Here is the text of an email that I sent to Doug Keenan on 25th January 2013. It sets out some of my personal thoughts on statistical modelling of trends in global mean temperature (or many of the other timeseries in the Earth system).
I believe it has some bearing on this post at Bishop Hill.
Dear Doug Keenan,
Thanks for clarifying your concerns somewhat; hopefully I can set your mind at rest on some of
your worries. I suspect that parliamentary questions are not the best route through which to
discuss some of the technical aspects of statistical theory.
Please rest assured that we do not use a linear statistical analysis of global temperature trends
in anything other than a descriptive manner. I believe that the trends were reported as linear as a
method of description for consistency with the IPCC Fourth Assessment Report. The report
itself acknowledges the deficiencies of the statistical model that it uses, while arguing that the
analysis still gives some useful information. I’m inclined to agree.
These trend lines are just there to summarise the data, and shouldn’t be used (for example) to
extrapolate a forecast of global temperatures in the future. They shouldn’t be taken as a
representing the “true” statistical model for global warming. Perhaps you might suggest other
methods for summarising the data?
I think the problem is the proxy war being fought around the word “significant”, and the difference
between its scientific and formal statistical use. We should be careful to separate out their use.
There appear to be individuals who would like to say that the evidence for warming is “not
statistically significant”, and others who would like to say that it is. This is to misunderstand the
nature of “significance”.
Of course, we both know that the correct answer to the question “is the trend of global mean
temperature statistically significant?” is not “yes”, or even “no” but, “that is not a valid question”.
This is because the appropriate statistical model to use for the timeseries is not known perfectly,
and any statistical significance test uses assumptions about that timeseries that may turn out to
be invalid. Cohn and Lins (2005) for example, state “significance depends critically on the null
hypothesis which in turn reflects notions about what one expects to see.”
This does not mean that there is no evidence of a significant increasing temperature trend
there is plenty of scientific evidence for this just that using naive statistical tests in the absence
of other information is inappropriate.
A significance test attempts to answer the question “given that there was no anthropogenically
driven global warming, what is the probability that we would see these temperatures?” This is
interesting, but not really what we are looking for. As you and others have noted, this kind of test
does not allow you to distinguish between forced trends, and the degree of long term persistence
in the system. I would suggest a Bayesian solution to this problem. I think the appropriate
question is “given that we see this these temperatures, what is the probability they are
anthropogenically driven?” Using Bayes theorem, this combines the likelihood (from the first
question), with the prior probability that global temperatures are anthropogenically driven, to
some degree. Of course, this prior probability contains subjective judgements and information
from elsewhere, including fundamental physics.
Our judgements about the probability that temperatures are anthropogenically driven, contain
information not just from the global temperature trend, but also our knowledge of the way that the
system works. The global temperature series in isolation simply does not contain the information
we are looking for.
The Bayesian solution cuts through the problem that you identified, of being able to tell the
difference in the validity of (as in your example), a linear trend with an autoregressive element,
and a driftless autoregressive integrated model. The model that we choose has to be consistent
with the understanding of the system that we gain from other data, and our basic knowledge of
the physics of the system. We encode our knowledge about physics, and the rest of the system
in climate models, and their simulations. Again, I stress that a great deal of work has been done
in this field, but that it is more likely to be found in the detection and attribution literature (and
corresponding IPCC chapter), than in the observations literature.
In conclusion, my suggestion is that when asked if there has been a “significant” change in
temperatures since the 1880s, we should say “yes”. If we are asked if there has been a
statistically significant change in temperatures since the 1880s, we do not say “yes” or “no”, we
say “that is not a valid question”. A difficulty may be in getting people to understand the reasoning
sufficiently to accept the latter as the correct answer to the question. In this, I welcome your help.
Cohn, T.A. and Lins, H.F. (2005) Nature’s style: Naturally trendy, Geophys. Res. Lett., 32,