inside_binom function

Check Whether Choice Frequencies are in Polytope

Computes relative choice frequencies and checks whether these are in the polytope defined via (1) A*x <= b or (2) by the convex hull of a set of vertices V.

inside_binom(k, n, A, b, V)

inside_multinom(k, options, A, b, V)

Arguments

• k: choice frequencies. For inside_binom: per item type (e.g.: a1,b1,c1,..) For inside_multinom: for all choice options ordered by item type (e.g., for ternary choices: a1,a2,a3, b1,b2,b3,..)
• n: only for inside_binom: number of choices per item type.
• A: a matrix with one row for each linear inequality constraint and one column for each of the free parameters. The parameter space is defined as all probabilities x that fulfill the order constraints A*x <= b.
• b: a vector of the same length as the number of rows of A.
• V: a matrix of vertices (one per row) that define the polytope of admissible parameters as the convex hull over these points (if provided, A and b are ignored). Similar as for A, columns of V omit the last value for each multinomial condition (e.g., a1,a2,a3,b1,b2 becomes a1,a2,b1). Note that this method is comparatively slow since it solves linear-programming problems to test whether a point is inside a polytope (Fukuda, 2004) or to run the Gibbs sampler.
• options: only for inside_multinom: number of response options per item type.

Examples

############ binomial
# x1<x2<x3<.50:
A <- matrix(c(
  1, -1, 0,
  0, 1, -1,
  0, 0, 1
), ncol = 3, byrow = TRUE)
b <- c(0, 0, .50)
k <- c(0, 1, 5)
n <- c(10, 10, 10)
inside_binom(k, n, A, b)

############ multinomial
# two ternary choices:
#     (a1,a2,a3,   b1,b2,b3)
k <- c(1, 4, 10, 5, 9, 1)
options <- c(3, 3)
# a1<b1, a2<b2, no constraints on a3, b3
A <- matrix(c(
  1, -1, 0, 0,
  0, 0, 1, -1
), ncol = 4, byrow = TRUE)
b <- c(0, 0)
inside_multinom(k, options, A, b)

# V-representation:
V <- matrix(c(
  0, 0, 0, 0,
  0, 0, 0, 1,
  0, 1, 0, 0,
  0, 0, 1, 1,
  0, 1, 0, 1,
  1, 1, 0, 0,
  0, 1, 1, 1,
  1, 1, 0, 1,
  1, 1, 1, 1
), 9, 4, byrow = TRUE)
inside_multinom(k, options, V = V)

See Also

inside