You Ask, I Answer: Causation Without Correlation?

In today's episode, Vito asks, let's assume that we have the joint probability distributions of A and B. In that scenario, is it possible that A causes B, but A and B are not correlated? So this is a comment that was left on my website about a post I did a number of years ago on correlation and causation. It is generally accepted that correlation is not causation just because two variables are correlated does not mean that one causes the other. Ice cream death uh constant consumption of ice cream and uh number of deaths from drowning are strongly correlated in a lot of data sets.

Why? Well, logically, we know that there's uh this thing called summertime, and as people uh the weather gets warmer, people eat more ice cream, people go swimming more, except in years of pandemics, uh, and uh you have an increase in drowning deaths. So what about the reverse, which is what Vito is asking? Uh can the reverse be true? Can you have causation without correlation?

The answer is yes. Uh it is possible. In fact, it is probable in some cases where uh you have you know hidden data or missing data, things like that. So uh let's talk about a few of these situations, uh five of them. Uh the first is anytime you have a uh distribution of data that is not a normal distribution, uh, there may be something causal in it, but it may end up having uh uh a statistical correlation of zero.

So if you think about um your typical plots, right? You know, there's dots scattered all over the places, uh, or there may be a line of dots, or you can draw a line through the dots. Uh that's your typical Pearson correlation. If you have a shape, for example, of dots that looks like a big square, guess what? You have a statistical Pearson score of zero, even though there may be something very causal in that data.

You could have something that looks like a smiley face, right? Again, that would have a score of zero. Um, but you could there could very well be something causal happening there. So that's an example where uh you have non-normal distributions, and you still have a correlation, a mathematical correlation of zero, uh, even if those things are causal in nature. Uh you can have hidden data, hidden data that is uh unobserv the unobserved.

You didn't see it. Uh there may be a pathway to um the cause, but it's not but if you're used to measuring in stages, it may not make sense. So for example, let's say you have A, B, and C. Call them like I don't know, paths to conversion, your Google Analytics. Uh A leads to B, B leads to C.

And you may have correlations along those. There may be a D in there somewhere, right? Maybe A, D C, and then B has no correlation to D. You may end up having um a break in correlation, even though that fourth interfering factor there that you didn't measure or you didn't know about was playing a role. Uh that's where things like for example, especially with the analytics like propensity score modeling come in really handy to be able to tease out, oh, there's something else at play here.

Uh even if the the regression score is zero uh net across your chain of conversion, uh, there may be interfering factors along the way. A third way this can happen is when you have something that's unobservable that cannot be measured. Again, textbook example here. We know there are that gravity exists, right? There's there's no debate about this by anybody who has even a grade school education.

Um I'm sure there's some people out there who believe that because the earth is flat, there's no gravity, but they're idiots. Um gravity has no particle that we've been able to find yet in quantum physics. So even though we know it exists, it is causal, we cannot measure it, and therefore there is no correlation because there's you can't correlate something that you don't have data for. Um so that's an example that's very obvious. Oh, there's there's a cause, gravity, but there's no data to back it up.

Uh a fourth situation, one that happens a lot in marketing, uh, is when you have say A and B, uh, and you're looking for a relationship, um, and B has many, many, many, many causes. A may be causal, but very weakly causal and may not show a relationship, uh, especially if there's a lot of noise. Uh, again, in marketing, this is you see this a lot with attribution analysis. Uh what's the impact of Facebook, of Twitter, of email, of referrals, of direct traffic, of SEO, of SEM, all these different channels. And any one channel may have a very strong or weak relationship to the um the outcome that you're looking for, like conversions.

You may not be able to show a correlation between A and B, because there's like D that's just making all the noise. But that doesn't mean that A is not causal to B. It just means that you can't measure it because it's that you've got too much interference. Um then the last situation where this is likely to happen is when you have um causes that collide. Um again, the textbook example is here is uh things like uh illness, right?

Illness and mortality are are two variables, and and there may be a negative correlation there, and then or a positive correlation, then you may have treatment and mortality, and may have a negative correlation. The more treatment, the less mortality. And if you put those together, if you were grouping them together, you would get with a net of zero, right? Because they effectively cancel each other out. And so in that instance, your R is zero, but that doesn't mean that there's no causal relationship.

In fact, they you would have to break up the data to figure out that oh, actually, um the illness and mortality is positively correlated, the treatment and the mortality is negatively correlated, and you separate those two out. A and B and B and C should not be grouped together, because if you're if you're trying to measure uh illness and treatment and mortality together, yeah, you you they cancel each other out from correlation perspective. So those are uh five examples. Um and then they're just weird things. I I guess the weird things would be like stuff that would fall in the first category.

Like if you if you're trying to measure, for example, uh performance on a a test, an academic test, and you have like uh stress or fatigue or something like that, is again not gonna be a normal uh a normal linear distribution. Uh it could be all over the place, and you might not find a a mathematical relationship, even though there is a causal relationship. Like a little bit of stress for a test is good, motivates you to study. A lot of stress before test that keeps you up all night, not so good, right? Because you go into the test a zombie.

Um so there are there are instances where causation and correlation, mathematical correlation, uh do not line up. Um they are much less rare, and obviously the the case where correlation is not causation is much more common, but it does exist in cases where you've got pieces of data either missing or unobservable or uh lots of interference. So be aware of them. Uh a couple of the examples we talked about, like in attribution analysis are real problems that marketers may have to face, especially if you're doing more and more complex attribution models. Uh, you may need to use different techniques than just regression analysis if you've got a lot of either you know contributing causes or canceling causes.

Uh so being aware of how you're doing your computations is really important. So uh that's a set of answers. Uh interesting question, a tough one to to dig through. Hopefully you made sense. If you got follow up questions, leave them in the comments box below.

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