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Causation and Correlation

If you’ve ever taken a statistics course, you’ve probably heard an instructor warn that “correlation does not imply causation”. After hearing it, I definitely knew to be wary of making such associations, but never had a good answer to the question: “If there are no causal relationships at play, why am I seeing correlations?”

In this post, I’m going to talk about three ways in which you could find correlations between two variables even though there’s in fact no causal relation between them. Though my examples are completely made up, I’ll illustrate the following:

1. Why countries introduced to social media might see rises in extremism,
2. Why we might find that people who are vaccinated are more likely to be diagnosed with autism, and,
3. Why an amateur dietician might come to the conclusion that weight gain is an appropriate remedy for bad skin.

First, let’s define correlation and causation a bit better.

Let’s start with correlation. If I randomly picked a person from the population, and found out that they had a college degree, I’d find that they were more likely than the average person to own a Tesla. This means that a person’s having a college degree is correlated with them owning a Tesla, i.e. knowing if they have a college degree changes the likelihood that they also own a Tesla.

Now does this mean that having a college degree causes them to own a Tesla? Perhaps, but not necessarily.  If I randomly picked a person from the population, and put them through college to the point where they got a college degree, and then found that they became more likely to own a Tesla, this would be an instance of causation. More precisely, if having some property A (like a college degree), causes you to have some property B (like owning a Tesla), then forcing someone to have A will make them more likely to have B.

From now on, I’m going to refer to these properties (e.g. having a college degree, owning a Tesla, etc.) as variables, because they can take on one of many values (e.g. True or False). When talking about causal relationships between variables, it’s convenient to use a Causal Diagram.

In the diagram above, we’ve represented a situation in which A causes B. If there were no arrow, we would represent a situation in which A does not cause B, i.e., even if we had forced A to be true, it would have no influence on whether B were true.

I’m now going to describe three ways in which, despite there being no causal relationship between two variables, we still observe correlations between them. Before we jump in, let me remind you again that all these examples are completely made up for the sake of illustration.

The Mediator

I’m going to start off with the most intuitive but least surprising way in which two variables could be correlated despite having no direct causal relationship between them. Let’s say that we’re looking into whether we can blame social media for a rise in extremism. Consider the Causal Diagram below:

Here, we’re assuming that social media doesn’t directly cause a rise in extremism in a country, but rather causes the country to have a platform for Fake News, and that this platform in turn causes a rise in extremism. Let’s put some numbers on things. Let’s say 60% of all countries have widespread use of social media. Since certain social media companies don’t do a great job of filtering out Fake News, let’s say:

1. If social media has widespread use in a country, there’s a 95% chance it has a platform for Fake News, and,
2. If social media doesn’t have widespread use in a country, there’s a 30% chance it has a platform for Fake News (perhaps a messaging app or some other medium).

Furthermore, let’s say:

1. If a country has a platform for Fake News, there’s a 70% chance it experiences a rise in extremism, and,
2. If a country doesn’t have a platform for Fake News, there’s a 20% it experiences a rise in extremism.

Lets look at an imaginary world of 400 countries at a glance:

In the picture above, each triplet of dots is a country. If the dot is shaded dark, then that variable is true for that country. For example, if a red circle is shaded dark, then that country has a platform for Fake News.

What we’re interested in seeing is whether countries with widespread use of social media (blue dots) are more likely to see a rise in extremism (yellow dots). We study this by splitting the population by whether they have widespread use of social media:

In the left column, we’ve grouped countries that don’t have widespread use of social media, and in the right column, we’ve grouped countries that do. It’s easy to see from this grouping the effect we suspected, only 35% of countries without social media have seen a rise in extremism, while 67.5% of countries with social media have seen it.

This clearly suggests that if a country wanted to curb the rise in extremism, they should ban social media, right? Well, sort of. What they would be missing is that widespread use of social media caused a rise in extremism only because it created a platform for Fake News. You can see this more clearly in the picture below, where we first split the population by whether or not they have a platform for Fake News, and then see if countries with social media have any influence on the rise in extremism for each group separately:

What we see is that the effect almost completely disappears, and any residual effect is only due to the fact that I’m using a population of only 400 countries as opposed to larger number.

So would you have been wrong in saying social media causes a rise in extremism? No, because in this case, social media is an indirect cause. But understanding the mediator leads to the insight that if you want to control the rise in extremism, you’re better off preventing social media platforms from becoming platforms for Fake News, than banning them altogether.

So what’s the takeaway? If you’re pretty confident that one variable causes another, consider whether that causation acts through a mediator. Controlling the mediator would be a more effective way of controlling the variable downstream.

The Confounder

In the case of the mediator, you’d still be right in technically saying that A caused C, even if it caused it indirectly. We’re now going to look at the most common source of spurious correlation, the confounder, a.k.a. the ‘common cause’. Here, you would not be correct in assuming the correlation you’re observing indicates causation.

We’re going to see why we might find that people who are vaccinated are more likely to be diagnosed with autism, despite there being no actual causal relationship between them. Let’s say we have population of a million people. Of these million people, 30% of them happen to be children. Since its more common for a person to get vaccines around when they’re children, lets say that:

1. If you’re a child, there is a 70% chance you got vaccinated in the last 6 months.
2. If you aren’t a child, there is a 10% chance you got vaccinated in the last 6 months.

Now, since its more common for someone to be diagnosed with autism when they’re a child rather than when they aren’t, lets say that:

1. If you’re a child, there is a 15% chance that you get diagnosed with autism.
2. If you aren’t a child, there is a 2% chance that you get diagnosed with autism.

The causal diagram above reflects these relationships, but notice that it doesn’t explicitly reflect any causal relationship between being recently vaccinated and having autism. Nevertheless, if one were to just look at these two variables, you’d find that people who had just been vaccinated were almost four times more likely to be diagnosed with autism that those who hadn’t!

Heres why. The picture below on the left represents the population, where each triplet of dots is a person. As before, if the dot is shaded dark, then that variable is true for that person.

Let’s rearrange the population a bit in the picture below, separating people who haven’t been vaccinated (on the left) and those who have (on the right).

The astute anti-vaxxer might look at the number of people diagnosed with autism in each group. On the left, you’d see that only 3.5% of people are diagnosed with autism, while on the right, 12.5% of people are. This might lead a person to the conclusion that getting a vaccine makes you almost four times as likely to be diagnosed with autism! Note that this is despite the fact that in the Causal Diagram I used to generate this population, there is no actual causal relationship between vaccines and autism.

So what would an epidemiologist do in this situation? Assuming they suspected that being a child is a confounding variable, i.e. a likely cause of both getting vaccinated and being diagnosed with autism, they would control for whether the person is a child. In the picture below, we see how this is done.

We first split our population based on whether the person is a child (on the right) or not (on the left). Then, we ask separately for each population: does having been vaccinated make you more likely to have autism? Looking at the numbers, you’d see that people are no longer four times more likely to be diagnosed with autism if they’d been recently vaccinated. As it turns out, the only reason you see any difference at all is because I’m representing a million people with 400 people.

So what’s the takeaway? If you see a statistic saying “people who do A are X times more likely to have condition B”, think about whether there might be a common cause responsible for that correlation.

The Collider

Next up, what I think is the most interesting but least obvious source of spurious correlation: the collider. Let’s imagine an amateur dietician who has seen about 200 clients. In the population, let’s say:

1. 40% of people are overweight, and,
2. Independently, 40% of people have acne.

Now let’s assume that both being overweight and having acne might be reasons a person would see a dietician. Specifically, let’s say:

1. If a person is not overweight and doesn’t have acne, there’s a 10% chance they’d see the dietician,
2. If a person is either overweight or has acne, but not both, there’s a 60% chance they’d see the dietician, and,
3. If a person is both overweight and has acne, there’s a 70% chance they’d see the dietician.

Now lets say we look at the population as a whole, as in the picture below:

And now, as before, we check if being overweight makes you any more or less likely to have acne by splitting the population into those who are overweight, and those who aren’t:

As expected, we see that being overweight has no effect on having acne, i.e. a person’s chances of having acne are the same (40%) whether or not they are overweight.

But, let’s now imagine that the dietician made the mistake of assuming that their clients were representative of the population, i.e., they assume that trends they observe amongst their clients are trends present in the population. The figure below illustrates this. The right column is the subset of the population that has visited the dietician. On the top of the column, we have the people that are overweight, and at the bottom are the people that aren’t.

What would be apparent to the dietician is that, of their clients, those who are overweight are only half as likely to have acne. This might make them want to draw the conclusion that weight gain is a good remedy for acne. As we saw in the earlier figure, this conclusion would be wrong! In the population as a whole, having acne and being overweight were independent.

Is this something only a dietician should worry about? Well, it comes up more often than you might think. Imagine that you leave a sheltered childhood to join a big university. You might notice people who are intelligent are less likely to be athletic. Though, since being intelligent and being athletic are both sufficient causes for getting into the university, you might be incorrect in generalizing this observation to the population at large.

So what’s the takeaway? If you’re making observations about a subset of people, be careful about whether the variables you’re observing caused them to be in subset. Scientists studying psychedelics in the 60’s had a particularly hard time with this because the people who signed up to participate in their studies did so because they were inclined to try psychedelics and, and so were not very representative of the population at large.

Summary

I showed three ways in which you could observe a correlation between two variables (say, A and B) even though one didn’t directly cause the other. In the case of the mediator and confounder, we saw that there might be a third variable you’re missing, and had you controlled for that variable C, the correlation you see between A and B would disappear. However, in the case of a collider, controlling for that third variable C is the source of the correlation, so you want to make sure you’re not controlling for it. We saw that people tend to introduce correlations through colliders by unintentionally assuming a subset of the population is representative of the whole, thereby unintentionally controlling for a variable.

The next time you come across a statistical correlation used to argue for some causal relationship or another, I hope you use these lenses to consider ways in which that correlation might be spurious, and what else you might need to know to see if it is.

References

Want to know more about probabilistic modeling and causality? Check out the resources below:

• The Book of Why: The New Science of Cause and Effect by Judea Pearl (2018), which is a great popsci read and what inspired me to write this post,

• Chapter 2 of Decision Making Under Uncertainty by Mykel J. Kochenderfer (2015), which is a fantastic textbook written by my PhD advisor at Stanford, and,
• This Coursera course on Probabilistic Graphical Models, if you want to know how to use this stuff at scale.

A Brave New World

Imagine you are given the funds to support five underprivileged 18-year old students through college. You hold a call for applicants, and receive one hundred applications. In addition to asking for quantitative metrics such as grades and extracurricular achievements, you also interview each student. On what basis would you allocate the funds?

Presumably, you would try to elicit an estimate of how likely the student was to succeed in college if given the scholarship. For simplicity, let’s call this the student’s aptitude. A sensible allocation of the scholarships would be to the students who you estimate have developed the highest aptitude.

Now consider an alternative allocation strategy—a lottery that randomly picks five students from a hat and gives them the scholarships. Again, for simplicity, let’s assume that all one hundred applicants came from similar socioeconomic backgrounds. The second strategy would disregard merit in favor of assuring equality of opportunity.

Personally, I would lean toward favoring the first strategy. Given that all applicants are from similar backgrounds, a demonstration of aptitude likely correlates with the applicant having put in hard work, and is the best predictor I have that they will maximize their use of the opportunity.

If your intuitions agree with mine, let’s make this more interesting. Suppose you are given scholarships to support a 5-year old through a superior education and upbringing. Again, you get a hundred applications, again of students from similar underprivileged backgrounds. Would you try to assess these five year olds’ aptitudes, and give the scholarship to the one with the highest aptitude?

If you’re like me, you certainly didn’t display your aptitude for anything you thrive at today when you were five. So, it would seem that assigning based on the aptitude of a five year old is not a good, or fair strategy, because of how poorly aptitude at that age predicts aptitude in the future. However, if you had access to an oracle that could perfectly estimate how likely the five year old was to succeed at the superior school, then perhaps you would want to give the scholarship to the one with the highest chance of succeeding. If this intuition doesn’t sit right with you, it’s worth thinking about what you think changes between when the child is five and eighteen, aside from them simply fulfilling the oracle’s prophecy.

This oracle doesn’t exist, but what factors might it have considered? Our aptitude at the age of 18 is presumably a function of three things—nature (i.e. one’s genetic predispositions), nurture (i.e. environmental factors such as whether one endured an abusive home, or whether their drinking water was contaminated with lead), and depending on your philosophical inclinations, possibly free will.

I was surprised to learn in this podcast with Robert Plomin, that through genome wide associated studies, an individual’s genes can be surprisingly good at predicting their aptitude for certain skills. Let’s call a person’s genetic predisposition to succeed at a given task their genetic aptitude. As the science progresses, we will find that for many skills, genetic aptitude will be poor predictors of overall aptitude, and in others, they might be highly predictive.

Consider, hypothetically, that the ability to play the violin at a professional level is almost entirely governed by genetic aptitude. Let’s say you’re given a scholarship to provide a single, underprivileged baby the finest possible violinist education, but you’ve received 100 applications. Suppose they were also screened for genetic aptitude, and we find that only one of them has a noteworthy genetic aptitude. What this means is if we were to give the scholarship to all but one of them, we would be nurturing little more than a nice hobby in them, but if we give it to the one with noteworthy aptitude, we might set them up for a thriving career in violin. Would you give it to the baby with the highest genetic aptitude?

If you’re inclined to say yes, the implication is pretty dire—we are tacitly saying that allocating resources based on ones predetermined aptitudes would create a better society than one in which these aptitudes were ignored.

Suppose in addition to an applicant’s genetic aptitude, we also received an evaluation of their environmental aptitude, i.e. their upbringing, neighborhood, family, and probably friends, were all considered to estimate how likely it is that they could be nurtured to succeed in a particular field. Imagine, hypothetically, that the ability to play guitar is governed little by genetic aptitude but almost entirely by environmental aptitude. Again, suppose we have 100 babies’ applications, and only one has noteworthy environmental aptitude for playing guitar. Would we give a guitar scholarship to her, or give it to one at random?

If we’re being consistent, we should probably give it to the one with the highest environmental aptitude. This, for me at least, sounds a moral alarm bell—something seems unfair here. Those babies didn’t choose their environments, so why should one be privileged with a scholarship after already winning a lottery in terms of the environment they’re born into? Household income is probably a very relevant environmental factor, and so we would end up giving scholarships primarily to the rich! Presumably, a fair society would do all that is possible to counteract the influence of environment on aptitude, it would try to ensure every baby an equal opportunity to succeed at a particular task.

But is an assignment on genetic aptitude any less fair than one on environmental aptitude? The genes a baby is born with is just as much a luck of the draw as their environment. If a fair society tries to counteract the effects of environment on aptitude, and the technology were available, then it would try to also counteract the effects of genes, right?

So now we have a society in which all babies are equally likely to succeed at any particular task. But, since the process merely spread out likelihood over all babies as opposed to concentrating them in a few, we end up having no one particularly likely to be exceptional at anything.

It’s possible that this homogenous society is able to sustain itself, but would one that deliberately fostered exceptional individuals be better off? Would a society with exceptional scientists be more likely to find the cures for their cancers than one without.

So, would you choose a society in which aptitudes were equalized, or one where we deliberately distribute aptitudes so that we have 5% of babies with a very high aptitude for thriving as doctors, another 5% as artists, and so on? Note that in such a society, since aptitudes are predetermined, we would have no reason to monetarily compensate those who are exceptional doctors and more than those who are exceptional janitors. Everyone would be playing their part, and would earn the same standard of living for doing so.

The world we’ve ended up with is essentially the world described by Aldous Huxley in the book Brave New World. When one reads it, it appears to be a dystopia, but in this blogpost I’ve tried to illustrate that we might want such a world if we buy into one pretty benign premise: that if given a 100 applications for college scholarships from underprivileged individuals, we would give the scholarships to those with the highest aptitudes, and not assign them at random.

In this post, I’ve made a couple of grave oversimplifications. Firstly, I’ve tacitly assumed that we are doing the most good for society by rewarding success, instead of passion for passion’s sake. The second is that I’ve implicitly assumed that it is even possible perfectly predict what an individual could bring to a given field based on their genes and environment. This firstly ignores the implicit value of their lived experience, and assumed an omniscient level of knowledge about the possible futures an individual could encounter in their lives.

These simplifications aside, I do believe there is a question worth considering here. Supposing we have two individuals, Alice and Bob, living in similar environmental conditions, and we genome sequence them. We find Alice’s genes make her 10% more likely to not only be better at violin than guitar, but also to enjoy it more. Say the opposite is true for Bob—he is 10% more likely to enjoy guitar and to be good at it. Would we use this information, and encourage them toward the two instruments asymmetrically? Or are we better off ignoring this information?

On the other hand, the naive strategy has an expected loss of a much larger $1.6, with a standard deviation of 1¢ on that estimate. Using ttest2, we can assert that the MDP strategy is better with near perfect confidence. While the difference between the MDP performance and that of the cheat sheet is very marginal, the likeliest for why is performs worse is that I made a small error in painstakingly filling out the almost two million state transitions probabilities in $P$. However, what is clearly evident is that the exercise does generate a near optimal strategy for playing Blackjack, one that performs much better than a naive decision policy. Conclusion The purpose of this project was to get familiar with using Markov Decision Processes to provide optimal strategies in discrete, finite state stochastic environments. Blackjack seemed to be a perfect candidate to try the algorithm on. To formulate the game in a manner acceptable to an MDP solver, I first specified the probabilities for transitioning from each gamestate to every other under all actions using either explicit analysis or Monte Carlo simulation. I then specified the rewards for transitioning from game states to terminal states under the different actions. With these matrices specified, I used the MDP solver to provide the optimal strategy. I compared this strategy to a widely available “Blackjack Cheat Sheet”, which claims to be the optimal blackjack strategy. Repeatedly simulating the strategies in the game of Blackjack, I found that both the MDP’s strategy and that of the cheat sheet lose only about 5¢ per$10 bet on average, while a more naive strategy loses \$1.6 on the same bet. According to wizardofodds.com, the “house advantage” for a game of Blackjack with the assumed rules is around 0.5%. The house advantage indicates the expected value of the dealer assuming the player is playing optimally, and thus provides evidence that the strategy given by the MDP is indeed close to optimal.

I did this project with the help of Dilip Ravindran, a close friend and graduate student at Columbia Economics.

Spatial Heterodyne Infrared Raman Spectrometry

As part of the Harvey Mudd College Clinic Program, I worked on a project for the Kinohi Institute. Over the course of a year, we designed and built the first spatial heterodyne Raman spectrometer capable of operating at near-infrared excitation light. The team also tested and validated the performance of this device.

Raman spectroscopy is a common technique used to determine the material composition of a substance. Currently, the most common Raman spectrometers are slit spectrometers (which have low light throughput), and Fourier Transform spectrometers (which require sensitive moving parts). Spatial Heterodyne Spectrometry (SHS) has no moving parts and high light throughput, which means it is robust and non-destructive to the sample. The Spatial Heterodyne Infrared Spectroscopy (SHIRS) technique developed in this project can be employed to detect cancer, observe protein folding in real time, and could even be used for the non-destructive detection of extraterrestrial life.

In this post, I will describe how the device excites Raman signals, isolates the signal from noise, and identifies the Raman signal through spectroscopy. I will then discuss the results of the effort.

Exciting Raman Signals (A)

Raman signals are a form of light that reflects of molecules with a change in energy that acts as a signature for the molecule. Hence, if we shine light of a single, pure wavelength onto any molecule and identify the spectrum of the resulting Raman signals, we can correctly identify the molecule. However, this isn’t quite so easy. For one, Raman shifts are a very low probability event, which results in their signals being extremely weak. In addition, the unchanged reflection of the input laser (Rayleigh scattering) and fluorescent reflections are sources of noise that flood out the Raman signals. The method for dealing with this noise is discussed in the next section.

Isolating Raman Signals (B)

Aside from the advantages of causing little damage to organic samples, using a near infrared excitation laser has the additional advantage of causing much less fluorescence than the more conventionally used visible or UV excitation lasers in Raman or SHS systems. Additionally, since Raman shifts are known to be longer wavelength than the excitation laser, a Long-Pass Filter (LP on Figure 1) and Long-Pass Dichroic Mirror (D) are used to ensure that all light of lower than the excitation wavelength are heavily attenuated. A Notch Filter (N) is used to heavily attenuate the Rayleigh scattering from the excitation laser. The remaining light is mostly composed of just the Raman signal at a sufficiently high Signal-to-Noise ratio. The next section will discuss how a Spatial Heterodyne Spectrometer recovers the spectra of this light.

Identifying Raman Signals (C)

With reference to Figure 1, the SHS works by first using a 50:50 Beam Splitter (BS) to direct the light towards two Diffraction Gratings (DG). The light then reflects of these gratings at a wavelength-dependent angle. Since the angle of reflection depends on wavelength, there is a wavelength-dependent separation between the the two reflected wavefronts that recombine on the Camera’s Sensor (C). From the theory of interference, we know that the frequency of oscillations in an interference pattern depends on the angle separating the wavefronts. Thus, the wavelength of light is encoded in the frequency of oscillations produced on the camera’s sensor. Performing a Fourier Transform on the resulting image recovers the spectrum of the input light.

The figure above shows examples of fringe patterns created by single-frequency sources (top), and the spectra that are generated from the fringe patterns (bottom). The generated spectra are compared to spectra gathered from an off-the-shelf spectrometer. We can see that for two test cases, the device functions as an accurate and precise spectrometer compared to an off-the-shelf alternative.

Results

The complete SHIRS system was validated as a standalone spectrometer on various test sources such as Hydrogen, Neon, and Sodium Vapor Lamps. The spectrometer demonstrated a bandwidth of 131 nm and a resolution of 0.1-0.6 nm, which satisfy the constraints for the desired applications of the Kinohi Institute.

The figure above shows the spectrum of the sodium vapor lamp captured by the team’s SHS and the ScienceSurplus slit-based spectrometer. Notice that only the team’s SHS is able to resolve sodium’s characteristic double peak.

The complete system was validated by testing elemental Sulfur, which is known to have a strong Raman signal and is a typical first-test case.

This figure shows the Raman spectrum gathered using the team’s prototype from a sulfur sample. The peak seen matches the peak expected from published Raman spectra to within 1 cm-1. This spectrum was reproducible across multiple trials.

I completed this project on a team with Alex Alves, Achintaya Bansal, Erica Martelly, and Apoorva Sharma. The team was advised by Professors Philip Cha and Gregory Lyzenga of Harvey Mudd College. If you would like to apply this technology to any of your work please contact me or the liaison of this project from the Kinohi Institute, Dr. Michael Storrie-Lombardi (mike@kinohi.org).

Magnetic Levitation with an FPGA

In this project, a small disc magnet is levitated by a solenoid that is PID controlled by an FPGA at 156 kHz sample rate. A slower open loop is driven by a Raspberry Pi to stably move the magnet between set points. The Pi communicates the set point, steady state voltage, and gains, thus slowly making big transitions by commanding smaller jumps with accurate linearization. A video of the system in action can be found here.

Figure 2 shows a block diagram of the system components. As shown in Figure 1, the physical system is composed of a solenoid above a disc magnet. A PID controller is implemented on an FPGA to control the position of the disc magnet. The 10-bit position of the magnet is measured by the Hall-effect sensor, which is located about an inch below the solenoid so it avoids saturation from the solenoid’s magnetic field but still detects the disc-magnet’s field. The voltage from the sensor is passed through an ADC and then through a pre-calibrated lookup-table to estimate the position of the magnet. The position is then compared to the desired setpoint and used to compute a voltage across the solenoid. The voltage is computed by a PID controller that implements discrete-time derivative filtering for improved performance. This voltage is then passed through a PWM to generate a binary control signal. This control signal leaves the FPGA and is applied to a transistor which controls the current flow to the solenoid.

The linearized gains and feed-forward voltages most applicable to a given setpoint were found in calibration and stored on a Raspberry Pi. The Raspberry Pi takes a desired setpoint and passes the associated gains and feed-forward voltage to the FPGA’s PID module over SPI. In order to create the up-and-down motion seen in the video, the Pi takes small, discrete steps to march the magnet to the endpoints, holding each command for a time long enough for the magnet to reach the desired intermediate location and dissipate its transient response.

The project was done as a joint final project for the Harvey Mudd College courses Microprocessor Design (E155) and Systems Simulation (E205). The project was done in collaboration with Vaibhav Viswanathan.