Example of Simple Linear Regression - global warming trends

Here is a simple statistical analysis using linear regression showing global warming of 0.2C this decade (as projected in the IPCC Fourth Assessment Report 2007) is “unlikely”.

Below are graphs for the last ten years and the trend line for global temperatures for four sources from Anthony Watts over the period January 1998 to February 2008. The simple linear regression line through the points shows the 10 year trend.

One of the main claims of the theory of global warming is that greenhouse gases in the atmosphere cause increasing temperatures. If temperatures stop increasing for long enough, while greenhouse gases such as CO2 continue to rise, then we could be justified in not believing the theory.

The basic numeracy skill from statistics is the hypothesis test. To set up the test we assume no difference between the datum being tested (called a null hypothesis or H0) and estimate the probability of assuming incorrectly, based on the data. The hypothesis test on these data would be as follows:

First, we estimate the expected variability of the trend. The standard deviation of the 10 year trend was estimated directly from the trends over the entire period of data collection according to the robust procedure described here. We also calculate the difference between the IPCC expected trend of 0.2C per decade and observed trend per decade. We then estimate the probability of the observed trend, given an expected trend of 0.2C by comparing the difference and the SD. This gives us the probability of the null hypothesis is rejected, that there is no difference between the predicted and the observed trends.

We can estimate the actual probability for each of the global temperature indices with the R function pnorm() as follows:

> pnorm(0.135,sd=0.155)
[1] 0.8081141

Below is a table of data and results for all the four main global temperature indices.

Source	10yr_trend(t)	SD	0.2-t	p	IPCC Confidence
UAH	+0.045	0.135	0.155	0.81	Likely
RSS	+0.009	0.122	0.191	0.94	Very Likely
CRU	+0.098	0.216	0.102	0.68	Likely
GISS	+0.173	0.142	0.027	0.58	Medium Likelihood

As of this month, the trend in temperatures for the last 10 years is so low, that an increase of 0.2C per decade could be rejected in 3 out of 4 indices with some level of confidence. In one case, using the IPCC terminology, these results suggest IPCC projection of global warming this century are very unlikely (1-10% chance) to be correct. This is a controversial result contradicting the IPCC ‘consensus’ position.

To any controversial result there are objections. One is that ten years is too short a time to test climate trends. This is of course a statistical nonsense as a trend of any length in a time series can be tested providing appropriate uncertainty is used. With short trends the SD simply becomes much larger. For example, the SD for the 5 year trend is closer to 0.5C, so a trend deviation of more than twice as much would be needed to reject the null hypothesis at the same level as a 10 year trend.

Another objection here is that a ten year trend is cherry picking. However, to cherry pick one needs options to pick from. Treating data from the present day 10 years back is not cherry picking as we are talking about temperatures now, and it is impossible to cherry pick the present, there is only one present. The choice of length of trend is the only free variable.

Another objection is that increasing temperatures may being masked by factors such as volcanic eruptions, El Niños, sulphates, and trade winds. However, the masking effect was not part of the IPCC predictions of 0.2C per decade, and the results concern those predictions. Also, the masking argument tries to convince us that the models’ magnitude of warming is certain, and explains failure by reference to uncertain factors. Such excuses for lack of predictive skill get tired very quickly.

One could also argue that uncertainty limits may be wider than classical statistics and this finite empirical process for estimating SD suggest. Climatic processes exhibit a scaling invariant behavior, also known as long-range persistence (LTP) or the Hurst phenomenon which produces random long term trends. A combination of analytical and Monte Carlo methods such as described in Uncertainty assessment of future hydroclimatic predictions: A comparison of probabilistic and scenario-based approaches (2007) may extend uncertainty limits more and reduce the confidence in the judgements.

I think it is a remarkable testament to the power of numbers, that one of the most complex and contentious issues of the time could potentially be brought down by such a simple statistical analysis. The IPCC model projections were only published in 2001 and are already looking very shaky. These projections are central to the IPCC mission. If the current stable temperature trend continues, put the AGW agenda on hold.

Appendix:

In the Summary for Policymakers and in the Technical Summary, the following words have been used where appropriate to indicate judgmental estimates of confidence: virtually certain (greater than 99% chance that a result is true); very likely (90-99% chance); likely (66-90% chance); medium likelihood (33-66% chance); unlikely (10-33% chance); very unlikely (1-10% chance); exceptionally unlikely (less than 1% chance).