$PRIOR with normal for OMEGA?

Is it possible to use a normal prior for OMEGA? The default is inverse Wishart, but I'd be interested in using Normal (insuring that it is positive definite) Any ideas? thanks Mark Sale M.D. Vice President, Modeling and Simulation Nuventra Pharma Sciences, Inc. 2525 Meridian Parkway, Suite 280 Durham, NC 27713 Phone (919)-973-0383 [email protected]<[email protected]> CONFIDENTIALITY NOTICE The information in this transmittal (including attachments, if any) may be privileged and confidential and is intended only for the recipient(s) listed above. Any review, use, disclosure, distribution or copying of this transmittal, in any form, is prohibited except by or on behalf of the intended recipient(s). If you have received this transmittal in error, please notify me immediately by reply email and destroy all copies of the transmittal.

Mark, Without getting too technical, recall that the inverse wishart is a distribution on matrices, and is naturally conjugate to the multivariate normal distribution (i.e. on vectors of random variables.) Generally, this is chosen because posterior computations are simplified and it is easy to sample from this posterior distribution. What you're suggesting is the Matrix Normal distribution. Since it is not conjugate to the multivariate normal, posterior computations can cause headaches. So while it is *possible* to use such a distribution as a prior, it is cumbersome to work with in practice and requires thinking about some things like correlation and scale in non straightforward ways. -------------- Andy Gewitz, PhD Bioengineering and Therapeutic Sciences University of California, San Francisco

Hi Mark, Not sure about this but how about estimating THETAS scaling fix OMEGAs (e.g. CL = THETA(1)*EXP(THETA(2)*ETA(1)) ; $OMEGA 1 fix)? Then you can implement the prior in the same way for random effect parameters as you do for fixed effect parameters. Ps. Probably not be compatible with MU-parameterization. Best regards, Martin Bergstrand, Ph.D. Senior Consultant Pharmetheus AB +46(0)709 994 396 [email protected] www.pharmetheus.com +46(0)18 513 328 U-A Science Park, Dag Hammarskjölds v. 52b 752 37 Uppsala, Sweden Skickat från min iPhone 11 nov. 2016 kl. 04:57 skrev Mark Sale <[email protected]>: > Is it possible to use a normal prior for OMEGA? The default is inverse > Wishart, but I'd be interested in using Normal (insuring that it is positive > definite) Any ideas? > > thanks > > > > > Mark Sale M.D. > Vice President, Modeling and Simulation > Nuventra Pharma Sciences, Inc. > 2525 Meridian Parkway, Suite 280 > Durham, NC 27713 > Phone (919)-973-0383 > [email protected] > CONFIDENTIALITY NOTICE The information in this transmittal (including > attachments, if any) may be privileged and confidential and is intended only > for the recipient(s) listed above. Any review, use, disclosure, distribution > or copying of this transmittal, in any form, is prohibited except by or on > behalf of the intended recipient(s). If you have received this transmittal in > error, please notify me immediately by reply email and destroy all copies of > the transmittal. > >

Hi Mark, As I’m sure you know NONMEM has the TNPRI functionality for that. If you want to use it without the TNPRI functionality, you can estimate OMEGA as THETA and use a multivariate normal for that. If you have off-diagonal elements, you may want to do a Cholesky transformation (you can get that automatically in PsN). As Andy writes there are pros and cons with different priors. While IW has a better shape to its prior for variances, it is problematic that there is no correlation between the typical value estimates and their variances. Best regards, Mats Mats Karlsson, PhD Professor of Pharmacometrics Dept of Pharmaceutical Biosciences Faculty of Pharmacy Uppsala University Box 591 75124 Uppsala Phone: +46 18 4714105 Fax + 46 18 4714003 http://www.farmbio.uu.se/research/researchgroups/pharmacometrics/

Hi Mark, Indeed there is: As an alternative to NWPRI, there is the TNPRI subroutine that you can use with $PRIOR (frequentist prior). This functionality is tripple normal, with regards to thetas, omegas and sigma(s). I will describe this more in detail than Mark would need (hopefully for the benefit of others). I used to think that TNPRI was an appealing alternative when the standard error of population parmeters were all modest. The implementation appears to be appealing at a first glance (less error prone): Simply plug in the MSFO file from a previous run (generating the prior), as a prior representing the covariance matrix from that previous run. In addition, if from that previous run one has reported SEs based on the covariance matrix, it may be appealing to use the same distribution when simulating with uncertainty in population parameters (what I call simulation mode, below), or as a prior in the next analysis with a new analysis data set (what I call estimation mode, below). However, over the years I have been using it less and less due to various limitations and “features”. I am not sure if Marks question was with regards to estimation with support of a prior (estimation mode), or simulation with uncertainty in population parameters based on a prior distribution (simulation mode), but separate the list of bugs/features/limitations we have come across, below. Some of these features are documented, whereas others I believe are not. In estimation mode (using TNPRI) there are only a few limitations that comes to my mind: Any thetas that are fixed must appear as the last thetas in your model (already when generating the prior) When generating the prior, do not use the UNCONDITIONAL option for the covariance step. Even in cases where the estimation is successful (so that the UNCONDITIONAL option is not needed), the subsequent estimation with TNPRI will fail (If I recall correctly, it will run forever). If you use PsN: TNPRI is not supported by all programs, in particular, you can not use scm. Some may raise their eyebrows, thinking that the prior does not allow testing for (new) covariates, so I will adress that comment right away. With a new patient population at hand, you may want to use scm to test whether there is a significant difference in any of the population parameters, as compared to the prior (prior not including the new patient population). In simulation mode (using TNPRI) there are additional limitations that I would tend to call bugs, and I will only mention a few: From the TABLE output you can use IPRED (and the distribution of population parameters), but other PRED defined variables can not be trusted, including PRED itself: so any clever calculations you may do in your control stream (e.g. change from baseline): Do not use it! The output may have been generated based on the initial estimates (i.e. prior mode, despite TRUE=PRIOR), rather than based on the simulations that include uncertainty in population parameters Limitations on which parameters needs to be fixed is even greater. If I remember correctly, the whole model must be re-formulated in case you have any terminal thetas: SIGMAs and OMEGAs must then also be fixed (to 1), and magnitudes estimated as fixed effects (representing e.g. standard error of IIV, or the covariance) - these additional thetas must then also appear before the fixed thetas. But this is when generating the prior (in estimation mode, before the subsequent simulation). Possibly, when using the prior in simulation mode, then all previously fixed thetas must be unfixed again. When generating the prior, do not use the UNCONDITIONAL option for the covariance step. Even in cases where the estimation is successful (so that the UNCONDITIONAL option is not needed), the subsequent simulation step will fail (If I recall correctly, it will run forever). At Pharmetheus, we have not used TNPRI widely and tend to use it less and less (favouring NWPRI), and we have never had the time to fully characterise these bug/features: as soon as we have concluded it works for the task at hand, we leave it without further exploring situations where TNPRI may provide an unexpected/erroneous output. Consequently, you may find my bug/feature description above a bit unclear. I do not know exactly what situations trigger these bugs, and I could list additional vague descriptions of bugs/features we have come across, if I look back into previous projects. But I think if I do that it would raise more questions than it answers... However, this discussion is mainly on simulations, and maybe missess out entirely on Marks question? Hopefully, someone will find it useful, still. Finally, back more towards Marks question, if SE is large in the sense that the normal (uncertainty) distribution would go outside the boundaries (e.g. OMEGA<0), for any population parameter (fixed and random), then there is functionality to handle this. I have never used TNPRI with any large SE:s, but Mats Karlsson once mentioned to me that the functionality does not really handle this situation the way you would expect: the tail of the distribution that goes outside the boundary will be moved to the other end of the distribution. Obviously, this is not what you want in case that tail constitutes a large fraction, but if it is only a question of 1 out of 10 000 sample, this may be harmless (in most cases). Maybe someone can complement with a fuller description on the limitations with TNPRI than what I could provide? Otherwise, I leave you with the following short summary, for when how to use TNPRI: In generating the prior try to avoid fixing any theta (e.g. do not use a fixed theta to represent allometric constants), and if the purpose of TNPRI is simulation try to avoid fixing any omega as well Do not use UNCONDITIONAL in the covariance step In estimation with support of the TNPRI prior Be aware you can not use PsN scm, for covariate selection (NWPRI works fine with the new nonmem notation THETAP, etc. But you need to be aware of the issues of testing covariates with suport of a prior that did not include that covariate) In simulation using TNPRI From the TABLE output, you may use IPRED (and other variables that are not simulated, like ID, trial-replicate nr, time and dose). THETAS, OMEGAS and SIGMAS can be used to check the distribution across replicate simulations, but pred-defined variables should not be used for anything! It used to be quite hairy and error prone to manually set up a complicated nonmem control stream for using the NWPRI. However, if you can use automatic functionality for adding the NWPRI distribution in the control stream and just check it has been implemented correctly, this is not a big hurdle. For example, PsN has such a functionality as an option to the “update_inits” program. In addition, the new nonmem notation makes it easier for others to understand the control stream (THETAP, etc). Therefore, in most cases where I need to use a (frequentist) prior, NWPRI is currently my first option. (But I leave you with the cheery reservation that I do not mention much around limitations/features with NWPRI, since that is clearly out of scope for the topic in this tread :>) Best regards Jakob Jakob Ribbing, Ph.D. Senior Consultant, Pharmetheus AB Cell/Mobile: +46 (0)70 514 33 77 [email protected] www.pharmetheus.com Phone, Office: +46 (0)18 513 328 Uppsala Science Park, Dag Hammarskjölds väg 52B SE-752 37 Uppsala, Sweden This communication is confidential and is only intended for the use of the individual or entity to which it is directed. It may contain information that is privileged and exempt from disclosure under applicable law. If you are not the intended recipient please notify us immediately. Please do not copy it or disclose its contents to any other person.

Hi To add on Martin’s suggestion, one item to think of when using theta for estimating the variances of eta is to use log scaled STD as your model parameter, so instead of THETA*ETA (with OMEGA fixed) use exp(THETA)*ETA. This way you would assume a log-normal prior for the standard deviation of ETA which is more reasonable. Moreover the log-normal is quite close to the gamma distribution which I believe is the (univariate) distribution of diagonal elements of the inverse Wishart. Kind Regards Magnus Åstrand Principal Clinical Pharmacometrician, Ph.D. _____________________________________________________________________________________________ AstraZeneca Innovative Medicines | Quantitative Clinical Pharmacology SE-431 83 Mölndal, Sweden T: +46 (0)31 776 23 41 Mob: +46 (0)708 467 667 magnus.astrand_at_astrazeneca.com Please consider the environment before printing this e-mail

Hi To add on Martin’s suggestion, one item to think of when using theta for estimating the variances of eta is to use log scaled STD as your model parameter, so instead of THETA*ETA (with OMEGA fixed) use exp(THETA)*ETA. This way you would assume a log-normal prior for the standard deviation of ETA which is more reasonable. Moreover the log-normal is quite close to the gamma distribution which I believe is the (univariate) distribution of diagonal elements of the inverse Wishart. Kind Regards Magnus Åstrand Principal Clinical Pharmacometrician, Ph.D. _____________________________________________________________________________________________ AstraZeneca Innovative Medicines | Quantitative Clinical Pharmacology SE-431 83 Mölndal, Sweden T: +46 (0)31 776 23 41 Mob: +46 (0)708 467 667 [email protected] Please consider the environment before printing this e-mail

(Sorry for raking up a discussion that's a month old!) It's worth pointing people in the direction of Andrew Gelman's work on specifying priors for between individual variance terms. http://www.stat.columbia.edu/~gelman/research/published/taumain.pdf Short summary: Inverse-Gamma (so by extension Inverse-Wishart) may lead to bias in estimates, if the observed between individual variance is small. Half-Cauchy is a more robust choice. (I'm not sure about the additional influence of shrinkage on top of that though...) Mike