Bob, thanks for your reply and pointer to the source code for tdev2 . I have looked at it but perhaps you can clarify what the source of tdev2 is and how it is used (I cannot see any calls made to it in your source code, so I am not really sure how it is being used). I can see for DF =1, tdev2 returns a Cauchy distributed random variable, which is the 1D t-distribution for DF=1, and for t=2 it looks like it returns the 1D t-distribution for DF=2. For the other DF values it is not so clear, but I am guessing it returns a 1D t-distribution with the specified DF.
If you simply assemble a p-dimensional sample vector with components which are independent 1D t random variables with a specified DF , indeed it will avoid the dimensional mismatch and work OK without bias in the QR case, but it will not be a true multivariate t-distribution (the components of a multivariate t-distribution are not independent). The usual argument for using a multivariate t as an importance sampler is that a) It has fatter tails than a multivariate normal b) It preserves the radial symmetry of the multivariate normal in the case where the covariance matrix is the unit matrix (i.e. the density is only a function of the distance from the origin) Assembling a p-dimensional sample vector whose individual components are 1D t distributions with a specified DF satisfies a), but not b). It’s a perfectly valid importance sampling distribution, and indeed has been suggested in the literature as much simpler alternative to the multivariate t, particularly in low dimensions, but it is not a multivariate t (which is the more usual meaning of ‘t-distribution’ in the multivariate importance sampling context) . It is not radially symmetric (which may or may not be important, depending on the posterior multidimensional shape and the dimensionality. There is a directionality bias which increases with dimensionality) The documentation is somewhat ambiguous – DF=4 The proposal density is to be t distribution with 4 degrees of freedom. Based on the code I have seen, I now guess this means that each component of the proposal is an independent t random variable with DF=4. But I am only guessing – perhaps you could clarify what distribution is really being referred to by ‘t-distribution’ in the p-dimensional case – multivariate p-dimensional t with specified DF or Cartesian product of p independent 1D t random variables with specified DF? From: [email protected] [mailto:[email protected]] On Behalf Of Bauer, Robert Sent: Thursday, May 15, 2014 9:57 PM To: [email protected] Subject: RE: [NMusers] SAEM and IMP Bob: Yes, when I was developing the code for using the Sobol method in NONMEM, I too found that using the t-distribution with Sobol/Quasi normal method resulted in biased assessments of the objective function, when I used the unmodified standard technique of generating t-samples. So, I experimented with various alternative methods like you did, and the final algorithms used in NONMEM 7.2 and 7.3 appear to avoid the bias, at least when I tried some simple models, such as the example1 problem in the NONMEM ..\examples directory. For such a problem, I tried the following DF settings, with and without using the SOBOL method (RANMETHOD=3 (default), or RANMETHOD=3S2), DF=0, 1, 2, 3, 4, 5, 6, 7, 8, 10, 11, and 12. I think beyond 12 DF is sufficiently normal like, and not much use. In all cases, the OBJF was within 1 unit of each other, and always within a STD of the Monte Carlo noise to the OBJF (I used EONLY=1, 20 iterations to obtain OBJFV replicates for each setting). By the way, the .cnv file publishes the mean and standard deviation of the last CITER objective function values (see nm730.pdf manual on root.cnv). Certainly it is worth assuring this lack of bias under other kinds of, and more complex, models, just to make sure, so it is worth while for modelers to use caution when mixing Sobol with DF>0. I should point out that odd-number DF’s such as 7 makes the corrective algorithm for Sobol with t-distributions work harder, whereas even-numbered DF’s are a little cleaner. So I recommend even-numbered DF’s, or 1, up to 10 to work with Sobol, at least as far as my algorithm is concerned (so DF=1,2,4,6,8,10 seem most efficient to use with Sobol). The algorithm I use for t-distributions is in the subroutine TDEV2, and is in ..source\general.f90. I publish it because I borrowed freely from the literature, and I consider it therefore “open-source”. Just a note for orienting you to the code, IRANM=5 refers to the Sobol method. I am afraid my theoretical expertise in the Sobol/Quasi random methods is not as good as yours, so please feel free to review the code, and perhaps let me and the modeling community know if there may be any potential pitfalls in using this t-distribution algorithm with Sobol. I am also happy to consider improvements you may come up with on this matter. Robert J. Bauer, Ph.D. Vice President, Pharmacometrics, R&D ICON Development Solutions 7740 Milestone Parkway Suite 150 Hanover, MD 21076 Tel: (215) 616-6428 Mob: (925) 286-0769 Email: [email protected]<mailto:[email protected]> Web: www.iconplc.com<http://www.iconplc.com/> ICON plc made the following annotations. ------------------------------------------------------------------------------ This e-mail transmission may contain confidential or legally privileged information that is intended only for the individual or entity named in the e-mail address. If you are not the intended recipient, you are hereby notified that any disclosure, copying, distribution, or reliance upon the contents of this e-mail is strictly prohibited. If you have received this e-mail transmission in error, please reply to the sender, so that ICON plc can arrange for proper delivery, and then please delete the message. Thank You, ICON plc South County Business Park Leopardstown Dublin 18 Ireland Registered number: 145835 NOTICE: The information contained in this electronic mail message is intended only for the personal and confidential use of the designated recipient(s) named above. This message may be an attorney-client communication, may be protected by the work product doctrine, and may be subject to a protective order. As such, this message is privileged and confidential. If the reader of this message is not the intended recipient or an agent responsible for delivering it to the intended recipient, you are hereby notified that you have received this message in error and that any review, dissemination, distribution, or copying of this message is strictly prohibited. If you have received this communication in error, please notify us immediately by telephone and e-mail and destroy any and all copies of this message in your possession (whether hard copies or electronically stored copies). Thank you. buSp9xeMeKEbrUze
