Guest analysis by JAMAL MUNSHI (h/t to Adam Wildavsky)
ABSTRACT: Homicides in England and Wales 1898-2003 are studied against the atmospheric carbon dioxide data for the same period. The Charney Equilibrium Sensitivity of homicides is found to be λ=1.7 thousands of additional annual homicides for each doubling of atmospheric CO2. The sensitivity estimate is supported by a strong correlation of ρ=0.95 and detrended correlation of ρ=0.86. The analysis illustrates that spurious proportionalities in time series data in conjunction with inadequate statistical rigor in the interpretation of empirical Charney climate sensitivity estimates impedes the orderly accumulation of knowledge in this line of research.
The so called greenhouse effect of atmospheric carbon dioxide is based on the theory that solar irradiance reaches the surface of the earth relatively unhindered but the longer wavelength re-radiated by the surface does not escape to outer space unhindered but is absorbed by carbon dioxide and re- radiated at a lower frequency such that much of it is returned to the surface causing the surface temperature to be higher than it would have been without this absorption effect (Arrhenius, 1896) (Tyndall, 1865) (Tyndall, 1861). This idea, initially studied to explain ice ages over long time span of more than a million years, has been applied to the much shorter time span of 100-200 years to propose that fossil fuel emissions since the Industrial Revolution have introduced extraneous carbon as an unnatural perturbation of a delicately balanced carbon cycle and climate system (Callendar, 1938) (Keeling, 1977) (Revelle, 1957).
It is proposed that emissions of extraneous CO2 increases atmospheric CO2 concentration to unnatural levels such as to cause catastrophic anthropogenic global warming and climate change (AGW) by virtue of a direct linear proportionality between surface temperature and the logarithm of atmospheric CO2 (Hansen, 1981) (Lacis, 2010) (Charney, 1979) (IPCC, 2007) (IPCC, 2013). In the Charney/IPCC convention, this proportionality is described in terms of the amount of warming in Celsius units for a doubling of atmospheric carbon dioxide concentration (Charney, 1979).
Described as the “Equilibrium Climate Sensitivity” to atmospheric carbon dioxide (ECS) and denoted by the Greek letter λ, its theoretical value is stated as λ=3oC but with a large confidence Interval of 1.5oC to 4.5oC (IPCC, 2013). The relationship is considered to be sufficiently deterministic such that “CO2 is the
control knob” (the so called “Lacis Control Knob”) that determines surface temperature according to atmospheric CO2 (Lacis, 2010).
The Lacis Control Knob and its implied climate sensitivity λ form the fundamental underpinnings for the theory of AGW. However, empirical validation of this principle has been hampered by methodological issues. The large spread in the theoretical value of ECS proposed by the IPCC as [1.5-4.5] combined with insufficient statistical discipline has led to the understanding among most empirical researchers that any positive value for the ECS constitutes empirical support for the theory regardless of other statistical parameters such as the strength of the proportionality. Without falsifiability limits, climate sensitivity research has evolved into a large and undisciplined fishing expedition.
Active research in this area has generated a large collection of empirical ECS values over a large range from λ<2 to λ>9 (Danabasoglu, 2009) (Dessler, 2018) (Ehlert, 2017) (Gregory, An observationally based estimate of the climate sensitivity, 2002) (Hegerl, 2006) (Johansson, 2015) (Mitchell, 1995) (Roe, 2007)
(Rogelj, 2012) (Senior, 2000) (Sherwood, 2014) (Stevens, 2016) (WEN, 2013). A summary of these results is provided in prior works (Munshi, Uncertainty in Empirical Climate Sensitivity Estimates 1850-2017, 2018) (Munshi, From Equilibrium Climate Sensitivity to Carbon Climate Response, 2018). However, the net result of these works has not been an orderly accumulation of knowledge and advancement of understanding but rather a sense of confusion and frustration at what has been termed the “uncertainty issue” in climate sensitivity (Caldeira, 2003) (Roe, 2007) (Jones, 2003) (Stainforth, 2005) (Curry, 2011).
The unsettled state and apparent disarray in climate sensitivity research has turned many researchers to turn to other approaches for the critical relationship between fossil fuel emissions and warming.
Notable among these is the so called Carbon Climate Response (CCR) and the similar Transient Climate Response to Cumulative Emissions (TCRE) (Allen, 2009) (Matthews, 2009) (Zickfeld, 2009) (Solomon, 2009) (Ehlert, 2017). Both measures propose cumulative emissions as the agent of warming. The CCR/TCRE approach proposes a linear proportionality between cumulative fossil fuel emissions and surface temperature. A strong positive linear proportionality has been shown and it has gained widespread adoption in climate science. It is now thought that the ECS climate sensitivity should be abandoned for the more straightforward and precise CCR/TCRE parameter (Knutti, 2017).
The editor of Nature magazine, in accepting the Matthews 2009 paper wrote: “To date, efforts to describe and predict the climate response to human CO2 emissions have focused on climate sensitivity: the equilibrium temperature change associated with a doubling of CO2. But recent research has suggested that this ‘Charney’ sensitivity, so named after the meteorologist Jule Charney who first adopted this approach in 1979, may be an incomplete representation of the full Earth system response, as it ignores changes in the carbon cycle, aerosols, land use and land cover. Matthews et al. propose a new measure, the carbon-climate response, or CCR. Using a combination of a simplified climate model, a range of simulations from a recent model intercomparison, and historical constraints, they find that independent of the timing of emissions or the atmospheric concentration of CO2 emitting a trillion tonnes of carbon will cause 1.0 – 2.1 C of global warming, a CCR value that is consistent with model predictions for the twenty-first century.”
However, a statistical flaw in the CCR/TCRE proportionality may limit the utility of cumulative emissions (Munshi, 2018). In this work we identify similar defects in ECS research methodology and propose refinements that may revive the ECS as a usable metric in empirical research. The refinement contains falsifiability because it must be possible for empirical research to prove the theory wrong.
2. DATA AND METHODS
Global and regional monthly mean temperature reconstructions from 1850 to 2017 are provided as deseasonalized temperature anomalies by the Hadley Centre Climate Research Unit of the Met Office, Government of the UK (Hadley Centre, 2018). Monthly means are converted to annual means with a simple average across the twelve calendar months for use with atmospheric CO2 data that are available only as annual means prior to 1959.
Atmospheric CO2 concentrations measured at Mauna Loa from 1959 to 2017 are provided as monthly mean ppm CO2 by the Scripps Institution (Scripps, 2018). They are converted to annual means with a simple average across calendar months. Atmospheric CO2 concentrations older than 1959 are taken from the Law Dome ice core data provided by Commonwealth Scientific and Industrial Research Organization as annual means (CSIRO, 2018). These data are available only as annual means and that necessitated the conversion of all other data into the annual format.
Homicide data for England and Wales 1898-2003 are provided by the Home Office of the Government of the UK as the number of homicides per year (HomeOffice, 2018). These data are used to demonstrate spurious correlations in time series data. Spurious correlations are an issue in all data of course but this problem is particularly prevalent in time series data (Theiler, 1986). A common pattern is that incidental trends in the data, unrelated to the variables under study can create an illusory correlation. Procedures to study correlation net of such trend effects include detrended correlation analysis and detrended fluctuation analysis (Shumway, 2011) (Podobnik, 2008) (Chen, 2002). Proportionalities in time series data must therefore be tested with these procedures but, as we shall see in our spurious correlation example, these procedures reduce but not eliminate the possibility of a spurious correlation in time series data.
Proportionality in time series data implies both a non-zero regression coefficient and a strong correlation and so both of these parameters are evaluated. Empirical tests of theory are meaningful only if there exists a condition or conditions under which the theory is falsified by the test. In the case of Equilibrium Climate Sensitivity (ECS) research, this constraint has been relaxed and that has created a large number of inconsistent findings all presented as evidence of ECS but without statistical rigor. In this work the required proportionality is tested for statistical significance and robustness and the potential for spurious results is demonstrated. The Bowley procedure for estimating the standard deviation of the correlations coefficient is used in this work (Bowley, 1928). Hypothesis tests are carried at α=0.001 in accordance with “Revised Standards for Statistical Evidence” proposed in view of an unacceptable rate of irreproducible results in published research (Johnson, 2013). The research question is whether the object variable is responsive to the logarithm of atmospheric CO2 at an annual time scale in the linear relationship implied by the Charney Sensitivity parameter. All data and computational details including Excel files used in this work are available for download from an online data archive https://drive.google.com/open?id=1E0FSRsB2CU4R4t_XvqVnH2rvGQuQh0vZ
3 DATA ANALYSIS
Figure 1: Annual homicides in England and Wales: Full Span 1898-2003
Figure 1 presents the results for the sensitivity of the annual homicide rate in England and Wales to the logarithm of atmospheric carbon dioxide for the full span of the time series 1898-2003. The graphic consists of two panels (top and bottom) and three frames in each panel. The left and middle frames show the data and the right frame displays their proportionality in terms of linear regression and correlation. The top panel displays the proportionality in the source data and the bottom panel likewise shows proportionality in the detrended series.
The proportionality in the source data can be described as a linear regression coefficient of β=2.45 thousand homicides per unit increase in Ln (CO2). The coefficient of determination ρ2=0.8923 indicates that 89.23% of the variance in homicides is explained by the regression and implies a strong correlation coefficient is ρ=0.945. These metrics taken together provide good evidence of proportionality between Ln (CO2) and homicides. Multiplication of β by Ln (2) converts the regression coefficient into the Charney Sensitivity format as λ=1.7 thousand more homicides for each doubling of atmospheric CO2. The 95% confidence interval for the Charney Sensitivity is 95%CI=[1.58<λ<1.81].
Proportionality of this nature between time series data can derive from at least two sources: (1) a responsiveness of homicides to Ln(CO2) at an annual time scale and (2) a proportionality imposed by a shared and possibly incidental long term trend that is unrelated to theory and that does not imply a responsiveness at any finite time scale. As a general principle, proportionalities at relevant finite time scales provide stronger indication of causal relationships than those derived from long term trends (Podobnik, 2008). To assess the relative contributions of these two sources of correlation, the two series are detrended and the detrended correlation is estimated. Although correlation in the data does not in itself imply causation, such correlation is nevertheless a necessary though not sufficient condition for causation.
The results of detrended analysis appear in the bottom panel of Figure 1. They show a larger regression coefficient of β=3.68 and Charney Sensitivity of λ=2.55 but these parameters are not interpreted in the detrended test. The coefficient of determination and corresponding correlation coefficient are somewhat lower at ρ2=0.7385 and ρ=0.8594 indicating a weaker but still a statistically significant proportionality between the two detrended series. Detrended analysis thus establishes that the proportionality seen in the source data is not a spurious figment created by incidental shared long term trends but that it derives mostly from a responsiveness of homicides to Ln (CO2) at an annual time scale.
These results taken together appear to serve as empirical evidence in support of the proposition that homicides are driven by atmospheric CO2. Had there been a theoretical basis for such a proposition, the data and analysis presented would fail to refute or falsify such a claim. Yet the proposition is preposterous and clearly without merit. These results therefore serve only as an example of spurious correlations in time series data that are often accepted as empirical evidence in climate science as demonstrated in a prior parody (Munshi, 2018-3).
In this case, the spuriousness of the proportionality is revealed when the finding is examined with split- half analysis to determine whether the linearity and proportionality implied by the 106-year full span results 1898-2003 are uniform across the sample period or whether they are artifacts of violations of OLS assumptions in the behavior of the time series data (Shumway, 2011) (VonStorch, 1999) (Kantz/Schreiber, 2004). The results of the split half analysis are displayed graphically in Figures 3&4 and tabulated in Figure 2. The spuriousness of the statistics supporting the proposition that homicides in England and Wales is driven by atmospheric CO2, is exposed in these results.
In the first 53 years of the data, 1898-1950 we find a very low regression coefficient of β=0.858 in the source data which implies a Charney Sensitivity of λ=0.595. The correlation in the source data is also low (ρ=0.325) and not statistically significant, consistent with the graphic in Figure 3. The results of detrended analysis for the first half are equally unimpressive with negative sensitivity and absence of correlation (ρ=0.281).
A different kind of pattern is seen in the last 53 years of the data 1951-2003. Though the source data shows a strong correlation (ρ=0.962) and Charney Sensitivity (β=2.998, λ=2.078) that are statistically significant, all is lost, including statistical significance, when the data are detrended (β=1.998, λ=1.385, ρ=0.299). The results imply that the proportionality seen in the source data is driven mostly by a shared long term trend and not by annual responsiveness of homicides to atmospheric CO2.
Based on the failure to find consistency in the split half results, we can now state that the data do not provide convincing evidence that homicides in England and Wales are driven by atmospheric carbon dioxide concentration at an annual time scale in a Charney model such that the annual homicides are proportional to the logarithm of atmospheric CO2. Yet, it is easy to see that if the proposed relationship had seemed reasonable, and if therefore sufficient statistical rigor was not followed, we might have accepted the results in Figure 1 as confirmation of the theory, and reported the existence of Charney Homicide Sensitivity as λ=1.7 thousand additional homicides per doubling of atmospheric CO2.
Figure 2: Summary of results for the Homicide series
Figure 3: Annual homicides in England and Wales: First Half 1898-1950
Figure 4: Annual homicides in England and Wales: Second Half 1951-2003
We now demonstrate the application of these diagnostics in the same sample period to the more generally accepted proposition by climate science, based on the Arrhenius effect, that surface temperature is responsive to atmospheric CO2 concentration in accordance with a Charney Sensitivity parameter that describes a linear relationship between temperature and Ln (CO2). The HadCRU global mean temperature reconstructions for the period 1898-2003 are used for this demonstration.
Results for the HadCRU global mean temperature anomalies are displayed graphically in Figure 6 and summarized in Figure 5. The data show a Charney Climate Sensitivity of λ=2.15oC of warming for each doubling of atmospheric CO2 concentration. The correlation between temperature and Ln (CO2) in this period is strong at ρ=0.849 and this correlation is statistically significant.
However, detrended correlation analysis yields a much lower correlation of ρ=0.269 and this correlation is not statistically significant. This result implies that the strong statistical significance of the correlation in the source data derives mostly from shared long term trends and not from responsiveness at an annual time scale. Thus, the data do not support the existence of a climate sensitivity that works at an annual time scale. In this sense the evidence for a Charney Climate Sensitivity is weaker than that for the Charney Homicide Sensitivity presented in Figures 1 through 3. The value of λ=2.15 is therefore spurious because it has no interpretation.
Further weakness in the climate sensitivity estimate is found in split half analysis. It shows a much higher climate sensitivity in the first half of the sample period 1898-1950 of λ=5.64oC of warming for each doubling of atmospheric CO2. The proportionality in the data is supported by a strong statistically significant correlation of ρ=0.796. However, detrended analysis reveals a low detrended correlation of ρ=0.039 which implies that the proportionality seen in the source data derives not from responsiveness at an annual time scale but from a shared and perhaps incidental upward trend in both CO2 and temperature that has no interpretation in terms of climate sensitivity.
The second half of the data 1951-2003 yields a very low climate sensitivity of λ=1.92oC of warming for each doubling of atmospheric CO2 with a strong proportionality indicated by a correlation coefficient of ρ=0.807 and a statistically significant detrended correlation of ρ=0.657. These results can be interpreted in terms of global warming theory (Charney, 1979) (Lacis, 2010) because the results imply a responsiveness of temperature to atmospheric CO2 at an annual time scale. However, its interpretation is limited in the context of the overall results that are indicative of spuriousness. The temptation in such cases is to change the theory midstream and thereby to make a circular reasoning claim that “the
climate sensitivity is apparent since 1951”. The logical fallacy here is that a data from which a theory is derived does not serve as evidence for it. In the AGW context the relevant time span is “since the
Industrial Revolution” that has been described as being well after 1850 and perhaps since 1880 (Callendar, 1938) (Hansen, 1981). That the effect is found only for the brief period 1951-2003 is inconsistent with the theory being tested.
Figure 5: HadCRU global mean temperature reconstruction 1898-2003
Figure 6: HadCRU mean annual global surface temperature Full span: 1898-2003
4. SUMMARY AND CONCLUSIONS
The theory that fossil fuel emissions since the Industrial Revolution have caused global warming is based on the proposition that such emissions increase atmospheric carbon dioxide concentration which in turn increases surface temperature according to the Arrhenius effect. A testable implication of the theory is the Charney Climate Sensitivity equal to the increase in surface temperature for a doubling of atmospheric CO2 and based on a linear relationship between the logarithm of atmospheric CO2 and surface temperature. However, the large body of empirical research in climate sensitivity has not produced an orderly accumulation of knowledge but instead created confusion and mistrust of the climate sensitivity parameter by virtue of the range of sensitivity values that are reported (Curry, 2011).
The frustration of climate science with this so called “uncertainty issue in climate sensitivity” has motivated proposals to abandon the climate sensitivity approach in favor of the “Climate Response” to cumulative emissions contained in the CCR (Carbon Climate Response) or TCRE (Transient Climate Response to Cumulative Emissions) (Knutti, 2017) (Matthews, 2009).
This state of affairs in climate sensitivity research is likely the result of insufficient statistical rigor in the research methodologies applied. This work demonstrates spurious proportionalities in time series data that may yield climate sensitivities that have no interpretation. A parody of the Charney sensitivity with data for homicides in England and Wales 1898-2003 is used for the demonstration. A parallel analysis of global mean temperature reconstructions for the same period show that such spurious results also occur under conditions where they are more likely to be accepted at face value. The results imply that the large number of climate sensitivities reported in the literature are likely to be mostly spurious and without an interpretation in terms of the Charney climate sensitivity. Sufficient statistical discipline is likely to settle the Charney climate sensitivity issue one way or the other, either to determine its hitherto elusive value or to demonstrate that the assumed relationships do not exist in the data.
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