1、Approximations of the population Fisher information matrix- differences and consequences Joakim Nyberg, Sebastian Ueckert, Andrew C. Hooker,2,Background,At PODE 2009 all Population Optimal Design (OD) Software should evaluate the same simple Warfarin problem1-compartment model, 1st order absorption,
2、 oral dose 70 mgProportional error model (2=0.01)32 subjects with 8 measurements at0.5, 1, 2, 6 ,24, 36, 72,120 hours (evaluation),3,Fisher Information Matrix (FIM),FIM can be calculated in different ways:,A* is somewhat modified/updated if full is used, i.e.,Assuming var(y) w.r.t. the fixed effects
3、0,Assuming var(y) w.r.t. the fixed effects=0,Different between full and reduced,4,Fisher Information Matrix (FIM),The FIM,If we have correlation between fixed effectsand random effects in like the FULL is the “theoreticallycorrect” method.If not, the Reduced is “correct theoretically”but this is sel
4、dom the case in Pharmacometrics,5,Results from last PODE 2009,The “truth”,* Retout, Mentr Further developments of FIM in NLME-models. J. BioPharm. Stat 2003,*,*,6,Results from last PODE 2009,The “truth”,7,Results from last PODE 2009,Possibly issues with the Cramr-Rao inequality,8,Results from last P
5、ODE 2009 summary,Software gave similar results with similar approximationsReduced superior to Full in terms of predicting the “truth”Even less predictive performance with higher orderFOCE-based FIM.,9,Possible reasons Initial ideas,The derivation of Full or Red is wrongDerive FIM with simulations, i
6、.e. integrate over observed FIMFO-approximation too poor - FOCE is obviously not enough, try high order approximationsAsymptotic behavior (FIM-1COV) - Increase data set x 2 = SE should decrease by 2(1/2)Numerical instability in Full but not in Red FIM - Using automatic differentiation (AD) to avoid
7、step length issuesEstimation software is not true ML-estimator,i.e. efficiency of estimator not accurate- NM hard to know how the parameter search is performedbut Monolix well documented,10,Investigations Reducing the complexity,ln-transform model to have additive res-error(avoiding interaction term
8、s)Check that the problem holds for prop IIV structure (FO approximation = proportional IIV = exp IIV)Fix all parameters except fixed effect Ka,11,Results Reducing the complexity,* 100 000 bootstrap samples,ln model, add error, exp IIV = prop IIV,Issues still remaining = work with simplified model,*
9、Retout, Mentr Further developments of FIM in NLME-models. J. BioPharm. Stat 2003,12,Results Full vs Reduced,Asymptotic behavior (FIM-1COV)Increase data set x 2 = SE should decrease by 2(1/2)Numerical instability in Full but not in Red FIM - Using automatic differentiation (AD) to avoid step length i
10、ssues,None of this affected the results (2 down 3 to go),13,Next things to try.,The derivation of Full or Red is wrongDerive FIM with simulations, i.e. integrate over observed FIMFO-approximation too poor - FOCE is obviously not enough, try high order approximations,SO,14,Results High order approxim
11、ations & simulation based derivations,* 100 000 bootstrap samples,15,Results Full vs Red,The derivation of Full or Red is wrong Integration FIM Analytic FIM = Not the answerFO-approximation too poorSO shrinks the differences but still to poor of an approx. FOCE is worse but NONMEM integrated FOCE FI
12、M is good?!Possibly issues with the FOCE method?,16,FOCE FIM Differences & Improvements,NONMEM FOCE assumes linearization around the modeof the distribution = correlation between the individualparameters and the population parameters.Analytic FIMFOCE * does not assume this To calculate individual mo
13、de data is needed,Update Analytic FIMFOCE to include the correlation:Calculate Expected Empirical Bayes Estimates (EEBE)EEBE are not data dependentWhenever PopED differentiates pop parameters; differentiate EEBE as well,* Retout, Mentr Further developments of FIM in NLME-models. J. BioPharm. Stat 20
14、03,17,Results Updated FIMFOCE,The new FOCE method solves the problem!,18,The answer,The Full FIM does not always work with the FO-approximation,19,When to use which method?,20,Does Full/Red affect the optimal design?,Full,Reduced,1,2,2,2,1,6,1,1,3 support points,5 support points,21,Does linearizatio
15、n method affect the optimal design?,Optimal SOCE,SOCEMC,SOCE=MC,SOCEMC,Example from PAGE 2009 Nyberg et al,Surface of |FIM| for SOCE-MC,Optimal MC,22,Always use reduce FIM instead?,Reduced |FIM|,Results from Nyberg et al PODE 2008,Full |FIM|,Similar results with the transit compartment model,23,Surf
16、ace of RSE(%) Full FIM only different from reduced in some regions,Full Ka RSE(%),Reduced Ka RSE(%),S3-S7: (7.79h),S3-S7: (7.79h),S8: (120h),S8: (120h),Full Red,24,Conclusions,The FO-approximation is not always enough for Full FIM Possibly also too poor approx. for ReducedReduced FIM collapses occas
17、ionallyHigh order approximations stabilize differencesDifferent approximations give different optimal designs, e.g. different sampling times and different number of support points,25,Suggestions,If runtime allows Use high order approximations FOCE, SO, SOCE, MC etc. If Red is stable Use reduced to o
18、ptimize but evaluate with both If Red is unstable Optimize with Full but evaluate with RedBeware: - No “golden” solution is presented- The Cramer-Rao inequality does not hold comparingdifferent methods when optimizing / estimating- To get “correct” SE from the FIM either sim/est needsto be performed or high order FIMs need to be evaluated,26,Thank you,I would like to acknowledge Sergei Leonov for our interesting emails discussing these issues.,
