.htm, https://data.oecd.org/healthres/doctors.htm - I copy/pasted this into Excel) and I found data on the life expectancy over here (https://data.worldbank.org/indicator/SP.DYN.LE00.IN). I am learning how to use R - I downloaded both of these datasets and made this quick graph to investigate the correlation:

# NOTE: This is not a Doctoral Thesis, but a very informal learning assignment where the data doesn't need to be fully accurate - please take it easy on me.. thank you!
# PART 1: DOCTOR DATA

#PART 2: LIFE EXPEC DATA
#chop off the first few rows in excel to make the data manipulation easier
life_e = data.frame(life_e$$Country.Name, life_e$$X2020)
colnames(life_e)[1] <- "Country"
colnames(life_e)[2] <- "Life_Expec"

#PART 3: JOIN FILE
graph_file = merge(x = life_e, y = doctors, by = "Country", all.y = TRUE)
graph_file = na.omit(graph_file)

# PART 4: PLOT GRAPH
par(mfrow=c(1,2))
plot(graph_file$$Life_Expec, graph_file$$Doctors, main = "Correlation Between Life Expectancy and Number of Doctors Per 100 People (2020)" , xlab = "Life Expectancy (Years)", ylab = "Number of Doctors Per 1000 People")
text(graph_file$$Life_Expec, graph_file$$Doctors, labels=graph_file\$Country)


Based on this (very hurried) analysis that I performed, it would appear that there is somewhat of a linear correlation between these two concepts (we can also calculate the Correlation Coefficient to find out how strong, i.e. Rho Value). It appears that the more doctors there are relative to the population, the life expectancy tends to also increase. Hurray!

But here is my question: How Do I Know That This Relationship Is Not Spurious? Maybe for generations, there is a secret undiscovered vitamin in the water of Norway and Austria that makes people live longer - and this increase in life expectancy has nothing to do with the high ratio of doctors/people!!??

In a previous question (Does Correlation "Sometimes" Imply Causality?), I started learning about "Causality vs Correlation" and found out that it is very difficult to ascertain the presence of Causality.

However in this question (i.e. Doctors vs Life Expectancy), is there truly a way to determine whether or not this correlation is spurious or not? It seems to me that there is no straightforward answer to this question - such a question can only be answered by rigorous experiments, subject matter knowledge and science. I am guessing that extensive reviews of the healthcare systems, exogeneous and endogenous factors, life style habits, epidemiological patterns, review of academic intuitions ... all would have to be comprehensively studied over decades to find out if in fact this doctor/life expectancy correlation is spurious or non-spurious. And this extensive research will likely not be finished by next Wednesday (i.e. the date of my informal presentation).

Thus, we have no choice but to interpret our results under the following framework : Although philosophically, all correlations might be partly spurious - some correlations are likely to be far more spurious than other correlations (e.g. divorce rate in Maine/vowels in the name of hurricanes vs life expectancy/number of doctors). Although nothing can be ever be conclusively determined (e.g. I can not be 100% sure that a rhinoceros is standing outside my door - but I can still be more relatively certain that a pink rhinoceros is not standing outside my door, and even more relatively certain that a pink a rhinoceros riding a unicycle is not standing outside my door - but I can be pretty damn certain). However, we can not twiddle our thumbs indefinitely waiting on a universe of delusional and highly improbable events to manifest and risk jeopardizing and delay taking important decisions that can better mankind (e.g. a pharmaceutical drug performed very well in many clinical trials and has demonstrated the ability to save lives with minimum side effects - but we will never market this drug because technically all test patients might have just gotten lucky). Thus, using domain specific knowledge and rigorous statistical methodologies can help us attribute a level of confidence in the interim and provide reasonable beliefs if a specific correlation is spurious or not as we learn more about our data with each passing day and proceed to make important data based decisions to hopefully improve society.

Can someone again confirm if I am "out to lunch" (i.e. crazy) - or is my understanding somewhat correct?

• The term 'spurious correlation' can be used in different ways. What is meant with spurious correlation? Is it used to refer to spurious (causal) relation? Or does it refer to the inference of a correlation that is actually not statistically significant? Commented Oct 10, 2022 at 0:39
• Even though we cannot perform the experiment, it's fair to assume that removing all doctors from a country will measurably reduce its life expectancy. That is a causal relation. Commented Oct 10, 2022 at 9:23

In statistics, a spurious relationship or spurious correlation is a mathematical relationship in which two or more events or variables are associated but not causally related, due to either coincidence or the presence of a certain third, unseen factor (referred to as a "common response variable", "confounding factor", or "lurking variable").

You are correct that correlation does not imply causation, so many correlations are spurious. When do we know that the relation is spurious? It’s when there’s no causal relationship. We don’t use correlations to find them, so looking at correlations alone won’t tell you that.

Now the question boils down to “how do we detect causality?” It was answered in many treads tagged as , e.g. Interview question: If correlation doesn't imply causation, how do you detect causation?, while Introduction to causal analysis gives many good references to dive deeper.

• In OP's example, the confounding factors are probably something like median income/wealth and social security (over the last 80 years or so). Commented Oct 10, 2022 at 6:03
• @Roland in the OP's example there might even not be a clear correlation. It depends a bit on how one views the population that the sample represents. If one considers the countries from the image to be the entire population then there is definitely correlation, but possibly one regards the countries in the image to be only a subset/sample from a larger population and regards the correlation in the sample as reflecting some relationship in that larger population... Commented Oct 11, 2022 at 7:20
• ...In that case there might be a Yule-Simpson effect, which complicates how we look at that relationship. There seem to be three clusters when we look at the distribution on the x-axis (three countries with life expectancy << 75 years, 9 countries with life expectancy ~ 76 years and, 21 countries with life expectancy ~ 82 years) and between clusters there might be a positive correlation but within the clusters the correlation is not so clear and could be even negative. Commented Oct 11, 2022 at 7:21

I'd say your understanding is correct. Philosophically, it's not truly possible to "prove" causation with 100% certainty, but we can convince ourselves that the alternative is arbitrarily unlikely. Most methods of determining causation rely on manipulating pertinent factors and randomizing all other possibly contributing factors, but this requires knowing what other possibly contributing factors there are. No matter how we control or randomize, it's not possible to rule out all potential unknowns, by their nature of being unknowns.

For any well-described cause-effect relationship, I could posit that it's actually an unseen wizard that's manipulating the effect in just the right way such that it happens to line up perfectly with your suggested "cause". This of course quickly becomes very implausible when we look at realistic physical mechanisms for cause and effect, but you can't categorically disprove the existence of something by lack of evidence that it exists. We can become increasingly sure that a cause-effect relationship is real by appropriate experimental design and by using domain knowledge to describe realistic and well-understood mechanisms for cause and effect, but you can't truly rule out the possibility of an unknown cause that's never been observed or described, it just becomes vanishingly unlikely. At some point, the statistical and mechanistic support for the link between cause and effect are great enough to consider it "proven", even though it's technically not possible to rule out unseen causes with no understood mechanism.