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Roy's Risky Regression

Posted on 7 July 2012 by Dikran Marsupial

In my previous post Murry Salby's Correlation Conundrum I demonstrated why a correlation with a rate of increase says very little about the cause of the increase itself, because the long term increase is largely due to the mean value of the rate of increase, and correlations are insensitive to the mean.  In this post, I will attempt to explain why regression analysis is similarly prone to misinterpretation (which is not greatly surprising as regression is a correlation based method), using an example losely based on a blog post by Dr Roy Spencer, again questioning whether the observed rise in atmospheric CO2 is of anthropogenic origin.

Risky Regressions

The argument that the rise in atmospheric CO2 is due to increasing sea surface temperatures (SSTs), rather than anthropogenic emissions has previously been suggested by Dr Roy Spencer.  Dr Spencer demonstrates that the annual increase in atmopsheric CO2 is corelated with sea surface temperatures, with a lag of about six months, which is evident in the observations (Fig. 1).

Figure 1: Normalized net global emissions (inferred from Mauna Loa observations) and sea surface temperatures (HadSST2).  Click on the image for details.

As we saw in the preceding article, this is essentially uncontroversial, the link between ENSO and the annual increase in CO2 is well known.  The correlation coefficient between these two sets of observations is 0.75, which suggests that they are probably related somehow.  A straightforward regression analysis would aim to fit a straight line on a graph of net global emission (NGE) as a function of SST.  In this case, we get:

NGE = 0.2003*SST + 0.1050

We can then use this to predict NGE given the observed SSTs (n.b. this is not what Dr Spencer actually did, but we will get to that later):

Figure 2:  Regression model of net global emission as a function of sea surface temperature.

We can then compute the cumulatve sum to determine the effect of SSTs on the rise in atmospheric CO2 since 1959 (Figure 3).


Figure 3: Modelled and observed increase in atmospheric CO2.

and it looks like it does a pretty good job.  Unfortunately there are a couple of serious flaws in this line of reasoning.  Firstly, if we look at the regression equation

NGE = 0.1688*SST + 0.1079

The additive constant at the end is a component of net global emission that cannot be explained by SSTs so we ought to delete it before we evaluate the cumulative sum, giving us this result:

Figure 4: Decomposition of the modelled rise into components that are and are not explainable by SSTs.

Not quite so impressive!

The second problem with the regression analysis is an example of omitted variable bias; clearly anthropogenic emissions should be expected to have an effect on net global emission (all things being equal), and this was not included as an independent variable in the regression analysis.  The physics of climate also tells us that surface temperatures (including SSTs) should also be increasing due to global warming.  This means that SSTs are correlated with anthropogenic emissions.  As regression is a correlation based technique this means that SSTs may explain net global emissions either because SSTs actually do affect net global emission, or because they act as a proxy for anthropogenic emissions, or both, and a simply regression analysis cannot tell which of these is actually correct.  As a result, we cannot actually assert that the component due to SST actually represents a causal relationship between SSTs and net global emission.

Fortunately Dr Spencer includes anthropogenic emissions in his simple model of the carbon cycle,

?[CO2]/?[t] = a*SST + b*Anthro

where a and b are the coefficients, which in Dr Spencer's analysis are determined by manual experimentation, rather than by formal regression methods.  Here, we will use a regression based approach, which has the advantage of being objective, giving the results shown in Figure 5.


Figure 5: Regression analysis based on the simple model proposed by Prof. Spencer

These results are not that similar to those of Spencer, but here they are at least optimal in a least-square sense, rather than being manually chosen.  As SSTs and anthropogenic emissions are correlated, it is possible to construct a variety of models that explain the observations almost equally well, so manual search is vulnerable to unintended subjective bias.  We will use the regresion based approach, but use an offset term as well, i.e.

?[CO2]/?[t] = a*SST + b*Anthro + c

as this gives better performance, shown in Figure 6.


Figure 6: Model of net global emission based on SST and anthropogenic emissions.

Again, we can plot the contribution from each part of the model to the explanation of the increase in atmospheric CO2, giving:

 Figure 7: Attribution of the increase in atmospheric CO2 according to the regression model.

However, this model is still flawed, again due to omitted variable bias.  While it is certainly true that the solubility of CO2 in the oceans decreases with increasing temperature, Henry's law tells us that the solubilty of CO2 also increases with an increasing difference in the partial pressures of CO2 in the atmosphere and in the surface waters.  Thus as atmospheric CO2 increases, the net oceanic sink should increase, taking in more CO2.  Thus atmospheric CO2 itself ought to be a variable included in the model.  However, there is no real point in performing a yet more complex regression model.  We already know the net anthropogenic and natural contribution to the observed increase, with high certainty, via the mass balance analysis.

Figure 8 shows a more realistic (subjective) attribution of the observed increase in atmospheric CO2.  We know from the mass balance analysis that the annual rise is about twice the annual increase in atmospheric CO2, so in reality approximately 200% of the rise is anthropogenic.  However, the natural environment is known to be a net sink, again via the mass balance analysis, and has been taking up about half of anthropogenic emissions each year.  Thus the natural contribution to the observed increase is about -100%.  Changes in SSTs do affect the annual growth rate of atmospheric CO2, but its effect is cyclical, so overall it does not lead to a significant long term trend in atmospheric CO2 (as discussed in the previous post), but just modulates atmospheric CO2 up and down slightly.

Figure 8:  A more realistic attribution of the observed increase in atmospheric CO2

Key point: Trying to make causal arguments based on regression analysis is risky, especially if the underlying assumptions of regression analysis are violated, for instance because a relevant variable is omitted.  Regression analysis is often used to show how much of Y can be explained by X.  It is vital to remember that this does not imply that any of Y actually is explained by X.  To make that leap, a plausible physical explanation is required that both explains the correlation and can also explain the magnitude of the observed effect.  This is why we need to keep in mind that "correlation is not causation". Interesting correlations are a good stimulus to research, but at the end of the day we need the physics to support a causal link.

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Comments 1 to 24:

  1. Excellent post
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  2. Gosh, they literally try to deny anything they get their hands on. I wonder if someone out there doesn't deny the existence of CO2 - we can't even see it! How does Spencer explain the isotope signature of the atmospheric Carbon? Our emissions are more than double the amount that stays in the atmosphere annually. Does he have an explanation for the secret hole where all this anthropogenic Carbon gets hidden every year?
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  3. Alexandre The isotopic signature of the carbon provides good evidence that volcanoes are not responsible, but, as I understand it, the same isotopic ratio could be expected if the CO2 came from net emissions from, for instance, plants. Salby puts a lot of emphasis on the isotopic bit, which I see as a technique to distract attention from the much more formidable mass balance considerations.
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  4. @tristan cheers! @Alexandre Note Dr Spencer published his blog article some time ago and I suspect he has since understood his error as he has not mentioned it since, as far as I can tell. @ScepticalWombat I think is is more likely that Salby is simply unaware of the mass balance argument, as he he explicitly states in his Sydney Institute talk that the required observations are reliable and he gives the fundamental equation on one of his slides and explains how it is net emissions that matter at great length. He just hasn't seen the obvious conclusion that you get when you combine the observations and the equation. I suspect the same was true for Dr Spencer, he even included this plot for me it is obvious from that diagram that the natural environment is opposing the rise in atmospheric CO2; but apparently it doesn't seem so obvious to everbody!
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  5. Actually, the isotopes are just as damning to the idea that there is a natural CO2 source. Suess documented the decline in atmospheric C-14 back in the 50s, which indicated introduction of old or fossil carbon (>10,000 years). The source of this old carbon could have been due to plants or volcanoes. The later observation that C13 of atmospheric CO2 was changing as well indicated that plant material was contributing to the increase. So we know that the increase is in part due to fossil plant carbon.
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  6. I meant that a natural Co2 source was responsible for the recent increase. The stable isotope and mas balance arguments are particularly compelling because they reinforce each other.
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  7. Oh bother...You all know what what I mean, right? Anyway, DM, excellent post. These kinds of mistakes are pretty elementary, but also pretty common. Still, you'd think someone would have mentioned these problems to Spencer and Salby and that they would have listened or corrected themselves after the fact. I'd like an explanation there.
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  8. Cheers Srephen, sadly Salby and Spencer give a couple of good examples of what is wrong with the climate debate: Dr Spencer writes: "And just how strenuous and vehement the resulting objections are to what I have presented above will be a good indication of how politicized the science of global warming has become." Note the lack any suggestion that Dr Spencer may be simply mistaken (no big deal, it happens to us all evey now and again), and that any objections are politically motivated. Regarding the science being settled, Prof. Salby says (at about the 31 minute mark in his Sydney Institute talk) "Science is dynamic, it is predicated on discourse; questioning, that is how we get to the truth. If not for discourse we would still be in the dark ages. Excluding discourse from the equation isn't science, it is advocacy" Rather ironic then, that there have been no replies to my emails questioning the science presented in Prof. Salby's Sydney Institute talk. I fully agree with Prof. Salby that questioning and discourse are vital to progress in science.
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  9. Alexandre @2 - coincidentally, we do have a post on the works addressing, among other claims, the suggestion that CO2 isn't a significant contributor to global warming because it's not visible. Made by a Nobel Laureate no less. Similar to Dikran's comments @8, the Laureate felt that criticisms of his comments stemmed from a politization of science, rather than the fact that his arguments were just plain ignorant and deserved every bit of criticism they received. Climate contrarians like questioning science - just not the scientific arguments they put forth!
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  10. Said Nobel laureate wouldn't happen to have been involved with superconductors perchance?
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  11. That's the one, Tristan. I don't mean to take these comments off-topic though.
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  12. Some of the comments on this thread have left me a little confused. My understending is that the isotpoic changes in atmospheric co2 show that increases in co2 come from fossil carbon, either fossil fuels or volcanic. And that the decrease in atmospheric o2 in lockstep increases in co2 showed that co2 increases were coming fossil carbon being burned in the atmosphere and not coming from volcanoes ie. fossil fuels.
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  13. Sol - the 3 carbon isotope ratios tell us that the increase in atmospheric CO2 is due to fossil fuel emissions. Carbon 14 forms in the upper atmosphere due to bombardment by cosmic rays (high energy particles from distant, but powerful, stars). This isotope is radioactive so it emits and loses energy over time. Its half-life is about 6000 years. Fossil fuels are depleted in Carbon 14 because they formed over hundreds of millions of years underground. Burn fossil fuels and we should see a decline in the ratio of Carbon 14 in the atmosphere relative to the two other isotopes. This is what is observed. Plants favour the lighter Carbon 12 over Carbon 13 because it is energetically more efficient. Fossil fuels are made of dead organic matter, therefore a fossil fuel source should see an increase in the relative concentration of Carbon 12. Once again, this is observed. So that discounts volcanoes as a source. And most critically of all we actually have detailed inventories of the coal, oil, natural gas etc that we are burning and these match the observations too.
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  14. Alexandre @ 2. Further to Dana's reply, the connoisseur of 'how far will they' go may also enjoy Tim Curtin's '[CO2 and H2O] do not [trap heat] because they cannot, given the 2nd Law. Remember that photons are largely a fiction as they have no mass.' Well, that would also explain how they've escaped the Higgs Boson! Here, (comments page 6), June 23, 12.42pm (post NG's takeover ScienceBlogs no longer features permalinks, sadly.)
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  15. Rob Painting @ 13 Thanks for your comment. Why would carbon from volcanic activity be less depleted of c14 than fossil fuel carbon?
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  16. Sol, The volcanic carbon and the fossil plant carbon are both depleted in C14, because the carbon is so old the radioactive C14 has decayed. Plant carbon is depleted in a different isotope, C-13, for a different reason, because plants accumulate C-12 preferentially over the heavier C-13 during photosynthesis. During the recent increase in atmospheric CO2, C14:C12 and C13:C12 ratios have both declined. Together, these two patterns indicate that the CO2 added to the atmosphere is 1) old and 2) derived from plants. Thus, the source must be fossil plant material, ie fossil fuels. The increases also agree with what is expected based on known emission rates and natural exchange rates, although the C-14 has been complicated by addition atomic bomb testing in the 60s. You are correct that the observation that O2 has declined in proportion to increasing CO2 is one more line of evidence that oxidation of organic material. There are so many consistent lines of evidence that it still beggars belief that people question the anthropogenic source of the recent CO2 increase.
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  17. DM @8 Ugg... Well, that Spencer comment is a classic case of poisoning the well, and does not reflect well on his intentions. You do not judge the merit of an argument by its tone, but its content and you cannot assess content by ignoring everything else we know about a suject. The "strength and vehemence" of the objections to his ideas largely reflects frustration at Spencers unwillingness to engage in rational discourse through peer reviewed channels. To his credit, he did not dismiss "well-reasoned" objections as politically motivated. I think your approach here and in several posts on this issue are safe by his standards! The Salby comment is fine as it is. Of course, by ignoring evidence accumulated over a century of research on the issue, he is essentially doing exactly what he warns against. How is holding your hands over your ears less distructive to discourse than telling someone to shutup? The fact that he has not responded to you also hoists him by his petard.
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  18. Stephen Baines @ 16 Thanks. So a relative decrease in C14 = old and a relative increase in C12 = plant Therefore, if the rise were due to volcanoes then C14 would still decline but C12 would stay relatively constant. Sorry for dragging this off-topic
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  19. Exactamundo.
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  20. sol @18: Declining C14 ratio indicates old, hence fossil fuel or volcanic (ie, not oceanic outgasing or deforestation; Declining C13 ration indicates organic, hence not volcanic; Declining O2 concentration indicate combustion, hence not volcanic; Measured CO2 emissions from all (surface and beneath the sea) volcanoes are 1 hundredth of antrhopogenic CO2 emissions; hence not volcanic; Partial pressure of CO2 in the ocean is increasing, hence not oceanic outgassing; Known changes in biomass too small by a factor of 10, hence not deforestation; and as the icing on the cake, The start of the growth in CO2 concentration coincides with the start of the industrial revolution, hence anthropogenic; and Increase in CO2 concentration over the long term almost exactly correlates (corr.: 0.998; R^2: 0.997) with cumulative anthropogenic emissions: (Source) and finally, Annual CO2 concentration growth is less than Annual CO2 emissions, hence anthropogenic. In all, six independent lines of evidence preclude a volcanic source for the increased CO2; five independent lines of evidence preclude a source from the non-fossil biosphere; and three lines of evidence are only consistent with an anthropogenic source.
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  21. Rob, Stephen & Tom Thanks Sol
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  22. Tom Curtis at 08:22 AM on 8 July, 2012 That's a great summary! Do you have any link with that information so I can keep it as a reference?
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  23. Alexandre, an introduction to the basic information can be found in the article What is causing the increase in atmospheric CO2?. For the data on volcanic emissions, the article Do volcanoes emit more CO2 than humans? is a good start point. Anthropogenic Emissions from Land Use change and deforestation represent 10% of all human emissions (0.9 PgC of 10 PgC). Over the last century, human caused deforestation and other land use changes have been by far the largest cause of change in land cover, and hence natural changes cannot be significantly larger than that (and in fact are known to be a net sink of CO2 by the mass balance argument).(Source) Evidence that CO2 rise coincides with the start of industrialization is found in ice cores (Scroll to bottom for thousand year record), and also in other markers, including sponges: (Source) A little discussion of the O2 decline may be helpful. Because the change in solubility of O2 in water with change in temperature significantly differs from that of CO2, the change in O2 concentration is not effected by other possible CO2 sinks. That means the decline in CO2 concentration means any large unknown natural sources of CO2 must not come from a source of combustion but must come from a low C14 source generated by photosynthesis. These facts together almost completely preclude the existence of such putative natural sources. Because of the importance of the O2 decline, it is worthwhile lookinig at the chart below from the IPCC TAR which shows it: The observed decline in O2 is straighforward. The diagonal arrow from the start point marked "fossil fuel burning" represents the expected change in CO2 and O2 concentrations from known fossil fuel consumption. The arrow marked "ocean uptake" represents the uptake of CO2 by the ocean, which does not effect the O2 level. The arrow marked "land uptake" is the uptake of O2 by photosynthesis, which also reduces O2 concentration. Finally, the small arrow marked "outgasing" represents outgasing from of O2 from the ocean, which does not effect CO2 concentration. That outgasing is partly the result of a warming ocean, and partly a result of the very slight decrease in the partial pressure of oxygen in the atmosphere. These factors are reasonably well, but not exactly known. The most important fact is that because the fall in O2 concentration is significantly less than that predicted from known combustion of fossil fuels, the uptake of CO2 by photosynthesis must exceed the combustion or decay of modern organic material from either anthropogenic (Land Use Changes) or natural sources.
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    Moderator Response: TC: word "corals" deleted and "sponges" added.
  24. "Figure 8 shows a more realistic (subjective) attribution of the observed increase in atmospheric CO2. " Again, I wholeheartedly agree that the anthropogenic causation of the CO2 rise is (well, "should be") indisputable. Figure 8 isn't, however, an attribution so much as it is a mass-balance. A proper attribution regarding the anthropogenic contribution should tell me what the world would look like if we zeroed out anthropogenic emissions. There are 2 ways of doing that: 1) run a full carbon-cycle-climate model with no human emissions. We'd probably see approximately constant concentration since preindustrial. That gets us about 100% anthropogenic. 2) Run a carbon-cycle model with observed temperatures but no human emissions. I predict you'd see a small rise in CO2 due to the rise in ocean temps - maybe on the order of 5 to 20 ppm. So that would leave about 90% of the rise as anthropogenic (if we ignore the anthropogenic component of ocean heating). -MMM
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