From: "Cawley Gavin Dr \(CMP\)"
To:
Subject: RE: Possible error in recent IJC paper
Date: Fri, 31 Oct 2008 11:01:46 -0000
Cc: "Jones Philip Prof \(ENV\)" , "Gavin Schmidt" , "Thorne, Peter" , "Tom Wigley"
Dear Ben,
many thanks for the full response to my query. I think my confusion arose from the
discussion on RealClimate (which prompted our earlier communication on this topic), which
clearly suggested that the observed trend should be expected to lie within the spread of
the models, rather than neccessarily being close to the mean as the models are stochastic
simulations (which seemed reasonable). I've just re-read that post, the key paragraph from
[1]http://www.realclimate.org/index.php/archives/2007/12/tropical-troposphere-trends/ is as
follows:
"The interpretation of this is a little unclear (what exactly does the sigma refer to?),
but the most likely interpretation, and the one borne out by looking at their Table IIa, is
that sigma is calculated as the standard deviation of the model trends. In that case, the
formula given defines the uncertainty on the estimate of the mean - i.e. how well we know
what the average trend really is. But it only takes a moment to realise why that is
irrelevant. Imagine there were 1000's of simulations drawn from the same distribution, then
our estimate of the mean trend would get sharper and sharper as N increased. However, the
chances that any one realisation would be within those error bars, would become smaller and
smaller. Instead, the key standard deviation is simply sigma itself. That defines the
likelihood that one realisation (i.e. the real world) is conceivably drawn from the
distribution defined by the models."
I had therefore expected the test to use the standard deviations of both the models and the
observations (which would give a flat plot in 5B and there would be an obvious overlap of
the uncertainties in 6a at say 500hPa).
best regards
Gavin
-----Original Message-----
From: Ben Santer [[2]mailto:santer1@llnl.gov]
Sent: Fri 10/31/2008 4:06 AM
To: Cawley Gavin Dr (CMP)
Cc: Jones Philip Prof (ENV); Gavin Schmidt; Thorne, Peter; Tom Wigley
Subject: Re: Possible error in recent IJC paper
Dear Gavin,
Thanks very much for your email, and for your interest in our recent
paper in the International Journal of Climatology (IJoC). There is no
error in equation (12) in our IJoC paper. Let me try to answer the
questions that you posed.
The first term under the square root in our equation (12) is a standard
estimate of the variance of a sample mean - see, e.g., "Statistical
Analysis in Climate Research", by Francis Zwiers and Hans von Storch,
Cambridge University Press, 1999 (their equation 5.24, page 86). The
second term under the square root sign is a very different beast - an
estimate of the variance of the observed trend. As we point out, our d1*
test is very similar to a standard Student's t-test of differences in
means (which involves, in its denominator, the square root of two pooled
sample variances).
In testing the statistical significance of differences between the model
average trend and a single observed trend, Douglass et al. were wrong to
use sigma_SE as the sole measure of trend uncertainty in their
statistical test. Their test assumes that the model trend is uncertain,
but that the observed trend is perfectly-known. The observed trend is
not a "mean" quantity; it is NOT perfectly-known. Douglass et al. made a
demonstrably false assumption.
Bottom line: sigma_SE is a standard estimate of the uncertainty in a
sample mean - which is why we use it to characterize uncertainty in the
estimate of the model average trend in equation (12). It is NOT
appropriate to use sigma_SE as the basis for a statistical test between
two uncertain quantities. The uncertainty in the estimates of both
modeled AND observed trend needs to be explicitly incorporated in the
design of any statistical test seeking to compare modeled and observed
trends. Douglass et al. incorrectly ignored uncertainties in observed
trends.
I hope this answers your first question, and explains why there is no
inconsistency between the formulation of our d1* test in equation (12)
and the comments that we made in point #3 [immediately before equation
(12)]. As we note in point #3, "While sigma_SE is an appropriate measure
of how well the multi-model mean trend can be estimated from a finite
sample of model results, it is not an appropriate measure for deciding
whether this trend is consistent with a single observed trend."
We could perhaps have made point #3 a little clearer by inserting
"imperfectly-known" before "observed trend". I thought, however, that
the uncertainty in the estimate of the observed trend was already made
very clear in our point #1 (on page 7, bottom of column 2).
To answer your second question, d1* gives a reasonably flat line in
Figure 5B because the first term under the square root sign in equation
(12) (the variance of the model average trend, which has a dependence on
N, the number of models used in the test) is roughly a factor of 20
smaller than the second term under the square root sign (the variance of
the observed trend, which has no dependence on N). The behaviour of d1*
with synthetic data is therefore dominated by the second term under the
square root sign - which is why the black lines in Figure 5B are flat.
In answer to your third question, our Figure 6A provides only one of the
components from the denominator of our d1* test (sigma_SE). Figure 6A
does not show the standard errors in the observed trends at discrete
pressure levels. Had we attempted to show the observed standard errors
at individual pressure levels, we would have produced a very messy
Figure, since Figure 6A shows results from 7 different observational
datasets.
We could of course have performed our d1* test at each discrete pressure
level. This would have added another bulky Table to an already lengthy
paper. We judged that it was sufficient to perform our d1* test with the
synthetic MSU T2 and T2LT temperature trends calculated from the seven
radiosonde datasets and the climate model data. The results of such
tests are reported in the final paragraph of Section 7. As we point out,
the d1* test "indicates that the model-average signal trend (for T2LT)
is not significantly different (at the 5% level) from the observed
signal trends in three of the more recent radiosonde products (RICH,
IUK, and RAOBCORE v1.4)." So there is no inconsistency between the
formulation of our d1* test in equation (12) and the results displayed
in Figure 6.
Thanks again for your interest in our paper, and my apologies for the
delay in replying to your email - I have been on travel (and out of
email contact) for the past 10 days.
With best regards,
Ben
Cawley Gavin Dr (CMP) wrote:
>
>
> Dear Prof. Santer,
>
> I think there may be a minor problem with equation (12) in your paper
> "Consistency of modelled and observed temperature trends in the tropical
> trophosphere", namely that it includes the standard error of the models
> 1/n_m s{}^2 instead of the standard deviation s{}^2. Firstly
> the current formulation of (12) seems at odds with objection 3 raised at
> the start of the first column of page 8. Secondly, I can't see how the
> modified test d_1^* gives a flat line in Figure 5B as the test statistic
> is explicitly dependent on the size of the model ensemble n_m. Thirdly,
> the equation seems at odds with the results depicted graphically in
> Figure 6 which would suggest the models are clearly inconsistent at
> higher levels (400-850 hPa) using the confidence interval based on the
> standard error. Lastly, (12) seems at odds with the very lucid
> treatment at RealClimate written by Dr Schmidt.
>
> I congratulate all 17 authors for an excellent contribution that I have
> found most instructive!
>
> I do hope I haven't missed something - sorry to have bothered you if
> this is the case.
>
> best regards
>
> Gavin
>
--
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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) 422-3840
FAX: (925) 422-7675
email: santer1@llnl.gov
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References
1. http://www.realclimate.org/index.php/archives/2007/12/tropical-troposphere-trends/
2. mailto:santer1@llnl.gov