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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)

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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 HistoryTransformations 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 History3. 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

-4.0 -3.2 -2.5 -1.8 -1.1 -0.4 0.3 1.0 1.7 2.4 3.1

x

0.0

0.1

0.2

0.3

0.4rnorm(1000,0,1)skewness = 0.04

0.0 2.3 4.6 6.9 9.1 11.4 13.7 16.0 18.2 20.5 22.8

x

0.0

0.1

0.2

0.3

0.4

0.5

exp(rnorm(1000,0,1))Skewness = 3.6

Right Skewed

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.7 0.8 0.9 1.0

x

0

2

4

6

8

10

Left Skewed

rbeta(1000,2,0.3)Skewness = -1.8

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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.

-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

Mesokurtic

rnorm(1000,0,1)Kurtosis = -0.2

0.0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.0

x

0.0

0.4

0.8

1.2

Platykurtic

rbeta(1000,1.5,1.5)Kurtosis = -1

-7.3 -5.8 -4.3 -2.8 -1.3 0.2 1.7 3.2 4.7 6.2 7.6

x

0.0

0.1

0.2

0.3

Leptokurtosis (Heavy Tailed)

rt(1000,4)Kurtosis = 3.8

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Transformations for Skewness Removal

0,0: ET

0,0;1

ET

Power Family:

Box - Cox (1964)

Manly (1976)

0;1

eET

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Kurtosis Removal

0:)1|ln(|)(

0:1)1|(|

)(

signET

signET

John - Draper (1980):

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An Example, Finally!Back to our second example: PopPK!

C1.TXT DATA1.TXT

-0.6 -0.4 -0.2 0.0 0.1 0.3 0.5 0.7 0.9 1.0 1.2

ETA2

0.0

0.5

1.0

1.5

2.0

Eta bar: -0.012

p-value: 0.97

Skewness: 1.23

Kurtosis: 2.77

Example 1. Density of Modal Values for Random Effect on Elimination Rate: One Sub-population:Conditional Estimation with nteraction

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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 nm6C6.TXT

Drop in MOF of ~ 16 points.

Estimate = 0.9

-0.7 -0.5 -0.4 -0.2 -0.1 0.1 0.2 0.4 0.5 0.7 0.8

ETA2

0.0

0.5

1.0

1.5

Eta bar = 0.002

p-value = 0.095

Skewness = 0.079

Kurtosis = 0.70

Example 1. Density of Modal Values for Random Effect on Elimination Rate:

One Sub-population:Conditional Estimation with nteraction

:Transformed Random Effect