Gregory et al 2019: Unsound claims about bias in climate feedback and climate sensitivity estimation

Reposted from Dr. Judith Curry’s Climate Etc.

Posted on October 18, 2019 by niclewis |

By Nic Lewis

The recently published open-access paper “How accurately can the climate sensitivity to CO2 be estimated from historical climate change?” by Gregory et al.[i] makes a number of assertions, many uncontentious but others in my view unjustified, misleading or definitely incorrect.

Perhaps most importantly, they say in the Abstract that “The real-world variations mean that historical EffCS [effective climate sensitivity] underestimates CO2 EffCS by 30% when considering the entire historical period.” But they do not indicate that this finding relates only to effective climate sensitivity in GCMs, and then only to when they are driven by one particular observational sea surface temperature dataset.

However, in this article I will focus on one particular statistical issue, where the claim made in the paper can readily be proven wrong without needing to delve into the details of GCM simulations.

Gregory et al. consider a regression in the form R = α T, where T is the change in global-mean surface temperature with respect to an unperturbed (i.e. preindustrial) equilibrium, and R = α T is the radiative response of the climate system to change in T. α is thus the climate feedback parameter, and F2xCO2 /α is the EffCS estimate, F2xCO2 being the effective radiative forcing for a doubling of preindustrial atmospheric carbon dioxide concentration.

The paper states that “that estimates of historical α made by OLS [ordinary least squares] regression from real-world R and T are biased low”. OLS regression estimates α as the slope of a straight line fit between R and T data points (usually with an intercept term since the unperturbed equilibrium climate state is not known exactly), by minimising the sum of the squared errors in R. Random errors in R do not cause a bias in the OLS slope estimate. Thus in the below chart, with R taken as plotted on the y-axis and T on the x-axis, OLS finds the red line that minimizes the sum of the squares of the lengths of the vertical lines.

However, some of the variability in measured T may not produce a proportionate response in R. That would occur if, for example, T is measured with error, which happens in the real world. It is well known that in such an “error in the explanatory variable” case, the OLS slope estimate is (on average) biased towards zero. This issue has been called “regression dilution”.

Regression dilution is one reason why estimates of climate feedback and climate sensitivity derived from warming over the historical period often instead use the “difference method”.[ii] [iii] [iv] [v] The difference method involves taking the ratio of differences, ΔT and ΔR, between T and R values late and early in the period. In practice ΔT and ΔR are usually based on differencing averages over at least a decade, so as to reduce noise.

I will note at this point that when a slope parameter is estimated for the relationship between two variables, both of which are affected by random noise, the probability distribution for the estimate will be skewed rather than symmetric. When deriving a best estimate by taking many samples from the error distributions of each variable, or (if feasible) by measuring them each on many differing occasions, the appropriate central measure to use is the sample median not the sample mean. Physicists want measures that are invariant under reparameterization[vi], which is a property of the median of a probability distribution for a parameter but not, when the distribution is skewed, of its mean. Regression dilution affects both the mean and the median estimates of a parameter, although to a somewhat different extent.

So far I agree with what is said by Gregory et al. However, the paper goes on to state that “The bias [in α estimation] affects the difference method as well as OLS regression (Appendix D.1).” This assertion is wrong. If true, this would imply that observationally-based estimates using the difference method would be biased slightly low for climate feedback, and hence biased slightly high for climate sensitivity. However, the claim is not true.

The statistical analyses in Appendix D consider estimation by OLS regression of the slope m in the linear relationship y(t) = m x(t), where x and y are time series the available data values of which are affected by random noise. Appendix D.1 considers using the difference between the last and first single time periods (here, it appears, of a year), not of averages over a decade or more, and it assumes for convenience that both x and y are recentered to have zero mean, but neither of these affect the point of principle at issue.

Appendix D.1 shows, correctly, that when only the endpoints of the (noisy) x and y data are used in and OLS regression, the slope estimate for m is Δyx, the same as the slope estimate from the difference method. It goes on to claim that taking the slope between the x and y data endpoints is a special case of OLS regression and that the fact that an OLS regression slope estimate is biased towards zero when there is uncorrelated noise in the x variable implies that the difference method slope estimate is similarly so biased.

However, that is incorrect. The median slope estimate is not biased as a result of errors in the x variable when the slope is estimated by the difference method, nor when there only two data points in an OLS regression. And although the mean slope estimate is biased, the bias is high, not low. Rather than going into a detailed theoretical analysis of why that is the case, I will show that it is by numerical simulation. I will also explain how in simple terms regression dilution can be viewed as arising, and why it does not arise when only two data points are used.

The numerical simulations that I carried out are as follows. For simplicity I took the true slope m as 1, so that the true relationship is y = x, and that true value of each x point is the sum of a linearly trending element running from 0 to 100 in steps of 1 and a random element uniformly distributed in the range -30 to +30, which can be interpreted as a simulation of a trending “climate” portion and a non-trending “weather” portion.[vii]  I took both x and y data (measured) values as subject to zero-mean independent normally distributed measurement errors with a standard deviation of 20. I took 10,000 samples of randomly drawn (as to the true values of x and measurement errors in both x and y) sets of 101 x and 101 y values.

Using OLS regression, both the median and the mean of the resulting 10,000 slope estimates from regressing y on x using OLS were 0.74 – a 26% downward bias in the slope estimator due to regression dilution.

The median slope estimate based on taking differences between the averages for the first ten and the last ten x and y data points was 1.00, while the mean slope estimate was 1.01. When the averaging period was increased to 25 data points the median bias remained zero while the already tiny mean bias halved.

When differences between just the first and last measured values of x and y were taken,[viii] the median slope estimate was again 1.00 but the mean slope estimate was 1.26.

Thus, the slope estimate from using the difference method was median-unbiased, unlike for OLS regression, whether based on averages over points at each end of the series or just the first and last points.

The reason for the upwards mean bias when using the difference method can be illustrated simply, if errors in y (which on average have no effect on the slope estimate) are ignored. Suppose the true Δx value is 100, so that Δy is 100, and that two x samples are subject to errors of respectively +20 and –20. Then the two slope estimates will be 100/120 and 100/80, or 0.833 and 1.25, the mean of which is 1.04, in excess of the true slope of 1.

The picture remains the same even when (fractional) errors in x are smaller than those in y. On reducing the error standard deviation for x to 15 while increasing it to 30 for y, the median and mean slope estimates using  OLS regression were both 0.84. The median slope estimates using the difference method were again unbiased whether using 1, 10 or 25 data points at the start and end, while the mean biases remained under 0.01 when using 10 or 25 data point averages and reduced to 0.16 when using single data points.

In fact, a moment’s thought shows that the slope estimate from 2-point OLS regression must be unbiased. Since both variables are affected by error, if OLS regression gives rise to a low bias in the slope estimate when x is regressed on y, it must also give rise to a low bias in the slope estimate when y is regressed on x. If the slope of the true relationship between y and x is m, that between x and y is 1/m. It follows that if regressing x on y gives a biased low slope estimate, taking the reciprocal of that slope estimate will provide an estimate of the slope of the true relationship between y and x that is biased high. However, when there are 2 data points the OLS slope estimate from regressing y on x and that from regressing x on y and taking its reciprocal are identical (since the fit line will go through the 2 data points in both cases). If the y-against-x and x-against-y OLS regression slope estimates were biased low that could not be so.

As for how and why errors in the x (explanatory) variable cause the slope estimate in OLS regression to be biased towards zero (provided there are more than two data points), but errors in the y (dependent) variable do not, the way I look at it is this. For simplicity, I take centered (zero-mean) x and y values.  The OLS slope estimate is then Σxy / Σxx, that is to say the weighted sum of the y data values divided by the weighted sum of  the x data values, the weights being the x data values.  An error that moves a measured x value further from the mean of zero not only reduces the slope y/x for that data point, but also increases the weight given to that data point when forming the OLS slope estimate. Hence such points are given more influence when determining the slope estimate. On the other hand, an error in x that moves the measured value nearer to zero mean x value, increasing the y/x slope for that data point, reduces the weight given to that data point, so that it is less influential in determining the slope estimate. The net result is a bias towards a smaller slope estimate. However, for a two-point regression, this effect does not occur, because whatever the signs of the errors affecting the x-values of the two points, both x-values will always be equidistant from their mean, and so both data points will have equal influence on the slope estimate whether they increase or decrease the x-value. As a result, the median slope estimate will be unbiased in this case.  Whatever the number of data points, errors in the y data points will not affect the weights given to those data points when forming the OLS slope estimate, and errors in the y-data values will on average cancel out when forming the OLS slope estimate Σxy / Σxx.

So why is the proof in Gregory et al. AppendixD.1, supposedly showing that OLS regression with 2 data points produces a low bias in the slope estimate when there are errors in the explanatory (x) data points, invalid? The answer is simple. The Appendix D.1 proof relies on the proof of low bias in the slope estimate in Appendix D.3, which is expressed to apply to OLS regression with any number of data points. But if one works through the equations in Appendix D.3, one finds that in the case of only 2 data points no low bias arises – the expected value of the OLS slope estimate equals the true slope.

It is a little depressing that after many years of being criticised for their insufficiently good understanding of statistics and lack of close engagement with the statistical community, the climate science community appears still not to have solved this issue.

Nicholas Lewis ……………………………………………….. 18 October 2019


[i] Gregory, J.M., Andrews, T., Ceppi, P., Mauritsen, T. and Webb, M.J., 2019. How accurately can the climate sensitivity to CO₂ be estimated from historical climate change?. Climate Dynamics.

[ii] Gregory JM, Stouffer RJ, Raper SCB, Stott PA, Rayner NA (2002) An observationally based estimate of the climate sensitivity. J Clim 15:3117–3121.

[iii] Otto A, Otto FEL, Boucher O, Church J, Hegerl G, Forster PM, Gillett NP, Gregory J, Johnson GC, Knutti R, Lewis N, Lohmann U, Marotzke J, Myhre G, Shindell D, Stevens B, Allen MR (2013) Energy budget constraints on climate response. Nature Geosci 6:415–416

[iv] Lewis, N. and Curry, J.A., 2015. The implications for climate sensitivity of AR5 forcing and heat uptake estimates. Climate Dynamics, 45(3-4), pp.1009-1023.

[v] Lewis, N. and Curry, J., 2018. The impact of recent forcing and ocean heat uptake data on estimates of climate sensitivity. Journal of Climate, 31(15), pp.6051-6071.

[vi] So that, for example, the median estimate for the reciprocal of a parameter is the reciprocal of the median estimate for the parameter. This is not generally true for the mean estimate. This issue is particularly relevant here since climate sensitivity is reciprocally related to climate feedback.

[vii] There was an underlying trend in T over the historical period, and taking it to be linear means that, in the absence of noise, linear slope estimated by regression and by the difference method would be identical.

[viii] Correcting the small number of negative slope estimates arising when the x difference was negative but the y difference was positive to a positive value (see, e.g., Otto et al. 2013). Before that correction the median slope estimate had a 1% low bias. The positive value chosen (here the absolute value of the negative slope estimate involved) has no effect of the median slope estimate provided it exceeds the median value of the remaining slope estimates, but does materially affect the mean slope estimate.

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34 thoughts on “Gregory et al 2019: Unsound claims about bias in climate feedback and climate sensitivity estimation

    • Regression dilution and its effect on climate sensitivity is an issue I have been pointing out for years , both here and on Climate Etc. see my article there on this subject. Many thanks to Nick Lewis for picking up this issue.

      https://judithcurry.com/2016/03/09/on-inappropriate-use-of-least-squares-regression/

      It is interesting that Forster & Gregory 2006 was the only paper which seemed to even recognise the issue but chose to bury it in an appendix and not cover it is the paper itself, or its conclusions.

      They invoked questionable logic for using OLS which gave exaggerated climate sensitivity reported in the paper.

  1. Sure,sure,sure, only in the world of GCM simulations can Co2 control the “Global Warming” changed to “Climate Change”.

  2. It is a little depressing that after many years of being criticised for their insufficiently good understanding of statistics and lack of close engagement with the statistical community, the climate science community appears still not to have solved this issue.”

    “It is difficult to get a man to understand something when his salary depends upon his not understanding it.”
    – Upton Sinclair.

    • Exactly.

      Nic is being cautious, I understand, like McIntyre. But it’s blatantly clear, after decades, that the climate science consensus crew isn’t interested in proper statistics. They understand them perfectly well, well enough to abuse them.

  3. Anyone with a basic knowledge of radiative heat exchange will realize that an equation such as R = α T has the implicit assumption that a Stephan-Boltzmann radiation law applies, based on a blackbody-type radiation spectrum and holding CONSTANT both effective emissivity of the radiating surface and total area of radiation. Under such assumptions, this simplified equation can be derived by dR/R = C*dT/T, where R is radiated power, T is the absolute temperature of the radiator, and C is a numeric constant.

    The fundamental mistakes in applying the simplified equation to an atmosphere-covered Earth is that:
    a) the average spectrum of back radiation from Earth’s surface is nowhere near blackbody,
    b) the effective emissivity of Earth’s surface for both incoming radiation and outgoing radiation is nowhere near constant due to variable atmospheric humidity and cloud coverage and to changes in snow, ice and plant coverage.
    c) the area of Earth’s “surface” that can be considered as either receiving or emitting radiation at a given temperature T is highly variable over time, such as the variation from the tops of clouds (when present) to the Earth’s land surfaces to the Earth’s sea surfaces.

    R = α T is useful, perhaps, for Gedanken-Experiments, but not for developing quantitive predictions for Earth.

    • Yes Gordon. Another factor is that Earth’s Atmosphere is not a surface (the basis of the the S-B equation is between two surfaces in a vacuum) Prof Hoyt Hottel made some estimates for radiation to and from gases in an enclosure (see Chemical Engineering Handbook). The absorptivity of a gas mixture can be determined from the partial pressures of the absorbing gases (in flames water vapor and CO2), the temperature of the gas, the wavelength absorption factors for the gases, and the path length through the gas. (note the absortivity and emissivity in the same surrounds and at the same temperature are equal according to Kerchoff’s law) Making that calculation for the atmosphere which needs a bit of maths gives that the absorptiviy over a path length of 5 kms is very small and so the radiation absorption by CO2 is close to zero. Then going further cold CO2 in the atmosphere can not radiate back to the Earth surface. The supposed sensitivity is then zero which coincides with measurements that surface temperature changes (particularly in oceans ) leads CO2 changes at the surface at all ages from daily to 1000’s of years. (for daily and seasonal look at the measurements by Kreutz)

  4. The “Gregory grab” can be summed up very simply.

    Claiming ALL the global temperature increase in the last two centuries, as entirely a function of climate sensitivity to CO2, is not enough.

    They now claim all the warming – and then some.

    So out of every 2 degrees of warming, 3 are caused by CO2.

  5. The determination of ECS by looking at changes in global temperature over time is based on the unproven (and essentially unprovable) assumption that changes in global temperature are caused only by changes in atmospheric CO2 content.

    I can re-phrase that statement in a more contentious-sounding way:

    The determination of ECS by looking at changes in global temperature over time is based on the unproven (and essentially unprovable) assumption that natural climate variability ceased in 1850.

    Critiquing of the statistical methods used is (in my opinion) deflecting attention away from the circular reasoning that is so deeply embedded in modern climate science. It’s a bit like complaining that the style of the Emperor’s clothes is rather outmoded because those bell-bottoms went out with the 1970s, when the real question is – do they exist or are they just a hologram?

    And as for the adequacy and reliability of historical temperature records and the question of how they have been adjusted (“tampered with”), there’s nothing I can add to countless posts here on WUWT or on Tony Heller’s site.

    • Agreed, 100%. The first thing that jumped out at me was the whole discussion being based on “the change in global-mean surface temperature with respect to an unperturbed (i.e. preindustrial) equilibrium” – WHAT?!

      There IS NO “equilibrium,” and to the extent there was any “stability,” it would not be changed measurably by CO2 level changes. They have once again started with the UNPROVEN and UNSUPPORTED assumptions that (1) CO2 “drives” temperature and (2) that ALL temperature change since the beginning of the industrial age is CAUSED by CO2. All bullshit, as usual.

      “Critiquing of the statistical methods used is (in my opinion) deflecting attention away from the circular reasoning that is so deeply embedded in modern climate science.” – A perfect summary of the problem. It’s like we’re complaining about the attire of a terrorist trying to board an aircraft while ignoring his baggage full of guns, knives and explosives.

      • AGW climatology is a subjectivist narrative, equivalent to the cultural studies and critical theory pseudo-intellectual projects that dominate academic Sociology and Humanities departments.

        Subjectivist narratives assume what should be proved, give their assumptions the weight of data, and every study is confirmatory. AGW climate non-science to a ‘T.’

        The whole AGW enterprise is pseudo-science.

      • That sentence (“change in global-mean surface temperature with respect to an unperturbed (i.e. preindustrial) equilibrium”) also immediately jumped out at me.

        They are literally denying climate change.

      • Gah, the reasoning is more circular than it appears. If one claims that the climate temperature has risen from t=1850 to t=today, and then further claim the the variation (vaguely claimed to be “error”) causes the resulting regression to underestimate the true slope over that time interval, then the end-point variables, T(1850) and T(today), have equal uncertainty/error.

        Consequently one can then calculate a standard error (pick your poison) and then check to see whether the claimed temperature rise exceeds that which could be expected from the error (i.e.: the error bars overlap). If that be the case, then you’ve got bupkis.

        I haven’t done the math, but since the claim is that the global temperature has risen about 1.2 C during that period, then me thinks one is up the proverbial creek without the canoe if the year-to-year variability in climate temperature is on the order of 0.6 C.

        Ancillary Question: If the global temperature record under-reports global warming, does that make it a climate denier?

      • AGW is not science
        Agreed – how can there be equilibrium sensitivity?
        Or equilibrium anything?
        In a far-from-equilibrium system such as climate?

        Even to call current catastrophist climate science an “ill-posed question” would be to give it far too much respect.

  6. There is no such thing as CS except as a concept for comparing models tween themselves and with past observations. CS is no good as a predictive number. Or, if you say it is, show me.

  7. Basically, the u.n. and its sister company Ipcc, has lied so it can take billions from governments, because it believes the world has to many poor ungrateful people in it. The plan is to cause mass anxiety and fear among the planets population, then they will take control of everything everybody needs, Oil! Once they control that they control the remaining population… All of this is of cause for the greater good of humankind 😐 Not forgetting XR and greta the co2 seeing puppet (has anybody found how rich greta and her family has become)??

  8. According to these statistical machinations, global warming is the only thing keeping us out of the next glaciation. Go! Drive! Breathe out fearlessly!

    • I printed up some bumper stickers that say ‘Save the Planet/Emit CO2’, and have had two of them broadly displayed on the back of my car for about 4 months. I was inspired to do this by the ‘Maxim of our resolve” quote at the upper-right of this website. The bumper stickers were a small step towards the fire.

      I was prepared to take a lot of flack, but so far, there has been very little reaction. I believe this is because most people don’t understand what it means. I have had no opportunities to expound on my pro CO2 stance out in the real world.

  9. The grand blinds
    It is that CO2 cannot absorb vast amounts of heat by vaporizingg and condensing and freezing as water does, transporting vast amounts of heat from surface to space. See paullitely.com

  10. Here’s the Connolly’s power point presentation in Tucson Arizona USA in July. Well worth an hour of your time.
    A lot to take in but this is all about the actual balloon data over a long period of time.
    No modelling or theories or guesses, just the results of millions of balloon flights over decades.
    There’s a very short Q&A at the end. I hope those interested have the time to look at the video and perhaps have a friend who understands the chemistry + data etc involved?
    Way beyond my capabilities.
    https://www.youtube.com/watch?v=XfRBr7PEawY

  11. Check out the Australian Bureau of Meterologly website data on line for the Mawson and Casey bases in the Antarctic. There hasn’t been any warming in 65 years.

  12. “However, when there are 2 data points the OLS slope estimate from regressing y on x and that from regressing x on y and taking its reciprocal are identical (since the fit line will go through the 2 data points in both cases)”

    Nic,
    there is only one possible straight line connecting two points, however you turn it.
    What is needed with >2 points is an errors-in-variables model. Don’t know which software package offers this. Posssibly an OLS model can be adapted (you need the ratio of variances if I remember well but I forgot the details.)

    As a heuristic way to get at least a less biased estimate of the slope,
    the “GM (geometric mean) regression” was proposed:

    GM slope = sqrt[slope(yx)*slope(xy)^-1]

    Using this, one might investigate what change it generates in the physical model.

    [I’ve not yet read your text until the end, nor followed Greg’s link, so this might be redundant]

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