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This message in a Reuter's article from 25.02.2019 is currently all over the news:

Evidence for man-made global warming hits 'gold standard'

[Scientists] said confidence that human activities were raising the heat at the Earth’s surface had reached a “five-sigma” level, a statistical gauge meaning there is only a one-in-a-million chance that the signal would appear if there was no warming.

I believe that this refers to this article "Celebrating the anniversary of three key events in climate change science" which contains a plot, which is shown schematically below (It is a sketch because I could not find an open source image for an original, similar free images are found here). Another article from the same research group, which seems to be a more original source, is here (but it uses a 1% significance instead of $5\sigma$).


The plot presents measurements from three different research groups: Remote Sensing Systems, the Center for Satellite Applications and Research, and the University of Alabama at Huntsville.

The plot displays three rising curves of signal to noise ratio as a function of trend length.

anthropogenic signal

So somehow scientists have measured an anthropogenic signal of global warming (or climate change?) at a $5\sigma$ level, which is apparently some scientific standard of evidence.

For me such graph, which has a high level of abstraction, raises many questions$^{\dagger}$, and in general I wonder about the question 'How did they do this?'. How do we explain this experiment into simple words (but not so abstract) and also explain the meaning of the $5\sigma$ level?

I ask this question here because I do not want a discussion about climate. Instead I want answers regarding the statistical content and especially to clarify the meaning of such a statement that is using/claiming $5 \sigma$.


$^\dagger$ What is the null hypothesis? How did they set up the experiment to get a anthropogenic signal? What is the effect size of the signal? Is it just a small signal and we only measure this now because the noise is decreasing, or is the signal increasing? What kind of assumptions are made to create the statistical model by which they determine the crossing of a 5 sigma threshold (independence, random effects, etc...)? Why are the three curves for the different research groups different, do they have different noise or do they have different signals, and in the case of the latter, what does that mean regarding the interpretation of probability and external validity?

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    $\begingroup$ @MattF. My expectation is that it will be possible to make a simple exposition that explains the statistical concept of the $5\sigma$ threshold that has been used here (at least the high energy particle physicists, who also use $\sigma$ discrepancies/effects to describe signal to noise ratio's in counts of events, have no problem with this). With simple I mean something stripped away from the climatology jargon, but sophisticated enough to contain the essence. Say, it would be something written for professional statisticians and mathematicians such that they can understand the $5\sigma$ here. $\endgroup$ – Martijn Weterings Mar 6 at 14:50
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    $\begingroup$ To stress the contrast with high energy physics: for this field statisticians can understand that the $5\sigma$ level is basically meaningless and the bar is set high because the computation is technically wrong (1. the look elsewhere effect 2. wrong assumptions about the error distribution ignoring systematic effects 3. implicitly doing a Bayesian analysis, 'extraordinary claims require extraordinary evidence'). $\endgroup$ – Martijn Weterings Mar 6 at 15:11
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    $\begingroup$ The question is how much these three effects are present in the case of this man-made global warming article. I think it is important to make this clear, to demystify the sciency claims. It is so common to just throw some numbers into an argument to make it sound rigorous, and most people stop questioning it. $\endgroup$ – Martijn Weterings Mar 6 at 15:15
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    $\begingroup$ Have you seen this critique: judithcurry.com/2019/03/01/… ? $\endgroup$ – Robert Long Mar 7 at 20:29
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    $\begingroup$ Coincidentally I was reading these papers just a few days ago, and now noticed your new bounty. I might write something up now. $\endgroup$ – amoeba Oct 6 at 14:46
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It is not always about statistical testing. It can also be about information theory.

The term 5σ is what it says it is: a ratio of "signal" to "noise." In hypothesis testing we have an estimate of a distribution parameter, and standard error of the estimate. The first is a "signal," the second is "noise," and the ratio of the statistics and its standard error is the z-statistics, t-statistics, F-statistics, you name it.

Nevertheless signal-to-noise ratio is useful everywhere where we receive/perceive some information through some noise. As the cited link explains

Signal-to-noise ratio (often abbreviated SNR or S/N) is a measure used in science and engineering to quantify how much the signal is corrupted by noise.

In our case the "signal" is the measured actual change in the temperature of some strata of the atmosphere and the "noise" are predictions of the change from the simulations without the known anthropogenic influences. It so happens that these simulations predicted more or less stationary temperature with a certain standard deviation σ.

Now back to statistics. All test statistics (z, t, F) are the ratios of the estimate to its standard error. So when we statisticians hear of something like S/N, we think a z-statistics and equip it with the probability. The climatologists obviously don't do this (there is no mention of the probability anywhere in the article). They simply find out that the change is "roughly three to eight" times bigger as expected, the S/N is 3σ to 8σ.

What the article is reporting is, that they made two kinds of simulations: ones with the known anthropogenic influences included in the model and the others with the known anthropogenic influences excluded. The first simulations were similar to the measured actual satellite data, while the second were way off. If this is probable or not, they don't say and obviously don't care.

To answer other questions. They didn't set any experiments, they made simulations according to their models. So there is no explicit null hypothesis except the obvious one, that the change is similar to the expected (S/N is 1).

The effect size of the signal is a difference between the actual data and the simulations. It is a signal 5 times larger then expected (five times the usual variability of the temperatures). It seems that the noise is decreasing because of the amount and possibly accuracy of the measurements.

Contrary to our expectations from the "real scientists," there is no statistical model that we could talk about, so the question about the assumptions made is vacuous. The only assumption is that their models enable them to predict climate. This is as valid as saying that the models used for the weather forecasts are solid.

There are much more then three curves. They are the simulation results from different models. They simply have to be different. And yes, have different noise. The signal, as far as it is different, are different sets of measurements, which have their measurement error, and also should be different. What this means regarding the interpretation? Probability interpretation of the S/N is not a good one. However the external validity of the findings is sound. They simply assert that the climate changes in the period from 1979 to 2011 are comparable to simulations when the known anthropogenic influences are accounted for and roughly five times bigger than the ones calculated by simulation when the known anthropogenic factors are excluded from the model.

So there is one question left. If the climatologists would ask the statisticians to make a model, what should it be? In my opinion something in the line of Brownian motion.

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  • $\begingroup$ So what constitutes the "signal," what is the nature of the "noise," and to what unseen process(es) may we attribute it? $\endgroup$ – Josh Mar 7 at 2:12
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    $\begingroup$ Sory @Josh, I prematurely hit the send button. Now you can read my full answer. More ore less, the "signal" are the actual measurements, and the "noise" are the results of the simulations when the known anthropogenic factors are excluded from the model. And in my opinion this is Very unstatistical... $\endgroup$ – Nino Rode Mar 7 at 2:24
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    $\begingroup$ Second, what I get from your post is that the S/N statistic is determined by the signal: the difference between two theoretical models (human effect versus baseline), and noise: the deviation within those theoretic models. But this can be enormously influenced by systematic effects. The distribution of the random effects is not well determined by simply averaging over the variance in monte carlo simulations (see the Vivianonium particle). If there is a systematic error, then you can make the $n\sigma$ discrepancy as large as you want just by gathering more data. $\endgroup$ – Martijn Weterings Mar 7 at 8:24
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    $\begingroup$ @NinoRode Maybe I'm missing something, but since the "noise" model without anthropogenic influences is evidently wrong due to the fact that the mean temperature has risen based on empirical measurements, how does that model provide a relevant baseline? Since it is understood that temperatures fluctuate due to natural processes (en.wikipedia.org/wiki/Little_Ice_Age) in addition to anthropogenic ones, what is the basis for the assumption that the "noise" model should have a mean-zero temperature increase over the analysis period? $\endgroup$ – Josh Mar 7 at 17:10
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    $\begingroup$ @Scott, the issue with the clever cartoon is that there is no noise shown through the time series because the measurements likely aren't refined enough to determine what the temperature was in a certain century, let alone a specific year. So it looks smooth and gradual until the advent of modern measurement devices. In fluid mechanics this would be like comparing an instantaneous observation of a velocity field to a Reynolds-averaged one; it's not an appropriate comparison. Unless you really think there was essentially zero volatility in global temperatures until Greta Thunberg was born. :) $\endgroup$ – Josh Sep 20 at 15:13
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Caveat: I am NOT an expert on climatology, this is not my field. Please bear this in mind. Corrections welcome.


The figure that you are referring to comes from a recent paper Santer et al. 2019, Celebrating the anniversary of three key events in climate change science from Nature Climate Change. It is not a research paper, but a brief comment. This figure is a simplified update of a similar figure from an earlier Science paper of the same authors, Santer et al. 2018, Human influence on the seasonal cycle of tropospheric temperature. Here is the 2019 figure:

enter image description here

And here is the 2018 figure; panel A corresponds to the 2019 figure:

enter image description here

Here I will try to explain the statistical analysis behind this last figure (all four panels). The Science paper is open access and quite readable; the statistical details are, as usual, hidden in the Supplementary Materials. Before discussing statistics as such, one has to say a few words about the observational data and the simulations (climate models) used here.


1. Data

The abbreviations RSS, UAH, and STAR, refer to reconstructions of the tropospheric temperature from the satellite measurements. Tropospheric temperature has been monitored since 1979 using weather satellites: see Wikipedia on MSU temperature measurements. Unfortunately, the satellites do not directly measure temperature; they measure something else, from which the temperature can be inferred. Moreover, they are known to suffer from various time-dependent biases and calibration problems. This makes reconstructing the actual temperature a difficult problem. Several research groups perform this reconstruction, following somewhat different methodologies, and obtaining somewhat different final results. RSS, UAH, and STAR are these reconstructions. To quote Wikipedia,

Satellites do not measure temperature. They measure radiances in various wavelength bands, which must then be mathematically inverted to obtain indirect inferences of temperature. The resulting temperature profiles depend on details of the methods that are used to obtain temperatures from radiances. As a result, different groups that have analyzed the satellite data have obtained different temperature trends. Among these groups are Remote Sensing Systems (RSS) and the University of Alabama in Huntsville (UAH). The satellite series is not fully homogeneous – the record is constructed from a series of satellites with similar but not identical instrumentation. The sensors deteriorate over time, and corrections are necessary for satellite drift in orbit. Particularly large differences between reconstructed temperature series occur at the few times when there is little temporal overlap between successive satellites, making intercalibration difficult.

There is a lot of debate about which reconstruction is more reliable. Each group updates their algorithms every now and then, changing the whole reconstructed time series. This is why, for example, RSS v3.3 differs from RSS v4.0 in the above figure. Overall, AFAIK it is well accepted in the field that the estimates of the global surface temperature are more precise than the satellite measurements. In any case, what matters for this question, is that there are several available estimates of the spatially-resolved tropospheric temperature, from 1979 until now -- i.e. as as a function of latitude, longitude, and time.

Let us denote such an estimate by $T(\mathbf x, t)$.

2. Models

There are various climate models that can be run to simulate the tropospheric temperature (also as a function of latitude, longitude, and time). These models take CO2 concentration, volcanic activity, solar irradiance, aerosols concentration, and various other external influences as input, and produce temperature as output. These models can be run for the same time period (1979--now), using the actual measured external influences. The outputs can then be averaged, to obtain mean model output.

One can also run these models without inputting the anthropogenic factors (greenhouse gases, aerosols, etc.), to get an idea of non-anthropogenic model predictions. Note that all other factors (solar/volcanic/etc.) fluctuate around their mean values, so the non-anthropogenic model output is stationary by construction. In other words, the models do not allow the climate to change naturally, without any specific external cause.

Let us denote the mean anthropogenic model output by $M(\mathbf x,t)$ and the mean non-anthropogenic model output by $N(\mathbf x, t)$.

3. Fingerprints and $z$-statistics

Now we can start talking about statistics. The general idea is to look at how similar the measured tropospheric temperature $T(\mathbf x, t)$ is to the anthropogenic model output $M(\mathbf x, t)$, compared to the non-anthropogenic model output $N(\mathbf x, t)$. One can quantify the similarity in different ways, corresponding to different "fingerprints" of anthropogenic global warming.

The authors consider four different fingerprints (corresponding to the four panels of the figure above). In each case they convert all three functions defined above into annual values $T(\mathbf x, i)$, $M(\mathbf x, i)$, and $N(\mathbf x, i)$, where $i$ indexes years from 1979 until 2019. Here are the four different annual values that they use:

  1. Annual mean: simply average temperature over the whole year.
  2. Annual seasonal cycle: the summer temperature minus the winter temperature.
  3. Annual mean with global mean subtracted: the same as (1) but subtracting the global average for each year across the globe, i.e. across $\mathbf x$. The result has mean zero for each $i$.
  4. Annual seasonal cycle with global mean subtracted: the same as (2) but again subtracting the global average.

For each of these four analyses, the authors take the corresponding $M(\mathbf x, i)$, do PCA across time points, and obtain the first eigenvector $F(\mathbf x)$. It is basically a 2D pattern of maximal change of the quantity of interest according to the anthropogenic model.

Then they project the observed values $T(\mathbf x, i)$ onto this pattern $F(\mathbf x)$, i.e. compute $$Z(i) = \sum_\mathbf x T(\mathbf x, i) F(\mathbf x),$$ and find the slope $\beta$ of the resulting time series. It will be the numerator of the $z$-statistic ("signal-to-noise ratio" in the figures).

To compute the denominator, they use non-anthropogenic model instead of the actually observed values, i.e. compute $$W(i) = \sum_\mathbf x N(\mathbf x, i) F(\mathbf x),$$ and again find its slope $\beta_\mathrm{noise}$. To obtain the null distribution of slopes, they run the non-anthropogenic models for 200 years, chop the outputs in 30-year chunks and repeat the analysis. The standard deviation of the $\beta_\mathrm{noise}$ values forms the denominator of the $z$-statistic:

$$z = \frac{\beta}{\operatorname{Var}^{1/2}[\beta_\mathrm{noise}]}.$$

What you see in panels A--D of the figure above are these $z$ values for different end years of the analysis.

The null hypothesis here is that the temperature fluctuates under the influence of stationary solar/volcanic/etc inputs without any drift. The high $z$ values indicate that the observed tropospheric temperatures are not consistent with this null hypothesis.

4. Some comments

The first fingerprint (panel A) is, IMHO, the most trivial. It simply means that the observed temperatures monotonically grow whereas the temperatures under the null hypothesis do not. I do not think one needs this whole complicated machinery to make this conclusion. The global average lower tropospheric temperature (RSS variant) time series looks like this:

enter image description here

and clearly there is a very significant trend here. I don't think one needs any models to see that.

The fingerprint in panel B is somewhat more interesting. Here the global mean is subtracted, so the $z$-values are not driven by the rising temperature, but instead by the the spatial patterns of the temperature change. Indeed, it is well-known that the Northern hemisphere warms up faster than the Southern one (you can compare the hemispheres here: http://images.remss.com/msu/msu_time_series.html), and this is also what climate models output. The panel B is largely explained by this inter-hemispheric difference.

The fingerprint in panel C is arguably even more interesting, and was the actual focus of the Santer et al. 2018 paper (recall its title: "Human influence on the seasonal cycle of tropospheric temperature", emphasis added). As shown in Figure 2 in the paper, the models predict that the amplitude of the seasonal cycle should increase in mid-latitudes of both hemispheres (and decrease elsewhere, in particular over the Indian monsoon region). This is indeed what happens in the observed data, yielding high $z$-values in panel C. Panel D is similar to C because here the effect is not due to the global increase but due to the specific geographical pattern.


P.S. The specific criticism at judithcurry.com that you linked above looks rather superficial to me. They raise four points. The first is that these plots only show $z$-statistics but not the effect size; however, opening Santer et al. 2018 one will find all other figures clearly displaying the actual slope values which is the effect size of interest. The second I failed to understand; I suspect it is a confusion on their part. The third is about how meaningful the null hypothesis is; this is fair enough (but off-topic on CrossValidated). The last one develops some argument about autocorrelated time series but I do not see how it applies to the above calculation.

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    $\begingroup$ (+1) This is a great answer! If you don't mind: could you expand on the "PCA across time points" step? I don't understand the thinking behind doing a PCA there instead of analysing each dimension separately. $\endgroup$ – mkt Oct 8 at 7:05
  • $\begingroup$ +1 This is a wonderfull explanation. Very close to what I expected (I did not really know what to expect actually and my question was vague) and worthy of bounty (I will leave it till the end to draw attention). I'll need to read the fingerprint section a few more times and have it simmer for a while in my brain, I still desire a bit more intuition and better grasp behind the $\beta_{\text {noise}}$ and the connection to noise in data and what underlying principle of probability is causing this (in high energy particle physics this is more obvious). But this answer will help me sufficiently. $\endgroup$ – Martijn Weterings Oct 8 at 7:23
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    $\begingroup$ @mkt I am sure there are many different ways to do a similar analysis. This is not my field and I would not know why the authors made these particular analysis choices. That said, they do PCA do reduce what I called $N(x,i)$ to $F(x)$, i.e. to remove the time-dependency. This is because they want to project the observed values in each year (what I called $T(x,i)$) onto this $F(x)$. For this purpose, it should be time independent. I suspect that instead of doing PCA, they could have used $N(x, 2019)$ or the average over the last several years. But why not PCA. $\endgroup$ – amoeba Oct 8 at 21:18
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    $\begingroup$ Yeah, this stuff can be discussed from all kinds of angles. I am personally often without much judgement about any side, but I do like that arguments are crisp and clear. The reporting about climate is currently very fuzzy. $\endgroup$ – Martijn Weterings Oct 8 at 22:07
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    $\begingroup$ What still puzzles me about the technical treatement is the meaning of $F(x)$ (you can describe the theoretic time series as a sum of components and this is the one with largest variance?) But why correlate the measured signal with this component and relate it to the variance of the correlation of the anthropogenic model with this component ? (did you maybe switch anthropogenic and non anthropogenic model?) All this stuff (hidden analysis) makes it very difficult to see whether they truely discovered a bump with 5 sigma or whether they just found that measurements do not fit the model. $\endgroup$ – Martijn Weterings Oct 8 at 22:16

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