mult_bf_equality {multibridge}

Computes Bayes Factors For Equality Constrained Multinomial Parameters

Description

Computes Bayes factor for equality constrained multinomial parameters using the standard Bayesian multinomial test. Null hypothesis H_0 states that category proportions are exactly equal to those specified in p. Alternative hypothesis H_e states that category proportions are free to vary.

Usage

mult_bf_equality(x, a, p = rep(1/length(a), length(a)))

Arguments

x

numeric. Vector with data

a

numeric. Vector with concentration parameters of Dirichlet distribution. Must be the same length as x. Default sets all concentration parameters to 1

p

numeric. A vector of probabilities of the same length as x. Its elements must be greater than 0 and less than 1. Default is 1/K

Details

The model assumes that data follow a multinomial distribution and assigns a Dirichlet distribution as prior for the model parameters (i.e., underlying category proportions). That is:

x ~ Multinomial(N, \theta)

\theta ~ Dirichlet(\alpha)

Value

Returns a data.frame containing the Bayes factors LogBFe0, BFe0, and BF0e

Note

The following signs can be used to encode restricted hypotheses: "<" and ">" for inequality constraints, "=" for equality constraints, "," for free parameters, and "&" for independent hypotheses. The restricted hypothesis can either be a string or a character vector. For instance, the hypothesis c("theta1 < theta2, theta3") means

• theta1 is smaller than both theta2 and theta3

• The parameters theta2 and theta3 both have theta1 as lower bound, but are not influenced by each other.

The hypothesis c("theta1 < theta2 = theta3 & theta4 > theta5") means that

• Two independent hypotheses are stipulated: "theta1 < theta2 = theta3" and "theta4 > theta5"

• The restrictions on the parameters theta1, theta2, and theta3 do not influence the restrictions on the parameters theta4 and theta5.

• theta1 is smaller than theta2 and theta3

• theta2 and theta3 are assumed to be equal

• theta4 is larger than theta5

References

Damien P, Walker SG (2001). “Sampling truncated normal, beta, and gamma densities.” Journal of Computational and Graphical Statistics, 10, 206–215.

Gronau QF, Sarafoglou A, Matzke D, Ly A, Boehm U, Marsman M, Leslie DS, Forster JJ, Wagenmakers E, Steingroever H (2017). “A tutorial on bridge sampling.” Journal of Mathematical Psychology, 81, 80–97.

Frühwirth-Schnatter S (2004). “Estimating marginal likelihoods for mixture and Markov switching models using bridge sampling techniques.” The Econometrics Journal, 7, 143–167.

Sarafoglou A, Haaf JM, Ly A, Gronau QF, Wagenmakers EJ, Marsman M (2021). “Evaluating Multinomial Order Restrictions with Bridge Sampling.” Psychological Methods.

See Also

Other functions to evaluate informed hypotheses: binom_bf_equality(), binom_bf_inequality(), binom_bf_informed(), mult_bf_inequality(), mult_bf_informed()

Examples

data(lifestresses)
x <- lifestresses$stress.freq
a <- rep(1, nrow(lifestresses))
mult_bf_equality(x=x, a=a)