Transcript Mixture Modeling with NONMEM V

Between Subject Random Effect
Transformations with NONMEM
VI
Bill Frame
09/11/2009
Wolverine Pharmacometrics Corporation
Between Subject Random Effect ()
Transformations.
• Why bother with transformations?
• What is a transformation?
• Examples and Brief History.
• Implementation and examples in NONMEM (V
or VI)
Wolverine Pharmacometrics Corporation
Why Bother with Transformations?
Variance stabilization (Workshop 7).
NONMEM assumes that ~ N(0,)
A better statistical fit to the data?
Perhaps simulations can be improved upon, as opposed to a model
with no eta transformation?
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Q: What is an ETA transformation?
A: A one to one function that maps ETA to a new random effect
ET, as a function of a fixed effect parameter ().
Q: What are desirable properties of such a transformation?
• Invertible, this means one to one.
• Domain = Real line, the same as ETA.
• Differentiable with respect to argument and parameter, more of a
theoretical issue than a practical one.
• Null value for lambda is not on boundary of parameter space.
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Examples and Brief History
Transformations can be applied to:
1. Statistics i.e.
Fisher’s Z transformation for the Pearson product moment
correlation coefficient ().
Z = ½*loge((1+)/(1-))
2. The response (Y=DV):
Change Y to Z=Y1/2 if E(Y)  Var(Y) and model Z, this is
sometimes done for Poisson data.
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Examples and Brief History
3. Predictors (i.e. SHOE):
Consider the simple linear (in the random effects) mixed model
with the usual assumptions:
Y = THETA(1) + THETA(2)*SHOE**THETA(3) + ETA(1) + EPS(1)
4. Random effects (): The rest of workshop 6.
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What is Skewness?
A number? This is pulled from the S-Plus 6.1 help API.
If y = x - mean(x), then the "moment" method computes the skewness value
as mean(y^3)/mean(y^2)^1.5
Right Skewed
rnorm(1000,0,1)
skewness = 0.04
0.4
0.5
exp(rnorm(1000,0,1))
Skewness = 3.6
0.4
0.3
0.3
0.2
0.2
0.1
0.1
0.0
-4.0
-3.2
-2.5
-1.8
-1.1
-0.4
0.3
1.0
1.7
2.4
0.0
3.1
x
0.0
Left Skewed
10
6
4
2
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0
0.1
0.2
0.3
0.4
0.5
0.6
x
0.7
4.6
6.9
9.1
11.4
13.7
x
rbeta(1000,2,0.3)
Skewness = -1.8
8
2.3
0.7
0.8
0.9
1.0
16.0
18.2
20.5
22.8
What is Kurtosis?
A number? This is pulled from the S-Plus 6.1 help API. If y = x - mean(x),
then the "moment" method computes the kurtosis value as
mean(y^4)/mean(y^2)^2 - 3.
Platykurtic
Mesokurtic
rnorm(1000,0,1)
Kurtosis = -0.2
0.4
rbeta(1000,1.5,1.5)
Kurtosis = -1
1.2
0.3
0.8
0.2
0.4
0.1
0.0
0.0
-2.8
-2.2
-1.6
-1.1
-0.5
0.0
0.6
1.1
1.7
2.2
2.8
x
0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0
Leptokurtosis (Heavy Tailed)
rt(1000,4)
Kurtosis = 3.8
0.3
0.2
0.1
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0.0
-7.3
-5.8
-4.3
-2.8
-1.3
0.2
1.7
x
3.2
4.7
6.2
7.6
x
Transformations for Skewness
Removal
Power Family:
ET    :   0,  0
Box - Cox (1964)
  1
ET 
;   0,  0

Manly (1976)
ET 
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e
 

1
;  0
Kurtosis Removal
John - Draper (1980):
ET  sign( ) 
(| | 1)   1

:  0
ET  sign( )  ln(| | 1) :   0
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An Example, Finally!
Back to our second example: PopPK!
C1.TXT DATA1.TXT
Example 1. Density of Modal Values for Random Effect on Elimination Rate:
One Sub-population:Conditional Estimation with nteraction
Eta bar:
p-value:
Skewness:
Kurtosis:
2.0
-0.012
0.97
1.23
2.77
1.5
1.0
0.5
0.0
-0.6
-0.4
-0.2
0.0
0.1
0.3
ETA2
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0.5
0.7
0.9
1.0
1.2
Much Data/Subject + Conditional
Estimation =
$PK
KA=THETA(1)*EXP(ETA(1))
ET2=(EXP(ETA(2)*THETA(4))-1)/THETA(4) ;THETA(4) = LAMBDA
K=THETA(2)*EXP(ET2)
S2=THETA(3)*WT
$THETA
(0,1) ;KA
(0,.12) ;K
(0,.4) ;VD
(.5) ;LAMBDA TRANSFORM PARAMETER
$OMEGA .25 ;INTER-SUBJECT VARIATION KA
$OMEGA BLOCK(1) .05 ;INTER-SUBJECT VARIATION K
$ERROR
Y=F*(1+EPS(1))
$SIGMA .013 ;PROPORTIONAL ERROR
$ESTIMATION MAXEVALS=9000 PRINT=1 METHOD=1 INTERACTION
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Results with nmv or nm6
C6.TXT
Drop in MOF of ~ 16 points.
 Estimate = 0.9
Example 1. Density of Modal Values for Random Effect on Elimination Rate:
One Sub-population:Conditional Estimation with nteraction
:Transformed Random Effect
Eta bar
= 0.002
p-value
= 0.095
Skewness = 0.079
Kurtosis = 0.70
1.5
1.0
0.5
0.0
-0.7
-0.5
-0.4
-0.2
-0.1
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0.1
0.2
ETA2
0.4
0.5
0.7
0.8