BIOEQUIVALENCE and Statistics in Clinical Pharmacology Villanova University, Summer III, 2010 Dr. Scott Patterson ([email protected]) http://www.homepage.villanova.edu/scott.patterson Weeks 2-4

1

Objectives for Week 2 BE data from a 2 × 2 cross-over. Modelling BE data from a 2 × 2 cross-over. Testing for Average Bioequivalence. Sensitivity Assessments. The TOST.

2

2 × 2 Cross-over Designs and Average Bioequivalence with Examples 1.

The two-period two-treatment design

2.

Example 1

3.

Plotting and describing the data

4.

Formal analysis via a linear model

5.

Testing for ABE: Confidence interval approach

6.

Example 2

7.

Outlier Identification

8.

Sensitivity Analysis

9.

Two-one-sided-test (TOST) procedure

3

Two-Period Cross-over Design Sequence

Period 1

2

1

R

T

2

T

R

Assume: n1 subjects on sequence 1 (RT) n2 subjects on sequence 2 (TR) and a washout of at least 5 half-lives between periods 1 and 2

4

Example 1: Data Sequence RT AUC

Cmax

Period

Period

Subject

1

2

1

2

1

2849

2230

499

436

4

2790

2864

733

416

.

.

.

.

.

.

.

.

.

.

34

1737

1655

425

319

36

2040

2199

464

384

Sequence TR Subject

1

2

1

2

2

2025

2000

438

361

3

2090

1826

535

558

.

.

.

.

.

.

.

.

.

.

30

2519

1941

537

400

35

1560

1629

463

372

5

Bivariate Normal Data We will ignore sequence and period effects for now. Let x be the response for ln-AUC for a subject following the Test formulation, and let y be the response be the response for ln-AUC for a subject following the Reference formulation. Then  we assume:    2 2 x µT σB + σW   ∼ BV N  , 2 y µR σB

6

2 σB 2 2 σB + σW

 

Bivariate Normal Data(cont’d) BVN is the bivariate normal density. µT , µR are the means for the test and reference formulations, respectively. 2 2 and σB , σW are the between-subject (inter) variation and within-subject (intra) variation, respectively. These are assumed to be homogeneous across formulations (for now).

7

Expected value and variance of individual differences  Then, recall that

h

1

−1

i 

x y

 =x−y

2 ∼ N (µT − µR , 2σW )

.

8

SAS code for calculating individual differences Note that subjects are assumed to be independent and identically distributed BV N in this fashion. Reduces data to the level of a paired t-test (when we ignore sequence and period effects). We’ll get to sequence and period effects later. See example3.1.sas for the data set. Note that Tmax is NOT lognormal, and we’ll look at methods for that data later.

9

Example 3.1 Using data set twobytwo, apply the following code: data indt;set twobytwo;if formula=’T’; lnauct=log(auc);lncmaxt=log(cmax); keep subject lnauct lncmaxt;run; proc sort;by subject;run; data indr;set twobytwo;if formula=’R’; lnaucr=log(auc);lncmaxr=log(cmax); keep subject lnaucr lncmaxr;run; proc sort;by subject;run; data indiv;merge indt indr;by subject; d_lnauc=lnauct-lnaucr;d_lncmax=lncmaxt-lncmaxr; run;

10

Table of individual differences for Example 3.1 Table 1: Individual differences for Example 3.1 Subject

ln-AUC T-R

ln-Cmax T-R

1

-0.24497

-0.13496

2

0.01242

0.19334

... T=Test R=Reference

11

Summary Statistics Apply the following code: proc univariate data=indiv; var d_lnauc d_lncmax; run; 2 Estimates for µ ˆT − µ ˆR , 2ˆ σW are -0.02, 0.029 for ln −AUC and -0.03, 0.077 for ln −Cmax.

12

Plotting of individual differences Before we formally analyse BE data we should plot the data in various ways. This will alert us to any unusual features and help us get a good idea of what the formal analysis will report.

13

NPP of ln −AUC individual differences

● ●

0.2

● ● ●

●

0.0

lnauc

●●●

●

●

●

●

● ● ● ● ●●●

●

● ● ● ● ●

-0.2

● ●

● ●

●

-0.4

●

-0.4

-0.2

0.0

Expected Normal Order Statistics

14

0.2

NPP of ln −Cmax individual differences

0.6

0.4 ● ●

lncmax

0.2

● ●

0.0

● ● ● ● ●

●

●

●

●

●

● ● ● ● ● ● ● ●

●

●

-0.2 ● ● ●

-0.4

●

●

●

●

●

-0.6

-0.6

-0.4

-0.2

0.0

Expected Normal Order Statistics

15

0.2

0.4

0.6

Histogram of ln −AUC individual differences

2.0

Probability

1.5

1.0

0.5

0.0 -0.6

-0.4

-0.2

0.0 d.lnauc

16

0.2

0.4

Histogram of ln −Cmax individual differences

1.5

Probability

1.0

0.5

0.0 -0.6

-0.4

-0.2

0.0 d.lncmax

17

0.2

0.4

Further plotting the data In the following we will see some additional useful plots. These somewhat begin to account for sequence and period effects. We begin with • Subject-profiles plot. • Paired-agreement plot. Both are useful for displaying the variation in results between subjects and any differences between the two results on the same patient.

18

Subject-profiles plot The subject-profiles plot is constructed for each sequence by 1. plotting on the vertical axis, for each subject, the Period 1 and Period 2 responses against the values 1 and 2, respectively, on the horizontal axis, and then 2. joining the two responses for each subject with a line.

19

Example 1: Subject profiles plot.

8 7

logCmax

5

6

7 6 5

logAUC

8

9

logCmax

9

logAUC

R T Sequence 1

T R Sequence 2

R T Sequence 1

20

T R Sequence 2

Paired-agreement plot Here the Test response is plotted against the Reference response for each subject. Figure 2 shows, for simulated sets of data, the patterns that might be see in such a plot. These patterns correspond to: 1. no difference between the two responses on a subject (Identity), 2. a period difference in the absence of a formulation difference (Period difference), 3. a formulation difference in the absence of a period difference (Formulation difference) 4. when there is both a period difference and a formulation difference. To emphasis the underlying pattern in each plot, we have removed the within-subject variability. 21

Examples of patterns in a paired-agreement plot.

15 5

10

Test 5

10

Test

15

20

Period difference

20

Identity

5

10

15

20

5

Reference

10

15

20

Reference

15 5

10

Test 5

10

Test

15

20

Formulation and Period differences

20

Formulation difference

5

10

15

20

5

Reference

10

15

Reference

22

20

Example 1: Paired-agreement plots.

8 5

6

7

Test

7 6 5

Test

8

9

logCmax

9

logAUC

5

6

7

8

9

5

Reference

6

7 Reference

23

8

9

Review Each subject’s data in a BE trial is BVN (on ln −scale). This enables us to use the large variety of tools for statistical analysis of such data. We understand how to derive estimates for the difference in means and the relevant variance (intra-subject) ignoring sequence/period effects and how such data ‘looks’ when period or sequence effects are present. We now turn to how to derive estimates for the difference in means when sequence and period effects are presumed to be present.

24

Formal Analysis of the Data In order to produce a more formal analysis we need to specify a model for the observed data. This linear model needs to account for those features that: • vary systematically between the two responses on a subject (the fixed effects) • vary randomly between the two responses on a subject and between responses on different subjects (the random effects).

25

Linear Model for Data • Let yijk denote the (log-transformed) response obtained from subject k in period j in sequence group i; i = 1, 2; j = 1, 2; k = 1, 2, . . . , ni . • Model will contain: – fixed effects for sequences, periods and formulations – random effects for subjects and within-subject variability.

26

Linear Model for Data - cont’d Table 2:

Fixed effects

Group Period 1 1(RT) γ1 + µR + π1 2(TR) γ2 + µT + π1

Period 2 γ1 + µT + π2 γ2 + µR + π2

Note: we assume that there are no carry-over effects of formulation from Period 1 to Period 2

27

Linear Model for Data - cont’d Table 3:

Random effects

Group Period 1 1(RT) ξk(1) + ε11k 2(TR) ξk0 (2) + ε21k0

Period 2 ξk(1) + ε12k ξk0 (2) + ε22k0

Random effect for subject k in Group 1 and subject k 0 in Group 2.

28

Linear Model for Data - cont’d Putting this all together we have: yijk = γi + µd[i,j] + πj + ξk(i) + εijk , where • γi , πj , and µd[i,j] are fixed effects for sequence, period, and formulation, • d[1, 1] = R, d[1, 2] = T , d[2, 1] = T and d[2, 2] = R, • ξk(i) and εijk are random effects for subjects and within-subjects, respectively, 2 • Var(ξk(i) )=σB , the between-subject variance, 2 , the within-subject variance. • Var(εijk )=σW

29

Estimation of µT − µR Let y¯ij.

nij X 1 = yijk nij k=1

denote the mean response of the subjects in period j in sequence group i. Table 4:

Group-by-Period Means

Group 1(RT) 2(TR)

Period 1 y¯11. y¯21.

30

Period 2 y¯12. y¯22.

Estimation of µT − µR , cont’d For Group 1: E(¯ y11. − y¯12. ) = π1 − π2 + µR − µT . For Group 2: E(¯ y21. − y¯22. ) = π1 − π2 + µT − µR . Hence  E

 1 [(¯ y21. − y¯22. ) − (¯ y11. − y¯12. )] = µT − µR . 2

31

Estimation of µT − µR , cont’d That is, 1 µ ˆT − µ ˆR = (¯ y21. − y¯22. − y¯11. + y¯12. ) 2 and Var(ˆ µT − µ ˆR ) =

1 4



2 σW

n1

+

2 σW

n1

+

2 σW

n2

+

2 σW

n2

 =

2 σW

2

If n1 = n2 = n/2, then Var(ˆ µT − µ ˆR ) =

2 σW

2

32



2 2 2 2σW + = . n n n







1 1 + . n1 n2

Estimation of µT − µR , cont’d Table 5: Example 1: Group-by-Period Means logAUC Group

Period 1

Period 2

Mean

1(RT)

y¯11. = 7.60(17)

y¯12. = 7.50(17)

y¯1.. = 7.55

2(TR)

y¯21. = 7.55(15)

y¯22. = 7.48(15)

y¯2.. = 7.51

Mean

y¯.1. = 7.58

y¯.2. = 7.49

y¯... = 7.53

logCmax 1(RT)

y¯11. = 5.99(17)

y¯12. = 5.91(17)

y¯1.. = 5.95

2(TR)

y¯21. = 6.02(15)

y¯22. = 5.99(15)

y¯2.. = 6.01

Mean

y¯.1. = 6.01

y¯.2. = 5.95

y¯... = 5.98

(sample size in brackets) 33

Groups-by-Periods Plot The difference between the means of T and R over the two periods in each group can be displayed in the groups-by-periods plot. Here, the four mean y¯11. , y¯12. , y¯21. and y¯22. are plotted against their respective period labels and the two means from the same group are joined up.

34

Example 1. Groups-by-Periods plot logCmax

6.2 6.0

Mean logCmax

5.8

7.6 7.4

5.6

7.2 7.0

Mean logAUC

7.8

6.4

8.0

logAUC

R T

R T Period 1

Period 2

Period 1

35

Period 2

PROC GLM Apply the following code: proc glm data=twobytwo alpha=0.1; class sequence subject period formula; model ln_auc=sequence subject(sequence) period formula; lsmeans formula/pdiff cl; estimate ’T-R’ formula -1 1; output out=aucglmout predicted = pred rstudent=stdres residual=res; run;

36

Selected Outputs Intra-subject variation: Source

DF

Mean Square

F Value

Pr > F

Model

33

0.10829876

9.84

F

sequence subject(sequence) period formula

1 30 1 1

0.02055193 0.11413275 0.12157143 0.00436775

0.1818