failure of statistical methods to prove … · farmacia, 2011, vol. 59, 3 367 failure of...

FARMACIA, 2011, Vol. 59, 3

367

FAILURE OF STATISTICAL METHODS TO PROVE BIOEQUIVALENCE OF TWO MELOXICAM BIOEQUIVALENT FORMULATIONS. II. NON-PARAMETRIC METHODS ROXANA SANDULOVICI1, ANCA VATASESCU2, FLORIN ENACHE3, CONSTANTIN MIRCIOIU2* 1Biopharmacy & Pharmacol Res S.A., Bucharest 2Carol Davila University of Medicine & Pharmacy, Bucharest 3Institute of Statistics, Romanian Academy, Bucharest *corresponding author: [email protected]

Abstract Applying the “official” parametric methods to analyze the results of a clinical

bioequivalence (BE) study concerning two suppository formulations containing meloxicam as active substance, the bioequivalence couldn’t be proved, tested drug appearing to have a greater bioavailability than the reference drug. Since the reference drug presented an important intervariability and the tested drug proved a greater bioavailability than the reference drug, it was considered that the products could be bioequivalent, but the official statistical test failed to prove this. Following mainly the high variability of reference drug and a distribution of plasma levels of reference drug far from normality, the failure was thought as a consequence of the application of statistical parametric tests beyond the field of their validity.

Statistical models for building non-parametric confidence intervals for the ratios of means of pharmacokinetic parameters were less restrictive that in the case of parametric analysis. In a first approximation there were neglected the sequence effects and further, both sequence and period effects. The results lead to the same failure of proving BE, like parametric methods. The conclusion was that non-parametric methods lead to the same conclusion concerning BE but are more efficient in rejecting the effects of outliers. Suspicion remains that even non-parametric methods are not efficient in correcting the bias induced by partition of data in some different classes as in the case of pharmacokinetic parameters of the reference drug.

Rezumat Aplicând metodele “oficiale” parametrice pentru a analiza rezultatelor unui

studiu clinic de bioechivalenţă (BE) a două formulări de supozitoare care conţin ca substanţă activă meloxicam, BE nu a putut fi demonstrată, medicamentul testat dovedind o biodisponibilitate (BD) mai mare decât medicamentul de referinţă. Deoarece medicamentul de referinţă a prezentat o intervariabilitate foarte mare, iar medicamentul testat a avut o BD mai mare decât cel de referinţă, s-a luat în considerare faptul că produsele ar putea fi BE, dar testele statistice oficiale nu au demonstrat acest lucru. Urmărind în principal variabilitatea mare şi distribuţia nivelelor plasmatice a medicamentului de referinţă, departe de normalitate, s-a considerat că eşuarea demonstrării BE este consecinţa aplicării testelor statistice parametrice în afara câmpului lor de aplicare.


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Modelele statistice pentru construirea intervalelor de încredere non-parametrice pentru rapoartele mediilor parametrilor farmacocinetici au fost mai puţin restrictive decât în cazul analizei parametrice. Ca o primă aproximare, s-au neglijat efectele de secvenţă, iar apoi atât efectele de secvenţă cât şi efectele de perioadă. Rezultatele au dus la acelaşi eşec în a demonstra BE ca şi metodele parametrice. Rezultatul a fost că metodele nonparametrice conduc la aceeaşi concluzie în ceea ce priveşte BE dar sunt mai eficiente în respingerea efectelor outlier-ilor.

Este posibil ca şi metodele non-parametrice să fie ineficiente în deplasarea indusă de partiţia datelor în clase diferite cum a fost cazul parametrilor farmacocinetici ai medicamentului de referinţă.

Keywords: bioequivalence, meloxicam, non-parametric methods. Introduction The present paper looked for alternative non-parametric methods in

testing bioequivalence (BE). Starting from both maxC (maximum concentration) and total 0AUC −∞ (area under curve) data, non-parametric methods applied [1] to compare the differences in means with official acceptance limits indicated that cannot conclude the bioequivalence of tested and reference formulations. Since pharmacokinetic parameters found for the reference drug were far from a normal distribution, being shared in 3 different classes, application of parametric methods was in fact not justified and results were not reliable.

The problem of establishing BE is a problem of building a confidence interval (CI) for the ratio of means of main pharmacokinetic parameters. Statistical model for the main pharmacokinetic parameters area under curve and maximum concentration ( 0AUC −∞ or maxC ) is:

( ) ( ), 1,ijk ik j ijkj k j kY S P F C eµ −= + + + + +

where: µ = general mean, ikS =the effect of the i-th subject within the k-th sequence, which, for the sake of testing hypotheses, we must assume to be a normally distributed random variable with mean 0 and variance 2

sσ ,

jP = the effect of the j-th period, ( ),j kF = the direct effect of the drug,

( )1,j kC − = the residual effect of the drug, ijkε = the random fluctuation which

is normally distributed with mean 0 and variance 2eσ , and is independent of

the ikS . The conditions for the application of parametric methods in testing

bioequivalence were not fulfilled. Data concerning the reference drug (both untransformed and logarithmically transformed) were not normally


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distributed and the variability of the reference drug looked to be much higher than that of the tested drug.

Since the decision concerning bioequivalence or non-bioequivalence is based on statistical calculus, a solution is to look for in this field. Current regulations allow the use of alternative non-parametric methods. An important change appeared in the regulations concerning bioequivalence rules [2] of European Medicine Agency (EMEA). The new regulations specify “a non-parametric analysis is not acceptable”, no other alternative being indicated in cases of clear alteration of the normality and assumptions.

American guidelines and statisticians recommend criteria scaled with the variance of the reference drug for failure of proving BE clearly following reference drug deficiencies. Even EMEA initiated in 2008 a discussion about scaled criteria for Highly Variable Drugs, but the 2010 guidelines missed completely the subject.

A solution to this problem would be to run another study with a greater number of subjects, but this is difficult to be accepted by the Ethics Committees. Consequently, a further statistical analysis had to be undertaken.

Materials and methods

In the study, there were enrolled 24 healthy volunteers, 18 of them completing the study. The data obtained for areas under curve (AUC) and their ratios are presented in table I.

Table I Area under curve (AUC) for Reference ( R ) and Tested ( T ) drug

Subject 1 6 7 10 11 15 19 20 21 secv 1 RT RT RT RT RT RT RT RT RT

1P 36721 3494 24163 21584 40403 21322 48654 19776 313879

2P 44936 12608 42293 39924 77951 40642 38428 30948 24932

T/R 1.22 3.61 1.75 1.85 1.93 1.91 0.79 1.56 0.08 Subject 3 5 8 12 13 16 17 18 23 secv 2 TR TR TR TR TR TR TR TR TR

1P 65279 50454 25033 34553 25217 37065 37007 19996 35726

2P 64049 47631 21132 24823 24918 29063 25463 17423 23702

T/R 1.02 1.06 1.18 1.39 1.01 1.28 1.45 1.15 1.51 secv. 1 and secv. 2 – the first sequence of drug administration (the order of drug administration: RT (reference drug and afterwards the tested drug); TR (the tested drug first and afterwards the reference drug)


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One subject had to be eliminated since he showed zero absorption after administration of the reference drug. Also some concentrations in the final curve of reference drug were under the limit of quantification. Since neglecting such data induces a bias estimation of area under curve [3] we shortened the time interval taken into calculus.

Results and discussion

Non - parametric analysis

Wilcoxon- Man Whitney two one-sided test for bioequivalence [4] For the standard 2x2 crossover design consisting of a pair of dual

sequences (i.e. RT and TR), the distribution – free rank sum test can be applied directly to the two one-sided tests concerning bioequivalence [5,6,7]

Using the above standard notations it results

RT µµθ −= .

The usual set of unilateral hypotheses concerning bioequivalence are:

01 1: 0 : 0L A L L LH vs H whereθ θ θ θ θ∗ ∗ ∗≤ 〉 = − and

02 2: 0 : 0U A U U UH vs H whereθ θ θ θ θ∗ ∗ ∗≥ 〈 = − The estimated values of Lθ

∗ and Uθ∗ can be obtained from a linear

combination (contrast) of period differences ikd , 1,..,i n= ; 1,2k = .

; , 1; 2

ik hhik

ik

d h L U for subjects in sequenceb

d for subjects in sequenceθ− =⎧

= ⎨⎩

,

where: kni ,1= , 2,1=k , ( )2 1 / 2ik i k i kd Y Y= − , and h L or U= index for lower respectively upper limit of confidence interval

When there are no carry-over effects, the expected value and variance of hikb , are given by the equations:

( )( ) ( )

( )

2 1

2 1

1 2 121 22

h

hik

P P for kE b

P P for k

θ θ

θ

⎧ − + − =⎡ ⎤⎣ ⎦⎪⎪= ⎨⎪ − + =⎡ ⎤⎣ ⎦⎪⎩


371

and ( ) ( )2

2

2 e

dikhikdDbD σσ ===

The difference between the means of hikb in the two

sequences equals the formulation effect as follows: ( ) ( ) ( ) ∗=−=− hhhihi bEbE θθθ21

Considering LR as the sum of the ranks of the responses for subjects

in sequence 1: ( )∑==

1

11

n

iLiL bRR and ( )

2111 +

−=nnRW LL .

For testing the second hypothesis we consider similarly

( )∑==

1

11

n

iUiU bRR and ( )

2111 +

−=nnRW UU

We reject 01H if 1LW w α−〉 , where 1w α− is the ( )1 thα− quantile of the distribution of LW which can be found in tables for Mann – Whitney test, and 02H if UW wα〈 (

ααwnnw −=

− 211 ).

Hence, bioequivalence is concluded if both 01H and 02H are rejected; that is:

1LW w α−〉 and ( )UW w α〈

Analysis of totAUC data for the two MELOXICAM formulations

We estimate the L Uandθ θ− as 20% of the mean AUC for R (estimation of Rµ )

1 2

44900RR

AUCAUC

n n= =

+∑ ; !!L =!U = 0.2 "AUCR = 8980

It results: 11 11 4108 8980 13088L Lb d θ= − = + = , etc.

The complete set of obtained values are given in table II. We ordered the values of Likb ( Uikb )

( )1

11

54n

L Lii

R R b=

= =∑ ; ( )1 1 19

2L L

n nW R

+= − =

( )1

11

81n

U Uii

R R b=

= =∑ ; ( )1 1 136

2U U

n nW R

+= − =


372

Table II Calculus of ranks of period differences

secv1 1P 2P ikd 1Lib ( )LikbR 1Uib ( )UikbR RT 36721 44936 4108 13088 7 -4872 14 RT 3494 12608 4557 13537 6 -4423 12 RT 24163 42293 9065 18045 4 85 4 RT 21584 39924 9170 18150 3 190 3 RT 40403 77951 18774 27754 1 9794 1 RT 21322 40642 9660 18640 2 680 2 RT 48654 38428 -5113 3867 8 -14093 17 RT 19776 30948 5586 14566 5 -3394 10 RT 313879 24932 -144474 -135494 18 -153454 18

secv2 1P 2P ikd 2Lib ( )LikbR 2Uib ( )UikbR

TR 65279 64049 -615 -615 10 -615 6 TR 50454 47631 -1411 -1411 12 -1411 8 TR 25033 21132 -1950 -1950 13 -1950 9 TR 34553 24823 -4865 -4865 15 -4865 13 TR 25217 24918 -149 -149 9 -149 5 TR 37065 29063 -4001 -4001 14 -4001 11 TR 37007 25463 -5772 -5772 16 -5772 15 TR 19996 17423 -1287 -1287 11 -1287 7 TR 35726 23702 -6012 -6012 17 -6012 16

In the tables it was found

9;9;0.05 22w Wα = = and 1 1 2 59w n n wα α− = − = Since22 36 Uw Wα〈 ⇒ 〈 and 159 9 Lw Wα−〉 ⇒ 〉 , it means that the

products are not bioequivalent. Furthermore, we checked the hypothesis concerning bioequivalence,

based on enlarged acceptance interval 0.67 ; 1.33( ) .

In this case !!L =!U = 0.33"AUCR =14817 and 11 11 4108 14817 18925L Lb d θ= − = + = , etc Ordering descending the absolute values of Likb , respective Uikb it results the following:

( )1

11

54n

L Lii

R R b=

= =∑ , ( )1 1 19

2L L

n nW R

+= − =

( )1

11

111n

U Uii

R R b=

= =∑ ; ( )1 1 166

2U U

n nW R

+= − =


373

Bioequivalence is concluded if 1U LW w w Wα α−〈〈〈 ⇒ 66 22 59 54and〉〈 which is not the case.

Figure 1 Order of quantiles and results of test required to accept BE

Once again we couldn’t establish bioequivalence. In order to

understand better the statistical phenomena appearing in comparison of the two drugs we determined 90% confidence intervals for the ratio of the mean values of areas under curve for the tested and the reference drugs.

Distribution-free confidence intervals

An estimator and confidence interval associated with Wilcoxon signed rank statistic.

Let us consider the Wilcoxon signed rank test [8]. We can compare n pairs of values ( ),i iX Y , i=1,…,N. Put ( )i i iZ X Y= − . We computed the

statistics T + and the absolute values 1 ,..., nZ Z and ordered them in

ascending order. Let ir denote the rank of iZ and d the random variable 1id = if 0iZ > and 0id = if 0iZ <

Let be T + the sum of positive ranks. Then iT d i+ =∑ .

The mean of T + is ( ) ( ) ( )1 1

N Ni iE T E d i iE d+ = =∑ ∑

But E di( ) =1! 12+ 0 ! 1

2= 12

and consequently

( ) ( )1

112 4

N N NE T i+ +

= =∑

We can consider two sided test for the effect of treatment θ : 0 : 0 : 0aH Hθ θ= ≠ at the α level of significance, by comparison

of T + with 1w and wα α− , quantiles of Wilcoxon repartition function.

WU

C

WL wα w1- α


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An estimator of the treatment effect θ and confidence interval for differences of means was given by Hollander and Wolfe [9,10] based on the the Hodges-Lehmann [11] estimator !! , defined by the following equation:

!! = medianZ i + Z j

2, i ! j =1,...,n

"

#$

%

&'

The ( )1 / 2n n+ averages ( ) / 2i jZ Z+ , 1,...,i j n≤ = , are called

Walsh [12] averages. If we define W + the number of positive Walsh averages, then (when there are no ties among the [Z]’s and none of the Z’s is zero) W + is identical with T+. This result is in accordance with Tukey [13].

When 0θ = , the distribution of the statistic T + is symmetric about its mean, ( )1 / 4n n+ . A natural estimator of θ is the amount !! that should

be subtracted from each iZ so that the value of T + , when applied to the

shifted sample Z1 !!! ,...,Zn !!

! , is as close to ( )1 / 4n n+ as possible.

Roughly speaking, we estimate θ by the amount (!! ) that the Z sample

should be shifted in order that Z1 !!! ,...,Zn !!

! appears (when “viewed” by the signed rank statistic T + ) as a sample from a population with median 0.

The estimator !! is relatively insensitive to outliers. This is not the

case with the classical estimator 1

/n

ii

Z Z n=

=∑ . Thus the use of !! provides

protection against important errors. This procedure was applied by Steinijens and Diletti [14] in order to

differentiate the pharmacokinetic data. In fact their statistical model is very simple, neglecting both sequence and period effects.

We applied the method for AUC. We can calculate for each subject

( ) ( )ln ln ln lni

i i Ti T R ii

R

AUCd AUC AUC rAUC

⎛ ⎞= − = =⎜ ⎟

⎝ ⎠ and

ln ln ln2

i ji j i j

d dr r rr

+= + =

The method calculates the geometric means of ratios for all possible

pairs of subjects, id est for ( )21+NN pairs, including the pair (R, R), for the


375

same subject. The values of geometric means are ordered. The inferior and superior limits for 90 % confidence intervals are found in Wilcoxon tables.

Ratio RT for the subject number 1 is AUC T( )

AUC R( ) =4493636720.9

=1.22

Further more, we need the geometric means of the pair of ratios. For the first subject combined with itself, it results 1.22 !1.22 "1.22

For subject 1 combined with subject 2, it results 1.397 !1.067 "1.221

For the entire group of 18 subjects there are N N +1( )2

= 18 !192

=171

combinations presented in table III. Table III

Geometric means of pairs of T/R ratios 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18

1 1.22 2 1.12 1.02 3 1.14 1.04 1.06 4 2.10 1.92 1.96 3.61 5 1.46 1.34 1.36 2.51 1.75 6 1.20 1.10 1.12 2.07 1.44 1.18 7 1.50 1.37 1.40 2.58 1.80 1.48 1.85 8 1.54 1.40 1.43 2.64 1.84 1.51 1.89 1.93 9 1.31 1.19 1.21 2.24 1.56 1.28 1.60 1.64 1.39

10 1.11 1.02 1.04 1.91 1.33 1.09 1.37 1.40 1.19 1.01 11 1.53 1.39 1.42 2.62 1.83 1.50 1.88 1.92 1.63 1.39 1.91 12 1.25 1.14 1.16 2.15 1.49 1.23 1.54 1.57 1.33 1.14 1.56 1.28 13 1.33 1.22 1.24 2.29 1.59 1.31 1.64 1.67 1.42 1.21 1.66 1.36 1.45 14 1.19 1.08 1.10 2.04 1.42 1.17 1.46 1.49 1.26 1.08 1.48 1.21 1.29 1.15 15 0.98 0.90 0.91 1.69 1.18 0.97 1.21 1.23 1.05 0.89 1.23 1.00 1.07 0.95 0.79 16 1.38 1.26 1.29 2.38 1.66 1.36 1.70 1.74 1.48 1.26 1.73 1.41 1.51 1.34 1.11 1.56 17 0.99 0.90 0.92 1.69 1.18 0.97 1.21 1.24 1.05 0.90 1.23 1.01 1.07 0.95 0.79 1.11 0.79 18 1.36 1.24 1.26 2.33 1.62 1.34 1.67 1.71 1.45 1.24 1.70 1.39 1.48 1.32 1.09 1.54 1.09 1.51

Table IV Non-parametric confidence intervals based on Wilcoxon’s test upper and lower

ranks Number of subjects

(N) Rank for lower limit Rank for upper limit

95% 90% 95% 90% 12 14 18 65 61 13 18 22 74 70 14 22 26 84 80 15 26 31 95 90 16 30 36 107 101 17 35 42 119 112 18 41 48 131 124 19 47 54 144 137 20 53 61 158 150


376

As it can be seen in table IV, the lower and upper cutoff points for a 90% confidence interval (CI) are the values ranked 48 and 124. For our data these values correspond to the ratios 1.18 and 1.54.

Figure 2

Lower and upper ranks for CIs

We used the Excell function small(array;k) for calculating the k ranking value in a given set (array).

• =small(F3:Q20;48)=1.15 • =small(F3:Q20;124)=1.54

CI 90% = 1.18 ;1.54( ) Consequently we had to reject the hypothesis of bioequivalence

Hauschke – Steinijans – Diletti method. The previous method was criticized by the same authors later [15]

since it is based on the assumption of equal period effects. Both methods lead to the reject of bioequivalence and the result is

correct. Considering the many possible outliers in the set of data (subject outliers, subject-by-formulation outliers, single data point outliers [16]) concerning the reference drug, the results are not reliable and not ethically correct. Following high variability of the reference drug, a more general approach should be considered, regarding the differences in manufacturing processes [17], but we had no information about the manufacturing of the reference drug. Another problem of the above method is that the confidence interval is based on a theorem of Lehmann [18] in obtaining the confidence interval, which includes the comparison of all possible pairs between subjects. But bioequivalence is essentially connected with intravariability of the pharmacokinetic parameters. Interchangeability means that we are interested in comparing the plasma levels reached after administration of different drugs to the same subject. We further applied another non-parametric test, based on the comparison of the plasma levels to the same subject.


377

The a method which the differences of iko variables from sequences 1 and 2.

1 1 2 11 1 1( ) ( ) ( ) ( ) ( )2 2 2i i T R T RE o E d P P F F F F= = − + − = −

2 2 2 11 1 1( ) ( ) ( ) ( ) ( )2 2 2i i R T T RE o E d P P F F F F= − = − − − − = −

1 2( ) ( )i j T RE o o F F+ = − Since 2 2i io d= − the sums equal the differences

1 2 1 2, ( 1,..., , 1,..., )i jd d i n j n− = = . In this case we don’t need to suppose equal period effects. Procedure

comes from methods of Moses and Wolfe, once again by intermediate of Hollander and Wolfe.

The examination of individual profiles reveals a clear splitting in three classes of reference drug curves (a subject with aberrant small values, five subjects with high values and the “homogeneous distributed” group), which cannot be considered as a normal situation [19].

Figure 3

Individual concentration profiles for the reference drug

The fact that plasma levels of the tested drug are greater than plasma levels of the reference drug represents a sign that the tested drug is better than the reference drug or something is out of order with the reference


378

drug but, since the variability of the reference is very high, the second hypothesis is much more reliable. In this conditions, the rejecting of the tested drug is both incorrect and unfair since the deficiencies belong to the reference drug.

Comparative analysis of results obtained with parametric and non-parametric methods

Both parametric and non-parametric methods indicate a lack of bioequivalence. The result of the nonparametric order tests is an expected one, since for almost all subjects.

0 0R TAUC AUC−∞ −∞〈

The result is essentially incorrect since variability and distribution of the reference drug data influence substantially the estimation of intravariability and, consequently, the result of the tests concerning bioequivalence.

For such separation, it is necessary to repeat the administration of the drugs, which is possible, but less acceptable by the Ethic Committees.

Table V 90 % Confidence intervals (CI) calculated for all subjects (HSD - Hauschke,

Steinjans, Diletti method, SD - Steinjans, Diletti method, L - lower limit, U - upper limit) Method L CI 90 U CI 90 Length L

Parametric 1.19 1.59 0.40 HSD 1.09 1.56 0.47 SD 1.15 1.54 0.39

Table VI 90 % Confidence intervals calculated after exclusion of an outlier subject

Method L CI 90 U CI 90 Length L Parametric 1.15 1.47 0.31 HSD 1.11 1.50 0.39 SD 1.12 1.48 0.36

It can be observed that all three methods are influenced by outliers

but the most sensitive seems to be the parametric one. It can also be observed that the addition of an outlier subject increases the length of intervals.

Table VII Decrease of the length of CIs following elimination of an outlier subject

Method ΔL Parametric 0.09 HSD 0.08 SD 0.03


379

The parametric method is actually less sensitive than the non-parametric method, but the difference is not important.

Conclusions Starting from 0AUC −∞ data, non-parametric methods indicated that

cannot conclude bioequivalence of tested and reference formulations. Confidence interval for differences of means T R

AUC AUCµ µ− was calculated using Walsh means (method given by Hollander and Wolfe [9, 10] based on the Hodges-Lehmann [1] estimator), i.e. comparison of period differences both between sequences and inside sequences which is a valid procedure in the hypothesis of equal period effects. The lack of period effects was evidenced using ANOVA calculus. Since ANOVA is also based on normal distribution, the results become relative.

It is largely accepted that non-parametric tests are less sensitive to outliers, which was confirmed, but the differences were not important.

The rejection of non-parametric methods by the EMEA recent guidelines is not justified. When data are normally or log-normally distributed there are not significant differences between the results. When normality hypothesis is altered parametric methods are no more reliable and no other alternative is indicated.

Application of non-parametric methods is fully justified if sequence effects are absent. The lack of sequence effects can be assured by an adequate washing period and verified by the assay of plasma concentrations at the beginning of the second period.

References

1. Vatasescu A., Enache F., Mircioiu C., Miron D.S., Sandulovici R., Failure of statistical methods to prove bioequivalence of two meloxicam bioequivalent formulations. I. Parametric methods, Farmacia, 2011 (to be published)

2. Guideline of the investigation of bioequivalence, http://www.emea.europa.eu/pdfs/human/qwp/140198enrev1.pdf (row 504)

3. Tomuţă I., Iovanov R., Bodoki E., Leucuţa S.E., Quantification of meloxicam and excipients on intact tablets by near infrared spectrometry and chemometry, Farmacia, 2010, 58 (5), 559-571.

4. Chow S.C., Liu J.P., Design and Analysis of Bioavailability and Bioequivalence Studies, 2nd edition, Marcel Dekker, 1999, 110-113.

5. Hauschke D., Steinijans V.W., Dileti E., A distribution free procedure for the statistical analyses of bioequivalence studies. Int J Clin Pharmacol Ther Toxicol, 1990, 28, 72-78.

6. Cornell R.G., The evaluation of bioequivalence using nonparametric procedures, Commun Stat Theory Methods, 1990, 19, 4153-65.

7. Liu J.P., Bioequivalence and Intrasubject variability, J Biopharm Stat, 1991, 1, 205-219. 8. Mircioiu C., Sandulovici R., Statistică aplicată în farmacie şi studii clinice, ed.II, Ed. Univ

“Carol Davila”, Bucureşti 2009, 63-73.


380

9. Hollander M., Wolfe D.A., Non-parametric Statistical Methods, Wiley, 1973. 10. Hollander M., Wolfe D.A., Non-parametric Statistical Methods, 2nd edition,Wiley, New

York, 1999, 51-55. 11. Hodges J.L., Lehman E.L., Hodges Lehmann estimators - In Enciclopedia of Statistical

Sciences, volume 3, J. Wiley, 1983, 463-465. 12. Walsh J.E., Some significance tests for the median which are valid under very general

conditions, Ann Mat Stat, 1949, 20, 64-81. 13. Tukey J.W., The simplest signed-rank tests. Memo 17, Statistical Reseach Group, Princeton

University, 1949. 14. Steinijens V.W., Diletti E., Statistical Analysis of Bioavailability Studies: Parametric and

Non-parametric Confidence Intervals, Eur. J. Clin. Pharmacol, 1983, 24, 127-136 15. Hauschke D., Steinijans V.W., Diletti E., A distribution-free procedure for the statistical

analysis of bioequivalence studies, Clin Pharmacol Ther Toxicol, 1990, 28(2), 72-78. 16. Schall R., Endrenyi L., Ring A., Residuals and outliers in replicate design crossover studies,

J Biopharm Stat., 2010, 20(4), 835-849. 17. Chow S.C., Liu J.P., Statistical assessment of biosimilar products, J Biopharm Stat., 2010,

20(1), 10-30. 18. Lehman E.L., Non-parametrics: Statistical Methods Based on Ranks, Holden-Day, San

Francisco, 1975, 78. 19. Medvedovici A., Albu F., Georgita C., Mircioiu C., David V., A non-extracting procedure

for the determination of meloxicam in plasma samples by HPLC-diode array detection, Arzneimittel Forschung/Drug Research, 2005, 55(6), 326-331.

__________________________________ Manuscript received: December 10th 2010

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