Oh, I forgot about the std error formula. Is where the sqrt(2) comes from? Doh, that would be retarded. Then I know someone who would require sqrt(3) for the NOE spectra! Is that really what Palmer meant, that std error is the same as "the standard deviation of the differences between the heights of corresponding peaks in the paired spectra" which "is equal to sqrt(2)*sigma" (Palmer et al., 1991)?
I'm pretty sure though that the standard error is not the measure we want for the confidence interval of the peak intensity. The reason is because I think that the std error is a measure of how far the sample mean is from the true mean (ignore this, this is a quick reference for myself: http://en.wikipedia.org/wiki/Standard_error_(statistics) ). (Warning, from here to the end of the paragraph is a rant!) This is similar in concept to AIC model selection (see http://en.wikipedia.org/wiki/Akaike_information_criterion and http://en.wikipedia.org/wiki/Model_selection if you haven't heard about the advanced statistical field of model selection before). AIC is a little more advanced though as it estimates the Kullback-Leibler discrepancy (http://en.wikipedia.org/wiki/Kullback–Leibler_divergence) which is a measure of distance between the true distribution and the back-calculated distribution using all information about the distribution. Ok, that wasn't too relevant. Anyway, the std error as a measure of the differences in means of 2 different distributions is not a measure of the spread of either the true, measured, or back-calculated distributions (or the 4th distribution, the back-calculated from the fit to the true model). The std error is not the confidence intervals of any of these 4 distributions, just the difference between 2 of them using only a small part of the information of those distributions. It's the statistical measure of the difference in means of the true and measured distributions. As an aside, for those completely lost now a clearer explanation of these 4 distributions fundamental to data analysis, likelihood, discrepancies, etc. can be read in section 2.2 of my PhD thesis at http://dtl.unimelb.edu.au:80/R/-?func=dbin-jump-full&object_id=67077&current_base=GEN01 or http://www.amazon.com/Protein-Dynamics-Model-free-Analysis-Relaxation/dp/3639057627/ref=sr_1_6?ie=UTF8&s=books&qid=1219247007&sr=8-6 (sorry for the blatant plug ;). Oh, the best way to picture all of these concepts and the links between them is to draw and label 4 distributions on a piece of paper on the same x and y-axes with not too much overlap between them and connect them with arrows labelled with all the weird terminology. Sorry, that was just a long way of saying that the std error is the quality of how the sample mean matches the real mean and how the standard deviation is the spread of the distribution. That being so, I would avoid setting the standard error as the peak height uncertainty. Maybe it would be best to do as you say Chris, and also avoid the averaging of the replicated intensities. So, now to the dirty statistics required for the sparse sampling caused by lack of NMR time. We can use the standard deviation formula - the measure of the spread of the measured distribution which is the estimator of the spread of the true distribution that optimisation (which is chi-squared fitting and hence finding the maximal likelihood (http://en.wikipedia.org/wiki/Maximum_likelihood)), Monte Carlo simulations, and the frequentist model selection techniques all utilise as the foundation for their fundamental derivations. We would then square this to get the variance: sigma^2 = sum({Ii - Iav}^2) / (n - 1). In my previous posts I was saying sigma was the variance, but just ignore that. Next we need to average the variance across all spins, simply because the sample size is so low for each peak and hence the error estimate is horrible. Whether this estimator of the true variance is good or bad is debatable (well, actually, it's bad), but it is unavoidable. It also has the obvious disadvantage in that the peak height error is, in reality, different for each peak. Now, if not all spectra are replicated, then the approach needs to be modified to give us errors for the peaks of the single spectra. Each averaged replicated time point (spectrum) has a single error associated with it, the average variance. These are usually different for different time points, in some cases weakly decreasing exponentially. So I think we should average the average variances and have a single measure of spread for all peaks in all spectra. This estimator of the variance is again bad. Interpolation might be better, but is still not great. Cheers, Edward P.S. None of this affects an analysis using peak heights of non-replicated spectra and the RMSD of the baseplane noise. But if someone wants to use the peak volume as a measure of peak intensity, then we have a statistical problem. Until someone finds a reference or derives the formula for how the RMSD of the base plane noise relates to volume error, then peak heights will be essential for error analysis. I'm also willing to be corrected on any of the statistics above as I'm not an expert in this and may have missed some fundamental connections between theories. And there may be a less 'dirty' way of doing the dirty part of the statistics. On Fri, Oct 17, 2008 at 1:59 AM, Chris MacRaild <[EMAIL PROTECTED]> wrote: > On Thu, Oct 16, 2008 at 8:07 PM, Edward d'Auvergne > <[EMAIL PROTECTED]> wrote: >> On Thu, Oct 16, 2008 at 7:02 AM, Chris MacRaild <[EMAIL PROTECTED]> wrote: >>> On Thu, Oct 16, 2008 at 3:11 PM, Sébastien Morin >>> <[EMAIL PROTECTED]> wrote: >>>> Hi, >>>> >>>> I have a general question about curve fitting within relax. >>>> >>>> Let's say I proceed to curve fitting for some relaxation rates >>>> (exponential decay) and that I have a duplicate delay for error estimation. >>>> >>>> ======== >>>> delays >>>> >>>> 0.01 >>>> 0.01 >>>> 0.02 >>>> 0.04 >>>> ... >>>> ======== >>>> >>>> Will the mean value (for delay 0.01) be used for curve fitting and rate >>>> extraction ? >>>> Or will both values at delay 0.01 be used during curve fitting, hence >>>> giving more weight on delay 0.01 ? >>>> >>>> In other words, will the fit only use both values at delay 0.01 for >>>> error estimation or also for rate extraction, giving more weight for >>>> this duplicate point ? >>>> >>>> How is this handled in relax ? >>>> >>>> Instinctively, I would guess that the man value must be used for >>>> fitting, as we don't want the points that are not in duplicate to count >>>> less in the fitting procedure... Am I right ? >>>> >>> >>> I would argue not. If we have gone to the trouble of measuring >>> something twice (or, equivalently, measuring it with greater >>> precision) then we should weight it more strongly to reflect that. >>> >>> So we should include both duplicate points in our fit, or we should >>> just use the mean value, but weight it to reflect the greater >>> certainty we have in its value. >>> >>> As I type this I realise this is likely the source of the sqrt(2) >>> factor Tyler and Edward have been debating on a parallel thread - the >>> uncertainty in height of any one peak is equal to the RMS noise, but >>> the std error of the mean of duplicates is less by a factor of >>> sqrt(2). >> >> At the moment, relax simply uses the mean value in the fit. Despite >> the higher quality of the duplicated data, all points are given the >> same weight. This is only because of the low data quantity. As for >> dividing the sd of differences between duplicate spectra by sqrt(2), >> this is not done in relax anymore. Because some people have collected >> triplicate spectra, although rare, relax calculates the error from >> replicated spectra differently. I'm prepared to be told that this >> technique is incorrect though. The procedure relax uses is to apply >> the formula: >> >> sd^2 = sum({Ii - Iav}^2) / (n - 1), >> >> where n is the number of spectra, Ii is the intensity in spectrum i, >> Iav is the average intensity, sd is the standard deviation, and sd^2 >> is the variance. This is for a single spin. The sample number is so >> low that this value is completely meaningless. Therefore the variance >> is averaged across all spins (well due to a current bug the standard >> deviation is averaged instead). Then another averaging takes place if >> not all spectra are duplicated. The variances across all duplicated >> spectra are averaged to give a single error value for all spins across >> all spectra (again the sd averaging bug affects this). The reason for >> using this approach is that you are not limited to duplicate spectra. >> It also means that the factor of sqrt(2) is not applicable. If only >> single spectra are collected, then relax's current behaviour of not >> using sqrt(2) seems reasonable. >> > > Here is how I understand the sqrt(2) issue: > > The sd of duplicate (or triplicate, or quadruplicate, or ... ) peak > heights is assumed to give a good estimate of the precision with which > we can measure the height of a single peak. So for peak heights that > have not been measured in duplicate (ie relaxation times that have not > been duplicated in our current set of spectra), sd is a good estimate > of the uncertainty associated with that height. > > For peaks we have measured more than once, we can calculate a mean > peak height. The precision with which we know that mean value is given > by the std error of the mean ie. sd/sqrt(n) where n is the number of > times we have measured that specific relaxation time. I think this is > the origin of the sqrt(2) for duplicate data. > > A made up example: > T I > 0 1.00 > 10 0.90 > 10 0.86 > 20 0.80 > 40 0.75 > 70 0.72 > 70 0.68 > 100 0.55 > 150 0.40 > 200 0.30 > > The std deviation of our duplicates is 0.04 so the uncertainty on each > value above is 0.04 > > BUT, the uncertainty on the mean values for our duplicate time points > (10 and 70) is 0.04/sqrt(2) = 0.028 > > So if we use the mean values as points in our fit, we should use 0.028 > as the uncertainty on those values (while all other peaks have > uncertainty 0.04) > > Alternatively (and equivalently) we can use the original observations, > including all duplicate points. In this case, all points have the same > uncertainty of 0.04, as they are all the result of a single > measurement. > > To do anything else is to underestimate the precision with which we > have measured our relaxation rates. > > Chris > > > > >> Regards, >> >> Edward >> >> >> P.S. The idea for the 1.3 line is to create a new class of user >> functions, 'spectrum.read_intensities()', 'spectrum.set_rmsd()', >> 'spectrum.error_analysis()', etc. to make all of this independent of >> the analysis type. See >> https://mail.gna.org/public/relax-devel/2008-10/msg00029.html for >> details. >> > _______________________________________________ relax (http://nmr-relax.com) This is the relax-users mailing list relax-users@gna.org To unsubscribe from this list, get a password reminder, or change your subscription options, visit the list information page at https://mail.gna.org/listinfo/relax-users
