Non-Survival Endpoints: Continuous, Binary, and Count

This vignette describes Regional Consistency Probability (RCP) calculations for three non-survival endpoint types: continuous, binary, and count (negative binomial). For each endpoint, the statistical model, treatment effect scale, closed-form formulae, and worked examples are provided.

1. Continuous Endpoint

Statistical model

Let \(\hat{\mu}_j\) denote the sample mean for Region \(j\). Under the assumption that individual observations are independently and identically distributed as \(N(\mu, \sigma^2)\) within each region, the regional sample means are:

\[ \hat{\mu}_j \sim N\!\left(\mu,\; \frac{\sigma^2}{N_j}\right), \qquad j = 1, \ldots, J \]

independently across regions. The treatment effect relative to a historical control mean \(\mu_0\) is \(\delta = \mu - \mu_0 > 0\).

Consistency criteria

Method 1 (Effect Retention):

\[ \text{RCP}_1 = \Pr\!\left[\,(\hat{\mu}_1 - \mu_0) \geq \pi\,(\hat{\mu} - \mu_0)\,\right] \]

Defining \(D = (\hat{\mu}_1 - \mu_0) - \pi(\hat{\mu} - \mu_0)\), the condition \(D \geq 0\) is equivalent to:

\[ D = (1 - \pi f_1)\,(\hat{\mu}_1 - \mu_0) - \pi(1 - f_1)\,(\hat{\mu}_{-1} - \mu_0) \geq 0 \]

where \(\hat{\mu}_{-1}\) is the sample mean pooled over regions \(2, \ldots, J\). Under homogeneity:

\[ E[D] = (1 - \pi)\,\delta, \qquad \mathrm{Var}(D) = (1 - \pi f_1)^2\,\frac{\sigma^2}{N_1} + \bigl[\pi(1 - f_1)\bigr]^2\,\frac{\sigma^2}{N - N_1} \]

Therefore:

\[ \text{RCP}_1 = \Phi\!\left(\frac{(1 - \pi)\,\delta} {\sqrt{(1 - \pi f_1)^2\,\sigma^2/N_1 + \{\pi(1 - f_1)\}^2\,\sigma^2/(N - N_1)}}\right) \]

Method 2 (Simultaneous Positivity):

\[ \text{RCP}_2 = \Pr\!\left[\,\hat{\mu}_j > \mu_0 \;\text{ for all } j\,\right] = \prod_{j=1}^{J} \Phi\!\left(\frac{\delta\,\sqrt{N_j}}{\sigma}\right) \]

Example

Setting: \(\mu = 0.5\), \(\mu_0 = 0.1\), \(\sigma = 1\), \(N = 100\) (\(J = 3\) regions with \(N_1 = 20\)), \(\pi = 0.5\).

result_f <- rcp1armContinuous(
  mu       = 0.5,
  mu0      = 0.1,
  sd       = 1,
  Nj       = c(20, 40, 40),
  PI       = 0.5,
  approach = "formula"
)
print(result_f)
#> 
#> Regional Consistency Probability for Single-Arm MRCT
#> Endpoint : Continuous
#> 
#>    Approach    : Closed-Form Solution
#>    Target Mean : mu  = 0.5000
#>    Null Mean   : mu0 = 0.1000
#>    Std. Dev.   : sd  = 1.0000
#>    Sample Size : Nj  = (20, 40, 40)
#>    Total Size  : N   = 100
#>    Threshold   : PI  = 0.5000
#> 
#> Consistency Probabilities:
#>    Method 1 (Region 1 vs Overall)  : 0.8340
#>    Method 2 (All Regions > mu0)    : 0.9522
result_s <- rcp1armContinuous(
  mu       = 0.5,
  mu0      = 0.1,
  sd       = 1,
  Nj       = c(20, 40, 40),
  PI       = 0.5,
  approach = "simulation",
  nsim     = 10000,
  seed     = 1
)
print(result_s)
#> 
#> Regional Consistency Probability for Single-Arm MRCT
#> Endpoint : Continuous
#> 
#>    Approach    : Simulation-Based (nsim = 10000)
#>    Target Mean : mu  = 0.5000
#>    Null Mean   : mu0 = 0.1000
#>    Std. Dev.   : sd  = 1.0000
#>    Sample Size : Nj  = (20, 40, 40)
#>    Total Size  : N   = 100
#>    Threshold   : PI  = 0.5000
#> 
#> Consistency Probabilities:
#>    Method 1 (Region 1 vs Overall)  : 0.8338
#>    Method 2 (All Regions > mu0)    : 0.9479

Visualisation

plot_rcp1armContinuous(
  mu        = 0.5,
  mu0       = 0.1,
  sd        = 1,
  PI        = 0.5,
  N_vec     = c(20, 40, 100),
  J         = 3,
  nsim      = 5000,
  seed      = 1,
  base_size = 8
)

Line plot of RCP versus f1 for a continuous endpoint with mu = 0.5, mu0 = 0.1, sigma = 1, showing Method 1 and Method 2 across N = 20, 40, 100

2. Binary Endpoint

Statistical model

Let \(Y_j\) denote the number of responders in Region \(j\). Under independent Bernoulli trials with a common response rate \(p\):

\[ Y_j \sim \mathrm{Binomial}(N_j,\; p), \qquad j = 1, \ldots, J \]

independently across regions. The regional response rate estimator is \(\hat{p}_j = Y_j / N_j\), the overall estimator is \(\hat{p} = \sum_j Y_j / N\), and the treatment effect is \(\delta = p - p_0 > 0\).

Consistency criteria

Method 1 (Effect Retention) — Exact Enumeration:

\[ \text{RCP}_1 = \Pr\!\left[\,(\hat{p}_1 - p_0) \geq \pi\,(\hat{p} - p_0)\,\right] \]

By the additivity of independent binomials, \(Y_{-1} = \sum_{j \geq 2} Y_j \sim \mathrm{Binomial}(N - N_1,\; p)\). The formula approach enumerates all combinations \((y_1, y_{-1}) \in \{0, \ldots, N_1\} \times \{0, \ldots, N - N_1\}\) and sums the joint probabilities satisfying the consistency condition:

\[ \text{RCP}_1 = \sum_{y_1=0}^{N_1} \sum_{y_{-1}=0}^{N-N_1} b(y_1;\,N_1,\,p)\;b(y_{-1};\,N{-}N_1,\,p) \cdot \mathbf{1}\!\left[\frac{y_1}{N_1} - p_0 \geq \pi\!\left(\frac{y_1+y_{-1}}{N} - p_0\right)\right] \]

where \(b(y;\,n,\,p) = \binom{n}{y}p^y(1-p)^{n-y}\).

Method 2 (Simultaneous Positivity):

The condition \(\hat{p}_j > p_0\) is equivalent to \(Y_j \geq y_{j,\min}\) where \(y_{j,\min} = \lfloor N_j p_0 \rfloor + 1\). Denoting by \(F_{\mathrm{Bin}(n,\,p)}(k)\) the CDF of the binomial distribution with parameters \(n\) and \(p\) evaluated at \(k\):

\[ \text{RCP}_2 = \prod_{j=1}^{J} \left[1 - F_{\mathrm{Bin}(N_j,\,p)}(y_{j,\min} - 1)\right] \]

Example

Setting: \(p = 0.5\), \(p_0 = 0.2\), \(N = 100\) (\(J = 3\) regions with \(N_1 = 20\)), \(\pi = 0.5\).

result_f <- rcp1armBinary(
  p        = 0.5,
  p0       = 0.2,
  Nj       = c(20, 40, 40),
  PI       = 0.5,
  approach = "formula"
)
print(result_f)
#> 
#> Regional Consistency Probability for Single-Arm MRCT
#> Endpoint : Binary
#> 
#>    Approach      : Exact Solution
#>    Response Rate : p  = 0.5000
#>    Null Rate     : p0 = 0.2000
#>    Sample Size   : Nj = (20, 40, 40)
#>    Total Size    : N  = 100
#>    Threshold     : PI = 0.5000
#> 
#> Consistency Probabilities:
#>    Method 1 (Region 1 vs Overall) : 0.9234
#>    Method 2 (All Regions > p0)    : 0.9939
result_s <- rcp1armBinary(
  p        = 0.5,
  p0       = 0.2,
  Nj       = c(20, 40, 40),
  PI       = 0.5,
  approach = "simulation",
  nsim     = 10000,
  seed     = 1
)
print(result_s)
#> 
#> Regional Consistency Probability for Single-Arm MRCT
#> Endpoint : Binary
#> 
#>    Approach      : Simulation-Based (nsim = 10000)
#>    Response Rate : p  = 0.5000
#>    Null Rate     : p0 = 0.2000
#>    Sample Size   : Nj = (20, 40, 40)
#>    Total Size    : N  = 100
#>    Threshold     : PI = 0.5000
#> 
#> Consistency Probabilities:
#>    Method 1 (Region 1 vs Overall) : 0.9203
#>    Method 2 (All Regions > p0)    : 0.9933

Visualisation

plot_rcp1armBinary(
  p         = 0.5,
  p0        = 0.2,
  PI        = 0.5,
  N_vec     = c(20, 40, 100),
  J         = 3,
  nsim      = 5000,
  seed      = 1,
  base_size = 8
)

Line plot of RCP versus f1 for a binary endpoint with p = 0.5, p0 = 0.2, showing Method 1 and Method 2 across N = 20, 40, 100

3. Count Endpoint (Negative Binomial)

Statistical model

Count data are modelled by the negative binomial distribution. The total event count in Region \(j\) is:

\[ Y_j \sim \mathrm{NB}\!\left(\mu = N_j\,\lambda,\;\; \mathrm{size} = N_j\,\phi\right), \qquad j = 1, \ldots, J \]

independently across regions, where \(\lambda > 0\) is the expected count per patient under the alternative and \(\phi > 0\) is the dispersion parameter. The regional rate estimator is \(\hat{\lambda}_j = Y_j / N_j\), and the treatment effect is expressed as a rate ratio:

\[ \widehat{RR}_j = \frac{\hat{\lambda}_j}{\lambda_0} \]

Benefit is indicated by \(RR = \lambda / \lambda_0 < 1\).

By the reproducibility property of the negative binomial, the pooled count for regions \(2, \ldots, J\) follows \(\mathrm{NB}(\mu = (N - N_1)\lambda,\; \mathrm{size} = (N - N_1)\phi)\), enabling exact enumeration analogous to the binary case.

Consistency criteria

Method 1 (log-RR scale):

\[ \text{RCP}_{1,\log} = \Pr\!\left[\,\log(\widehat{RR}_1) \leq \pi\,\log(\widehat{RR})\,\right] \]

Since \(RR < 1\) (benefit), \(\log(RR) < 0\), so the condition requires \(\log(\widehat{RR}_1)\) to be sufficiently negative relative to the overall \(\log(\widehat{RR})\).

Method 1 (linear-RR scale):

\[ \text{RCP}_{1,\text{lin}} = \Pr\!\left[\,(1 - \widehat{RR}_1) \geq \pi\,(1 - \widehat{RR})\,\right] \]

Both Method 1 variants use exact enumeration over all \((y_1, y_{-1})\) combinations via the outer product of negative binomial PMFs.

Method 2:

Denoting by \(F_{\mathrm{NB}(\mu,\,\phi)}(k)\) the CDF of the negative binomial distribution with mean \(\mu\) and size \(\phi\) evaluated at \(k\), the condition \(\widehat{RR}_j < 1\) is equivalent to \(Y_j < N_j\lambda_0\), i.e., \(Y_j \leq \lfloor N_j\lambda_0 \rfloor - 1\) when \(N_j\lambda_0\) is not an integer (and \(Y_j \leq N_j\lambda_0 - 1\) otherwise). Therefore:

\[ \text{RCP}_2 = \prod_{j=1}^{J} \Pr\!\left(\widehat{RR}_j < 1\right) = \prod_{j=1}^{J} F_{\mathrm{NB}(N_j\lambda,\,N_j\phi)}\!\left(\lfloor N_j\lambda_0 \rfloor - 1\right) \]

Example

Setting: \(\lambda = 2\), \(\lambda_0 = 3\), \(\phi = 1\), \(N = 100\) (\(J = 3\) regions with \(N_1 = 20\)), \(\pi = 0.5\).

result_f <- rcp1armCount(
  lambda     = 2,
  lambda0    = 3,
  dispersion = 1,
  Nj         = c(20, 40, 40),
  PI         = 0.5,
  approach   = "formula"
)
print(result_f)
#> 
#> Regional Consistency Probability for Single-Arm MRCT
#> Endpoint : Count (Negative Binomial)
#> 
#>    Approach       : Exact Solution
#>    Expected Count : lambda     = 2.000000
#>    Control Count  : lambda0    = 3.000000
#>    Dispersion     : dispersion = 1.000000
#>    Sample Size    : Nj         = (20, 40, 40)
#>    Total Size     : N          = 100
#>    Threshold      : PI         = 0.5000
#> 
#> Consistency Probabilities:
#>    Method 1 (Region 1 vs Overall):
#>       Log-RR based    : 0.8186
#>       Linear-RR based : 0.8406
#>    Method 2 (All Regions Show Benefit):
#>       RR < 1          : 0.9320
result_s <- rcp1armCount(
  lambda     = 2,
  lambda0    = 3,
  dispersion = 1,
  Nj         = c(20, 40, 40),
  PI         = 0.5,
  approach   = "simulation",
  nsim       = 10000,
  seed       = 1
)
print(result_s)
#> 
#> Regional Consistency Probability for Single-Arm MRCT
#> Endpoint : Count (Negative Binomial)
#> 
#>    Approach       : Simulation-Based (nsim = 10000)
#>    Expected Count : lambda     = 2.000000
#>    Control Count  : lambda0    = 3.000000
#>    Dispersion     : dispersion = 1.000000
#>    Sample Size    : Nj         = (20, 40, 40)
#>    Total Size     : N          = 100
#>    Threshold      : PI         = 0.5000
#> 
#> Consistency Probabilities:
#>    Method 1 (Region 1 vs Overall):
#>       Log-RR based    : 0.8118
#>       Linear-RR based : 0.8331
#>    Method 2 (All Regions Show Benefit):
#>       RR < 1          : 0.9276

The output reports three RCP values: Method 1 on the log-RR scale (Method1_logRR), Method 1 on the linear-RR scale (Method1_linearRR), and Method 2 (Method2).

Visualisation

The count endpoint plot uses a grid layout: facet rows distinguish the two Method 1 scales (log-RR and \(1 - RR\)), and facet columns correspond to different total sample sizes.

plot_rcp1armCount(
  lambda     = 2,
  lambda0    = 3,
  dispersion = 1,
  PI         = 0.5,
  N_vec      = c(20, 40, 100),
  J          = 3,
  nsim       = 5000,
  seed       = 1,
  base_size  = 11
)

Grid plot of RCP versus f1 for a count endpoint with lambda = 2, lambda0 = 3, showing Method 1 on log-RR and linear-RR scales and Method 2 across N = 20, 40, 100

Summary

Endpoint Model Effect parameter Benefit direction Method 1 computation Method 2 computation
Continuous Normal \(\delta = \mu - \mu_0\) \(\hat{\mu}_j > \mu_0\) Closed-form (normal approximation) Product of normal tail probabilities
Binary Binomial \(\delta = p - p_0\) \(\hat{p}_j > p_0\) Exact enumeration (binomial) Product of binomial tail probabilities
Count Negative binomial \(\log(RR) = \log(\lambda/\lambda_0)\) (Method 1, log-RR scale); \(1 - RR = 1 - \lambda/\lambda_0\) (Method 1, linear-RR scale) \(\widehat{RR}_j < 1\) Exact enumeration (negative binomial) Product of NB tail probabilities

References

Homma G (2024). Cautionary note on regional consistency evaluation in multiregional clinical trials with binary outcomes. Pharmaceutical Statistics, 23(3):385–398. https://doi.org/10.1002/pst.2358