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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)
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
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: (Cmax)
▬ Time to reach the maximum concentration in blood: (Tmax)

Logarithm transformation of these outcomes is usually performed
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
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
Assessing ABE

   Tolerable mean difference between drugs T and R
▬ statistical parameters:
T   R                 Diff. in mean

▬ Confidence interval:
90%CI (T   R )  [ DLower, DUpper ]

▬ Criterion:                AL  [DLower, DUpper ]   AU

ABE lower limit,                     ABE upper limit,
ln(0.8) = -0.2231                    ln(1.25) = 0.2231
Assessing PBE
   Difference in the distribution between drugs
(assuming Normal distribution)

▬ Statistical parameters:

T   R ,  TT   TR
2      2

Difference between
total variance of T
and R
Assessing PBE (cont.)
▬ Criterion:
PBE limit,
(  T   R )  (   )
2          2         2                a constant
TT
p    TR

max( T 0 ,  )
2        2
TR

Parameter to control for total
variance (0.04 typically)
Assessing PBE (cont.)

▬ The linear scale of the criterion
 p  ( T   R ) 2  ( TT   TR )   p max( T 0 ,  TR )
2      2                2       2

▬ 95% CI of the scale
95%CI ( p )  [ pLower , pUpper]

▬ To satisfy
 pUpper  0
Assessing IBE
   Individual difference (similar effects of
same individual on both drugs)
Corr. (T, R)

 D  var(Tj   Rj )  ( BT   BR ) 2  2(1   ) BT BR
2

 BT   TT   WT
2      2      2
Between individual

 BR   TR   WR
2      2      2                        variation

Within individual
variance
Assessing IBE (cont.)
IBE limit, preset
constant
▬ Criterion
Parameter to control for
within-subj. variance
▬ Linear scale of the criterion
 I  (T   R ) 2   D  ( WT   WR )   I max( W 0 , WR )
2      2      2                2      2

▬ Calculate 95%CI of the scale and to satisfy
 IUpper  0
Limitations of FDA methods
 Estimators of Moment method (less efficient,
not necessarily sufficient)
 Complex design?
 Joint bioequivalence of AUC, Cmax and Tmax?
 Covariates effects?
FDA calculation of CI for IBE criteria scale

 I  ( T   R ) 2   D  ( WT   WR )   I  WR
ˆ                      ˆ2     ˆ2     ˆ2           ˆ2
1 2
 ˆ 2   I2  ( WT   WR )  ( WT   WR )   I  WR
ˆ        ˆ       ˆ2        ˆ2     ˆ2         ˆ2
2
 ˆ 2   I2  0.5 WT  (1.5   I ). WR
ˆ         ˆ2                  ˆ2

(ln 1.25) 2   I
I 
W0
2
FDA calculation of CI for IBE criteria scale (cont.)
   Assuming chi-square distribution for each var.
term
 WT 2
2
 WR 2
2
 I2 2
ˆ   2
~       ( N  2) ,  WR ~
ˆ 2
 ( N  2) ,  I ~
ˆ 2
 ( N  2) ,
N 2                     N 2                    N 2
WT

1
ˆ ~ N ( ,                  I2 )
4(n1  n2 )

1                              0.5( N  2) WT ˆ2        ( N  2) I2
H D  ( ˆ  t1 , N  s (                I2 )1/ 2 ) 2 , H T 
ˆ                                        ， HI                ,
4(n1  n2 )                              , N  2
2
 , N  2
2

 (1.5   I )( N  2) WR
ˆ2
H R1 
 , N  2
2
FDA calculation of CI for IBE criteria scale (cont.)

Let
ED  ˆ 2 , E I   I2 , ET  0.5 WT
ˆ              ˆ2       E R1  (1.5   I ) WR ,
ˆ2
U q  ( H q  Eq ) 2 , q  D, I , T , RI

95%CI upper limit:

H  ( E D  E I  ET  E R1 )  (U D  U I  U T  U R1 )1 / 2
Alternative method?
Data structure of cross-over designs

2  2 for a sequence/block

Period
1 2
Sequence     1   T   R

2   R   T
Data structure of cross-over design (cont.)

2  4 for a sequence/block

Period
1   2 3      4
Sequence    1   T   R T      R
2   R   T T      R
Data structure of cross-over design (cont.)

Jth
individual

p1           p2                p3           p4

R        T   R        R       T         T   T        R
Sources of variation

 Between sequences/individuals
 Within sequence/individual
Between periods (repeated measures over time)
Between treatment groups (treatment effect)
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)
Basic 2-level model for repeated measures

Model 1
yij   0  1 x1ij  u0 j  eij  ith time point for jth individual,
eij ~ N (0,  e2 )                x = 0 for drug R, 1 for drug T
 Between individual variance       u20
u0 j ~ N (0,  u 0 )
2

 Within individual variance  2
u0 j   j  0   Intercept: mean for drug R
e

 Slope: mean diff. between T & R

Mean diff. of jth              u0j residuals at individual level
individual from
population                     eij residuals at time level
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 (SSt = SSb + SSw), Intra-Class Correlation (ICC) =
SSb/SSt
How MLM works for BE evaluation?
Assessing ABE under multilevel models (MLM)
yij   0  1 x1ij  u0 j  eij
eij ~ N (0,  e2 )
u0 j ~ N (0,  u20 )
                                       ˆ
Estimate and test the slope estimate  1
 Calculate 90% CI of the estimate
 Compare with ABE limit [-0.2231, 0.2231]
Two-level model for PBE (Model 2)
yij   0  1 x1ij  (u0 j  u1 j x1ij )  (e0ij  e1ij x1ij )

    Between individuals (level 2) variance:
var(u 0 j  u1 j x1ij )   u20   u21 x12ij  2 u 01 x1ij

    Within individual (level 1) variance:
var(e0ij  e1ij x1ij )   e20   e21 x12ij  2 e01 x1ij
Two-level model for PBE (cont.)
Total variance of drug T:
 TT  ( u20   e20 )  ( u21   e21 )  2( u01   e 01 )
2

Total variance of drug R:
 TR   u 0   e 0
2      2       2
Assessing PBE (cont.)
   The linear scale of the FDA criterion
 p  ( T   R ) 2  ( TT   TR )   p max( T 0 ,  TR )
2      2                2       2

   95% CI of the scale
95%CI ( p )  [ pLower , pUpper]

   To satisfy
 pUpper  0
Two-level model for IBE

   Linear scale of FDA criteria for IBE:
 I  ( Tj   Rj ) 2   D  ( WT   WR )   I max( W 0 ,  WR )
2      2      2                2       2

 D  var iance of (  Tj   Rj )
2

 ( BT   BR ) 2  2(1   ) BT  BR

The difference of within-individual variance and the
interaction of individual and drug effects: random
effects of drug effect between individuals.
Variance components in Model 2
Drug R                   Drug T                         Diff. (T-R)
Between
individuals    u20 ( BR )
2
 u20   u21  2 u01 ( BT )
2
 u21  2 u 01
(Level 2)
Within
individual    e20 ( WR )
2
 e20   e21  2 e01 ( WT )
2
 e21  2 e01
(Level 1)
Total          TR
2
 TT
2
 u21   e21  2( u01   e01)
Two-level model for IBE (cont.)
   Diff. of within-individual var.
( WT   WR ) estimated by  e21  2 e01
2      2

   Interactive term
 D estimated by
2              u1
2
Assessing IBE

   Linear scale of the FDA criterion
 I  (T   R ) 2   D  ( WT   WR )   I max( W 0 , WR )
2      2      2                2      2

   Calculate 95%CI of the scale, to satisfy
 IUpper  0
An example of anti-hypertension drug trial*
Period
Sequence
1              2              3                4
1(RTTR)          6.928195        7.186318       6.802861           7.06784
N=16             7.080717        7.273086         7.31402         7.300655
:               :              :               :
:               :              :               :
2(TRRT)          6.857083        7.401054       7.638559          7.303796
N=16               6.65214       6.420956       6.686185          6.650939
:               :              :               :
:               :              :               :
* Chen (2004). Chinese Clinical Pharmacology and Treatment, 9(8): 949-953
ABE between FDA method and MLM
(Model 1)
FDA                MLM

Mean difference        -0.040              -0.040

SE (mean diff.)        0.0614              0.0614

90%CI         [-0.1407, 0.0607]   [-0.1407, 0.0607]

Tolerance limit   [-0.2231, 0.2231]   [-0.2231, 0.2231]
Model estimates
Model 2          Model 3
Est. (SE)       (Est. (SE)
Fixed effects   0         7.6615(0.1064)   7.8705(0.3328)

1         -0.0400(0.0614) -0.0400(0.0614)
Period                      0.0448(0.0210)
Sequence                    -0.1841(0.2092)
Random effects
Level 2        u20      0.3708(0.0964)   0.3726(0.0961)

 u 01     -0.0104(0.0398) -0.0072(0.0405)

 u21      0.0509(0.0351)   0.0543(0.0349)

Level 1        e20      0.0734(0.0173)   0.0671(0.0158)

 e01      0.0116(0.0143)   0.0145(0.0138)

 e21      0.0000(0.0000)   0.0000(0.0000)
Variance components between FDA & MLM
Variance                          2-level model est.
FDA est.
component              Without covariates       With covariates
 TT
2
0.5102          0.4975                  0.5088

 TR
2
0.4407          0.4442                  0.4397

 WT
2
0.09997         0.0966                  0.0961

 WR
2
0.0691          0.0734                  0.0671

 BT
2
0.4102          0.4009                  0.4127

 BR
2
0.3716          0.3708                  0.3726

D 2
0.0507          0.0509                  0.0543
PBE parameters between FDA & MLM

FDA               MLM
Mean diff.         -0.040              -0.040

Variance diff.      0.0695             0.0691

Criteria scale      -0.698              -0.704

95%CI of Criteria     -0.048                ???
scale: upper limit                  Bootstrap, MCMC??

Tolerance limit      pUpper  0
IBE parameters between PDA & MLM
FDA                MLM
Mean diff.         -0.040              -0.040

Variance diff.      0.0309             0.0290
Interaction        0.0507             0.0509

Criteria scale     -0.0892             -0.0859

95%CI of Criteria     0.0750                ???
scale: upper limit                  Bootstrap, MCMC??
Tolerance limit      IUpper  0
Merits of MLM
   Straightforward estimation of the criterion scale for ABE,
PBE or IBE
   Expandable to cover complex cross-over designs
   Capacity in assessing multiple outcomes jointly (multilevel
multivariate models)
   Missing data (MAR) was not an issue due to ‘borrowing
force’ in model estimation procedure
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
Thank you!
(ln 1.25) 2   p          (ln 1.25) 2   I
p                        I 
   2
T0                     W0
2

```
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