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Sample Size Calculations for the Development of Biosimilar Products Based on Binary Endpoints

Seung-Ho Kang1,a, Ji-Yong Junga, and Seon-Hye Baika

aDepartment of Applied Statistics, Yonsei University, Korea
Correspondence to: Seung-Ho Kang
Department of Applied Statistics, Yonsei University, 50 Yonsei-ro, Seodaemun-gu, Seoul 120-749, Korea. E-mail: seungho@yonsei.ac.kr
Received June 17, 2015; Revised July 14, 2015; Accepted July 14, 2015.
Abstract

It is important not to overcalculate sample sizes for clinical trials due to economic, ethical, and scientific reasons. Kang and Kim (2014) investigated the accuracy of a well-known sample size calculation formula based on the approximate power for continuous endpoints in equivalence trials, which has been widely used for Development of Biosimilar Products. They concluded that this formula is overly conservative and that sample size should be calculated based on an exact power. This paper extends these results to binary endpoints for three popular metrics: the risk difference, the log of the relative risk, and the log of the odds ratio. We conclude that the sample size formulae based on the approximate power for binary endpoints in equivalence trials are overly conservative. In many cases, sample sizes to achieve 80% power based on approximate powers have 90% exact power. We propose that sample size should be computed numerically based on the exact power.

Keywords : equivalence trial, power, sample size formula, follow-on biologics
1. Introduction

Many best-selling biological products are set to lose their patents over the next few years; constantly, the assessment of biological product biosimilarity for regulatory approval has received significant attention (Chow, 2014; Chow and Liu, 2010; Chow et al., 2009; Chow et al., 2013; Hsieh et al., 2013; Kang and Chow, 2013; Li et al., 2013; US FDA, 2012; World Health Organization, 2009). It is therefore necessary to demonstrate similar qualities, efficacy, and safety for biosimilar products and renovator biological products in order to obtain regulatory approval. Consequently, the characterization of both products are examined by comparing physicochemical properties, biological activities, impurities, and stability. Immunogenicity tests, preclinical studies, and clinical trials are also conducted to demonstrate no clinically significant differences in safety and efficacy. A phase III comparative study (often designed as an equivalence study) is an important step in systematic studies.

The sample size calculation for a phase III clinical trial is important due to economic, ethical, and scientific considerations (Altman, 1980; Moher et al., 1994). This paper emphasize the drawbacks of oversized studies that calculate sample size based on approximate powers. An oversized study results in an unnecessary waste of resources with the potential to expose unnecessarily large number of subjects to potentially harmful or ineffective treatments (Altman, 1980). Therefore, it is important to compute the minimal sample size needed to achieve a pre-specified power, such as 80% or 90%.

Kang and Kim (2014) investigated the accuracy of a well-known sample size calculation formula for continuous endpoints in equivalence trials. The formula is given as (Chow et al., 2003, p.60)

$nT=knR,?????????n2=(zα+zβ/2)2σ2(δ-?μT-μR?)2?(1+1k)$

to test H0 : |μT ? μR| ≥ δ against Ha : |μT ? μR| < δ, where zα is the upper α quartile of the standard normal distribution (for example, z0.05 = 1.645); μT and μR represent the population means of primary endpoints for a biosimilar product and a renovator biological product, respectively; σ2 is the population variance of the primary endpoint; k is the allocation ratio; nT and nR are the sample sizes of a biosimilar product group and the renovator biological product group, respectively. Kang and Kim (2014) found that the sample size calculation based on (1.1) is very conservative, requiring unnecessarily large samples.

The primary endpoint is often binary; therefore, it is important to investigate the accuracy of sample size calculation formulae for binary endpoints in equivalence trials. This paper extends the results of Kang and Kim (2014) to binary endpoints for three popular metrics: the risk difference, the log of the relative risk, and the log of the odds ratio.

This paper is organized as follows. Section 2 reviews the hypotheses of equivalence trials for binary endpoints. Section 3 provides the sample size calculation formulae based on the approximate and exact powers. Section 4 numerically compares approximate powers with exact powers. Section 5 presents the conclusions.

2. Equivalence Trials for Binary Endpoints

Let XT and XR denote the number of events of interest from the biosimilar product group and the renovator biological product group, respectively. It is assumed that XT and XR follow binomial distributions B(nT, pT) and B(nR, pR), respectively. There are three popular metrics that can be used to assess the treatment effect estimated from an equivalence trial. The first is the risk difference, RD = pT ? pR, which is the difference between the test and control groups in proportions of outcomes. The second is the relative risk, or risk ratio (RR = pT/pR), which is the ratio of the rates of unfavorable events in the test and control groups. The third is the odds ratio, which is the ratio of the odds of success (or failure) of the test product relative to the control product. The characteristics of these three metrics are shown in Sinclair and Bracken (1994) and Walter (2000). In this paper, the log of the relative risk and the log of the odds ratio are investigated instead of the relative risk and the odds ratio, as the former allow metrics that are normally distributed and easier to evaluate in the analysis.

The hypotheses of equivalence for the three metrics are given as follows. For the risk difference, it is given as

$H0D:?pT-pR?≥δ?????????vs.?????????HaD:?pT-pR?<δ,$

where δ(> 0) is a pre-specified equivalence margin. For the log of the relative risk, it is given as

$H0R:|log(pTpR)|≥δ?????????vs.?????????HaR:|log(pTpR)|<δ.$

For the log of the odds ratio, it is given as

$H0O:|log?(pT/(1-pT)pR(1-pR))|≥δ?????????vs.?????????HaO:|log?(pT/(1-pT)pR(1-pR))|<δ.$

The hypotheses in (2.1), (2.2), and (2.3) can be decomposed into two one-sided hypotheses. Specifically, the hypothesis in (2.1) can be re-expressed into two one-sided hypotheses as:

$H01D:pT-pR≤-δ?????????vs.?????????Ha1D:pT-pR>-δ$

and

$H02D:pT-pR≥δ?????????vs.?????????Ha2D:pT-pR<δ.$

The hypotheses in (2.2) and (2.3) can also be decomposed into two one-sided hypotheses. For the log of the relative risk, they are given as

$H01R:log(pT)-log(pR)≤-δ?????????vs.?????????Ha1R:log(pT)-log(pR)>-δ$

and

$H02R:log(pT)-log(pR)≥δ?????????vs.?????????Ha2R:log(pT)-log(pR)<δ.$

For the log of the odds ratio, they are given as

$H01O:log?(pT1-pT)-log?(pR1-pR)≤-δ?????????vs.?????????Ha1O:log?(pT1-pT)-log?(pR1-pR)>-δ$

and

$H02O:log?(pT1-pT)-log?(pR1-pR)≥δ?????????vs.?????????Ha2O:log?(pT1-pT)-log?(pR1-pR)<δ.$

If two null hypotheses ( $H01D$ and $H02D$ for the risk difference, $H01R$ and $H02R$ for the log of the relative risk, $H01O$ and $H02D$ for the log of the odds ratio) in two one-sided hypotheses for each metric are rejected at the significance level α, it can be concluded that the original null hypothesis for each metric ( $H0D$ for the risk difference, $H0R$ for the log of the relative risk, $H0O$ for the log of the odds ratio) can be rejected at significance level α. The biosimilar product and the renovator biological product in each case are claimed to be biosimilar.

3. Sample Size Calculation Based on the Approximate and Exact Powers

Section 2 introduces two one-sided hypotheses for the risk difference, the log of the relative risk, and the log of the odds ratio. Test statistics for each hypothesis can be constructed using the central limit theorem, Slutsky’s theorem, and the delta method. For the risk difference, the test statistics are given by

$ZLD=p^T-p^R-δp^T(1-p^T)nT+p^R(1-p^R)nR,?????????ZUD=p^T-p^R+δp^T(1-p^T)nT+p^R(1-p^R)nR.$

For the log of the relative risk, the test statistics are given by

$ZLR=log?(p^T)-log?(p^R)-δ1-p^TnTp^T+1-p^RnRp^R,?????????ZUR=log?(p^T)-log?(p^R)+δ1-p^TnTp^T+1-p^RnRp^R.$

For the log of the odds ratio, the test statistics are given by

$ZLO=log?(P^T1-P^T)-log?(P^R1-P^R)-δ1nTP^T(1-P^T)+1nRP^R(1-P^R),?????????ZUO=log?(P^T1-P^T)-log?(P^R1-P^R)+δ1nTP^T(1-P^T)+1nRP^R(1-P^R).$

For the risk difference, both $H01D$ and $H02D$ are rejected at the significance level α if $ZLD<-zα$ and $ZUD>zα$. Similar conclusions can be drawn for the log of the relative risk and the log of the odds ratio using ( $ZLR,ZUR$) and ( $ZLO,ZUO$), respectively.

Kang and Kim (2014) showed that, under the alternative hypothesis Ha : |pT ? pR| < δ, the power of the test for the risk difference is given by

$P(ZLD<-zα?and?ZUD>zα?Ha)$$=P(ZLD<-zα?Ha)+P(ZUD>zα?Ha)-P(ZLD<-zα?????????or?????????ZUD>zα?Ha)=P(ZLD<-zα?Ha)+P(ZUD>zα?Ha)-[1-P(ZLD≥-zα?????????and?????????ZUD≤zα?Ha)]≤P(ZLD<-zα?Ha)+P(ZUD>zα?Ha)-1$$=Ψ(δ-(pT-pR)pT(1-pT)nT+pR(1-pR)nR-zα)+Ψ(δ+(pT-pR)pT(1-pT)nT+pR(1-pR)nR-zα)-1≥2Ψ(δ-?pT-pR?pT(1-pT)nT+pR(1-pR)nR-zα)-1,$

where Ψ is the cumulative distribution function of the standard normal distribution. The powers in (3.1) and (3.3) are exact and approximate powers, respectively. An advantage of the approximate power is that a closed form of the sample size calculation can be obtained. The sample size needed to achieve power 1 ? β based on the approximate power can be obtained by solving the following equation.

$1-β=2Ψ(δ-?pT-pR?pT(1-pT)nT+pR(1-pR)nR-zα)-1.$

Then we have

$zβ/2=δ-?pT-pR?pT(1-pT)nT+pR(1-pR)nR-zα.$

Therefore, the sample size to achieve power 1 ? β based on the approximate power to test the hypothesis in (2.1) is

$nT=(zα+zβ2)2[pT(1-pT)k+pR(1-pR)1](δ-?pT-pR?)2,?????????nT=knR,$

where k is an allocation ratio. The sample size calculation formula in (3.4) can be found in Chow et al. (2003, p.89). Similarly, the sample size to test the hypothesis in (2.2) is

$nT=(zα+zβ2)2[1-pTkpT+1-pRpR](δ-?log?(pT/pR)?)2,?????????nT=knR$

and the sample size to test the hypothesis in (2.3) is

$nT=(zα+zβ2)2[1kpT(1-pT)+1pR(1-pR)](δ-?log?(pT/(1-pT)pR/(1-pR))?)2,?????????nT=knR.$

Wang et al. (2002) obtained the sample size calculation formula in (3.6).

4. Comparison of the Exact and Approximate Power

The closed forms of the sample size calculation formulae based on the approximate power in equivalence trials for binary endpoints were derived in Section 3 and given by (3.4), (3.5), and (3.6). However, the approximate power might be smaller than the exact power because the two inequalities in (3.2) and (3.3) are used to derive the approximate power. Hence, it is important to compare the exact power obtained from (3.1) and the approximate power calculated from (3.3). Both the exact and approximate powers were calculated numerically with R code based on (3.1) and (3.3) as presented in Tables 1?3 (the R code is available from the authors upon request). In all cases, the exact powers are always greater than the approximate powers.

Figure 1 is a graphical representation of the differences between the two powers for the risk difference when α = 5%, n1 = n2 = 100, and δ = 0.2. As the value of p1 ? p2 increases, the differences between the two curves also increase and means that the accuracy of the approximate power drops rapidly. When the value of p1 ? p2 is greater than 0.12, the approximate power drops below zero, which is unacceptable because powers should be positive. Figures 2 and 3 show similar patterns of differences between two powers for the relative risk and the odds ratio.

In order to investigate how many sample size differences are produced by two different powers, the R code was made to compute sample sizes based on exact and approximate powers. Tables 4?6 display sample sizes needed to achieve 80% and 90% power using two different powers for risk difference, the log of the relative risk, and the odds ratio log, respectively. Sample sizes based on approximate powers are greater than those based on exact powers in all investigated cases. For example, when the risk difference is used for β = 0.2, δ = 0.2, p1 = 0.2, and p2 = 0.1, the sample size based on the approximate power is 215, but the sample size based on the exact power is 155. The two powers produce a difference for 60 patients which may lead to substantial extra costs and ethical concerns.

Kang and Kim (2014) discovered an interesting phenomenon that the sample sizes needed to achieve 80% approximate power are the same as those needed to achieve 90% exact power for a continuous endpoint. Similar phenomena are also observed for a binary endpoint. Such phenomena occur in 34 of 75 cases in Tables 4?6. For example, such an event occurs when δ = 0.2, p1 = 0.18, and p2 = 0.1 in Table 4 (nT = nR = 142). In Kang and Kim (2014) Theorem 1 for a continuous endpoint explains why such phenomena occur. A similar theorem can be derived for a binary endpoint as follows.

### Theorem 1

Let nT = nR and

$w=zα-[pT-pR]+δpT(1-pT)nT+pR(1-pR)nR,?????????for?the?risk?difference,w=zα-[log(pT)-log(pR)]+δ1-pTnTpT+1-pRnRpR,?????????for?the?log?of?the?relative?risk,w=zα-log?(PT1-PT)-log?(PR1-PR)+δ1nTPT(1-PT)+1nRPR(1-PR),?????????for?the?log?of?the?odds?ratio.$

When w is so small that Ψ(w) is negligible, the exact power with the sample size to achieve 1 ? β approximate power is actually 1 ? β/2.

Proof

The proof of this theorem is the same as Theorem 1 in Kang and Kim (2014).

Some cases in which the phenomenon described in Theorem 1 occurs were chosen from Tables 4?6, and the values of w were examined (Table 7). All values of w in Table 7 are small and negligible.

5. Conclusion

In this paper, we studied the accuracy of sample size calculation formulae based on the approximate power for binary endpoints in equivalence trials. The risk difference, the log of the relative risk, and the log of the odds ratio were investigated. Formulae were very conservative because the two inequalities derived the closed form of the sample size calculation based on approximate power. In many practical cases, equivalence trials are planned to achieve 80% power. However, this paper shows that the sample sizes to achieve 80% approximate power often have 90% exact power. Therefore, sample size calculation based on the approximate power may produce unnecessary costs and ethical concerns.

This paper proposes that sample sizes for binary endpoints in equivalence trials should be calculated based on the exact power. The R code to calculate the sample sizes based on the exact power is available from the authors upon request.

Figures
Fig. 1. Comparison of exact and approximate power (risk difference) (p1 = 0.1?0.3, p2 = 0.1, δ = 0.2, α = 0.05, n1 = n2 = 100).
Fig. 2. Comparison of exact and approximate power (relative risk) (p1 = 0.4?0.8, p2 = 0.4, δ = 0.6, α = 0.05, n1 = n2 = 100).
Fig. 3. Comparison of exact and approximate power (odd ratio) (p1 = 0.4?0.7, p2 = 0.4, δ = 1, α = 0.05, n1 = n2 = 100).
TABLES

### Table 1

Risk difference: the exact and approximate powers (α = 0.05)

n1 = n2δp1p2p1 ? p2ExactApprox
1000.10.0520.050.0020.88270.8677

0.10.0540.050.0040.87340.8422

0.10.0560.050.0060.86250.8139

0.10.0580.050.0080.85000.7827

0.10.0600.050.0100.83580.7487

1000.20.1200.10.0200.99190.9847

0.20.1400.10.0400.96720.9347

0.20.1600.10.0600.90490.8100

0.20.1800.10.0800.79300.5861

0.20.2000.10.1000.63880.2775

1000.250.2200.20.0200.98940.9812

0.250.2400.20.0400.97360.9481

0.250.2600.20.0600.93990.8802

0.250.2800.20.0800.88140.7629

0.250.3000.20.1000.79420.5884

1000.250.3200.30.0200.96290.9389

0.250.3400.30.0400.93550.8768

0.250.3600.30.0600.88720.7769

0.250.3800.30.0800.81590.6329

0.250.4000.30.1000.72260.4456

1000.30.4300.40.0300.98620.9744

0.30.4600.40.0600.96300.9264

0.30.4900.40.0900.91230.8247

0.30.5200.40.1200.82320.6464

0.30.5500.40.1500.69270.3854

### Table 2

Relative risk: the exact and approximate powers (α = 0.05)

n1 = n2δp1p2p1/p2ExactApprox
1000.80.320.31.0670.95980.9340

0.80.340.31.1330.94500.8946

0.80.360.31.2000.91880.8389

0.80.380.31.2670.88190.7642

0.80.400.31.3330.83440.6689

1000.60.420.41.0500.93080.8910

0.60.440.41.1000.91140.8343

0.60.460.41.1500.87690.7579

0.60.480.41.2000.82930.6598

0.60.500.41.2500.76970.5398

1000.50.520.51.0400.94090.9066

0.50.540.51.0800.92360.8568

0.50.560.51.1200.89250.7882

0.50.580.51.1600.84860.6983

0.50.600.51.2000.79260.5854

1000.40.620.61.0330.93080.8907

0.40.640.61.0670.91090.8327

0.40.660.61.1000.87470.7531

0.40.680.61.1330.82390.6488

0.40.700.61.1670.75930.5188

1000.40.720.71.0290.99220.9864

0.40.740.71.0570.98770.9759

0.40.760.71.0860.97920.9585

0.40.780.71.1140.96530.9307

0.40.800.71.1430.94410.8882

### Table 3

Odd ratio: the exact and approximate powers (α = 0.05)

n1 = n2δp1p2$p1/(1-p1)p2/(1-p2)$ExactApprox
1001.00.420.41.0860.92180.8776

1.00.430.41.1320.90860.8406

1.00.440.41.1790.88990.7956

1.00.450.41.2270.86570.7419

1.00.460.41.2780.83610.6791

1001.00.520.51.0830.93100.8918

1.00.530.51.1280.91790.8566

1.00.540.51.1740.89930.8128

1.00.550.51.2220.87500.7596

1.00.560.51.2730.84490.6962

1001.00.620.61.0880.91670.8698

1.00.630.61.1350.89990.8255

1.00.640.61.1850.87620.7703

1.00.650.61.2380.84530.7029

1.00.660.61.2940.80710.6225

1001.20.730.71.1590.95250.9131

1.20.740.71.2200.93410.8735

1.20.750.71.2860.90830.8201

1.20.760.71.3570.87390.7500

1.20.770.71.4350.82970.6608

1001.40.820.81.1390.96480.9391

1.40.830.81.2210.94670.8996

1.40.840.81.3130.91780.8397

1.40.850.81.4170.87500.7526

1.40.860.81.5360.81520.6320

### Table 4

Risk difference: sample size calculations based on exact and approximate powers (α = 0.05)

δp1p2p1 ? p2PowerExactApproxPowerExactApprox
0.10.0520.050.00280%848790%105110
0.10.0540.050.004
8592
108116
0.10.0560.050.006
8898
111123
0.10.0580.050.008
90104
115131
0.10.0600.050.010
93110
119139

0.20.1200.10.02080%445290%5666
0.20.1400.10.040
5471
7289
0.20.1600.10.060
7299
99124
0.20.1800.10.080
103142
142179
0.20.2000.10.100
155215
215271

0.250.2200.20.02080%475490%6068
0.250.2400.20.040
5267
6985
0.250.2600.20.060
6284
84106
0.250.2800.20.080
78108
108136
0.250.3000.20.100
102141
141178

0.250.3200.30.02080%617090%7788
0.250.3400.30.040
6685
87107
0.250.3600.30.060
77105
105133
0.250.3800.30.080
96133
133167
0.250.4000.30.100
124172
172217

0.30.4300.40.03080%485790%6273
0.30.4600.40.060
5573
7492
0.30.4900.40.090
6996
96121
0.30.5200.40.120
94130
130164
0.30.5500.40.150
134186
186235

### Table 5

Relative risk: sample size calculations based on exact and approximate powers (α = 0.05)

δp1p2p1/p2PowerExactApproxPowerExactApprox
0.80.320.31.06780%627190%7990
0.80.340.31.133
6481
83102
0.80.360.31.200
6993
93117
0.80.380.31.267
78107
107136
0.80.400.31.333
91126
126159

0.60.420.41.05080%718290%90103
0.60.440.41.100
7394
96118
0.60.460.41.150
80109
109137
0.60.480.41.200
92127
127161
0.60.500.41.250
109151
151191

0.50.520.51.04080%687890%8799
0.50.540.51.080
7089
92112
0.50.560.51.120
76103
103130
0.50.580.51.160
87120
120151
0.50.600.51.200
103142
142179

0.40.620.61.03380%718290%90103
0.40.640.61.067
7494
97119
0.40.660.61.100
81110
110138
0.40.680.61.133
94129
129163
0.40.700.61.167
113156
156197

0.40.720.71.02980%455190%5764
0.40.740.71.057
4657
5972
0.40.760.71.086
4864
6480
0.40.780.71.114
5372
7291
0.40.800.71.143
6082
82104

### Table 6

Odd ratio: sample size calculations based on exact and approximate powers (α = 0.05)

δp1p2$p1/(1-p1)p2/(1-p2)$PowerExactApproxPowerExactApprox
1.00.420.41.08680%738590%93107
1.00.430.41.132
7592
98117
1.00.440.41.179
79101
104128
1.00.450.41.227
84112
113141
1.00.460.41.278
91124
124156

1.00.520.51.08380%718290%90103
1.00.530.51.128
7389
94113
1.00.540.51.174
7798
101124
1.00.550.51.222
82108
109137
1.00.560.51.273
89120
121152

1.00.620.61.08880%758690%95109
1.00.630.61.135
7895
101121
1.00.640.61.185
82106
109134
1.00.650.61.238
89119
120150
1.00.660.61.294
99135
135170

1.20.730.71.15980%637790%8197
1.20.740.71.220
6786
88108
1.20.750.71.286
7297
97122
1.20.760.71.357
81110
110139
1.20.770.71.435
92127
127161

1.40.820.81.13980%597090%7688
1.40.830.81.221
6480
83101
1.40.840.81.313
7093
94117
1.40.850.81.417
81110
110138
1.40.860.81.536
96133
133168

### Table 7

Further investigation on Theorem 1 (α = 0.05)

Metricδp1p2Sample size with 80% approx powerSample size with 90% exact powerw
Risk difference0.20.180.101421429.9519 × 10?8
Risk difference0.20.200.102152154.2490 × 10?13
Risk difference0.250.280.201081082.4720 × 10?5
Risk difference0.250.300.201411411.0651 × 10?7

Relative risk0.80.380.301071079.2356 × 10?5
Relative risk0.80.400.301261262.2052 × 10?6
Relative risk0.60.480.401271276.1416 × 10?5
Relative risk0.60.500.401511511.0050 × 10?6

Odd ratio1.20.760.701101105.0446 × 10?4
Odd ratio1.20.770.701271277.0076 × 10?5
Odd ratio1.40.850.801101105.9863 × 10?4
Odd ratio1.40.860.801331335.1409 × 10?5

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