bf_binom {multinomineq}

Bayes Factor for Linear Inequality Constraints

Description

Computes the Bayes factor for product-binomial/-multinomial models with linear order-constraints (specified via: A*x <= b or the convex hull V).

Usage

bf_binom(k, n, A, b, V, map, prior = c(1, 1), log = FALSE, ...)

bf_multinom(
  k,
  options,
  A,
  b,
  V,
  prior = rep(1, sum(options)),
  log = FALSE,
  ...
)

Arguments

k

vector of observed response frequencies.

n

the number of choices per item type. If k=n=0, Bayesian inference is relies on the prior distribution only.

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.

map

optional: numeric vector of the same length as k with integers mapping the frequencies k to the free parameters/columns of A/V, thereby allowing for equality constraints (e.g., map=c(1,1,2,2)). Reversed probabilities 1-p are coded by negative integers. Guessing probabilities of .50 are encoded by zeros. The default assumes different parameters for each item type: map=1:ncol(A)

prior

a vector with two positive numbers defining the shape parameters of the beta prior distributions for each binomial rate parameter.

log

whether to return the log-Bayes factor instead of the Bayes factor

...

further arguments passed to count_binom or count_multinom (e.g., M, steps).

options

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

Details

For more control, use count_binom to specifiy how many samples should be drawn from the prior and posterior, respectively. This is especially recommended if the same prior distribution (and thus the same prior probability/integral) is used for computing BFs for multiple data sets that differ only in the observed frequencies k and the sample size n. In this case, the prior probability/proportion of the parameter space in line with the inequality constraints can be computed once with high precision (or even analytically), and only the posterior probability/proportion needs to be estimated separately for each unique vector k.

Value

a matrix with two columns (Bayes factor and SE of approximation) and three rows:

• `bf_0u`: constrained vs. unconstrained (saturated) model

• `bf_u0`: unconstrained vs. constrained model

• `bf_00'`: constrained vs. complement of inequality-constrained model (e.g., pi>.2 becomes pi<=.2; this assumes identical equality constraints for both models)

References

Karabatsos, G. (2005). The exchangeable multinomial model as an approach to testing deterministic axioms of choice and measurement. Journal of Mathematical Psychology, 49(1), 51-69. doi:10.1016/j.jmp.2004.11.001

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. doi:10.1037/dec0000007

See Also

count_binom and count_multinom for for more control on the number of prior/posterior samples and bf_nonlinear for nonlinear order constraints.

Examples

k <- c(0, 3, 2, 5, 3, 7)
n <- rep(10, 6)

# linear order constraints:
#             p1 <p2 <p3 <p4 <p5 <p6 <.50
A <- matrix(
  c(
    1, -1, 0, 0, 0, 0,
    0, 1, -1, 0, 0, 0,
    0, 0, 1, -1, 0, 0,
    0, 0, 0, 1, -1, 0,
    0, 0, 0, 0, 1, -1,
    0, 0, 0, 0, 0, 1
  ),
  ncol = 6, byrow = TRUE
)
b <- c(0, 0, 0, 0, 0, .50)

# Bayes factor: unconstrained vs. constrained
bf_binom(k, n, A, b, prior = c(1, 1), M = 10000)
bf_binom(k, n, A, b, prior = c(1, 1), M = 2000, steps = c(2, 4, 5))
bf_binom(k, n, A, b, prior = c(1, 1), M = 1000, cmin = 200)