Two Binary Co-Primary Endpoints (Exact Methods)

Overview

This vignette demonstrates exact sample size calculation and power analysis for clinical trials with two co-primary binary endpoints. The methodology is based on Homma and Yoshida (2025), which provides exact inference methods using the bivariate binomial distribution.

library(twoCoprimary)
library(dplyr)
library(tidyr)
library(knitr)

Background

When to Use Exact Methods

Exact methods are recommended when:

• Small to medium sample sizes ($N < 200$)
• Extreme probabilities ($p < 0.10$ or $p > 0.90$)
• Strict Type I error control is required
• Regulatory requirements for exact inference

Asymptotic methods may not maintain the nominal Type I error rate in these situations.

Advantages of Exact Methods

1. Accurate Type I error control: Exact tests guarantee $\alpha \leq$ nominal level
2. Better small-sample performance: No reliance on asymptotic approximations
3. Valid for extreme probabilities: No restrictions on $p$ values
4. Regulatory acceptance: Often preferred by regulatory agencies

Disadvantages

1. Computational intensity: Requires enumeration of possible outcomes
2. Conservatism: Discrete nature can lead to conservatism
3. Implementation complexity: More complex than asymptotic methods

Statistical Framework

Model and Assumptions

Consider a two-arm parallel-group superiority trial comparing treatment (group 1) with control (group 2). Let $n_{1}$ and $n_{2}$ denote the sample sizes in groups 1 and 2, respectively.

For patient $i$ in group $j$ ($j = 1$: treatment, $j = 2$: control), we observe two binary outcomes:

Endpoint $k$ ($k = 1, 2$): $$X_{i,j,k} \in \{0, 1\}$$

where $X_{i,j,k} = 1$ if patient $i$ in group $j$ is a responder for endpoint $k$, and 0 otherwise.

True response probabilities: $$p_{j,k} = \text{P}(X_{i,j,k} = 1)$$

where $0 < p_{j,k} < 1$ for each $j$ and $k$.

Joint Distribution of Binary Outcomes

The paired binary outcomes $(X_{i,j,1}, X_{i,j,2})$ for patient $i$ in group $j$ follow a multinomial distribution with four possible outcomes:

Per-trial probabilities:

• $p_{j}^{(1,1)} = \phi_{j}$: Both endpoints successful
• $p_{j}^{(1,0)} = p_{j,1} - \phi_{j}$: Only endpoint 1 successful
• $p_{j}^{(0,1)} = p_{j,2} - \phi_{j}$: Only endpoint 2 successful
• $p_{j}^{(0,0)} = 1 - p_{j,1} - p_{j,2} + \phi_{j}$: Both endpoints unsuccessful

where $\phi_{j} = \text{P}(X_{i,j,1} = 1, X_{i,j,2} = 1)$.

Let $Z_{j}^{(\ell,m)}$ denote the random variable representing the number of times $\{(X_{i,j,1}, X_{i,j,2}) : i = 1, \ldots, n_{j}\}$ takes the value $(\ell, m)$ for $\ell, m \in \{0, 1\}$. Then:

$$(Z_{j}^{(0,0)}, Z_{j}^{(1,0)}, Z_{j}^{(0,1)}, Z_{j}^{(1,1)}) \sim \text{Multinomial}(n_{j}; p_{j}^{(0,0)}, p_{j}^{(1,0)}, p_{j}^{(0,1)}, p_{j}^{(1,1)})$$

Number of Responders

Let $Y_{j,k} = \sum_{i=1}^{n_{j}} X_{i,j,k}$ represent the number of responders in group $j$ for endpoint $k$. Then:

• $Y_{j,1} = Z_{j}^{(1,1)} + Z_{j}^{(1,0)}$
• $Y_{j,2} = Z_{j}^{(1,1)} + Z_{j}^{(0,1)}$

Bivariate Binomial Distribution

Following Homma and Yoshida (2025), the joint distribution of $(Y_{j,1}, Y_{j,2})$ can be expressed as a bivariate binomial distribution:

$$(Y_{j,1}, Y_{j,2}) \sim \text{BiBin}(n_{j}, p_{j,1}, p_{j,2}, \gamma_{j})$$

where $\gamma_{j}$ is a dependence parameter related to the correlation $\rho_{j}$ between $X_{i,j,1}$ and $X_{i,j,2}$.

Probability mass function (Equation 3 in Homma and Yoshida, 2025):

$$\text{P}(Y_{j,1} = y_{j,1}, Y_{j,2} = y_{j,2} \mid n_{j}, p_{j,1}, p_{j,2}, \gamma_{j}) = f(y_{j,1} \mid n_{j}, p_{j,1}) \times g(y_{j,2} \mid y_{j,1}, n_{j}, p_{j,1}, p_{j,2}, \gamma_{j})$$ For more details, please see Homma and Yoshida (2025).

Correlation Structure

The correlation $\rho_{j}$ between $X_{i,j,1}$ and $X_{i,j,2}$ is:

$$\rho_{j} = \text{Cor}(X_{i,j,1}, X_{i,j,2}) = \frac{\phi_{j} - p_{j,1} p_{j,2}}{\sqrt{p_{j,1}(1 - p_{j,1}) p_{j,2}(1 - p_{j,2})}}$$

The dependence parameter $\gamma_{j}$ is related to $\rho_{j}$ through (Equation 4 in Homma and Yoshida, 2025):

$$\gamma_{j} = \gamma(\rho_{j}, p_{j,1}, p_{j,2}) = \rho_{j} \sqrt{\frac{p_{j,2}(1 - p_{j,2})}{p_{j,1}(1 - p_{j,1})}} \left(1 - \rho_{j} \sqrt{\frac{p_{j,2}(1 - p_{j,2})}{p_{j,1}(1 - p_{j,1})}}\right)^{-1}$$

Important property: The correlation between $Y_{j,1}$ and $Y_{j,2}$ equals $\rho_{j}$, the same as the correlation between $X_{i,j,1}$ and $X_{i,j,2}$.

Marginal distributions: $$Y_{j,k} \sim \text{Bin}(n_{j}, p_{j,k})$$

Correlation bounds: Due to $0 < p_{j,k} < 1$, the correlation $\rho_{j}$ is bounded:

$$\rho_{j} \in [L(p_{j,1}, p_{j,2}), U(p_{j,1}, p_{j,2})] \subseteq [-1, 1]$$

where:

$$L(p_{j,1}, p_{j,2}) = \max\left\{-\sqrt{\frac{p_{j,1} p_{j,2}}{(1 - p_{j,1})(1 - p_{j,2})}}, -\sqrt{\frac{(1 - p_{j,1})(1 - p_{j,2})}{p_{j,1} p_{j,2}}}\right\}$$

$$U(p_{j,1}, p_{j,2}) = \min\left\{\sqrt{\frac{p_{j,1}(1 - p_{j,2})}{p_{j,2}(1 - p_{j,1})}}, \sqrt{\frac{p_{j,2}(1 - p_{j,1})}{p_{j,1}(1 - p_{j,2})}}\right\}$$

Special cases: - If $p_{j,1} = p_{j,2}$, then $U(p_{j,1}, p_{j,2}) = 1$ - If $p_{j,1} + p_{j,2} = 1$, then $L(p_{j,1}, p_{j,2}) = -1$

Hypothesis Testing

Superiority Hypotheses

Since higher values of both endpoints indicate treatment benefit, we test:

For endpoint 1: $$\text{H}_{0}^{(1)}: p_{1,1} \leq p_{2,1} \text{ vs. } \text{H}_{1}^{(1)}: p_{1,1} > p_{2,1}$$

For endpoint 2: $$\text{H}_{0}^{(2)}: p_{1,2} \leq p_{2,2} \text{ vs. } \text{H}_{1}^{(2)}: p_{1,2} > p_{2,2}$$

Co-Primary Endpoints (Intersection-Union Test)

The trial succeeds only if superiority is demonstrated for both endpoints simultaneously:

Null hypothesis: $\text{H}_{0} = \text{H}_{0}^{(1)} \cup \text{H}_{0}^{(2)}$ (at least one null is true)

Alternative hypothesis: $\text{H}_{1} = \text{H}_{1}^{(1)} \cap \text{H}_{1}^{(2)}$ (both alternatives are true)

Decision rule: Reject $\text{H}_{0}$ at level $\alpha$ if and only if both $\text{H}_{0}^{(1)}$ and $\text{H}_{0}^{(2)}$ are rejected at level $\alpha$ without multiplicity adjustment.

Statistical Tests

Homma and Yoshida (2025) consider five exact test methods:

Method 1: One-sided Pearson Chi-squared Test (Chisq)

For endpoint $k$, the test statistic is:

$$Z(y_{1,k}, y_{2,k}) = \frac{\hat{p}_{1,k} - \hat{p}_{2,k}}{\sqrt{\hat{p}_{k}(1 - \hat{p}_{k})\left(\frac{1}{n_{1}} + \frac{1}{n_{2}}\right)}}$$

where:

• $\hat{p}_{j,k} = y_{j,k} / n_{j}$ is the sample proportion
• $\hat{p}_{k} = \frac{n_{1} \hat{p}_{1,k} + n_{2} \hat{p}_{2,k}}{n_{1} + n_{2}}$ is the pooled proportion

Reject $\text{H}_{0}^{(k)}$ if $Z(y_{1,k}, y_{2,k}) > z_{1-\alpha}$, where $z_{1-\alpha}$ is the $(1-\alpha)$-quantile of the standard normal distribution.

Method 2: Fisher’s Exact Test (Fisher)

Conditional test: Conditions on the total number of successes $y_{1,k} + y_{2,k}$.

Under $\text{H}_{0}^{(k)}$, $Y_{1,k}$ follows a hypergeometric distribution given $Y_{1,k} + Y_{2,k} = y_{k}$.

One-sided p-value:

$$p_{k}^{\text{Fisher}} = \sum_{y=y_{1,k}}^{\min(n_{1}, y_{k})} \frac{\binom{n_{1}}{y} \binom{n_{2}}{y_{k} - y}}{\binom{n_{1} + n_{2}}{y_{k}}}$$

Reject $\text{H}_{0}^{(k)}$ if $p_{k}^{\text{Fisher}} < \alpha$.

Method 3: Fisher’s Mid-P Test (Fisher-midP)

Reduces conservatism by adding half the probability of the observed outcome:

$$p_{k}^{\text{mid-p}} = p_{k}^{\text{Fisher}} - \frac{1}{2} \times \frac{\binom{n_{1}}{y_{1,k}} \binom{n_{2}}{y_{k} - y_{1,k}}}{\binom{n_{1} + n_{2}}{y_{k}}}$$ Note: The twoCoprimary package can implement the Fisher’s Mid-P Test, but Homma and Yoshida (2025) has not investigated this test.

Method 4: Z-pooled Exact Unconditional Test (Z-pool)

Unconditional test: Maximizes the p-value over all possible values of the nuisance parameter (common success probability $p_{k}$ under $\text{H}_{0}$).

Uses the $Z$-test statistic and finds the maximum $p$-value across all possible values of $p_{k}$.

Method 5: Boschloo’s Exact Unconditional Test (Boschloo)

Similar to Z-pooled, but based on Fisher’s exact $p$-values. Maximizes Fisher’s exact $p$-value over the nuisance parameter space.

Most powerful of the exact unconditional tests, but computationally intensive.

Exact Power Calculation

Power Formula

The exact power for test method $A$ is (Equation 9 in Homma and Yoshida, 2025):

$$\text{power}_{A}(\boldsymbol{\theta}) = \text{P}\left[\bigcap_{k=1}^{2} \{p_{A}(y_{1,k}, y_{2,k}) < \alpha\} \mid \text{H}_{1}\right]$$

$$= \sum_{(a_{1,1}, a_{2,1}) \in \mathcal{A}_{1}} \sum_{(a_{1,2}, a_{2,2}) \in \mathcal{A}_{2}} f(a_{1,1} \mid n_{1}, p_{1,1}) \times f(a_{2,1} \mid n_{2}, p_{2,1}) \times g(a_{1,2} \mid a_{1,1}, n_{1}, p_{1,1}, p_{1,2}, \gamma_{1}) \times g(a_{2,2} \mid a_{2,1}, n_{2}, p_{2,1}, p_{2,2}, \gamma_{2})$$

where:

• $\boldsymbol{\theta} = (p_{1,1}, p_{2,1}, p_{1,2}, p_{2,2}, n_{1}, n_{2}, \gamma_{1}, \gamma_{2})$ is the parameter vector
• $\mathcal{A}_{k}$ is the rejection region for endpoint $k$
• $\mathcal{A}_{k} = \{(y_{1,k}, y_{2,k}) : p_{A}(y_{1,k}, y_{2,k}) < \alpha\}$

Sample Size Calculation

The required sample size $n_{2}$ to achieve target power $1 - \beta$ is (Equation 10 in Homma and Yoshida, 2025):

$$n_{2} = \arg\min_{n_{2} \in \mathbb{Z}} \{\text{power}_{A}(\boldsymbol{\theta}) \geq 1 - \beta\}$$

This cannot be expressed as a closed-form formula due to:

1. Discreteness of binary outcomes
2. Non-monotonic “sawtooth” power curve

Algorithm: Sequential search starting from asymptotic normal approximation (AN method) as initial value.

Replicating Homma and Yoshida (2025) Table 4

Table 4 from Homma and Yoshida (2025) shows sample sizes for various correlations using the Chisq, Fisher, Z-pool, and Boschloo. Note that the following sample code compute only scenario for $\alpha=0.025$.

The notation used in the function is: p11 = $p_{1,1}$, p12 = $p_{1,2}$, p21 = $p_{2,1}$, p22 = $p_{2,2}$, where the first subscript denotes the group (1 = treatment, 2 = control) and the second subscript denotes the endpoint (1 or 2).

# Recreate Homma and Yoshida (2025) Table 4
library(dplyr)
library(tidyr)
library(readr)

param_grid_bin_exact_ss <- tibble(
  p11 = 0.54, 
  p12 = 0.54,
  p21 = 0.25,
  p22 = 0.25
)

result_bin_exact_ss <- do.call(
  bind_rows,
  lapply(c("Chisq", "Fisher", "Z-pool", "Boschloo"), function(test) {
    do.call(
      bind_rows,
      lapply(1:2, function(r) {
        design_table(
          param_grid = param_grid_bin_exact_ss,
          rho_values = c(0, 0.3, 0.5, 0.8),
          r = r,
          alpha = 0.025,
          beta = 0.1,
          endpoint_type = "binary",
          Test = test
        ) %>% 
          mutate(alpha = 0.025, r = r, Test = test)
      })
    )
  })
) %>% 
  pivot_longer(
    cols = starts_with("rho_"),
    names_to = "rho",
    values_to = "N",
    names_transform = list(rho = parse_number)
  ) %>% 
  select(r, rho, Test, N) %>% 
  pivot_wider(names_from = Test,  values_from = N) %>% 
  as.data.frame()

kable(result_bin_exact_ss,
      caption = "Table 4: Total Sample Size (N) for Two Co-Primary Binary Endpoints (α = 0.025, 1-β = 0.90)^a,b^",
      digits = 1,
      col.names = c("r", "ρ", "Chisq", "Fisher", "Z-pool", "Boschloo"))

Table 4: Total Sample Size (N) for Two Co-Primary Binary Endpoints (α = 0.025, 1-β = 0.90)^a,b

r	ρ	Chisq	Fisher	Z-pool	Boschloo
1	0.0	142	152	144	144
1	0.3	142	150	142	142
1	0.5	140	150	140	140
1	0.8	128	144	134	134
2	0.0	162	174	180	162
2	0.3	159	174	180	159
2	0.5	156	171	177	156
2	0.8	147	159	168	150

^a Chisq denotes the one-sided Pearson chi-squared test. Fisher stands for Fisher’s exact test. Z-pool represents the Z-pooled exact unconditional test. Boschloo signifies Boschloo’s exact unconditional test.

^b The required sample sizes were obtained by assuming that $p_{1,1} = p_{1,2} = 0.54$ and $p_{2,1} = p_{2,2} = 0.25$.

Practical Examples

Example 1: Basic Exact Power Calculation

# Calculate exact power using Fisher's exact test
result_fisher <- power2BinaryExact(
  n1 = 50,
  n2 = 50,
  p11 = 0.70, p12 = 0.65,
  p21 = 0.50, p22 = 0.45,
  rho1 = 0.5, rho2 = 0.5,
  alpha = 0.025,
  Test = "Fisher"
)

print(result_fisher)
#> 
#> Power calculation for two binary co-primary endpoints
#> 
#>              n1 = 50
#>              n2 = 50
#>     p (group 1) = 0.7, 0.65
#>     p (group 2) = 0.5, 0.45
#>             rho = 0.5, 0.5
#>           alpha = 0.025
#>            Test = Fisher
#>          power1 = 0.46345
#>          power2 = 0.46196
#>  powerCoprimary = 0.297231

Interpretation:

• power1: Power for endpoint 1 alone
• power2: Power for endpoint 2 alone
• powerCoprimary: Exact power for both co-primary endpoints

Example 2: Sample Size Calculation

# Calculate required sample size using Boschloo's test
result_ss <- ss2BinaryExact(
  p11 = 0.70, p12 = 0.65,
  p21 = 0.50, p22 = 0.45,
  rho1 = 0.5, rho2 = 0.5,
  r = 1,
  alpha = 0.025,
  beta = 0.2,
  Test = "Boschloo"
)

print(result_ss)
#> 
#> Sample size calculation for two binary co-primary endpoints
#> 
#>              n1 = 120
#>              n2 = 120
#>               N = 240
#>     p (group 1) = 0.7, 0.65
#>     p (group 2) = 0.5, 0.45
#>             rho = 0.5, 0.5
#>      allocation = 1
#>           alpha = 0.025
#>            beta = 0.2
#>            Test = Boschloo

Example 3: Comparison of Test Methods

# Compare different exact test methods
test_methods <- c("Chisq", "Fisher", "Fisher-midP", "Z-pool", "Boschloo")

comparison <- lapply(test_methods, function(test) {
    result <- ss2BinaryExact(
        p11 = 0.50, p12 = 0.40,
        p21 = 0.20, p22 = 0.10,
        rho1 = 0.7, rho2 = 0.6,
        r = 1,
        alpha = 0.025,
        beta = 0.2,
        Test = test
    )
    data.frame(
        Test = test,
        n2 = result$n2,
        N = result$N
    )
})

comparison_table <- bind_rows(comparison)

kable(comparison_table,
      caption = "Sample Size Comparison Across Test Methods",
      col.names = c("Test Method", "n per group", "N total"))

Sample Size Comparison Across Test Methods

Test Method	n per group	N total
Chisq	42	84
Fisher	49	98
Fisher-midP	43	86
Z-pool	43	86
Boschloo	43	86

Impact of Correlation

Example 4: Correlation Effect

# Calculate sample size for different correlation values
rho_values <- c(0, 0.3, 0.5, 0.8)

correlation_effect <- lapply(rho_values, function(rho) {
    result <- ss2BinaryExact(
        p11 = 0.70, p12 = 0.60,
        p21 = 0.40, p22 = 0.30,
        rho1 = rho, rho2 = rho,
        r = 1,
        alpha = 0.025,
        beta = 0.2,
        Test = "Fisher"
    )
    data.frame(
        rho = rho,
        n2 = result$n2,
        N = result$N
    )
})

rho_table <- bind_rows(correlation_effect)

kable(rho_table,
      caption = "Impact of Correlation on Sample Size (Fisher's Test)",
      col.names = c("ρ", "n per group", "N total"))

Impact of Correlation on Sample Size (Fisher’s Test)

ρ	n per group	N total
0.0	61	122
0.3	60	120
0.5	59	118
0.8	56	112

Key finding: Higher positive correlation reduces required sample size.

Comparison: Exact vs Asymptotic

Example 5: Exact vs AN Method

# Exact method (Chisq)
exact_result <- ss2BinaryExact(
    p11 = 0.60, p12 = 0.40,
    p21 = 0.30, p22 = 0.10,
    rho1 = 0.5, rho2 = 0.5,
    r = 1,
    alpha = 0.025,
    beta = 0.1,
    Test = "Chisq"
)

# Asymptotic method (AN)
asymp_result <- ss2BinaryApprox(
    p11 = 0.60, p12 = 0.40,
    p21 = 0.30, p22 = 0.10,
    rho1 = 0.5, rho2 = 0.5,
    r = 1,
    alpha = 0.025,
    beta = 0.1,
    Test = "AN"
)

comparison_exact_asymp <- data.frame(
  Method = c("Exact (Chisq)", "Asymptotic (AN)"),
  n_per_group = c(exact_result$n2, asymp_result$n2),
  N_total = c(exact_result$N, asymp_result$N),
  Difference = c(0, asymp_result$N - exact_result$N)
)

kable(comparison_exact_asymp,
      caption = "Comparison: Exact vs Asymptotic Methods",
      col.names = c("Method", "n per group", "N total", "Difference"))

Comparison: Exact vs Asymptotic Methods

Method	n per group	N total	Difference
Exact (Chisq)	59	118	0
Asymptotic (AN)	60	120	2

Practical Recommendations

Test Method Selection

1. Fisher’s exact test:
   • Most widely used and accepted
   • Conservative but guarantees Type I error control
   • Recommended for regulatory submissions
2. Boschloo’s test:
   • Most powerful among exact tests
   • Best choice when computational resources permit
   • Recommended for final analysis
3. Chi-squared test:
   • Less conservative than Fisher
   • May be anti-conservative for small samples
   • Use with caution for $N < 200$
4. Z-pooled and Fisher-midP:
   • Intermediate between Fisher and chi-squared
   • Reduce conservatism while maintaining validity

When to Use Each Method

Sample size guidelines:

1. $N < 100$: Always use exact methods

2. $100 \leq N < 200$: Exact methods preferred, especially if:

   • Extreme probabilities ($p < 0.1$ or $p > 0.9$)
   • Strict Type I error control required

3. $N \geq 200$ and $0.1 < p < 0.9$: Asymptotic methods acceptable

Correlation Estimation

• Use pilot data or historical information
• Be conservative if uncertain (use $\rho = 0$)
• Consider sensitivity analysis across plausible range

Allocation Ratio

• Balanced design ($r = 1$) generally most efficient
• Unbalanced designs may be justified by:
  • Limited control group availability
  • Ethical considerations
  • Cost constraints

Computational Considerations

Modern computers handle all methods efficiently for typical clinical trial sample sizes ($N < 300$).

Software Implementation

The twoCoprimary package implements all methods efficiently using:

• Bivariate binomial distribution (dbibinom)
• Rejection region calculation (rr1Binary)
• Vectorized computations for speed

References

Homma, G., & Yoshida, T. (2025). Exact power and sample size in clinical trials with two co-primary binary endpoints. Statistical Methods in Medical Research, 34(1), 1-19.