1. Slide no 1 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 Dealing with censored data in linear and non-linear models Wan Hui Ong Clausen Birgitte Biilmann Rønn DSBS/FMS 26 Apr 2006

2. Slide no 2 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 Overview Background Model Estimation Implementation Examples Simulation Conclusion

3. Slide no 3 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 Censored PK data PK data: Pharmacokinetic data Concentration of drug/preparation over time Disposition of the drug/preparation Example 1: Biphasic insulin Three subcutaneous injections a day Concentrations measured over 24 hours

4. Slide no 4 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 Biphasic insulin concentration over time – three subcutaneous injections Censored at 13pmol/l

5. Slide no 5 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 Censored PD data PD data: Pharmacodynamic data Effect of the drug/preparation Measurements of the effect over time Example 2: Dose-response trial with inhaled insulin 5 dose levels given in iso-glycaemic clamp Glucose infusion rate measured over 10 hours

6. Slide no 6 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 Cumulated glucose infusion rate versus dose

7. Slide no 7 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 Censored GIR observations The method (manual clamp) might not be sufficiently sensitive, when the ’true’ glucose need is very low AUC(0-10h)GIR valued 0 are instead included in the analysis as being less than a treshold value, (e.g. 3.5).

8. Slide no 8 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 Analysis with censored data: ’Usual’ solution: Treat observations as missing Problem: Biased estimate of mean Biased estimated of variance Simple solution: Obtain original data when possible μ censored c σ σ censored μ

9. Slide no 9 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 Model with normal distributed error Linear or non-linear mean structure and general covariance structure: where Y i is the observation vector for subject i, β is the vector of fixed parameters, b i is the vector of random effects, b i ~N(0,Ψ) mutually independent and independent of ε i, the residual error vector, ε i ~N(0,Σ).

10. Slide no 10 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 Marginal likelihood function for fixed effects parameters with full data:

11. Slide no 11 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 Marginal likelihood function for fixed effects parameters with censored data:

12. Slide no 12 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 Approximate likelihood inference The intergral can rarely be solved explicitly for repeated measurements For non-linear mean function (in the random effects) Intergral approximations must be used Laplace approximation or Adaptive Gaussian quadrature See eg.Wolfinger, R.D. (93) Laplace’s approximation for nonlinear mixed effects models, Biometrica 80:791-795, Davidian,M., Giltinan, D.M. (95) Nonlinear Models for Repeated Measurements Data. London: Chapman & Hall, Pinheiro, J.C., Bates, D.M. (1995). Approximations to the log-likelihood function in nonlinear mixed-effects model. J.Computat.Graph.Statist. 4:12-35, or Vonesh, E.F. Chinchilli, V.M. (97). Linear and Nonlinear Models for the Analysis of Repeated Measurements. New York: Marcel Decker, Inc.

13. Slide no 13 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 Cumulated glucose infusion rate -after dosing with 5 different doses Primary interest: regression on log(dose) 6 out of 13 subjects recieving the lowest dose level are non- responders wrt GIR Treshold C=3.5

14. Slide no 14 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 Cumulated glucose infusion rate Linear mixed model: with intercept α, slope β, random subject effect, U i ~N(0,ω 2 ) and residual ε ij ~N(0,σ 2 dose ) with variace depending on dose level

15. Slide no 15 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 Estimation with PRC NLMIXED in SAS proc nlmixed data=PDdata; parms intercept=9 slope=1 vlow=13 vnlow=0.1 s1randsubj=-1.9 s1s2=-2 s2randsubj=- 1.4; if (treatment=1) then randsubj = rand1; else randsubj = rand2; m = intercept + slope*logdose + randsubj; if (low_dose=0) then ll = -(lauc-m)**2/(2*vnlow) - 0.5*log(2*3.14159*vnlow); if (low_dose=1) then do; if cens=0 then ll = -(lauc-m)**2/(2*vlow) - 0.5*log(2*3.14159*vlow); if cens=1 then ll = log(probnorm((3.5-m)/sqrt(vlow))); end; model lauc ~ general(ll); random rand1 rand2 ~ normal([0,0],[exp(s1randsubj), exp(s1s2), exp(s2randsubj)]) subject=subj_id; run;

16. Slide no 16 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 Estimates -from analysis of log(AUC(0-10h)GIR) AUCGIR InterceptSlopeCV: Between subjects CV: Higher doses CV: Low dose (REML) Imputed values 8.92 [8.62; 9.22] 1.15 [1.01; 1.28] 38%41%372% (ML) Imputed values 8.92 [8.63; 9.22] 1.15 [1.02; 1.28] 37%40%372% (ML) Censored values 8.94 [8.63; 9.25] 1.16 [1.03; 1.30] 37%40%167%

17. Slide no 17 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 Example: PK data

18. Slide no 18 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 Biphasic insulin concentration over time – three subcutaneous injections 70 out of 873 serum insulin concentrations were reported as < LLoQ at 13pmol/l

19. Slide no 19 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 PK Example: Compartment model

20. Slide no 20 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 Nonlinear PK models Two-level random effects model Level 1: between-subject variations on all parameters, diagonal variance structure Level 2: For K pf, Vary Fixed effects, estimates (log-scale) Between subject variance Between injection (within subject) variance Variance (residuals) K pf (min -1 ) 0.0087 (-4.7496) 0.5966 2 0.3948 2 25.1830 2 K fs (min -1 ) 0.0056 (-5.1916) 2.1094 2 K xp (min -1 ) 0.0190 (-3.9610) 0.5726 2 V i (L Kg -1 )0.9584 (-0.0425) 0.4522 2 Clausen W.H.O., De Gaetano A. & Vølund A. (2005) Pharmacokinetics of Biphasic Insulin Aspart Administered by Multiple Subcutaneous Injections: Importance of Within-subject Variation. Research report 09/05

21. Slide no 21 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 Does this approximate approach leads to better estimates?

22. Slide no 22 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 Simulation study: Theophylline data

23. Slide no 23 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 Simulation study: First-order open-compartment model Central compartment V=Cl/K e KeKe KaKa D: Dose K a :Absorption rate K e : Elimination rate Cl: Clearance

24. Slide no 24 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 Simulation study – cont’ 1000 simulations 12 subjects 10 concentrations at t = 0, 0.25, 0.5, 1, 2, 3.5, 7, 9, 12, 24h Dose = 4.5mg lK a = 0.5, lCl = -3, lK e = -2.5 lK a and lCl are allowed to vary randomly, b i ~ N(0, ψ), where ψ is diagonal, 0.36 and 0.04 respectively 36% of the simulated data <LLoQ (3mg/l)

25. Slide no 25 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 Mean estimates – Laplacian Approx ___________________________________________________________ lK a lCllK e ψ lK a ψ lCl σ 2 ___________________________________________________________ True value0.500-3.000-2.5000.3600.0400.490 Full data0.498-3.016-2.5050.2800.0360.480 LLoQ=3mg/l Suggested method 0.498-3.015-2.5040.2790.0360.475 Omit data 0.661-3.154-2.6800.2540.0290.442 ___________________________________________________________ Clausen, W.H.O., Tabanera, R., Dalgaard, P. (2005) Solvng the bias problem for censored pharmacokinetic data. Research report 05/05 University of Copenhagen.

26. Slide no 26 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 Mean estimates – AGQ (5 abscissae) ___________________________________________________________ lK a lCllK e ψ lK a ψ lCl σ 2 ___________________________________________________________ True value0.500-3.000-2.5000.3600.0400.490 Full data0.495-3.011-2.4980.2860.0370.476 LLoQ=3mg/l Suggested method 0.491-3.008-2.4920.2870.0370.471 Omit data 0.635-3.142-2.6610.2660.0300.432 ___________________________________________________________ Clausen, W.H.O., Tabanera, R., Dalgaard, P. (2005) Solvng the bias problem for censored pharmacokinetic data. Research report 05/05 University of Copenhagen.

27. Slide no 27 Wan Hui Ong Clausen and Birgitte B. Rønn 26/4-2006 Conclusion Models with closed-form representation The method could be applied using PROC NLMIXED available in SAS Models without closed-form representation a differential equation solver is necessary With censored data, the same approach can be applied – need some programming work The results from simulation study shows that bias introduced by left censoring is almost fully removed.