Transform Vertex/Inequality Representation of Polytope
For convex polytopes: Requires rPorta (https://github.com/TasCL/rPorta) to transform the vertex representation to/from the inequality representation. Since rPorta cannot be compiled with R versions >=4.0.0 anymore, the function is currently deprecated.
V_to_Ab(V) Ab_to_V(A, b, options = 2)
V: a matrix with one vertex of a polytope per row (e.g., the admissible preference orders of a random utility model or any other theory). Since the values have to sum up to one within each multinomial condition, the last value of each multinomial is omitted (e.g., the prediction 1-0-0/0-1 for a tri and binomial becomes 1-0/0).
A: a matrix defining the convex polytope via A*x <= b. The columns of A do not include the last choice option per item type and thus the number of columns must be equal to sum(options-1)
(e.g., the column order of A for k = c(a1,a2,a2, b1,b2)
is c(a1,a2, b1)).
b: a vector of the same length as the number of rows of A.
options: number of choice options per item type. Can be a vector options=c(2,3,4) if item types have 2/3/4 choice options.
Choice models can be represented as polytopes if they assume a latent mixture over a finite number preference patterns (random preference model). For the general approach and theory underlying binary and ternary choice models, see Regenwetter et al. (2012, 2014, 2017).
The function is currently deprecated since the package rPorta cannot be compiled with R>=4.0.0!
For binary choices (options=2), additional constraints are added to A and b
to ensure that all dimensions of the polytope satisfy: 0 <= p_i <= 1. For ternary choices (options=3), constraints are added to ensure that 0 <= p_1+p_2 <=1 for pairwise columns (1+2, 3+4, 5+6, ...). See Ab_multinom.
Regenwetter, M., & Davis-Stober, C. P. (2012). Behavioral variability of choices versus structural inconsistency of preferences. Psychological Review, 119(2), 408-416. tools:::Rd_expr_doi("10.1037/a0027372")
Regenwetter, M., Davis-Stober, C. P., Lim, S. H., Guo, Y., Popova, A., Zwilling, C., … Messner, W. (2014). QTest: Quantitative testing of theories of binary choice. Decision, 1(1), 2-34. tools:::Rd_expr_doi("10.1037/dec0000007")
Regenwetter, M., & Robinson, M. M. (2017). The construct–behavior gap in behavioral decision research: A challenge beyond replicability. Psychological Review, 124(5), 533-550. https://doi.org/10.1037/rev0000067