1. 臨床試驗 Analysis of 2x2 Crossover Design
授課老師: 劉仁沛教授
國立台灣大學 與 國家衛生研究院
臨床試驗 Analysis of 2x2 Crossover Design
【本著作除另有註明外，採取創用CC「姓名標示－非商業性－相同方式分享」台灣3.0版授權釋出】

2. Inference for 2x2 Crossover Design
References:
Chow, SC, and Liu, JP (2014) Design and Analysis of Clinical Trials, 3rd Ed. Wiley
Chow, SC and liu, JP (2008) Design and Analysis of Bioavailability and Bioequivalence, 3rd Ed., Chapman & Hall/ CRC, Chapter 3
Chow, SC and Liu, JP (1998) Design and Analysis of Animal Studies in Pharmaceutical Development, Marcel Dekker.
PhRMA (2007) Drug Discovery and Development: understanding the R&D process, The Pharmaceutical Research and Manufactures of America

3. Inferences for the 2x2 Design
3.1 Introduction Data Structure Sequence Period I Period II 1 (RT) Reference Test Data: Yi11 Data:Yi21 2 (TR) Test Reference Data: Yi12 Data:Yi22

4. Inferences for the 2x2 Design
The general model for the standard 2x2 crossover design Yijk =  + Sik + Pj + F(j,k) + C(j-1,k) + eijk where i(subject) = 1,…,nk, j(period) = 1,2, k(sequence) = 1,2.

5. Inferences for the 2x2 Design
FR, if k=j F(j,k) = { k=1,2; j=1,2 FT, if kj CR, if k=1, j=2 C(j-1,k) = { CT, if k=2, j=2

6. Inferences for the 2x2 Design
Fixed effects
Sequence Period I Period II
1 (RT) 11=  +P1+FR 21 =  +P2+FT+CR
2 (TR) 12=  +P1+FT 22=  +P2+FR+CT
where P1 + P2 = 0, FR + FT = 0, and CR + CT =0.

7. Inferences for the 2x2 Design
Assumptions Sik iid ~ N(0, S2) eijk iid ~ N(0, e2) Sik and eijk are mutually independent

8. Inferences for the 2x2 Design
3.2 The Carryover Effects
Subject totals: Uik = Yi1k + Yi2k
2 + CR, if sequence 1
E(Uik) = {
2 + CT, if sequence 2
V(Uik) = 2(e2 + 2s2) = u2
{U11…,Un11} and {U12…,Un22} are independent
samples with the same variance u2

9. Inferences for the 2x2 Design
Define C = CT - CR Ho: C = 0 vs. Ha: C  0  U.k = (1/nk)Uik, k=1,2. The MVUE OF C is given as       Ĉ =U.2 – U.1 = (Y.12+ Y.22) - (Y.11+ Y.21)

10. Inferences for the 2x2 Design
V(Ĉ) = u2 [(1/n1) + (1/n2)]  V(Ĉ) = su2 [(1/n1) + (1/n2)], where  su2 =  (Uik - U.k)2 /(n1+n2 –2) Tc = c/v(c)

11. Inferences for the 2x2 Design
Reject Ho if Tc > t(/2, n1+n2 –2) Confidence interval c  t(/2, n1+n2 –2)v(c) The carryover effect is confounded with the sequence effect

12. Inferences for the 2x2 Design
2.3 The Direct Drug Effect
Period differences
dik = (Yi2k – Yi1k)/2
[(P2 - P1) + (FT - FR) + CR]/2, if sequence = 1
E(dik) = {
[(P2 - P1) + (FR - FT) + CT]/2, if sequence = 2
V (dik) = d2 = e2/2

13. Inferences for the 2x2 Design
{d11…,dn11} and {d12…,dn22} are independent
samples with the same variance d2
Sample means of period differences 
d.k = (1/nk)dik, k=1,2
Define F = FT - FR
 
E(d.1 - d.2) = F – C/2.

14. Inferences for the 2x2 Design
Unless C = 0, no unbiased estimator for F based on the data from both period exists
     
f = d.1 – d.2 = (Y.21 - Y.11) - (Y.22 - Y.12)
   
= (Y.21+ Y.12) - (Y.11+ Y.12)
 
= Y.T - Y.R
f is a linear combination of the sequence-by-period means and the least squares estimator of F

15. Inferences for the 2x2 Design
V(f) = d2 [(1/n1) + (1/n2)] v(f) = sd2 [(1/n1) + (1/n2)], where  sd2 =  (dik - d.k)2 /(n1+n2 –2) Ho: F = 0 vs. Ha: F  0 Test Statistic Td = f/v(f)

16. Inferences for the 2x2 Design
Under the assumption of C=0
Reject Ho if
TF > t(/2, n1+n2 –2)
Confidence interval
f  t(/2, n1+n2 –2)v(f)

17. Inferences for the 2x2 Design
When C  0, f is not unbiased for F.
However, unbiased estimator can be obtained as the difference of sample means of the first period between the two sequences:
 
f C = Y Y.12
V(f C) = (e2 + s2) [(1/n1) + (1/n2)]
V(f C) – V(f) = (e2/2 + s2) [(1/n1) + (1/n2)]

18. Inferences for the 2x2 Design
3.4 The Period Effect Crossover difference dik , if sequence = 1 Oik = { -dik , if sequence = 2

19. Inferences for the 2x2 Design
[(P2 - P1) + (FT - FR) + CR]/2, if sequence = 1
E(Oik) = {
[(P1 – P2) + (FT - FR) - CT]/2, if sequence = 2
V (Oik) = d2 = e2/2
{O11…,On11} and {O12…,On22} are independent samples with the same variance d2.
The inference for the period effect can be performed as that for the carryover effects and direct effect. (homework)

20. Inferences for the 2x2 Design
3.5 The Analysis of Variance 
SStotal =  (Yijk - Y…)2
  
=  (Yijk - Yi.k + Yi.k - Y... )2
  
=  (Yijk - Yi.k)2 + (Yi.k - Y... )2
= SSwithin + SSbetween

21. Inferences for the 2x2 Design
SSbetween = SScarry + SSinter
   
SScarry = [2n1n2/(n1+n2)][(Y.12+ Y.22) - (Y.11+ Y.21)]2
df = 1
SSinter = Y2i.k/2 - Y2..k/2nk, df = n1+n2-2
E(MScarry) = [2n1n2/(n1+n2)](CT - CR)2 + e2 + 2s2
E(MSinter) = e2 + 2s2

22. Inferences for the 2x2 Design
SSwithin = SSdrug + SSperiod + SSintra
   
SSdrug = [2n1n2/(n1+n2)]{1/2[(Y.21 - Y.11) - (Y.22+ Y.12)]}2
df =1
SSperiod = [2n1n2/(n1+n2)]{1/2[(Y.21 - Y.11) - (Y.12+ Y.22)]}2
df = 1
SSintra = Y2ijk - Y2i.k/2 - Y2.jk /nk - Y2..k/2nk,
df = n1+n2-2

23. Inferences for the 2x2 Design
E(MSdrug)=[2n1n2/(n1+n2)][(FT - FR)+(CT - CR)/2]2+e2 E(MSperiod) = [2n1n2/(n1+n2)](P2 – P1)2 + e2 E(MSintra) = e2

24. Inferences for the 2x2 Design
Carryover effect: Fc = MScarry/MSinter
Direct Drug Effect: Fd = MSdrug/MSintra
Period Effect : FP = MSperiod/MSintra
Intersubject variability
Fv = MSinter/MSintra

25. Inferences for the 2x2 Design
3.6 Example (Chow and Liu, 2008) Sequence Sequence Period I Period II Mean 1 (RT) Y.11= Y.21= (TR) Y.12= Y.22= Period mean

26. Inferences for the 2x2 Design
Summary of Inference for the Fixed Effects Estimated Effect MVUE Variance 95% C.I. T p-value Carryover (-42.10, 22.91) Direct Drug (-10.03, 5.46) Period ( -9.47, 6.01)

27. Inferences for the 2x2 Design
The Analysis of Variance Table SOV df SS MS F p-value Intersubject Carryover Residual Intrasubject Direct Drug Period Residual Total

28. Inferences for the 2x2 Design

29. Inferences for the 2x2 Design

30. 臨床試驗 Evaluation of Equivalence and Non-inferiority
授課老師: 劉仁沛教授
國立台灣大學 與 國家衛生研究院
【本著作除另有註明外，採取創用CC「姓名標示－非商業性－相同方式分享」台灣3.0版授權釋出】

31. Design and Analysis of Bioequivalence Studies
References: Chow, S.C. and Liu, J.P. (2008)Design of Bioavailability and Bioequivalence Studies, 3rd Ed., CRC/Chapman & Hall, Chapter 4 and 5 FDA (2003) Guidance on Bioavailability and Bioequivalence Studies for Orally Administered Drug Products – General Considerations FDA(2001) Guidance on Statistical Approaches to Establishing Bioequivalence

32. Methods for Average Bioavailability
4.1 Introduction 4.2 Interval Hypothesis Testing 4.3 The Confidence Interval Approach 4.4 Log-transformation 4.4 Discussion

33. Methods for Average Bioavailability
4.1 Introduction
The general model for the 2x2 crossover design without carryover effect:
Yijk =  + Sik + Pj + F(j,k) + eijk
where i(subject) = 1,…,nk, j(period), k(sequence) = 1,2.
Define T =  + FT and R =  + FR
     
Y.R = (Y.11+ Y.22) and Y.T = (Y.21+ Y.12)
 
E(Y.T) = T and E(Y.R) = R

34. Methods for Average Bioavailability
Issues on Assessment of ABE
Selection of ABE measures
difference T - R vs ratio T/R
Raw Data vs Log-transformed Data
Determination of ABE limits
20/20 in difference or 80/125 in ratio
4. Hypotheses
Differences vs equivalence

35. Methods for Average Bioavailability
Raw Data:
L < T - R < U
20/20 rule: L= -0.2R and U = 0.2R, R is unknown
L < T/R < U  (L – 1)R < T - R < (U – 1)R,
R is unknown.
Log-transformed data:
L < T/R < U  ln(L)< ln(T) – ln(R) < log(U)
Limits are not functions of unknown R

36. Methods for Average Bioavailability
4.2 Interval Hypotheses Testing
Difference:
Ho: T - R  L or T - R  U
vs. Ha: L < T - R < U
Ratio:
Ho: T/R  L or T/R  U
vs. Ha: L < T/R < U

37. Methods for Average Bioavailability
Two one-sided hypotheses
HoL: T - R  L vs. HaL: T - R > L
and
HoU: T - R  U vs. HaU:T - R < U
The parameter space of Ho is the union of the parameter spaces of HoLand HoU.
The parameter space of Ha is the intersection of the parameter spaces of HaLand HaU.

38. Methods for Average Bioavailability
4.2.2 Schuirmann’s Two One-sided Tests (TOST) Procedure Conclude ABE if TL = (f - L)/v(f) > t(, n1+n2 –2) and TU = (f - U)/v(f) < -t(, n1+n2 –2). TOST procedure control the type I error rate at its nominal  level. Nonparametric TOST is also available.

39. Methods for Average Bioavailability
Example 4.2.1:
 
Y.R = , and Y.T =
t(0.05, 22) = and sd = 9.145
U = -L = 0.2(82.559) = 16.51
TL = [( ) ]/9.145sqrt(1/6) = 3.810
TU = [( ) ]/9.145sqrt(1/6) =
Issue: U = -L is actually an estimate.

40. Methods for Average Bioavailability
4.3 Confidence Interval Approach
Data: Raw and untransformed data
If a (1-2)100% confidence interval for the difference T - R or the ratio T/R is within the acceptance limits as recommended by the regulatory agency, then accept the test formulation; otherwise reject it.
The confidence interval approach is operationally equivalent to the TOST procedure based on the difference T - R
 = 5%  90% C.I.
T - R: 20%
T/R: (80%, 120%).

41. Methods for Average Bioavailability
4.3.1 The Classical Confidence Interval:
 
L1= (Y.T - Y.R) - t(, n1+n2 –2)sd2 [(1/n1) + (1/n2)],
 
U1= (Y.T - Y.R) + t(, n1+n2 –2)sd2 [(1/n1) + (1/n2)].
and
 
L2= (L1/Y.R + 1)x100%, and U2= (U1/Y.R + 1)x100%
Conclude ABE if
(1) (L1, U1)  (L, U), L,= -0.2R and U = 0.2R or
(2) (L2, U2)  (80%, 120%)

42. Methods for Average Bioavailability
Example 4.3.1
 
Y.R = , and Y.T =
t(0.05, 22) = and sd = 9.145
U= -L = 0.2(82.559) = 16.51
(L1, U1) = (-8.698, 4.123) (-16.51, 16.51)
(L1, U1) = (89.46%, %) (80%, 120%)
Issue: U = -L is actually an estimate.

43. Methods for Average Bioavailability
4.3.2 CI Based on Fieller’s Theorem
1. For ratio T/R based on raw data
2. Take variability of estimator of R into account
 = T/R = ( + FT)/( + FR)
Define
(Yi21 - Yi11) , if sequence 1
U*ik = {
(Yi12 - Yi22) , if sequence 2

44. Methods for Average Bioavailability
{U*11,…, U*n11} are iid normal with mean (P2 - P1) and variance 2,
{U*12,…, U*n22} are iid normal with mean -(P2 - P1) and variance 2,
where 2 = (T2+S2) - 2S2+ 2(R2+S2).
 
T = (U*.1 + U*.2)/sqrt(wS2U)  t(n1+n2 –2).
   
U*.1 + U*.2 = Y.T - Y.R, w = [(1/n1) + (1/n2)]/4
S2U = S2TT - 2S2TR + 2S2RR.

45. Methods for Average Bioavailability
 
S2RR = [1/(n1+n2-2)][(Yi11- Y.11)2 + (Yi22- Y.22)2]
 
S2TT = [1/(n1+n2-2)][(Yi21- Y.21)2 + (Yi12- Y.12)2]
 
SSTR = [1/(n1+n2-2)][(Yi11- Y.11)(Yi21 - Y.21) +
 
(Yi12- Y.12)(Yi22 - Y.22)]

46. Methods for Average Bioavailability
 
T = (Y.T - Y.R.)/sqrt[w(S2TT - 2STR + 2S2RR)]
A (1-2)100% CI for  is the set that
{| T2 < t2(, n1+n2 –2)}.
To solve the following quadratic equation
(Y.T - Y.R.)2 - t2(,n1+n2 –2)[w(S2TT - 2STR + 2S2RR)]
Conditions for the limits are positive real numbers: 
Y.R/sqrt(wS2RR) > t(,n1+n2 –2) and
Y. T/sqrt(wS2TT) > t(,n1+n2 –2).

47. Methods for Average Bioavailability
Example 4.3.2: 
Y.R/sqrt(wS2RR) = > t(0.05, 22) = 1.717
Y. T/sqrt(wS2TT) = > t(0.05, 22) = 1.717
90% CI for  is given as (89.78%, %)

48. 4.4 Log-transformation The distributions of PK measures such as AUC and Cmax are skewed. FDA 2003 guidance suggest that statistical evaluation for BE should be performed based on the log-transformation of AUC and Cmax.

49. Log-transformation
The multiplicative (log-transformed) model Xijk = ’S’ikP’jF’(j,k)C’(j-1,k)e’ijk or Yijk = ln(Xijk) = +Sik+Pj+F(j,k)+C(j-1,k)+eijk Xijk = exp( + Sik + Pj + F(j,k) + C(j-1,k) + eijk)

50. Log-transformation (Xi1k, Xi2k) follows a bivariate lognormal distn.
Sequence-by-period means and medians in log-transformed model
Seq Period I Period II
1 Mean exp[(+P1+FR)+(R2+S2)/2] exp[(+P2+FT)+(T2+S2)/2]
Median exp(+P1+FR) exp(+P2+FT)
2 Mean exp[(+P1+FT)+(T2+S2)/2] exp[(+P2+FR)+(R2+S2)/2]
Median exp(+P1+FT) exp(+P2+FR)

51. Log-transformation
BE measures 1 Ratio of Medians  = ’T/’R = (’ + F’T)/(’ + F’R) = exp[( + FT)/( + FR)] = exp[(FT - FR)] = exp(F) 2. Ratio of Means M = exp[F +(T2-R2)/2].

52. Log-transformation
Ho: ’T/’R  L or ’T/’R  U vs. Ha: L < ’T/’R < U Ho: ln(’T) – ln(’R) ln(L) or ln(’T)-ln(’R) ln(U) vs. Ha: ln(L) < ln(’T) - ln(’R) < ln(U) Ho: T - R  L or T - R  L vs. Ha: L < T - R < U L and U are known nonrandom constants, i.e., ± for 80/125.

53. Log-transformation
The Confidence interval approach and Schuirmann’s TOST can be directly applied to the log-transformed data for assessment of ABE.
The period difference on the log-scale is the period ratio on the original scale
dik = (Yi2k – Yi1k)/2 = ln (Xi2k/Xi1k)/2 = ln(rik)/2.

54. Log-transformation         f = d.1 – d.2 = Y.T - Y.R
   
f = d.1 – d.2 = Y.T - Y.R
The MLE under the normality assumption for the log-transformed data is given as
   
f’ = exp(f) = exp(d.1 – d.2) = exp(Y.T - Y.R)

55. Log-transformation
Let Rk be the geometric mean of period ratios obtained from sequence k, k =1,2. f’ = (R1/R2)1/2 f’ follows an univariate log-normal distribution with mean exp(md2/2). f’is biased for  The (1-2)100% CI for  is given as [exp(L1), exp(U1)]

56. Log-transformation
Under 80/125 rule, ABE is concluded if exp(L1) > 0.8 and exp(U1) < 1.25, or L1 > and U1 < Reasons of selection of 80% and 125% as ABE limits 1. Symmetric about 0 on log-scale. 2. Maximal power occurs when ratio is Maximal power for 80/120 limit occurs at 0.98.

57. Log-transformation
The MVUE for  is give as f’MVUE = f’(-mSSD), where SSD is the pooled SS of period differences on the log-transformed data,  is df, (-mSSD)={(/2)/[(/2+j)j!}[(-m/4)SSD]j and (.) is gamma function. E[(cSSD)] = exp[(c/2)d2] E{[(cSSD)]2} = exp[cd2](c d4)

58. Log-transformation
Theorem Bias(f’) = [exp(md2/2) –1]. MSE(f’) = 2[exp(md2/2) –1]2 + 2exp(md2/2)[exp(md2/2) –1] V(f’MVUE) = 2{exp(md2/2)[(md2)2] – 1} v(f’MVUE) = exp(2f){[(-mSSD)]2 - (-4mSSD)}

59. Log-transformation Example Data: p181 of Chow and Liu (2008)
ANOVA Table based on log-transformed data
SS df SS MS F p-value
Intersubject
Carryover
Residuals
Intrasubject
Formulation
Period l
Residual

60. Log-transformation
  Y.T = , Y.R= , f = SSD = 2.472, sd2 = (L1, U1) = ( , ) f’ = exp( ) = [exp( ), exp( )] = (0.5145, ) Bias (f’) = , MSE(f’) = = f’MVUE = (0.7110)(0.9831) = v(f’) = v(f’MVUE) = [( – ] = Eff(f’MVUE, f’) = / = %

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《Design and Analysis of Bioavailability and Bioequivalence》， 作者:Chow, SC, Liu, JP ，出版社: Chapman & Hall/ CRC (third edition)，p.78。本作品依據著作權法第 46、52、65 條合理使用。 8
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《Design and Analysis of Bioavailability and Bioequivalence》， 作者:Chow, SC, Liu, JP ，出版社: Chapman & Hall/ CRC (third edition)，p.81。本作品依據著作權法第 46、52、65 條合理使用。 13
《Design and Analysis of Bioavailability and Bioequivalence》， 作者:Chow, SC, Liu, JP ，出版社: Chapman & Hall/ CRC (third edition)，p.82。本作品依據著作權法第 46、52、65 條合理使用。 17
《Design and Analysis of Bioavailability and Bioequivalence》， 作者:Chow, SC, Liu, JP ，出版社: Chapman & Hall/ CRC (third edition)，p.84。本作品依據著作權法第 46、52、65 條合理使用。 18
《Design and Analysis of Bioavailability and Bioequivalence》， 作者:Chow, SC, Liu, JP ，出版社: Chapman & Hall/ CRC (third edition)，p.86。本作品依據著作權法第 46、52、65 條合理使用。 20
《Design and Analysis of Bioavailability and Bioequivalence》， 作者:Chow, SC, Liu, JP ，出版社: Chapman & Hall/ CRC (third edition)，p.88。本作品依據著作權法第 46、52、65 條合理使用。

63. 版權聲明 頁碼 作品
版權圖示
來源/作者 25
《Design and Analysis of Bioavailability and Bioequivalence》， 作者:Chow, SC, Liu, JP ，出版社: Chapman & Hall/ CRC (third edition)，p.92。本作品依據著作權法第 46、52、65 條合理使用。 26
《Design and Analysis of Bioavailability and Bioequivalence》， 作者:Chow, SC, Liu, JP ，出版社: Chapman & Hall/ CRC (third edition)，p.96。本作品依據著作權法第 46、52、65 條合理使用。 27
《Design and Analysis of Bioavailability and Bioequivalence》， 作者:Chow, SC, Liu, JP ，出版社: Chapman & Hall/ CRC (third edition)，p.97。本作品依據著作權法第 46、52、65 條合理使用。 28
《Design and Analysis of Bioavailability and Bioequivalence》， 作者:Chow, SC, Liu, JP ，出版社: Chapman & Hall/ CRC (third edition)，p 。本作品依據著作權法第 46、52、65 條合理使用。 29

64. 版權聲明 頁碼 作品
版權圖示
來源/作者 33
《Design and Analysis of Bioavailability and Bioequivalence》， 作者:Chow, SC, Liu, JP ，出版社: Chapman & Hall/ CRC (third edition)，p.101。本作品依據著作權法第 46、52、65 條合理使用。 36
《Design and Analysis of Bioavailability and Bioequivalence》， 作者:Chow, SC, Liu, JP ，出版社: Chapman & Hall/ CRC (third edition)，p.119。本作品依據著作權法第 46、52、65 條合理使用。 38
《Design and Analysis of Bioavailability and Bioequivalence》， 作者:Chow, SC, Liu, JP ，出版社: Chapman & Hall/ CRC (third edition)，p.120。本作品依據著作權法第 46、52、65 條合理使用。 39
《Design and Analysis of Bioavailability and Bioequivalence》， 作者:Chow, SC, Liu, JP ，出版社: Chapman & Hall/ CRC (third edition)，p.121。本作品依據著作權法第 46、52、65 條合理使用。 41
《Design and Analysis of Bioavailability and Bioequivalence》， 作者:Chow, SC, Liu, JP ，出版社: Chapman & Hall/ CRC (third edition)，p.123。本作品依據著作權法第 46、52、65 條合理使用。

65. 版權聲明 頁碼 作品
版權圖示
來源/作者 42
《Design and Analysis of Bioavailability and Bioequivalence》， 作者:Chow, SC, Liu, JP ，出版社: Chapman & Hall/ CRC (third edition)，p.124。本作品依據著作權法第 46、52、65 條合理使用。 43
《Design and Analysis of Bioavailability and Bioequivalence》， 作者:Chow, SC, Liu, JP ，出版社: Chapman & Hall/ CRC (third edition)，p.110。本作品依據著作權法第 46、52、65 條合理使用。 48
《Design and Analysis of Bioavailability and Bioequivalence》， 作者:Chow, SC, Liu, JP ，出版社: Chapman & Hall/ CRC (third edition)，p.184。本作品依據著作權法第 46、52、65 條合理使用。 49
《Design and Analysis of Bioavailability and Bioequivalence》， 作者:Chow, SC, Liu, JP ，出版社: Chapman & Hall/ CRC (third edition)，p.185。本作品依據著作權法第 46、52、65 條合理使用。 50

66. 版權聲明 頁碼 作品 版權圖示 來源/作者 51-52 53-54 55-56 57-58 59-60
《Design and Analysis of Bioavailability and Bioequivalence》， 作者:Chow, SC, Liu, JP ，出版社: Chapman & Hall/ CRC (third edition)，p.186。本作品依據著作權法第 46、52、65 條合理使用。
53-54
《Design and Analysis of Bioavailability and Bioequivalence》， 作者:Chow, SC, Liu, JP ，出版社: Chapman & Hall/ CRC (third edition)，p.188。本作品依據著作權法第 46、52、65 條合理使用。
55-56
《Design and Analysis of Bioavailability and Bioequivalence》， 作者:Chow, SC, Liu, JP ，出版社: Chapman & Hall/ CRC (third edition)，p.189。本作品依據著作權法第 46、52、65 條合理使用。
57-58
《Design and Analysis of Bioavailability and Bioequivalence》， 作者:Chow, SC, Liu, JP ，出版社: Chapman & Hall/ CRC (third edition)，p.190。本作品依據著作權法第 46、52、65 條合理使用。
59-60
《Design and Analysis of Bioavailability and Bioequivalence》， 作者:Chow, SC, Liu, JP ，出版社: Chapman & Hall/ CRC (third edition)，p.203。本作品依據著作權法第 46、52、65 條合理使用。