Simulating different populations

From: "Tsai, Max" max.tsai@spcorp.com Subject: [NMusers] Simulating different populations Date: Wed, 10 Aug 2005 10:35:42 -0400 Can NONMEM perform simulations to incorporate variability associated with fixed effects across populations? When one develops a NM model, NONMEM output lists the final parameter estimates and the standard error estimates for those parameters. Can NONMEM use those standard errors to simulate different populations so that the fixed effects are not so "fixed"? For instance, if SUBPROBS=2, I would like to see one population have one set of fixed effects and a second population have a different set of fixed effects. For a given population, the fixed effects would be determined by the population parameter estimates plus some population variability whose distribution is specified by the standard error. This would be on the population level between replicate studies and thus a step higher than BSV and WSV. Is this capability built into NONMEM? If so, how would it be done? Thanks. -Max Max Tsai, Ph.D. Senior Scientist (DM/PK) Schering-Plough Corporation 2015 Galloping Hill Road K-15-2-2650 Kenilworth, NJ 07033-0530 *: (908) 740-3911 *: (908) 740-2916 *: max.tsai@spcorp.com

From: "Gastonguay, Marc" Subject: Re: [NMusers] Simulating different populations Date: Wed, 10 Aug 2005 11:56:36 -0400 Max, If I understand you correctly, it sounds like you want to simulate from the posterior (or uncertainty) distributions on the population parameters (THETA, OMEGA and SIGMA). NONMEM is an approximate maximum likelihood method, so you don't get true posteriors but you could use the var-cov matrix of the estimates as an approximation. Implementation is not so straightforward. To do this correctly you need to simulate from a multivariate normal dstribution for THETAs and an Inverse Wishart distribution for OMEGA and SIGMA. There is an experimental, unsupported and undocumented subroutine in NONMEM, known as the PRIOR subroutine, which could implement this sort of simulation, but this is not publicly available and I have no idea if/when it may be included in future versions on NONMEM. You could also do this by sampling THETAs OMEGAs and SIMGAs from a posterior distribution generated from a parametric or non-parametric bootstrap - but you'd have to write your own program to do so. We (and I believe others) have been working on our own utilities to implement these types of simulations in NONMEM and will share results with the user community in the near future. Another option is to move to a full Bayesian MCMC method, such as WinBUGS, which generates posterior distributions for any parameters and derived quantities you're interested in. You could then simulate from these distributions. In the meantime, you could try this bit of code (below) in the NMTRAN control stream, which allows for a simple simulation from independent posterior distributions for any population fixed effects parameters (could also be applied to variance terms if you use a log-normal distribution - but this is a crude approximation). The result is a simulation with inter-trial uncertainty in the population fixed effects parameters. Hope this helps. Marc ========================= Marc R. Gastonguay, Ph.D. Metrum Research Group LLC www.metrumrg.com ;To define inter-trial uncertainty in a parameter,use code similar to this in $PRED or $PK... ; This is from $PRED example where uncertainty in relative bio fraction (FR1) is simulated IF(IREP.EQ.1.AND.NEWIND.EQ.0) COUNT = 0 IF(ICALL.EQ.4) THEN IF(IREP.NE.COUNT) THEN ;SHOULD ONLY HAPPEN AT THE START OF EACH SIMULATION REPLICATE FR1 = -1 DOWHILE(FR1.LE.0) CALL RANDOM(2,R) FR1 = THETA(2) + THETA(3)*R ENDDO ENDIF COUNT = IREP ;Define model using FR1... AUCR = THETA(1) + ETA(1) AUCT = AUCR * FR1 + ETA(2) Y = AUCR*(1-FLG) + AUCT*FLG + EPS(1) ENDIF TRL=IREP ; trial replicate number $THETA 100 ; TYPICAL REFERENCE AUC 0.8 ; MODE OF POSTERIOR DISTRIBUTION ( POINT ESTIMATE OF FR1) 0.075 ;THETA(3) IS SD OF UNCERTAINTY, OR STANDARD ERROR FROM NONMEM $COV $OMEGA 100 100 $SIGMA 10 1 $SIMULATION (437565 NEW) (47003 NORMAL) ONLYSIM SUBPROBLEMS=100 $TABLE ID DAY FLG AUCR AUCT F1 TRL DV FILE = SIMUNCERT.tab NOAPPEND NOPRINT NOHEADER

From: Liping Zhang ZHANG_LIPING@lilly.com Subject: Re: [NMusers] Simulating different populations Date: Wed, 10 Aug 2005 11:09:16 -0500 Hi, Max, what I typically do, is to sample (let's say 1000 replicates) from the distribution of "fixed effects" parameters in Splus, given estimated population parameter value and their variance-covariance matrix generated by NONMEM. So now you have a 1000 different "population", each has different "fixed effects", but overall they reflect the population parameter estimates and their uncertainty. Then through some simple Unix C-shell script I simulate the outcome from each "population" (with fixed population parameters and BSV and WSV) and do statistical summary according to modeling needs. As Ken Kowalski pointed out not long ago, this procedure is computer-labor intensive, you need to simulate large number of population, and within each population, a sufficient number trials. I would not go for this route unless I know for my modeling need it requires this kind of "truthful representation". best regards, Liping Liping Zhang, PhD Global PK/PD Modeling and Trial Simulation ELi Lilly & Company Work: 317-277-8687 Fax: 317-433-6661 DC 0734 Email: zhang_liping@lilly.com

From: "Kowalski, Ken" Ken.Kowalski@pfizer.com Subject: RE: [NMusers] Simulating different populations Date: Wed, 10 Aug 2005 13:12:09 -0400 Liping, Just to clarify. I indicated that simulating different sets of population estimates based on the var-cov matrix from NONMEM and then assuming multivariate normal and possibly inverse-Wishart (for the variance parameters in Omega and/or Sigma) distributions for the parameter estimates is NOT computationally intensive. However, it is laborious to use these population estimates and feed them back in to NONMEM to perform simulations that take into account this parameter uncertainty because it requires custom coding. Based on Marc's comments it sounds like he and others are working on developing utilities that will automate this process in NONMEM which would make it a lot easier to do this as a general practice. If on the other hand we do not wish to make a parametric assumption using the var-cov matrix and multivariate Normal/Inverse-Wishart distribution we could perform parametric or nonparametric bootstrap simulations and re-fit the model in NONMEM for each of say 1000 bootstrap datasets to obtain 1000 estimates of the population parameters from the posterior distribution and use these in subsequent simulations to account for parameter uncertainty. This approach IS computationally intensive because of having to re-fit the model in NONMEM. My feeling is that it is better to routinely do something to take into account parameter uncertainty using the var-cov matrix and parametric assumptions than to doing nothing to take into account parameter uncertainty. Hopefully, individuals working on utilities to automate this parametric approach will be able to share them with us. Ken

From: Liping Zhang ZHANG_LIPING@lilly.com Subject: RE: [NMusers] Simulating different populations Date: Wed, 10 Aug 2005 12:20:37 -0500 Yes, Ken. I understand what you are saying, that is why I did not say "computationally intensive", but says "computer-labor intensive", indicating the long time it takes to get the final results and the custom coding. I did not follow up with the previous posting closely. I would be interested to know who are the "he (Marc) and others" working on developing utilities that will automate this process in NONMEM. best regards, Liping Liping Zhang, PhD Global PK/PD Modeling and Trial Simulation ELi Lilly & Company Work: 317-277-8687 Fax: 317-433-6661 DC 0734 Email: zhang_liping@lilly.com

From: "GIRARD PASCAL" PASCAL.GIRARD@adm.univ-lyon1.fr Subject: RE : [NMusers] Simulating different populations Date: Thu, 11 Aug 2005 12:43:13 +0200 Hi Marc and Max, My 20 cents... In order to do what Marc suggests, you make the assumptions: (1) that NONMEM fixed effects posterior distribution is multivariate normal (MVN) (2) parameterized with mean equal to the THETA estimates and variance to $COV var-cov matrix from which you exclude random effects (co)variance lines and columns, and (3) that you have identified all covariates and/or multimodal distributions using mixture models. Those assumptions are probably not true for many parameters. For example clearance is more likely to be log-normally distributed. F bioavailability is classically bounded by 0 and 1. So in order to use your MVN assumption, you will need first to re-parameterize your model with exponential, logistic functions, ... (for more details, see http://www.page-meeting.org/page/page2002/PascalGirardPage2002.pdf ) Assuming that you have a function or subroutine to resample from MVN (R and Splus do it very easily), second difficulty is to pass var-cov matrix from NONMEM output to your MVN resampling function. Depending on the complexity of your model, this matrix can be very large and keying all those numbers a real pain in ... and source of errors! So for doing this, I have developed an INFN subroutine (to use with PREDPP or an equivalent for $PRED) that outputs THETAs and var-cov matrix in an ASCII file. Probably, "Marc and others" have developed similar tools. Anyway, I am still dreaming of an integrated tool doing all this without extra-coding... Hope it helps, Dr Pascal Girard EA 3738, Ciblage Thrapeutique en Oncologie Fac Mdecine Lyon-Sud, BP12 69921 OULLINS Cedex France Tel +33 (0)4 26 23 59 54 / Fax +33 (0)4 26 23 59 49

From: "Gastonguay, Marc" marcg@metrumrg.com Subject: Re: [NMusers] Simulating different populations Date: Thu, 11 Aug 2005 08:59:42 -0400 Hi, Pascal. Thanks for pointing out these limitations and assumptions. I think your point about fixed effects parameters that have natural bounds is an excellent one; we do we have to think carefully about the parameterization and choice of posterior distributions. For this reason it may be simpler (but more computationally intensive) to simulate from empirical distributions, such as those that might result from a nonparametric bootstrap. For the implementation part, I agree that an automated extraction of relevant components is mandatory. It sure would be nice to have an all-in-one tool that did this for us, but I think that this discussion also illustrates the wonderful flexibility of an ASCII text-based, command-line program like NONMEM, when coupled with a useful programming language (we use R and Perl). Thanks for the helpful suggestions. Marc _______________________________________________________