20 Correlation and Cause

Section 1: Introduction

The ability to identify causes is one of our most fundamental intellectual skills.  It is because we can identify the likely causes of events that we can accurately predict how objects and people will behave, and this allows us to exercise control over our circumstances.  More specifically, it allows us to pursue desired effects and to prevent undesirable ones.  We will use the term causal inference to refer to a family of arguments all of which ultimately conclude that one thing causes or caused another.  Most causal inferences are Inferences to the Best Explanation, and although we have already talked about such inferences it will be useful to take a closer look.  In this chapter we will focus specifically on inferences that draw a causal conclusion on the basis of a correlation.

Section 2: Causal Language

We talk about causes constantly, though we do not always use the word ‘cause’.  The word ‘cause’ has many synonyms: for example, to ‘make’, ‘create’, ‘generate’, or ‘produce’, and many action terms express causation, for example: ‘push’, ‘pull’, ‘hit’, ‘kick’, and ‘move’.  Similarly, it is important to recognize that ‘prevention’ is a causal term, and so are its synonyms, ‘hinder’, ‘hamper’, ‘impede’, and so on.  Before we look at causal reasoning itself, we need to clarify what we mean when we use causal terms.  As it turns out, our everyday causal language tends to oversimplify the relations between events, and we often forget that virtually all events are the effects of a complicated series of prior events or circumstances.

Consider a very simple case.  Suppose your computer is off and you want to turn it on so you can start working on your homework.  What do you do?  Presumably, you would simply push the ‘on’ button.  Does pushing this button cause your computer to turn on?  Well, yes, but this is a qualified ‘yes’.  After all, pushing the ‘on’ button does not, all by itself, turn on the computer.  Pushing this button causes the computer to turn on only if it is also connected to a power source of some kind, the button functions correctly, the CPU is functional, and so forth.  In an important sense, then, the cause of the computer’s turning on is actually this whole set of conditions taken together.  None of these circumstances alone are responsible for turning on the computer; rather it is the combination that brings about the desired effect.   Let us call a group or set of conditions which together causes an effect, a complete cause of the effect, and let us call any individual circumstance or event that is part of a complete cause, a partial cause of the effect.

Despite the fact that pushing the ‘on’ button is only a partial cause of the computer’s turning on, it is not surprising that in everyday language we tend to say that it is the cause of the computer’s turning on.  After all, most of the time, you only need to push the button.  In this way, pushing the button is a reliable way of bringing about the desired effect.  Let us say in general that a reliable cause of some effect is an event the occurrence of which makes it very likely the effect will occur.  Though pushing the button is only a partial cause of the computer’s turning on, it is reliable because the other partial causes in the complete cause are largely fixed—they do not fluctuate much (the computer is normally plugged in, functional, etc.).  Since these other conditions tend to hold, all you need to do is push the button.

Knowing the reliable cause of some effect is incredibly useful, and when engineers create machines or other tools they normally design them so users can easily and reliably use the device.  In other cases, however, we are not so lucky.

Often, many of the partial causes themselves vary or fluctuate.  Take exposure to asbestos, for example.  Asbestos is a mineral that, for many years, was mined and used as a building material.  However, asbestos is now known to cause cancer.  More specifically, however, asbestos is one among an incredibly complex array of bio-chemical factors which, taken together, can result in cancer.  Further, many of the circumstances that figure into a complete cause of a person’s developing cancer are not fixed.  In other words, many of these factors are variable to some degree, and may not hold, or may not hold in the right way, to figure into the complete cause.  Since a person develops cancer only if all the causal factors that figure into a complete cause of cancer hold, an exposure to asbestos alone will not reliably cause cancer.  Indeed, many people who have been exposed to asbestos do not develop cancer.  Of course, this is not to say that asbestos isn’t dangerous—it is!  Given this, it will be useful to introduce the notion of a probabilistic cause.  Let us say that a probabilistic cause of some effect is a partial cause that raises the likelihood the effect will occur, but does not reliably bring about the effect.  Indeed, it need not even raise the likelihood substantially—and an appropriately related event that raises the likelihood of the effect only 10% is still a probabilistic cause.  Overall, probabilistic causes are common—especially when it comes to the kinds of complex systems found in organisms, ecosystems, or the atmosphere (to give a few examples).

In sum, our everyday causal language is remarkably imprecise.  We often simply refer to one event as the cause of another, but this way of speaking glosses over different ways the events might be related.  Indeed, in most cases what we actually mean when we express a causal claim is that one thing is the partial and reliable/probabilistic cause of another.  Again, in both cases the cause makes the effect more likely, the difference between the two has to do with how likely.  A reliable cause makes it highly likely the effect will occur, whereas a probabilistic one merely raises its chances to some degree.

Section 3: Correlations

We spend a lot of time trying to identify causes.  This happens in an informal way all the time in our daily lives: “I wonder what caused my raised taste bud,” “why did I sleep so poorly the last few nights,” “how come there are so many accidents as this intersection,” and so on.  Scientists and researchers pursue the same kinds of questions in a much more formal way, looking for the causes of a fish die-off, foods that contribute to heart disease, or whether a tax cut stimulated the economy, for example.

Either way, when looking for causes a common place to start is to look for statistical patterns, and finding the right kind of pattern can indeed be a good jumping off point towards this end.  But what counts as the right kind of pattern, and why can it be a good jumping off point?  The answer to both questions is grounded in the fact, noted above, that causes make their effects more likely.  This fact means that the effect occurs more commonly when the cause is present than when it is absent.  That is, when one thing is the cause of another, then this statistical pattern will hold between them (though this pattern holds in other cases as well).

To illustrate, smoking causes lung cancer, and consequently we find that the percentage of smokers who develop lung cancer is higher than the percentage of non-smokers who do.  Similarly, studying for exams causally contributes to getting good grades on them, and we see that the percentage of students who get good grades is higher among people who study than people who do not.  Finally, consuming a lot of sugar causes people to get cavities, and we accordingly observe that the percentage of people with many cavities is higher among those who consume a lot of sugar than it is among those who do not.  The statistical pattern common to all of these causal relationships is called a positive correlation.  This relation is also sometimes called an association or link (particularly in news reporting).  Regardless of how we refer to it, in general we can say:

If X causes Y, then X is positively correlated with Y.

We will take two points from this.  First, when you are trying to identify the cause of some phenomena it makes sense to look for a correlation involving it, because the cause—whatever it turns out to be—will be something that is correlated with the effect you are wondering about.  Importantly, however, a correlation does not imply or entail a causal relationship (for reasons we will discuss in a bit).  Nevertheless, a correlation is a place to start—it is a clue to causation.

Second, when you are looking for causes, and you find that a suspected cause is not correlated with the effect, then you can rule it out.  Say that you get a raised taste bud after you eat sometimes, and you are trying to figure out what is causing it.  You read online that for some people spicy foods can cause raised taste buds, and you’ve been watching what you eat lately.  However, you’ve found you don’t get a raised taste bud when you eat spicy foods any more commonly than when you don’t eat spicy foods.  Given this, spicy food is not correlated with your raised taste buds, and you can rule it out and move on (by Modus Tollens!).

Before we move on, it is important to clarify the relation a correlation expresses.  In general terms, a positive correlation is a comparative statistic.  We will use X and Y to stand for different categories or groups.  Although correlations can hold between other kinds of variable as well (e.g. quantitative or continuous variables), we will focus on categorical correlations as they are both common and somewhat easier to understand.  Given this, we will say:

X is positively correlated with Y when (and only when) the percentage of Xs that are Ys is greater than the percentage of non-Xs that are Ys.

To illustrate, let us look at a couple of new cases so you can clearly see the pattern.  Being a professional basketball player is positively correlated with being 6’4 or taller, in that the percentage of professional basketball players who are 6’4 or taller is greater than the percentage of people who are not professional basketball players who are 6’4 or taller.

Importantly, a positive correlation is not saying that the number of Xs that are Ys is higher than the number of non-Xs that are Ys.  There are, after all, only around 600 professional basketball players in the United States, an although most of them are 6’4 or taller, there are hundreds of thousands of people who are not professional basketball players who are 6’4 or over within United States (pop. 330 million).  Again, a positive correlation is saying that the percentage is higher—not the number.  Here is one more example: in the United States being a Democrat is positively correlated with being a vegetarian.  That is, the percentage of Democrats who are vegetarians is greater than the percentage of non-Democrats (Independents and Republicans) who are vegetarians.^[1]

As we noted at the beginning of the chapter, prevention is a causal relation as well.  This means that we might be interested in identifying cases where the presence of some object or event makes the occurrence of another less likely, and this would show up in a statistical pattern called a negative correlation.  We will say that:

X is negatively correlated with Y when (and only when) the percentage of Xs that are Ys is less than the percentage of non-Xs that are Ys.

For example, getting the vaccine for COVID-19 is negatively correlated with getting a severe infection.  That is, the percentage of people who got the vaccine and suffered severe infections by the virus is less than the percentage of people who did not get the vaccine and suffered severe infections.  Importantly, this is not inconsistent with the point made earlier that causes make their effects more likely.  In the case of prevention, a cause produces, and therefore makes more likely, an effect that subsequently interferes or disrupts a distinct set of causes.  By interfering in this second chain of events, the cause thereby makes the effect in question less likely.

It is important to add that correlations can be stronger or weaker.   In everyday informal thinking, it is difficult to be precise about the degree of correlation.  We tend to have only a vague idea of how much more commonly Y occurs given X than non-X.  However, scientists and researchers are normally in a position to be much more precise, and measure the degree of correlation on a scale between -1 and 1.  On this scale, 0 says that there is no meaningful relationship between the two variables, whereas 1 means there is perfect positive correlation between two variables, and -1 refers to a perfect negative correlation.  This matters because the weaker the correlation the more likely it is mere coincidence, whereas the stronger the correlation the more likely it captures a genuine relation between two things.

Finally, although we have been thinking about correlation in the context of identifying causes, they are useful for other purposes as well.  This is because a correlation allows you to make predictions.  If you know that X is correlated with Y, and you know that X is present, then you know that Y is more likely than not to be present as well.  For example, say you work in marketing.  Knowing that consumers who are interested in your product tend to watch the game show Jeopardy! at a greater rate than other consumers, gives you important information about where to spend your advertising dollars.  In this case, you aren’t looking to identify cause/effect relationships, but simply to predict what potential consumers are watching.  In any case, now that we have a sense for what a correlation says, let us turn to its use in causal inferences.

Section 4: Inferring Cause from Correlation

Suppose that you have discovered a correlation.  You learn from a reputable source that teenagers that spend more than 3 hours a day on social media tend to have higher rates of anxiety and depression than teenagers who spend less time on social media.  This makes you start thinking about your own social media usage, and to wonder whether you should try to limit it as a way to prevent depression or anxiety.  This train of thought embodies a move from correlation to cause, and implies that extended social media use is a partial cause of depression/anxiety.  But can we really draw such a conclusion from the information provided?

Well, we need to be careful.  Let’s take a look at this inference:

1. Teens who spend more than 3 hours per day on social media have higher rates of anxiety and depression than those that spend less time.
2. So probably extended use of social media causally contributes to anxiety and depression among teens.

This argument begins with an observation about the world, and draws a conclusion about the likely cause or explanation for this state of affairs.  That is, this is an Inference to the Best Explanation, and the link between the stated premise and stated conclusion is the assumption that the best explanation for this correlation is a causal relation.

[picture—teens and social media]

Given this, let us recall and apply the steps for evaluating an Inference to the Best Explanation.

Three Questions to Ask of Inferences to the Best Explanation:

• How likely is the proposed explanation?
• Are there other plausible explanations? (Failure: Hasty Explanation)
• Would the truth of the proposed explanation be less surprising than the truth of any competitor? (Failure: Poor Explanation)

Step one: does the proposal that social media use causally contributes to depression and anxiety seem like a likely explanation for the correlation?  Well, if one really causes the other, then we would certainly find a correlation between the two.  Whether a person regards this as plausible will depend on their relevant background knowledge, but this will likely seem plausible to many people, at least on its face.

Step two: are there other plausible explanations for the correlation?  This is a little more difficult to answer, but yes there are (after all, there are many possible explanations for any phenomena—including correlations).  For example, it might be the other way around—it might be that anxiety and depression among teens leads them to spend a lot of time on social media.  Alternatively, it may be that there is no causal relation at all—perhaps there is some distinct third factor that causes anxiety and depression among teens and pushes them toward extended social media use at the same time.  For example, perhaps interpersonal conflicts with friends or family can have these effects.  Finally, this correlation may be sheer coincidence.  In this case, there will be no deeper explanation for the correlation—it is just a statistical anomaly.  This is important to emphasize.  Although coincidences are not common (that is part of what makes them coincidental!), they do happen—especially in cases where correlation is based on relatively few observations.

Given this, we are now in a position to understand the warning you might have heard that “correlation does not imply causation.”  The idea behind this catchphrase is that we cannot jump from the observation of a correlation to a conclusion about causation without first considering and comparing alternative explanations.  That is, because there are always many explanations for any given correlation, we cannot simply pick out one cause and claim it is THE explanation unless we have reason to believe that it is the best one available.  To do otherwise is to draw a Hasty Explanation.

This brings us to step 3: what is the most likely explanation among the options out there?  In order to think through this, we will need to evaluate and ultimately rule out, or at least cast doubt upon, some of these explanations. How do we begin?  Luckily, the possible explanations for a correlation between X and Y can be separated into only 4 types.

Type 1:            X causally contributes to Y.

Type 2:            Y causally contributes to X.

Type 3:            This is some underlying causal factor or factors which relate X to Y.

Type 4:            The correlation is accidental or coincidental.

Again, we can only conclude that one of these explanations is correct to the extent that we have ruled out, or at least cast doubt on its alternatives.

This raises the question: how do we rule out alternative causes?  The answer to this question is complicated, but we can offer some general strategies.  First, we know that causes come before their effects, thus if we know that X occurred before Y, we can rule out the Type 2 explanation that Y caused X.  Returning to the example, we’d want to know which came first—extensive social media use or depression/anxiety?  Second, the more evidence we have supporting the correlation, the less likely the correlation is merely coincidental (Type 4).  That is, a correlation that is based on more observations or data is less likely to be coincidental than one based on fewer observations or less data.  So, we might ask: how many teens did researchers look at in conducting the study?  This correlation is more likely to be mere coincidence if they looked at 10 teens than if they looked at 1000. A third, and related, point is that a correlation is less likely to be accidental or coincidental to the extent that we can envision a possible mechanism connecting X and Y (though our ability to do depends a lot on our background knowledge).  Thus we might ask: are there some known connections between extended social media use and people’s emotional state that might account for the connection?  Fourth, what about the possibility of underlying causal factors (Type 3)?  This is the most difficult possibility to rule out or cast doubt upon.  After all, there are always factors we might not be aware of.  In this case, we can only proceed by considering possible underlying causes and trying to rule them out one by one (although we will discuss this point further in the next chapter).

In any case, once we’ve identified one explanation as the most likely by at least casting doubt on the others, we can claim to have identified a correlation that is likely to capture a causal relation.  Such correlations are called significant correlations.  There are other techniques that we can use to rule out possible causes, and thereby identify significant correlations.  In the next chapter we will take a look at one of the best methods: a controlled experiment.

Exercises

Exercise Set 20A:

#1:

Give an example of a reliable cause (other than the examples given in the reading).

#2:

Give an example of a probabilistic cause (other than the examples given in the reading).

#3:

Give an example of a correlation that you think is significant or important (other than the examples given in the reading).

#4:

Suppose that being a Bollog is positively correlated with being a Trollog.  Does it thereby follow that most Bollogs are Trollogs?  Why or why not?  (Note: you do not need to know what a Bollog or Trollog are in order to answer this question).

Exercise Set 20B:

Directions: Using the following chart to answer the questions below.

A college is thinking about changing its mascot, and wants to gauge community attitudes. So the school conducts a poll.  As the poll is set up, students must reply either ‘In Favor’ or ‘Not in Favor’. Here is the part of the information they collected that compares responses from students involved in athletics at the college and students who are not.

#1:

Is there a correlation between being an athlete and being not in favor of the change?  If so, explain, and say whether it is positive or negative.

#2:

Is there a correlation between being an athlete and being in favor of the change?  If so, explain, and say whether it is positive or negative.

#3:

Is there a correlation between being not in favor and being an athlete?  If so, explain, and say whether it is positive or negative.

Exercise Set 20C:

Directions: for each of the following (i) identify the correlation on the basis of which the causal inference is drawn, (ii) the causal conclusion, and (iii) using appropriate questions think through the argument, and comment on the strength of the inference.  In each case, assume the correlation is true.  

#1:

Do you want healthy teeth?  Buy a speedboat!  According to a recent study, people who own speedboats have fewer serious dental problems than those who do not.

#2:

People with allergies to animal dander are less likely to have pets than those who do not have these allergies.  Therefore, people should avoid getting pets if they want to avoid developing allergies to pet dander.

#3:

We should seriously consider putting limits on the amount of cheese a person can buy, since per capita cheese consumption almost perfectly matches the number of people who die by becoming tangled in their bedsheets each year!^[2]

#4:

College entrance exams aren’t fair since you can effectively buy a high score.  Look at the stats: students from high income families disproportionately score in the top 10% of all SAT scores.

#5:

You should encourage your grandpa to walk more.  I read the other day that elderly adults who walk for at least 180 minutes per week have lower rates of dementia than those who walk less.

#6:

Teenagers who are classified as heavy uses of marijuana are more likely to have strained relations with at least one parent than students who use marijuana less often or not at all.  We should recommend limiting marijuana use among teenagers as a way to improve relationships within families.

#7:

During the 20th century, Washington D.C.’s professional football team had an amazing degree of control over presidential elections.  Over the interval from 1940-2000 the outcome of the final Washington D.C football team’s home game perfectly matches presidential election results.  If they won, the incumbent party held on to the presidency; if they lost, the opposition party took it.^[3]

#8:

You have slept very poorly the last few nights.  You are wondering what is causing it, and notice that there have been poor air quality advisories the last few days too.  Moreover, prior to this stretch of poor air quality you were sleeping fine.  You reason that the poor air quality is probably disrupting your sleep and think about getting an air purifier for your room.