cc: Steven Sherwood <Steven.SherwoodatXYZxyze.edu>, "Thorne, Peter" <peter.thorneatXYZxyzoffice.gov.uk>, Leopold Haimberger <leopold.haimbergeratXYZxyzvie.ac.at>, Karl Taylor <taylor13atXYZxyzl.gov>, Tom Wigley <wigleyatXYZxyz.ucar.edu>, John Lanzante <John.LanzanteatXYZxyza.gov>, "'Susan Solomon'" <ssolomonatXYZxyznoaa.gov>, Melissa Free <Melissa.FreeatXYZxyza.gov>, peter gleckler <gleckler1atXYZxyzl.gov>, "'Philip D. Jones'" <p.jonesatXYZxyz.ac.uk>, Thomas R Karl <Thomas.R.KarlatXYZxyza.gov>, Steve Klein <klein21atXYZxyzl.llnl.gov>, carl mears <mearsatXYZxyzss.com>, Doug Nychka <nychkaatXYZxyzr.edu>, Gavin Schmidt <gschmidtatXYZxyzs.nasa.gov>, Frank Wentz <frank.wentzatXYZxyzss.com>

date: Mon, 02 Jun 2008 09:32:01 -0700

from: Ben Santer <santer1atXYZxyzl.gov>

subject: Re: Our d3* test

to: Carl Mears <mearsatXYZxyzic.net>

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Dear Carl,

This issue is now covered in the version of the manuscript that I sent

out on Friday. The d2* and d3* statistics have been removed. The new d1*

statistic DOES involve the standard error of the model average trend in

the denominator (together with the adjusted standard error of the

observed trend; see equation 12 in revised manuscript). The slight irony

here is that the new d1* statistic essentially reduces to the old d1*

statistic, since the adjusted standard error of the observed trend is

substantially larger than the standard error of the model average trend...

With best regards,

Ben

Carl Mears wrote:

> Hi

>

> I think I agree (partly, anyway) with Steve S.

>

> I think that d3* partly double counts the uncertainty.

>

> Here is my thinking that leads me to this:

>

> Assume we have a "perfect model". A perfect model means in this context

> 1. Correct sensitivities to all forcing terms

> 2. Forcing terms are all correct

> 3. Spatial temporal structure of internal variability is correct.

>

> In other words, the model output has exactly the correct "underlying"

> trend, but

> different realizations of internal variability and this variability has

> the right

> structure.

>

> We now run the model a bunch of times and compute the trend in each case.

> The spread in the trends is completely due to internal variability.

>

> We compare this to the "perfect" real world trend, which also has

> uncertainty due

> to internal variability (but nothing else).

>

> To me either one of the following is fair:

>

> 1. We test whether the observed trend is inside the distribution of

> model trends. The uncertainty in the

> observed trend is already taken care of by the spread in modeled trends,

> since the representation of

> internal uncertainty is accurate.

>

> 2. We test whether the observed trend is equal to the mean model trend,

> within uncertainty. Uncertainty here is

> the uncertainty in the observed trend s{b{o}}, combined with the

> uncertainty in the mean model trend (SE{b{m}}.

>

> If we use d3*, I think we are doing both these at once, and thus double

> counting the internal variability

> uncertainty. Option 2 is what Steve S is advocating, and is close to

> d1*, since SE{b{m}} is so small.

> Option 1 is d2*.

>

> Of course the problem is that our models are not perfect, and a

> substantial portion of the spread in

> model trends is probably due to differences in sensitivity and forcing,

> and the representation

> of internal variability can be wrong. I don't know how to separate the

> model trend distribution into

> a "random" and "deterministic" part. I think d1* and d2* above get at

> the problem from 2 different angles,

> while d3* double counts the internal variability part of the

> uncertainty. So it is not surprising that we

> get some funny results for synthetic data, which only have this kind of

> uncertainty.

>

> Comments?

>

> -Carl

>

>

>

>

> On May 29, 2008, at 5:36 AM, Steven Sherwood wrote:

>

>>

>> On May 28, 2008, at 11:46 PM, Ben Santer wrote:

>>>

>>> 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) ]

>>>

>>> where

>>>

>>> 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)

>>

>> Shouldn't the first term under sqrt be the standard deviation of the

>> estimate of <<b(m)>> -- e.g., the standard error of <b(m)> -- rather

>> than the standard deviation of <b(m)>? d3* would I think then be

>> equivalent to a z-score, relevant to the null hypothesis that models

>> on average get the trend right. As written, I think the distribution

>> of d3* will have less than unity variance under this hypothesis.

>>

>> SS

>>

>>

>> -----

>> Steven Sherwood

>> Steven.SherwoodatXYZxyze.edu <mailto:Steven.Sherwood@yale.edu>

>> Yale University ph: 203

>> 432-3167

>> P. O. Box 208109 fax: 203

>> 432-3134

>> New Haven, CT 06520-8109

>> http://www.geology.yale.edu/~sherwood

>>

>>

>>

>>

>>

>>

>

--

----------------------------------------------------------------------------

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

email: santer1atXYZxyzl.gov

----------------------------------------------------------------------------

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