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I have had three semesters of college statistics as part of my BSBA degree. From what I recall from regression analysis the graph seems to show a very high coefficient of determination between CO2 and temperature would tend to be be a reliable predictor of future changes in one variable based on changes to the other.

People here are saying that climate science does not predict the one-to-one relationship between global CO2 and global temperature that the graph seems to clearly show. How is this discrepancy explained?

This is the closest to the original question. The answer seems to come from climate science rather than statistics. Even though the graph has an R-squared value of 0.752, showing a consistent linear relationship between CO2 and temperature, climate science seems to indicate that increases in CO2 have a logarithmic relation to changes in temperature. The best numbers that I could find are for every doubling of CO2 global temperature increases 3C.

The average coefficient of determination (R-squared) turns out to be 0.752
https://www.ncbi.nlm.nih.gov/pmc/articles/PMC5840396/
For the entire record, the 90% confidence ranges of the correlations (R-squared) of CO2 with δD, ΔTsite and ΔTsource are 0.68–0.73, 0.76–0.80 and 0.75–0.79, respectively.

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polcott
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    You will need to remove the confounding effect of time to avoid high spurious correlation. – Michael M Oct 02 '19 at 05:15
  • As a first step you should difference the two series and check the plots again. Also, $R^2$ can be misleading. – user2974951 Oct 02 '19 at 08:18
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    You might be interested in some of the questions under the [tag:causality] tag, such as [Under what conditions does correlation imply causation?](https://stats.stackexchange.com/questions/534/under-what-conditions-does-correlation-imply-causation) – mkt Oct 02 '19 at 08:39
  • I reject the paraphrasing of the answers added in the edit. It is not a reasonable summary of what @DikranMarsupial and I have written. Although 'spike' in temperatures is undefined here, the OP refers to 8 degrees C in other comments, which is unsupported by the graph and - to my knowledge - is several degrees higher than the most commonly cited estimates from climate science. – mkt Oct 03 '19 at 08:38
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    @mkt acknowledged. Since the graph consistently showed a linear one-to-one relationship I did not understand how that would not continue. The current science seems to be that temps raise CO2 on a one-to-one basis but CO2 does not raise temps at this same rate. The fact that CO2 has spiked at 123-fold faster than its next fastest rate in at least 800,000 years may indicate that temps may rise at a different rate than they have historically. I show these calculations here: https://www.researchgate.net/publication/336145637_Proving_the_Reality_of_Global_Warming – polcott Oct 04 '19 at 14:54
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    @polcott I don't think the ice-core data has the temporal resolution to claim that the rise is 123 times the next fastest in the last 800,000 years. As I said, you need physics for this, not just statistics. CO2 is not the only thing that causes warming (nor is warming the only thing that causes changes in CO2). It just isn't that simple. – Dikran Marsupial Oct 05 '19 at 08:18
  • @DikranMarsupial I asked the physics people and they kept dodging the question. This graph has the finest temporal resolution that I have seen: http://joannenova.com.au/globalwarming/graphs/VostokIceCores400000Kmed.jpg – polcott Oct 05 '19 at 12:50
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    @polcott download the data, read the papers describing the data. There are no short cuts. Probably best to get your information from the scientists that collect the data rather than climate skeptic blogs. – Dikran Marsupial Oct 05 '19 at 15:32
  • This question keeps changing and is veering off into non-statistical territory. The original question was about how to interpret a graph and was on topic. The latest iteration "*What temperature increase can we expect from the current 410 PPM spike in CO2*?" is not a question that we have the expertise to answer here, @DikranMarsupial notwithstanding. I recommend reverting the question to the original focus and taking the new question to a different SE that would be more suited to it, such as Earth Science or Skeptics. – mkt Oct 07 '19 at 08:32
  • @DikranMarsupial I provided that actual ice core data. This data has a rate of 84.9 PPM / 6664 years = 0.0127 PPM per year. The current rate is 2.11 PPM per year, is 15,700% faster than the ice core data. – polcott Oct 07 '19 at 13:18
  • @mkt It looks like every time the global CO2 doubles the global temps increase 3C. I had initially assumed form the graph that the one-to-one rate would continue because it has continued for 800,000 years. – polcott Oct 07 '19 at 13:21
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    @polcott each of those slices in the core each cover several centuries. How do you know that the CO2 change in each slice took place gradually over the course of the period, or whether it happened suddenly at some point in the middle, or that is oscillated and that is just the mean value? The ice core does not have the temporal resolution to support that claim. Self-skepticism is vital in science and in statistics. – Dikran Marsupial Oct 07 '19 at 13:26
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    BTW "84.9 PPM / 6664 years = 0.0127 PPM per year. " just using the difference bewteen the first two readings in the list gives a rate of 0.0272 PPM per year. If you are going to correct climate skeptics, then this sort of "nuanced" use of statistics is likely to backfire. – Dikran Marsupial Oct 07 '19 at 13:31
  • The substantial changes in the question are going to leave a set of widely varying answers, unsuitable for the focused aims of this site. – whuber Oct 07 '19 at 13:53
  • @whuber What can I do to fix it? I will try to revert. – polcott Oct 07 '19 at 14:29
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    This is not the original question. And "*How is this discrepancy explained?*" is still a question about climate science, not statistics. – mkt Oct 07 '19 at 14:50
  • @mkt The answer to the original question would be "yes" based on the high "r-squared" value alone. Because this answer is misleading to make sure that others are not mislead by this answer we should probably add a little climate science. – polcott Oct 07 '19 at 14:54
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    @polcott "*The answer to the original question would be "yes" based on the high "r-squared" value alone*" Two answers have argued extensively that this is not a reasonable conclusion to draw from that graph. In extended comment discussions afterwards, this has been reiterated. And yet you persist with this claim. I'm done with this comment chain and question, this is not a good use of anyone's time. – mkt Oct 07 '19 at 14:58
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    Me too. The reason why the naive statistical analysis is misleading has been explained in sufficient detail, and it has been explained why physics is a better approach to answering questions/misunderstandings about this, and I have repeated myself often enough already. – Dikran Marsupial Oct 07 '19 at 15:01
  • @DikranMarsupial I just wanted to restore the question back to its original form well enough that the hold could be taken off. – polcott Oct 07 '19 at 15:16

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If you had no information at all other than this relationship, you should not have a lot of confidence in your prediction of a spike in temperature soon. That's because it might be a spurious correlation, with some other variable driving changes in both temperature and CO2, or there might be reverse causation, with temperature driving CO2 changes (checking for Granger causality could help a bit EDIT: but see Dikran Marsupial's concerns about this). But this graph certainly shouldn't make you comfortable that there's no relationship at all! It should mildly move your priors in the direction of expecting a rise in temperature soon.

But more importantly: this is NOT the only information we have available. We have good earth systems models that predict that temperature should be associated with CO2, and this is based on a deep scientific understanding of how CO2 interacts with the atmosphere, oceans and biosphere. In fact, we've understood the basic science behind it since the late 1800s:

In 1896 Svante Arrhenius used Langley's observations of increased infrared absorption where Moon rays pass through the atmosphere at a low angle, encountering more carbon dioxide (CO 2), to estimate an atmospheric cooling effect from a future decrease of CO 2. He realized that the cooler atmosphere would hold less water vapor (another greenhouse gas) and calculated the additional cooling effect. He also realized the cooling would increase snow and ice cover at high latitudes, making the planet reflect more sunlight and thus further cool down, as James Croll had hypothesized. Overall Arrhenius calculated that cutting CO 2 in half would suffice to produce an ice age. He further calculated that a doubling of atmospheric CO 2 would give a total warming of 5–6 degrees Celsius.[24]

Further, Arrhenius' colleague Arvid Högbom, who was quoted in length in Arrhenius' 1896 study On the Influence of Carbonic Acid in the Air upon the Temperature of the Earth[25] had been attempting to quantify natural sources of emissions of CO 2 for purposes of understanding the global carbon cycle. Högbom found that estimated carbon production from industrial sources in the 1890s (mainly coal burning) was comparable with the natural sources.[26] Arrhenius saw that this human emission of carbon would eventually lead to warming. However, because of the relatively low rate of CO 2 production in 1896, Arrhenius thought the warming would take thousands of years, and he expected it would be beneficial to humanity.[26][27]

Scientific questions are almost never resolved by a single piece of information, such as this graph. Evidence accumulates through many different sources and methods, and the high confidence that most scientists have in climate science is because it has a great deal of support from multiple sources.

mkt
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    CO2 both causes warming (the greenhouse effect) AND changes in temperature cause changes in CO2 levels (Henry's Law), so the causation goes in both directions. Personally I am rather skeptical of statistical concepts of causality when we actually understand the physics reasonably well. – Dikran Marsupial Oct 02 '19 at 08:59
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    @DikranMarsupial Agreed - that paragraph addresses a hypothetical scenario in which we had no information other than the graph presented. – mkt Oct 02 '19 at 09:00
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    Yes, the spike is an indication that there is a different process involved now compared to most of the last 800,000 years covered by the Vostok core. – Dikran Marsupial Oct 02 '19 at 09:13
  • @DikranMarsupial The 800 year lag between changes in temperature and changes in CO2 level trend reversal seems to be quite telling: http://joannenova.com.au/global-warming-2/ice-core-graph/ – polcott Oct 02 '19 at 16:42
  • Is my above paraphrase of your answer correct? Would the huge spike plausibly be 8C (the dashed line at the top of the graph)? – polcott Oct 02 '19 at 22:46
  • @polcott Not quite. The climate projections I'm familiar with are for quite a lot less than 8C. If you assumed that you had only this figure and no other information, a prediction of 8C would still be poorly supported. That's because it is well outside the range of the earlier data, so the 1-to-1 relationship between CO2 and temperature anomaly that you seem to be assuming need not hold. I don't understand why you are trying to draw these conclusions from this one graph, though, especially when we've made clear that the science is built on a lot more than this. – mkt Oct 02 '19 at 22:54
  • @mkt Because we must get everyone on the same page ASAP, I am aiming to refute climate change deniers in 60 seconds or less. If the above data could be reasonably construed as plausibly predictive of a huge spike in global temperatures the climate emergency deniers could be equally refuted by the same graph. – polcott Oct 02 '19 at 23:27
  • @polcott you would be better off sticking to the physics if you are talking to skeptics (I don't use "denier" as it results in them evading the science by asking why you are linking them with the Holocaust and why are you so mean to them). The reason why the extrapolation is not reasonable is that the balance of the two mechanism mentioned in my answer is much different. – Dikran Marsupial Oct 03 '19 at 06:10
  • @DikranMarsupial My original graph shows that an increase from 200 PPM to 300 PPM of CO2 corresponds to a 9 to 14 change in temperature. That works out to be a 50% change in CO2 corresponds to a 55% change in temperature. – polcott Oct 03 '19 at 06:24
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    @polcott I agree with Dikran Marsupial. I also think that this goal of convincing people about a complex forecast in 60 seconds seems misguided. But this discussion seems unproductive because you seem quite set on doing things this way. – mkt Oct 03 '19 at 06:27
  • @mkt Although conventional science indicates a smaller increase in temperature relative to the increase in CO2 the actual numbers themselves seem to directly contradict this conventional science. How do we explain that? – polcott Oct 03 '19 at 06:33
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    @polcott, I have already explained that in my answer. It isn't one physical mechanism that you are looking at, it is two. To resolve the contradiction, you need to understand the basics of the phsyics (and I would suggest against arguing with climate skeptics if you don't have a good grasp of the basic physics - most skeptic arguments are misunderstandings of basic physics). – Dikran Marsupial Oct 03 '19 at 06:52
  • You might want to try this online course https://www.edx.org/course/making-sense-of-climate-science-denial-2 (I did a couple of the carbon cycle videos, but IIRC I think there is one about the meaning of the Vostok ice core data from one of the other contributors). – Dikran Marsupial Oct 03 '19 at 06:55
  • @polcott What do you mean by '*the actual numbers themselves seem to directly contradict this conventional science'*? – mkt Oct 03 '19 at 06:57
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    @polcott Also, this is not the best forum to debate climate science broadly. If you raise specific examples that you wanted statistical opinions on (such as this graph), we can address them, as has been done in these two answers. But the field of climate science is too broad and outside most of CV's expertise (though you are fortunate that Dikran Marsupial appears to be an exception) for a worthwhile debate here. – mkt Oct 03 '19 at 07:03
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    Indeed, it would be well worth asking on https://earthscience.stackexchange.com/ where you will find people with more expertise than I have on this topic. Ah, I can see you have already tried there https://earthscience.stackexchange.com/questions/18111/does-the-following-graph-make-an-extremely-compelling-case-for-global-warming . – Dikran Marsupial Oct 03 '19 at 07:11
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    @polcott I still think you would benefit from reading some of the [tag:causality] questions here, and also perhaps taking a look at the [spurious correlations site](https://www.tylervigen.com/spurious-correlations). I will also add that however good your intentions are, you seem to be asking for validation of your position rather than honest opinions, both here and at the Earth Science SE. This is not the best approach to using these sites, especially when your goal is to refute a group notable for rejecting expertise and evidence. – mkt Oct 03 '19 at 08:35
  • @mkt I am just trying to cut-to-the chase of the most compelling analysis of the key facts such that a whole litany of counter-arguments can be discredited in one fell swoop. It looks like the most compelling argument that is totally supported by the key facts that the recent 2.11 PPM per year rate of increase of CO2 is 15,700% faster than the next fastest rate in the ice-core data of 0.0127 PPM per year. Furthermore there is no other rate as fast as this fastest ice core rate in any of the less granular geological data. – polcott Oct 07 '19 at 14:24
  • @polcott And as I've said before - **this is a forum for questions and answers about statistics**. We can answer your questions about graphs and statistical analysis, but you don't seem entirely receptive to our opinions, possibly because your priorities are different. Your basic question seems to be '*What is the fastest/most effective way to answer climate sceptics?*' which may be valuable - but this is not the place for it. – mkt Oct 07 '19 at 14:36
  • @mkt The other places on stack exchange seem to be far less receptive to climate change questions. – polcott Oct 07 '19 at 14:43
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This is a situation where you need physics rather than statistics. The causal relationship between CO2 and Global Mean Surface Temperature (GMST) is bi-directional, changes in CO2 cause changes in temperature (the [rather inaptly named] greenhouse effect) but Henry's law means ocean CO2 solubility depends on ocean temperature, so changes in temperature cause changes in atmospheric CO2. For most of the last 800,000 years, most of what we see is due to the latter. Milankovic cycles (changes in orbital parameters) cause a small amount of warming, which leads to a reduction in CO2 solubility and hence an increase in CO2, which then provides positive feedback, adding to the warming via the greenhouse effect. In the post industrial era, it is dominated by the greenhouse effect resulting from anthropogenic emissions. This is clearly seen if you plot CO2 as a function of temperature (caveat: it is regional Antarctic temperature for the Vostok data, not GMST). The red data points are the Vostok ice core data, and the green are modern instrumental measurements and from the Law Dome ice core. It is clear that there is a difference in the data generating processes invoved.

enter image description here

In short, where we have physics, we should use it rather than rely on statistics... or better still, use both! ;o)

IIRC there have been some deeply misleading statistical analyses of this particular correlation in the past, I think including Grainger causality. I'll see if I can find them (and the responses to them).

Dikran Marsupial
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  • Is my above paraphrase of your answer correct? Would the huge spike plausibly be 8C (the dashed line at the top of the graph)? – polcott Oct 02 '19 at 22:47
  • @polcott is this a coursework exercise? No, you wouldn't get an 8C spike, but you would get a sharp increase in temperature. The temperature increase is logarithmic in CO2 concentration; the transient response of the climate system is about 1-2.5 C per *doubling* of CO2, and the equilibrium sensitivity (once the surface oceans have equilibriated) is about 2-4.5 C per doubling. We haven't had a doubling of CO2 yet, but termperatures have risen by nearly 1C, so that is consistent with the TCR range. – Dikran Marsupial Oct 03 '19 at 06:04
  • My original graph shows that an increase from 200 PPM to 300 PPM of CO2 corresponds to a 9 to 14 change in temperature. That works out to be a 50% change in CO2 corresponds to a 55% change in temperature. – polcott Oct 03 '19 at 06:22
  • "The temperature increase is logarithmic in CO2 concentration" How do we account for the graph's linear (thus not logarithmic) relationship? – polcott Oct 03 '19 at 06:40
  • @polcott it is the CO2->temperature relationship (greenhouse effect) that is logarithmic, the temperature->CO2 relationship (Henry's law) is closer to being linear. Over the last 800,000 years, the latter has been dominant, in the post-industrial era it is the former. If you are going to argue with climate skeptics, use physics, not statistics. Statistics without physics in such circumstances can easily be deeply misleading. – Dikran Marsupial Oct 03 '19 at 06:50
  • What is the projected increase in global temperature based on the current 410 PPM? – polcott Oct 03 '19 at 22:24
  • @polcott I don't have the figure to hand, but as I said, the relationship is logarithmic and from 2-4.5C per doubling. – Dikran Marsupial Oct 04 '19 at 06:52
  • Yes I found those same kind of figures too. Things may have changed this time. I calculated that atmospheric CO2 since 1950 has increased at 123-faster than any time in the last 800,000 years. Here are my calculations: https://www.researchgate.net/publication/336145637_Proving_the_Reality_of_Global_Warming – polcott Oct 04 '19 at 15:01
  • @polcott, there are better ways of showing that the rise in atmospheric CO2 is due to anthropogenic emissions (need to include land use change), see https://skeptics.stackexchange.com/questions/3435/does-co%e2%82%82-cause-global-warming/17707#17707 which includes a link to my paper on this subject (a peer-reviewed comment rebutting a climate skeptic paper that made it into a journal). The fact that the rate of increase is less than the rate of anthropogenic emissions ought to be sufficient to show that the natural environment is opposing the rise, rather than causing it. – Dikran Marsupial Oct 05 '19 at 07:21
  • So it looks like you agree that the 123-fold faster rise in global CO2 could not have any other explanation besides human emissions. – polcott Oct 05 '19 at 12:54
  • @polcott I have already pointed out that the data don't support the 123-fold faster claim, I don't think there is sufficient temporal resolution to make that claim. Only the most extreme of climate skeptics would argue that the rise in CO2 is not anthropogenic, it is a very well established result, with multiple lines of physical evidence. There really is no need for questionable statistical arguments. – Dikran Marsupial Oct 05 '19 at 15:38
  • "Only the most extreme of climate skeptics would argue that the rise in CO2 is not anthropogenic" Going by Facebook (where half the population gets their news) about 1/3 disagree with anthropogenic cause. My purpose is to find the most efficient way to utterly refute them. – polcott Oct 05 '19 at 18:05
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    @polcott No, most disagree that the cause of the warming is anthropogenic or that the effect is minor. It *is* only a handful of extreme skeptics that doubt the rise in CO2 is anthropogenic. I know this because I have been trying to stop that particular canard from promulgating for a decade. – Dikran Marsupial Oct 07 '19 at 06:26
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    @polcott how many times does it need to be said, the way to utterly refute them is physics, not statistics. There are no short cuts - if you want to master this particular topic, you need to learn the physics. – Dikran Marsupial Oct 07 '19 at 06:28
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Just to focus on the basics of visual data analysis, a time series plot looking at "level" variables is not the best way to study relationships between two variables. The more accurate way to look at this is use a scatter plot. The X-axis would be a First Difference change in CO2 concentration (the independent X variable), and the Y-axis would be a First Difference change in Temperature (the dependent Y variable).

If you use Excel, implementing such a graph would allow you to fit various types of regression and polynomial to that scatter plot graph. From that framework you could very readily examine the strength of the relationship between the two variables. Don't be entirely surprised if when looking at it this way, the relationship between X and Y is a lot weaker than what your eye interprets it to be when looking at your time series with "level" variables.

This visual framework is also important if you want to model or estimate Y using X. If you use "level" variables, your model may end up with a very high R Square (same as coefficient of determination). But, that is a classic "spurious regression" (look at the seminal paper on the subject by Clive Granger). You should instead first-difference all your variables (so they represent periodic change not levels); and, then you are in better shape to run your model. "Spurious" regressions very often have very high R Square (close to 1.). Very often this is a diagnosis that you have indeed a "spurious" regression.

Sympa
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