Has Warming in the Lower Troposphere been Statistically Significant?

“Has there been any warming yet?”

This question should have been the touchstone where every policy analyst should have started from the beginning. The corresponding scientific question should have started with, “Has the observed warming been statistically significant? Astonishingly, that did not happen.

This article examines lower troposphere temperature anomalies in conjunction with carbon dioxide levels using classical regression methods. These techniques are accessible to anyone with no more than a minor in undergraduate statistics. The technical question to be asked and answered is, has the observed warming to date been statistically significant?

Conclusion. At first blush, the empirical evidence appears to support the assertion that there has been warming. However, after conventional model diagnostics and reformulation, the statistical significance disappears completely and we must conclude that the observed warming does not meet any reasonable criterion of statistical significance. The observed warming could easily be the result of simple chance.

– – –

Why examine temperatures of the lower troposphere when it is the surface temperatures we experience?

  • There is no reason to expect any substantive temperature divergence between the surface and the lower troposphere that lies directly above it.
  • Temperatures for the lower troposphere have been consistently and regularly measured by satellites over the past thirty-five years. This is very high quality data.

CO2 and Trend Models. Let’s start with a picture. To anyone who has investigated global warming, this image should be familiar.


The trend line certainly looks convincing.

Next, let’s look at the regression of temperature anomaly on carbon dioxide levels.


The regression looks strong. 42% R-Squared. With a t-statistic of 17.69 and a significance <0.0001, the coefficient for CO2 certainly looks significant.

Diagnostics. What’s left to check? Model assumptions. Are the model’s errors (aka residuals) normally distributed, constant (homoskedastic) and independent?

To cut to the chase, the Durbin-Watson statistic of 0.49 tells us that there is something seriously amiss with this CO2 model. The D-W falls well below the lower bound of 1.65 for D-W from a standard table for >=100 observations. Therefore, we are forced to consider the implications of significant auto-correlation in the model’s residuals.

A check of the ACF (autocorrelation function) and PACF (partial autocorrelation function) on the residuals strongly confirms significant and substantial, even profound, violations of residuals independence at both the first and second lags. The t-statistic for the PACF for the first lag is 15.8 and the second is 5.1. Both have positive signs.


What does this mean? Estimates or inferences that depend on error variance are suspect, at best. That includes any tests of statistical significance. The errors are not independently and identically distributed (iid). We often push the limits on statistical assumptions for normality and constant variance, but not independence.

There is a related point to consider. Carbon dioxide and temperature have both been increasing over this time interval.  So, they are correlated. However, does CO2 level do a better job than a trivial time trend model? If CO2 were a useful explanatory variable, we would expect it to perform at least a little better than a trivial trend model.

Does CO2 do any better than case number? No. Model testing shows that the standard error for the CO2 model is 0.1727 while the standard error for a trivial trend (case number) model is 0.1728. Note: lower is better. An improvement of 0.0001 is no difference. This is a tell to experienced modelers that the CO2’s correlation relationship with temperature is spurious.

Can we even answer the question? All is not lost, of course. We can still move forward and develop a model to learn whether the warming has been statistically significant. In general, how do we reformulate models when our error terms are riven with autocorrelation? We dust off our Box-Jenkins text and try out an ARIMA model.

Cutting through a pleasant afternoon of model exploration, this parsimonious ARIMA(1,1,0) model emerged as adequate for our purposes.


ARIMA(1,1,0) is a simple change model. The middle number ‘1’ means that this is a first difference (simple change) model. The first ‘1’ means that there is a single autoregressive term. That is to say, each observation is closely related to the previous. The ‘0’ means that there is no moving average (MA) term.

In the output table, the ‘Overall Constant’ can also be referenced as ‘drift’. This term corresponds to the trend coefficient in the simple trend model. The (AR)P(1) term is the autoregression coefficient. In mathematical form, this model says:

  • dX(t) = Drift + AR * dX(t-1) + A(t,0,SE)
  • dX(t) = 0.0017 – 0.3359 * dX(t-1) + A(t,0,0.1131)


  • dX(t) is the change for time period t.
  • Drift is the amount of expected change for every time increment. In this model, that would be 0.0017 degrees per month or 0.20 degrees per decade.
  • dX(t-1) references the previous change
  • AR is the coefficient that adjusts the previous change. The -0.3359 value indicates strong reversion or rebound.
  • A(…) refers to white noise for time period t with mean of zero and standard error of 0.1131.

Compare the much improved ARIMA standard error of 0.1131 with standard error of 0.1728 for the CO2 and simple trend models. The ARIMA model passes model diagnostics for normality, independence and constant variance.

Where’s the Warming? The ‘Overall Constant’ is the drift term that corresponds to global warming. If there were warming, this term would show significance. In this model, the monthly change is 0.0017 degrees (0.20/decade) with a standard error of 0.0135. The t-ratio is the ratio of the constant to the standard error.

What does the t-ratio tell us? From the t-statistic we infer the likelihood that a result came about as the consequence of chance. A t-ratio greater than 1.96 (~2.0) indicates a likelihood of <.05 (<5%) that a result was the result of randomness. This is called p-value. Interpretation of t gets more involved when there are fewer than thirty observations.

In academia, the lowest standard for a t-ratio to be considered significant has traditionally been 1.96. In the business world, I have used terms in models with t-ratios as low as 1.2.

The t-ratio for the drift term for this reformulation is  (0.0017/0.0135) = 0.1224. That is to say, the drift is not significantly different from zero. No self-respecting analyst who wanted to keep their job would ever consider retaining a term in a model with a t-ratio of 0.1224.

Does this mean that there has been no global warming? Not at all. What this does mean, however, is that in the thirty six years since we started taking temperature measurements from satellites, there has been no statistically significant warming in the lower troposphere. This is not even a close call. The observed warming could very very easily be the mere consequence of random variation. That is to say, nothing out of the ordinary with respect to lower troposphere temperature changes has occurred.

Nevertheless, this flatly contradicts the models put forward by warming activists. Over the past thirty-six years, carbon dioxide levels in the atmosphere have increased by nearly twenty percent. If that change hasn’t produced statistically significant changes in temperature, then their models lack validity. Their predictions have zero basis in the empirical evidence.

Yes, it really is that simple.

– – –

Data Notes. The data used was acquired from two sources. Anyone can recreate this analysis with these data tables.

Carbon dioxide levels were downloaded from the Earth System Research Laboratory.



Lower Troposphere temperature anomaly records were obtained from National Space Science and Technology Center, hosted at the University of Alabama, Huntsville.



The CO2 data is in weekly form while the satellite temperature anomaly data is aggregated monthly by the NSSTC. The carbon dioxide data is labeled ‘CO2 molefrac’, while the temperature data is labeled ‘Globe’. To get the weekly CO2 data to mesh with the monthly temperature data, the average was taken of the weeks over the month. While there were several missing weeks, there were no entire months with missing data. There was no additional processing or transformation of the data.

  • Nanker Phelge

    Nice little blog. pretty graphs. But thanks, I think I’ll listen to the National Academy of Sciences on this topic.

    • http://risquant.com James Carson

      Perhaps you can point us to a specific analysis that demonstrates statistical significance.

      • Lakota in Austin

        You’re an uninformed amateur performing simple regression analysis on climate data. The fact that you consider this ridiculous blog to be anything other than an exercise in sophomoric hubris is sad and egomaniacal. But congratulations for learning how to abuse your trivial understanding of “statistical significance” as a means to support your pre-existing beliefs.

        Edit: Oops, I assumed you were the author of this page. My remarks are directed at that individual. If you happen to trust in the analysis on this page, all I can tell you is that you can’t do regression analysis on a massively complex chaotic system such as climate and expect to come up with much that is statistically significant at the level of analysis required in order to detect signals amidst the noise. Teasing out the spectacular number of influences on climate to detect which factors are doing what is an art as well as a science, and the mathematical modeling, along with the size of data sets, requires supercomputers to perform runs for many hours to many days just top come up with a single estimate, which by itself is only a data point for additional analysis.

        Anyone who tries to demonstrate an alternative analysis in a few pages is either intentionally misleading you or they’re just not very sophisticated in their understanding of climate science and atmospheric dynamics. The danger here is that the general public isn’t smart (and this applies to those who do and those who don’t accept the scientific consensus re: global warning), so simple, intuitively appealing arguments that are in accord with our own biases readily gain acceptance.

        • http://GreenHeretic.com/ GreenHeretic

          I am the author of this page. Sorry that I missed your post and did not respond earlier.

          Your overly personal attack on myself is mistaken.

          I am no “amateur” wrt statistical analysis of weather data. I’ve performed such analyses since 1998 in the energy sector. This includes original work in modeling temperatures.
          As a professional in the energy (electricity) sector, I know a lot of meteorologists. I have passed this analysis by several of them, a couple say that I am right and none say that I am mistaken.
          Einstein said in an article entitled “Induction and Deduction” (translated and paraphrased from German): No amount of experimentation can ever prove me right; a single experiment can prove me wrong. Central to the practice of real science (even relativity) is the notion that any theory can be ‘falsified’ empirically. If the lack of statistical significance after thirty five years does not falsify AGW theory, then it cannot be falsified, and therefore is not science. Regardless of the complexity of a chaotic system, if you are going to claim that there has been warming, then that statement should be able to withstand basic scrutiny.
          If I am wrong, please cite the specific study (with tests of significance and i-i-d) that contradicts me. I suspect that you tried, but could not find anything.

          • http://GreenHeretic.com/ GreenHeretic

            Please note that the deleted comments were deleted by the writer, not by myself.

    • American_Sharecropper

      Nice little blog. pretty graphs. But thanks, I get my propaganda straight from the Kings alchemists on this topic.

      • Lakota in Austin

        How is it possible for a literate person, such as yourself, to embrace such a bizarre world view? You somehow trust people who spout opinion without any basis in fact, while the professionals who devote their lives to studying complex phenomena are the people you least trust? I get that there’s an emotional appeal to the way you’re reacting to the world around you, because science is cold and fact-oriented and it often overturns our most cherished traditional beliefs. Is that what this is about, that you’re frightened and feel threatened by the findings of science? Evolution debunks creation, genetics proves homosexuality has a biological basis, and scientists are always finding out that the very technological inventions that scientific progress makes possible often come with disastrous consequences.

        So perhaps you want to return to traditional values, while keeping the technological innovation. This is a catastrophic combination, like putting deadly weapons in the hands of innocent children. We can’t afford to ignore the consequences of our actions. Aren’t conservatives always harping on the need for people to take personal responsibility? Then let’s do it, in every aspect of our lives. Let’s be responsible, and stop running away from the truth by pretending that scientists are engaged in a fanciful conspiracy to enrich themselves or impose a world government or whatever it is that you believe is motivating them.

        • http://GreenHeretic.com/ GreenHeretic

          Why would anyone ‘trust’ anyone else when they can verify the analyses themselves? The fact is that the AGW theory does not withstand empirical scrutiny. QED.

          • Kevin O’Neill

            One reason to ‘trust’ experts is that they typically have knowledge that amateurs do not. Those that *think* they have the same knowledge as experts usually suffer from Dunning-Kruger. We might consider the case of Richard Muller and BEST as a case in point.

            As Tamino points out, your analysis is flawed. That you didn’t realize it was flawed is simply a sign that you lacked the required knowledge. Not a lack of intelligence, but more a sign of hubris. Any time one comes up with a result that flies in the face of accepted scientific knowledge one should *immediately* suspect one’s own understanding. It is rare that a scientific breakthru comes from an amateur. Science is highly competitive. Almost every result is scrutinized by others looking to poke holes in it. The more basic or simple the ‘mistake’ made by experts the less likelihood there is that it is actually a mistake, but instead a blunder on the part of the amateur playing around outside his area of expertise.

            The internets are filled with amateurs telling us scientists and experts are all wrong. Add one more to the list. And add one more to the list of those obviously suffering from D-K.

          • http://GreenHeretic.com/ GreenHeretic

            Wasn’t that the foolish argument that silenced Galileo? You’re an idiot.

          • Kevin O’Neill

            No, Galileo was one of the experts of his day. He was a professor of mathematics, geometry, mechanics, and astronomy. He was as close to a professional scientist as existed in the late 1500s/1600s. Not only that, he was silenced by the science deniers of his day – the Christians.

          • http://GreenHeretic.com/ GreenHeretic

            You know less about history than you know about science.
            Neverthless, by your standard, we should have ignored the contributions of two bicycle mechanics and a patent clerk.

          • Kevin O’Neill

            Do you dispute he was a professor of the subjects cited?

            Do you dispute that he was silenced by those that denied the truth of his scientific findings?

            Do you deny that group of deniers was christian?

            What exactly do you disagree with? If my history is wrong, please cite some references that say otherwise – else we might consider who is and is not an idiot and who actually knows something about history. Wikipedia seems to back me up.“>

          • http://GreenHeretic.com/ GreenHeretic

            Answered above.

          • Kevin O’Neill

            No, you didn’t answer anything – you managed to misuse the term ‘solar parallax’ when the actual argument against heliocentrism is ‘stellar parallax.’ Galileo didn’t discover heliocentrism – that was Copernicus. Scientists of the day were divided – even within the Catholic church. As pointed out above, Copernicus’s works were well known, but weren’t banned until a group came to power within the church that opposed the moral/theological implications of heliocentrism that Gaileo was expounding many decades later.

            I don’t find it surprising that you have a very suspect knowledge of history.

          • http://GreenHeretic.com/ GreenHeretic

            See above.

          • Kevin O’Neill

            Why? Einstein was a PhD physicist! So he worked for awhile in a patent office – not every post-doc immediately gets a job in academia. Einstein was always recognized as being gifted by his teachers (though some also thought he was a bit obnoxious). Einstein graduated from college at 21, had a Ph.D. by 26, and was a professor at the top university in the German-speaking world by the time he was 35. No one claimed Einstein was a whack nut-job. And he built on the work of many other scientists before him. His work was accepted very quickly considering the difficulty in gathering physical evidence to ‘prove’ it.

          • http://GreenHeretic.com/ GreenHeretic

            Einstein hadn’t received his PhD at the time he submitted his paper on special relativity. They were nearly coincident, but the PhD was awarded a month (July) after he submitted that paper in June. LOL.

          • Kevin O’Neill

            I’m sorry – I missed your point? Did I say otherwise? I said he received his PhD by the age of 26. Many people publish papers before they receive their PhDs. Some are – as was Einsteins – highly regarded papers. Again, did you have a point? LOL might have looked cute when you wrote it, but it’s not much of an argument.

          • http://GreenHeretic.com/ GreenHeretic

            Yes, you did ‘say otherwise’. He received his PhD AFTER submitting his paper on Special Relativity. Was he any sort of acknowledged expert? No.

          • Kevin O’Neill

            “No flying machine will ever fly from New York to Paris …[because] no known motor can run at the requisite speed for four days without stopping.” – Orville Wright

          • http://GreenHeretic.com/ GreenHeretic

            And Bill Gates said that we would never use 640 of RAM in a computer. POINT?

          • Kevin O’Neill

            I would ask the same. What was your point? Many people were attempting to build self-powered flying machines capable of carrying a human at the time. The Wrights were not unique. They had as much experience as anyone else around at the time. Their backgrounds in mechanics was well-suited to building one of the first human-flight capable machines. They were as much ‘experts’ as anyone around at the time. It was a wide open field and within a few years many people were flying. Their importance is probably over-stated.

          • http://GreenHeretic.com/ GreenHeretic

            My point should be obvious, even to you. Is this misdirection a strategy of yours? To belabor my obvious point, the Wright brothers were not acknowledged experts, yet they achieved first powered flight. Your original point, which you seem to have lost, was that we should trust the experts. YOUR view applied to many other situations is just dumb.

          • Kevin O’Neill

            And as *I* pointed out – there were no experts on human powered flight – it didn’t exist yet. So it’s hard to claim they weren’t as expert as anyone else. In fact probably more so since they read everything the Smithsonian had on flying machines. This alone probably made them more expert than anyone. What is *obvious* is that you haven’t thought this through.

          • http://GreenHeretic.com/ GreenHeretic

            While Galileo was ultimately correct (mostly), the scientists of his day contradicted heliocentrism by citing the absence of solar parallax. The Church relied on their analysis in large part to silence him. Geocentrism was the prevailing scientific view of his time.

            That is what I meant when I said that you know less about history than you know about science. Your understanding is merely cursory.

          • Kevin O’Neill

            No, you said, “Wasn’t that the foolish argument that silenced Galileo? You’re an idiot.”

            Science was divided at the time on heliocentrism vs geocentrism. The Catholic church had generally backed science and astronomers. Heliocentrism was no exception. Copernicus’ work was well known and his books had never been banned. In fact, De revolutionibus orbium coelestium opens with a letter from Copernicus’ friend, Nikolaus von Schönberg, Cardinal Archbishop of Capua, urging Copernicus to publish his theory.

            Galileo had cultivated influential friends in the church, but when they fell out of power he was left at the mercy of those he had previously scorned. His theological views, which he believed were a direct result of heliocentrism, are what got him in trouble. In fact, Copernicus books were allowed to be published and Galileo allowed to discuss heliocentrism as long as it was passed off as certain. I.e., as long as it was kept as a hypothesis it was OK.

            And how does your original statement – “Wasn’t that the foolish argument that silenced Galileo?” apply to any of this?

            Galileo was silenced by people that denied the truth of scientific discovery. He wasn’t the first or the best known proponent of heliocentrism. Galileo, vis a vis heliocentrism, was following in the footsteps of Copernicus and several Islamic astronomers. Rather than defying the ‘experts’ he was defying the church, not so much on the science of heliocentrism, but the theological/moral implications of heliocentrism.

            Most people that try to cite Galileo in the context you used have no idea what they’re talking about. You’re no exception.

          • http://GreenHeretic.com/ GreenHeretic

            Galileo was silenced by those who believed that the scientific fact of the absence of stellar parallax disproved him scientifically, NOT by Church dogma as you falsely claim. YOUR point was to ‘trust experts’. The scientific experts at the time of Galileo were able to contradict his claims with science and used that to shut him down, very much like what YOU are trying to do with your silly belittling of a point of view that differs from your own. Get it now?

          • Kevin O’Neill

            Copernicus was an acknowledged scientific expert – as was Galileo (hence the professorship teaching astronomy). The church – as i pointed out – was not a monolithic organization with one set of views. You do not answer why a Cardinal of the church would *urge* Copernicus to publish his views. You do not answer why the church would continue to allow Galileo to continue discussing and writing about the heliocentric theory – as long as he kept it in the context of one possible explanation – and not *the* truth.

            You say you’re not an astronomer. I can also deducee you don’t have a degree in any of the physical sciences. For instance you write: “There is no reason to expect any substantive temperature divergence between the surface and the lower troposphere that lies directly above it.” Oh, really?

            1) The lower troposphere as measured by satellites is a proxy measurement – measuring microwave radiation at various frequencies and calculating an implied temperature. (surface measurements are direct measurements). As with all proxy measurements they are rarely more reliable than direct measurement.{1}

            2) The lower troposphere as measured by satellites is a vertically weighted average of the atmosphere that – depending on the weighting scheme – averages to about 4000m. (do you expect no divergence in temperatures between the surface and 13,000 ft altitude?)

            3) Consider the arctic ocean in summer – the surface temperature is basically pegged to the melting point of ice; for months at a time each year the surface temperatures north of 80 degrees varies by a couple degrees C. The lower troposphere over this same region has no physical constraint within this same window.

            4) Ocean currents typically move much slower than air currents. Warm Kelvin waves can take months to cross the ocean. The air thousands of feet above them moves much more quickly.

            Should we go on?

            {1}”As a data scientist, I am among the first to acknowledge that all climate datasets likely contain some errors. However, I have a hard time believing that both the satellite and the surface temperature datasets have errors large enough to account for the model/observation differences. For example, the global trend uncertainty (2-sigma) for the global TLT trend is around 0.03 K/decade (Mears et al. 2011). Even if 0.03 K/decade were added to the best-estimate trend value of 0.123 K/decade, it would still be at the extreme low end of the model trends. A similar, but stronger case can be made using surface temperature datasets, which I consider to be more reliable than satellite datasets (they certainly agree with each other better than the various satellite datasets do!). So I don’t think the problem can be explained fully by measurement errors.” – Carl Mears, Senior Research Scientist and Vice-President of Remote Sensing Systems (otherwise known as the producers of the RSS satellite dataset).

          • http://GreenHeretic.com/ GreenHeretic

            Your points about Copernicus support MY view. Why would I ‘answer’ them? You originally made the foolish statement that it was the religious men of the Catholic Church that silenced Galileo. On the contrary, it was his fellow scientists of his age who fully participated in that effort by providing the scientific basis for shutting him down.

            Are you saying that the lower troposphere doesn’t closely track what is going on underneath it? That’s the only way your point has any relevance to this discussion. If you want to make a fool of yourself with that claim, be my guest.

          • Kevin O’Neill

            No, my point was that Copernicus was the acknowledged expert. Galileo read and followed Copernicus. You seem to forget that it was the *church* that silenced Galileo – not an army of sciientific experts. In fact, as I pointed out, they did so on theological grounds – not even claiming that heliocentrism was wrong, just that it wasn’t proven. It was *theology* that caused Galileo to fall under the eye of the Inquisitors.

            And again, I repeat myself because you fail to answer arguments – somehow believing that a sharp elbow constitutes an effective rebuttal. Copernicus was urged by the Cardinal of Capua to publish his paper on heliocentrism. There was widespread debate over whether it was correct or not – and few saw it as an issue the church would look aghast at. This buttresses my argument that Galileo was silenced on theological grounds. As does the fact he was allowed to continue discussing and writing about heliocentrism as long as he didn’t pass it off as proven fact.

            Now I realize most of this is over your head. You have shown an innate ability to jump to insults rather than reason or logic. You should really try to read up on a subject and understand it from multiple sources before jumping in clueless.

            As for divergence of the LT and the surface, there is this: “The new LT trend of +0.114 C/decade (1979-2014) is 0.026 C/decade lower than the previous trend of +0.140 C/decade, but about 0.010 C/decade of that difference is due to lesser sensitivity of the new LT weighting function to direct surface emission by the land surface, which surface thermometer data suggests is warming more rapidly than the deep troposphere.” – Dr. Roy Spencer. Think he knows something about the UAH dataset? I agree that you are making a fool of yourself.

          • Kevin O’Neill

            I know the difference between stellar parallax and solar parallax. An absence of solar parallax would be really, really, interesting. Interesting how the person who thinks he knows so much is consistently wrong. Do you ever self-reflect?

          • http://GreenHeretic.com/ GreenHeretic

            You got me. I should have said, ‘stellar parallax’. I’m not an astronomer.

          • http://GreenHeretic.com/ GreenHeretic

            Who is ‘Tamino’ and where does he/she point out any such thing? I don’t see any posts from that person. FYI, I did not delete the comment above.

            Why do you think that I haven’t had my analysis checked by meteorologists and other professional statisticians? As a matter of fact, I have. As a professional statistician, myself, with decades of experience, I know better than to stake out a position like this without having my methodology checked. They all say that it is right. Why do you ignore the growing number of professionals in climatology and meteorology who disagree with the so-called consensus?

          • Kevin O’Neill

            The first thing to do is try Google. Tamino occupies most of the first few results. Or you can follow this link: https://tamino.wordpress.com/2015/09/07/more-mathturbation/

          • http://GreenHeretic.com/ GreenHeretic

            Thanks for letting me know that I received a mention in his blog TWO DAYS ago! He should have let me know that so that I could respond to him. I will.

          • http://GreenHeretic.com/ GreenHeretic

            FYI, I posted a response to Tamino’s stupidity over on his site. I also posted it as an article here in case he decides to not approve it.


  • http://GreenHeretic.com/ GreenHeretic

    FYI, Clean Technica has blocked me from commenting on their Disqus feed SOLELY for the reason that they disagree with my posts. That makes them hypocrites.