Mixed Count and Continuous Co-Primary Endpoints

Clinical Context

In clinical trials for treating asthma and chronic obstructive pulmonary disease (COPD), regulatory agencies require evaluation of both:

1. Count endpoint: Number of exacerbations over follow-up period (overdispersed count data)
2. Continuous endpoint: Lung function measure such as \(\text{FEV}_{1}\) (forced expiratory volume in 1 second)

Why Negative Binomial Distribution?

Count outcomes in clinical trials often exhibit overdispersion, where the variance substantially exceeds the mean. The Poisson distribution assumes variance = mean, which is often violated.

Negative binomial (NB) distribution accommodates overdispersion:

\[X \sim \text{NB}(\lambda, \nu)\]

• Mean: \(\text{E}[X] = \lambda = r \times t\) (rate \(\times\) time)
• Variance: \(\text{Var}[X] = \lambda + \lambda^{2}/\nu\)
• Dispersion parameter: \(\nu > 0\) controls overdispersion
  • As \(\nu \to \infty\): \(\text{NB} \to \text{Poisson}\) (no overdispersion)
  • Small \(\nu\): High overdispersion (variance \(\gg\) mean)

Variance-to-mean ratio (VMR): \[\text{VMR} = \frac{\text{Var}[X]}{\text{E}[X]} = 1 + \frac{\lambda}{\nu}\]

When \(\text{VMR} > 1\), data are overdispersed and NB is more appropriate than Poisson.

Model and Assumptions

Consider a two-arm superiority trial with sample sizes \(n_{1}\) (treatment) and \(n_{2}\) (control), with allocation ratio \(r = n_{1}/n_{2}\).

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

Outcome 1 (Count): \[X_{i,j,1} \sim \text{NB}(\lambda_{j}, \nu)\]

where \(\lambda_{j}\) is the expected number of events in group \(j\), and \(\nu > 0\) is the common dispersion parameter.

Outcome 2 (Continuous): \[X_{i,j,2} \sim \text{N}(\mu_{j}, \sigma^{2})\]

where \(\mu_{j}\) is the population mean in group \(j\), and \(\sigma^{2}\) is the common variance across groups.

Correlation Structure

The correlation between count and continuous outcomes within subjects is measured by \(\rho_{j}\) in group \(j\). This correlation must satisfy feasibility constraints based on the Fréchet-Hoeffding copula bounds, which depend on the marginal distributions.

Use the corrbound2MixedCountContinuous() function to check valid correlation bounds for given parameters.

Hypothesis Testing

We test superiority of treatment over control for both endpoints. Note: In COPD/asthma trials, treatment benefit is indicated by lower values for both endpoints (fewer exacerbations, less decline in lung function).

For count endpoint (1): \[\text{H}_{01}: r_{1} \geq r_{2} \text{ vs. } \text{H}_{11}: r_{1} < r_{2}\]

Equivalently, testing \(\beta_{1} = \log(r_{1}) - \log(r_{2}) < 0\).

For continuous endpoint (2): \[\text{H}_{02}: \mu_{1} - \mu_{2} \geq 0 \text{ vs. } \text{H}_{12}: \mu_{1} - \mu_{2} < 0\]

Co-primary endpoints (intersection-union test): \[\text{H}_0 = \text{H}_{01} \cup \text{H}_{02} \text{ vs. } \text{H}_1 = \text{H}_{11} \cap \text{H}_{12}\]

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

Count endpoint (Equation 7 in Homma and Yoshida, 2024):

\[Z_{1} = \frac{\hat{\beta}_{1}}{\sqrt{\text{Var}(\hat{\beta}_{1})}}\]

where:

• \(\hat{\beta}_{1} = \log(\bar{X}_{1,1}) - \log(\bar{X}_{2,1})\) is the log rate ratio
• \(\text{Var}(\hat{\beta}_{1}) = \frac{1}{n_{2}}\left[\frac{1}{t}\left(\frac{1}{\lambda_{2}} + \frac{1}{r\lambda_{1}}\right) + \frac{1+r}{\nu r}\right] = \frac{V_{a}}{n_{2}}\)

Continuous endpoint:

\[Z_{2} = \frac{\bar{X}_{1,2} - \bar{X}_{2,2}}{\sigma\sqrt{\frac{1+r}{r n_{2}}}}\]

When \(\sigma\) is a common known standard deviation.

Joint Distribution and Correlation

Under \(\text{H}_{1}\), the test statistics \((Z_{1}, Z_{2})\) asymptotically follow a bivariate normal distribution (Appendix B in Homma and Yoshida, 2024):

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

where:

• \(\omega_{1} = \frac{\sqrt{n_{2}}\beta_{1}}{\sqrt{V_{a}}}\) with \(\beta_{1} = \log(r_{1}) - \log(r_{2})\)
• \(\omega_{2} = \frac{\delta}{\sigma\sqrt{\frac{1+r}{r n_{2}}}}\) with \(\delta = \mu_{1} - \mu_{2}\)

Correlation between test statistics (Equation 11 in Homma and Yoshida, 2024):

\[\gamma = \sum_{j=1}^{2} \frac{n_{2}\rho_{j}\sqrt{1+\lambda_{j}/\nu}}{n_{j}\sqrt{\lambda_{j}V_{a}(1+r)/r}}\]

For balanced design (\(r = 1\)) with common correlation (\(\rho_{1} = \rho_{2} = \rho\)):

\[\gamma = \frac{\rho}{\sqrt{2}}\frac{\sqrt{1/\lambda_{2} + 1/\nu} + \sqrt{1/\lambda_{1} + 1/\nu}}{\sqrt{(1/\lambda_{2} + 1/\lambda_{1} + 2/\nu)}}\]

Power Calculation

The overall power is (Equation 10 in Homma and Yoshida, 2024):

\[1 - \beta = \text{P}(Z_{1} < z_{\alpha} \cap Z_{2} < z_{\alpha} \mid \text{H}_{1})\]

Using the bivariate normal CDF \(\Phi_{2}\):

\[1 - \beta = \Phi_{2}\left(z_{\alpha} - \frac{\sqrt{n_{2}}(\log r_{1} - \log r_{2})}{\sqrt{V_{a}}}, z_{\alpha} - \frac{\sqrt{n_{2}}(\mu_{1}-\mu_{2})}{\sigma\sqrt{\frac{1+r}{r}}} \Bigg| \gamma\right)\]

Sample Size Determination

The sample size is determined by solving the power equation numerically. For a given allocation ratio \(r\), target power \(1 - \beta\), and significance level \(\alpha\), we find the smallest \(n_{2}\) such that the overall power equals or exceeds \(1 - \beta\).

Sequential search algorithm:

1. Calculate initial sample size based on single-endpoint formulas
2. Compute power at current sample size
3. If power \(\geq\) target: decrease \(n_{2}\) until power \(<\) target, then add 1 back
4. If power \(<\) target: increase \(n_{2}\) until power \(\geq\) target

Example: Calculate Correlation Bounds

# Scenario: lambda = 1.25, nu = 0.8, mu = 0, sigma = 250
bounds1 <- corrbound2MixedCountContinuous(lambda = 1.25, nu = 0.8, mu = 0, sd = 250)
cat("Correlation bounds for NB(1.25, 0.8) and N(0, 250²):\n")
#> Correlation bounds for NB(1.25, 0.8) and N(0, 250²):
cat("Lower bound:", round(bounds1[1], 3), "\n")
#> Lower bound: -0.846
cat("Upper bound:", round(bounds1[2], 3), "\n\n")
#> Upper bound: 0.846

# Higher dispersion (smaller nu) typically restricts bounds
bounds2 <- corrbound2MixedCountContinuous(lambda = 1.25, nu = 0.5, mu = 0, sd = 250)
cat("Correlation bounds for NB(1.25, 0.5) and N(0, 250²):\n")
#> Correlation bounds for NB(1.25, 0.5) and N(0, 250²):
cat("Lower bound:", round(bounds2[1], 3), "\n")
#> Lower bound: -0.802
cat("Upper bound:", round(bounds2[2], 3), "\n\n")
#> Upper bound: 0.802

# Higher mean count
bounds3 <- corrbound2MixedCountContinuous(lambda = 2.0, nu = 0.8, mu = 0, sd = 250)
cat("Correlation bounds for NB(2.0, 0.8) and N(0, 250²):\n")
#> Correlation bounds for NB(2.0, 0.8) and N(0, 250²):
cat("Lower bound:", round(bounds3[1], 3), "\n")
#> Lower bound: -0.863
cat("Upper bound:", round(bounds3[2], 3), "\n")
#> Upper bound: 0.863

Important: Always verify that the specified correlation \(\rho\) is within the feasible bounds for your parameters.

Example 1: Balanced Design

Calculate sample size for a balanced design with moderate effect sizes:

# Balanced design: n1 = n2 (i.e., r = 1)
result_balanced <- ss2MixedCountContinuous(
  r1 = 1.0,              # Count rate in treatment group
  r2 = 1.25,             # Count rate in control group
  nu = 0.8,              # Dispersion parameter
  t = 1,                 # Follow-up time
  mu1 = -50,             # Mean for treatment (negative = benefit)
  mu2 = 0,               # Mean for control
  sd = 250,              # Standard deviation
  r = 1,                 # Balanced allocation
  rho1 = 0.5,            # Correlation in treatment group
  rho2 = 0.5,            # Correlation in control group
  alpha = 0.025,
  beta = 0.2
)

print(result_balanced)
#> 
#> Sample size calculation for mixed count and continuous co-primary endpoints
#> 
#>              n1 = 705
#>              n2 = 705
#>               N = 1410
#>              sd = 250
#>            rate = 1, 1.25
#>              nu = 0.8
#>               t = 1
#>              mu = -50, 0
#>             rho = 0.5, 0.5
#>      allocation = 1
#>           alpha = 0.025
#>            beta = 0.2

Example 2: Effect of Correlation

Demonstrate how correlation affects sample size:

# Fixed effect sizes
r1 <- 1.0
r2 <- 1.25
nu <- 0.8
mu1 <- -50
mu2 <- 0
sd <- 250

# Range of correlations
rho_values <- c(0, 0.2, 0.4, 0.6, 0.8)

ss_by_rho <- sapply(rho_values, function(rho) {
  result <- ss2MixedCountContinuous(
    r1 = r1, r2 = r2, nu = nu, t = 1,
    mu1 = mu1, mu2 = mu2, sd = sd,
    r = 1,
    rho1 = rho, rho2 = rho,
    alpha = 0.025,
    beta = 0.2
  )
  result$n2
})

result_df <- data.frame(
  rho = rho_values,
  n_per_group = ss_by_rho,
  N_total = ss_by_rho * 2,
  reduction_pct = round((1 - ss_by_rho / ss_by_rho[1]) * 100, 1)
)

kable(result_df,
      caption = "Effect of Correlation on Sample Size",
      col.names = c("ρ", "n per group", "N total", "Reduction (%)"))

# 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(600, max(ss_by_rho) * 1.1))
grid()

Interpretation: Higher positive correlation reduces required sample size. At \(\rho = 0.8\), sample size is reduced by approximately 5-7% compared to \(\rho = 0\).

Example 3: Effect of Dispersion Parameter

Compare sample sizes for different levels of overdispersion:

# Fixed design parameters
r1 <- 1.0
r2 <- 1.25
mu1 <- -50
mu2 <- 0
sd <- 250
rho <- 0.5

# Range of dispersion parameters
nu_values <- c(0.5, 0.8, 1.0, 2.0, 5.0)

ss_by_nu <- sapply(nu_values, function(nu) {
  result <- ss2MixedCountContinuous(
    r1 = r1, r2 = r2, nu = nu, t = 1,
    mu1 = mu1, mu2 = mu2, sd = sd,
    r = 1,
    rho1 = rho, rho2 = rho,
    alpha = 0.025,
    beta = 0.2
  )
  result$n2
})

result_df_nu <- data.frame(
  nu = nu_values,
  VMR = round(1 + 1.125/nu_values, 2),  # VMR at lambda = 1.125 (average)
  n_per_group = ss_by_nu,
  N_total = ss_by_nu * 2
)

kable(result_df_nu,
      caption = "Effect of Dispersion Parameter on Sample Size",
      digits = c(1, 2, 0, 0),
      col.names = c("ν", "VMR", "n per group", "N total"))

Key finding: Higher overdispersion (smaller \(\nu\), larger VMR) requires larger sample sizes.

Example 4: Unbalanced Allocation

Calculate sample size with 2:1 allocation ratio:

# Balanced design (r = 1)
result_balanced <- ss2MixedCountContinuous(
  r1 = 1.0, r2 = 1.25, nu = 0.8, t = 1,
  mu1 = -50, mu2 = 0, sd = 250,
  r = 1,
  rho1 = 0.5, rho2 = 0.5,
  alpha = 0.025,
  beta = 0.2
)

# Unbalanced design (r = 2, i.e., n1 = 2*n2)
result_unbalanced <- ss2MixedCountContinuous(
  r1 = 1.0, r2 = 1.25, nu = 0.8, t = 1,
  mu1 = -50, mu2 = 0, sd = 250,
  r = 2,
  rho1 = 0.5, rho2 = 0.5,
  alpha = 0.025,
  beta = 0.2
)

comparison_allocation <- data.frame(
  Design = c("Balanced (1:1)", "Unbalanced (2:1)"),
  n_treatment = c(result_balanced$n1, result_unbalanced$n1),
  n_control = c(result_balanced$n2, result_unbalanced$n2),
  N_total = c(result_balanced$N, result_unbalanced$N)
)

kable(comparison_allocation,
      caption = "Comparison: Balanced vs Unbalanced Allocation",
      col.names = c("Design", "n₁", "n₂", "N total"))

cat("\nIncrease in total sample size:", 
    round((result_unbalanced$N - result_balanced$N) / result_balanced$N * 100, 1), "%\n")
#> 
#> Increase in total sample size: 11.1 %

Design Considerations

1. Estimating dispersion parameter: Use pilot data or historical studies to estimate \(\nu\). Underestimating \(\nu\) (overestimating overdispersion) leads to conservative sample sizes.

2. Estimating correlation: Use pilot data; be conservative if uncertain (\(\rho = 0\) is conservative).

3. Direction of benefit: For COPD/asthma trials, ensure test directions are correct (lower is better for both endpoints).

4. Balanced allocation: Generally most efficient (\(r = 1\)) unless practical constraints require otherwise.

5. Sensitivity analysis: Calculate sample sizes for range of plausible \(\nu\), \(\rho\), and effect sizes.

When to Use This Method

Use mixed count-continuous methods when:

• One endpoint is an overdispersed count (exacerbations, events)
• Other endpoint is continuous (lung function, quality of life)
• Both endpoints are clinically meaningful co-primary endpoints
• Negative binomial distribution is appropriate for count data

Challenges and Considerations

1. Overdispersion estimation: Requires historical data or pilot studies; misspecification affects sample size

2. Correlation estimation: Correlation between count and continuous outcomes is challenging to estimate

3. Direction specification: Ensure treatment benefit direction is correctly specified for both endpoints