# Correlation does not imply causation; but what about when one of the variables is time?

I know this question has been asked a billion times, so, after looking online, I am fully convinced that correlation between 2 variables does not imply causation. In one of my stats lectures today, we had a guest lecture from a physicist, on the importance of statistical methods in physics. He said an astounding statement:

correlation does not imply causation, UNLESS one of the variables is time. So, if there is a strong correlation between some independent variable and time, then this implies causation as well.

I had never heard this statement before. Do physicists/relativists see "Causation" differently than stats people?

• This is a vague statement, and probably untrue. Time doesn't cause much of anything except radioactive decay. Vocabulary tends to improve with age but it is entirely mediated by socialization and education. Can you describe the context and problem in which this statement was asserted? – AdamO Jun 7 '17 at 18:43
• @AdamO The conditions for causality are simpler when you know temporal precedence, but they're not as simple as in this question. – Neil G Jun 7 '17 at 18:46
• It almost sounds like they are describing Granger causality. – Barker Jun 7 '17 at 21:04
• Just noting that if you really want to know how physicists see causality, you're more likely to get those answers on Physics. A modified version of this question could be on topic there. – David Z Jun 8 '17 at 17:14
• I've heard it said that adding time to a model as an independent variable just means you have not spent much time trying to model the data generating process producing your dependent variables. – Alexis Jun 9 '17 at 1:39

I'll provide another answer, since I think the ones currently provided miss an important point of the statement the physicist made. The quoted statement is:

"correlation does not imply causation, UNLESS one of the variables is time. So, if there is a strong correlation between some independent variable and time, then this implies causation as well."

The physicist is not saying:

"If X and Y are correlated, and X comes before Y, then the correlation implies causation."

That would be incorrect. What the physicist is saying is:

"If X and time are correlated, then that correlation implies that increasing time causes an increase (or decrease) in X."

An example might be entropy. If we have a strong correlation between time passing and entropy increasing, then we might say that increasing time causes an increase in entropy. Note that this ignores what the physical causes of the increasing entropy might be (particle decay, expanding universe, etc.).

One of the traditional requirements for causation is time progression, namely that X can only cause Y if X comes before Y. But if one of your variables IS time, then time progression is already built into the relationship (if a relationship exists).

EDIT: Based on a variety of comments, I'm going to add the following. I think that the physicist may be using a different idea of the word "causation" here. He seems to be saying that if there is a correlation between an independent variable and time, you can conclude that the independent variable changes predictably as time passes. Some people might say the changes are "caused" by time passing, this isn't really how statisticians use the words "cause" or "causation", so that may be causing some of the confusion.

• +1 Exactly, that's how I interpreted the statement as well (see my earlier comments & answer) – Ruben van Bergen Jun 7 '17 at 19:20
• If you're going to make time a variable in your graphical model, then time has no causes and is the cause of everything. It is therefore a vacuous claim to suggest that time causes any particular thing since time causes everything. – Neil G Jun 7 '17 at 19:23
• Vacuous or not, this is the interpretation that seems consistent with what the physicist purportedly said. Don't shoot the messenger ;). Also, I think it's a point worth making if the purpose is to educate people about the relationship between correlation and causation, even if you think it's trivial to actually consider time causing things in practice. – Ruben van Bergen Jun 7 '17 at 19:36
• @GeoMatt22 - I'd disagree with the "time causes everything" idea. Consider flipping a coin a bunch of times - even if I flip for hours, I should still get about a 1/2 ratio of heads, so time doesn't "cause" the probability of heads to go up or down. Put an ice cube out in a room and its temperature will rise and it will melt as time passes - time "causes" temperature equilibrium in this case. This may be a different sense of the word "cause" that statisticians use, but I think it's a functional interpretation from the perspective of physics. – Duncan Jun 7 '17 at 19:38
• The point is that you would never consider a graphical structure whereby any variable causes the passage of time. Therefore the only graphical structure is that time is the cause of all other variables. It may have absolutely no influence on them (as in your example), but the causal arrows are claims about the causal graphical structure, which implies conditional independence relationships given observations and interventions. The strength of influence is a separate question. – Neil G Jun 7 '17 at 20:07

We don't know what the physicist meant. Two different interpretations follow.

The claim that $X$ preceding $Y$ and being correlated with $Y$ implies that $X$ causes $Y$ is wrong. It's not enough for $X$ and $Y$ to be dependent even if $X$ precedes $Y$. For example, $X$ and $Y$ can both be caused by some other variable $W$: $X \leftarrow W \rightarrow Y$. Or, an even more complicated pattern could arise: $X \leftarrow V \rightarrow Z \leftarrow W \rightarrow Y$ where $Z$ is observed. Now $X$ and $Y$ are dependent and have no common cause, but neither causes the other.

However, temporal precedence greatly simplifies the conditions for asserting a causal relationship, which you can find in Pearl's Causality book Chapter 2.7 "Local criteria for causal relations".

A variable $X$ has a causal influence on $Y$ if there is a third variable $Z$ and a context $S$, both occurring before $X$, such that:

1. $(Z\; \not\perp\!\!\!\!\perp Y \mid S)$;
2. $(Z \perp\!\!\perp Y \mid S \cup X)$

Essentially, (1) implies that $Z$ is a potential cause of $Y$ given the temporal precedence, and (2) implies that $X$ is able to break that relationship, which can only happen if $X$ causes $Y$.

This condition is much simpler than Pearl's definition for a genuine cause without temporal information.

Another possibility outlined in some of the other answers is that the physicist meant that if $X$ is the passage of time and it's correlated with $Y$, then $X$ causes $Y$. This statement is correct, but vacuous since the passage of time is the cause of all other variables, by which I mean that the causal graphical structure is this way. A causal graphical structure is a set of claims about independence relationships given observations and interventions.

• As I mentioned in comments to GeoMatt22's answer, I don't think the physicist's statement has anything do to with precedence. – Ruben van Bergen Jun 7 '17 at 19:21
• @RubenvanBergen As I explained in another answer, that interpretation is vacuous. Time causes everything. – Neil G Jun 7 '17 at 19:24
• In your example $X \leftarrow V \rightarrow Z \leftarrow W \rightarrow Y$, $X$ and $Y$ would be dependent, but not correlated (unless $V$ and $W$ are correlated through a connection you didn't specify). – Ruben van Bergen Jun 7 '17 at 19:53
• @RubenvanBergen They could be correlated. It depends on the nature of the dependencies. By the way, I said $X$ and $Y$ are dependent given $Z$ observed. – Neil G Jun 7 '17 at 19:54
• @RubenvanBergen I think you're misunderstanding the arrows. These are causal arrows, and information can flow from $V$ to $W$ because of explaining away at $Z$. Consider $V$ is "Rain", $W$ is "Sprinkler is off", $Z$ is wet ground, $X$ is the sound of the rain, and $Y$ is an indicator for the sprinkler being off. Now given that the ground is wet, $X$ is correlated with $Y$ due to explaining away. – Neil G Jun 7 '17 at 20:01

I speculate that your guest lecturer meant that in physics the only correlations that survive replication are the ones where there is an underlying causal relationship. Time variable is an exception because it is the only variable that is not controlled by the physicist. Here's why.

In physics we usually deal with repeatable phenomena and experiments. As a matter of fact it's almost a given that any experiment is repeatable, and can be replicated by you at later time or by other researchers. So, let's say you observe a sample where $y_i,x_{ki}$ are observations of the variable of interest and independent variables $x_k$. As I mentioned above we fully control the variables $x_{k}$, and can set them to any value we wish.

Your physicist guy is saying that in this setup you will not see any correlation $Corr[y,x_{k}]$ unless there's a causal link. Why? Because someone else or even you yourself will repeat the experiment with any combination and sequence of $x_{kj}$, and only the correlations with causal relationships will survive the replications of an experiment. All other (spurious) correlations will disappear once you collect enough data in all possible combinations of an experiment.

This situation is in stark contrast with social sciences and some business application where you can't do experiments. You observe only one sequence of GDP of a country, and can't change the unemployment holding all else equal and observe the correlations.

Now, time is the only variable that a physicist cannot control. There's only one Jan 1 2017. He can't repeat this day. He can repeat any other variable, but not time. That's why when it comes to time (not lapsed time or age), a physicist is in the same boat as everyone else: correlation does not imply causation for him.

I have not heard this before, and it would not be true according to the conceptions of causality that I am familiar with (though I am not a physicist).

Typically, for $X$ to cause $Y$ it is necessary that $X$ precedes $Y$ in time. So if $Y$ precedes $X$ then it cannot be "caused" by $X$, regardless of any correlation. Moreover, $X$ preceding $Y$ is not a sufficient condition for causation (also regardless of any correlation).

• I think you misunderstand what this physicist meant. I think they were referring to a situation where two variables are correlated with each other, and one of these variables is time. You're assuming neither variable is time, but that where time comes in is that one variable precedes the other. – Ruben van Bergen Jun 7 '17 at 18:12
• I was trying to indicate that passage of time is typically required for some change in $Y$ to be "caused" by something, but a correlation of $Y_t$ vs. $t$ is not typically spoken of as "causality" (a $\Delta{t}$ is necessary but not sufficient). I meant to communicate that I do not know if this is what the physicist meant or not. I imagine a physicist would typically say "decrease in carbon 14 through time is caused by radioactive decay)" rather than "... caused by passage of time". (Though perhaps "requires passage of time".) – GeoMatt22 Jun 7 '17 at 18:35
• @RubenvanBergen perhaps the lecturer was trying to express a simplified version of something like what Wikipedia seems to call "causal structure"? Correlation with time (on sufficiently fine scales) would imply differentiability in the "time like direction". I may be misreading it, but skimming Wikipedia suggests a usage similar to what I wrote above: "causal structure" defines what "precedes" means. But it still seems like "necessary but not sufficient" to me. – GeoMatt22 Jun 7 '17 at 18:49
• I'm just going by the quote in the question: "correlation does not imply causation, UNLESS one of the variables is time. So, if there is a strong correlation between some independent variable and time, then this implies causation as well." To me this implies that we have some variable X which is correlated with time. We conclude that the passage of time causes X, rather than that X causes the passage of time, because the latter is nonsensical. – Ruben van Bergen Jun 7 '17 at 19:18

I don't think time is necessarily unique in this, but it's certainly a good example. The point is that typically if A & B are correlated, you can surmise that there is some common causality, but you don't know whether A causes B or B causes A, or perhaps a third variable C causes both A & B. However, in certain cases, you can rule out that any other variable caused A, and so it must be that A caused B. One such example is a controlled experiment, where you, the experimenter, control A. Then if the change you make in A "correlates" with a change in B, you know it must have been A that caused B to change, not the other way around.

Another type of scenario, which is the one this example with time falls into, is if you simply know that no other variable could have caused A because you know that nothing whatsoever can influence A. Since time just flows by one second at a time regardless of any other variable in the world, then if time correlates with changes in some variable your interested in (say, the number of people on the planet), you know for sure that the passage of time must have caused that variable to change, rather than your variable causing time to pass or otherwise change (i.e. time didn't go forward because more people were born, it has to be the other way around).

What you still don't know, of course, is whether the causality is direct. Presumably the passage of time itself doesn't automatically produce more human beings. Rather, history unfolding causes progress in various aspects of society, and this causes the population to increase in size (and even that is a simplification of many little causative relationships). But regardless of the precise factors at play, you do definitely know that A (ultimately) leads to B and not the other way around.

• In your first paragraph, your three cases aren't exhaustive. There are other graphical structures compatible with correlation. – Neil G Jun 7 '17 at 19:28
• The ruling out of all other variables causing $A$ is not possible for any realistic problem. There is a method whereby you rule out a flow of information via causes of $A$ that lead to $B$, which is called the back door method. This can establish causality. – Neil G Jun 7 '17 at 19:28
• As I said in another answer, the idea of interpreting "the passage of time" as variable and claiming that it must be the cause of some other variable is vacuous. This time variable is the cause of everything. – Neil G Jun 7 '17 at 19:29
• Broadly speaking, I'm fairly sure the options I listed are all the possibilities. We can either have A causing B or B causing A (directly or indirectly), or we can have something else causing both A & B. Of course combinations of these are also possible, where e.g. A has some causal effect on B, but at the same time a third factor C also is causally affecting both A & B. And then I guess there is coincidence as another option, but that's boring. But I'd be curious to learn of any other possibilities. – Ruben van Bergen Jun 7 '17 at 19:46
• Check out my answer. I illustrated a fourth case, although there are many more cases. – Neil G Jun 7 '17 at 19:48

Actually, correlation does imply a causal relationship.

Perhaps A caused B, or C caused A and B.

However, correlation does not prove causation.

This is self-evident.

• A look around the answers and comments suggests the conversation here has gone way beyond such trivialities. I recommend reviewing some of the posts as an aid to appreciating the issues. – whuber Jun 8 '17 at 22:26

I would interpret this as a semantic rather than mathematical/statistical argument. I would also take it as a rather severe generalization.

The Bradford Hill Criteria, often used in epidemiology, provide a good framework for thinking about causation. Nothing can definitively prove causation, whether or not time is a factor, and I suspect that the lecturer was not trying to make such a strong assertion. However, many different factors can be used as reasonable arguments for causation.

For example, the Bradford Hill criteria suggest that strength of association between variables can provide evidence for causation, but is not on its own sufficient. Similarly, an association that is consistent with other known/believed facts may suggest causation more strongly than an association that is inconsistent with prevailing knowledge. Temporality is also one of the criteria -- a cause should precede its effect. An association, and the inferences we make about causation, must make temporal sense. I recommend reviewing the other criteria. Some are specific to epidemiology and aren't as applicable to physics but it's still a useful way of thinking.

The main point is that, while no single piece of evidence is going to definitively prove causation, you can build a good case for it based on a number of different logical checks. I would argue that giving absolute precedence to any one criterion, such as time, is not appropriate, but temporality can be an important factor when making a case that causation is plausible.

This leads to a broader point about statistics: generally speaking, we use statistics to make an argument. We use data and statistical tools to make a certain point. Often, the same data (and even the same tools) can be used to make conflicting points. We can't locate the definitive proof of causation in the math itself, but we can deploy our statistical tools as part of a broader argument. For more on that, I recommend Abelson's Statistics as Principled Argument.

To loop this back to the original situation, let's say you've done an experiment about the effect of a the concentration of a certain chemical in a solution on the temperature of that solution. You suspect that adding more of this chemical will result in a reaction that increases temperature. You add more gradually over time. You can look at temperature against time and see an increase. All this shows is that temperature is increasing with time; it doesn't prove that time itself (or anything else, for that matter) has some causal effect. It does, however, provide some evidence in a broader argument that increased concentration of this chemical results in a reaction that increases temperature.

• It's a peculiar thought to apply the Hill criterion of "temporal precedence" to an exposure of time itself. Certainly time preceded time itself. Trends as we know are rarely causal but reflect other simultaneous phenomenon. In this example, I don't think time caused anything, but summarized global shifts in settings which happened to affect both exposure and outcome. – AdamO Jun 7 '17 at 20:31
• I'm not arguing that we apply the argument to time itself so much as saying that, if we have time as part of our data, we can use that to make one part of a broader argument for causation. By demonstrating that our observations make temporal sense, we're closer to having a reasonable causal argument. Hopefully, we would have considerably more than that to work with in order to create a stronger argument. – djlid Jun 8 '17 at 11:27

The sentence is quite simple and not worth overthinking (and has nothing to do with precedence).

If there is an established correlation between a variable and time (i.e. we know that an increase in time is accompanied by an increase in the variable, and this is a given), then we know the "causal" direction: i.e. time increasing, causes the variable to increase.

Because the alternative hypothesis of "nah-uh, it could be that time only increased because the variable increased first" simply cannot stand given the way time works.

This might sound like a silly observation, but it has important implications for study design trying to prove a causal direction. An important example in medicine is the difference between doing a cross-sectional and a cohort study.

E.g., a cross-sectional study trying to find a link between smoking and cancer might take a group of people, divide it into smokers vs non-smokers, and see how many in each group have cancer vs no-cancer. However, this is weak evidence because a correlation between smoking and cancer could also be interpreted as "people who have cancer are more likely to enjoy taking up smoking".

However, if you perform a cohort study, i.e. take a group of smokers and a group of non-smokers, and follow them up through time, and measure the variable "cancer in smokers minus cancer in non-smokers", and establish a positive correlation of this variable with time, (under reasonable assumptions, such that smoking amount once started is constant and independent of time etc) then you know that "time" is the cause of the cancer difference, since you cannot claim that increased rates of cancer caused time to pass more in the smoking group. Therefore you can claim a causation between time passing and a positive cancer difference related to higher rates in the smoker group. (or, more simply stated, time spent belonging to the smoking group causes a proportional increase in cancer risk).

Furthermore, the weakness of the cross-sectional study, i.e. the possibility that "people with cancer are more likely to take up smoking" has now gone out the window, since smoking as a variable has been taken out of the "time vs cancer" equation (here assumed to be constant and therefore unaffected by time). In other words, by formulating the study in this way, we have examined a very specific causal direction. If we wanted to examine the extent to which the reverse causal direction applies (i.e. how likely it is that people who will eventually get cancer are to take up smoking as time goes by), then we would necessarily have to design a cohort study split into "future cancer vs no-future cancer" and measure the uptake of smoking over time.

Note that this is a discussion over a causal direction rather than one of finding a direct causal link. The question of confounding is a separate one. (i.e. there is nothing to suggest that there isn't an independent third variable that both makes you more likely to be a smoker and increases your chances of cancer with time). I.e., in terms of counterfactual causality, we have not definitively shown that "had it not been for smoking these people would not have gotten cancer". But we have shown that "the association between smoking group and cancer would not have increased had time not passed". (i.e. the association is not down to a snapshot of cancer sufferers mere preference for being in the smoking group or not, but is stengthened over time).

• "Therefore you can claim a causation between time passing and more cancer developing by virtue of being a smoker. (or, more simply stated, time spent smoking causes a proportional increase in cancer risk)." — No, you can't do that. The cigarette companies backed by Sir Ronald Fisher (!) argued for years that genetic predisposition was a potential common cause of smoking and cancer. This very example is in the back of Pearl's book (p. 353). – Neil G Jun 7 '17 at 20:56
• @NeilG no, I stand by my example as it is formulated. The point you're making is not one of reverse causality, but of confounding. My example, as it stands, shows that time spent in the smoking group is associated with an increase in cancer rates. Bu this in itself does not prove that "genetic predisposition" isn't the driving force behind the increased rates in the smoking group. Two different things. The point here is that introducing the causal direction as a time variable eradicates the "reverse causality" argument (i.e. cancer makes you take up smoking), but not the "confounding" one. – Tasos Papastylianou Jun 7 '17 at 21:19
• Your comment is correct, but it doesn't seem consistent with what you wrote. You wrote that "time spent smoking causes a proportional increase in cancer risk". That's unjustified. – Neil G Jun 7 '17 at 21:29
• @NeilG fair enough, you're right. I didn't expect such scrutiny, hahah. I'll rephrase a bit more accurately. – Tasos Papastylianou Jun 7 '17 at 21:36
• Why is it necessarily the case that the advancement of time is uncaused? How can we so conclusively rule out the possibility that there is something that causes time to advance? That seems to me to be a most extraordinary claim that would require exceptionally strong evidence. – David Schwartz Jun 7 '17 at 21:52

This is really a question of how to establish causality, because events which are related but not causative WILL likely be correlated in time or space. So looking at some correlated data, how can we determine if the relationship is dependent? A wise research adviser once told me, "correlation does not imply causation, it just tells you where to look".

Let's consider the situation where events A and B are found to be temporally or spatially correlated. If we would like to investigate the preposition that A causes B, the traditional line of thinking is to introduce tests of necessity and sufficiency -- which is what causality really means.

• If the absence of event A leads to the absence of event B, it can be called necessary.
• If only event A alone leads to event B, it can be called sufficient.

If not having milk causes me to go to the store, what we are saying is not that I get into my empty milk and drive. Absolute causality would mean that whenever I still have milk, I can't be bothered to go to the store; and conversely whenever I'm at the store, it's because I don't have any milk. Now it's easy to see the problem with positively establishing causality in the rigorous sense: most things are not absolutely causal. There's lots of other reasons I might go to the store which aren't related to the milk-state.

This is an easy way to tell a great paper from an alright paper. In careful research, you will see sufficiency and necessity tests everywhere. Making the claim that small-molecule drug A may lead to disassembly of protein complex B? You will immediately see the tests:

necessity  ----test---- ----result---- everything but B --> [nothing] (check for false positive) everything but A --> assembled everything with A-like compound --> assembled (control group) 

sufficiency  A + B alone (in vitro) --> disassembled (check for false negative) A + B + everything --> disassembled (trial group) 

This is the traditional way you would build an inductive argument for causality experimentally USING correlation, which is what I am confident your lecturer was eluding to!

• Just to emphasize the point, time dependence doesn't imply causality either. We may have that event A leads to event B often and it is B that causes C and not A. Yet A will be correlated with C but not cause it. – Michael R. Chernick Jun 7 '17 at 22:46