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