Chapter 8: Correlations and Correlation Arguments
Chapter 8: Correlations and Correlation Arguments
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What is a correlation?
We now know the basics of scientific reasoning and we’ve seen some of the ways arguments for single statistical conclusions fit into that method. Now we can explore correlations. A correlation is a statistical association between two properties in a population. Recall the example of Ignaz Semmelweis, the doctor in Vienna in the 1840s who suspected that there was a connection between women dying in maternity clinics and their doctors not washing their hands after working on cadavers. A few cases or anecdotes might have led him to be suspicious, but what he needed to do is identify two different statistical statements and their relationship. If there’s a connection between doctors’ handwashing and maternity mortality, he would need to answer two questions: What is the rate of mothers dying with doctors who washed their hands, and what is the rate of mothers dying with doctors who did not wash their hands? That is, we want to know the relationship between maternal mortality AND clean doctor’s hands. If the rate of women dying with doctors with dirty hands was the same as the rate of women dying with doctors with clean hands, then it would appear that there’s no relationship of interest between the two. But if the rate of women dying with doctors with dirty hands is higher than the rate of women dying with doctors with clean hands, then we’ve discovered a correlation. And that was Semmelweiss’ landmark discovery.
Correlations defined
A correlation is an association between two properties in a population. When a positive correlation is present in a population, there are two properties that are connected such that if a member of that population has one property, then it is more likely (for positive correlations) to have the other property. Semmelweis discovered that doctor’s hand washing is positively correlated with survival among mothers.
When a negative correlation is present in a population, then a member with the first property is less likely to have the other property. Consider: Exercising regularly is negatively correlated with obesity among humans. A way to understand this sort of claim is that if we consider the population of humans and divide them into the ones that exercise regularly and the ones who do not, we are less likely to find obesity present among the exercisers than the non-exercisers.
As part of our general project to improve scientific literacy, we need to be able to recognize when a correlation is being stated, and then put it into standard form:
Standard form for correlation statements:
Property A is positively (or negatively) correlated with Property B among population P.
Swimming is positively correlated with drowning among humans.
This claim means that if we divide humans into the ones who are swimming and the ones who are not, we will find a higher percentage of the swimmers who drown than the non-swimmers.
Here are more correlation claims expressed informally and then put into standard form:
Sleeping with your shoes on is associated with waking up with a headache
Sleeping with your shoes on is positively correlated with waking up with a headache among people.
People who visit the hospital are more likely to die.
Hospital visits are positively correlated with mortality among people.
School children have a higher chance of having asthma if their school is near a hazardous waste site.
Going to school near a hazardous waste site is positively correlated with asthma among school children.
Babies born to poor families tend to have lower birth weights.
Poverty is negatively correlated with birth weight among babies.
What does A is positively/negatively correlated with B mean?
A positive correlation just means that the rate is higher; a negative correlation just means that the rate is lower. Notice that "higher" does not mean "Most." "Most" means "greater than 50%." Swimming is positively correlated with drowning does not mean that most swimmers drown. Most do not. But the percentage of them that drown is higher than the percentage of the non-swimmers who drown. The claim also does not mean that people do not drown in other circumstances. There are people who drown in bathtubs and hot tubs too. But the rates of drowning in the swimmer and non-swimmer groups is different.
Correlation does not imply causation. When we find correlations in the world, the most important question we can ask is why? Why are the rates of the things that have property A that also have property B higher than in the non-A group? What is the connection? If the connection is causal because A causes B, B causes A, or some other third factor causes them both, then we want to identify that causal relationship and understand it. Knowing what causes what in the world is at the center of the scientific enterprise. Consider these correlations. In each case the presence of the first property predicts the second, but it does not cause it:
Being at the doctor’s office is positively correlated with being sick among humans.
Being in the NBA is positively correlated with having big shoes among humans.
Windshield wiper use is positively correlated with car wrecks among drivers.
Ice cream consumption is positively correlated with murder among people.
Doctor’s office visits don’t cause illness; the causal arrow goes the other way. Being drafted into the NBA doesn’t lead to big shoes; a third cause, genetics, contributes to both. Windshield wipers don’t cause wrecks; rain causes both. And eating ice cream isn’t dangerous, but consumption and murder go up in hot weather.
Correlations are symmetrical. If A is correlated with B among P, then it’s also true that B is correlated with A among P. These sentences are equivalent in meaning with the position of the two properties are switched:
Ice cream consumption is positively correlated with murder among Americans
means the same thing as
Murder is positively correlated with ice cream consumption among Americans.
And these two sentences are equivalent in meaning:
Poverty is negatively correlated with health among humans.
means the same thing as
Health is negatively correlated with poverty among humans.
That is, if we were to randomly sample some humans in poverty and some humans who are not impoverished, we would find that the health rates are lower among the impoverished ones. And if we were to sample some healthy people and some unhealthy people, we would find that the healthy people were less likely to be impoverished.
Every positive correlation is equivalent to a negative correlation. When we find a correlation, it can be stated a positive or negative.
Less education is positively correlated with religiousness among people
Education is negatively correlated with religiousness among people.
Birth weight is positively correlated with poverty among babies.
Birth weight is negatively correlated with affluence among babies.
More Examples of Correlations.
Here are some correlations. See if you can recognize the two properties being correlated and the population. Think about what they are claiming and what is not being claimed. Do not make any assumption of a causal connection.
People with efficient public sanitation systems live longer than people without.
People exposed to air pollution are more likely to suffer from cognitive decline
The more firefighters sent to a fire, the more damage the property sustains.
SAT scores are higher if you have more family income
People who smoke pot are more likely to try heroin.
New gun buyers are 57 times more likely to commit suicide.
Women who buy guns are much more likely to be murdered.
Real Examples, Putting the Correlation into Standard Form
Consider these real examples. What correlations can we pull from these reports?
Master the elements of correlations, and correlation arguments as explained in this chapter. Then go to this custom ChatGPT agent to practice:
Master the elements of correlations, and correlation arguments as explained in this chapter. Then go to this custom ChatGPT agent to practice:
What this tool is: You have access to a custom argument-analysis practice tool designed specifically for this course. Think of it as a logic gym: you do reps, it checks your form, and it tells you precisely where you went wrong.
What this tool is: You have access to a custom argument-analysis practice tool designed specifically for this course. Think of it as a logic gym: you do reps, it checks your form, and it tells you precisely where you went wrong.
What skills you are training
What skills you are training
This tool is designed to help you master correlations and correlation arguments
How to practice
How to practice
Go to the tutor and ask it to quiz you on correlations, and correlation arguments. It will give you examples like the ones in this chapter and have you answers questions. It will give you feedback and explanations.
How to use the tool effectively
How to use the tool effectively
To get the full benefit:
Practice until you are consistently correct with the questions about the examples.
Aim for at least ~80% correctness across multiple sessions.
If you can do that, you have the skills the course is testing.
One of the statistical conclusions that are central to this graph, in standard form, is:
62% of 18-29 Americans use TikTok.
And we can see that the statistical claim is part of a correlation claim, in standard form:
Being 18-29 is positively correlated with using TikTok among Americans.
What is an important correlation being presented here?
There are many claims being presented here, particularly in the change of rates over time between age groups. But one important correlation we can see from the chart about 18-29 year olds is:
Using ChatGPT is pos. Corr. with being 18-29 among Americans.
Consider this example:
What is the negative correlation in this passage?
Being a man is negatively correlated with having a 4 year college degree among Americans.
What is the positive correlation in this passage?
Being a woman is pos. Corr. with having a 4 year college degree among Americans.
Practice identifying correlations in science reporting: One of the essential skills we will need to master for this chapter is being able to analyze ordinary science reporting, articles, and headlines like these, and to correctly identify the correlation (positive or negative) being presented. To practice and master this skill, prompt our AI agent with “Quiz me on identifying correlations.” or “Give me real examples where I have to identify the correlation.”
Correlation Arguments
A good argument for a correlation starts with strong arguments for statistical conclusions, like the ones we mastered in the last chapter. Once we have found good evidence for a difference in the rates of two properties in a sample population, then we have found a measured correlation. From there, we can argue for accuracy and representation, like we did for statistical arguments. Here is a template for a deductively valid correlation argument. And here is an example to illustrate.
Correlation arguments are similar to arguments for statistical conclusions. Often the research or study is conducted on a sample population and then those results are taken to be representative of what is true in the target population. And the research will use some instrument or test or measurement to see if there is a measured correlation in the sample population. And then, if there is good reason to think that the measurement is accurate, then it is taken to be evidence that there is a target correlation in the sample population.
For example, here’s a correlation that we have evidence to think is true:
Covid is positively correlated with cooler weather among Americans.
That is, the rate of covid in the population goes up during winter. How do we know this? How do we measure the properties and what is the sample population? One method that public health officials like the CDC use is to run hospital inpatient surveillance in large cities. So the measured correlation in the sample is something like:
Reported Covid hospital cases are positively correlated with winter dates listed in the data among large cities in the CDC database.
And that correlation is taken to be good evidence for “Covid is positively correlated with cooler weather among people,” because they think that measuring or counting the reported hospital cases of Covid is an accurate measure of cases of Covid. And they believe that the listed dates on the data set in the data base are accurate; that is, the dates indicating when the patients came into the hospital are accurate. And they believe that what’s happening in the large cities being surveilled in the study is representative of what’s happening in the country as a whole.
Research that finds a measured correlation in the sample population is taken to be an accurate and representative measure that there is a target correlation in the target population.
Here’s another example. Creatine use is positively correlated with strength gains among athletes. Why do we think so? Research has been conducted where previously resistance trained volunteers were assigned to a creatine or a placebo group. The creatine group were instructed to consume creatine monohydrate. And then they were given a periodized heavy resistance training program for 8 weeks. They strength on the bench press and squat were measured in the lab before and after. The researchers found this measured corelation in the sample:
Self-reported creatine consumption is positively correlated with measured bench press and squat weights among the study participants.
And that measured correlation was taken to be good evidence for:
Creatine consumption is positively correlated with strength gains among athletes.
We can now see the important issues. In order to reasonably draw this conclusion, there are questions that the study authors would need to answer: Is “self-reported creatine consumption” an accurate measure of creatine consumption? That is, did the participants actually take the creatine on the days and in the amounts that they were instructed? If they didn’t or if they took more, then that would undermine the accuracy. Is “measured bench press and squat weights” an accurate measure of strength gains? We might wonder about other muscle groups, other ways of measuring strength, variations of performance under lab testing pressures, and so on. And finally, are the resistance trained volunteers in the study representative of “athletes”? There has been a lot of research on creatine and these sorts of studies have been repeated many times at this point, and many of those studies are well designed. Generally, good scientific researchers are well aware of these issues and they purposely work on addressing them in their study design. The peer review process during journal publication also confirms that these sorts of issues get addressed. What’s important for us here is to be able to recognize in a broad sense, what are the major issues in a correlation argument? What makes those arguments strong? And how can we charitably reconstruct them?
Templates for a Correlation Argument
Here’s a template for a generic correlation argument that captures these issues more carefully, and in a deductively valid form.
There is a measured correlation in the sample population.
If there is a measured correlation in the sample population, then there is a target population in the sample population.
__________________________________________
Therefore, there is a target correlation in the sample population.
If there is a target correlation in the sample population, then there is a target correlation in the target population.
___________________________________________
Therefore, there is a target correlation in the target population.
And here’s a report about the creatine research from above that we can adapt into the template:
Case Study: Creatine Supplementation and Neuromuscular Adaptation
Case Study: Creatine Supplementation and Neuromuscular Adaptation
In a controlled investigation, researchers recruited 30 resistance-trained men to participate in a double-blind, placebo-controlled trial spanning eight weeks. The participants were split into two cohorts: one receiving a daily dose of creatine monohydrate (including an initial five-day loading phase), and the other receiving an inert maltodextrin placebo. Participants were given the supplements to take home, and they logged their own consumption in an app. To isolate the effects of the supplement, both groups adhered to a standardized, high-intensity resistance training program consisting of compound lifts like the squat and bench press, performed four days per week. Participants were given the program, access to a gym, and were asked to log their workouts. The study aimed to determine if a measurable correlation existed between increased intramuscular phosphocreatine stores and the rate of 1RM (one-repetition maximum) strength progression.
Subjects were tested for their one rep max on bench press and squat at the beginning and at the end. By the end of the eight-week period, data analysis revealed a clear divergence between the two groups. While both cohorts experienced strength improvements due to the training stimulus, the creatine group demonstrated a statistically significant advantage, outperforming the placebo group in bench press and back squat maxes by an average of 5 kg and 7 kg, respectively. The researchers concluded that the correlation was driven by the physiological role of creatine in Adenosine Triphosphate (ATP) regeneration. By facilitating faster energy recovery between sets, the creatine group was able to sustain a higher total volume of work, which acted as the primary driver for superior neuromuscular adaptation and hypertrophy compared to the control group.
Template for Correlation Arguments Applied to Case Study
Reported creatine consumption is positively correlated with one rep max measurements in bench and squat among the subjects in the study. (EP)
If reported creatine consumption is positively correlated with one rep max measurements in bench and squat among the subjects in the study, then creatine consumption is positively correlated with strength gains among the subject in the study. (IP) .
__________________________________________
Therefore, creatine consumption is positively correlated with strength gains among the subjects in the study. (1, 2, MP)
If creatine consumption is positively correlated with strength gains among the subjects in the study, then creatine consumption is positively correlated with strength gains among athletes. (IP)
___________________________________________
Therefore, creatine consumption is positively correlated with strength gains among athletes. (3, 4, MP)
Recognizing the Elements of Correlation Arguments
In order to be scientifically literate, and to be a thoughtful and skeptical consumer of information, we need to be able to quickly and correctly identify the important elements of a correlation argument when we encounter science reporting. Another way to address the issues and concepts that were brought up in the reconstruction of a correlation argument above is to be able to answer several central questions after reading a scientific report. Consider this example of a report on some correlation research:
A new report finds that in recent decades, having more money has become increasingly associated with greater happiness for Americans
The GSS asked, “Taken all together, how would you say things are these days? Would you say that you are very happy, pretty happy, or not too happy?” The new study divided respondents into quintiles and deciles on the basis of income and looked at how they answered that question over several decades.
Adults who self reported that they were in the top decile of inflation-adjusted household income ($108,410 and higher) were 5 percent more likely to say they were “very happy” than people in the ninth decile.
The new study found no evidence that happiness tapers off after a certain income point, though it did not study incomes within the top decile to see if the happiness-income correlation continued to rise for those earning over $108,410.
Analysis: The 1% are much more satisfied with their lives than everyone else, survey finds
“The link [between income and happiness] is stronger now than in previous decades,” Jean Twenge, the paper’s lead author, told The Washington Post, adding that the decrease in happiness among lower-income people may be a result of rising inequality, increasing real estate values and decreased ability to pay for education.
What is the target correlation or the conclusion being argued for here?
Having money is positively correlated with happiness among Americans.
What is the measured correlation?
Answering “very happy,” to the question“Taken all together, how would you say things are these days? Would you say that you are very happy, pretty happy, or not too happy?” is positively correlated with self-reporting household income of $108k+ among people in the study.
What is the sample population?
People in the study (We don’t get more details here.)
What is the target population?
Americans
Is the correlation they measured in the sample population accurate? Can we trust “saying you are happy” as a measure of “Is happy?” and can we trust “reported income” as an accurate measure of actual income?
It’s hard to say. People probably report more happiness because it’s socially desirable to say. People probably over report their income too. If the study was anonymous or private, that might help. We might be somewhat agnostic about the accuracy premise.
Representativeness of sample? Is the sample population, the people in the study, composed in such a way that they are representative of the target population, Americans?
We don’t have enough information to be sure. There’s no information about how sample was selected.
Overall strength of the argument? Since we have doubts about both the accuracy and the representation of this argument, we might suspend judgment. But individual results may vary here.
Why Astrology Doesn’t Work; A Correlation Argument
Now that we understand correlations, it’s possible to see an interesting and important argument regarding astrology. One of the foundational claims of astrology is that what sign you are is associated with features of your personality. That is, astrology advocates appear to be asserting that there is a correlation between having a particular astrological sign and behaviors or character. So, unless those advocated are just committing confirmation bias and picking out only the cases where it appears to be true, if they are right, we would expect to find evidence for these associations or correlations. The problem is that this alleged set of correlations has been investigated over and over and no evidence has been found that it is real. The evidence that astrology works just isn't there. Here's the argument put more formally:
If astrology works, then we would find correlations between personality traits and astrological signs.
a. Being a Leos would be positively correlated with being a leader
b. Being a Gemini would be positively correlated with being intellectual
c. Being a Cancer would be positively correlated with being intuitive, and so on.
There are no correlations between personality traits and astrological signs.
a. The rate of leadership qualities are the same among Leos and non-Leos
b. The rate of being intellectual is the same among Geminis and non-Geminis
c. The rate of being intuitive is the same among Cancers and non-Cancers.
______________________________________
Therefore, astrology doesn’t work. (1, 2, MT)
Here are two studies from peer reviewed journals that investigate the question and give us evidence for believing premise 2.
That is, people who advocate for astrology believe that your astrological sign predicts or gives us information about what sort of person you are. They will often give examples of someone who is a Leo who is brave, or a Cancer who is intuitive. But it would take more than these anecdotes to make the case. What they really need to prove their conclusion is evidence for a correlation between astrological signs and these personality traits. But these two studies, and many others, suggest that no such correlation exists. So we should not conclude that astrology works. Nevertheless, people spend tens of billions per year on astrology forecasts, apps, merchandise, and products. And they spend their time reading astrology forecasts, and making decisions on the basis of them. They generalize about others and make judgments about who they do and don't want to be around on the basis of astrology, and so on. There's no correlation betwen astrology signs and personalities, so we shouldn't believe in astrology or spend our time and money on it.
Summary of Correlation Arguments
Standard form for a correlation statement: A is positively/negatively correlated with B among P
A positive correlation means that when the members of the population have property A, then they are more likely to have property B, or the rate of property B in that group is higher.
Heart disease is positively correlated with a high fat diet among humans.
A negative correlation means that when the members of the population have property A, then they are less likely to have property B, or the rate of property B in that group is lower.
Heart disease is negatively correlated with exercise among humans.
No correlation: If the rate of B among the A and non-A members of the population is the same, then there is no correlation.
Being brave is not correlated with being a Taurus among humans.
Correlations are symmetrical.
Every positive correlation is equivalent to a negative correlation.
All causal connections are correlations, but not all correlations are causal. Correlation does not imply causation.
A correlation does not mean that all or most As have property B. It only means that the rate is higher (positive) or lower (negative).
The measured correlation is the actual correlation that is found in the sample population in a correlation study/argument.
Reported creatine consumption is positively correlated with one rep max measurements in bench and squat among the subjects in the study.
Answering “very happy,” to the question“Taken all together, how would you say things are these days? Would you say that you are very happy, pretty happy, or not too happy?” is positively correlated with self-reporting household income of $108k+ among people in the study.
Template for Correlation Arguments:
There is a measured correlation in the sample population.
If there is a measured correlation in the sample population, then there is a target population in the sample population.
__________________________________________
Therefore, there is a target correlation in the sample population.
If there is a target correlation in the sample population, then there is a target correlation in the target population.
___________________________________________
Therefore, there is a target correlation in the target population.
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