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Causation And Correlation

Wearing a tie is cv.correlated with making logically true statements.
Our friend Christopher Monckton of Brenchley wrote a piece over at Anthony Watt’s place in which he said (Reader KA Rodgers asked me to have a look):

CO2 concentration continues to climb. Global temperature doesn’t. Absence of correlation necessarily implies absence of causation. Game over, logically speaking.

The key proposition—Absence of correlation necessarily implies absence of causation—pleased some and set others off into a standard internet tizzy. Some said, “Right!”, others “Wrong!” Since it’s Advent, a joyous time, I’m happy to report both sides are right and can cease fighting. Although, those who said “Right!” were righter than those who said “Wrong!”

Correlation has a technical definition in statistical theory and roughly means that two “variables” when plotted one against the other show something resembling a straight line. Imagine a cosine wave from 0 to 2π: one variable time, the other amplitude. The statistical, technical correlation of this series is zero. But any civilian who looks at this plot would say that something is going on; i.e. that these two variables are “correlated.”

To distinguish these two meanings of the word, let st.correlation represent statistical, technical correlation, and let cv.correlation be the civilian’s “something is going on.” Keep in mind that cv.correlation encompasses st.correlation: if you have st.correlation you automatically have cv.correlation, but (as we have just seen) the converse is not true.

Obviously, one can have causation (if we insist time causes the amplitude to take the values it does) and no st.correlation. Lord Monckton’s modified phrase “Absence of st.correlation necessarily implies absence of causation” is therefore false. Why? Because absence of st.correlation does not imply absence of causation.

But given causation, we see cv.correlation, and vice versa. However, as is well known, we can have cv.correlation without causation, i.e. cv.correlation does not imply causation.

Causation in this case says the probability the amplitude takes the value y given that time equals t is either 1 or 0, and that this condition holds for all values of y and t. In other words, if we know the value of t we know with certainty the value of y.

But don’t forget we’re insisting that time makes amplitude take its values. In most real-life examples, it wouldn’t be time causing something, but the state of some other thing at time t which is causing the thing of interest (here amplitude) to take the value it does. Time is merely a proxy for this other variable. Thus knowledge of causation is something greater than knowledge of probability.

Lord Monckton’s example is thus just like ours, using time and temperature, with time being a proxy for carbon dioxide. I think it is also clear he had in mind cv.correlation and not st.correlation because nobody expects carbon dioxide to everywhere and always cause linear effects on temperature.

Now some logic. “If causation implies correlation” is true, then (via elementary rules) the contrapositive “If no correlation then no causation” is also true.

We have already seen that “causation implies st.correlation” is not necessarily true, which is why “no st.correlation then no causation” is not necessarily true. But if—as the full logical statement goes—“causation implies st.correlation” is true, then “If no st.correlation then no causation” must be true. So Lord Monckton was on solid logical (but perhaps not solid physics) ground if he had meant st.correlation—which, again, I don’t think he did.

Now “causation implies cv.correlation” is true. Causation, by definition, has some effect which must be evinced in cv.correlation. Thus it must be so that “if no cv.correlation then no causation”. Monckton scores again.

Could anything else booger up the works? Well, there could be some third (or third plus forth, plus fifth…) thing which somehow (we needn’t know how) blocks carbon dioxide from causally changing temperature. But since this third thing is blocking carbon dioxide, carbon dioxide is not causing changing temperatures.

Or it could be that carbon dioxide does cause temperature to change but that some third (etc.) thing immediately (or at least before we can measure temperature) changes temperature back to where it was before carbon dioxide changed it. Kind of like an evil anti-carbon dioxide. But how could we prove this? This brings us back to the epistemology (probability)—ontology (actual cause) distinction made above. If all we have are the measurements, it would look like there was no causality. That is, our knowledge of causality would be missing, even though it’s there.

The third (fourth, etc.) thing could also cause temperatures to change but to all sorts of other values and not just to the exact place to where it was before being affected by carbon dioxide. This is the normal case, as seen in nature. The cv.correlation will not be exact, and again we musn’t confuse epistemology with ontology. In any case, the man from Brenchley is still right.

29 thoughts on “Causation And Correlation Leave a comment

  1. This came up on another blog and there was an insistence that causation could occur without correlation, based on something like the evil anti-carbon dioxide example. However, in all cases, there had to be a second variable involved and there was correlation with that variable. Is there ever a case where causation involves no correlation? No hidden variables, etc. Does one not need correlation somewhere in the example to have causation?

  2. Amen and all that.

    Wearing a tie is cv.correlated with making logically true statements.

    I wonder which is the cause. If it’s the tie, I suspect it’s not the only cause.

  3. insistence that causation could occur without correlation

    Depends on what is meant by correlation. Statistical correlation is counter-intuitive as it means linear correlation. Using the other meaning, cause can only be determined through correlation. In fact, how can X be a cause if it doesn’t change something (i.e., cause it).

    When the trigger on a gun is pulled, the trigger is in the causal chain but itself does not make the bullet come out of the gun as many things happen after the trigger is pulled. It’s fair to label anything in the chain as a cause.

  4. It is my understanding that historically the proxies datum indicate that the rise in the co2 level lagged the rise of temperature by several hundred or more years, indicating that the temperature was driving the co2 increase throughout history up until now. My question is how can something that has apparently always been an affect become the cause of what it was previously an affect of?

  5. Rexx Vernon Shelton,,

    Determining which came first can be tricky when there are many cycles. Also, some things are mutually causative.

  6. “nobody expects carbon dioxide to everywhere and always cause linear effects on temperature”

    This statement is false, because this is exactly how the global climate models are written.

  7. Your friend,Christopher Monckton of Brenchley is wrong.

    My daughter just told me yesterday that her grades hadn’t changed much but the number of hours she spent in study continued to climb. She didn’t say “game over” though.

  8. Correlation has a technical definition in statistical theory and roughly means that two “variables” when plotted one against the other show something resembling a straight line. Imagine a cosine wave from 0 to 2π: one variable time, the other amplitude. The statistical, technical correlation of this series is zero. But any civilian who looks at this plot would say that something is going on; i.e. that these two variables are “correlated.”

    Note that for the function y=f(t)=a cos(t), the amplitude is |a| and the period is 2Π. I think you meant to say the Pearson correlation coefficient between y and time is zero.

    There are other statistical measures of correlation. To quantify the correlation between two variables with a nonlinear relationship, e.g., one can run a regression of y on cos(x), and the resulting coefficient of determination (which is 1 for the example of cosine function) is a measure on the correlation between the two variables.

  9. As usual, etymology helps. Co-relation is, well, it’s clear what it is. It is a relation that involves both members of the pair of concepts under study. As per dictionary.com, first definition: “mutual relation of two or more things, parts, etc.”

    Causation is a relation. It involves both members of a pair. Therefore, it is impossible that causation should not involve correlation in the broader (linguistic) sense. It would be like saying that red does not involve colors. Causation is a kind of correlation. It is distinguished from other kinds by its specific features, but it is still a kind of correlation.

    For anyone to suppose that the species can be present while the genus is absent is just sad.

  10. 1. Suppose you have a chamber with oxygen in it and you strike a match. Result, a flame.
    2. Now evacuate the oxygen and strike a match. Result, no flame.
    3. Now introduce hydrogen into the chamber and strike a match. Result, no flame.
    Conclusion: flaming correlates with oxygen; but hydrogen has no causative connection.

    But wait, there is a fourth combination: introduce hydrogen and oxygen into the chamber and strike a match. Result, KABLOOEY! The mixture of hydrogen and oxygen is what destroyed the Hindenburg.

    That makes water a very dangerous explosive, right?
    + + +
    Sometimes, there are interesting correlations with combinations of variables: like charcoal, sulfur, and saltpeter. All three must be present in the right quantities for the explosion to occur.
    + + +
    An example of causation with no apparent correlation will occur when the X-variable does not vary beyond a narrow range; esp. if the variance in Y is high.
    + + +
    In addition to multi-linear correlations, there are numerous examples of non-linear correlations: e.g., light intensity correlates with the square of the distance from the light source.

    The correlation of temperature to CO2 concentration is logarithmic. That is, successive doublings of CO2 produce smaller and smaller increases in temperature. (That’s why the models need feedback mechanisms to produce the desired effects.) Absent these feedbacks, the temperature increase due to CO2 will grow flatter.

  11. YOS,

    “That makes water a very dangerous explosive, right?”

    No, technically that would make water an explosive residue. 🙂

    “Sometimes, there are interesting correlations with combinations of variables: like charcoal, sulfur, and saltpeter. All three must be present in the right quantities for the explosion to occur.”

    Black Powder burns rapidly, it does not explode. Creating an explosion with black powder requires burning it in a pressure vessel not quite strong enough for the pressure generated by the hot gasses from the burning gun powder.

  12. YOS, I assume that this is theoretical reasoning and that you didn’t do the experiment. I would advise against the latter since match heads usually contain their own oxidizing agents.

  13. Lance Armstrong won seven Tour de France races. He never failed a PED or doping test. The correlation between artificial enhancement and performance was absent, yet witness testimony and eventually his own confession confirmed causation had in fact occurred. Moral of the story: a perfect masking agent can make one believe lack of correlation implies lack of causation. Therefore one must be diligent in looking deeper than the simple cause/effect connection.

  14. “The correlation of temperature to CO2 concentration is logarithmic.” When was this established and by what method? Just saying it is so does not make it so. I read this assertion over at skepticalscience.com Link: http://www.skepticalscience.com/exponential-increase-CO2-warming.htm
    And I read their explanation here: http://www.skepticalscience.com/empirical-evidence-for-co2-enhanced-greenhouse-effect-advanced.htm

    They just skipped over the fact that in all of history the co2 increase has lagged warming until the present situation. Just how do you explain an effect becoming a cause?

  15. “The correlation of temperature to CO2 concentration is logarithmic.” When was this established and by what method?

    I cheated: it’s the Arrhenius equation, not a statistical sampling.

    ln(k)= -Ea/RT + ln(A)

    And as I said, one can only achieve the desired temperature increase by adding various feedback loops, some of which are far less supported than the direct Arrhenius effect. For example: that warming will increase evaporation which will accelerate cloud formation and clouds will increase the warming effect. It’s like the way Ptolemy added epicycles to the pure Platonic circles, the difference being that Ptolemy’s model actually made correct predictions.

  16. all of the proxy datum shows co2 lagging warming by hundreds of years

    As the oceans warm, they can hold less CO2 in solution, and so CO2 gasses out. The interesting question is what makes the oceans warmer. Unless you think you can warm up a pot of water by turning a hair-dryer on it, a warmer atmosphere does not necessarily imply a warmer ocean. Direct insolation, radiative heating may be a factor. I don’t know. These things seem to follow solar cycles rather well, but it may not be insolation, as such.

    On the time scale of those proxies, I doubt a great deal of precision. In fact, I wonder a bit about their proxitude.

  17. Mariner, well said.

    In addition, I think Sheri asks an excellent question at #1.

    —–

    Furthermore, if we consider the context of Monckton’s statement (CAGW alarmism), then it becomes more difficult to see how his statement could be wrong. Specifically, the claim of CAGW is that increased CO2 will cause warming. Despite all the silly and misguided discussions about CO2 causing warming “all things being equal,” the central claim remains: more CO2 -> warming.

    So when a claim is “x causes y,” and when the result of x is not y, then x didn’t “cause” y, no matter how you look at it. Yes, we could backpedal and say, “Well, x would have caused y, if not for a, b and c.” But that is just an admission that “x causes y” is false; instead the proposition should have been stated “x might lead to y.” So it is true — quite trivially true in fact — that if a claim is made that “x causes y,” and then x occurs but doesn’t result in y, then the statement was false. No if’s and’s or but’s.

    Now what is a committed alarmist to do? Well, one possibility would be to abandon the storyline altogether, but few are inclined to do that. Instead, the tactic is to say, in essence,

    “We didn’t really mean that CO2 would always cause warming. No, what we really meant to say was that CO2 would cause warming over a particular period of time, say 10 years. I mean 17 years. Oops, I mean 20 years. Well, darn it, perhaps 30 years — You’d better believe that warming is coming sometime!”

    or alternatively,

    “What we really mean is that CO2 will cause warming only if the rest of the system behaves like we think it should. You know, no unexpected volcanoes, no changes in cloud cover, no quiet Sun, no long-term oscillations to “mask” the warming that we just know is there. None of that kind of stuff.”

    Monckton, for whatever other faults he may have, has properly put pressure on the painful issue: We’ve had more CO2, but the prophesied warming hasn’t come. What to make of this state of affairs?

    Well, maybe the models haven’t adequately taken into account the full complicated climate system. Maybe net feedbacks aren’t positive after all. Maybe CO2 can only be called a “contributing factor,” rather than a “cause.” Or maybe, perish the heresy, in our particular climate system a few hundred ppm of additional CO2 doesn’t do much of anything measurable or noticeable with respect to temperature.

  18. proposition should have been stated “x might lead to y.”So it is true — quite trivially true in fact — that if a claim is made that “x causes y,” and then x occurs but doesn’t result in y, then the statement was false. No if’s and’s or but’s.

    Falling from a great height causes death — but not if you’re wearing a parachute or otherwise slowed before hitting the ground.

    If no exceptions are allowed, then any causal relationship which can have an inhibitor can never be labeled a cause. There are many such relationships. Pulling the trigger on a gun fires the gun but not if the safety is on; the gun mechanism is nonfunctional; the gun isn’t loaded; or has been loaded with faulty ammunition. Still, it’s generally accepted that pulling the trigger will cause the gun to fire.

    I think you’re being a bit pedantic.

  19. “Still, it’s generally accepted that pulling the trigger will cause the gun to fire.”

    True, but this is known from the design of the mechanics of the gun. This knowledge has nothing to do with statistical correlations.

  20. True but a lot to do with correlation and causation.Even if you didn’t know how a gun works, the trigger pull precedes the discharge which, if nothing else, is highly suggestive of it being the cause not to mention the large number of times the pull and discharge are associated with each other.

  21. @Ye old Statistican: “1. Suppose you have a chamber with oxygen in it and you strike a match. Result, a flame.”
    Oxygen by itself doesn’t burn. Our planet would be a dangerous place to live on.

  22. DAV on 18 December 2013 at 9:36 pm said:

    “I think you’re being a bit pedantic.”

    The “x causes y” statement is unambiguous: it is not the same as “x causes y IF a equals b”.

    Alarmist claims about CO2 were at first explicitly framed as the former. Since then, as a result of the Pause, they have been slyly re-framed as the latter – without mentioning what has been done, obviously.

    Insisting on honest argument is not being pedantic.

  23. See Siegenthaler U, Stocker T F, Monnin E, Lüthi D, Schwander J, Stauffer B, Raynaud D, Barnola J, Fischer H, Masson-Delmotte V & Jouzel J. 2005. Stable Carbon Cycle-Climate Relationship During the Late Pleistocene. Science. Vol. 310 No. 5752. pp. 1313-1317 in regards to the lag time in which they say, “Ice core analysis does reveal that increasing atmospheric carbon dioxide levels did lag behind increasing temperatures at the end recent ice ages. Although there is a certain degree of uncertainty regarding the lag time, it is generally accepted that at the end of an ice age, temperatures began to increase, and then carbon dioxide levels began to increase.”
    There exist no records to show if the seas are warming or cooling since 1951 and then only for surface temperature. They started using drifting buoys have been in 1979. During the 1980s and 1990s, a network of buoys in the central and eastern tropical Pacific ocean helped study the El Niño-Southern Oscillation. The first weather buoy in the Southern Ocean was deployed by the Integrated Marine Observing System (IMOS) on March 17, 2010.

    The point being they have no idea if the oceans are warming or not, but if they as as the ocean warms and oceanic algae grows and prospers. This emits millions times more CO2 than humans could possibly do. It takes around 800 years for the oceans (which cover 2/3rds of this planet) to warm and therefore the relationship between a rise in CO2 and the air temperature is that the increase in CO2 will come 800 years after any significant warming in air temperature. This again is a proved fact.

    Regarding you statement “Unless you think you can warm up a pot of water by turning a hair-dryer on it, a warmer atmosphere does not necessarily imply a warmer ocean.” See the post I wrote on this subject at

    http://carolinacowboy.wordpress.com/2013/12/19/speaking-of-heat/

  24. If something results from a change elsewhere, it is connected and visibly so (else how would anyone know?). If no connection is observed, the “change elsewhere” cannot be called a/the cause.

    The result could be reverse, zero, or positive, without distinction. Knowledge of the “cause” would be, at the very least, useless.

  25. Rexx: “if they as as the ocean warms”

    I can usually guess what the author meant in such a case, but I’m lost here.

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