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Short Subject Presentation Unbounded Likelihoods with NM6

Short Subject Presentation Unbounded Likelihoods with NM6. B Frame 9/15/2009. Background. Mentioned in Gelman’s text “ Bayesian Data Analysis ”. Also known as “ variance escape ” to some frequentists. Dealt with in at least two NONMEM based papers.

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Short Subject Presentation Unbounded Likelihoods with NM6

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  1. Short Subject PresentationUnbounded Likelihoods with NM6 B Frame 9/15/2009 Wolverine Pharmacometrics Corporation

  2. Background • Mentioned in Gelman’s text “Bayesian Data Analysis”. • Also known as “variance escape” to some frequentists. • Dealt with in at least two NONMEM based papers. Wolverine Pharmacometrics Corporation

  3. Conditioning on Certain Random Events Associated with Statistical Variability in PK/PD, Stuart L. Beal Volume 32, Number 2 / April, 2005 This paper discusses several interesting topics. Mixed effects modeling of weight change associated with placebo and pregabalin administration Bill Frame1 , Stuart L. Beal2, Raymond Miller1, Jeannette Barrett3 and Paula Burger1 Volume 34, Number 6 / December, 2007 This paper discusses a particular method of dealing with unbounded likelihoods. Wolverine Pharmacometrics Corporation

  4. So what is an unbounded likelihood? Consider the Gaussian kernel… Wolverine Pharmacometrics Corporation

  5. Consider the limit as 2  0 • There are two cases here… • One when y   • And one when y =  • See the homework! Wolverine Pharmacometrics Corporation

  6. NONMEM Symptomatology • -2LL rapidly decreases then NONMEM crashes without any output or error messages. • Diagonsed by Professor Stuart L. Beal as being caused by a group of subjects with all their observations equal to baseline. Wolverine Pharmacometrics Corporation

  7. Specifics of the Problem • The pregabalin weight change data set. • Baseline weight not modeled, treated as a covariate. • The model giving rise to the Problem is not the one that was publised. Wolverine Pharmacometrics Corporation

  8. But… • A very similar model is described in my chapter on Finite Mixtures in Ene Ette’s Pharmacometrics text book. • A model of this type was stable with the pregabalin data, then at one point variance escape occurred. Wolverine Pharmacometrics Corporation

  9. Knowing what causes the problem… • It should be easy to cook up an example so you can watch NONMEM crash and burn. • After nearly a day of simulating and estimating I could not re-create the problem with data that I can share. Wolverine Pharmacometrics Corporation

  10. So, to get the technique out… • I simulated some data that does not crash NONMEM but does drive a sigma to zero. • I will show the technique developed by Stuart. Wolverine Pharmacometrics Corporation

  11. Wolverine Pharmacometrics Corporation

  12. On the surface • There are two types of subjects. • Those whogain and those that do not. • But there are really two types of “stay the samers”, ones that do not move at all, and ones that bounce around baseline. Wolverine Pharmacometrics Corporation

  13. Model/Data c1.txt / nmdata100.csv $PRED AS=THETA(1)*EXP(ETA(1)) K=THETA(2) Y=BSLN*EXP(AS*(1-EXP(-K*TIME)))+EPS(1) $THETA (0,0.2) ;1 ASYMPTOTE (0,0.1) ;2 RATE Subjects are modeled as gainers, or possibly stayers if ETA(1) << 0. -2LL = 638.753 $COV = YES Wolverine Pharmacometrics Corporation

  14. Model/Data c2.txt / nmdata100.csv $PRED AS=THETA(1)*EXP(ETA(1)) K=THETA(2) IF (MIXNUM.EQ.1) THEN Y=BSLN*EXP(AS*(1-EXP(-K*TIME)))+EPS(1) ;GAINERS GO HERE ELSE Y=BSLN+EPS(2) ; BOTH TYPES OF STAYERS GO HERE ENDIF 0PARAMETER ESTIMATE IS NEAR ITS BOUNDARY THIS MUST BE ADDRESSED BEFORE THE COVARIANCE STEP CAN BE IMPLEMENTED -2LL = -1144.884 $COV = NO Wolverine Pharmacometrics Corporation

  15. SIGMA - COV MATRIX FOR RANDOM EFFECTS - EPSILONS **** EPS1 EPS2 EPS1 + 9.68E-01 EPS2 + 0.00E+00 1.00E-05 Here is the problem, a sigma has gone to zero. Wolverine Pharmacometrics Corporation

  16. Stu’s Solution • Weights were recorded to the nearest 0.1kg. • Initial observations equal to baseline are discarded. • The likelihood for the first non-baseline observation is adjusted to reflect that it cannot be in [baseline -0.05,baseline+0.05) Wolverine Pharmacometrics Corporation

  17. That is… For an arbitrary subject, with baseline=b, and random effects vector , and initial observation x, let… L0(x) be the un-adjusted likelihood of the observation under the model. Then the adjusted likelihood for the first non-baseline observation is… L0(x)/1-p0(b)), where p0(b) is the probability that x is in [b-0.05, b+0.05 ) Wolverine Pharmacometrics Corporation

  18. Do your homework! Wolverine Pharmacometrics Corporation

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