Two Binary Co-Primary Endpoints (Asymptotic Methods)

Clinical Context

Binary co-primary endpoints are common in:

• Migraine trials: Relief from headache (yes/no) + Relief from the most bothersome migraine-related symptom (yes/no)

• Psoriasis trials: At least 75% improvement in the score on the psoriasis area-and-severity index (PASI) (yes/no) + Score on the physician’s global assessment of 0 or 1 (yes/no)

• Primary myelofibrosis trials: Clinically relevant complete haematological response (yes/no) + Lack of progression of clinical symptoms (yes/no)

When to Use Asymptotic Methods

Asymptotic (normal approximation) methods are appropriate when:

• Sample sizes are large (typically \(N > 200\))

• Probabilities are not extreme (\(0.10 < p < 0.90\))

• Computational efficiency is important

For small to medium sample sizes or extreme probabilities, use exact methods based on Homma and Yoshida (2025), which can be computed using exact approaches without any approximations (see vignette("two-binary-endpoints-exact")).

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 the two groups, with allocation ratio \(r = n_{1}/n_{2}\) (i.e., total sample size is \(N=n_{1}+n_{2}\)).

For subject \(i\) (\(i = 1, \ldots, n_j\)) in group \(j\) (\(j = 1\): treatment, \(j = 2\): control), we observe two binary outcomes:

\[X_{i,j,k} \in \{0, 1\}, \quad k = 1, 2\]

where \(X_{i,j,k} = 1\) indicates success and \(X_{i,j,k} = 0\) indicates failure for outcome \(k\).

Marginal probabilities: Let \(p_{j,k}\) denote the success probability for outcome \(k\) in group \(j\):

\[p_{j,k} = \text{P}(X_{i,j,k} = 1), \quad k = 1, 2\]

For details on the joint distribution of \((X_{i,j,1}, X_{i,j,2})\), see Homma and Yoshida (2025) Section 2.1 or Sozu et al. (2010).

Correlation Structure

The correlation \(\rho_{j}\) between the two binary outcomes in group \(j\) is defined as (Homma and Yoshida, 2025):

\[\rho_j = \text{Cor}(X_{i,j,1}, X_{i,j,2}) = \frac{\phi_j - p_{j,1}p_{j,2}}{\sqrt{p_{j,1}(1-p_{j,1})p_{j,2}(1-p_{j,2})}}\]

where \(\phi_j = \text{P}(X_{i,j,1} = 1, X_{i,j,2} = 1)\) is the joint probability that both outcomes are successful.

Practical interpretation: \(\rho_{j} > 0\) means subjects who succeed on outcome 1 are more likely to succeed on outcome 2.

Valid correlation range: Because \(0 < p_{j,k} < 1\), the correlation \(\rho_{j}\) is not free to range over \([-1, 1]\), but is bounded by (Homma and Yoshida, 2025, Equation 2):

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

where:

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

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

If \(p_{j,1} = p_{j,2}\), then \(U(p_{j,1}, p_{j,2}) = 1\), whereas \(L(p_{j,1}, p_{j,2}) = -1\) for \(p_{j,1} + p_{j,2} = 1\).

These bounds can be computed using the corrbound2Binary() function in this package.

Hypothesis Testing

We test superiority of treatment over control for both endpoints. The endpoints for testing are the risk differences:

For endpoint \(k\): \[\text{H}_{0k}: p_{1,k} - p_{2,k} \leq 0 \text{ vs. } \text{H}_{1k}: p_{1,k} - p_{2,k} > 0\]

For co-primary endpoints (intersection-union test):

Null hypothesis: \(\text{H}_{0} = \text{H}_{01} \cup \text{H}_{02}\) (at least one null is true)

Alternative hypothesis: \(\text{H}_{1} = \text{H}_{11} \cap \text{H}_{12}\) (both alternatives are true)

Decision rule: Reject \(\text{H}_{0}\) at level \(\alpha\) if and only if both \(\text{H}_{01}\) and \(\text{H}_{02}\) are rejected at level \(\alpha\).

Test Statistics

Several test statistics are available for binary endpoints. We focus on asymptotic normal approximation methods following Sozu et al. (2010).

Joint Distribution and Correlation

Under \(\text{H}_{1}\), \((Z_{1}, Z_{2})\) asymptotically follows a bivariate normal distribution:

\[\begin{pmatrix} Z_1 \\ Z_2 \end{pmatrix} \sim \text{BN}\left(\begin{pmatrix} \omega_1 \\ \omega_2 \end{pmatrix}, \begin{pmatrix} 1 & \gamma \\ \gamma & 1 \end{pmatrix}\right)\]

The correlation \(\gamma\) between test statistics depends on \(\rho_{1}\), \(\rho_{2}\), and the marginal probabilities. For details on the derivation and approximation formulas, see Sozu et al. (2010) and Homma and Yoshida (2025). This approximation is implemented in the package functions.

Power Calculation

The overall power (probability of rejecting both null hypotheses) is:

\[1 - \beta = \text{P}(Z_1 > z_{1-\alpha} \text{ and } Z_2 > z_{1-\alpha} \mid \text{H}_1)\]

Using the bivariate normal distribution:

\[1 - \beta = \Phi_2(-z_{1-\alpha} + \omega_1, -z_{1-\alpha} + \omega_2 \mid \gamma)\]

where \(\Phi_{2}(a, b \mid \gamma)\) is the CDF of the standard bivariate normal distribution with correlation \(\gamma\).

Sample Size Calculation

For target power \(1 - \beta\) and balanced design (\(n_{1} = n_{2} = n\)), we solve the power formula shown above numerically for \(n_{2}\). Then, the total sample size is obtained as \(N = (1+r)n_{2}\).

Key Findings

Effect of correlation:

• Positive correlation (\(\rho > 0\)) reduces required sample size
• Negative correlation (\(\rho < 0\)) increases required sample size
• Zero correlation (\(\rho = 0\)) provides intermediate sample size

Test method comparison:

• AN: Generally smallest sample size, most efficient
• ANc: Slightly larger than AN due to continuity correction
• AS: Similar to AN for moderate probabilities
• ASc: Largest sample size, most conservative

Example 1: Equal Effect Sizes

Calculate sample size when both endpoints have equal effect sizes:

# Both endpoints: 70% vs 50% (20% difference)
# p_{1,1} = p_{1,2} = 0.7 (treatment group)
# p_{2,1} = p_{2,2} = 0.5 (control group)
ss_equal <- ss2BinaryApprox(
  p11 = 0.7, p12 = 0.7,  # Treatment group
  p21 = 0.5, p22 = 0.5,  # Control group
  rho1 = 0.5,            # Correlation in treatment group
  rho2 = 0.5,            # Correlation in control group
  r = 1,                 # Balanced allocation
  alpha = 0.025,
  beta = 0.2,
  Test = "AN"
)

print(ss_equal)
#> 
#> Sample size calculation for two binary co-primary endpoints
#> 
#>              n1 = 116
#>              n2 = 116
#>               N = 232
#>     p (group 1) = 0.7, 0.7
#>     p (group 2) = 0.5, 0.5
#>             rho = 0.5, 0.5
#>      allocation = 1
#>           alpha = 0.025
#>            beta = 0.2
#>            Test = AN

Note: In the function, p11 corresponds to \(p_{1,1}\) (success probability for endpoint 1 in treatment group), p12 to \(p_{1,2}\), p21 to \(p_{2,1}\), and p22 to \(p_{2,2}\).

Example 2: Unequal Effect Sizes

When effect sizes differ, the endpoint with smaller effect dominates:

# Endpoint 1: 75% vs 65% (10% difference)
# Endpoint 2: 80% vs 60% (20% difference)
ss_unequal <- ss2BinaryApprox(
  p11 = 0.75, p12 = 0.80,
  p21 = 0.65, p22 = 0.60,
  rho1 = 0.3, rho2 = 0.3,
  r = 1,
  alpha = 0.025,
  beta = 0.2,
  Test = "AN"
)

print(ss_unequal)
#> 
#> Sample size calculation for two binary co-primary endpoints
#> 
#>              n1 = 329
#>              n2 = 329
#>               N = 658
#>     p (group 1) = 0.75, 0.8
#>     p (group 2) = 0.65, 0.6
#>             rho = 0.3, 0.3
#>      allocation = 1
#>           alpha = 0.025
#>            beta = 0.2
#>            Test = AN

# Compare with individual endpoint sample sizes
ss_ep1 <- ss1BinaryApprox(p1 = 0.75, p2 = 0.65, r = 1, 
                          alpha = 0.025, beta = 0.2, Test = "AN")
ss_ep2 <- ss1BinaryApprox(p1 = 0.80, p2 = 0.60, r = 1,
                          alpha = 0.025, beta = 0.2, Test = "AN")

cat("Single endpoint 1 sample size:", ss_ep1$n2, "\n")
#> Single endpoint 1 sample size: 329
cat("Single endpoint 2 sample size:", ss_ep2$n2, "\n")
#> Single endpoint 2 sample size: 82
cat("Co-primary sample size:", ss_unequal$n2, "\n")
#> Co-primary sample size: 329

The co-primary design requires approximately the same sample size as the endpoint with smaller effect size.

Example 3: Effect of Correlation

Demonstrate how correlation affects sample size:

# Fixed effect sizes: p_{1,1} = p_{1,2} = 0.7, p_{2,1} = p_{2,2} = 0.5
p11 <- 0.7
p21 <- 0.5
p12 <- 0.7
p22 <- 0.5

# Range of correlations
rho_values <- c(-0.3, 0, 0.3, 0.5, 0.8)

ss_by_rho <- sapply(rho_values, function(rho) {
  ss <- ss2BinaryApprox(
    p11 = p11, p12 = p12,
    p21 = p21, p22 = p22,
    rho1 = rho, rho2 = rho,
    r = 1,
    alpha = 0.025,
    beta = 0.2,
    Test = "AN"
  )
  ss$n2
})

result_df <- data.frame(
  rho = rho_values,
  n_per_group = ss_by_rho,
  N_total = ss_by_rho * 2
)

kable(result_df,
      caption = "Sample Size vs Correlation (p₁,₁ = p₁,₂ = 0.7, p₂,₁ = p₂,₂ = 0.5)",
      col.names = c("ρ", "n per group", "N total"))

# Plot
plot(rho_values, ss_by_rho, 
     type = "b", pch = 19,
     xlab = "Correlation (ρ)", 
     ylab = "Sample size per group (n)",
     main = "Effect of Correlation on Sample Size",
     ylim = c(min(ss_by_rho) * 0.9, max(ss_by_rho) * 1.1))
grid()

Interpretation: Higher positive correlation substantially reduces required sample size.

Example 4: Unbalanced Allocation

Calculate sample size with unequal group allocation:

# 2:1 allocation (treatment:control)
ss_unbalanced <- ss2BinaryApprox(
  p11 = 0.7, p12 = 0.7,
  p21 = 0.5, p22 = 0.5,
  rho1 = 0.5, rho2 = 0.5,
  r = 2,  # 2:1 allocation
  alpha = 0.025,
  beta = 0.2,
  Test = "AN"
)

print(ss_unbalanced)
#> 
#> Sample size calculation for two binary co-primary endpoints
#> 
#>              n1 = 172
#>              n2 = 86
#>               N = 258
#>     p (group 1) = 0.7, 0.7
#>     p (group 2) = 0.5, 0.5
#>             rho = 0.5, 0.5
#>      allocation = 2
#>           alpha = 0.025
#>            beta = 0.2
#>            Test = AN
cat("Total sample size:", ss_unbalanced$N, "\n")
#> Total sample size: 258

Design Considerations

1. Estimate correlation: Use pilot data or historical information; be conservative if uncertain

2. Consider asymmetric effects: Use actual effect sizes rather than assuming equality

3. Balanced allocation: Generally more efficient unless practical constraints require otherwise

4. Test method selection: AN is most common; use continuity correction for added conservatism

5. Sample size sensitivity: Calculate for range of plausible correlations and effect sizes

When to Use Exact Methods Instead

Use exact methods (ss2BinaryExact) based on Homma and Yoshida (2025) when:

• Small to medium sample sizes (\(N < 200\))
• Extreme probabilities (\(p < 0.1\) or \(p > 0.9\))
• Strict Type I error control required
• Regulatory preference for exact tests

Exact methods using Fisher’s exact test can provide more accurate sample sizes and better Type I error control in these situations.

Asymptotic Validity

These asymptotic methods are appropriate when:

• \(N > 200\)
• \(0.1 < p < 0.9\)
• Computational efficiency is important