1. Short Subject Presentation Unbounded Likelihoods with NM6
B Frame
9/15/2009
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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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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.
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4. So what is an unbounded likelihood?
Consider the Gaussian kernel…
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5. Consider the limit as 2  0
There are two cases here…
One when y  
And one when y = 
See the homework!
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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.
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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.
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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.
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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.
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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.
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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.
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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 = $COV = YES
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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 = $COV = NO
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SIGMA - COV MATRIX FOR RANDOM EFFECTS - EPSILONS ****
EPS EPS2
EPS1
E-01
EPS2
E E-05 Here is the problem, a sigma has gone to zero.
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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)
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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 )
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Do your homework!
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