Sample Size Estimation for (Bio)equivalence Testing ... - LexJansen

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Sample Size Estimation for (Bio)equivalence Testing Between Two Treatments Madan G. Kundu, i3 Statprobe, Gurgaon, India ABSTRACT Standard 2X2 and replicated 2X2m crossover designs are recommended in the regulatory guidelines to establish bioequivalence of generic drug with off patent brand-name drug. Estimation of sample size, in any clinical trial, targets to optimize the resource usage with assurance of having adequate probability (power) to get significant result. This paper discusses the statistical concept behind sample size estimation for 2X2 and 2X2m bioequivalence trials along with its implementation in SAS. Procedure of sample size estimation in SAS has been discussed through some simple datastep functions (%BESS, %BESSm and %BESSg) and PROC POWER, a procedure for calculation of sample size. To assess the accuracy of these macros and PROC POWER, estimated sample size obtained through these two procedures have been compared with those estimated by Diletti et. al.(1991). It has been observed that sample size obtained through macros and PROC POWER is matching almost perfectly with those obtained by Diletti et. al.

1. INTRODUCTION Once the brand-name drug goes off patent, filing for generic drug can be made by providing evidence of average bioequivalence between the generic and brand-name drug. Two drugs are said to be bioequivalent if they reaches systemic circulation at the same rate and to the same extent, i.e. the plasma concentration – time profiles of both the drugs are comparable. Extent of Absorption is measured by AUC whereas rate of absorption is measured by Cmax and Tmax. As indicated in regulatory guidance, average bioequivalence can be established using either standard 2X2 crossover design (for less variable drugs) or replicated 2X2m crossover design (for highly variable drugs). Sample size, in any clinical trial, is a critical issue because it determines whether the trial has adequate probability (power) to get significant result, and at the same time larger sample size involves increased cost and increased concern about ethical issues. In this paper, statistical concept behind sample size estimation and its implementation in SAS has been discussed through some datastep functions. A simpler sample size estimation approach through PROC POWER, a feature for calculating sample size available only from SAS version 9, has also been discussed. Finally, these sample size estimates have been compared with those provided by Diletti et. al. (1991) to assess their accuracy. This paper is limited to the discussion of average bioequivalence (ABE) testing of Test drug (T) with Reference drug (R) using either 2X2 or 2X2m crossover trial design, and also throughout this paper the term bioequivalence or equivalence refers to average bioequivalence only, unless otherwise specified. The material discussed in this paper is applicable to those trials also where the purpose is to establish equivalence in any parameter through Two One Sided Tests (TOST) in a 2X2 or 2X2m trial design situation.

2. STATISTICAL CONCEPT In a bioequivalence trial main goal is to demonstrate that a function of population measures lies in between the bioequivalence limits. A comprehensive detail of statistical concept about bioequivalence can be found in Berger and Hsu (1996) and Chow and Liu (2000). Often, we denote function of population measures by θ and θ L and θ U are used to denote lower and upper bioequivalence limits. Statistically, we can state null (H0) and alternative (Ha) hypothesis of any bioequivalence trial in the following way:

H 0 :θ ≤ θ L or θ ≥ θU H1 :θ L < θ < θU

(1)

In practice, θ can be difference ( θT − θ R ) or ratio ( θ T / θ R ) of two population parameters, θ T and θ R. θ T and θ R are the measures pertaining to Test and Reference populations, respectively. In bioequivalence trials, pharmacokinetic parameters such as Area Under the Curve (AUC), Maximum Concentration (Cmax), or Time to achieve Cmax (Tmax) are used as population measures. (In other equivalence trials population measures such as pharmacodynamic parameters or any other clinical outcomes can be considered.) Generally BE limits are set to -20% and 20% of θ R when θ is difference. In case θ is the ratio, lower and upper BE limits are considered as 0.80 and 1.25. However, any other BE limits which are determined in consensus with researcher (or sponsor) and drug regulatory agency can be considered.

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When population measure is distributed lognormally, θ i s considered as ratio and data is analyzed after logarithmic transformation. After logarithmic transformation ratio hypotheses is converted to difference hypotheses in the following way:

log θ = log

θT = log θ T − log θ R θR

To keep integrity, BE limits are also converted to their logarithmic values when θ is ratio. For example, when the actual BE limits are 0.80 and 1.25, one should use log(0.80)= -0.223 and log(1.25)=0.223 as lower and upper BE limits, respectively for testing purpose. It is clear from the above discussion that ratio hypotheses can be well converted into the hypotheses of difference. Thus, without any loss of generality, now onwards it will be assumed that θ is difference between the population parameters. For the ease of discussion, we will refer the corresponding bioequivalence hypotheses as ratio bioequivalence hypotheses and difference bioequivalence hypotheses when θ is ratio and difference, respectively. 2.1. STANDARD 2X2 CROSSOVER TRIAL

Consider a 2X2 crossover trial where we wish to compare R and T using two sequences of treatment (RT and TR) given in two periods. Let n1 and n2 subjects be allocated to the two sequences, respectively. Also assume that

yT and y R are the Test and Reference means, respectively, estimated from these (n1 + n2) subjects. Two statistical approaches are suggested in literature for testing bioequivalence between T and R. These are: • Two One Sided Hypothesis Tests or TOST procedure at α significance level (Westlake,1981 and Schuirmann,1981) • (1 – 2α)X100% Confidence Interval procedure. The derivation of the sample size formula can be based either on confidence interval approach or two one sided hypothesis tests (TOST). Both lead to the same formula, and here we will consider the approach based on hypothesis testing. Before approaching to the sample size estimation procedure, we will take a quick look on TOST procedure of bioequivalence testing. 2.1.1. TOST PROCEDURE

In TOST procedure two sided bioequivalence test is divided into the two one sided tests with the following hypotheses: Test1:

H 0+ : θ ≤ θ L vs. H 1+ : θ > θ L

(2)

Test2:

H : θ ≥ θ U vs. H : θ < θ U

(3)

− 0

− 1

Test statistics for Test1 and Test2 would be

T+ =

(y

T

)

− y R − θL

(

) Vˆ (y

Vˆ y T − y R where,

T− =

and

T

)

− yR =

(y

)

− y R − θU Vˆ y − y

T

(

T

)

R

σ ⎛1

1⎞ ⎜⎜ + ⎟⎟ , is the estimate of variance of mean treatment difference. 2 ⎝ n1 n2 ⎠ 2 e

σ e2 is the MSE (Mean Square Error) from ANOVA of population measures (or its logarithmic transformation in ratio hypotheses) considering Sequence, Period and Treatment as fixed factors and subject as random factor.

T−

(= n1 + n2 − 2 ) d.f. and non-centrality parameters, θU − θˆ θˆ = yT − y R ; δ− = ˆ V yT − y R

follows non-central t distribution with v

δ+ =

θˆ − θ L

(

Vˆ y T − y R

)

and

Bioequivalence is claimed if both hypotheses ( H

(

+ 0

)

and H

− 0

) are rejected at level α, that is,

T > t v ,0 (α ) and T < −t v ,0 (α ) +

−

tν , 0 (α ) is the 100(1-α)th quantile of the central t distribution with v d.f.

2

T + and

2.1.2. SAMPLE SIZE ESTIMATION

Concept of sample size estimation discussed in this paper has been adopted from Siqueira et. al.(2005) and Wang and Chow (2002). For the sample size estimation we make the assumption that the design is balance one which means number of subjects in each sequence is equal, that is,

n1 = n2 = n . In that case,

σ ⎛1 1⎞ σ ⎜ + ⎟= Vˆ yT − y R = n 2 ⎜⎝ n1 n2 ⎟⎠

(

)

2 e

2 e

From the theory of inference Power for TOST procedure would be: Prob [Bioequivalence will be claimed| θ , θL t v ,0 (α ) | θ ] + P[ T − < −t v ,0 (α ) | θ ] – P[ T + > t v , 0 (α ) U T − < −t v , 0 (α ) | θ ]

+ − + − = {1 – P[ T < t v ,0 (α ) | θ ]} + P [ T > t v ,0 (α ) | θ ] – {1 – P[ T < t v ,0 (α ) I T > −t v , 0 (α ) | θ ]} + − = {1 – P[ T < t v ,0 (α ) | θ ]} + {1 – P[ T < t v ,0 (α ) | θ ] } – {1 – + − P[ T < t v ,0 (α ) I T > −t v , 0 (α ) | θ ]} + − + − = 1 – P[ T < t v ,0 (α ) | θ ] – P[ T < t v ,0 (α ) | θ ] + P[ T < t v ,0 (α ) I T > −t v ,0 (α ) | θ ]

The last term of the above expression is very small and can be ignored (Wang & Chow, 2002). Considering this we get the following simpler expression of power: + − = 1 – P[ T < t v ,0 (α ) | θ ] – P[ T < t v ,0 (α ) | θ ]

(

)

(

)

= 1 – λ v ,δ − t v , 0 (α ) – λ v ,δ + t v ,0 (α )

(4)

λ v ,δ (t v , 0 (α )) is the CDF of non-central t distribution with v d.f. and non-centrality parameter δ . If the desired power is

1-β, then 1-β ≤ 1 - λ v ,δ − t v , 0 (α ) - λ v ,δ + t v ,0 (α )

(

or,

λv ,δ (t v ,0 (α )) + λv ,δ −

+

) ( ) (t (α )) ≤ β

(5)

v ,0

The gold standard sample size can be obtained by finding the smallest value of inequality.

n which satisfies the above

2.1.3. IMPLEMENTATION IN SAS: %BESS

In Appendix I, a macro %BESS to estimate sample size required in a 2X2 bioequivalence trial is presented. %BESS calculates sample size using the inequality (5). %BESS has following six macro parameters: DIFF Treatment to Reference Mean difference MSE MSE from ANOVA ALPHA Probability of TypeI error POWER Power LL Lower Bioequivalence Limit UL Upper Bioequivalence Limit Out of these only ALPHA and POWER are optional whose default values have been set to 0.05 and 0.80, respectively, and the remaining parameters are mandatory to specify. Example1: Consider a prospective 2X2 bioequivalence trial where following hypotheses are planned to be tested:

H 0 :θT − θ R ≤ −0.2 × θ R or θT − θ R ≥ 0.2 × θ R

H1 : − 0.2 × θ R < θT − θ R < 0.2 × θ R We have an estimate of 18, 28, and 1548 for treatment difference, Reference mean, and MSE of ANOVA, respectively. In this case bioequivalence Limit will be (-0.2X28, 0.2X28) or (-56, 56). %BESS can be used to calculate sample size for this trial in the following way: %BESS(diff=18, mse=1548, alpha=0.05, power=0.80, LL=-56, UL=56);

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Upon invocation of the above macro, it will produce the following output: Estimation of sample size for 2X2 BE trial (Ref: Pharmaceutical Statistics (2005), 4: 233-243) INPUT : MSE = 1548 Difference (T-R) = 18 BE limits = -56 - 56 Alpha = 0.05 Power = 0.80 RESULTS : No. of iteration = 7 Estimated sample size (per sequence) = 8

Example2: Consider a second 2X2 bioequivalence trial where following hypotheses are planned to be tested:

H 0 :θT / θ R ≤ 0.8 or θT / θ R ≥ 1.25

H1 : − 0.8 < θT / θ R < 1.25 We have an estimate of 1.15, and 0.225 for treatment ratio, and CV (Coefficient of variation; usually reported instead of MSE when ratio hypotheses is tested), respectively. Prior to sample size estimation, since this is a ratio test of hypothesis, we have to transform this ratio to difference of logarithmic population values taking the logarithm of treatment ratio. Similarly lower and upper limits should also be transformed into their logarithm values. Therefore, in this case input for DIFF, LL and UL would be 0.140 (=log1.15), -0.223(=log0.80), and 0.223(=log1.25), respectively. Another point to be noted that here CV has been reported instead of MSE. For Lognormal population MSE (from 2 ANOVA of log-transformed data) and CV are interrelated using the relation MSE = log(1+CV ). For this particular example, MSE would be 0.049. %BESS can be invoked to calculate sample size for this trial in the following way: %BESS(diff=0.14, mse=0.049, alpha=0.05, power=0.80, LL= -0.223, UL= 0.223); Upon invocation of the above macro, it will produce the following output: Estimation of sample size for 2X2 BE trial (Ref: Pharmaceutical Statistics (2005), 4: 233-243) INPUT : MSE = 0.049 Difference (T-R) = 0.14 BE limits = -0.223 - 0.223 Alpha = 0.05 Power = 0.80 RESULTS : No. of iteration = 44 Estimated sample size (per sequence) = 45 2.1.4. SAMPLE SIZE ESTIMATION USING PROC POWER

The POWER procedure in SAS can be used to estimate sample size for a variety of statistical tests . This procedure is available with version 9.x onwards, but not in previous versions. Details about PROC POWER can be found in [6]. Following are the code to estimate sample size for testing bioequivalence hypotheses in 2X2 crossover set up. When θ is the difference ( θT

−θR )

4

Code to estimate total sample size for testing bioequivalence hypotheses described in section 2.1.3 as Example 1 is proc power; twosamplemeans test=equiv_diff lower = -56 upper = 56 meandiff = 18 stddev = 39.34 npergroup = . power = 0.8; run; When

θ is the ratio ( θ T / θ R )

Code to estimate total sample size for testing bioequivalence hypotheses described in section 2.1.3 as Example 2 is proc power; twosamplemeans test=equiv_ratio lower = 0.80 upper = 1.25 meanratio = 1.15 cv = 0.225 npergroup = . power = 0.8; run; The above procedure provides total sample size estimate which should be equally distributed to the each of two sequences. In case, total sample size estimate comes as an odd number we should consider the next even number as the total sample size. 2.1.5. ACCURACY LEVEL OF %BESS & PROC POWER

Diletti, et. al. (1991) suggested an exact estimation method of sample size for bioequivalence trial using bivariate noncentral t-distribution. They have also calculated sample size for 2X2 bioequivalence trial over a range of ratio, CV and power using their method. To check the accuracy of %BESS and PROC POWER, sample size obtained from these two methods will be compared to sample size estimate provided by Diletti et. Al. (1991). Sample size estimates, for testing ratio hypotheses with bioequivalence limits as (0.80, 1.25), obtained using %BESS and PROC POWER along with the sample size provided by Diletti, et. al. (1991) are tabulated in Appendix II. From the table it is quiet evident that sample size estimated from %BESS and PROC POWER are matching almost perfectly with the estimates of Diletti, et. al. (1991). However, in case of difference from the estimate of Diletti et. al. (1991), total sample size estimates is always higher in %BESS and lower in PROC POWER as comparison to the estimates of Diletti et. al. (1991). 2.2. REPLICATED CROSSOVER TRIAL (2X2m)

Often for the highly variable drugs (CV>30%), we prefer Replicated crossover design over the standard 2X2 to establish bioequivalence between two drugs. This is because the usual 2X2 may not be sufficient to provide enough power to establish bioequivalence; otherwise a larger sample size (and thus not economical) would be required. Another advantage of this kind of trial is that it enables us to test subject-by-formulation effect. In a replicated crossover design each subject is administered with both the drugs m times, and such type of designs can be denoted as 2X2m crossover design. 2X4 (m=2) is the most popular replicated crossover design in this class. TOST and the iterative sample size estimation formula described for 2X2 crossover design are still applicable in this case for testing of bioequivalence. The only exception is in the variance of the mean treatment difference which is,

1 2 2 (σ wT + σ wR ) m(n1 + n 2 ) 2 2 Here σ wT and σ wR are within Test and Reference treatment variances respectively. These two within variances can

be estimated by

2 2 and σ wR σ e2 , MSE of ANOVA. Replacing σ wT

the estimate of variance of the mean treatment difference as,

(

)

Vˆ yT − y R =

1 1 σ e2 (2 × σ e2 ) = m × 2n m×n

5

by σ e

2

, and also assuming n1=n2=n, we can have

Note that here for m=1, the replicated crossover design reduces to standard 2X2 design and so the sample size estimation formula. 2.2.1. IMPLEMENTATION IN SAS: %BESSm

In the Appendix III, The macro %BESSm to estimates the sample size required in a 2X2m bioequivalence trial is presented. Note that this macro is almost similar to that of %BESS with the only difference in calculation of variance of treatment difference. %BESSm contains all the six key parameters as in %BESS, additionally contains a key parameter M. M indicates the order of replication. For 2X4 replicated crossover design M=2. Note that, for M=1, %BESSm converts to %BESS only. Example3: Consider a 2X4 crossover design (TRTR, RTRT) (i.e. m=2). Suppose we have an estimate of CV as 42% along with the estimate of Test to Reference treatment ratio as 1.05. In this case,

σ e2

will be log(1+CV2) or 0.16. In

this situation, following invocation of %BESSm will estimate the required sample size for testing ratio bioequivalence hypotheses: %BESSm(diff=0.05, mse=0.16, M=2, alpha=0.05, power=0.80, LL= -0.223, UL=.223); Estimation of sample size for 2X2m (m=2) BE trial (Ref: Jr. of Biopharmaceutical Statistics (2002), 12(3): 295-309) INPUT : MSE = 0.16 Difference (T-R) = 0.05 BE limits = -0.223 - 0.223 Alpha = 0.05 Power = 0.80 RESULTS : No. of iteration = 17 Estimated sample size (per sequence) = 18

3. A GENERAL MACRO For the sake of convenience, a general macro %BESSg has been provided in Appendix IV. This macro can be used to estimate sample size for bioequivalence testing between two treatments for: 1. both standard and replicated crossover design 2. both ratio and difference hypotheses The advantages of %BESSg over the other two macros described earlier are, we can pass either of ratio estimate (for testing ratio bioequivalence hypotheses) or estimate of difference (for testing difference bioequivalence hypotheses), and also either of CV or MSE. There is no need to perform any pre-calculation (e.g., Logarithmic transformation or CV to MSE conversion, see section 2.1.3) before supplying the value to the macro. The macro itself performs all these calculations. 3.1. BRIEF DESCRIPTION

This macro employs the statistical formula discussed in this paper for sample size estimation. It contains following macro parameters: DIFF Mean difference between treatments RATIO Ratio of Treatment mean to Reference mean MSE Mean Square Error from ANOVA CV Coefficient of Variation M Order of the replication of each treatment in a sequence ALPHA Probability of Type I error POWER Power LL Lower Bioequivalence Limit UL Upper Bioequivalence Limit

6

Usage note: • Input for either of DIFF or RATIO should be provided. In case both are provided, the value supplied in DIFF will be used for sample size estimation. • When DIFF is specified macro will assume that the equivalence test is based on difference hypotheses. • When RATIO is specified macro will assume that the equivalence test is based on ratio hypotheses. • Input for either of MSE or CV should be provided. In case both are provided, MSE will be preferred over CV. • Use of M is optional. Default value of M is 1. • Use of parameter ALPHA and POWER is optional. Default value of ALPHA and POWER is 0.05 and 0.80 to provide 5% level of significance and 80% power. 3.2. ILLUSTRATION

To calculate sample size for the situation described in Example1 (section 2.1.3) %BESSg can be invoked in the following way (output is shown in box): Estimation of sample size for 2X2m (m=1) BE trial (Ref: Jr. of Biopharmaceutical Statistics (2002), 12(3): 295-309) HYPOTHESES : H0: H1:

%BESSg(diff=18, mse=1548, alpha=0.05, power=0.80, LL= -56, UL= 56);

Mu(T)-Mu(R)56 -560 %then %do; CV=&cv.; mse=log(1+cv*cv); %end; %else mse=&mse.;; M=&M.; Alpha=α Beta=1-&power.; /*Running iteration*/ N=1; i=0; do until (beta ge probsum); i=i+1; N=N+1; nct1=sqrt(N*M)*(diff - LL)/sqrt(mse); nct2=sqrt(N*M)*(UL - diff)/sqrt(mse); quantile=tinv(1-alpha,2*N-2); prob1=probt(quantile, 2*n-2, nct1); prob2=probt(quantile, 2*n-2, nct2); probsum=prob1+prob2; end; /*Taking all the informations in NEW variable*/ length new $ 200; new="HYPOTHESES : "; output; new=" "; output; %if %length(&diff.)=0 and %length(&ratio.)>0 %then %do; new=" H0: &LL.