Leonid, Nick, Plotting the uniform distribution w/o exponentation was useful to me (R code): hist(runif(100)) hist(runif(1000)) hist(exp(runif(100))) hist(exp(runif(1000))) hist(exp(runif(10000))) - Also after exponentation, the uniform distribution has very sharp edges. I have never encountered such data distributions myself. And such sharp edges seem pretty difficult to capture in a continuous model. - You need an excessive amount of data to pinpoint the shape of a distribution exactly
On a more general note: the more informative a dataset is on a distribution, the less assumptions you have to make about it. From limited to very rich informativeness one could go from untransformed via exponential (*), semi-parametric and splines to non-parametric approaches in order to describe the distribution, if needed. My guess is that in most real-life cases we will have to live with making assumptions about the shape of the distribution. Best regards, Jeroen Modeling & Simulation Expert Pharmacokinetics, Pharmacodynamics & Pharmacometrics (P3) - DMPK MSD PO Box 20 - AP1112 5340 BH Oss The Netherlands jeroen.elassa...@merck.com T: +31 (0)412 66 9320 M: +31 (0)6 46 101 283 F: +31 (0)412 66 2506 www.msd.com (*) or vice versa, from exponential via untransformed, as exponential transformation often makes more sense and describes data better in PK-PD analyses -----Original Message----- From: owner-nmus...@globomaxnm.com [mailto:owner-nmus...@globomaxnm.com] On Behalf Of Leonid Gibiansky Sent: Monday, 31 May, 2010 23:31 To: Nick Holford; nmusers Subject: Re: [NMusers] distribution assumption of Eta in NONMEM Nick, I think, transformation idea is the following: Assume that your (true) model is CL=POPCL*exp(ETAunif) where ETAunif is the random variable with uniform distribution. Assume that you have transformation TRANS that converts normal to uniform. Then ETAunif can be presented (exactly) as ETAunif=TRANS(ETAnorm). Therefore, the true model can be presented (again, exactly) as CL=POPCL*exp(TRANS(ETAnorm)) This model should be used for estimation and according to Mats, should provide you the lowest OF Leonid -------------------------------------- Leonid Gibiansky, Ph.D. President, QuantPharm LLC web: www.quantpharm.com e-mail: LGibiansky at quantpharm.com tel: (301) 767 5566 Nick Holford wrote: > Leonid, > > The result is what I expected. NONMEM just estimates the variance of > the random effects. It doesn't promise to tell you anything about the > distribution. > > It is indeed bad news for simulation if your simulation relies heavily > on the assumption of a normal distribution and the true distribution > is quite different. > > I think you have to be very careful looking at posthoc ETAs. They are > not informative about the true ETA distribution unless you can be sure > that you have low shrinkage. If shrinkage is not low then a true > uniform will become more normal looking because the tails will collapse. > > The approach that Mats seems to suggest is to try different > transformations of NONMEM's ETA variables to try to lower the OFV. > What is not clear to me is why these transformations which lower the > OFV will make the simulation better when the ETA variables that are > used for the simulation are required to be normally distributed. > > Imagine I use this for estimation: > CL=POPCL*EXP(ETA(1)) where the true ETA is uniform If I now use the > estimated OMEGA(1,1) which will be a good estimate of the uniform > distribution variance, uvar, for simulation then I am using > CL=POPCL*EXP(N(0,uvar)) > which will be wrong because I am now assuming a normal distribution > but using the variance of a uniform. > > Now suppose I try: > CL=POPCL*TRANS(ETA(1)) where TRANS is some transformation that lowers > the OFV to the lowest I can find but the true ETA is still uniform If > I now use the same transformation for simulation with an OMEGA(1,1) > estimate of the variance transvar > CL=POPCL*TRANS(N(0,transvar)) which uses a normal distribution then > why should I expect the simulated distribution of CL to resemble the > true distribution with a uniform ETA? > > Nick > > Leonid Gibiansky wrote: >> Hi Nick, >> I think, I understood it from your original e-mail, but it was so >> unexpected that I asked to confirm it. >> >> Actually, not a good news from your example. >> >> Nonmem cannot distinguish two models: >> with normal distribution, and >> with uniform distributions >> as long as they have the same variance. >> >> So if you simulate from the model, you will end up with very >> different >> results: either simular to the original data (if by chance, your >> original problem happens to be with normal distribution) or very >> different (if original distribution was uniform). >> >> This shows the need to investigate normality of posthoc ETAs very >> carefully. >> >> Very interesting example >> Thanks >> Leonid >> >> -------------------------------------- >> Leonid Gibiansky, Ph.D. >> President, QuantPharm LLC >> web: www.quantpharm.com >> e-mail: LGibiansky at quantpharm.com >> tel: (301) 767 5566 >> >> >> >> >> Nick Holford wrote: >>> Leonid, >>> >>> I meant by OMEGA(1) the OMEGA value estimated by NONMEM. I suppose I >>> should have written OMEGA(1,1) to be more precise -- sorry! >>> >>> Nick >>> >>> Leonid Gibiansky wrote: >>>> Nick, Mats >>>> >>>> I would guess that nonmem should inflate variance (for this >>>> example) trying to fit the observed uniform (-0.5, 0.5) into some >>>> normal N(0, ?). This example (if I read it correctly) shows that >>>> Nonmem somehow estimates variance without making distribution assumption. >>>> Nick, you mentioned: >>>> >>>> "the mean estimate of OMEGA(1) was 0.0827" >>>> >>>> does it mean that Nonmem-estimated OMEGA was close to 0.0827 or you >>>> refer to the variances of estimated ETAs? >>>> >>>> Thanks >>>> Leonid >>>> >>>> >>>> -------------------------------------- >>>> Leonid Gibiansky, Ph.D. >>>> President, QuantPharm LLC >>>> web: www.quantpharm.com >>>> e-mail: LGibiansky at quantpharm.com >>>> tel: (301) 767 5566 >>>> >>>> >>>> >>>> >>>> Mats Karlsson wrote: >>>>> Nick, >>>>> >>>>> >>>>> >>>>> It has been showed over and over again that empirical Bayes >>>>> estimates, when individual data is rich, will resemble the true >>>>> individual parameter regardless of the underlying distribution. >>>>> Therefore I don't understand what you think this exercise contributes. >>>>> >>>>> >>>>> >>>>> Best regards, >>>>> >>>>> Mats >>>>> >>>>> >>>>> >>>>> Mats Karlsson, PhD >>>>> >>>>> Professor of Pharmacometrics >>>>> >>>>> Dept of Pharmaceutical Biosciences >>>>> >>>>> Uppsala University >>>>> >>>>> Box 591 >>>>> >>>>> 751 24 Uppsala Sweden >>>>> >>>>> phone: +46 18 4714105 >>>>> >>>>> fax: +46 18 471 4003 >>>>> >>>>> >>>>> >>>>> *From:* owner-nmus...@globomaxnm.com >>>>> [mailto:owner-nmus...@globomaxnm.com] *On Behalf Of *Nick Holford >>>>> *Sent:* Monday, May 31, 2010 6:05 PM >>>>> *To:* nmusers@globomaxnm.com >>>>> *Cc:* 'Marc Lavielle' >>>>> *Subject:* Re: [NMusers] distribution assumption of Eta in NONMEM >>>>> >>>>> >>>>> >>>>> Hi, >>>>> >>>>> I tried to see with brute force how well NONMEM can produce an >>>>> empirical Bayes estimate when the ETA used for simulation is >>>>> uniform. I attempted to stress NONMEM with a non-linear problem >>>>> (the average DV is 0.62). The mean estimate of OMEGA(1) was 0.0827 >>>>> compared with the theoretical value of 0.0833. >>>>> >>>>> The distribution of 1000 EBEs of ETA(1) looked much more uniform >>>>> than normal. >>>>> Thus FOCE show no evidence of normality being imposed on the EBEs. >>>>> >>>>> $PROB EBE >>>>> $INPUT ID DV UNIETA >>>>> $DATA uni1.csv ; 100 subjects with 1 obs each $THETA 5 ; HILL >>>>> $OMEGA 0.083333333 ; PPV_HILL = 1/12 $SIGMA 0.000001 FIX ; EPS1 >>>>> >>>>> $SIM (1234) (5678 UNIFORM) NSUB=10 $EST METHOD=COND MAX=9990 SIG=3 >>>>> $PRED IF (ICALL.EQ.4) THEN >>>>> IF (NEWIND.LE.1) THEN >>>>> CALL RANDOM(2,R) >>>>> UNIETA=R-0.5 ; U(-0.5,0.5) mean=0, variance=1/12 >>>>> HILL=THETA(1)*EXP(UNIETA) >>>>> Y=1.1**HILL/(1.1**HILL+1) >>>>> ENDIF >>>>> ELSE >>>>> >>>>> HILL=THETA(1)*EXP(ETA(1)) >>>>> Y=1.1**HILL/(1.1**HILL+1) + EPS(1) ENDIF >>>>> >>>>> REP=IREP >>>>> >>>>> $TABLE ID REP HILL UNIETA ETA(1) Y ONEHEADER NOPRINT FILE=uni.fit >>>>> >>>>> I realized after a bit more thought that my suggestion to >>>>> transform the eta value for estimation wasn't rational so please >>>>> ignore that senior moment in my earlier email on this topic. >>>>> >>>>> Nick >>>>> >>>>> >>>>> -- >>>>> >>>>> Nick Holford, Professor Clinical Pharmacology >>>>> >>>>> Dept Pharmacology & Clinical Pharmacology >>>>> >>>>> University of Auckland,85 Park Rd,Private Bag 92019,Auckland,New >>>>> Zealand >>>>> >>>>> tel:+64(9)923-6730 fax:+64(9)373-7090 mobile:+64(21)46 23 53 >>>>> >>>>> email: n.holf...@auckland.ac.nz <mailto:n.holf...@auckland.ac.nz> >>>>> >>>>> http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford >>>>> >>> >>> -- >>> Nick Holford, Professor Clinical Pharmacology Dept Pharmacology & >>> Clinical Pharmacology University of Auckland,85 Park Rd,Private Bag >>> 92019,Auckland,New Zealand tel:+64(9)923-6730 fax:+64(9)373-7090 >>> mobile:+64(21)46 23 53 >>> email: n.holf...@auckland.ac.nz >>> http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford >>> > > -- > Nick Holford, Professor Clinical Pharmacology Dept Pharmacology & > Clinical Pharmacology University of Auckland,85 Park Rd,Private Bag > 92019,Auckland,New Zealand tel:+64(9)923-6730 fax:+64(9)373-7090 > mobile:+64(21)46 23 53 > email: n.holf...@auckland.ac.nz > http://www.fmhs.auckland.ac.nz/sms/pharmacology/holford > This message and any attachments are solely for the intended recipient. 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