# Confidence Region Approach for Assessing Bioequivalence and Biosimilarity Accounting for Heterogeneity of Variability

Authors: Jianghao Li; Shein-Chung Chow
Journal: Journal of Probability and Statistics
 (2015)

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## Abstract

For approval of generic drugs, the FDA requires that evidence of bioequivalence in average bioequivalence in terms of drug absorption be provided through the conduct of a bioequivalence study. A test product is said to be average bioequivalent to a reference (innovative) product if the 90% confidence interval of the ratio of means (after log-transformation) is totally within (80%, 125%). This approach is considered a one-parameter approach, which does not account for possible heterogeneity of variability between drug products. In this paper, we study a two-parameter approach (i.e., confidence region approach) for assessing bioequivalence, which can also be applied to assessing biosimilarity of biosimilar products. The proposed confidence region approach is compared with the traditional one-parameter approach both theoretically and numerically (i.e., simulation study) for finite sample performance.


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## Body

## 1. Introduction

For approval of generic drug products, the United States (U.S.) Food and Drug Administration (FDA) requires that evidence of average bioequivalence (ABE) in terms of drug absorption in blood stream be provided. The evidence can only be obtained through the conduct of a bioequivalence study. Two drug products are claimed to be average bioequivalent if the 90% confidence interval for the ratio of means of the two drug products (i.e.,μT/μR, where μT and μR are means of the test product and the reference product, resp.) based on log-transformed data is totally within the bioequivalence limits of (80%, 125%). We will refer to this approach as the one-parameter approach. One of the major criticisms of this approach is that it ignores possible heterogeneity of variability between the two drug products.Alternatively, Chow and Shao [1] proposed considering a confidence region approach (which we will refer to as a two-parameter approach) for assessing average bioequivalence. In other words, we consider confidence region (ellipse) for (μT,μR) (two-parameter approach) rather than confidence interval for μT/μR (one-parameter approach). Hsu and Lu [2] studied the relationship between the one-parameter approach and the two-parameter approach. As indicated by Chow and Shao and Hsu and Lu, the one-parameter approach is a special case of the two-parameter approach. In this paper, the proposed confidence region method is compared with the one-parameter approach by comparing the slopes of the tangent lines that hit the confidence region (ellipse). In addition, the probability of consistency (i.e., the probability of claiming bioequivalence based on the proposed method given that T and R  are claimed to be bioequivalent based on the traditional one-parameter approach) will be evaluated.In the next section, the traditional one-parameter approach is briefly outlined. The proposed confidence region approach is described in Section3. Section 4 compares the proposed method with the traditional one-parameter approach both theoretically and via simulation study for finite sample performance. Also included in this section is the evaluation of the probability of inconsistency. An example is presented to illustrate the proposed method in Section 5. Some concluding remarks are given in the last section.


## 2. Confidence Interval Approach for Assessing Average Bioequivalence

In practice, the classic (shortest) confidence interval (CI) ofμT/μR is usually derived using the one-parameter method. According to the 2003 FDA guidance [3], bioequivalence should be assessed based on the log-transformed data of primary pharmacokinetic (PK) parameters such as AUC (area under the blood concentration-time curves) and Cmax (maximum concentration). This means the calculation of CI for μT/μR can be obtained from the CI of log⁡μT/μR=log⁡μT-log⁡μR. Assuming that the data follow a lognormal distribution, let X-T and X-R be the sample means for the test and reference products, respectively (after log-transformation); the classic (1-2α)×100% confidence interval can then be obtained based on the following statistics:(1)T=X-T-X-R-μT-μRSnTnRnT+nR-2nT+nR,where nT and nR are the numbers of subjects in the test and reference group, respectively. Define S as S2≜∑i(XTi-X-T)2+∑i(XRi-X-R)2, where XTi and XRi are the responses of subject i who receives test formulation and reference product, respectively. Under the assumption of normality and homogeneity, it can be verified that T follows a central t distribution with (nT+nR-2) degrees of freedom. Thus, the 90% confidence interval for log⁡μT-log⁡μR=log⁡μT/μR can be obtained,(2)L1=X-T-X-R-QSnT+nRnTnRnT+nR-2,U1=X-T-X-R+QSnT+nRnTnRnT+nR-2,where Q is the 95th quantile of a central t distribution with (nT+nR-2) degrees of freedom. The exact confidence interval for μT/μR can thus be obtained by taking exponential of L1 and U1 as follows:(3)L2=exp⁡X-T-X-R-QSnT+nRnTnRnT+nR-2,U2=exp⁡X-T-X-R+QSnT+nRnTnRnT+nR-2.Let L and U be the respective lower and upper bioequivalence limits. Then average bioequivalence is claimed if(4)L2,U2∈L,U,or(5)L1,U1∈log⁡L,log⁡U.Meanwhile, we could also derive the criteria of confidence interval approach based on raw data. Assuming thatX-R=μR, nT=nR=n, and we could get(6)X-TX-R-QSX-R1nn-1>L,X-TX-R+QSX-R1nn-1<U.


## 3. Confidence Region Approach for Assessing Bioequivalence and Biosimilarity

Chow and Shao [1] proposed a confidence region approach in assessing average bioequivalence. They performed bioequivalence testing under a crossover design and derived a confidence region for (μT,μR) based on the raw data. Their method is similar to statistics derived in the third case below. In what follows, we will further expand Chow and Shao’s notion more specifically working on the raw data and then discuss the relationship between using raw data and log-transformed data. We will mainly focus on the results in parallel design and crossover design whose period effects and sequence effects have been ruled out.We will first look into the cases of raw data before eventually moving on to log-transformed data in Scenario6. From Scenarios 1 to 5, we suppose that nT subjects receive the test product (T) and nR subjects receive the reference product (R). Assume that the response of T (XT) follows a normal distribution, that is, XT~N(μT,σT2), and the response of R also follows a normal distribution, that is, XR~N(μR,σR2). Denote by X-T and X-R the sample mean responses of T and R, respectively. We will consider the following possible scenarios.Scenario 1 (XT and XR are independent with σT and σR known).
X
T
i (i=1,2,…,nT) and XRi (i=1,2,…,nR) are independent and identically distributed (i.i.d.); that is, X-T~N(μT,σT2/nT), X-R~N(μR,σR2/nR). Thus, we have(7)X-≜X-TX-R~Nμ,Σ,μ=μTμR,Σ=σT2nT00σR2nR.It can be verified that X--μ′Σ-1X--μ follows a chi-square distribution with 2 degrees of freedom. Suppose q is the (1-α)th quantile of the distribution. Then X--μ′Σ-1X--μ≤q is the (1-α)×100% confidence region for μ. Thus, we have(8)μT-X-T2qσT2/nT+μR-X-R2qσR2/nR≤1.In order to compare with the one-parameter approach, that is, 90% confidence interval of μT/μR is totally within (80%, 125%) after log-transformation, consider the tangent line with the form of μT=kμR intercepting the ellipse. With some calculation, it can be verified that(9)k=X-TX-R±qσTσRX-T2/qσT2/nT+X-R2/qσR2/nR-11/nRnTX-R2-qσR2/nR.That gives the upper bound and lower bound of μT/μR. Bioequivalence is concluded if k falls within the given bioequivalence limits (L,U). Besides, the center of the ellipse (X-T,X-T) should be within y=Lx and y=Ux. With some calculation (the details are shown in the Appendix), the criteria are the same as the following:(10)LX-R-X-T2>qσT2nT+qσR2nRL2,UX-R-X-T2>qσT2nT+qσR2nRU2,L<X-TX-R<U.Figure 1 shows the criteria for bioequivalence of the confidence region method. The red lines are μT=LμR and μT=UμR, which are the lower and upper bound of the tangent lines. Bioequivalence is concluded only when the confidence region (the ellipse) is completely within the boundaries. That is, the center of the ellipse is within the area bounded by the two red lines while the slope of the tangent lines is within [L,U]. As is shown in Figure 1, the two red lines indicate μT=LμR and μT=UμR, respectively. The dotted lines indicate two tangent lines of the central ellipse that pass through the origin. Unlike the other two ellipses, which intersect with at least one of the two red lines, the central ellipse is right inside the area bordered by the red lines. Bioequivalence can only be claimed in this situation.
In order to calculate the possibility of inconsistency, we need to evaluate the distribution of relevant statistics. AsL and U are constants, that is, L=80% and U=125%, respectively, we could evaluate two statistics (LX-R-X-T) and (UX-R-X-T). It can be verified that both statistics follow normal distributions LX-R-X-T~N(LμR-μT,σT2/nT+σR2/nRL2), UX-R-X-T~N(UμR-μT,σT2/nT+σR2/nRU2). This is useful in the next section.
If we want to use mean difference (μT-μR) to test bioequivalence, we can use the test statistics described in Scenario 6.Figure 1
Bioequivalence criteria for two-parameter approach.Scenario 2 (XT and XR are dependent with known covariance matrix).
Suppose the test product and the reference product have the same sample size and they have pairwise correlation; that is, suppose the correlation coefficient ofXTi and XRi is ρ; then cov⁡(X-T,X-R)=ρσTσR/n. Thus, we have(11)X-≜X-TX-R~Nμ,Σ,μ=μTμR,Σ=σT2nρσTσRnρσTσRnσR2n.Similarly, it can be verified that X--μ′Σ-1X--μ follows a chi-square distribution with 2 degrees of freedom and the (1-α)×100% confidence region is given by X--μ′Σ-1X--μ≤q. This leads to(12)μT-X-T2qσT2/n+μR-X-R2qσR2/n-2ρμR-X-RμT-X-TqσTσR/n≤1-ρ2n2,Δ≜-X-TX-R+ρqσTσRn2-X-R2-qσR2nX-T2-qσT2n,k=X-RX-T-ρqσTσR/n±ΔX-R2-qσR2/n.Similarly, bioequivalence is concluded if k is within (L,U). That is,(13)LX-R-X-T2>qσT2n+qσR2nL2-2ρqσTσRLn,UX-R-X-T2>qσT2n+qσR2nU2-2ρqσTσRLn,L<X-TX-R<U.
In this case,LX-R-X-T~N(μT-yμR,σT2/n+σR2/nL2-2ρσTσRL/n), RX-R-X-T~N(μT-yμR,σT2/n+σR2/nR2-2ρqσTσRL/n).Scenario 3 (covariance matrix is unknown).
In this case, the covariance could only be replaced by its unbiased and consistent estimate. For simplicity, supposenT=nR=n; that is, the responses XTi and XRi (i=1,2,…,n) can be expressed as Xi≜XRiXTi. A2≜∑i(Xi-μ)(Xi-μ)′ is the deviation matrix. Intuitively 1/n-1A is an unbiased and consistent estimate. Thus the following test statistic T2≜nX--μ′(1/n-1A)-1X--μ follows a Hotelling distribution with dimension equal to 2 and degree of freedom equal to n-1. Thus it can be shown that n-1+1-2/2(n-1)T2~F(2,n-2). Similarly the (1-α)×100% confidence region for μ can be obtained as(14)nn-22X--μ′A-1X--μ≤q,where q is the (1-α)th quantile of F(2,n-2). As A is symmetric, we assume its form as a2ρabρabb2. Thus, after transformation the confidence region can be obtained as(15)μT-X-T2qb2+μR-X-R2qa2-2ρμR-X-TμT-X-Tqab≤21-ρ2nn-2.
Suppose the tangent line isμT=kμR; solving the above equation, k can be obtained as follows:(16)Δ≜-X-RX-T+2ρabqnn-22-X-R2-2a2qnn-2X-T2-2b2qnn-2,k=X-RX-T-2ρa2b2q/nn-2±ΔX-R2-2a2q/nn-2.Bioequivalence is claimed when k is within (L,U); that is,(17)LX-R-X-T2>2qb2nn-2+2qa2nn-2L2-4ρabqnn-2L,UX-R-X-T2>2qb2nn-2+2qa2nn-2U2-4ρabqnn-2U,L<X-TX-R<U.Scenario 4 (variance comparison).
As most biosimilar products are highly variable, it is crucial to compare the variances of test and reference products. Comparing variabilities using the one-parameter approach has been studied in the literature. See, for example, Pitman; Morgan; Chow and Tse; Lee et al. [4–7]. Comparing variabilities using the two-parameter approach is also feasible, which is briefly described below.
As an example, consider the problem of testing homogeneity. In variances, supposeX is a p×1 vector that contains n responses for all the reference and test products and X~N(μ,Σ). The null hypothesis of interest is that H0:Σ=Ip, where Ip is a p×p identity matrix (the matrix with ones on the main diagonal and zeros elsewhere). Define A≜∑i(Xi-μ)(Xi-μ)′ and λ=exp⁡-1/2tr(A)An/2e/nnp/2. Define ε=-2ln⁡(λ). Then it can be verified that ε approximately follows chi-square distribution with the degree of freedom of p(p+1)/2.
In more general cases, we could test multiple reference products with multiple test products at the same time. In such cases,H0:Σ=Σ0, where Σ0 may not be an identity matrix as the variance of different reference products need not be the same. Then defining(18)λ′=exp⁡-12tr⁡AΣ0-1AΣ0-1n/2ennp/2,it can be proved that -2ln⁡(λ′) also follows χ2(pp+1/2).Scenario 5 (test for biosimilar products).
As indicated in Chow and Liu [8], biosimilar products can be very variable due to their susceptibility to environment conditions such as light and temperature. So they are not identical to reference products. Following similar idea, the biosimilarity of biosimilar products can be assessed using the two-parameter approach by taking the possible variation into account. In other words, it can be tested whether(19)LμR+d<μT<UμR+eholds true, where m and n are intercepts that may not be zeros. Note that these may not exist; this case may not be an easy solution with the one-parameter approach. However, we can formulate the criteria for testing bioequivalence as follows.
Adopting the notations in Scenario3 as described above, we have(20)nn-22X--μ′A-1X--μ≤qas the confidence region for (μR,μT). Bioequivalence is claimed when this ellipse region is within the area bounded by μT=UμR+e and μT=LμR+d. With some calculation, we could formulate the calculation as follows:(21)LX-R-X-T+d2>2qb2nn-2+2qa2nn-2L2-4ρabqnn-2L,UX-R-X-T+e2>2qb2nn-2+2qa2nn-2U2-4ρabqnn-2U,LX-R+d<X-T<UX-R+e,which is of approximately the same form as what have been derived in Scenario 3.
Figure2 shows the criteria for biosimilarity. It is similar to Figure 1, but both lines in Figure 2 may not pass through the origin.Figure 2
Biosimilarity criteria for two-parameter approach.Scenario 6 (test based on log-transformed data).
According to the 2003 FDA guidance [3], bioequivalence testing should be conducted based on the log-transformed data. Thus, the assumptions above should be modified so that the response (after log-transformation) of T (XT) follows XT~N(log⁡μT,σT2) and the response of R follows XR~N(log⁡μR,σR2). The bioequivalence criterion is(22)L<μTμR<U.This indicates that(23)log⁡μR+log⁡L<log⁡μT<log⁡μR+log⁡U.Note that this is simply a special case for what was derived in Scenario 5 by making the slopes of both lines equal 1. The equation then becomes(24)X-R-X-T+log⁡L2>2qb2nn-2+2qa2nn-2-4ρabqnn-2,X-R-X-T+log⁡U2>2qb2nn-2+2qa2nn-2-4ρabqnn-2,log⁡L<X-T-X-R<log⁡U.


## 4. Comparison between One-Parameter Approach and Two-Parameter Approach

In this section, we will compare the proposed method with the traditional one-parameter method. For simplicity, assume thatnT=nR=n, and we only need to be concerned with the case where covariance matrix is unknown in the two-parameter approach. According to the results is Section 2, the criteria for bioequivalence in one-parameter approach are equivalent to(25)X-T-LX-R>Q2S21nn-1,UX-R-X-T>Q2S21nn-1.Similarly, in the two-parameter approach, bioequivalence criteria are equivalent to the following form:(26)X-T-LX-R>2qb2nn-2+2qa2nn-2L2-4ρabqnn-2L,UX-R-X-T>2qb2nn-2+2qa2nn-2U2-4ρabqnn-2U.Compared with the two-parameter approach, it is obvious thatS2=a2+b2. Given that T and R  are claimed to be bioequivalent based on one-parameter approach, then we could suppose that σT2=σR2=σ2 due to the homogeneity assumption in the one-parameter model. Then LX-R-X-T~N(LμR-μT,1+L2/nσ2), UX-R-X-T~N(UμR-μT,1+U2/nσ2). So UX-R-X-T,X-T-LX-R′~N(μ,Σ), where(27)μ=μ1,μ2′=UμR-μT,μT-LμR′,Σ=1+U2nσ2-1nσ2-U2L2nσ2-1nσ2-U2L2nσ21+L2nσ2.Figure3 shows the relationship of the two approaches. The area in the upper right position of the red lines is the bioequivalent area of the two-parameter approach, while the upper right area of the blue lines is that of the one-parameter approach. The two approaches are inconsistent; that is, bioequivalence cannot be concluded by the two-parameter approach given the fact that T and R are bioequivalent based on the one-parameter approach, if the vector (UX-R-X-T,X-T-LX-R) falls into the area between the red line and the blue line. Defining the area between the red lines and blue lines as inconsistency area, we can calculate the possibility of the random vector not falling into this region.Figure 3
Comparison of two approaches’ criteria.We will briefly compare these two approaches. On the one hand, whenX-T and X-R vary greatly, for example, X-T is 1.5 times bigger than X-R, bioequivalence would not be claimed under either approach, as UX-R-X-T is smaller than zero. Thus both approaches are rigorous in the sense that bioequivalence can only be claimed when X-T and X-R are sufficiently close. On the other hand, when X-T and X-R are close, the influence of bigger variation differs in two approaches. Noting that Q2<2q, it can be easily estimated that when ρ is relatively small, 2qb2/n(n-2)+2qa2/n(n-2)L2-4ρabq/n(n-2)L>Q2S21/n(n-1). That means the two-parameter approach is more rigorous than the one-parameter approach. But when ρ is rather big, the situation is more sophisticated, as it is difficult to tell which method is more conservative. Thus, we try to delineate the relationship of the two approaches by calculating the following conditional probabilities.SupposeP1=P (two-parameter approach claims bioequivalence ∣ one-parameter approach claims bioequivalence), and Φx is the cumulative distribution function of standard normal distribution; then we could get the possibility of UX-R-X-T,X-T-LX-R′ falling into the inconsistency area P1UL:(28)P1UL=∬y,x∈I112πΣ-1/2exp⁡-12x-μ1,y-μ2′Σ-1x-μ1,y-μ2+∬y,x∈I212πΣ-1/2exp⁡-12x-μ1,y-μ2′Σ-1x-μ1,y-μ2,where(29)I1=y,x∣y=X-T-LX-R,x=UX-R-X-T,y∈Q2S21nn-1,2qb2nn-2+2qa2nn-2L2-4ρabqnn-2L,x∈Q2S21nn-1,+∞,I2=y,x∣y=X-T-LX-R,x=UX-R-X-T,y∈2qb2nn-2+2qa2nn-2L2-4ρabqnn-2L,+∞,x∈Q2S21nn-1,2qb2nn-2+2qa2nn-2U2-4ρabqnn-2U.Thus the possibility of consistency equalsP1=1-P1UL.On the other hand, given the fact that two products are claimed to be bioequivalent under the two-parameter approach, then we could not assume homogeneity in this model. SoUX-R-X-T,X-T-LX-R′~N(μ,Σ′), where(30)Σ∗=U2nσR2+1nσT2-1nσT2-U2L2nσR2-1nσT2-U2L2nσR2L2nσR2+1nσT2.AssumeP2=P (one-parameter approach claims bioequivalence ∣ two-parameter approach claims bioequivalence); we could similarly derive the value of P2 as follows:(31)P2UL=∬y,x∈I312πΣ∗-1/2exp⁡-12x-μ1,y-μ2′Σ∗-1x-μ1,y-μ2=∬y,x∈I412πΣ∗-1/2exp⁡-12x-μ1,y-μ2′Σ∗-1x-μ1,y-μ2,where(32)I3=y,x∣y=X-T-LX-R,x=UX-R-X-T,y∈2qb2nn-2+2qa2nn-2L2-4ρabqnn-2L,Q2S21nn-1,x∈2qb2nn-2+2qa2nn-2U2-4ρabqnn-2U,+∞,I4=y,x∣y=X-T-LX-R,x=UX-R-X-T,y∈Q2S21nn-1,+∞,x∈2qb2nn-2+2qa2nn-2U2-4ρabqnn-2U,Q2S21nn-1.Thus,(33)P2=1-P2UL.In order to evaluate the influence of variance on both approaches, we try to plotP1 and P2 against the variance. We set the sample means of reference and test products as constants. First, we suppose the variances of T and R are the same, that is, σR2=σT2, and change at the same time. The covariance coefficient ρ equals 0.6, referring to the example in the next section. We further suppose that the sample variances equal the actual variances of T and R. We could calculate approximate values of P1 and P2 to draw the two plots shown in Figure 4.Figure 4
Changes ofP1 and P2 against variance.
(a)
(b)From Figure4, it can be seen that P1 declined little when σR2 and σT2 change from 100 to 1000, staying close to 100%. But P2 drops rather dramatically, which coincides with what was predicted earlier in this section. That is, the one-parameter approach tends to be more conservative when the variation becomes greater.Besides, we are interested in what would happen when the variance of test products increases. This is an important question when it comes to assessment of biosimilarity. Thus we draw the plot shown in Figure5.Figure 5
Change ofP2 against σT2.In the calculation,σR2 is set as 400 (i.e., σR=20) and σT2 varies from 300 to 2500. From Figure 5, we could see that P2 has a single nadir and tends to increase when σT2 becomes way bigger than σT2. Eventually P2 equals 100%. This can be proved by noticing 2qb2/n(n-2)+2qa2/n(n-2)L2-4ρabq/n(n-2)L-Q2(a2+b2)1/n(n-1) is a quadratic function of b, with its quadratic coefficient bigger than zero. So the function value is above zero when b is large enough. Thus P2 turns 100% in that situation.We could also discuss the relationship of two approaches based on log-transformed data. According to the results in Section2 and Section 3 Scenario 6, the criteria for the CI approach and proposed approach can be rewritten with simple transformation as(34)X-T-X-R>Q2S21nn-1+log⁡L,(35)X-T-X-R<-Q2S21nn-1+log⁡U,(36)X-T-X-R>2qb2nn-2+2qa2nn-2-4ρabqnn-2+log⁡L,(37)X-T-X-R<-2qb2nn-2+2qa2nn-2-4ρabqnn-2+log⁡U,respectively. If bioequivalence or biosimilarity is claimed based on the one-parameter approach, then homoscedasticity is assumed; that is, σT2=σR2=σ2. So X-T-X-R~N(μT-μR,2/nσ2). The probability of inconsistency can be written as(38)P1UL′=max⁡Φ2qb2/nn-2+2qa2/nn-2-4ρabq/nn-2+log⁡U-μT-μR2/nσ2-ΦQ2S21/nn-1+log⁡U-μT-μR2/nσ2,0+max⁡ΦQ2S21/nn-1+log⁡L-μT-μR2/nσ2-Φ2qb2/nn-2+2qa2/nn-2-4ρabq/nn-2+log⁡L-μT-μR2/nσ2,0.⁡Then P1=1-P1UL′.Similarly, we could obtainP2 based on log-transformed data. Under the hypothesis of heterogeneity, X-T-X-R~N(μT-μR,1/nσT2+1/nσR2), thus(39)P2UL′=max⁡ΦQ2S21/nn-1+log⁡U-μT-μR1/nσT2+1/nσR2-Φ2qb2/nn-2+2qa2/nn-2-4ρabq/nn-2+log⁡U-μT-μR1/nσT2+1/nσR2,0+max⁡Φ2qb2/nn-2+2qa2/nn-2-4ρabq/nn-2+log⁡L-μT-μR1/nσT2+1/nσR2-ΦQ2S21/nn-1+log⁡L-μT-μR1/nσT2+1/nσR2,0.Then P2=P2UL′.In our simple simulation, we changeσT2 and σR2 from 10 to 100 and choose n as 500 in order to ensure that bioequivalence can be claimed under both approaches. Then we found that P2 constantly equals 1 while P1 declines with the growth of variances, as is shown in Figure 6.Figure 6
P
2 against variance based on log-transformed data.This means that, in cases where the confidence interval approach claims bioequivalence, our proposed approach is also likely to get to the same conclusion. But the two-parameter approach gives narrower confidence interval forX-T-X-R, so it provides greater power.


## 5. An Example

To illustrate the effects of the proposed approach, consider an example from Chow and Liu [8]. The study is a standard 2 × 2 crossover design, conducted on 24 volunteers. Each subject would take either 50 mg tablets (test formulation) or 5 mL of an oral suspension (reference formulation). AUC values were calculated with the blood samples obtained at different times. The data is given in Table 1.Table 1
Data of the crossover trial.
Sequence
Subject number
Period
Subject total
P.D.
C.D.
I
II
1
RT
1
74.675
73.675
148.350
−1.000
−1.000
RT
4
96.400
93.250
189.650
−3.150
−3.150
RT
5
101.950
102.125
204.075
0.175
0.175
RT
6
79.050
69.450
148.500
−9.600
−9.600
RT
11
79.050
69.025
148.075
−10.025
−10.025
RT
12
85.950
68.700
154.650
−17.250
−17.250
RT
15
69.725
59.425
129.150
−10.300
−10.300
RT
16
86.275
76.125
162.400
−10.150
−10.150
RT
19
112.675
114.875
227.550
2.200
2.200
RT
20
99.525
116.250
215.775
16.725
16.725
RT
23
89.425
64.175
153.600
−25.250
−25.250
RT
24
55.175
74.575
129.750
19.400
19.400
2
TR
2
74.825
37.350
112.175
−37.475
37.475
TR
3
86.875
51.925
138.800
−34.950
34.950
TR
7
81.675
72.175
153.850
−9.500
9.500
TR
8
92.700
77.500
170.200
−15.200
15.200
TR
9
50.450
71.875
122.325
21.425
−21.425
TR
10
66.125
94.025
160.150
27.900
−27.900
TR
13
122.450
124.970
247.420
2.525
−2.525
TR
14
99.075
85.225
184.300
−13.850
13.850
TR
17
86.350
95.925
182.275
9.575
−9.575
TR
18
49.925
67.100
117.025
17.175
−17.175
TR
21
42.700
59.425
102.125
16.725
−16.725
TR
22
91.725
114.050
205.775
22.325
−22.325
P.D. = 2 × (period difference).
C.D. = 2 × (crossover difference).It can be tested that there is no influence of period and sequence effects. When adopting the one-parameter approach, it can be calculated thatS=20202.89, X-T=80.27, X-R=82.56, and Q=1.68; thus the three requirements can be satisfied in the one-parameter approach. On the other hand, a2=9948.28, b2=10254.6, ρabq=6386.05, and q=2.56, so it can also be verified that bioequivalence is concluded in the two-parameter approach.We could also calculate the inconsistent probability in this example. Suppose the unbiased estimates of mean and variation are the accurate value. It is pretty hard to get the exact value byP2UL, but it should be smaller than(40)PX-T-LX-R∈2qb2nn-2+2qa2nn-2L2-4ρabqnn-2L,Q2S21nn-1+PUX-R-X-T∈2qb2nn-2+2qa2nn-2U2-4ρabqnn-2U,Q2S21nn-1=ΦQ2S21/nn-1-LX-R-X-T1+L2σ2/n-Φ2qb2/nn-2+2qa2/nn-2L2-4ρabq/nn-2L-LX-R-X-T1+L2σ2/n+ΦQ2S21/nn-1-UX-R-X-T1+U2σ2/n-Φ2qb2/nn-2+2qa2/nn-2U2-4ρabq/nn-2U-UX-R-X-T1+U2σ2/n=0.10905,where Φ is the cumulative distribution function for standard normal distribution. So P2UL<0.10905, which means the possibility of consistency P2 should be greater than 89.095%. Similarly, we could calculate that P1 is 100%.


## 6. Concluding Remarks

The traditional one-parameter approach is convenient in examining bioequivalence and could obtain rather accurate results when applying to log-transformed data for generic drugs. However, when applying to raw data, this method lacks accuracy as it assumes the average sample response of the reference products equalsμR. This is especially the case when testing biosimilar products whose response value varies dramatically in different environments. Thus the constructed confidence interval is not the exact confidence interval desired. Besides, tests for biosimilar products may focus on the form μT=aμR+b, where the intercept is not zero. It is hard for one-parameter method to find good test statistics, but the two-parameter method shows its privilege. Meanwhile, the test statistic of the one-parameter approach requires homogeneity, which may only be true for generic drugs. It is not so convincing to overlook the influence of the variation difference when testing biosimilar products.Our proposed method handles the problems mentioned above. Besides, the two-parameter method can be generalized to higher dimensions circumstances, that is, multiple test products and multiple reference products tested at the same time, that is,(μT1,…,μTm,μR1,…,μRm). The procedure is similarly finding confidence region and calculates the slope of the hyperplane μTi=kμRi (i=1,2,…,m). Our proposed method can be applied to all kinds of biosimilar products with only a change of parameter.As for study design, both parallel and crossover design are viable. However, when performing parallel design, the sample sizes for both formations should be the same. The choice of study design mainly depends on intersubject and intrasubject variance. If intersubject variance is higher, it is more likely to choose crossover design. And parallel design should be priority if it is the other way around. Besides, possibility of ruling out carryover effects and sequence effects should also be taken into consideration.There are also problems for the two-parameter approach. The approach may not give reasonable result when the origin is within the confidence region. That is,(41)X-T2qb2+X-R2qa2-2ρX-RX-Tqab≤21-ρ2nn-2.In this situation lines that pass through the origin always intercept the ellipse, so bioequivalence can never be claimed with the two-parameter approach using raw data. For log-transformed data, such limitation does not exist because we can adjust the intercept to find the tangent lines.Another problem of the two-parameter approach is the restriction on sample size. In times when it is more reasonable to choose parallel design, calculation on sample size could be a new obstacle. The determination of sample size is then important. Methods prevailing nowadays all focus on the one-parameter approach. If performing crossover design, then the methods for one-parameter approach are also feasible. But when conducting parallel design, it is harder to determine. One approach may be requiringnT=nR at the beginning, and then Y/2S2/n follows t distribution. Thus we could determine sample size based on statistical power.The two-parameter approach shows its advantage when applying to biosimilarity test for biosimilar products, becauseμT=aμR+b is often required. The intercept b is usually a nonzero parameter, usually between 5% and 10%, thus providing difficulty for the one-parameter approach. But the criteria for biosimilarity can be easily derived in the two-parameter approach.In our calculation for consistency, our proposed approach is more rigorous than the one-parameter approach when the sample covariance coefficient is relatively small. Besides, bothP1 and P2 are nonincreasing functions of variance. That is, the two methods are more likely to be divergent when the variances are high. This is conceivable, because the one-parameter method using raw data needs to suppose X-T=μT. This assumption is more likely to be false when variances increase. When the test product has way larger variance than the reference products, that is, σT2≫σR2, we proved that the two-parameter approach will be a more rigorous method. Nevertheless, as the homogeneity assumption does not hold in this situation, we advise the two-parameter approach in assessing biosimilarity.



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*Source: 298647-2015-12-27.xml*