Sample Size Considerations in Clinical Trials when Comparing Two Interventions using Multiple Co-Primary Binary Relative Risk Contrasts

Abstract

The effects of interventions are multi-dimensional. Use of more than one primary endpoint offers an attractive design feature in clinical trials as they capture more complete characterization of the effects of an intervention and provide more informative intervention comparisons. For these reasons, multiple primary endpoints have become a common design feature in many disease areas such as oncology, infectious disease, and cardiovascular disease. More specifically in medical product development, multiple endpoints are utilized as co-primary to evaluate the effect of the new interventions. Although methodologies to address continuous co-primary endpoints are well-developed, methodologies for binary endpoints are limited. In this paper, we describe power and sample size determination for clinical trials with multiple correlated binary endpoints, when relative risks are evaluated as co-primary. We consider a scenario where the objective is to evaluate evidence for superiority of a test intervention compared with a control intervention, for all of the relative risks. We discuss the normal approximation methods for power and sample size calculations and evaluate how the required sample size, power and Type I error vary as a function of the correlations among the endpoints. Also we discuss a simple, but conservative procedure for appropriate sample size calculation. We then extend the methods allowing for interim monitoring using group-sequential methods.

1. Introduction

Traditionally in clinical trials, one important and clinically relevant outcome is selected as the primary endpoint. This endpoint is then used as the basis for the trial design including sample size determination, interim monitoring, final analyses, and the reporting of trial results. However, many recent clinical trials have utilized more than one endpoint as co-primary. “Co-primary” in this setting means that the trial is designed to evaluate if the new intervention is superior to the control on all endpoints.

The need for new approaches to the design and analysis of clinical trials with co-primary endpoints has been noted. When designing the trial to evaluate the joint effects on all of the endpoints, no adjustment is needed to control the Type I error rate if each endpoint is tested at the common prespecified significance level. However, the Type II error rate increases as the number of endpoints to be evaluated increases. Thus sample size adjustment is needed to maintain the overall power for detecting the joint effects on all of the endpoints, often resulting in large and impractical sample sizes. In order to provide a more practical sample size, methods for clinical trials with co-primary endpoints have been discussed for fixed sample size designs. Methodologies to address multiple co-primary continuous endpoints are well-developed (Chuang-Stein et al., 2007; Dmitrienko et al., 2010; Eaton and Muirhead, 2007; Hung and Wang, 2009; Julious and McIntyre, 2012; Kordzakhia et al., 2010; Offen et al., 2007; Senn and Bretz, 2007; Sozu et al., 2006; Sugimoto et al., 2012; Xiong et al., 2005). When evaluating relative risks with time-to-event outcomes, Hamasaki et al. (2013) and Sugimoto et al. (2013) have developed methods for sizing clinical trials, focusing on the hazard ratio and logrank test statistics. However methodology for multiple binary endpoints is limited (Song, 2009; Sozu at al., 2010, 2011, 2015; Xu and Yu, 2013).

The lack of availability of appropriate methodology for multiple binary endpoints is problematic since clinical trials are often conducted with the objective of comparing the effect of a test intervention to that of a control intervention based on several binary outcomes. For example, PLACIDE is a randomized double-blinded, parallel group, placebo-controlled clinical trial evaluating lactobacilli and bifidobacteria in the prevention of antibiotic-associated diarrhea in older people admitted to hospital (Allen et al., 2012; Allen et al., 2013). The trial was designed to demonstrate that the administration of a probiotic comprising two strains of lactobacilli and two strains of bifidobacteria alongside antibiotic treatment, prevents antibiotic associated diarrhea. The co-primary outcomes were (1) the occurrence of antibiotic-associated diarrhoea (AAD) within 8 weeks and (2) the occurrence of C difficile diarrhoea (CDD) within 12 weeks of randomization. Another example can be seen in irritable bowel syndrome (IBS), one of the most common gastrointestinal disorders characterized by symptoms of abdominal pain, discomfort, and altered bowel function (American College of Gastroenterology, 2013; Grundmann and Yoon, 2010). The comparison of the interventions to treat IBS is based on the proportions of participants with: (1) adequate relief of abdominal pain and discomfort, and (2) improvements in urgency, stool frequency, and stool consistency. The U.S. Food and Drug Administration (FDA) recommends the use of two endpoints for assessing IBS signs and symptoms: (1) pain intensity, and (2) stool frequency (Food and Drug Administration, 2012). Meanwhile the Committee for Medicinal Products for Human Use (2013) also recommends the use of two endpoints for assessing IBS signs and symptoms: (1) global assessment of symptoms, and (2) assessment of symptoms of abdominal discomfort/pain. Offen et al. (2007) provides other examples.

Methodology for sizing trials when evaluating the absolute difference in proportions can be found in Sozu et al. (2010). The objective of this paper is to describe methodology for the power and sample size determination for clinical trials with multiple co-primary binary endpoints when evaluating relative risks contrasts. Methodology for the odds ratio is briefly discussed in the Appendix.

The paper is structured as follows: in Section 2, we describe a normal approximation method for sample size determination that incorporates the correlations among the endpoints into the calculations. We then evaluate the practical utility of the normal approximation method via Monte-Carlo simulation. In Section 3, we discuss a conservative procedure for sample size calculation by treating the endpoints as if they are not correlated. In Section 4, we extend the methodology to a group-sequential setting. In Section 5, we summarize the findings.

2. Methods for calculating the sample size with relative risks

2.1 Statistical Settings

Consider a randomized clinical trial comparing two interventions with K binary endpoints being evaluated as co-primary (K ≥ 2). There are N total participants, with rN participants assigned to the test (T) and (1 − r)N participants to the control (C) intervention groups, where r is the allocation ratio (0 < r < 1). Let the rN responses to the test intervention be denoted by Y_Tki and the (1 − r)N responses to the control intervention by Y_Ckj (k = 1, …, K; i = 1, …, rN; j = 1, …, (1 − r)N). Suppose that Y_Tki and Y_Ckj are independently distributed as Bernoulli distributions with probabilities of success p_Tk and p_Ck, but the K binary endpoints are correlated with common pairwise correlations of ρ^(kk′) = corr[Y_Tki, Y_Tk′i] = corr[Y_Ckj, Y_Ck′j](1 ≤ k < k′ ≤ K). Let $Y_{\mathrm{T}k}={\sum }_{i=1}^{rN}{Y}_{\mathrm{T}ki}$ and $Y_{\mathrm{C}k}={\sum }_{j=1}^{(1-r)N}{Y}_{\mathrm{C}kj}$ denote the number of successes under the test and the control interventions respectively. Then they are distributed as binomial distributions, i.e., Y_Tk ~ B(rN, p_Tk) and Y_Ck ~ B((1 −r)N, p_Ck).

We now have the K log-transformed (observed) relative risks logR̂_k = log(p̂_Tk/p̂_Ck), where p̂_Tk = (rN)⁻¹Y_Tk and p̂_Ck = ((1 −r)N)⁻¹Y_Ck. For large samples, by applying the delta-method, we assume that the joint distribution of (logR̂₁, …, logR̂_K) is approximately K-variate normal with mean vector μ = (logR₁, …, logR_K)^T and covariance matrix Σ determined by

$$\{\begin{array}{cc}\hfill {\sigma }_k^2=\frac{1}{N}(\frac{1-R_k{p}_{\mathrm{C}k}}{r{R}_k{p}_{Ck}}+\frac{1-p_{\mathrm{C}k}}{(1-r)p_{\mathrm{C}k}}),\hfill & \hfill k=k\prime ,\hfill \\ \hfill {\sigma }_{kk\prime }=\frac{{\rho }^{(kk\prime )}}{N\sqrt{p_{\mathrm{C}k}{p}_{\mathrm{C}k\prime }}}(\frac{\sqrt{(1-R_k{p}_{\mathrm{C}k})(1-R_{k\prime }{p}_{\mathrm{C}k\prime })}}{r\sqrt{R_k{R}_{k\prime }}}+\frac{\sqrt{(1-p_{\mathrm{C}k})(1-p_{\mathrm{C}k\prime })}}{1-r}),\hfill & \hfill k\ne k\prime ,\hfill \end{array}$$

where R_k = p_Tk/p_Ck. Since it is assumed that the correlations among the endpoints are common between the two intervention groups, for large samples, ${\rho }_R^{(kk\prime )}=\text{corr}[\text{log}{\hat{R}}_k,\text{log}{\hat{R}}_{k\prime }]$ is approximately given by ${\rho }_R^{(kk\prime )}={\rho }^{(kk\prime )}A/\sqrt{A^2+r(1-r)B^2}$, where $A=(1-r)\sqrt{(1-R_k{p}_{\mathrm{C}k})(1-R_{k\prime }{p}_{\mathrm{C}k\prime })}+r\sqrt{R_k(1-p_{\mathrm{C}k})R_{k\prime }(1-p_{\mathrm{C}k\prime })}$ and $B=\sqrt{R_k(1-p_{Ck})(1-R_{k\prime }{p}_{\mathrm{C}k\prime })}-\sqrt{R_{k\prime }(1-p_{\mathrm{C}k\prime })(1-R_k{p}_{\mathrm{C}k})}$. It follows that $|{\rho }_R^{(kk\prime )}|\le {\rho }^{(kk\prime )}$. A detailed calculation for ${\rho }_R^{(kk\prime )}$ is provided in the Appendix.

There are three measures of the correlation ρ^(kk′), i.e., the correlation coefficient of a multivariate Bernoulli distribution, the odds ratio, and the correlation coefficient of a latent normal distribution (Pearson 1900; Bahadur, 1961; Dale 1986; Le Cessie and van Houwelingen, 1994; Prentice, 1988). We consider the correlation from a bivariate Bernoulli distribution because it is intuitively attractive and interpretable even though the range of ρ^(kk′) is restricted, depending on the marginal probabilities (Prentice, 1988). However, the results for the power and sample size determination are provided in a general form and can be straightforwardly applicable to the other two correlation measures.

2.2 Power and sample sizes calculations

When evaluating the joint effects for multiple correlated binary relative risks, e.g., to establish a risk reduction in the test intervention compared with the control intervention, the hypotheses H₀: logR_k ≥ 0 for at least one k versus H₁: logR_k < 0 for all k are tested using the test statistics

$$Z_k=\sqrt{r(1-r)N}\text{log}{\hat{R}}_k/\sqrt{(1-{\hat{\overline{p}}}_k)/{\hat{\overline{p}}}_k},$$ (1)

where ${\hat{\overline{p}}}_k=r{\hat{p}}_{\mathrm{T}k}+(1-r){\hat{p}}_{\mathrm{C}k}$. If each endpoint is evaluated using a one-sided test at the same significance level of a, then the rejection regions of H₀ are [{Z₁ < −z_α} ∩ … ∩ {Z_K < −z_α}], where z_α is the (1 − α)th percentile of the standard normal distribution. Therefore, for the true relative risks R_k, for large samples, using straightforward algebra and substituting population parameters for estimates, provides the approximate overall power:

$$1-\beta =\text{Pr}[\underset{k=1}{\overset{K}{\cap }}\{Z_k<-z_{\alpha }\}|{\mathrm{H}}_1]\approx \text{Pr}[\underset{k=1}{\overset{K}{\cap }}\{Z_k^{*}<-c_k^{*}\}\left|{\mathrm{H}}_1\right],$$ (2)

which is referred to as “conjunctive (or complete) power” (Senn and Bretz, 2007), where

$$Z_k^{*}=(\text{log}{\hat{R}}_k-\text{log}{R}_k)/\sqrt{\frac{1}{N}(\frac{1-R_k{p}_{\mathrm{C}k}}{r{R}_k{p}_{\mathrm{C}k}}+\frac{1-p_{\mathrm{C}k}}{(1-r)p_{\mathrm{C}k}})}\text{and}{c}_k^{*}=(z_{\alpha }\sqrt{\frac{1-{\overline{p}}_k}{N{\overline{p}}_k}(\frac{1}{r}+\frac{1}{1-r})}+\text{log}{R}_k)/\sqrt{\frac{1}{N}(\frac{1-R_k{p}_{\mathrm{C}k}}{r{R}_k{p}_{\mathrm{C}k}}+\frac{1-p_{\mathrm{C}k}}{(1-r)p_{\mathrm{C}k}})}$$

with p̄_k = rp_Tk + (1 − r)p_Ck. The overall power (2) can be evaluated by using the distribution function of the K-variate normal distribution with zero mean vector and correlation matrix ${\mathit{\rho }}_R=\{{\rho }_R^{(kk\prime )}\}$, i.e., ${\mathrm{\Phi }}_K(c_1^{*},\dots ,c_K^{*}|{\mathit{\rho }}_R)$. So that the total required sample size for detecting the joint reduction in all relative risks with the overall power 1 − β at the significance level α, using the normal approximation, N_AN is the smallest integer satisfying Equation (2). In addition, we can simplify Equation (2) using the pooled variance estimate under the null hypothesis or the unpooled variance estimate under the alternative hypothesis. When using the unpooled variance estimate, the total sample size N_UP is the smallest integer satisfying the power function 1 −β, where

$$c_k^{**}=z_{\alpha }+\text{log}{R}_k/\sqrt{\frac{1}{N}(\frac{1-R_k{p}_{\mathrm{C}k}}{r{R}_k{p}_{\mathrm{C}k}}+\frac{1-p_{\mathrm{C}k}}{(1-r)p_{\mathrm{C}k}})}.$$

Similarly, when using the pooled variance estimate, the total sample size N_PL is the smallest integer satisfying the power function 1 −β, where

$$c_k^{\#}=z_{\alpha }+\text{log}{R}_k/\sqrt{\frac{1-{\overline{p}}_k}{N{\overline{p}}_k}(\frac{1}{r}+\frac{1}{1-r})}.$$

The relationship of these three sample sizes is N_PL ≤ N_AN ≤ N_UP as the relationship between the pooled and unpooled variances is given by, assuming p_T < p_C and 0 < r < 1,

$$\frac{1-{\overline{p}}_k}{{\overline{p}}_k}(\frac{1}{r}+\frac{1}{1-r})\le \frac{1-R_k{p}_{\mathrm{C}k}}{r{R}_k{p}_{\mathrm{C}k}}+\frac{1-p_{\mathrm{C}k}}{(1-r)p_{\mathrm{C}k}}.$$

Note that the simplified formula for the sample size with a pooled variance of N_PL results in the smallest sample size, but our simulation studies suggest that the methodology may not achieve the targeted power in many practical situations as given in supplemental document. Thus it is not recommended for use in practice. No closed form expression is available for calculating N_AN (or N_UP, N_PL) and an iterative procedure is required to solve for the power. We provide a practical algorithm for calculating the sample size in Appendix.

Figure 3.

Behaviors of the overall powers as a function of the effect size(s) for a given sample size in the case of two co-primary relative risks (A) and three co-primary relative risks (B). For the case of two co-primary relative risks, the sample size (equally sized groups; r =0.5) was calculated to detect a reduction in the relative risk R₁ with the individual power of (0.8)^1/2 =0.894 for a one-sided test at the significance level of α =0.025. For the case of three co-primary relative risks, the sample size (equally sized groups: r =0.5) was calculated to detect a reduction in the relative risk R₁ with the individual power of (0.8)^1/3 =0.928 for a one-sided test at the significance level of α =0.025.

There are more direct ways of calculating the sample size without using a normal approximation (e.g., see Sozu et al. (2010)). However such methods are computationally difficult and often impractical particularly for a large number of outcomes. On the other hand, the normal approximation discussed here may not work well with extremely small event rates or with small sample sizes from the analogy of discussions on a single binary endpoint. We evaluate the utility of using the normal approximation in Section 2.3.

2.3 Evaluation of the methodology utility

In order to evaluate the utility of the normal approximation described in the previous sections, the Type I error rate and power under given sample sizes N_AN were evaluated using Monte-Carlo simulation. Consider a clinical trial with two interventions being compared using two relative risks as co-primary contrasts. The required total sample size N_AN is calculated to detect the joint effect on both relative risks R₁ and R₂ with p_C1 and p_C2 with the desired overall power of 1 − β =0.80 for a one-sided test at the significance level of α =0.025. The total numbers of 1,000,000 and 100,000 datasets with a total sample size of N_AN for each parameter configuration, are generated for the assessments of the Type I error rate and the power respectively. The sample sizes calculated using the normal approximation are compared with the sample sizes derived by the Monte-Carlo simulation-based approach, where 100,000 replications were selected for the power evaluation. Bivariate Bernoulli data for Monte-Carlo simulation were generated using the method described in Emrich and Piedmonte (1991).

Figure 1 displays how the empirical powers and Type I error rates for the tests based on the test statistics (1) with common correlation ρ¹² =0.0, 0.3, 0.5, 0.8, and 0.99, in the cases of R₁ =0.8, 0.7, 0.6, 0.5 and 0.4 with p_C1 =0.5, 0.2, 0.10 and 0.05, and R₁ = R₂ = R and p_C1 = p_C2 = p_C, for K = 2, under a given sample size for detecting a joint effect on the two relative risks (using the above parameter configuration) with the desired overall power of 1 − β =0.80 for a one-sided test at the significance level of α =0.025. For evaluation of the Type I error rates, we consider two null hypotheses: (i) H₀: logR₁ = 0 and logR₂ = 0, and (ii) H₀: logR₁ < 0, but logR₂ = 0.

Figure 1.

Behaviors of the empirical powers and Type I error rates for the tests based on the test statistics (1), under a given sample size N_AN, with correlation ρ¹² =0.0, 0.3, 0.5, 0.8 and 0.99, in the cases of R₁ =0.8, 0.7, 0.6, 0.5 and 0.4 with p_C1 =0.5, 0.2, 0.10 and 0.05, R₁ = R₂ = R and p_C1 = p_C2 = p_C, and equally sized groups (r = 0.5) for K =2. The given sample size of N_AN was calculated to detect the join effect on all of the endpoints for each parameter setting with the desired overall power of 1 − β =0.80 for a one-sided test at the significance level of α =0.025. For evaluation of the Type I error rates, two situation were considered, i.e. (i) H₀: logR₁ =0 and logR₂ =0 and (ii) H₀: logR₁ =0, but logR₂ <0.

Use of N_AN achieves the targeted power of 1 − β =0.80, but is larger than the targeted power as R decreases and p_C increases, especially when R ≤ 0.5 and p_C =0.5. When R ≤ 0.5 and p_C =0.5, the required sample size is less than 154 (77 per group), where the sample size was calculated to detect a joint effect in R = 0.5 and p_C =0.5 with the targeted power of 0.80 at the significance level of 0.025 for a one-sided test, assuming zero correlation ρ¹² =0.0. The Type I error rate increases as the correlation goes toward one, but becomes greater than the nominal significance level of α =0.025 as R decreases and p_C increases, especially when R ≤0.5 and p_C =0.5 in both hypothesis settings (i) and (ii). As shown in the supplemental document, the empirical power for N_UP is larger than the targeted power, while the empirical power for N_PL is less than the targeted power. However, the Type I error rates for N_UP and N_PL behave similarly to those seen for N_AN. They are greater than the nominal significance level of α =0.025 when R is small and p_C is large, especially when R ≤0.5 and p_C =0.5.

Figure 2 displays the ratio of the required sample sizes calculated using the normal approximation to that resulting from the Monte-Carlo simulation-based approach, with a common correlation of ρ¹² =0.0, 0.3, 0.5, 0.8 and 0.99, in the cases of R₁ =0.8, 0.7, 0.6, 0.5 and 0.4 with p_C1 =0.5, 0.2, 0.10 and 0.05, and R₁ = R₂ = R and p_C1 = p_C2 = p_C, for K = 2, with the desired overall power of 1 − β =0.80 for a one-sided test at the significance level of α =0.025. The ratio is about one when R =0.8 and 0.7. The ratio is slightly larger than one when R =0.6, however, it is much larger than one when R ≤0.5. The most obvious difference occurs when p_C =0.5. The absolute difference in the two sample sizes per intervention group lies between 0 and 32.

Figure 2.

Behaviors of the ratio of the required sample size calculated by the normal approximation (N_AN) to that by the Monte-Carlo simulation-based approach, with correlation ρ¹² =0.0, 0.3, 0.5, 0.8 and 0.99 in the cases of R₁ =0.8, 0.7, 0.6, 0.5 and 0.4 with P_C1 =0.5, 0.2, 0.10 and 0.05, R₁ = R₂ = R and p_C1 = p_C2 = p_C, and equally sized groups (r =0.5) for K =2, with the desired overall power of 1 − β =0.80 for a one-sided test at the significance level of α =0.025.

2.4 Example

We illustrate the methodology with an example from the PLACIDE study (Allen et al., 2012; Allen et al., 2013) described in the Introduction. Recall that the study was designed to evaluate if the administration of a probiotic comprising two strains of lactobacilli and two strains of bifidobacteria alongside antibiotic treatment prevents antibiotic associated diarrhea. The co-primary outcomes were (1) the occurrence of antibiotic-associated diarrhoea (AAD) within 8 weeks and (2) the occurrence of C difficile diarrhoea (CDD) within 12 weeks of recruitment. The contrast measures for efficacy analysis were the binary relative risks of these two outcomes. The original sample size of 2,478 participants (1,239 in each group) was derived to detect a 50% reduction in CDD in the probiotic group compared with the placebo group, with a desired power of 0.80 using a two-sided test at 5% significance level, assuming CDD frequencies of 4% in placebo group. Allen et al. (2013) also mention that a trial of this size would provide a power of more than 0.99 to detect a 50% reduction in AAD (assuming AAD frequencies of 10% in placebo group) and a power of 0.90 to detect a 25% reduction in AAD (assuming AAD frequencies of 15% in placebo group). However, if the objective was to evaluate the efficacy of the new intervention based on both endpoints of CDD and AAD, then the power for detecting the joint effects for both endpoints was 0.792 (0.8×0.99) when assuming a 50% reduction in AAD frequencies with 10% in placebo group, and 0.72 (0.8×0.9) when assuming a 25% reduction in AAD frequencies with 15% in placebo group, when the correlation between the two endpoints was assumed to be zero. Thus the planned sample size would be insufficient to detect a joint effect with the overall power of 0.80.

Table 1 displays the required sample size for detecting a joint effect for both endpoints with the overall power of 1 − β =0.80 and 0.90 at the one-sided significance level of α =0.025 based on their original assumptions A1) 1. CDD: 50% reduction with CDD frequencies of 4% in placebo and 2. AAD: 50% reduction with ADD frequencies of 10% in placebo, and A2) 1. CDD: 50% reduction with CDD frequencies of 4% in placebo and 2. AAD: 25% reduction with ADD frequencies of 15% in placebo, when a common correlation between two endpoints in the two intervention groups was assumed.

Table 1.

Total sample size (equally sized groups; r =0.5) required for evaluating the joint effects of both relative risks with an overall power of 1 − β =0.80 and 0.90 for a one-sided test at the significance level of α =0.025

Assumption		Overall power	Sample Size	Correlation
				0.0	0.3	0.5	0.8	0.99
A1	1. CDD 50% reduction with CDD frequencies of 4% in placebo	0.80	Normal approximation	2,222	2,212	2,204	2,198	2,198
			(Empirical power)	(0.796)	(0.796)	(0.794)	(0.793)	(0.793)
			Simulation-based	2,260	2,240	2,238	2,234	2,234
			(Empirical power)	(0.802)	(0.800)	(0.801)	(0.802)	(0.802)
	2. AAD 50% reduction with ADD frequencies of 10% in placebo	0.90	Normal approximation	2,978	2,976	2,974	2,972	2,972
			(Empirical power)	(0.902)	(0.898)	(0.900)	(0.900)	(0.900)
			Simulation-based	2,968	2,972	2,974	2,972	2,972
			(Empirical power)	(0.900)	(0.900)	(0.900)	(0.900)	(0.900)
A2	1. CDD: 50% reduction with CDD frequencies of 4% in placebo	0.80	Normal approximation	3,130	3,052	3,014	3,014	3,014
			(Empirical power)	(0.799)	(0.799)	(0.798)	(0.798)	(0.798)
			Simulation-based	3,134	3,060	3,022	3,022	3,022
			(Empirical power)	(0.802)	(0.802)	(0.800)	(0.800)	(0.800)
	2. AAD: 25% reduction with ADD frequencies of 15% in placebo	0.90	Normal approximation	3,940	3,886	3,858	3,858	3,858
			(Empirical power)	(0.902)	(0.900)	(0.900)	(0.900)	(0.900)
			Simulation-based	3,928	3,886	3,856	3,856	3,856
			(Empirical power)	(0.900)	(0.900)	(0.900)	(0.900)	(0.900)

The empirical powers under the given sample size using 100,000 Monte-Carlo trials. Monte-Carlo simulation-based sample size was calculated using 100,000 Monte-Carlo trials.

Based on A1, when ρ¹² =0.0, 2,222 participants are required for providing a power of 0.80 (a conservative sample size). The required total sample size decreases from 2,212 to 2,198 participants as correlation varies from ρ¹² =0.3 to 0.99. Based on A2, 3,130 participants are required when ρ =0 and the required sample size decreases from 3,052 to 3,014 participants as correlation varies from ρ¹² =0.3 to 0.99. The empirical powers under the calculated sample size achieve the desired overall power and there is no notable difference in the required sample size resulting from the normal approximation and the Monte-Carlo simulation-based approach as the maximum difference in total sample size is modest (i.e., A1: 38 and 10 for 1 − β =0.8 and 0.9; A2: 8 and 12 for 1 − β =0.8 and 0.9).

3. A conservative procedure for sample size calculation

We consider a conservative sample size strategy when evaluating significance for all of the relative risks by using a suggestion from Hung and Wang (2009). Consider a scenario where there are K relative risks as co-primary. Assuming an unpooled variance estimate for simplicity, with a common value of $c^{*}=c_1^{*}=\cdots =c_K^{*}$ (i.e., R = R₁ = ⋯ = R_K and p_C = p_C1 = ⋯ = p_CK) and ${\rho }_R^{(12)}=\cdots ={\rho }_R^{(K,K-1)}=0$, if letting γ = 1 − (1 − β)^1/K, then a conservative sample size N_C is

$$N_C\ge \frac{{(z_{\alpha }+z_{\gamma })}^2}{{(\text{log}R)}^2}(\frac{1-R{p}_{\mathrm{C}}}{rR{p}_{\mathrm{C}}}+\frac{1-p_{\mathrm{C}}}{(1-r)p_{\mathrm{C}}})$$

where z_α and z_γ are (1 − α)th and (1 − γ)th percentiles of the standard normal distribution, respectively.

Furthermore, we discuss guidance on when and how the simplified equation can be used. Now calculate a total sample size N_AN (R₁) required to detect the reduction in relative risk R₁, with the individual power (1 − γ) at the significance level of α, assuming ${\rho }_R^{(12)}=\cdots ={\rho }_R^{(K,K-1)}=0$. The overall power 1 − β under N_AN(R₁) is $\beta =\mathrm{\Phi }(c_1^{*})\cdots \mathrm{\Phi }(c_K^{*})$, where $c_1^{*}=z_{\gamma },c_k^{*}=E_k/E_1(z_{\alpha }+z_{\gamma })-z_{\alpha }$ and Φ(·) is the cumulative distribution function of the univariate standard normal distribution. In addition, E_k(k = 1, …,K) is the standardized effect size given by

$$E_k=\text{log}{R}_k/\sqrt{\frac{1-R_k{p}_{\mathrm{C}k}}{r{R}_k{p}_{\mathrm{C}k}}+\frac{1-p_{\mathrm{C}k}}{(1-r)p_{\mathrm{C}k}}}.$$

Therefore, the overall power can be expressed as a function of the ratio of the standardized effect sizes.

Figure 3A illustrates the behavior of the overall power as a function of E₂/E₁ for a given sample size with two co-primary relative risks where the sample size (equally sized groups) was calculated to detect the reduction in relative risk R₁ with the individual power of 0.8^1/2 = 0.894 for a one-sided test at the significance level of α =0.025. The overall power increases toward 0.894 as the ratio of E₂/E₁ increases. In particular when the ratio of E₂/E₁ is greater than 1.639, the overall power reaches 0.894. This is because the individual power for R₂ is very close to one $(\mathrm{\Phi }(c_2^{*})\to 1)$ under the given sample size calculated for R₁.

Figure 3B illustrates the behavior of the overall power as a function of E₂/E₁ and E₃/E₁ for a given sample size with three co-primary relative risks, where the sample size (equally sized groups) was calculated to detect the reduction in relative risk R₁ with the individual power of 0.8^1/3 = 0.928 for a one-sided test at the significance level of α =0.025. The overall power increases toward 0.928 as both of the ratios of E₂/E₁ and E₃/E₁ increase. In particular when E₂/E₁ and E₃/E₁ are greater than 1.618, then the overall power reaches 0.928. This is because both of individual powers for R₂ and R₃ are very close to one $(\mathrm{\Phi }(c_2^{*})\to 1\text{and}\mathrm{\Phi }(c_3^{*})\to 1)$ under the given sample size calculated for R₁. In this situation, the required sample size greatly depends on the smallest reduction. If we observe a large difference among the values of E_k, then we could calculate the conservative sample size $N_C^{\prime }$,

$$N_C^{\prime }\ge \underset{k}{\text{max}}\left(\frac{{(z_{\alpha }+z_{\beta })}^2}{{(\text{log}{R}_k)}^2}\right(\frac{1-R_k{p}_{Ck}}{r{R}_k{p}_{Ck}}+\frac{1-p_{Ck}}{(1-r)p_{Ck}}\left)\right),$$

where z_β is the (1 − β)th percentile of the standard normal distribution.

4. An extension to group-sequential designs

As previously noted, the standard methods for sizing trials with co-primary endpoints often results in large sample sizes due to the conservative nature of the testing procedure even when the correlations among the endpoints is incorporated into the calculation. To improve efficiency, researchers may consider interim analyses to evaluate if the research questions can be answered with fewer trial participants or shorter follow-up. In this section, we extend the methods discussed in Section 2 to a group-sequential setting, allowing for the possibility of stopping a trial early when early evidence is overwhelming. We consider the scenario of a randomized clinical trial comparing two interventions with two binary endpoints being evaluated as co-primary. Suppose that a maximum of L analyses are planned and the two endpoints are analyzed at the same interim timepoint. For more flexible group-sequential designs for clinical trials with co-primary endpoints, please see Asakura et al. (2014, 2015), and Hamasaki et al. (2015).

Let N_l be the cumulative total number of participants at the lth analysis (l = 1, …, L). Hence, up to rN_L and (1 − r)N_L participants are recruited and randomly assigned to the test and the control intervention groups, respectively. We are interested in conducting a hypothesis test to evaluate if the test intervention is superior to the control, i.e., the hypotheses H₀: logR₁ ≥ 0 or logR₂ ≥ 0 versus H₁: logR₁ < 0 and logR₂ < 0. Let (Z_1l, Z_2l) be the statistics for testing the hypotheses at the lth analysis, given by $Z_{kl}=\sqrt{r(1-r)N_l}\text{log}{\hat{R}}_{kl}/\sqrt{(1-{\hat{\overline{p}}}_{kl})/{\hat{\overline{p}}}_{kl}}$, where R̂_kl = p̂_Tkl/p̂_Ckl, ${\hat{\overline{p}}}_{kl}=r{\hat{p}}_{\mathrm{T}kl}+(1-r){\hat{p}}_{\mathrm{C}kl}$, at the lth analysis, with ${\hat{p}}_{\mathrm{T}kl}={(r{N}_l)}^{-1}{\sum }_{i=1}^{r{N}_l}{Y}_{\mathrm{T}ki}$ and ${\hat{p}}_{\mathrm{C}kl}={((1-r)N_l)}^{-1}{\sum }_{j=1}^{(1-r)N_l}{Y}_{\mathrm{C}kj}$. The null hypothesis H₀ can be rejected if and only if superiority is achieved for the two endpoints simultaneously (i.e., at the same interim timepoint of the trial). If superiority is demonstrated on only one endpoint at a particular interim analysis, then the trial continues and the hypothesis testing is repeated for both endpoints until joint significance for the two endpoints is established simultaneously. The stopping rule is formally given as follows:

• At the lth analysis (l = 1, …, L − 1)

  • If Z_1l < −z_1l and Z_2l < −z_2l, then reject H₀ and stop the trial,
  • otherwise, continue to the (l + 1)th analysis,
• at the Lth analysis,

  • if Z_1L < −z_1L and Z_2L < −_2L, then reject H₀,
  • otherwise, do not reject H₀,

where z_1l and z_2l are the critical values, which are constant and selected separately, using any group-sequential methods such as the Lan-DeMets (LD) alpha-spending method (Lan and DeMets, 1983) to control an overall Type I error rate as if they were a single primary endpoint, ignoring the other co-primary endpoint. The power is approximately

$$1-\beta =\text{Pr}\left[{\cup }_{l=1}^L\right\{\{Z_{1l}<-Z_{1l}\}\cap \{Z_{2l}<-Z_{2l}\}\left\}\right|{\mathrm{H}}_1]\simeq \text{Pr}[{\cup }_{l=1}^L\{Z_{1l}^{*}<-c_{1l}^{*}\}\cap \{Z_{2l}^{*}<-c_{2l}^{*}\}|{\mathrm{H}}_1],$$ (3)

where

$$Z_{kl}^{*}=(\text{log}{\hat{R}}_{kl}-\text{log}{R}_k)/\sqrt{\frac{1}{N_l}(\frac{1-R_k{p}_{\mathrm{C}k}}{r{R}_k{p}_{\mathrm{C}k}}+\frac{1-p_{\mathrm{C}k}}{(1-r)p_{\mathrm{C}k}})}\text{and}{c}_{kl}^{*}=(z_{kl}\sqrt{\frac{1-{\overline{p}}_k}{N_l{\overline{p}}_k}(\frac{1}{r}+\frac{1}{1-r})}+\text{log}{R}_k)/\sqrt{\frac{1}{N_l}(\frac{1-R_k{p}_{\mathrm{C}k}}{r{R}_k{p}_{\mathrm{C}k}}+\frac{1-p_{\mathrm{C}k}}{(1-r)p_{\mathrm{C}k}})}.$$

For large samples, we assume that the joint distribution of $(Z_{11}^{*},Z_{21}^{*},\dots ,Z_{1L}^{*},Z_{2L}^{*})$ is approximately 2L multivariate normal with their correlations given by $\text{corr}[Z_{1l}^{*},Z_{2l}^{*}]={\rho }^{(12)}A/\sqrt{A^2+r(1-r)B^2},\text{corr}[Z_{1l}^{*},Z_{1l\prime }^{*}]=\text{corr}[Z_{2l}^{*},Z_{2l\prime }^{*}]=\sqrt{N_l/N_{l\prime }},\text{corr}[Z_{1l}^{*},Z_{2l\prime }^{*}]=\text{corr}[Z_{2l}^{*},Z_{1l\prime }^{*}]=\text{corr}[Z_{1l}^{*},Z_{2l}^{*}]\text{corr}[Z_{1l}^{*},Z_{1l\prime }^{*}]=\text{corr}[Z_{1l}^{*},Z_{2l}^{*}]\text{corr}[Z_{2l}^{*},Z_{2l\prime }^{*}](1\le l\le l\prime \le L)$, where $A=(1-r)\sqrt{(1-R_1{p}_{\mathrm{C}1})(1-R_2{P}_{\mathrm{C}2})}+r\sqrt{R_1(1-p_{\mathrm{C}1})R_2(1-p_{\mathrm{C}2})}$ and $B=\sqrt{R_1(1-p_{\mathrm{C}1})(1-R_2{p}_{\mathrm{C}2})}-\sqrt{R_2(1-R_1{p}_{\mathrm{C}1})(1-p_{\mathrm{C}2})}$

Based on the power (3), the two types of sample size, i.e., the maximum sample size (MSS) and the average sample number (ASN) can be calculated. The MSS is the sample size required for the final analysis to achieve the desired power 1 − β. The MSS is the smallest integer not less than N_L satisfying the power (3) for a group-sequential design at the prespecified R₁, R₂, p_C1, p_C2 and ρ⁽¹²⁾, with Fisher’s information time for the interim analyses N_l/N_L(l = 1, …, L). The ASN is the expected sample size under a specific hypothetical reference. Similarly as in the fixed sample size designs discussed in Section 2, the closed form solutions for the MSS and ASN are not available, and thus an iterative program is required to find these sample sizes. For more details, please see Asakura et al. (2014).

For illustration, consider the PLACIDE study again. Table 2 provides the MSS and ASN, based on the assumption A1. The MSS and ASN were calculated to detect the joint effect in the two endpoints with the overall power of 1 − β =0.80 at the one-sided significance level of α =0.025, where ρ⁽¹²⁾ =0.0, 0.3, 0.5, 0.8 and 0.99, L =2, 3, 4 and 5, and r =0.5. The critical values are determined by the O’Brien–Fleming testing procedure (O’Brien and Fleming, 1979) for both endpoints, with the LD alpha-spending method with equally spaced information time.

Table 2.

The MSS and ASN (equally-sized groups), and expected number of stops for detecting the joint effects for both endpoints based on the assumption A1 from the PLACIDE clinical trial, with the overall power of 1 − β =0.80 at the one-sided significance level of α =0.025. The critical values are determined by the O’Brien-Fleming testing procedure for both endpoints, with the LD alpha-spending method with equally spaced information time.

Correlation	Number of analyses	MSS	ASN	Expected number of stops
0.0	2	2,232	2,099	1.88
	3	2,250	1,950	2.60
	4	2,264	1,894	3.35
	5	2,270	1,858	4.09
0.3	2	2,220	2,056	1.85
	3	2,238	1,921	2.58
	4	2,248	1,859	3.31
	5	2,260	1,827	4.04
0.5	2	2,212	2,030	1.84
	3	2,226	1,899	2.56
	4	2,240	1,839	3.28
	5	2,250	1,805	4.01
0.8	2	2,200	1,997	1.82
	3	2,220	1,879	2.54
	4	2,232	1,816	3.25
	5	2,240	1,780	3.97
0.99	2	2,200	1,993	1.81
	3	2,214	1,872	2.54
	4	2,232	1,813	3.25
	5	2,240	1,777	3.97

Based on A1, i.e., L = 1 and ρ =0.0, the required sample size is 2,222 shown in Table 1. If four interim and one final analyses are planned (i.e., L =5), and conservatively assuming zero correlation between the endpoints, then the MSS is 2,270 and the ASN is 1,858. If the correlation is incorporated into the calculation when ρ⁽¹²⁾ =0.3, 0.5, 0.8, and 0.99, then the MSS are 2,260, 2,250, 2,240 and 2,240 respectively. The ASN are 1,827, 1,805, 1,780 and 1,777 respectively. The MSS for the four testing procedure combinations increases as the number of analyses increases and as the correlation decreases. The ASN decreases as the number of analyses increases and as the correlation increases.

Figure 4 summarizes the probability of rejecting/not rejecting the null hypothesis when ρ⁽¹²⁾ =0.0, 0.3, 0.5, 0.8 and 0.99 and L =2, 3, 4 and 5. The figure illustrates that the method offers the possibility to stop a trial early if early evidence is overwhelming and thus potentially requiring fewer patients than the fixed sample size designs. When ρ⁽¹²⁾ =0.0, it is difficult to reject the null hypothesis at the earlier analyses, but easier later on. On the other hand, as ρ⁽¹²⁾ goes toward one, it is easier to reject the null hypothesis at the earlier analyses.

Figure 4.

The probability of rejecting/not rejecting the null hypothesis when L =2, 3, 4 and 5. The MSS and ASN (equally-sized groups) for detecting the joint effect for both endpoints, with the overall power of 1 − β =0.80 at the one-sided significance level of α =0.025. The critical values are determined by the O’Brien-Fleming testing procedure for both endpoints with the LD alpha-spending method with equally spaced information time.

5. Summary

Traditionally in a clinical trial, a single primary endpoint is selected and is used as the basis for the design, interim data monitoring, final analyses, and publication of a clinical trial. However assessment of an intervention using a single endpoint may not provide a comprehensive picture of the important effects of the intervention. Co-primary endpoints offer an attractive design feature as they capture a more complete characterization of the effect of an intervention. For this reason, co-primary endpoints are becoming a common design feature in many clinical trials. However, these co-primary endpoints are potentially correlated creating complexities in the evaluation of power and sample size in the designing of such clinical trials, specifically relating to control of the Type I or Type II error rate (Gong et al., 2000; Sen and Bretz, 2007; Hung and Wang, 2009; Dmitrienko et al., 2010). It is important to note the distinction between multiple co-primary endpoints and multiple primary endpoints to which many researchers are unaware (Offen et al. (2007)).

In this paper, we discuss sample size determination for clinical trials using multiple correlated binary relative risks being evaluated as co-primary contrasts. We consider a scenario where the objective is to evaluate the superiority of the test intervention compared with the control intervention for all of the relative risks. We evaluate the normal approximation methodology and its utility via Monte-Carlo simulation. We then extend the methodology to group-sequential designs. We summarize our findings as follows:

• The sample size and power formulas presented here, address the challenges associated with sizing trials with potentially correlated co-primary binary endpoints when evaluating effects using relative risks. Incorporating the correlations among the endpoints into the power and sample size calculations leads to increased power and effectively reduces the required sample size. As the correlations are usually unknown, incorporating the correlation into the calculations should be done carefully. As the number of co-primary endpoints increases, the complicity associated with sample size calculations also increases. There may be range restrictions on the correlations resulting in adverse effects on the power and sample size.

• If the standardized effect size on one endpoint as measured by the relative risk is relatively smaller (roughly 33% smaller) than the others, then the advantage of incorporating the correlation into the sample size calculation is lessened. The required sample size is primarily determined by the smaller standardized effect size and does not greatly depend on the correlation. In this scenario, the equation for the sample size can be simplified using the equation for a single endpoint without adjustment for the power.

• The simulation results suggest that use of the normal approximation method may work well in most practical situations. But when the relative risk is larger than 0.5, then the empirical power is slightly above the targeted power and the Type I error rate is slightly inflated. In this scenario, the normal approximation does not work well and cannot be recommended as the Type I error rates are inflated. An alternative methodology is Monte-Carlo simulation, a computationally intensive approach for power evaluation and sample size calculation. Our experience suggests it may require users have considerable mathematical sophistication and appropriate programming knowledge of methods for generating data. When using Monte-Carlo simulation, the number of replications for simulations should be chosen carefully to control simulation error during calculation of the empirical power.

Appendix A: Calculation of the correlation of the log-transformed relative risks

Using the delta method, for large samples, we have an approximation of the covariance between the two log-transformed relative risks logR̂_k and logR̂_k′ (k = 1, …, K; 1 ≤ k < k′ ≤ K; K ≥ 2),

$$\text{cov}[\text{log}{\hat{R}}_k,\text{log}{\hat{R}}_{k\prime }]=\text{cov}[\text{log}{\hat{p}}_{\mathrm{T}k},\text{log}{\hat{p}}_{\mathrm{T}k\prime }]+\text{cov}[\text{log}{\hat{p}}_{\mathrm{C}k},\text{log}{\hat{p}}_{\mathrm{C}k\prime }]\approx \frac{{\rho }^{(kk\prime )}}{N}(\frac{\sqrt{(1-p_{\mathrm{T}k})(1-p_{\mathrm{T}k\prime })}}{r\sqrt{p_{\mathrm{T}k}{p}_{\mathrm{T}k\prime }}}+\frac{\sqrt{(1-p_{\mathrm{C}k})(1-p_{\mathrm{C}k\prime })}}{(1-r)\sqrt{p_{\mathrm{C}k}{p}_{\mathrm{C}k\prime }}}).$$

As the variance of logR̂_k is approximated by var[logR̂_k] ≈ (1 − p_Tk)/(rNp_Tk) + (1 − p_Ck)/{(1 − r)Np_Ck} (for example, please see Fleiss et al. (2003)), for large samples, we have the correlation between the two log-transformed relative risks

$$\text{corr}[\text{log}{\hat{R}}_{k,}\text{log}{\hat{R}}_{k\prime }]=\frac{{\rho }^{(kk\prime )}}{N\sqrt{p_{\mathrm{C}k}{p}_{\mathrm{C}k\prime }}}(\frac{\sqrt{(1-R_k{p}_{\mathrm{C}k})(1-R_{k\prime }{p}_{\mathrm{C}k\prime })}}{r\sqrt{R_k{R}_{k\prime }}}+\frac{\sqrt{(1-p_{\mathrm{C}k})(1-p_{\mathrm{C}k\prime })}}{1-r}).$$

Appendix B: Algorithm for the sample size calculation

When using the methods discussed in Section 2, an iterative procedure is required to identify the sample size that achieves the desired overall power. The easiest method involves a grid search that increases sample size gradually until the power exceeds the desired overall power. However, this method requires considerable computing time. Sugimoto et al. (2012) consider a faster Newton–Raphson algorithm with a convenient formula for N. Another faster but simpler method is to use linear interpolation to identify N (Hamasaki et al., 2013). We briefly describe this algorithm using linear interpolation as follows:

• Step 0: Select the values of the relative risks R_k (k = 1, …, K) with their correlations ρ^(kk′), and the significance level for the one-sided test α and the desired power 1 − β.

• Step 1: Select the two initial values N₀ and N₁. Then, calculate ${\mathrm{\Phi }}_K(N_0)={\mathrm{\Phi }}_K(c_1^{*},\dots ,c_K^{*}|N_0)$ and ${\mathrm{\Phi }}_K(N_1)={\mathrm{\Phi }}_K(c_1^{*},\dots ,c_K^{*}|N_0)$.

• Step 2: Update the value of N, using the following equation

  $$N_{m+1}=\frac{N_{m-1}\{{\mathrm{\Phi }}_K(N_m)-(1-\beta )\}-N_m\{{\mathrm{\Phi }}_K(N_{m-1})-(1-\beta )\}}{{\mathrm{\Phi }}_K(N_m)-{\mathrm{\Phi }}_K(N_{m-1})}$$

• Step 3: Evaluate ${\mathrm{\Phi }}_K(N_{m+1})={\mathrm{\Phi }}_K(c_1^{*},\dots ,c_K^{*}|N_{m+1})$ with the updated value N_m+1. Note that N_m+1 is rounded up if N_m+1 is not integer.

• Step 4: If N_m+1 − N_m = 0, then the iteration stops with N_m+1 as the final value. If not, then return to Step 2.

Options for the two initial values N₀ and N₁ are the smallest sample sizes among those calculated for detecting each relative risk R_k with the power 1 − β at the significance level of α using a sample size equation for a single relative risk (e.g., Chow et al. (2007)). Other options are the largest sample sizes among those calculated for detecting each relative risk R_k with the power 1 − (1 −β)^1/K at the significance level of α. This is because N lies between these options. In our experience, the iterative procedure tends to converge in a few steps.

Appendix C: Sample size calculation for the odds ratio

We outline methodology for the calculation of the sample size for the detection of the joint effects in K correlated odds ratios (K ≥ 2). We now have the K log-transformed (observed) odds ratios logψ̂_k (k = 1, …, K), where logψ̂_k = Ô_Tk/Ô_Ck with Ô_Tk = p̂_Tk/(1 − p̂_Tk) and Ô_Tk = p̂_Ck/(1 − p̂_Ck). For large samples, the joint distribution of (logψ̂₁, …, logψ̂_K) is approximately K-variate normal with mean vector μ = (logψ₁, …, logψ_K)^T and covariance matrix Σ determined by

$$\{\begin{array}{cc}\hfill {\sigma }_k^2=\frac{1}{N}(\frac{1}{r{O}_{\mathrm{T}k}{(1-p_{\mathrm{T}k})}^2}+\frac{1}{(1-r)O_{\mathrm{C}k}{(1-p_{\mathrm{C}k})}^2}),\hfill & \hfill k=k\prime \hfill \\ \hfill {\sigma }_{kk\prime }=\frac{1}{N}(\frac{{\rho }_{\mathrm{T}}^{(kk\prime )}}{r(1-p_{\mathrm{T}k})(1-p_{\mathrm{T}k\prime })\sqrt{O_{\mathrm{T}k}{O}_{\mathrm{T}k\prime }}}+\frac{{\rho }_{\mathrm{C}}^{(kk\prime )}}{(1-r)(1-p_{\mathrm{C}k})(1-p_{\mathrm{C}k\prime })\sqrt{O_{\mathrm{C}k}{O}_{\mathrm{C}k\prime }}}),\hfill & \hfill k\ne k\prime ,\hfill \end{array}$$

where ψ_k = O_Tk/O_Ck with O_Tk = p_Tk/(1 − p_Tk) and O_Tk = p_Tk/(1 − p_Tk). Thus, assuming ρ^(kk′) = corr[Y_Tki, Y_Tk′i] = corr[Y_Ckj, Y_Ck′j] (1 ≤ k < k ′ ≤ K), the correlation between the log-transformed odds ratios, ${\rho }_{\phi }^{(kk\prime )}=\text{corr}[\text{log}{\hat{\phi }}_k,\text{log}{\hat{\phi }}_{k\prime }]$ is given by

$${\rho }_{\psi }^{(kk\prime )}=\frac{{\rho }^{(kk\prime )}\{(1-r)\sqrt{p_{\mathrm{C}k}(1-p_{\mathrm{C}k})p_{\mathrm{C}k\prime }(1-p_{\mathrm{C}k\prime })}+r\sqrt{p_{\mathrm{T}1}(1-p_{\mathrm{T}1})p_{\mathrm{T}k\prime }(1-p_{\mathrm{T}k}\prime )}\}}{\sqrt{\{(1-r)p_{\mathrm{C}k}(1-p_{\mathrm{C}k})+r{p}_{\mathrm{T}k}(1-p_{\mathrm{T}k})\}\{(1-r)p_{\mathrm{C}k\prime }(1-p_{\mathrm{C}k\prime })+r{p}_{\mathrm{T}k\prime }(1-p_{\mathrm{T}k}\prime )\}}}$$

Let Z_k be the test statistics for the log-transformed odds ratios logψ̂_k, given by

$$Z_k=\sqrt{r(1-r)N}\text{log}{\mathrm{\psi ̂}}_k/\sqrt{1/\{{\hat{\overline{p}}}_k(1-{\hat{\overline{p}}}_k)\}}.$$

When requiring joint significance for all of the correlated odds ratios, the overall power is approximately given by

$$1-\beta =\text{Pr}[\underset{k=1}{\overset{K}{\cap }}\{Z_k<-z_{\alpha }\}|{\mathrm{H}}_1]\simeq \text{Pr}[\underset{k=1}{\overset{K}{\cap }}\{Z_k^{*}<-c_k^{*}\}\left|{\mathrm{H}}_1\right],$$

where

$$Z_k^{*}=(\text{log}{\mathrm{\psi ̂}}_k-\text{log}{\psi }_k)/\sqrt{\frac{1}{N}(\frac{1}{r{O}_{\mathrm{T}k}{(1-p_{\mathrm{T}k})}^2}+\frac{1}{(1-r)O_{\mathrm{C}k}{(1-p_{\mathrm{C}k})}^2})}{c}_k^{*}=(z_{\alpha }\sqrt{\frac{1}{N{\hat{\overline{p}}}_k(1-{\hat{\overline{p}}}_k)}(\frac{1}{r}+\frac{1}{1-r})}+\text{log}{\psi }_k)/\sqrt{\frac{1}{N}(\frac{1}{r{O}_{\mathrm{T}\mathrm{k}}{(1-p_{\mathrm{T}k})}^2}+\frac{1}{(1-r)O_{\mathrm{C}k}{(1-p_{\mathrm{C}k})}^2})}$$

The required total sample size for the detection of the joint effects in all odds ratios with the overall power 1 − β at the significance level α, N is the smallest integer satisfying this power.