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
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
I.
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,
ln(0.8) = -0.2231
ABE upper limit,
ln(1.25) = 0.2231
Assessing PBE

Difference in the distribution between drugs
(assuming Normal distribution)
▬ Statistical parameters:
2
T   R ,  TT2   TR
Difference between
total variance of T
and R
Assessing PBE (cont.)
▬ Criterion:
(T   R )  (   )
p
2
max( T 0 ,  )
2
2
TT
2
TR
2
TR
Parameter to control for total
variance (0.04 typically)
PBE limit,
a constant
Assessing PBE (cont.)
▬ The linear scale of the criterion
2
2
 p  (T   R ) 2  ( TT2   TR
)   p max( T20 , TR
)
▬ 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)
 D2  var(Tj  Rj )  ( BT   BR )2  2(1  ) BT BR
2
2
2
 BT
  TT
  WT
2
2
2
 BR
  TR
  WR
Between individual
variation
Within individual
variance
Assessing IBE (cont.)
IBE limit, preset
constant
▬ Criterion
▬ Linear scale of the criterion
Parameter to control for
within-subj. variance
2
2
2
 I  (T   R ) 2   D2  ( WT
  WR
)   I max( W2 0 , WR
)
▬ 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
2
2
2
ˆI  ( T   R ) 2  ˆ D2  (ˆWT
 ˆWR
)   I ˆWR
1 2
2
2
2
2
 ˆ 2  ˆ I2  (ˆWT
 ˆWR
)  (ˆWT
 ˆWR
)   I ˆWR
2
2
2
 ˆ 2  ˆ I2  0.5ˆWT
 (1.5   I ).ˆWR
I 
(ln1.25) 2   I
 W2 0
FDA calculation of CI for IBE criteria scale (cont.)

Assuming chi-square distribution for each var.
term
2
2
 WT
 WR
 I2 2
2
2
2
2
ˆ ~
 ( N  2) , ˆWR ~
 ( N  2) , ˆ I ~
 ( N  2) ,
N 2
N 2
N 2
1
ˆ ~ N ( ,
 I2 )
4(n1  n2 )
2
WT
H D  ( ˆ  t1 , N  s (
H R1 
2
1
0.5( N  2)ˆWT
( N  2) I2
，
,
ˆ I2 )1/ 2 ) 2 , HT 
H

I
4(n1  n2 )
2, N 2
2, N 2
2
 (1.5   I )(N  2)ˆWR
2, N 2
FDA calculation of CI for IBE criteria scale (cont.)
Let
2
ED  ˆ 2 , EI  ˆ I2 , ET  0.5ˆ WT
2
,
ER1  (1.5   I )ˆWR
U q  ( H q  Eq )2 , q  D, I , T , RI
95%CI upper limit:
H  (ED  EI  ET  ER1 )  (U D  U I  UT  U R1 )1/ 2
Alternative method?
Data structure of cross-over designs
2  2 for a sequence/block
BLK
Period
1 2
Sequence
1
T
R
2
R
T
P1
T
P2
R
T
R
Data structure of cross-over design (cont.)
2  4 for a sequence/block
Period
Sequence
1
1
T
2 3
R T
4
R
2
R
T T
R
Data structure of cross-over design (cont.)
Jth
individual
p1
R
p2
T
R
p3
R
T
p4
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,
 x = 0 for drug R, 1 for drug T
eij ~ N (0,  e2 )
2

2
u
0
 Between individual variance
u0 j ~ N (0,  u 0 )
 Within individual variance  2
e
u0 j   j  0
 Intercept: mean for drug R
 Slope: mean diff. between T & R
 u0j residuals at individual level
Mean diff. of jth
individual from
 eij residuals at time level
population
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 )
Estimate and test the slope estimate ˆ1
 Calculate 90% CI of the estimate
 Compare with ABE limit [-0.2231, 0.2231]
 In addition, adjusting for covariates if necessary.

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(u0 j  u1 j x1ij )   u20   u21 x12ij  2 u01 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:
 TT2  ( u20   e20 )  ( u21   e21 )  2( u01   e01 )
Total variance of drug R:
2
 TR
  u20   e20
Assessing PBE (cont.)

The linear scale of the FDA criterion
2
2
 p  (T   R ) 2  ( TT2   TR
)   p max( T20 , TR
)

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:
2
2
2
 I  (Tj   Rj ) 2   D2  ( WT
  WR
)   I max( W2 0 , WR
)
 D2  variance of (  Tj   Rj )
 ( 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
Between
individuals
(Level 2)
Within
individual
(Level 1)
Total
Drug R
Drug T
Diff. (T-R)
 u20 ( BR2 )
 u20   u21  2 u01 ( BT2 )
 u21  2 u01
 e20 (WR2 )
 e20   e21  2 e01 (WT2 )
 e21  2 e01
 TR2
 TT2
 u21   e21  2( u01   e01 )
Two-level model for IBE (cont.)

Diff. of within-individual var.
2
2
( WT
  WR
) estimated by  e21  2 e01

Interactive term
2
2

 D estimated by u1
Assessing IBE

Linear scale of the FDA criterion
2
2
2
 I  (T   R ) 2   D2  ( WT
  WR
)   I max( W2 0 , WR
)

Calculate 95%CI of the scale, to satisfy
 IUpper  0
An example of anti-hypertension drug trial*
Sequence
Period
1
2
3
1(RTTR)
6.928195
7.186318
6.802861
7.06784
N=16
7.080717
7.273086
7.31402
7.300655
:
:
:
:
:
:
:
:
6.857083
7.401054
7.638559
7.303796
6.65214
6.420956
6.686185
6.650939
:
:
:
:
:
:
:
:
2(TRRT)
N=16
4
* 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
Fixed effects
0
1
Model 2
Est. (SE)
Model 3
(Est. (SE)
7.6615(0.1064)
7.8705(0.3328)
-0.0400(0.0614) -0.0400(0.0614)
Period
Sequence
0.0448(0.0210)
-0.1841(0.2092)
Random effects
Level 2
 u20
 u 01
 u21
Level 1
 e20
 e01
 e21
0.3708(0.0964)
0.3726(0.0961)
-0.0104(0.0398) -0.0072(0.0405)
0.0509(0.0351)
0.0543(0.0349)
0.0734(0.0173)
0.0671(0.0158)
0.0116(0.0143)
0.0145(0.0138)
0.0000(0.0000)
0.0000(0.0000)
Variance components between FDA & MLM
Variance
component
 TT2
 TR2
 WT2
 WR2
 BT2
 BR2
 D2
FDA est.
2-level model est.
Without covariates
With covariates
0.5102
0.4975
0.5088
0.4407
0.4442
0.4397
0.09997
0.0966
0.0961
0.0691
0.0734
0.0671
0.4102
0.4009
0.4127
0.3716
0.3708
0.3726
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
scale: upper limit
-0.048
???
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
scale: upper limit
0.0750
???
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 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
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!
p 
(ln 1.25) 2   p

2
T0
I 
(ln1.25) 2   I
 W2 0