applying multilevel models in e valuation of bioequivalence in drug trials
Post on 26-Jan-2016
35 Views
Preview:
DESCRIPTION
TRANSCRIPT
Applying Multilevel Models in EApplying Multilevel Models in Evaluation of valuation of Bioequivalence in Drug Trials Bioequivalence in Drug Trials
Min YangProf of Medical Statistics
Nottingham Clinical Trials UnitSchool of Community Health Sciences
University of Nottingham(20/05/2010)
(min.yang@nottingham.ac.uk)
ContentsContents
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 trialsBioequivalence 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 Standard testing design (FDA guidance)(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 22 design
▬ Population bioequivalence (PBE) by 24 design
▬ Individual bioequivalence (IBE) by 24 design
Standard assessment criterionStandard 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 Assessing ABE
Tolerable mean difference between drugs T and R
▬ statistical parameters:
▬ Confidence interval:
▬ Criterion:
RT
AUUpperLowerAL DD ][ ,
ABE upper limit, ln(1.25) = 0.2231
ABE lower limit, ln(0.8) = -0.2231
Diff. in mean
][)(%90 , UpperLowerRT DDCI
Assessing PBEAssessing PBE
Difference in the distribution between drugs (assuming Normal distribution)
▬ Statistical parameters:
22, TRTTRT
Difference between total variance of T and R
Assessing PBE (cont.)Assessing PBE (cont.)
▬ Criterion:
p
TRT
TRTTRT
),max(
)()(22
0
222
Parameter to control for total variance (0.04 typically)
PBE limit, a constant
Assessing PBE (cont.)Assessing PBE (cont.)
▬ The linear scale of the criterion
▬ 95% CI of the scale
▬ To satisfy
),max()()( 220
222TRTpTRTTRTp
],[)(%95 pUpperpLowerpCI
0pUpper
Assessing IBEAssessing IBEIndividual difference (similar effects of
same individual on both drugs)
BRBTBRBTRjTjD )1(2)()var( 22
222
222
WRTRBR
WTTTBT
Within individual variance
Corr. (T, R)
Between individual variation
Assessing IBE (cont.)Assessing IBE (cont.)
▬ Criterion
▬ Linear scale of the criterion
▬ Calculate 95%CI of the scale and to satisfy
),max()()( 220
2222WRWIWRWTDRTI
IBE limit, preset constant
0IUpper
Parameter to control for within-subj. variance
Limitations of FDA methodsLimitations 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 scaleFDA calculation of CI for IBE criteria scale
2222
2222222
22222
ˆ).5.1(ˆ5.0ˆˆ
ˆ)ˆˆ()ˆˆ(2
1ˆˆ
ˆ)ˆˆ(ˆ)(ˆ
WRIWTI
WRIWRWTWRWTI
WRIWRWTDRTI
20
2)25.1(ln
W
II
FDA calculation of CI for IBE criteria scale (cont.)FDA calculation of CI for IBE criteria scale (cont.)Assuming chi-square distribution for each var.
term
)2(2
~ˆ 22
2
NN
WTWT
, )2(
2~ˆ 2
22
N
NWR
WR , )2(2
~ˆ 22
2
NN
II ,
))(4
1,(~ˆ 2
21Inn
N
22/12
21,1 ))ˆ
)(4
1(ˆ( IsND nn
tH ,
22,
2ˆ)2(5.0
N
WTT
NH
,
22,
2)2(
N
II
NH
,
22,
2
1
ˆ)2)(5.1(
N
WRIR
NH
FDA calculation of CI for IBE criteria scale (cont.)FDA calculation of CI for IBE criteria scale (cont.)
Let
95%CI upper limit:
2/111 )()( RTIDRTID UUUUEEEEH
2̂DE , 2ˆ IIE , 2ˆ5.0 WTTE 21 ˆ)5.1( WRIRE ,
2)( qqq EHU , RITIDq ,,,
Alternative method?
Data structure of cross-over designsData 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.)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.)Data structure of cross-over design (cont.)
Jth individual
p1 p2 p3 p4
R T R R T T T R
Sources of variationSources of variation
Between sequences/individualsWithin sequence/individual
Between periods (repeated measures over time)Between treatment groups (treatment effect)
Common methodological issuesCommon 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 Basic 2-level model for repeated measures
),0(~
),0(~200
2
0110
uj
eij
ijjijij
Nu
Ne
euxy
20u
Model 1 ith time point for jth individual,x = 0 for drug R, 1 for drug TBetween individual varianceWithin individual variance Intercept: mean for drug RSlope: mean diff. between T & Ru0j residuals at individual level
eij residuals at time level
2
e
Mean diff. of jth individual from population
00 jju
Lay interpretation of multilevel modellingLay 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)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.
1̂
),0(~
),0(~200
2
0110
uj
eij
ijjijij
Nu
Ne
euxy
Two-level model for PBE (Two-level model for PBE (Model 2Model 2))
)()( 110110110 ijijijijjjijij xeexuuxy
Between individuals (level 2) variance:
Within individual (level 1) variance:
ijuijuuijjj xxxuu 10121
21
20110 2)var(
ijeijeeijijij xxxee 10121
21
20110 2)var(
Two-level model for PBE (cont.)Two-level model for PBE (cont.)
Total variance of drug T:
Total variance of drug R:
2 2 2 2 20 0 1 1 01 01( ) ( ) 2( )TT u e u e u e
2 2 20 0TR u e
Assessing PBE (cont.)Assessing PBE (cont.)
The linear scale of the FDA criterion
95% CI of the scale
To satisfy
),max()()( 220
222TRTpTRTTRTp
],[)(%95 pUpperpLowerpCI
0pUpper
Two-level model for IBETwo-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.
),max()()( 220
2222WRWIWRWTDRjTjI
BRBTBRBT
RjTjD ofiance
)1(2)(
)(var2
2
Variance components in Model 2Variance components in Model 2
Drug R Drug T Diff. (T-R) Between
individuals (Level 2)
20u 2( )BR 01
21
20 2 uuu
2( )BT 0121 2 uu
Within individual (Level 1)
20e 2( )WR 01
21
20 2 eee
2( )WT 0121 2 ee
Total 2TR
2TT )(2 0101
21
21 eueu
Two-level model for IBE (cont.)Two-level model for IBE (cont.)
Diff. of within-individual var.
estimated by
Interactive term
estimated by
)( 22WRWT 01
21 2 ee
2D
21u
Assessing IBE Assessing IBE
Linear scale of the FDA criterion
Calculate 95%CI of the scale, to satisfy
),max()()( 220
2222WRWIWRWTDRTI
0IUpper
An example of anti-hypertension drug trialAn 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
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]
ABE between FDA method and MLM ABE between FDA method and MLM (Model 1)(Model 1)
Model estimatesModel estimates
Model 2 Est. (SE)
Model 3 (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 20u 0.3708(0.0964) 0.3726(0.0961)
01u -0.0104(0.0398) -0.0072(0.0405)
21u 0.0509(0.0351) 0.0543(0.0349)
Level 1 20e 0.0734(0.0173) 0.0671(0.0158)
01e 0.0116(0.0143) 0.0145(0.0138)
21e 0.0000(0.0000) 0.0000(0.0000)
Variance components between FDA & MLMVariance components between FDA & MLM
2-level model est. Variance component
FDA est. Without covariates With covariates
2TT 0.5102 0.4975 0.5088
2TR 0.4407 0.4442 0.4397
2WT 0.09997 0.0966 0.0961
2WR 0.0691 0.0734 0.0671
2BT 0.4102 0.4009 0.4127
2BR 0.3716 0.3708 0.3726
2D 0.0507 0.0509 0.0543
PBE parameters between FDA & MLMPBE 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 0pUpper
IBE parameters between PDA & MLMIBE parameters between PDA & MLM
FDA MLMMean 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 0IUpper
Merits of MLMMerits 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 MLMFurther 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!
2
0
2)25.1(ln
T
pp
2
0
2)25.1(ln
W
II
top related