Friday, June 15, 2012


date: Wed, 28 May 2008 20:46:53 -0700
from: Ben Santer <>
subject: Our d3* test
to: "Thorne, Peter" <>, Leopold Haimberger <>, Karl Taylor <>, Tom Wigley <>, John Lanzante <>, "'Susan Solomon'" <>, Melissa Free <>, peter gleckler <>, "'Philip D. Jones'" <>, Thomas R Karl <>, Steve Klein <>, carl mears <>, Doug Nychka <>, Gavin Schmidt <>, Steven Sherwood <>, Frank Wentz <>

Dear folks,

Just wanted to let you know that I did not submit our paper to IJoC.
After some discussions that I've had with Tom Wigley and Peter Thorne, I
applied our d1*, d2*, and d3* tests to synthetic data, in much the same
way that we applied the DCPS07 d* test and our original "paired trends"
test (d) to synthetic data. The results are shown in the appended Figure.

Relative to the DCPS07 d* test, our d1*, d2*, and d3* tests of
hypothesis H2 yield rejection rates that are substantially
closer to theoretical expectations (compare the appended Figure with
Figure 5 in our manuscript). As expected, all three tests show a
dependence on N (the number of synthetic time series), with rejection
rates decreasing to near-asymptotic values as N increases. This is
because the estimate of the model-average signal (which appears in the
numerator of d1*, d2*, and d3*) has a dependence on N, as does the
estimate of s{<b_{m}>}, the inter-model standard deviation of trends
(which appears in the denominator of d2* and d3*).

The worrying thing about the appended Figure is the behavior of d3*.
This is the test which we thought Reviewers 1 and 2 were advocating. As
you can see, d3* produces rejection rates that are consistently LOWER
(by a factor of two or more) than theoretical expectations. We do not
wish to be accused by Douglass et al. of devising a test that makes it
very difficult to reject hypothesis H2, even when there is a significant
difference between the trends in the model average signal and the
'observational signal'.

So the question is, did we misinterpret the intentions of the Reviewers?
Were they indeed advocating a d3* test of the form which we used? I will
try to clarify this point tomorrow with Francis Zwiers (our Reviewer 2).

Recall that our current version of d3* is defined as follows:

d3* = ( b{o} - <<b{m}>> ) / sqrt[ (s{<b{m}>} ** 2) + ( s{b{o}} ** 2) ]


b{o} = Observed trend
<<b{m}>> = Model average trend
s{<b{m}>} = Inter-model standard deviation of ensemble-mean trends
s{b{o}} = Standard error of the observed trend (adjusted for
autocorrelation effects)

In Francis's comments on our paper, the first term under the square root
sign is referred to as "an estimate of the variance of that average"
(i.e., of <<b{m}>> ). It's possible that Francis was referring to
sigma{SE}, which IS an estimate of the variance of <<b{m}>>. If one
replaces s{<b{m}>} with sigma{SE} in the equation for d3*, the
performance of the d3* test with synthetic data is (at least for large
values of N) very close to theoretical expectations. It's actually even
closer to theoretical expectations than the d2* test shown in the
appended Figure (which is already pretty close). I'll produce the
"revised d3*" plot tomorrow...

The bottom line here is that we need to clarify with Francis the exact
form of the test he was requesting. The "new" d3* (with sigma{SE} as the
first term under the square root sign) would lead to a simpler
interpretation of the problems with the DCPS07 test. It would show that
the primary error in DCPS07 was in the neglect of the observational
uncertainty term. It would also simplify interpretation of the results
from Section 6.

I'm sorry about the delay in submission of our manuscript, but this is
an important point, and I'd like to understand it fully. I'm still
hopeful that we'll be able to submit the paper in the next few days.
Many thanks to Tom and Peter for persuading me to pay attention to this
issue. It often took a lot of persuasion...

With best regards,


Benjamin D. Santer
Program for Climate Model Diagnosis and Intercomparison
Lawrence Livermore National Laboratory
P.O. Box 808, Mail Stop L-103
Livermore, CA 94550, U.S.A.
Tel: (925) 422-2486
FAX: (925) 422-7675


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