Enumerate All Multinomial Count Vectors for a Bivariate Binary Outcome

Generates the complete set of non-negative integer vectors \((x_{00}, x_{01}, x_{10}, x_{11})\) satisfying \(x_{00} + x_{01} + x_{10} + x_{11} = n_j\), where \(n_j\) is the sample size of group \(j \in \{t, c\}\). Each row of the returned matrix corresponds to one possible realisation of the aggregated bivariate binary counts. This enumeration is a prerequisite for exact operating characteristic assessment in pbayesdecisionprob2bin.

Usage

allmultinom(n)

Arguments

n

A single non-negative integer giving the sample size of the group (\(n_j\)).

Value

An integer matrix with \(\binom{n_j + 3}{3}\) rows and 4 columns named x00, x01, x10, x11. Each row is a distinct non-negative integer solution to \(x_{00} + x_{01} + x_{10} + x_{11} = n_j\). Rows are ordered by x00 (ascending), then x10 (ascending), then x01 (ascending).

Details

The number of non-negative integer solutions to \(x_{00} + x_{01} + x_{10} + x_{11} = n_j\) is the stars-and-bars count $$\binom{n_j + 3}{3} = \frac{(n_j + 1)(n_j + 2)(n_j + 3)}{6}.$$ For example, \(n_j = 10\) yields 286 rows and \(n_j = 20\) yields 1771 rows. The matrix is pre-allocated to avoid repeated memory reallocation, making the function efficient for the sample sizes typical in rare-disease proof-of-concept studies.

This function is an internal computational building block used by pbayesdecisionprob2bin to enumerate the sample space over which multinomial probabilities and decision indicators are summed when computing exact operating characteristics.

Examples

# Example 1: n = 2 (smallest non-trivial case)
allmultinom(2)
#>       x00 x01 x10 x11
#>  [1,]   0   0   0   2
#>  [2,]   0   1   0   1
#>  [3,]   0   2   0   0
#>  [4,]   0   0   1   1
#>  [5,]   0   1   1   0
#>  [6,]   0   0   2   0
#>  [7,]   1   0   0   1
#>  [8,]   1   1   0   0
#>  [9,]   1   0   1   0
#> [10,]   2   0   0   0

# Example 2: n = 10 (typical rare-disease PoC group size)
mat <- allmultinom(10)
nrow(mat)              # Should be choose(13, 3) = 286
#> [1] 286
all(rowSums(mat) == 10)  # Every row must sum to n
#> [1] TRUE

# Example 3: n = 0 (edge case - only the all-zero row)
allmultinom(0)
#>      x00 x01 x10 x11
#> [1,]   0   0   0   0

# Example 4: Verify column names
colnames(allmultinom(5))
#> [1] "x00" "x01" "x10" "x11"

# Example 5: Row counts match the stars-and-bars formula
n <- 15L
mat <- allmultinom(n)
nrow(mat) == choose(n + 3L, 3L)  # Should be TRUE
#> [1] TRUE