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Wolverine Pharmacometrics Corporation
Within Subject Random Effect Transformations with NONMEM
VI
B. Frame
9/11/2009
Wolverine Pharmacometrics Corporation
Dynamic Transform Both Sides (TBS)
• What is TBS?
• Why bother with TBS?
• Brief History, Jacobians, and Likelihoods.
• Implementation and examples in NONMEM (V or VI)
Wolverine Pharmacometrics Corporation
What is Transform Both Sides (TBS)?
Consider our usual set up: Yij = PRED(,)ij + ij
Where i indexes subject and j indexes the response
or prediction within subject i.
The assumption here is that ij ~ N(0,2)
In other words, the within subject variability does not depend on time, the PREDiction, or who the subject is (i).
Wolverine Pharmacometrics Corporation
What is Transform Both Sides (TBS)?
Consider an invertible transformation T, with a domain compatible with what is being transformed (response and prediction)...
Then in general TBS is...
T( Yij)= T(PRED(,)ij)+ ij
Once again, the assumption here is that ij ~ N(0,2)
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What is Transform Both Sides (TBS)?
A simple example (no transformation parameter)
ln( Yij)= ln(PRED(,)ij)+ ij ; Yij>0, PRED(,)ij>0
A dynamic example (with )
0;0;11
ijijijij Y
PREDY
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Why Bother with TBS?
0 20 40 60 80 100
Prediction
-20
-10
010
2030
Res
idua
l
Non-constant within subject variance example
-2 0 2 4 6
050
100
150
200
Residual
Rightly Skewed Residuals
-5 0 5
010
020
030
0
Residual
Heavy Tailed (Leptokurtic) Residual Distribution
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Useful Resourceshttp://www.stat.uconn.edu/~studentjournal/index_files/pengfi_s05.pdf
Carroll Rupert (1988) Transformation and Weighting in Regression
Estimating Data Transformations in Nonliner Mixed Effect Models; Oberg and Davidian; Biometrics 56,65-72;March 2000.
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Transformations, Likelihoods and Jacobians.
Suppose we have a continuous random variable, X whose
logarithm is distributed N(, 2). Letting Y=ln(X) we know
that the density for Y is...
2
)2/()(
2)(
22
y
Y
eyf
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But What is the Distribution for X?
To find fX(x) when X=g(Y), and g is monotone, we use the following change of variable formula...
|))((|))(()( 11 xgdx
dxgfxf YX
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OK, so we turn the crank!
• X=g(Y) = exp(Y)
• Y=g-1(X) = ln(X)
• d/dX(g-1(X)) = 1/X
2
)2/())(ln(
2)(
22
x
exf
x
X
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Now lets Focus on the Dynamic Box-Cox TBS
0,,);,0(~11 2
ijijijij PREDYN
PREDY
11)(1
ijij
ijij
PREDYYgX
)1(
2
)2/()/)1(/)1((
2)(
22
ij
PREDy
ijY ye
yfijij
ij
Our assumption is that ...
)1(1 ))(( ijij
ij
yygdy
d
Let,
so
and
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Example
• A new ‘patch’ has been developed for bromodrosis.
• The T/2 is short and we have 7 steady state serum concentrations on each of 100 subjects.
• This may be the simplest possible PK example!
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Initial Model (CWS7.TXT)$PROBLEM$DATA NMDATA7.CSV$INPUT ID DV ; JUST ID AND SERUM CONCENTRATION!$PRED W=THETA(2) ;ADDITIVE SD CL=THETA(1)*EXP(ETA(1)) ;CL/F WITH BETWEEN SUBJECT VAR PRE=1/CL ;@ SS ASSUMING INPUT RATE = 1 4 ALL RES1=(DV-PRE)/W ;FORM A WITHIN SUBJECT RESIDUAL Y=PRE+EPS(1)*W
$THETA (0,.1) ;CL/F (0,1) ;SD ADDITIVE$OMEGA .1$SIGMA 1 FIX$EST MAXEVALS=9999 METH=1 PRINT=1 ; JUST BECAUSE!$COV PRINT=E$TABLE ID RES1 ONEHEADER NOAPPEND NOPRINT FILE=TWS7.TXT
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Graphics.
-1.6 -1.0 -0.5 0.0 0.6 1.1 1.6 2.2 2.7 3.3 3.8
RES1
0.0
0.1
0.2
0.3
0.4
0.5
Density for CWS7.TXT Within Subject Residual
Skewness = 1.05
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CWS7L.TXT / CWS7L1.TXT$SUB CONTR=CONTR.TXT CCONTR=CCONTRA.TXT$PRED W=THETA(2) ;SD CL=THETA(1)*EXP(ETA(1)) ;CL/F PRE=1/CL ;@ SS ASSUMING INPUT RATE = 1 LAM=THETA(3) ;BOX COX LAMBDA PARAMETER PREL=(PRE**LAM-1)/LAM ;TRANSFORMED PREDICTION Y=PREL+EPS(1)*W ;ADDITIVE WITHIN SUBJECT ERROR IN ;THE TRANSFORMED SPACE RES1=((DV**LAM-1)/LAM-PREL)/W ;RESIDUAL IN THE T SPACE$THETA (0,.1) ;CL/F (0,1) ;SD ADDITIVE (0,1) ;BOX COX LAMBDA PARAMETER$OMEGA .1 ; THIS INIT WORKS FINE WITH NMV NM6??$SIGMA 1 FIX$EST MAXEVALS=9999 METH=1 PRINT=1$COV PRINT=E$TABLE ID RES1 ONEHEADER NOAPPEND NOPRINT FILE=TWS7L.TXT
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CCONTRA.TXT subroutine ccontr (icall,c1,c2,c3,ier1,ier2) parameter (lth=40,lvr=30,no=50) common /rocm0/ theta (lth) common /rocm4/ y double precision c1,c2,c3,theta,y,w,one,two dimension c2(*),c3(lvr,*) data one,two/1.,2./ if (icall.le.1) return w=y y=(y**theta(3)-one)/theta(3) call cels (c1,c2,c3,ier1,ier2) y=w c1=c1-two*(theta(3)-one)*log(y) return end
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Regression Engine Bake Off
nmv nm6 nm6 init2
SD in Transformed
Space
1.9 2.1 2.1
0.24 0.27 0.26
CL/F 0.01 0.01 0.01
$COV Yes No Yes!
0.004 1E-5 0.004
Wolverine Pharmacometrics Corporation
Last Slide
-3.2 -2.6 -2.1 -1.5 -0.9 -0.4 0.2 0.8 1.3 1.9 2.5
RES1
0.0
0.1
0.2
0.3
0.4
Density for Residual in T Space (nmv)
Skewness = -0.05Change in MOF = 187
