binom_bf_equality {multibridge} | R Documentation |
Computes Bayes factor for equality constrained binomial parameters.
Null hypothesis H_0
states that binomial proportions are exactly equal or
exactly equal and equal to p
.
Alternative hypothesis H_e
states that binomial proportions are free to vary.
binom_bf_equality(x, n = NULL, a, b, p = NULL)
x |
a vector of counts of successes, or a two-dimensional table (or matrix) with 2 columns, giving the counts of successes and failures, respectively |
n |
numeric. Vector of counts of trials. Must be the same length as |
a |
numeric. Vector with alpha parameters. Must be the same length as |
b |
numeric. Vector with beta parameters. Must be the same length as |
p |
numeric. Hypothesized probability of success. Must be greater than 0 and less than 1. Default sets all binomial proportions exactly equal without specifying a specific value. |
The model assumes that the data in x
(i.e., x_1, ..., x_K
) are the observations of K
independent
binomial experiments, based on n_1, ..., n_K
observations. Hence, the underlying likelihood is the product of the
k = 1, ..., K
individual binomial functions:
(x_1, ... x_K) ~ \prod Binomial(N_k, \theta_k)
Furthermore, the model assigns a beta distribution as prior to each model parameter (i.e., underlying binomial proportions). That is:
\theta_k ~ Beta(\alpha_k, \beta_k)
Returns a data.frame
containing the Bayes factors LogBFe0
, BFe0
, and BF0e
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
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.
Other functions to evaluate informed hypotheses:
binom_bf_inequality()
,
binom_bf_informed()
,
mult_bf_equality()
,
mult_bf_inequality()
,
mult_bf_informed()
data(journals)
x <- journals$errors
n <- journals$nr_NHST
a <- rep(1, nrow(journals))
b <- rep(1, nrow(journals))
binom_bf_equality(x=x, n=n, a=a, b=b)