Influence of missing data on analysis and power calculation in bioequivalence studies

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1 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th,2011 1/33 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel Ulm University in cooperation with Boehringer Ingelheim Pharma GmbH & Co. KG May 4 th, 2011

2 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th,2011 2/33 Outline 1 Motivation 2 Model 3 Simulation 4 SAS Procedures and evaluation 5 Conclusion

3 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th,2011 3/33 Outline 1 Motivation 2 Model 3 Simulation 4 SAS Procedures and evaluation 5 Conclusion

4 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th,2011 4/33 Bioequivalence studies (BE studies) Test if test- and reference formulations differ marginally with respect to pharmacokinetic (PK) characteristics Endpoints: PK-parameters Primary: AUC and C max Problem concerning analysis and power calculation: At how many missing values should further subjects be recruited?

5 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th,2011 5/33 Typical design of BE studies: 2x2-Crossover Test- (T) and reference (R) formulation are given in two periods N subjetcs are allocated randomly evenly to one of the following sequences: RT and TR Treatment according to sequence and period

6 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th,2011 6/33 Outline 1 Motivation 2 Model 3 Simulation 4 SAS Procedures and evaluation 5 Conclusion

7 Model (on log-scale) Y ijk = S ik + P j + F jk + C k + e ijk Y ijk S ik P j F jk C k e ijk logarithm of response (PK parameter) measured on subject i in sequence k in period j, i = 1,...,N, j=1, 2, k = 1, 2 random i-th subject effect in sequence k, N(0,τ 2 )i.i.d. fixed effect in period j fixed treatment effect in the k-th sequence in period j fixed effect in the k-th sequence residual (random) error associated with the i-th subject in sequence k in period j, N(0,σ 2 )i.i.d. Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th,2011 7/33

8 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th,2011 8/33 Why random subject effect? EMA: All effects should be considered fixed Subjects with one missing value excluded from analysis If applicable, further subjects have to be recruited! FDA: All avaible data should be included in the analysis Properties of REML can be used (e.g. in SAS with PROC MIXED) When should further subjects be recruited?

9 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th,2011 9/33 Model assumptions Number of subjects: N = 24 Y ijk = S ik + P j + F jk + C k + e ijk S ik : subject S ik N(0,τ 2 )withτ 2 = 0.25 P j : period P 1 =0, P 2 = 0.2 F jk : treatment F 11 = F 22 =0, F 12 = F 21 := F 2 = 0.1 C k : sequence C 1 = C 2 = 0 e ijk : residual error e ijk N(0,σ 2 )withσ 2 = 0.06 Modeling of response per sequence and period Sequence Period 1 Period 2 RT Y i11 = S i1 + e i11 Y i21 = S i e i21 TR Y i12 = S i e i12 Y i22 = S i e i22

10 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th, /33 Outline 1 Motivation 2 Model 3 Simulation 4 SAS Procedures and evaluation 5 Conclusion

11 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th, /33 Randomly missing values With N = 24 subjects: Number of missing values: 1, 2, 3, 4, 8, 12, 16, 20 Three alternatives to distribute missing values: - for both treatments and in both periods (case I) - only in period 2 (case II) - only for treatment R (case III)

12 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th, /33 Pseudo code DO 1 TO 1000 END 1. Simulate complete dataset for 24 subjects 2. Analyse complete dataset with PROC MIXED 3. FOR i = 1, 2, 3, 4, 8, 12, 16, 20 DO END PROC SURVEYSELECT to simulate incomplete datasets with i missing values for case I - III 4. Analyse incomplete dataset with PROC MIXED

13 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th, /33 One missing value for both treatments (case I)

14 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th, /33 Two missing values for both treatments (case I)

15 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th, /33 Three or more missing values for both treatments (case I)

16 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th, /33 Missing values in period 2 (case II) or for treatment R (case III)

17 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th, /33 PROC SURVEYSELECT

18 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th, /33 Outline 1 Motivation 2 Model 3 Simulation 4 SAS Procedures and evaluation 5 Conclusion

19 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th, /33 Is there a difference between the formulations? Test if: the 90% confidence interval, which covers the treatment quotient µ T µ R, lies within the equivalence domain [0.80, 1.25] the 90% confidence interval, which covers the treatment difference µ T µ R, lies within the equivalence domain [ 0.223, 0.223] (log-scale)

20 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th, /33 PROC MIXED Theory: Mixed model Y = X β + Zγ + fix random

21 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th, /33 Value of interest: gcv Among others considered: gcv := 100 exp(mse) 1 derived from CV of a log-normally distributed random variable X CV := Var(X ) E(X )

22 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th, /33 Evaluation with PROC BOXPLOT σ 2 =0.06 gcv = 25, missing values for treatment R (case III)

23 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th, /33 Summary statistics with PROC MEANS σ 2 =0.06 gcv = 25, missing values for treatment R (case III) Missing values Number of simulations Min P25 Mean P75 Max

24 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th, /33 Summary statistics graphic σ 2 =0.06 gcv = 25, missing values for treatment R (case III)

25 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th, /33 Power Goal: The probability to reject the hypothesis that the formulations are different when there is no relevent difference should lie e.g. between 80% and90% This probability is called Power gcv is a decisive factor here

26 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th, /33 PROC POWER

27 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th, /33 Power graphic: Mean of simulations σ 2 =0.06 gcv = 25, missing values for treatment R (case III)

28 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th, /33 Power graphic: P75 of simulations σ 2 =0.06 gcv = 25, missing values for treatment R (case III)

29 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th, /33 Outline 1 Motivation 2 Model 3 Simulation 4 SAS Procedures and evaluation 5 Conclusion

30 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th, /33 When should further subjects be recruited? For N = 24 subjects and 1000 simulations each (per number of missing values and case): In general, no further subjects have to be recruited Based on the observations in the 75% percentile: - at 10 to 12 missing values further subjects should be recruited - especially if the values are missing mainly for both treatments and in both periods

31 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th, /33 Outlook: Further questions cohorts and other covariates Variability of the rest variance σ 2

32 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th, /33 References [1] SAS/Stat 9.2 User s Guide: The Mixed Procedure (Book Excerpt). SAS Publishing, 2008 [2] S.C. Chow, J.P. Liu. Design and Analysis of Bioavailability and Bioequivalence Studies. Marcel Dekker, Inc., New York, 1992 [3] S. Patterson, B. Jones. Bioequivalence and Statistics in Clinical Pharmacology. Chapman & Hall / CRC, Boca Raton, 2006 [4] G.A. Milliken, D.E. Johnson. Analysis of Messy Data Volume I: Designed Experiments. Wadsworth, Inc., Belmont, 1984

33 Influence of missing data on analysis and power calculation in bioequivalence studies Henrike Häbel May 4 th, /33 Thank you for your attention! Special thanks to Julia Habeck and Michaela Mattheus from Boehringer Ingelheim Pharma GmbH & Co. KG

Bioavailability / Bioequivalence

Bioavailability / Bioequivalence Selection of CROs Selection of a Reference Product Metrics (AUC, C max /t max, Shape of Profile) Acceptance Ranges (0.80 1.25 and beyond) Sample Size Planning (Literature References, Pilot Studies) Steps