CONFIDENCE INTERVALS FOR BIOEQUIV ALENCE STUDIES.

P.Blood, D.J.Nichols and B.Jones, Upjohn Laboratories (Europe).

ABSTRACT An in-house system for pharmacokinetic data management and analysis was developed using SAS@

6.03 version software. This system was extended to perform bioequivalence testing using analysis of variance and calculation of confidence intervals for pharmacokinetic parameters. The bioequivalence tests included were conventional 90 or 95% confidence intervals, Westlake's symmetric confidence intervals and Schuirmann's two one-sided intervals. The Anderson & Hauck test and the Bayesian method of Mandallaz & Mau, using p values were also included (1).

Analysis of variance used PROC GLM, the confidence interval programs were written as macro's and all the programs were incorporated in a menu-driven system using SASI AP software. The user can choose which data set to study, which pharmacokinetic parameters to analyze, whether the parameters are transformed prior to analysis and which hypothesis test or tests are performed. Hardcopy output includes means values and the confidence intervals, expressed either as absolute values or as percentages of the reference. The system was tested with a variety of data sets and the bioequivalence results were comparable to the results obtained by analysis with two other commercial software packages.

1. INTRODUCTION. Upjohn Laboratories (Europe) is a

Research and Development establishment of The Upjohn Company, a pharmaceutical company based in Kalamazoo, Michigan, U.S.A.

An in-house PC based electronic data base PKDBASE for the storage and retrieval of pharmacokinetic data (PK) was developed, in order tospeed up PK data analysis; The part of the PK program for confidence intervals for bioequivalence grew out of the interplay between the pharmaco-kinetist, the SAS® programmer who was developing the PC based PK package and the statistician. The PC/PK system for pharmacokinetic analysis developed, uses SAS® 6.03 version on Compaq 386/25 machines run under MS DOS (3) within DESQview™ windows. This paper concentrates on the bioequivalence aspects.

2. PROGRAM MEmODOLOGY. 2.1 dBase Programs.

The pharmacokinetic analysis system was developed and constructed in dBase III PLUSTM (5) with 12 sub programs, which created and defined the drugs and studies used, imported data into the system data files, and would search, view, store, modify, and print any extracted data. It could also view a system log which recorded details of each use of the above functions. It identified any external files to the system and then extracted data and finally allowed it to be read by the SAS® system for statistical analysis. A flow chart of the whole PC/PK system is

shown in Figure 1.

2.2 SAS® Sub-System. The SAS® sub system was created to allow

data analysis all within the menu driven PKDBASE program. The SAS® analySis sub­system can use any data base system and does not require dBase III PLUS™ for a supply of data, as long as SAS® datasets are created. The SAS® programs were designed with the PC SAS version 6.03, although they have since been upgraded to version 6.04. The menu driven user interface uses SAS/ AP (2) software with occasionally the SAS@ screen control language SQL (2) for output of logs and listings.

2.3 Statistics Interface. The selection of this interface allows the

user to choose a specific statistic of interest to be produced on form filling screens or menus. Each screen is simply a front end for the built-in SAS/STA T3 (2) statistical PROCedure.

2.4 Bioequivalence Sub system. The bioequivalence sub-system is part of

the SAS® sub system and is called from the statistics menu. The function of this sub-system is to allow the user to perform bioequivalence testing upon data existing in the SAS® system either imported from the dBase m PLUS™ side of PK database or created in SAS®. Once processing has finished the results can be output in a number of different forms and most importantly it can be put into a text file for

306 '

import into WordPerfect8 for report production. Figure 2 shows the outline flow chart for these procedures.

3. STATISTICS SUBSYSTEM. The SAS statistics sub system gives the

user access to a number of different SAS procedures in an iterative manner and a number of procedures such as MEANS, SUMMARY, ANOVA, GLM, NUN, NPARIWAY, RANK, REG, TIEST are accessible through menus (see figure 1). Most of these PROC steps are commonly used in pharmacokinetic analysis. For example formal comparisons between drug treatments for AUC (area under plasma concentration curve) or CMAX (Maximum concentration) and other non-compartmental pharmacokinetic parameters are needed. Compartmental parameters are also determined within SIPHAR (6) (see Figure 1) and brought through the application menu for statistical analysis. The statistics sub system also has the further bioequivalence pseudo PROCedure (see figure 2) which gives the user the ability to test, the bioequivalence or otherwise, of pairs of drugs that have just under gone PK analysis in other parts of the PKDBASE program. If bio­equivalence tests are required, then within the Statistics sub system the PROCedure GLM is run, with the CLASS and MODEL statements being supplied through the statistics menu. The MODEL statement can have different forms according to what experimental design was used in the study, parallel, cross-over or other type.

4. SAS SYSTEM FOR BIOEQUIV ALENCE. 4.1 Introduction.

The bioequivalence sub-system uses a number of different testing methods (see figure 3) which do not currently exist in SAS, these are:

- Westlake asymmetric intervals (1,9,10). - Hauck and Anderson test (10,13). - Mandallaz & Mau (Bayesian) method (10,14). - Schuirmann's two one sided tests (11,12).

along with conventional confidence intervals. These tests have therefore been custom written for this purpose. The different tests can be selected, which means that they can be used together or separately depending upon the situation and conditions of the analysis.

The bioequivalence program uses the

results from the SASGD GLM PROCedure (see Figure 1). The SASGD GLM PROCedure is executed first in order to obtain the root mean square error (RMSE). The RMSE is then used on a later menu screen for the bioequivalence methods.

The other variables needed are the names of the test and reference drug responses which are entered via the menu screen (see figure 4) and then the means are collected via the SAS MEANS Procedure (see figure 1). If the data requires to be log transformed, this can also be achieved if the data is first run through the pre­processing menu choice. This option allows the user the flexibility to log transform, usually to base 10, any of the input data or carry out other operations upon it. However it should be added that the data should be studied at least as a histogram before deciding on an appropriate transformation.

Once the SAS· GLM procedure has been executed the main Bioequiva1ence menu, (figure 3) will invite the user to choose at least one confidence test, such as the Westlake or Schuirmann, to test the equiValence between the reference drug and the test drug. The other selection menu (figure 4) is then shown and this will contain the drug selections, and variable options for testing. Response variables such as AUC, CMAX or TMAX, can then be input. There are output options for printing or storage, a request for the size of the confidence intervals such as 90, 95, 99% (a = 0.10, 0.05, 0.01 respectively) and the degrees of freedom used to determine the RMSE. One can also indicate if the data has been log transformed, which ensures the results are converted back to the original scale on completion of the analysis. This paper assumes the tolerance specification is reference mean ± 20% In the case of lognormal data tolerance figures of -20 % and + 25 % , should be taken to ensure that the Conventional Confidence Interval on the logged and hence normal data, is symmetric about the ratio of the two means.

307.

4.2 Interval Types. This paper, brings a number of interval

methods together which are normally found separately. A prior study showed alternative new PC packages had some methods for interval determinations, though not all for our requirements. Some of these packages did not offer explanations as to how the intervals were

calculated and this was one of the reasons for our desire to create our own set of options. The intervals were based on published programs (10) and algorithms (1,9,11,12,13,14). However they have all been adapted to speed their calculations and add user interfaces. The mainframe program, originally written in version 5.18 is shown in the appendix.

It should be noted that there are inconsistencies in the notations used in the algorithms in the SAS program shown in the appendix. This reflects the diffe~nt sources of the algorithms. The separate methods were not given the same notation, in order Ithat the codes followed the original source papers. The program was later broken down into separate macro's for incorporation into the PKDBASE program.

4.3 Conventional Confidence Intervals The conventional limits are simply

calculated from

CI , C1 = (mt-m,) ± tf1/l*s*J(2!N)

where CI = -C1 .

mt is the mean test response and mr is the mean reference response of the two drug, treatments under consideration for bioequivalence. Where 'tal2' is the value from the student's 't' distribution with a/2 area lying in the upper tail, s is the root mean square error (RMSE) from the analysis of variance and N is the sample size ie number of patients or volunteers. s./ ( 2 IN) is the standard error of the difference between the mean test response and the mean reference response. Thus the test statistic determines if there is a significant difference between fit and mr. The conventional limits are then determined as percentages from the mean reference response.

L, = l(X),Ci 1m, +, 100

This is shown in the early part of the program in the appendix. However this approach suffers from a serious deficiency, namely it ignores the variability in mr the denominator term in the above equation.

4.4 Westlake Intervals. Briefly, Westlake (1), introduced the idea

of Confidence intervals which were symmetric about the reference mean (mr) rather than about

308

the difference (IDr-ffit). To achieve this, requires looking for asymmetric negative and positive 't' values. Thus an iterative procedure is needed to determine kl and kz from the 't' distribution, details are given else where (1,10).

The WESTLAKE program in PKDBASE uses the algorithm from Spriet & Beiler (9) and Metzler (10) and differs from the other programs in requiring an iterative approach in the algorithm. The mainframe version program in the appendix starts with a larger initial increment, INC=2 instead of 0.1 and a smaller real difference kl-k2 (9) than Metzler (10) and produces fewer steps in the iterative process on average. The program can output the results at each iterative step on the SASLOG file if required. Other confidence intervals such as Westlake's one sided interval were written in the mainframe version of SAS but were not eventually used in the PCIPK database system PKDBASE, as they do not appear to be currently used.

4.5 Hauck and Anderson Test. This test (la,13) determines the

bioequivalence by changing the usual position of null and alternative hypothesis. Thus consider

Ho : JoLt - JoL, ~ L V JoLt - JoL, ~ U

Where L and U are the specification 'limits ie ±0.2IDr. Thus a disjoint null hypothesis is involved, with fit-mr ¢ 0, so this approach therefore requires the non central 't' distribution and relies on an approximation to it for the calculations of the interval. Hauck and Anderson's test like the later Schuirmann method recognises that the appropriate test should consider a null hypothesis of a difference between the test and reference mean drug responses and should set out to disprove the null hypothesis and hence by inference indicate or "prove" bio-equivalence. ' Hauck and Anderson considered the test statistic

\

t "

t* = m -m -(U+L'''' , r '11111

Then they took

t = t*-a = m -m -U , r

as a good approximation to the central 't' distribution. An estimate for a (delta) is taken to be

a = (U-L)J2

s*J(2/N>

Note in the case of symmetric L and U then the Hauck and Anderson probability value is the sum of the Schuirmann probability tail areas.

As SAS® software now has the TPROB(x,df,nc), a three parameter function with 'nc' as the non central parameter, the algorithm (see appendix) could now be changed and the non central 't' distribution used directly.

4.6 Mandallaz and Mau Test. The Mandallaz and Mau method (14) uses

Rule 3 from the Metzler paper (10), for determination of a posterior probability, in other words applies a Bayesian analysis. Thus a posterior probability is computed that the test to reference ratio is contained within a certain interval, say (80% 120%). Usually symmetric intervals are required and that the response variables are normally distributed, and this is the approach taken in this paper. Now Fieller's rule indicates that for a ratio say

where both the numerator and denominator are subject to random error the variance for (lIlt - r*mr) should be found from the expression

Assuming we have independence between the test and reference drug responses then the equation can be simplified to

309

Where s2/N is taken as the variance for both the test and reference means. However it should be noted that independence is not true for the results from a cross-over study. The two student's 't' values are therefore produced from

tr = (m, - m,) - L, t, = (m, - m,) - U

s*V(I+r~/1V s*V(I+r~/1V

where rl = 0.8 (LOWTOL) and r2 = 1.2 (HIGTOL) for the tolerance limits for the ratio. View the next to last section in the appendix, on the Mandallaz & Mau program. The test is to accept bioequivalence if the probability tails from tr and ~ add to less than cx.

4.7 Schuinnann p values. The Schuirmann approach (11,12) is

however the current method that the FDA prefer, and hence was specifically included. The tests for bio-equivalence are basically of two types, the older type being the interval test like the WESTLAKE and the more modem approach of using p values with the null and alternative hypothesis reversed. The essence of the Schuirmann program, is shown in the last section in the SA~ program in the appendix. This program relies on the contrary concept of considering the two drugs of interest to be bio­INequivalent until the alternative hypothesis proves other wise, like the Anderson & Hauck method. This concept, to achieve its end result therefore has to use two simultaneous interval tests.

One needs to test the null hypothesis that the mean test drug response p, is less than I'r + L of the mean reference drug response or that the mean test drug (p.J is greater than I'r + U of the mean reference drug response. Thus test :

Hoi J.£t <= J.£r + L Hl1 J.£t > J.£r + L

or 11m J.£t => J.£r + U H12 J.£t < J.£r + U

The two student's t values are produced from

\

\

tr = (m, - m) - L t, = U - (me - m)

Where L is the lower limit say -O.2*nir ,

and U is the upper limit say +0.2*mr. N, s, lilt, mr are as defined earlier. The estimated 't' values are both then compared with the t(a/2,dt) value from the null hypothesis, where a is taken as 0.05 (5%). If both It and tr are > t(a/2,df) then the drugs can be considered bio-equivalent.

s. ANALYSIS For the purposes of this paper, and to

validate the bioequivalence sub-program, some simulated normally distributed data was created with the RANNOR statement in SAS-. This achieved 6 AUC variables using different seeds with the CALL sub routine as follows;

CALL RANNOR(SEEDbXJ; AUCi = MUij + SIGMAi *Xi ;

where i = 1 to 6 (variables) and j = 1 to 2 (drugs).

Three approximate levels of Coefficient of Variation (CV) 5%, 10% and. 20% from 50 patients in a 2 period cross-over study were involved. A summary of the data is given in Table I. This table shows the mean results from the simulated data which. was ntn through· the ANOV A analysis within thel SAS GLM procedure. The output data from this procedure was then run through the bioequivalence sub system within the PC PKDBASE program, with the determined RMSE and the means, test and reference responses, and with N =50 and 48 residual degrees of freedom (df), for all the above mentioned methods in section 4.

The raw data is in the file AUCSEUGI DATA which has 10 variables and its structure and bioequivalence analysis can be followed in the SAS® program in the appendix. The variable LIMIT, for example is the Confidence limit required, where in this paper it has been set to 95 %. V ARNAME is the name of the variables under test, AUCI-AUC6. The character variable LOGTRANF indicates whether the data has been log transformed (no). Finally LOWTOL and HIGHTOL indicate the specification limits, in this case 0.8 and 1.2 ie 80% and 120%. The PUT statements are currently inactive (with * in front of line). However the lines can become

310

active (removal of *) and show the intermediate variables and· in particular the iterative steps in some of the programs such as WESTLAKE. This can be useful during tracing of any errors during programming. The same number of subjects (N) were created on each drug treatment as it was a cross-over study.

Three variables, AUC1, AUC3 and AUC5 were· given reference means higher than the test means, and three variables, AUC2, AUC4 and AUC6, were set with test means higher than the reference means. Of the 6 variables, 2 were meant to be bio-equivalent, namely AUCI and AUC4, and the remaining 4 to be bio IN­equivalent. Two of the latter variables had intervals falling below the lower boundary (80% of reference), namely AUC3 and AUC5. Finally the remaining 2 failed above the upper boundary (120% of reference), namely AUC2 and AUC6. Of the failed variables, they were devised such that 1 of each pair failed when the difference between the reference mean and test mean was positive and the other of the pair to fail when the difference between the reference mean and test mean was negative.

The results were also compared with those produced by SIPHARlKinetics (6) and MKMODEL (I) programs.

6. RESULTS. 6.1 Conventional Limits

Tables IT-VI show the results of the comparison of the 6 AUC mean figures each with 3 different CV's, approximately 5%, 10%,20%. The tables are laid out for each test separately, starting with the Conventional Confidence Intervals in table 2. The test mean 34.429 in the AUCI variable is 82.11 % of the reference mean 41.930, with a 95% Confidence Interval from 80.14% to 84.09% which is well within 80% to 120% required for Bioequivalence. However for the AUC2 variable the test mean is higher than the reference mean at 116.21 % with a 95% confidence interval of 110.91 % to 121.52 %. Thus this interval fails to be wholly contained within the 80% to 120% interval criteria.

6.2 Westlake Intervals Study of table ill shows that the

confidence intervals for the AUC's instead of being symmetric about the ratio of the test/reference means, are designed to be symmetric about the 100% point, ie the reference

}.

mean. Thus the Westlake 95% interval for AUCI is 80.46% to 119.54% just within the 80%-120% specification range. Note however that on those AUC variables which fail bioequivalence namely AUC2, AUC3, AUC5 and AUC6, they do so at both ends. Whereas in the conventional Confidence intervals only one tail is outside the boundary points.

6.3 Hauck and Anderson probability values. For the Hauck and Anderson test, (see

table IV), AUCI has an overall p value of 0.01838 < 0.05 and therefore the test and reference means are considered bioequivalent. However for AUC2 the p value of 0.07877 > 0.05 and hence mr and mt are not bioequivalent for this variable. As before only AUCI and AUC4 are found to have test and reference means which are bioequivalent.

6.4 Mandallaz and Mau p values. Like the previous test, the Mandallaz &

Mau produce a single p value, (see table V), although the values are different from those produced by the Hauck and Anderson test. For AUCI the p value is 0.01086 whereas the value was 0.01838 for the previous test, for example.

6.5 Schuirmann probability values Turning now to the Schuirmann p values

in table VI the last but one column gives the probability values for both tails. For AUCI both p values, 0.01838 and 0.0000 are less than 0.025 (a/2). Thus mr and mt are considered bio­equivalent. For AUC2 the left tail p value 0.07877 > 0.025 and therefore the test fails and the test and reference drugs are not considered bio-equivalent.

In the case of the 6 variables used in this study all the 5 tests used in PKDBASE agreed in their conclusion that only AUCI and AUC4 had test and reference drug response means that were bioequivalent.

6.6 SIPIIAR output. The output from the bioequivalence

methods used in PKDBASE was checked with the same data set AUCSEUGI DATA, against the output produced by the SIPHARlKinetics package. Tables VII - IX give the results from the conventional confidence limits, the Westlake results and the Schuirmann figures respectively. For the conventional limits compare Table VII

with Table II which contain the results from the PKDBASE program. AUCl, AUC3, and AUC5 agree, but the other AUe values do not. AUC4 failed to prod~ce any results. AUC2 and AUC6 failed to agree at the lower tail. The output from SIPHAR gives the 90% C.I as well as the 95% and this suggests the problem lies with SIPHAR because the 90% C.I is not wholly contained within the 95% C.L Hence the C.I. is not symmetric about the ratio mean but the problem only occurs when the test mean is higher than the reference mean.

For the Westlake intervals in Table VIII, AUC4 again failed to produce output. However the other AUC values all agreed to 2 decimal places with the PKDBASE output.

For the comparison of the Schuirmann method with the two packages, it was necessary to collect the 't' values from the SASLOG file, as accurate p values are not given from the SIPHAR package. Thus Table X contains the 't' and p values from PKDBASE to compare with the output from SIPHAR which is given in Table IX. Again there are some discrepancies. Only AUCI and AUC4, which are the 2 measurements which have bioequivalence between the drug responses, have 't' and p values that agree between the packages. The other AUC measurements where the test and reference mean values are not bio-equivalent, have differences which need further analysis.

6.7 MKMODEL output. The raw data was also run through

MKMODEL (7) and the output agreed to 2 decimal places with all the output from PKDBASE, for the Conventional and Westlake Intervals and the Schuirmann t and p values. No tables are shown.

7. CONCLUSION. This paper has covered the bioequivalence

aspects of a new PK/PC software package PKDBASE, which has been developed in our research department for pharmacokinetic analysis. The results show that there is good agreement between the main methods chosen for the package, namely Conventional, Westlake and Schuirmann and with MKMODEL and agreement in parts with the SIPHAR output. Some discrepancies will need to be further investigated; however validation exercises are an important part of any software development.

311 .

Figure 1

Figure 2

Generic Flow Chart for PKDBASE Program.

• it Main Menu I

t i +

Read in data from dBase

SIPHAR interface

Applications Return to dBase

Preprocessing menu

1 Graphics

SAS Procedures

menu

t

Statistics menu

t

Bioequivalence Menu

Drop variables Add variables Merge datasets MEANS SUMMARY

ANOVA GLM I , Conventional Asymmetric confidence

interval

NLIN REG

Schuirmann's two 1 sided t test

Westlake Symmetric

interval

Hauck and Anderson test

Mandallaz and Mau

test

Generic Flow Chart for Bioequivalence Tests.

Start +

Import RMSE, OF and AlPHA required for

each variable selected

• 1

Clear existing 1 data from datasets

• 1 Is data to be I, ,------=log=-------;t,al;---Sform----,ed l

check variables requested for test

t Send request to table creation

procedure 1--.... ·0

Print. view or f:\ store tables '+---0

~------.-------~

E~d

312

o t

Create dataset with variables RMSE.MEAN .DIFFlolEAN

OF. AND ALPHA

perform appropriate . test upon dataset

Insert data and label variable for ref. set

I Insert data and label variable for test set ,

Produce amalgamated dataset for table

• o

FIGURE 3 +-----------------------------------------------------------------------+

PK Data~ase system statistics : Bioequivalence menu

Select an option : 1. Conventional Confidence intervals. 2. Westlake's symmetric intervals. 3. Hauck and Anderson's p values. 4. Schuirmann's 2 one sided p values. 5. Mandallaz & Mau's Bayesian method.

FlO. Return to previous menu +-----------------------------------------------------------------------+

FIGURE 4 +-----------------------------------------------------------------------+

PK Data~ase system Bioequivalence : Schuirmann 2 one si4e4 p values

Print log window [ & ] Print output window & Save to file [ & ]

variables : AUC [ & ] CMAX [ & ] TMAX [ & ] MRT [ & ] &VARN RMSE : &RMSE1 &RMSE2 &RMSE3 &RMSE4 &RMSES--

Confidence Interval 95% Degrees of freedom : &DF

Tolerance Limits

Treatment codes for comparison

& ] Data is log transformed & ] Additional information

WW 0.8

Reference &

HIGH 1.2

Test &

Press FlO to submit Press F1 to cancel

+-----------------------------------------------------------------------+

313

Table I

Start data fram sirolaticin exercise for PICDBASE program.

VARI- RMSE REFMEAN TESTMEAN ABLE

AUC1 2.059 41.930 34.429

AUC2 4.464 33.835 39.320

AUC3 84.656 416.014 354.757

AUC4 19.305 370.786 432.484

AUC5 9.148 105.836 86.147

AUC6 21.728 90.278 104.060

Table II

Conventional Confidence InterVals (95X) fram PICDBASE program.

EJ LOWER RATIO UPPER ABLE PERCENT 'of PERCENT

MEANS. (X) . (X) (X)

AUC1 SO.14 82.11 84.09

AUC2 110.91 116.21 121.52

AUC3 77.09 85.28. 93.46

AUC4 114.55 116.64 118.73

AUC5 77.92 81.40 84.87

AUC6 105.59 115.27 124.94

Table III

C.V

(X)

5.39

12.20

21.97

4.81

9.53.

22.36

BIO· EQUIV

?

Yes

No

No

Yes

No

No

Westlake Symmetric (about 10OX) Confidence Intervals (95X) fram PICDBASE program.

VARI LOWER RATIO UPPER BIO-ABLE PERCENT of PERCENT EQUIV

MEANS (X) (X) (X) ?

AUC1 80.46 82.11 119.54 Yes

AUC2 79.36 116.21 120.64 No

AUC3 78.45 85.28 121.55 No

AUC4 81.61 116.64 118.39 Yes

AUC5 78.50 81.40 121.50 No

AUC6 76.66 115.27 123.34 No

Mean OIFF' (abs)

-7.501

5.485

-61.257

61.698

-19.689

13.782

314

TabkIV

Hauck and Anderson probability values for' 95X POWER, fram PlCDBASE progr_.

VARI RATIO PROB' BIO-ABLE of AREA EQUIV

MEANS (X) ?

AUC" 82.11 0.01838 Yes

AUc2 116.21 0.07877 No

AUC3 85 .• 28 0.10056 No

AUC4 116.64 0.00113 Yes

AUC5 81.40 0.21161 No

AUC6 115.27 0.16517 No

Table V

'Mandallaz & Mau probability values for 95X POWER, fram PICDBASE program.

VARI RATIO PROB' BIO-ABLE of AREA EQUIV

MEANS (X) ?

AUC1 82.11 .0.01086 Yes·

AUC2 "6.2' 0.09992 No

AUC3 85.28 0.07940 No

AUC4 116.64 0.00265 Yes

AUC5 81.40 0.18842 No

AUC6 115.27. 0.18887 No

Table VI

SchulrEM Two 1 sided probability values for 95X POWER, fram PICDBASE program.

VARI RATIO PROB' AREA BIO· ABLE of left right EQUIV

MEANS tail tail (X) ?

AUC1 82.11 0.00000 0.01838 Yes

AUC2 "6.2' 0.07877 0.00000 No

AUt] 85.28 0.00000 0.10056 No

AUC4 116.64 0.00113 0.00000 Yes

AUC5 81.40 0.00000 0.21161 No

AUC6 115.27 0.16517 0.00000 No

Table VII

Conventional Confidence Intervals (95X) from SIPHAR program.

VARI LOWER UPPER ABLE PERCENT PERCENT

(X) (X)

AUCl 80.17 84.05

AUC2 89.01 121.44

AUC3 77.22 93.33

AUC4 f f

AUC5 77.97 84.82

AUC6 94.26 124.80

Table VIII

Westlake Synmetric (about 1001) Confidence Intervals (95X) from SIPHAR program.

EJ LOWER UPPER ABLE PERCENT PERCENT

(X) (X)

AUCl 80.46 119.54

AUC2 79.36 120.64

AUC3 78.45 121.55

AUC4 f f

AUC5 78.50 121.50

AUC6 76.66 123.34

Table IX

Schuirmann Two 1 sided probabil ity values for 95X POWER, from SIPHAR program.

VARI Left Left Right Right ABLE Tail Tail Tail Tail

t value p value t p value value

AUC1 38.58 < 0.001 2.15 < 0.05

AUC2 -0.86 N.S 0.91 N.S

AUC3 0.54 N.S ·0.51 N.S

AUC4 3.23 < 0.01 35.18 < 0.001

AUC5 1.59 N.S -1.52 N.S

AUC6 -0.45 N.S 0.47 N.S.

BIO-EQUIV

?

Yes

No

No

Yes

No

No

315

t and P values frOll Schuil'MM Two 1 sided probability values for 95X POWER, frOll PICDBASE progr_.

VARI Left Left right right ABLE Tail Tafl tail tail

t value II value t value II value

AUC1 38.574 0.000 2.148 0.018

AUC2 1.436 0.079 13.723 0.000

AUC3 8.532 0.000 1.296 0.101

AUC4 3.227 0.001 35.192 0.000

AUC5 22.331 0.000 0.808 0.212

AUC6 0.983 0.165 7.327 0.000

APPENDIX

/*****************************************************.****/ 1* 1* PROGRAM NAME=INTERVAL *1 1* SAS program by P.Blood:Upjohn Laboratories(Europe),U.K.*1 1* SAS Institute Inc, Cary, NC, USA •. versi~ 5.18 *1 1* run under the cms/xa operating system */ 1* use the sasplus COlllll8nd */ I****·**~*******************************~*****·**~*******/i %PUT &SYSDATE &SYS'rIME; OPTIONS NOCENTER DATE NUMBER DQUOTE NOTEXT82 NOTES; CMS FILEDEF RAW DISK AUCSEUGI DATA A; DATA RAWDATA; INFILE RAW MISSOVER;

INPUT N DFRMSE REFMEAN TESTMEAN LIMIT VARNAME£·LOGTRANF£ LOWTQL HIGTOL Ill;

IF LOGTRANF = 'E' OR LOGTRANF='10' THEN DELETE; ALPHA = OOO·LIMIT)/100;

PVAL = 1·(ALPHA/2); TV = TINV(PVAL,DF);

DlfFMEAN = TESTMEAN·REFMEAN; RATIOPCT = 100*DIFFMEAN/REFMEAN + 100;

FORMAT RATlOPCT 6.2; 1**********************************************************/ 1* calculation of conventional confidence interval */ 1* calculates student's t value from TlNV function· */ I*see sugi suppl'l library user's guide, v5, chap 29,p386 */ DATA CONLIMTS; ·SET RAWDATA;

UPPLIMIT = DIFFMEAN+TV*RMSE*SQRT(2/N); UPPPERCT = 100*UPPLIMIT/REFMEAN + 100; LOWLIMIT = DIFFMEAN-TV*RMSE*SQRT(2/N); LOWPERCT = 100*LOWLIMIT/REFMEAN + 100;

*PUT "TV PVAL RMSE OF DlfFMEAN"; *PUT TV PVAL RMSE DF DIFFMEAN;

LOWTOL = 100*LOWTOL; HIGTOL = 100*HIGTOL;

IF (LOWTOL < LOWPERCT) & (UPPPERCT < HIGTOL) THEN BIOEQUIV = "YES";

FORMAT UPPPERCT LOWPERCT 6.2; PROC PRINT; TITLE1 "RESULTS IN ORIGINAL UNITS"

"& PERCENTAGE fROM REFERENCE MEAN"; TlTLE2 "CONVENTIONAL CONFIDENCE INTERVALS";

VAR RMSE REFMEAN TESTMEAN LOWLIMIT DIFFMEAN UPPLIMIT LOWPERCT RATIOPCT UPPPERCT BIOEQUIV VARNAME;

FOOTNOTE1 "If Confidence Interval is wholly contained " "within the Tolerance Range, suggests"

"Bioequivalence between the test & reference means."; /**********************************************************/ 1* generate westlake intervals: see chap' 10, "statistical*/

1* methods for deciding bioequivalence of fOl"llLllations" *1 1* by c.metzler in "oral sustained release fOl"llLllations" */

1* edited by a.yacobi and e.halperin-walega 1988 *1 1* also on paper by a.spriet & d.beiler, 197B,n02,p129-132*/

1* european j. of drug metaboliSM & pharmacokinetics *1 DATA WESTLMTS; SET RAWDATA;

SUMK = -(DIFFMEAN/RMSE)*SQRT(2*N); INC = 2.0; IF SUMK > 0 THEN K1=-SUMK;

ELSE IF SUMK <= 0 THEN K1= SUNK; K2 = SUMK-K1;

AGAIN: I = 0; J = 0; LOWTAIL = PROBT(K1,DF); 1* PROBT is the cumulative *1

HIGHTAIL = 1-PROBT(K2,DF);I* t distribution function */ PROBAREA = LOWTAIl + HIGHTAIL;

IF LOWTAIL > ALPHA THEN DO; K1 = 2*K1· K2 = SUMK-K1· GOTO.AGAIN; END;

IF HIGHTAIL> ALPHA THEN DO; , K1 = K1+SUMK; K2 = SUMK-K1; GOTO AGAIN; END;

*PUT "J I SUMK K1 K2 LOWTAIL HIGHTAIL PROBAREA"; DO WHILE(INC > 0.0001);

K1 = K1 + INC; K2 = K2 - INC; *PUT J I SUMK K1 K2 LOWTAIL HIGHTAIL PROBAREA;

LOWTAIL = PROBT(K1,DF); HIGHTAIL = 1"PROBT(K2,DF); PROBAREA = LOWTAIL + HIGHTAIL;

IF PROBAREA > ALPHA THEN DO;

END;

K1 = K1 - INC; K2 = K2 + INC; INC = INC/5; J+1; END; 1+1;

WLLIMIT = DIFFMEAN+K1*RMSE*SQRT(2/N); WLLPCT = 100*WLLIMIT/REFMEAN + 100;

WULIMIT = DIFFMEAN+K2*RMSE*SQRT(2/N); WULPCT = 100*WULIMIT/REFMEAN + 100; LOWTOL = 100*LOWTOL; HIGTOL = 100*HIGTOL;

IF (LOWTOL < WLLPCT) & (WULPCT < HIGTOL) THEN BIOEQUIV = "YES";

FORMAT WLLIMIT WULIMIT 7.2 WLLPCT WULPCT 6.2; PROC PRINT; TlTLE2 "WESTLAKE CONFIDENCE INTERVALS";

VAR RMSE REFMEAN TESTMEAN WLLIMIT DIFFMEAN WULIMIT WLLPCT RATIOPCT WULPCT BIOEQUIV VARNAME;

/**********************************************************/ 1* generate anderson & hauck p values */ 1* see chapt 7,by W.J.Westlake in "Biopharm'! Statistics *1 I*for drug development",editor K.E.Peace,pubs Dekker 1988 *1 1* also paper "a new procedure for testing equivalence in *1

/* cOlllp8rative bioavallabil Ity & other clinical trials" *1 /*CaalUn' statfstfc-theor' meth'12(23),2,p2663-2692(19B3)*1

/* also biometrics, 41, p563, (1985) *1 /* see also c.metzler paper (1988) *1

DATA ANDERSON; SET RAWDATA; LOWY = (LOWTOL-1)*REFMEAN; HIGV = (HIGTOL-1)*REFMEAN;

TVL .. (DlfFMEAN-(HIGV+LOWY)/2)/(RMSE*SQRT(2/N»; DELTA" (HIGV-LOWY)f(2*RMSE*SQRT(2/N»;

TADJ_HIG. ABS(TVL)-DELTA; TADJ_LOW = -ABS(TVL)-DELTA;

PRHIG .. PROBT(TADJ_HIG,DF); PRLDW = PROBT(TADJ_LOW,DF);

PROBAREA .. PRHIG-PRLOW; IF PROBAREA <= ALPHA THEN BIOEQUIV""YES"; *PUT "LOWY HIGV TVL DELTA TADJ_HIG PRHIG TADJ_LOW PRLOW"; *PUT LOWY HIGV TVL DELTA TADJ_HIG PRHIG TADJ_LOW PRLOW;

fORMAT PROBAREA 7.5; PROC PRINT; TITLE2

"ANDERSON & HAUCK METHOD fOR TESTING BIOEQUIVALENCE"; VAR RMSE REFMEAN TESTMEAN DIFFMEAN RATIOPCT ALPHA

. PROBAREA BIOEQUIV VARNAME; FOOTNOTE1 "PROBability AREA OF <= ALPHA, suggests, "

"Bioequivalence between the test & reference means."; /**********************************************************/ 1* g_rate Mandallaz & Mau p values, equations */ /* are adapted from no 14 on page 215 of m & m paper *1 /*"cOlllp8rison of different IRethodS for decision-making in *1

1* bioequivalence assessment." */ 1* ref: mandallaz & MU: biometrics 37: 213-222 (1981) */

1* see also c.metzler paper (1988) */ 1* M & M paper uses diff' between cumulative t > 1-alpha */

1* but reverse test applied here *1 1* so slg' occurs when test statistic <- alpha */

DATA MAND; SET RAWDATA; TLEFT " (-HIGTOl*REFMEAN+TESTMEAN) /

(RMSE*SQRT( (1+HI GTOL **2)/N»; TRIGT - (-LOWTOL*REFMEAN+TESTMEAN) 1

(RMSE*SQRT«1+LOWTOL**2)/N»; PRHIG .. 1-PROBT(TRIGT ,OF); PRLOW = PROBT(TLEFT,DF);

PROBAREA = PRHIG+PRLOW; IF PROBAREA <- ALPHA THEN BIOEQUIV="YES"; *PUT "TLEFT TRIGT PRLOW PRHIG PROBAREA"; *PUT TLEFT TRIGT PRLOW PRHIG PROBAREA;

FORMAT PROBAREA 7.5; PROC PRINT; TITLE2

"MANDALLAZ & MAU METHOD FOR TESTING BIOEQUIVALENCE"; VAR RMSE REFMEAN TESTMEAN DIFfMEAN RATIOPCT ALPHA

PROBAREA BIOEQUIV VARNAME; /**********************************************************/ /* generate schufrmann 2 one sided intervals*1 /*from paper by d.j .schuirmann,j' of pharmaco' & biopharm'*/ 1* vol 15, no 6, 1987, p657-680 *1 1* also paper by k.f.phillips,j' of pharmaco' & biopharm' *1 /* . vol 18, no 2, 1990, p137- 144 */ /* Find 2 separate students t values for 1 sided tests *1 1* NB: program uses 10NY for theta1 and higv for theta2 *1 /* These are names of the var's used in schuirmann's paper*/ DATA SCHRLMTS; SET RAWDATA;

LOWY = (LOWTOL-1)*REFMEAN; HIGV = (HIGTOL-1)*REFMEAN;

TRIGT = (DIFFMEAN-LOWY)/(RMSE*SQRT(2/N»; TLEFT .= (HIGV-DIFFMEAN)/(RMSE*SQRT(2/N»;

IF (TLEFT => TV) & (TRIGT=> TV) THEN BIOEQUIV="YES"; PRHIG = 1 - PROBT(TRIGT ,DF); PRLOW = 1 - PROBT(TLEFT ,DF);

PUT HLOWY HIGV TV TLEFT TRIGT PRLOW PRHIG"; PUT LOWY HIGV TV TLEFT TRIGT PRLOW PRHIG ; FORMAT PRHIG PRLOW 7.5;

PROC PRINT; TITLE~ "SCHUIRMANN 2 ONE SIDED METHOD FOR TESTING BIOEQUIVALENCE";

VAR RMSE REFMEAN TESTMEAN DIFFMEAN RATIOPCT ALPHA PRLOW PRHIG·BIOEQUIV VARN~E;

FOOTNOTE 1 "Probability area (PRHIG) <= ALPHA/2 " Hand Probabil ity area (PRLOW) <= ALPHA/2,";

FOOTNOTE2."ie. Total PROBABILITY <= ALPHA, suggests" . .. "Bioequivalence between the test & reference means.";

/**********************************************************/

316

REFERENCES. 1a) W.J.Westlake (1988), "Biopharmaceutical Statistics for Drug Development", Chapter 7., Editor

K.E.Peace, Publishers M.Dekker. 1b) W.J.Westlake (1974), "Symmetric Confidence Intervals for Bioequivalence Trials", Biometrics, 32,

p741-744. 2) SAS/BASEiIP, SAS/GRAPHiIP, SASI AFiIP, SASIST A ~ are trademarks of the SAS Institute Inc,

N.C., U.S.A. 3) MS-DOS 3.3 by IDM Corporation and Microsoft, Inc. 4) DESQview™, v2 (1987), Quarterdeck Office systems, 150 Pico Boulevard, Santa Monica,

CA 90405, USA. 5) dBase III PLUS™, (1987), Aston-Tate (UK) Ltd, Aston-Tate Corporation, 'Oaklands', 1 Bath Rd,

Maidenhead, Berks, SL6 3BW. 6) SIPHAR/Kinetics, version 3.3, User Manual from SIMED, Centres d'Etudes et de Recherches en

Statistiques et Informatique Medicales. 9-11, rue G.-Enesco 94008 CRETEIL Cedex, France. 7) MKMODEL by N.Holford, version 4, 1990. User manual published by BIOSOFT, 22 Hills Road,

Cambridge, CB2 UP, U.K. or PO Box 580, Milltown, NJ 08850, U.S.A. 8) WordPerfect® Corporation, v5.1, (1989), 1555 N. Technology Way, Orem, Utah 84057, U.S.A. 9) A.Spriet & D.Beiler (1978), "Table to Facilitate Determination of Symmetric Confidence Intervals

in Bioavailability Trials with Westlake's Method", European J. of Drug Metabolism & Pharmacokinetics, N02, p129-132.

10) C.M.Metzler (1988), "Oral Sustained Release Formulations", Chapter 10 entitled "Statistical Methods for Deciding Bioequivalence of Formulations" , Book editors A. Yacobi and E. Halperin­Walega, Publishers Pergamon Press.

11) D.J.Schuirmann (1987), "Comparison of the Two 1 sided Tests Procedure and the Power Approach for Assessing the Equivalence of Average Bioavailability", J. of Pharmacokinetics and Biopharmaceutics, vol 15, No 6, p657-680.

12) K.F.Phillips (1990), "Power of the Two 1 sided Tests Procedures in Bioequivalence", J. of Pharmacokinetics and Biopharmaceutics, vol 18, no 2, p137-144.

13) W.W.Hauck and S.Anderson (1983), "A New Procedure for Testing Equivalence in Comparative Bioavailability and other Clinical Trials", Commun. Statist.- Theor. Meth., 12 (23), p2663-2692.

14) D.Mandallaz & J.Mau (1981), "Comparison of Different Methods for Decision Making in Bioequivalence Assessment", Biometrics 37, p213-222.

Acknowledgements: We would like to thank Chris Long and Carl Metzler for their previous work, from which this program PKDBASE and paper has taken some guidance.

Names & Address for correspondence: P.Blood or Dr D.J.Nichols, Upjohn Laboratories (Europe), Fleming Way, Crawley, West Sussex, RH10 2NJ, U.K.

317.