BIOEQUIVALENCE and Statistics in Clinical Pharmacology Villanova University, Summer III, 2010 Dr. Scott Patterson ([email protected]) http://www.homepage.villanova.edu/scott.patterson Week 5

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Contents Practical Planning for BE Trials Population and Individual BE Scaled Average BE Biosimilarity and Lot Consistency

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Practical Planning for BE Trials Restricted Maximum Likelihood Modelling Failing BE Carry-over Optional Designs Determining Sample Size Outliers

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Restricted Maximum Likelihood Why consider this? Drop-outs - can bias estimates of µT − µR and variances (or both). Very seldom do all subjects complete all sessions. Even when they do, something can happen at the lab... Need to plan for missing data.

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Restricted Maximum Likelihood Assume a normal distribution for the natural-log-transformed data of AUC and Cmax, appropriate to the study design. Express this as a mathematical equation (a likelihood). Use the sampled data to generate initial (starting) estimates for the parameters (µT , µR , etc.) of interest using summary statistics. If there are no missing data, and design is period and sequence balanced, then these estimates maximise the value of the likelihood and are unbiased minimum variance estimates for the parameters of interest, and no further action need be taken. These values maximise the likelihood indicating that no other estimates will improve the information available.

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Restricted Maximum Likelihood If data are missing at random, to find better estimates (best linear unbiased predictors), we will iterate in a restricted parameter space. First, account for the estimates of the fixed (designed) effects (i.e., sequence, period, formulation) in the data. By removing all these factors from design (the restriction), the remainder must presumably be due to inter- and intra-subject variances Estimate these variances, choosing parameter estimates for these which maximise the restricted value of the likelihood. Now, assuming these are the true values for the variances, estimate the design effects, choosing parameter estimates for these which maximise the value of the (again restricted) likelihood.

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Restricted Maximum Likelihood Continue iterations until fixed effect estimates and variance estimates estimates are stable (i.e., converge) at values maximising the likelihood. Obviously this can not be done by hand. The computations are complex and require differential calculus to perform. SAS code to perform these analyses on log-transformed data are as follows.

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REML in Standard Designs Standard Design: Cross-over BE designs where no formulation adminstration is replicated. We partition inter-subject variance from intra-subject variance using code like: proc mixed data=standard method=reml ITDETAILS maxiter=200; class sequence subject period formula; model lnauct=sequence period formula /ddfm=KENWARDROGER; random subject(sequence); estimate ’T-R’ formula -1 0 1 /cl alpha=0.10; estimate ’T-S’ formula 0 -1 1 /cl alpha=0.10;run; 8

Variance-Covariance in Replicate Designs Here we have two test measurements and two reference measurements, so we can do more with the variances. 2 2 2 We can now separate σB into σBT , σBR , and their correlation ρ. 2 2 2 Additionally, we can separate σW into σW T and σW R .

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Var-Cov in Replicate Designs (cont’d) Again ignoring period    µT x1     µ  x   T  2   ∼ MV N    µR  y1     µR y2

and sequence effects:     ,  

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Var-Cov in Replicate Designs (cont’d) 

2 σBT

ρσBT σBR ρσBT σBR   ρσ σ 2 σBT ρσBT σBR BT BR   2  ρσBT σBR ρσBT σBR σ BR  ρσBT σBR ρσBT σBR ρσBT σBR   2 σW 0 0 0 T    0 2 σW T 0 0      2  0 0 σW R 0    2 0 0 0 σW R

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ρσBT σBR



ρσBT σBR

   +  

ρσBT σBR 2 σBR

CSH REML in Replicate Designs How to do that in SAS (and account for period and sequence effects): proc mixed data=replicate method=reml ITDETAILS maxiter=200; class sequence subject period formula; model lnauct=sequence period formula /ddfm=KENWARDROGER; random formula/type=CSH subject=subject; repeated/group=formula subject=subject; estimate ’T-R’ formula -1 1/CL ALPHA=0.1;run;

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Selected Output - Variances Consider exam4.4.sas. Covariance Parameter Estimates Cov Parm

Subject

Var(1) Var(2) CSH Residual Residual

subject subject subject subject subject

Group

formula R formula T

Estimate 0.2471 0.2058 1.0000 0.1202 0.07563

Note that this is constrained REML as −1 ≤ ρˆ ≤ 1.

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Table 1: Example 4: TOST Procedure Results Endpoint

µ ˆT − µ ˆR

90% Confidence Interval

logAUC

0.1004

(0.0294, 0.1715)

logCmax

0.4120

(0.2827, 0.5414)

Endpoint

exp(ˆ µT − µ ˆR )

90% Confidence Interval

AUC

1.11

(1.03, 1.19)

Cmax

1.51

(1.33, 1.72)

Evidence for lack of ABE on Cmax.

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Subject-by-formulation interaction FDA specifies that such models be applied for several reasons. One reason is to be able to estimate a quantity known as p 2 2 σD = σBT + σBR − 2ρσBT σBR . Ourqestimate for AUC is ln −AUC is p 0.2471 + 0.2058 − 2 0.2471(0.2058) = 0.04. σD > 0.15 are held to be indicative of potential product failure on an individual basis as we will see later.

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Subject-by-formulation interaction The variance of µ ˆT − µ ˆR in such designs is standard error should be (for ln −AUC) q

0.0998 54

= 0.0429.

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2 σD

+

2 σD +

2 2 σW T +σW R , 2

2 2 σW T +σW R 2

n

.

so our

REML in Replicate Designs Different parameterizations of between-subject variances are readily possible. proc mixed data=replicate method=reml ITDETAILS maxiter=200; class sequence subject period formula; model lnauct=sequence period formula /ddfm=KENWARDROGER; random formula/type=UN subject=subject; repeated/group=formula subject=subject; estimate ’T-R’ formula -1 1/CL ALPHA=0.1; run;

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REML in Replicate Designs proc mixed data=replicate method=reml ITDETAILS maxiter=200; class sequence subject period formula; model lnauct=sequence period formula /ddfm=KENWARDROGER; random formula/type=FA0(2) subject=subject; repeated/group=formula subject=subject; estimate ’T-R’ formula -1 1/CL ALPHA=0.1; run;

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Kenward Roger SE and Degrees of Freedom This method for deriving the degrees of freedom accounts for the iterative nature of the REML procedure. For more see Kenward, M., Roger, J. (1997) Small sample inference for fixed effects from restricted maximum likelihood. Biometrics, 33, 983–997.

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Failing BE For some drug products, even if one tries time and time again to demonstrate bioequivalence, it may be that it just can not be done. A common misconception is that this means that the test and reference formulations are bioinequivalent. This is not necessarily the case.

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Potentially Bioinequivalent Products: Fitted Normal Densities

0

1

2

Density

3

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logAUC logCmax ABE Limits

-0.2

0.0

0.2

0.4

Estimated Mean T - R (log-scale)

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auc − ∆

and H02 :

µi − µj ≥∆

versus the alternative H12 :

µi − µj