Ab_max {multinomineq}

Automatic Construction of Ab-Representation for Common Inequality Constraints

Description

Constructs the matrix A and vector b of the Ab-representation A*x < b for common inequality constraints such as "the probability j is larger than all others (Ab_max)" or "the probabilities are ordered (Ab_monotonicity)").

Usage

Ab_max(
  which_max,
  options,
  exclude = c(),
  exclude_fixed = FALSE,
  drop_fixed = TRUE
)

Arguments

which_max

vector of indices refering to probabilities that are assumed to be larger than the remaining probabilities (e.g., which_max=c(1,2) means that p1>p3, p1>p4,..., p2>p3, ...). Note that the indices refer to all probabilities/categories (including one fixed probability within each multinomial distribution).

options

number of observable categories/probabilities for each item type/multinomial distribution, e.g., c(3,2) for a ternary and binary item.

exclude

vector of indices refering to probabilities that are excluded from the construction of the order constraints (including the fixed probabilities).

exclude_fixed

whether to exclude the fixed probabilities (i.e., the last probability within each multinomial) from the construction of the order constraints. For example, if options=c(2,2,3) then the probabilities/columns 2, 4, and 7 are dropped (which is equivalent to exclude=c(2,4,7)). This option is usually appropriate for binomial probabilities (i.e., if options = c(2,2,2,...)), e.g., when the interest is in the probability of correct responding across different item types.

drop_fixed

whether to drop columns of A containing the fixed probabilities (i.e., the last probability within each multinomial). after construction of the inequalities.

Value

a list with the matrix A and the vectors b and options

Examples

# Example 1: Multinomial with 5 categories
# Hypothesis: p1 is larger than p2,p3,p4,p5
Ab_max(which_max = 1, options = 5)

# Example 2: Four binomial probabilities
# Hypothesis: p1 is larger than p2,p3,p4
Ab_max(which_max = 1, options = c(2, 2, 2, 2), exclude_fixed = TRUE)