Short Subject Presentation Unbounded Likelihoods with NM6 B Frame 9/15/2009 Wolverine Pharmacometrics Corporation

Wolverine Pharmacometrics Corporation 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

Wolverine Pharmacometrics Corporation 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

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

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

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

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

Wolverine Pharmacometrics Corporation 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

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

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

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Wolverine Pharmacometrics Corporation On the surface There are two types of subjects. Those who gain 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

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

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

Wolverine Pharmacometrics Corporation 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

Wolverine Pharmacometrics Corporation 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

Wolverine Pharmacometrics Corporation 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

Wolverine Pharmacometrics Corporation Do your homework! Wolverine Pharmacometrics Corporation