No statistically significant warming since 1995: a quick mathematical proof

Watts Up With That?
Sunday, Dec 27th, 2009

Physicist Lubo? Motl of The Reference Frame demonstrates how easy it is to show that there is: No statistically significant warming since 1995

First, since it wasn?t in his original post, here is the UAH data plotted:

By: Lubo? Motl

Because there has been some confusion ? and maybe deliberate confusion ? among some (alarmist) commenters about the non-existence of a statistically significant warming trend since 1995, i.e. in the last fifteen years, let me dedicate a full article to this issue.

I will use the UAH temperatures whose final 2009 figures are de facto known by now (with a sufficient accuracy) because UAH publishes the daily temperatures, too:

Mathematica can calculate the confidence intervals for the slope (warming trend) by concise commands. But I will calculate the standard error of the slope manually.

x = Table[i, {i, 1995, 2009}]
y = {0.11, 0.02, 0.05, 0.51, 0.04, 0.04, 0.2, 0.31, 0.28, 0.19, 0.34, 0.26, 0.28, 0.05, 0.26};
data = Transpose[{x, y}]
(* *)
n = 15
xAV = Total[x]/n
yAV = Total[y]/n
xmav = x - xAV;
ymav = y - yAV;
lmf = LinearModelFit[data, xvar, xvar];
Normal[lmf]
(* *)
(*?http://stattrek.com/AP-Statistics-4/Estimate-Slope.aspx?Tutorial=AP?*)
;slopeError = Sqrt[Total[ymav^2]/(n - 2)]/Sqrt[Total[xmav^2]]

The UAH 1995-2009 slope was calculated to be 0.95 ?C per century. And the standard deviation of this figure, calculated via the standard formula on this page, is 0.88 ?C/century. So this suggests that the positivity of the slope is just a 1-sigma result ? a noise. Can we be more rigorous about it? You bet.

Mathematica actually has compact functions that can tell you the confidence intervals for the slope:

lmf = LinearModelFit[data, xvar, xvar, ConfidenceLevel -> .95];
lmf["ParameterConfidenceIntervals"]

The 99% confidence interval is (-1.59, +3.49) in ?C/century. Similarly, the 95% confidence interval for the slope is (-0.87, 2.8) in ?C/century. On the other hand, the 90% confidence interval is (-0.54, 2.44) in ?C/century. All these intervals contain both negative and positive numbers. No conclusion about the slope can be made on either 99%, 95%, and not even 90% confidence level.

Only the 72% confidence interval for the slope touches zero. It means that the probability that the underlying slope is negative equals 1/2 of the rest, i.e. a substantial 14%.

We can only say that it is ?somewhat more likely than not? that the underlying trend in 1995-2009 was a warming trend rather than a cooling trend. Saying that the warming since 1995 was ?very likely? is already way too ambitious a goal that the data don?t support.