cc: "'Philip D. Jones'" , Gavin Schmidt , "Thorne, Peter" , Tom Wigley date: Thu, 30 Oct 2008 21:06:31 -0700 from: Ben Santer subject: Re: Possible error in recent IJC paper to: "Cawley Gavin Dr (CMP)" 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 > -- ---------------------------------------------------------------------------- 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 ----------------------------------------------------------------------------