Two Binary Co-Primary Endpoints (Exact Methods)

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

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 \(L(p_{j,1}, p_{j,2}) = -1\)

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.

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.

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.

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"))

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"))

Key finding: Higher positive correlation reduces required sample size.

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"))

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

Software Implementation

The twoCoprimary package implements all methods efficiently using:

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