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Applying Multilevel Models in Evaluation of Bioequivalence in Drug Trials Min Yang Prof of Medical Statistics Nottingham Clinical Trials Unit School of Community Health Sciences University of Nottingham (20/05/2010) (min.yang@nottingham.ac.uk)

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Contents I. A review of FDA methods for ABE, PBE and IBE II. A brief introduction to multilevel-level models (MLM) III. MLM for ABE IV. MLM for PBE V. MLM for IBE VI. Comparison between FDA and MLM methods on an example of 2x4 cross-over design VII. Further research areas VIII. Questions

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Bioequivalence evaluation in drug trials Statistical procedure to assess inter-exchangeability between a brand drug and a copy of it Major outcome measures: Blood concentration of an active ingredient in the area under curve: (AUC) Maximum concentration of the ingredient in blood: (C max ) Time to reach the maximum concentration in blood: (T max ) Logarithm transformation of these outcomes is usually performed

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Standard testing design (FDA guidance) A generic copy of a drug for test (T) versus the established drug as reference (R) Cross-over experimental design (two drugs on same subject with washout periods) Assessing three types of bioequivalence Average bioequivalence (ABE) by 2 2 design Population bioequivalence (PBE) by 2 4 design Individual bioequivalence (IBE) by 2 4 design

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Standard assessment criterion Comprising of three parts: 1. A set of statistical parameters for specific assessment 2. Confidence interval (CI) of those parameters 3. Predetermined clinical tolerant limit

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Assessing ABE Tolerable mean difference between drugs T and R statistical parameters: Confidence interval: Criterion: ABE upper limit, ln(1.25) = 0.2231 ABE lower limit, ln(0.8) = -0.2231 Diff. in mean

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Assessing PBE Difference in the distribution between drugs (assuming Normal distribution) Statistical parameters: Difference between total variance of T and R

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Assessing PBE (cont.) Criterion: Parameter to control for total variance (0.04 typically) PBE limit, a constant

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Assessing PBE (cont.) The linear scale of the criterion 95% CI of the scale To satisfy

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Assessing IBE Individual difference (similar effects of same individual on both drugs) Within individual variance Corr. (T, R) Between individual variation

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Assessing IBE (cont.) Criterion Linear scale of the criterion Calculate 95%CI of the scale and to satisfy IBE limit, preset constant Parameter to control for within-subj. variance

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Limitations of FDA methods Estimators of Moment method (less efficient, not necessarily sufficient) Complex design? Joint bioequivalence of AUC, Cmax and Tmax? Covariates effects?

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FDA calculation of CI for IBE criteria scale

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FDA calculation of CI for IBE criteria scale (cont.) Assuming chi-square distribution for each var. term

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FDA calculation of CI for IBE criteria scale (cont.) Let 95%CI upper limit:

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Alternative method?

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Data structure of cross-over designs 2 2 for a sequence/block Period 12 Sequence1TR 2RT BLKP1TRP2TR

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Data structure of cross-over design (cont.) 2 4 for a sequence/block Period 1234 Sequence1TRTR 2RTTR

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Data structure of cross-over design (cont.) Jth individual p1 RT p2 RR p3 TT p4 TR

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Sources of variation Between sequences/individuals Within sequence/individual Between periods (repeated measures over time) Between treatment groups (treatment effect)

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Common methodological issues Cluster effect within individual (random effects analysis for repeated measures) Missing data over time (losing data) Imbalanced groups due to patient dropout or missing measures (analysis of covariate)

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Basic 2-level model for repeated measures Model 1 i th time point for j th individual, x = 0 for drug R, 1 for drug T Between individual variance Within individual variance Intercept: mean for drug R Slope: mean diff. between T & R u 0j residuals at individual level e ij residuals at time level Mean diff. of jth individual from population

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Lay interpretation of multilevel modelling Y=βX + τU = fixed effects + variance components An analytic approach that combines regression analysis and ANOVA (type II for random effects) in one model. It takes advantage of regression model for modelling covariate effects. It takes advantage of ANOVA for random effects and decomposing total variance into components: For a 2-level model, two variance components as between and within individual variances (SS t = SS b + SS w ), Intra-Class Correlation (ICC) = SS b /SS t

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How MLM works for BE evaluation?

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Assessing ABE under multilevel models (MLM) Estimate and test the slope estimate Calculate 90% CI of the estimate Compare with ABE limit [-0.2231, 0.2231] In addition, adjusting for covariates if necessary.

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Two-level model for PBE (Model 2) Between individuals (level 2) variance: Within individual (level 1) variance:

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Two-level model for PBE (cont.) Total variance of drug T: Total variance of drug R:

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Assessing PBE (cont.) The linear scale of the FDA criterion 95% CI of the scale To satisfy

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Two-level model for IBE Linear scale of FDA criteria for IBE: The difference of within-individual variance and the interaction of individual and drug effects: random effects of drug effect between individuals.

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Variance components in Model 2

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Two-level model for IBE (cont.) Diff. of within-individual var. estimated by Interactive term estimated by

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Assessing IBE Linear scale of the FDA criterion Calculate 95%CI of the scale, to satisfy

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An example of anti-hypertension drug trial *

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FDAMLM Mean difference-0.040 SE (mean diff.)0.0614 90%CI[-0.1407, 0.0607] Tolerance limit[-0.2231, 0.2231] ABE between FDA method and MLM (Model 1)

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Model estimates

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Variance components between FDA & MLM

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PBE parameters between FDA & MLM FDAMLM Mean diff.-0.040 Variance diff.0.06950.0691 Criteria scale-0.698-0.704 95%CI of Criteria scale: upper limit -0.048??? Bootstrap, MCMC?? Tolerance limit

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IBE parameters between PDA & MLM FDAMLM Mean diff.-0.040 Variance diff.0.03090.0290 Interaction0.05070.0509 Criteria scale-0.0892-0.0859 95%CI of Criteria scale: upper limit 0.0750??? Bootstrap, MCMC?? Tolerance limit

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Merits of MLM Straightforward estimation of the criterion scale for ABE, PBE or IBE Expandable to cover complex cross-over designs Capacity of adjusting covariates Capacity in assessing multiple outcomes jointly (multilevel multivariate models) Missing data (MAR) was not an issue due to borrowing force in model estimation procedure

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Further research areas in MLM Comparison of statistical properties of parameter estimates between FDA Moment approach and MLM (simulation study) Calculating CI of criteria scale point estimate for PBE and IBE (MCMC or Bootstrapping) assessing single outcome Calculating CI of criteria scale point estimates for multiple outcomes

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Thank you!

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