1. Wolverine Pharmacometrics Corporation Between Subject Random Effect Transformations with NONMEM VI Bill Frame 09/11/2009

2. 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)

3. 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?

4. Wolverine Pharmacometrics Corporation 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.

5. Wolverine Pharmacometrics Corporation 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*log e ((1+  )/(1-  )) 2. The response (Y=DV): Change Y to Z=Y 1/2 if E(Y)  Var(Y) and model Z, this is sometimes done for Poisson data.

6. Wolverine Pharmacometrics Corporation 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.

7. Wolverine Pharmacometrics Corporation 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

8. Wolverine Pharmacometrics Corporation 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.

9. Wolverine Pharmacometrics Corporation Transformations for Skewness Removal Power Family: Box - Cox (1964) Manly (1976)

10. Wolverine Pharmacometrics Corporation Kurtosis Removal John - Draper (1980):

11. Wolverine Pharmacometrics Corporation An Example, Finally! Back to our second example: PopPK! C1.TXT DATA1.TXT

12. Wolverine Pharmacometrics Corporation 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

13. Wolverine Pharmacometrics Corporation Results with nmv or nm6  C6.TXT Drop in MOF of ~ 16 points. Estimate = 0.9