San José State University
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A Statistical Test of the Global Warming Hypothesis

The change in temperature of a system is proportional to the net inflow of heat to that system. In the case of the Earth's surface the inflow is due to the influx of shortwave radiation from the Sun. The outflow is due to the longwave thermal radiation which is proportional to the fourth power of the absolute temperature of the surface. The level of greenhouse gases such as carbon dioxide (CO2) affects the proportion of the thermal radiation retained.

Let S be the sunspot number. This is used as a proxy for the intensity of solar radiation at the top of Earth's atmosphere. If T is the absolute temperature of the Earth's surface then its outgoing radiation is proportional to σT4, where σ is the Stefan-Boltzmann constant. The proportion of this thermal radiation which is not retained depends upon the concentration of greenhouses gases in the atmosphere, one of which is CO2. For the statistical analysis below it is presumed that the proportion not retained is a linear function of the concentration of CO2, pCO2.

The estimating equation is then

ΔT = a0 + a1S − (b0−b1pCO2)σT4
or, equivalently
ΔT = a0 + a1S − b0σT4 + (b1pCO2σT4)

where ΔT is the January-to-January chanage in temperature. It is important to use the change of temperature over an interval rather than the change in the annual average. When a variable is, as temperature is, the cumulative sum of disturbances the process of averaging introduces statistical artifacts that interfer with the statistical analysis. For more on this topic see stochastic structure. The temperature in σT4 is however the annual average.

All three of the coefficients, a1, b0 and b1, should be positive if the hypothesis that an increase in the level of CO2 contributes to an increase in global temperature.

The Data

The data on CO2 concentration are derived from air samples collected at Mauna Loa Observatory, Hawaii. The source is C.D. Keeling, T.P. Whorf, and the Carbon Dioxide Research Group at the Scripps Institution of Oceanography, University of California at La Jolla, May 2005. This data covers only from 1959 to 2004 so this is the interval of analysis.

The temperature data were constructed from the data set available from NASA. The temperature data goes back to 1880. It is worthwhile to look at some data scatter diagrams to get acquainted with the statistical characteristics of the data. First consider the times series for the January-to-January changes in temperatures.

What the diagram shows is that there were some extreme cases in the early years of the series that had more to do with the accuracy of the data than global climate. This is not a concern for the statistical work below because the analysis only covers the period for which there are data on CO2 concentrations.

The theory suggests that there should be an inverse correlation between temperature change and the level of temperature, or more precisely the fourth power of the absolute temperature.

First consider the scatter diagram for temperature change versus temperature.

There is a satifying downward slope to the data plot. When thermal radiation σT4 is used as the independent variable there is also a downward slope as seen below.

But what is clear is that the outliers, the extreme cases, will dominate the statistical results. This raises a note of caution in interpreting the statistical results.

Data for Global Temperature Change 1959-2004
Year Temp
Change
Abs. Temp pCO2 σ*T4 σ*T4*pCO2 Sunspots
1959 0.0 287.22 0.000316 385.8707 0.121935 159.0
1960 0.03 287.22 0.000317 385.8707 0.122286 112.3
1961 -0.02 287.25 0.000318 386.0319 0.122615 53.9
1962 -0.10 287.23 0.000318 385.9244 0.122901 37.6
1963 -0.05 287.13 0.000319 385.3873 0.122946 27.9
1964 0.04 287.08 0.000320 385.1189 0.123053 10.2
1965 -0.22 287.12 0.000320 385.3336 0.123341 15.1
1966 0.18 286.90 0.000321 384.1539 0.123444 47.0
1967 -0.13 287.08 0.000322 385.1189 0.124058 93.7
1968 0.01 286.95 0.000323 384.4218 0.124211 105.9
1969 0.27 286.96 0.000325 384.4754 0.124801 105.5
1970 -0.12 287.23 0.000326 385.9244 0.125676 104.5
1971 -0.31 287.11 0.000326 385.2799 0.125725 66.6
1972 0.58 286.80 0.000328 383.6186 0.125643 68.9
1973 -0.30 287.38 0.000330 386.7312 0.12747 38.0
1974 0.13 287.08 0.000330 385.1189 0.127201 34.5
1975 -0.14 287.21 0.000331 385.8169 0.127767 15.5
1976 0.16 287.07 0.000332 385.0652 0.127911 12.6
1977 -0.02 287.23 0.000334 385.9244 0.128852 27.5
1978 0.07 287.21 0.000336 385.8169 0.129449 92.5
1979 0.16 287.28 0.000337 386.1932 0.130105 155.4
1980 0.28 287.44 0.000339 387.0543 0.131084 154.6
1981 -0.49 287.72 0.000340 388.5646 0.132093 140.5
1982 0.36 287.23 0.000341 385.9244 0.131635 115.9
1983 -0.23 287.59 0.000343 387.8628 0.13294 66.6
1984 0.03 287.36 0.000344 386.6236 0.133169 45.9
1985 0.04 287.39 0.000346 386.7850 0.133773 17.9
1986 0.07 287.43 0.000347 387.0004 0.134343 13.4
1987 0.17 287.50 0.000349 387.3776 0.135191 29.2
1988 -0.44 287.67 0.000351 388.2946 0.136462 100.2
1989 0.37 287.23 0.000353 385.9244 0.136208 157.6
1990 -0.01 287.60 0.000354 387.9168 0.137396 142.6
1991 0.01 287.59 0.000356 387.8628 0.137932 145.7
1992 -0.15 287.60 0.000356 387.9168 0.138238 94.3
1993 0.05 287.45 0.000357 387.1081 0.138236 54.6
1994 0.12 287.50 0.000359 387.3776 0.139014 29.9
1995 -0.10 287.62 0.000361 388.02470 0.140038 17.5
1996 0.02 287.52 0.000363 387.4854 0.140494 8.6
1997 0.18 287.54 0.000364 387.5932 0.141022 21.5
1998 -0.02 287.72 0.000367 388.5646 0.142440 64.3
1999 -0.41 287.70 0.000368 388.4566 0.143069 93.3
2000 0.37 287.29 0.000369 386.2470 0.142707 119.6
2001 0.33 287.66 0.000371 388.2406 0.144049 111.0
2002 -0.05 287.99 0.000373 390.0252 0.145507 104.0
2003 -0.17 287.94 0.000376 389.7544 0.146396 63.7
2004 0.25 287.77 0.000377 388.8348 0.146758 40.4

The regression results are:


ΔT = 73.18 + 0.000810S − (0.201 − 33.961pCO2)σT4
             (6.68)    (1.60)           (6.7)         (5.87)                 
R2 = 0.522
 

The numbers shown in parentheses below the regression coefficients are the magnitudes of their t-ratios; i.e. the coefficients divided by the standard deviation of the regression coefficient. All but the coefficient for Sunspot number are significantly different from zero at the 95 percent level of confidence and they are of the right sign.

Shown below is a comparison of the observed temperature change and the temperature change predicted by the regression equation.

The observations are shown in black and the estimations from the regression equation are shown in fuchsia.

Another way of viewing the comparison is in the scatter diagram below of the actual and regression predicted temperature changes.

Although the t-ratios for the variables included in the regression equation are significant they only explain 52.2 percent of the variation in the year-to-year temperature change. Also the effect of the CO2 in the equation includes the effects of all variables influencing temperature change which are correlated with the general trend on CO2 concentration but are not in the equation. These would include the effects of anthropogenic water vapor and anthropogenic cloudiness.

Below is a comparison of the January global temperatures with the temperatures derived by accumulating the regression estimates of the temperature changes. Again the actual (observed) temperatures are shown in black and the ones based upon the regression estimates are shown in fuchsia.

(To be continued.)


For more statistical analysis of global temperatures see Climatology.


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