How to Think about Correlation?

Phase synchrony, where two signals rise and fall together. Cross correlation measures out of phase synchrony

I agree. For example no physicist pretends that any of the models of reality (newtonian physics, relativistic, quantum mechanics, ...) not even the latest models (say the standard model) are the Truth with a capital T. The absolutist picture of the comment you respond rely on the simplification that a model is either true or false (which is of course correct) but ignores the incredible success of abductive reasoning: the prior models can effectively correspond to the superceding models in the regimes that were explored previously while specifically disagree in a regime more recently explored or made accessible experimentally. This does typically correspond with errors / falsehoods in the prior model but they are typically concentrated in llmited prior false postulates that needed replacing.

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.

Actually under your experiment both could be still be causative factors. Determining Tar correlates with cancer does not eliminate the genetic factor.

The only assumption for direct control of the hypothesized causative factor that needs to be made is that your influence on the causative factor is 100% side effect free. This is not fully possible in reality but still assuming this in your experiments allows you to arrive at a conclusion without assuming other correlations.

Right I understand we're introducing new hypothesis, but it still stands.

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.

The comment you’re replying to addressed your concern about the genetic factor. As I said, by controlling for smoking behavior, you sever the statistical link between tar deposits and genetics in hypothesis (2+A), hence you also sever the indirect statistical link between tar deposits and cancer which is due to genetics.