How to Think about Correlation?
statmodeling.stat.columbia.edu
statmodeling.stat.columbia.edu
When observing the universe, humans can never prove any facts about the universe. We can only establish correlations. Correlations between events that occur in our universe is the furthest "truth" we can establish about the universe short of a full on proof.
What this means is that nothing in the physical universe can be proven. Proof is the domain of maths and logic, correlations is the domain of science. Science cannot prove anything, it can only establish correlations and causations.
The reason this occurs is because at any time in the future one can observe an event that contradicts a hypothesis. You can hypothesize that all birds have wings and observe 2 trillion birds with wings but you never know when one day you'll observe a bird without wings disproving your entire hypothesis. That is why nothing can be proven, you can only correlate things through observation.
The other interesting part about correlation is what it isn't: Causation. People often talk about how correlation is not causation but people never talk about what causation is and how to establish it. If I can't empirically use correlation to establish causation how on earth is causation ever formally established? People rarely question this disconnect.
The fact is, causation is rarely formally established but a method does exist and it's subtle. If I observe that whenever Bob flicks a switch the light comes on then I established that the light coming on is correlated with Bob flipping the switch. This is as far as I can go with just observation. To establish causation I must make myself both an observer and an entity that is part of the system itself. I have to take control and flip the switch randomly and observe that when I don't flip the switch nothing happens and when I do flip the switch the lights come on.
By doing this I establish causation. To establish causation to higher and higher degrees I need to Cause (keyword) random events and make sure that a cause influences an effect AND absence of a cause and therefore absence of an effect occurs.
Also note that establishing causation is not proof. At any point in time in the future I can flip the switch and the light may not come on which is contradictory evidence for causation. Causation in the statistical sense is like correlation, you establish it to a degree of confidence but you can never Prove that A caused B.
For example, suppose we wish to test the hypothesis that smoking causes lung cancer via the main mechanism of tar buildup in the lungs, against the alternative hypothesis that smoking is correlated with lung cancer because of a gene that predisposes people to both smoking and lung cancer (this example comes from Judea Pearl’s Book of Why).
If the former hypothesis is true, then we should see a correlation between smoking behavior and tar deposits, and we should also see a correlation between tar deposits and lung cancer even after controlling for smoking behavior. Composing the causal effects at each stage, we can then calculate the indirect causal effect of smoking on lung cancer. If this suffixes to explain the correlation, then that rules out the alternative genetic explanation for the correlation.
Of course, we can always propose ad-hoc further hypotheses that complicate the analysis: maybe the correlation between smoking and tar deposits is itself non-causative, etc. This only goes to show that scientific inquiry must be done with judgment and with expert domain knowledge, testing plausible hypotheses in good faith. But that’s true for detecting mere correlations too — we can always doubt our instruments, or claim the data is a statistical fluke. And it’s true in randomized controlled trials: it’s conceivable we didn’t properly randomize, etc.
It also ignores the power of multimodal measurements where the same or other factors are measured with independent means.
One of the most powerful assumptions is simply assuming no time travel is involved: my favorite example is post partum syndrome / suicides. If one creates a plot with a horizontal axis of relative time bins and a vertical axis of event counts then let the relative time of 0 represent time of giving birth, and for each mother that commits suicide as a function of say weeks since birth you increment the corresponding bin. Integrated over maternal suicides we observe that there is a background suicide rate (due to other reasons) and a childbirth related peak that falls off over time. This means that something(s) related to the time of childbirth is traumatizing. Obviously a maternal suicide does not cause childbirth a few weeks prior (assuming causation can not occur in reverse order), so while causation is hard to prove, it can be feasible to exclude hypothetical causation pathways. This still leaves the possibility of a prior event causing both: say the day of getting pregnant but this does not explain the maternal suicides of mothers that had shorter or longer duration pregnancies! The peak is more concentrated with respect to days since childbirth compared a broader smeared peak when plotted with respect to days since fertilization. The fact one can blind oneself of timing information, obviously frustrating cauation analysis, should not be construed as an impossibility proof to gradually tease out and map causation pathways.
One of the characteristics of post partum trauma is "hallucinatory conceptions" etcetera. I find it very suspicious that I never find a paper documenting these prolonged "figments of imagination" into classes, with examples on each. Except for the rarely abused class of deliriates the other drugs (LSD, THC, ...) don't cause prolonged concrete hallucinations. Hence I do not believe any endogenous hormonal imbalance is causing these "strange ideas" of a fraction of the women that recently gave child birth. Such a trait would obviously be selected against and disappear. More likely is that a fraction of them experience one or more common types of trauma during child birth. This could be many things: cutting the genitalia, caesarian section, ... but I believe one of the main factors is the female orgasm during childbirth, the contractions to squeeze the baby out of the vagina. If a person is unaware of this fact and then becomes aware during child birth, this feels like lack of consent and thus rape (with obvious suicide statistics of its own), betrayal, the discovery of prior taboo and censorship, and incomprehension why the information doesn't flow back to the medical (and school) system in order to adapt and better prepare future mothers of this natural function of the female orgasm. Instead they get assigned a shrink who shrugs off their frustration as "figments of their imagination", and openly document it as such in the literature. And then we collectively act surprised when some of them commit suicide? How many children must lose their mother, how many fathers must lose their wife, how family members must lose a relative, ... before we realize that this is too high a cost for the implicitly supposed "benefit" of "in sanity we trust"?
The original comment does have its merit in cautioning against causation analysis without any timing information whatsoever (sadly one of the most common analysis techniques), like correlatings between classes of people (smoker / non smoker) X (healthy / cancer).
But I think it would be more constructive for these run of the mill "correlation is not causation" rants to point out the value of multimodal measurement: for example measuring tar in your example, or measuring timing information as in the example I gave.
Like you said, you have to assume smoking causes the tar in addition to assuming that a correlation between tar and cancer indicates causation.
I feel that directly controlling the causative factor is the only way to verify the chain of causation.
(1) smoking → tar → cancer
(2) smoking ← genes → cancer
Note that (1) contains the sub-hypothesis (A) smoking → tar, which is also compatible with (2).
The super-graph containing both graphs is:
————genes————
↓ ↓In that super-graph, there is only one possible explanation for a correlation between smoking and tar: smoking causing tar. So if we see that correlation, then barring alternative hypotheses, we know the causation smoking → tar is true. (Again, we could always hypothesize other causal explanations, but we're presuming no one has yet submitted a plausible alternative explanation.)
Now once smoking → tar is established, we are still left with two possible explanations for the observed correlation between smoking and cancer:
(1) smoking → tar → cancer
(2+A) tar ← smoking ← genes → cancer
This is where controlling for smoking behavior comes in. Once we do that, we've severed the indirect statistical link between tar and cancer in graph (2+A). Thus if a correlation still persists between tar and cancer after this statistical control, we've ruled out (2+A), leaving only (1).
By controlling the causative factor no new hypothesis can undermine an established causation assuming that there are no side effects when we trigger a causation event. Think about it.
This means that causation is actually established versus the other method which is a flimsier verification. Another way to look at it is that rather then introducing a new "hypothesis" you had to introduce assumptions about causation.
You're thinking in terms of math where you can control your primitives and set a domain and range. In science none of these can be fully controlled and a "new hypothesis" does indeed apply.
Think about it. If I introduce a new hypothesis in the real world that actually lends good evidence in support of say the cancer outcome I cannot just discount it and say it's not good science. You can only restrict your universe like this in the world of math and logic, in the real world you have to except any reasonable evidence that comes your way.
In practice however, most scientists make a bunch of assumptions and will likely test things using your method because it's just 100x easier and more realistic in terms of setting up an experiment that's possible.
Also I mentioned in another comment that your experiment does not actually rule out the genetic factor. Both genetics and smoking and tar could all logically be causative factors for cancer. If the previous statement was true all your correlations established in your post would be exactly the same... there were many assumptions made. So in short your experimental evidence lends support for multiple conclusions that cannot be discounted.
Good science isn't restricting the entire universe to two possible hypothesis. Think about it logically. What scientist can publish a paper that says that genes don't predispose anyone to cancer or smoking because he just observed smoking correlates with tar?
You literally say smoking must cause tar because it's the only possibility in your two hypothesis... one must occur. Sure if this was math and you defined your universe to be one of two possibilities... but no person in the real world will take that seriously.
In math you don't introduce ad-hoc axioms and rules to your formal analysis, but the scientific method exists because all observation in the real universe is ad-hoc. Anything goes and that's why the best way to make sense of it is through correlation.
Which brings me full circle back to my original method. Can you introduce any "adhoc hypothesis" to render a causation established by directly controlling the causative event invalid? You can't. Therefore the actual formal scientific way to determining causation is by placing yourself in the system.
The only way the graph method can be applied to the real world is if you map out every possible scenario and assumption into that graph. Then you can say based off of what we know and assume A must cause B. But more often then not a scientist is likely just establishing some correlations and coming to a qualitative conclusion that takes into account all the assumptions and hypothesis that we're aware about in the real world.
Of course this is a simplified example to illustrate the causal logic at play. No one seriously believes there are only ever two possible hypotheses. But the point is the universe of plausible hypotheses is not infinite, either. Scientists in the real world are not dealing with the set of logically possible statements consistent with raw sensory experience. They're building models of the world, and testing and challenging a few of their beliefs at a time, to gradually come closer to the truth.
When they can conduct randomized controlled trials, that's great. But it's not always feasible or even possible, yet it's still often possible in those cases to test competing causal hypotheses via observation. The Eddington experiment tested the (causal) theory of General Relativity against the (causal) theory of Newtonian gravity, but there was no intervention. The discovery of iridium in a giant crater off the coast of Mexico was a key confirmation for the (causal) dinosaur meteor extinction theory, but there was no intervention there either. These were causal hypotheses tested by observation. A scientist is always free to propose an alternative hypothesis, but in the mean time, these were tests of their predictions.
The point is RCTs are a special case of a more general causal logic. RCTs work because they sever the statistical link to confounders, but only provided your assumptions about the RCTs are correct. In particular, your random number generator must not itself be confounded with what you're measuring (how do you know there's not a hidden variable simultaneously controlling your coin flip and the outcome of your experiment?), you must do the observations double-blinded, your results must not be a statistical fluke, etc. etc. All of these assumptions can be challenged, just like the assumptions in my smoking/lung cancer example. In the real world scientists often accept, at least tentatively, that these assumptions hold. For if they don't, they can never learn anything.
Absolutely. Science is about using mathematics and logic to abductively reason about and tease out underlying facts. In fact I see provaility in science and causation as orthogonal issues:
Consider a stable supplied voltage and a variable voltage divider (potentiometer). Suppose measuring the voltage once has intrinsic RMS voltage noise associated with it. So while we can never prove experimentally the exact value of the voltage at the voltage divider with zero error, we can drive the error arbitrarily low by oversampling, which means repeating the experiment over and over. In the general case we never actually reach the true value, but that doesn't mean "voltage can not be measured". Trying to measure this voltage inappropriately (say outside the reference range of an ADC, or with an ADC with too low a resolution such that the noise is systematically rounded in the same direction doesn't mean it's impossible to measure voltage, just that the measurement attempt is flawed.
Similar with causation. Of course caution is always advised with naive correlation. The problem is not propagation of such caution, but the pretense that no techniques exist to gather evidence of causation, and the omission of such techniques when sharing such a rant.
Sure, this operates on the assumption that all potential subsequent experiments that you could have conducted after your final observation also falls within your expected parameters.
You did 500 experiments that fit within your hypothesis but the next 10 trillion experiments you didn't do afterwords could have shown wildly different results rendering your 500 experiments as simply "noise."
This is a possibility and illustrates the flaw within the scientific method itself, it is however the best that we have, so we use the scientific method and its associated assumptions. Keep in mind the formal result of science has this flaw embedded within the correlation or causation number. However the qualitative conclusion is littered with assumptions.
Probably the best way to describe our different view points is optimism vs. pessimism. That's really it. I'm saying it's flawed but it's the best we have, you're saying it's been remarkably successful "so far."
You have to mention this explicitly. Because it's also presumed not controversial that genetics can predispose someone to cancer. Additionally if you restrict your universe to two hypothesis but then suddenly I can pick up adhoc "red herrings"/assumptions like this anywhere seems inconsistent with your point.
>But the point is the universe of plausible hypotheses is not infinite, either. Scientists in the real world are not dealing with the set of logically possible statements consistent with raw sensory experience. They're building models of the world, and testing and challenging a few of their beliefs at a time, to gradually come closer to the truth.
This is factually wrong. We don't know whether plausible hypothesis are finite or infinite. This is the very reason why nothing can actually be proven. The entire reason why nothing can be proven with science is because we can't specify a range and domain for the universe because we simply don't know.
Formally in the scientific world your method is flawed. This is what I'm saying. In the mathematical world your assumptions and logic can hold but not in science. I get that your toy example was used for illustration purposes but it's failure to be applicable to the real world is also an illustration of the flaw behind the method itself.
I can bring it back to the world of axioms and theorems so . Imagine the real world as a model where there the domain and range is infinite in size and entity types. The only thing that can be assumed is probability and logic. That is what the scientific method is operating in. Therefore your methods have to account for an infinite amount of hypothesis rendering your assumptions and deductions impossible without caveating your conclusion with a limiting assumption.
In the logical model above the best you can do to establish causality it by placing yourself into the system. Any other method requires assumptions (which is commonly done).
>When they can conduct randomized controlled trials, that's great. But it's not always feasible or even possible, yet it's still often possible in those cases to test competing causal hypotheses via observation. The Eddington experiment tested the (causal) theory of General Relativity against the (causal) theory of Newtonian gravity, but there was no intervention. The discovery of iridium in a giant crater off the coast of Mexico was a key confirmation for the (causal) dinosaur meteor extinction theory, but there was no intervention there either. These were causal hypotheses tested by observation.
Of course. I'm saying we have limited tools so we tend to jump to conclusions and make assumptions based off of our intuition and understanding of probable outcomes. But this does not discount the fact that these assumptions were made.
I am saying that the ONLY way to determine causality to a degree in a universe where the domain and codomain is unknowable is by placing your self into the system and making yourself the causal factor. But this is not how we operate in practice. Clearly many scientists assume causality based only off of correlating evidence do in no part to, like you said, how hard it is to actually create these experiments.
>The point is RCTs are a special case of a more general causal logic. RCTs work because they sever the statistical link to confounders, but only provided your assumptions about the RCTs are correct.
Ok we're probably on the same page then. This is exactly what I'm saying.
All of our science is founded off of a few foundational axioms/assumptions. The recursive assumption that logic is true, the assumption probability is true and maybe a few more intuitive adhoc foundational assumptions that the random number generator is random and that influencing the causal event does not produce side effects. The scientific method is foundational because it operates within these foundational assumptions and so does determining causality by placing yourself into the system.
The "RCT" thing you introduced is not foundational because you must create new assumptions outside of the foundational ones we use for all of science. You make up stuff. Which is fine, but it's no different then using correlation to imply causality. It's the same type of "assumption." We're so against the term correlation does not imply causation but these and similar leaps of assumptions are common place, often done and allow us to come to conclusions without actually proving anything.
I don't think you've understood my example. We may have philosophical disagreements left over even once you've understood my example. But so far you still don't seem to understand it. For example, you say:
You have to mention [that the smoking → tar causation is presumed not controversial] explicitly. Because it's also presumed not controversial that genetics can predispose someone to cancer.
Right, of course it's not controversial that genetics can predispose someone to cancer. That's the whole point. That's why it's among the two hypotheses being tested. It is ruled out (under the assumptions of the causal super-graph, which includes that hypothesis and an alternative hypothesis as the world of possibilities) by the statistical data. Please try to understand this point and acknowledge that it is the case.
You may question the usefulness of this model of science. But so far you don't seem to understand what I'm even saying with this.
This is factually wrong. We don't know whether plausible hypothesis are finite or infinite.
"Plausible", as I'm using it, is a partially subjective term: plausible in the view of subject matter experts. As you're using it, it appears to be an objective term, meaning something akin to "logically possible", perhaps with some ill-defined notion of probability (but what are your priors?). I'm talking about real scientists in the real world. You seem to be talking about some idealized scientist which doesn't, and can't, exist. General Relativity was not plausible in the 17th century. It was not even conceived of then. It became plausible in the 20th century. What's the plausible (as judged by actual scientists, not in some hypothetical abstract logic) alternative hypothesis to the meteor killing the dinosaurs? There is none, so far as I know. The set of hypotheses under consideration changes over time due to the imagination and skepticism of real flesh and blood scientists, not due to some Platonic logic which no scientist can follow.
Formally in the scientific world your method is flawed. This is what I'm saying. In the mathematical world your assumptions and logic can hold but not in science. I get that your toy example was used for illustration purposes but it's failure to be applicable to the real world is also an illustration of the flaw behind the method itself.
… The only thing that can be assumed is probability and logic. That is what the scientific method is operating in.
To be quite frank, I think it is your model which fails to be applicable to the real world. You'll be hard-pressed to find a single practicing scientist who sees themself as a logic/probability machine. In the real world we need to operate, provisionally, with certain assumptions about how the world works, or else we can understand nothing. Those assumptions can always be challenged. But they are held, provisionally, so that we can make progress in our understanding.
>The point is RCTs are a special case of a more general causal logic. RCTs work because they sever the statistical link to confounders, but only provided your assumptions about the RCTs are correct.
Ok we're probably on the same page then. This is exactly what I'm saying.
No, my whole point is that RCTs are just one special case of the more general theory of causal inference, and that more general theory can be applied elsewhere too. The assumptions underlying an RCT are not exceptional, and like everything else in science, they can be challenged.
If you have a population of 1000 people come in and take pills A or B, as determined by a random number generator, and then the ones who took pill A survive and the ones who took pill B die, the RCT says B killed them. But perhaps unbeknownst to you, there was a source of gamma rays due to an experiment in the next room which was simultaneously flipping bits on your random number generator and also irradiating your subjects with lethal doses. The gamma rays would then be the confounder.
"Science" does not rule out this possibility in theory. But in practice, it may be implausible to the experts. (Remember: plausibility is partially subjective, a function of the current state of scientific understanding! It can change over time!) So they are not worried about it when interpreting the results of the RCT.
Similarly, the universe of logically possible causal explanations consists mostly of hypotheses that no scientist currently takes seriously (i.e. considers plausible). Of the ones on the table, so to speak, the predictions can be tested to leave only one causal explanation (for the time being) standing.
Let's clarify something here. I never said it's not useful. I have a very clear definition of science, a very concrete model of what is should be. Usefulness is not what I'm talking about here, I'm more talking about the true nature of science and causation rather then how useful it is to do science or perform even your methodological technique of commuting a dependency graph (which I agree can be useful at times). But continue, I will reiterate your points to prove to you that I completely understand and that I always understood from the beginning.
>"Plausible", as I'm using it, is a partially subjective term: plausible in the view of subject matter experts. As you're using it, it appears to be an objective term, meaning something akin to "logically possible",
So let me describe what I'm thinking when you say "plausible hypothesis." I am thinking your talking about a some predicate statement that can either be true or false that is reasonable to predict or state due to your intuitive sense of how likely that hypothesis is to occur. I totally get it.
>I'm talking about real scientists in the real world. You seem to be talking about some idealized scientist which doesn't, and can't, exist. General Relativity was not plausible in the 17th century.
The irony here is that your initial example about smoking isn't even real world, it's highly idealized. You assume a fixed set of rules and hypotheses in your universe. I claimed that your idealized example can't function in the real world but your counter claim is that because the real world is imperfect your commuting model of drawing a dependency graph is valid because people in the "real" world make assumptions. Do you see the inconsistency here? Fear not, despite the inconsistency, I get your point and I will address it.
>What's the plausible (as judged by actual scientists, not in some hypothetical abstract logic) alternative hypothesis to the meteor killing the dinosaurs? There is none, so far as I know.
I assume you're saying this to point out how there's only a single "plausible hypothesis" about how the dinos became extinct, correct? I assume yes.
If so then this is my response: There was a period in time where a geocentric universe was the only hypothesis available with no alternative model in existence. There was a period of time where creationism was the only possible hypothesis for the origin of life. There was a period of time where a flat earth was the only plausible hypothesis about the world as we know it. These are all hypothesis that were at one time the sole single "plausible hypothesis" that was overturned at a later date.
"Plausibility" is usually defined relative to the amount of knowledge humans have rather then the total knowledge (aka mathematical domain) of the universe. So if humans have no other evidence about the spherical nature of the globe then a flat earth becomes a plausible hypothesis. But because the universe is not fully discovered or known the more accurate definition of "total knowledge of the universe" the right definition.
Because we currently have no way of knowing all the knowledge in the universe there is know true way to accurately gauge how plausible a hypothesis is. Because like the flat earth, like geocentrism, like creationism there is an infinite amount of models and hypothesis that can be discovered at any point in time that will render your current hypothesis, implausible.
Of course let me point out that I'm not saying that this does not make all of science not useful. It's the best tool we have. We can use it and assume the world is flat until we know better. My point in this case is to acknowledge your point and point out yes it's useful but here is the flaw and the problems with it.
>To be quite frank, I think it is your model which fails to be applicable to the real world. You'll be hard-pressed to find a single practicing scientist who sees themself as a logic/probability machine.
I think you're misunderstanding here. Statistics is founded on probability, the very fact that statistics is used in all of science indicates that all scientists assume probability to be true. I think you don't really understand the nature of probability.
Probability is just an axiomatic set of statements and theorems and formulas. You plug in a probability into this function and you get another numerical representation that we call probabilities.
By sheer coincidence probability happens to work in reality. If there is a raffle with 10 billion tickets where 10% of the tickets have my name on it, then when I draw from that raffle 1 million times around 10% of those tickets will have my name. The larger the numbers you work with the more and more reality seems to represent probability. Nobody to this day knows why this things are consistently like this. Currently we all just take this as an assumption and assume the mathematics of probability apply to the real world.
The same goes for logic. We assume logic is true, I call it a recursive statement because saying something is true is part of logic itself. These are assumptions that are the foundation of how we interpret reality as we know it.
>Those assumptions can always be challenged. But they are held, provisionally, so that we can make progress in our understanding.
Your saying this because you misunderstood me! not I you. I am saying 100% that we have to make foundational assumptions about our universe like probability and logic.
But I get where you're coming from. Your talking about adhoc assumptions like the sun in the sky exists and I don't have to use formal methodology to prove it because it's ludicrous. I get it. What I am saying goes deeper than this.
I am saying that any for any formal scientific method to truly establish causality should not need to include adhoc hypothesis and assumptions. It should as much as possible only include foundational axioms like probability and logic... Why? Because every additional assumption can be questioned as to whether it's true or not true.
I can hear what you're thinking. There's just a ton of obvious stuff in the universe that we can just assume to be true. For example the sun exists in the sky and the keyboard under my fingers exists. It's stupid to not use these assumptions in our "science." I would say all the hypothesis I mentioned fall under the category of "plausible hypothesis" the same category where the flat earth theory once belonged.
That is MY point. The scientific method itself is defined requiring only two assumptions: probability and logic is real. Your method is defined with the requirement of an adhoc amount of assumptions about the universe. It is not foundational. It is merely logical conjecture in a game of axioms and theorems.
>No, my whole point is that RCTs are just one special case of the more general theory of causal inference, and that more general theory can be applied elsewhere too. The assumptions underlying an RCT are not exceptional, and like everything else in science, they can be challenged.
Special case indeed, I wouldn't even call it a special case as it's more like a logic puzzle or a game where the entities are well defined. Rarely applicable to a universe composed of an infinite amount of possibilities.
>If you have a population of 1000 people come in and take pills A or B, as determined by a random number generator, and then the ones who took pill A survive and the ones who took pill B die, the RCT says B killed them. But perhaps unbeknownst to you, there was a sou.....
>"Science" does not rule out this possibility in theory. But in practice, it may be implausi....
Sure totally agreed. However in this experiment No causality was established! only correlation. The statement that correlation doesn't imply causation still holds. However you are assuming here that correlation implies causation.
Like I said earlier for the causal experiment truly be causal we must conduct the experiment by creating a cause with no side effects. Since it's basically impossible to confirm that your causal switch has no side effects are closest possible methodology is to account for it then assume no side effect occurs. Keep in mind, I mentioned that this assumption was a "foundational assumption" like probability and logic, this assumption is not the same as the adhoc assumptions you are bringing up to make your logic work correctly.
If this was an actual experiment this foundational assumption of no side effects would be an implicit part of the conclusion of the experiment: "Assuming there were no side effects when triggering the causal event, Pill B causes the person who eats it to die."
>Similarly, the universe of logically possible causal explanations consists mostly of hypotheses that no scientist currently takes seriously (i.e. considers plausible).
Right but the scientific method itself just uses foundational assumptions just like my controlling the causal event. In practice you take these foundational techniques and you adjust when needed. My argument and point was to say that your methodology is too adhoc and requires too many assumptions to be placed side by side with the scientific method.
Hopefully that makes it more clear because you said earlier that you were describing a "special case" which is exactly what I'm saying about your methodology. It's describing a special case that in actually you have to assume causality of other factors in order to assume causality itself, it's more like inference rather then building up the axiomatic components of causality from the ground up.
(Which is not to dismiss your point about their being a kind of savagery and philosophical philistinism among scientists. This was certainly Feyerabend's view.)
I believe this is where most of the disdain comes from. Yes philosophy has a lot to say about logic and science but then you get stuff like the philosophy of religion, education and of art which are purely human concepts unique to us.... similar to how the mating habits of the chimpanzee are unique to the chimpanzee it's easy to see a category error with philosophy.
There is a dichotomy here and philosophy does not respect it, science and logic apply to both the human and the chimpanzee (whether the chimpanzee chooses to understand it or not) but art and religion and mating behavior do not cross this divide and is unique to each species. This is the main issue with philosophy, by placing the philosophy of religion on the same level as the philosophy of logic it is saying that human centric concepts are no different then universal concepts and that humans are the center of the universe.
Rodgers & Nicewander 1988 "Thirteen ways to look at the correlation coefficient" https://www.jstor.org/stable/pdf/2685263.pdf
The only moving thing
Was the independent variable.
Correlations are meaningful because they indicate how strongly variables can predict other variables, in some circumstances. This is valuable because some variables are easier to use than others.
A positive correlation between two variables means they tend to move in the same direction (relative to their mean).
A negative correlation between two variables means they tend to move in opposite directions.
A strong correlation between two variables means that they tend to change in much the same kinds of relative magnitudes, and also that either variables can be used to generally predict the other.
A weak correlation between two variables means that the two variables tend to change in somewhat the same kinds of relative magnitudes, sometimes with some kinds of exceptions, and also that either variable can be used to somewhat predict the other.
Thus a strong positive correlation means that variables can predict each other effectively, and also move in the same direction. This is especially valuable for projects where one variable is easy to obtain, yet the other variable is hard to obtain.
And it turns out that a strong positive correlation can be a good place to hunt for possible causality, such as hunting for something that causing both variables to change together. However, correlation doesn't imply causation.
A strong positive correlation can be good evidence that a hypothesized causality is likely to be true. For example an experiment can test if X causes Y, by varying X, and discovering that X and Y tend to change together.
Edit: the replies below are great-- thank you-- well worth reading. I'm editing my post to make it clearer that it's a simplified answer, and to add suggestions by the replies.
> Corrleations are meaningful because they indicate how strongly variables can predict other variables.
This is not wrong per se, but may be wrongly interpreted.
1) I think it is important to stress that the reverse is not true. If a correlation is 0, it does not mean the variables are independent. It means they are linearly independent. There are a lot of things out there which are very much linked but not with a linear relationship.
2) This is only true if the correlation was done properly. For instance, correlating non-stationary variables will spuriously yield a high correlation, giving a false impression of link. The classic example is trying to correlate stock prices between themselves. An other example of spurious correlation is when not enough points are given to get an accurate result. Looking at tstat, r2, etc. is as important as looking at the actual correlation coefficient.
3) You use the word "predict" here, which I don't quite like. Correlation can predict outcomes only in the case where you have proven that past information will continue to occur. This is not always the case.
> A positive correlation between two variables means they tend to move in the same direction.
This is misleading also. The variables tend to move in the same direction _relatively to their mean_! I can plot you very different looking curves that are highly correlated just because their mean vary differently. Especially because of potential outliers.
> And it turns out that a strong positive correlation can be a good place to hunt for possible causality, such as hunting for something that causing both variables to change together.
I tend to dislike that thought process. It is very tempting to infer causality from correlation. Especially when automating correlation at scale is affordable, so generating spurious correlation is easy. To me correlation is more a way to reinforce a hypothesis of causality that is already suspected. It won't give you the direction, but may hint you that you are on the right track. I mean, correlation should not be a clue to discover causality, as there are too many false positives. Correlation should be an intermediary step in studying a potential causality.
From the NIST Engineer's Handbook:
"Generally, we first need to find and explore correlations and then try to establish causal relationships. It is much easier to find correlations as these are just properties of the data."
So I'm saying that correlation is often used (and misused) in an attempt to find good candidates for causation. This is how much science and engineering is practiced. If you disagree with this, you disagree with some fundamentals of how the scientific method is used (ie, hypothesis forming).
> To me correlation is more a way to reinforce a hypothesis of causality that is already suspected
It certainly can be used in this way, but that doesn't mean it's the only scientifically-valid way to use it.
The data may be flawed in many ways, so there is no automatic assumption that any conclusion is correct.
And - as this answer says - this is a completely different process to discovering a robust causal mechanism that can reliably explain a correlation and predict its persistence into the future.
I don't disagree, but I have yet to see a mathematically clean definition of "causality".
His intro to the topic bodes well: http://bayes.cs.ucla.edu/BOOK-2K/why.html
It’s not clear to me whether mathematical axioms would be outside of entropy driven time, or if the mere presence of operators in sequence requires entropy change to transmit the information and move the calculation forward.
y = x*b + e
b = cov(x,y)/var(x) = corr(x,y) * (std(y)/std(x))
Is it that an ommitted variable isn't part of a causal chain of an included variable, and a confounder is?
Omitted variable bias tells it like it is. A bias in a regression coefficient that results from an incorrectly specified model.
I wonder why Wikipedia has separate entries for them:
The picture at the top of the confounding article gives it away. Those kinds of diagrams are common in "hierarchical Bayesian models" like LDA.
In the simple linear setting, they are the same thing. Teaching people the "confounding variable" concept in a general setting before teaching them about "omitted variable" in a linear setting is like teaching people about Riemannian manifolds before teaching them about vector spaces.
Correlation isn't a particularly useful concept outside of simple linear models.
The main problem of correlation is that researchers often "start with a conclusion and then fill in the blanks".
The thing is that several items can fill in these blanks. With coherence.
Making predictions based on these techniques is often misleading & in some cases even dangerous.
You can attribute the irreplicability of several "scientific findings" to the use of correlation.
Correlation is great for " analysis" but not a way for drawing a conclusion
A p-value of 0.01 means that under the null hypothesis (generally that the variables are independent), a result at least this extreme (in this case, with at least this large a correlation) would be seen 1% of the time.
Note in general that R^2 = 10% does not imply p = 0.01, nor the converse. They are largely unrelated measures. In principle one could have a small correlation with high significance, or a large correlation with low significance.
That said convolution and correlation are not linear operators but bilinear operators, so I'm struggling a bit to see in what sense they're adjoint.
Maybe something like:
<Conv(x,y), z> = <y, Corr(x, z)>
which does actually work if you let the inner product be the integral of the product and the correlation the time-correlation (so the inner product is the correlation at time 0).
http://www.reproducibility.org/RSF/book/bei/conj/paper_html/...