Title:	Evaluating Multinomial Order Restrictions with Bridge Sampling
Version:	1.2.0
Date:	2022-10-28
Description:	Evaluate hypotheses concerning the distribution of multinomial proportions using bridge sampling. The bridge sampling routine is able to compute Bayes factors for hypotheses that entail inequality constraints, equality constraints, free parameters, and mixtures of all three. These hypotheses are tested against the encompassing hypothesis, that all parameters vary freely or against the null hypothesis that all category proportions are equal. For more information see Sarafoglou et al. (2020) <doi:10.31234/osf.io/bux7p>.
URL:	https://github.com/asarafoglou/multibridge/
BugReports:	https://github.com/asarafoglou/multibridge/issues
License:	GPL-2
Encoding:	UTF-8
LazyData:	true
Imports:	Brobdingnag, coda, mvtnorm, purrr, Rcpp (≥ 0.12.17), magrittr, progress, Rdpack, stringr
Suggests:	knitr, rmarkdown, testthat
SystemRequirements:	GNU make, mpfr (>= 3.0.0), gmp (>= 6.2.1_1)
RoxygenNote:	7.2.1
LinkingTo:	Rcpp
VignetteBuilder:	knitr
RdMacros:	Rdpack
Depends:	R (≥ 3.5.0)
NeedsCompilation:	yes
Packaged:	2022-10-28 15:43:48 UTC; alexandra.sarafoglougmail.com
Author:	Alexandra Sarafoglou [aut, cre], Frederik Aust [aut], Julia M. Haaf [aut], Joris Goosen [aut], Quentin F. Gronau [aut], Maarten Marsman [aut]
Maintainer:	Alexandra Sarafoglou <alexandra.sarafoglou@gmail.com>
Repository:	CRAN
Date/Publication:	2022-11-01 13:37:41 UTC

Adjusts Upper Bound For Free Parameters

Description

Corrects the upper bound for current parameter. This correction only applies for parameters that are free to vary within the restriction. Then the length of the remaining stick must be based on the largest free parameter value.

Usage

.adjustUpperBoundForFreeParameters(
  theta_mat,
  k,
  upper,
  nr_mult_equal,
  smaller_values,
  larger_values,
  hyp_direction
)

Arguments

Value

adjusted upper bound

Computes Length Of Remaining Stick

Description

When applying the probit transformation on the Dirichlet samples, this function is used as part of the stick-breaking algorithm. It computes the length of the remaining stick when the current element is broken off.

Usage

.computeLengthOfRemainingStick(theta_mat, k, hyp_direction)

Arguments

Value

stick length

S3 method for class 'bayes_factor.bmult'

Description

Extracts information about computed Bayes factors from object of class bmult

Usage

bayes_factor(x)

Arguments

Value

Returns list with three data.frames. The first dataframe bf_table summarizes information the Bayes factor for equality and inequality constraints. The second dataframe error_measures contains for the overall Bayes factor the approximate relative mean-squared error re2, the approximate coefficient of variation cv, and the approximate percentage error percentage. The third dataframe $bf_ineq_table summarized information about the Bayes factor for inequality constraints, that is, the log marginal likelihood estimates for the constrained prior and posterior distribution. In addition, it contains for each independent Bayes factor the approximate relative mean-squared error re2

Note

In case the restricted hypothesis is tested against H_0 four data.frames will be returned. The fourth dataframe $bf_eq_table summarizes information about the Bayes factor for equality constraints compared to the encompassing hypothesis.

Examples

# data
x <- c(3, 4, 10, 11)
n <- c(15, 12, 12, 12)
# priors
a <- c(1, 1, 1, 1)
b <- c(1, 1, 1, 1)
# informed hypothesis
factor_levels <- c('theta1', 'theta2', 'theta3', 'theta4')
Hr            <- c('theta1', '<',  'theta2', '<', 'theta3', '<', 'theta4')

## Multinomial Case
out_mult  <- mult_bf_informed(x=x, Hr=Hr, a=a, factor_levels=factor_levels,
niter=1e3, seed=2020)
bayes_factor(out_mult)

Extracts information about computed Bayes factors from object of class bmult

Description

Extracts information about computed Bayes factors from object of class bmult

Usage

## S3 method for class 'bmult'
bayes_factor(x)

Arguments

Value

Returns list with three data.frames. The first dataframe bf_table summarizes information the Bayes factor for equality and inequality constraints. The second dataframe error_measures contains for the overall Bayes factor the approximate relative mean-squared error re2, the approximate coefficient of variation cv, and the approximate percentage error percentage. The third dataframe $bf_ineq_table summarized information about the Bayes factor for inequality constraints, that is, the log marginal likelihood estimates for the constrained prior and posterior distribution. In addition, it contains for each independent Bayes factor the approximate relative mean-squared error re2

Note

In case the restricted hypothesis is tested against H_0 four data.frames will be returned. The fourth dataframe $bf_eq_table summarizes information about the Bayes factor for equality constraints compared to the encompassing hypothesis.

Examples

# data
x <- c(3, 4, 10, 11)
n <- c(15, 12, 12, 12)
# priors
a <- c(1, 1, 1, 1)
b <- c(1, 1, 1, 1)
# informed hypothesis
factor_levels <- c('theta1', 'theta2', 'theta3', 'theta4')
Hr            <- c('theta1', '<',  'theta2', '<', 'theta3', '<', 'theta4')

## Multinomial Case
out_mult  <- mult_bf_informed(x=x, Hr=Hr, a=a, factor_levels=factor_levels,
niter=1e3, seed=2020)
bayes_factor(out_mult)

Computes Bayes Factors For Equality Constrained Binomial Parameters

Description

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.

Usage

binom_bf_equality(x, n = NULL, a, b, p = NULL)

Arguments

Details

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)

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_inequality(), binom_bf_informed(), mult_bf_equality(), mult_bf_inequality(), mult_bf_informed()

Examples

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)

Computes Bayes Factors For Inequality Constrained Independent Binomial Parameters

Description

Computes Bayes factor for inequality constrained binomial parameters using a bridge sampling routine. Restricted hypothesis H_r states that binomial proportions follow a particular trend. Alternative hypothesis H_e states that binomial proportions are free to vary.

Usage

binom_bf_inequality(
  samples = NULL,
  restrictions = NULL,
  x = NULL,
  n = NULL,
  Hr = NULL,
  a = rep(1, ncol(samples)),
  b = rep(1, ncol(samples)),
  factor_levels = NULL,
  prior = FALSE,
  index = 1,
  maxiter = 1000,
  seed = NULL,
  niter = 5000,
  nburnin = niter * 0.05
)

Arguments

Details

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)

Value

List consisting of the following elements:

$eval

• q11: log prior or posterior evaluations for prior or posterior samples

• q12: log proposal evaluations for prior or posterior samples

• q21: log prior or posterior evaluations for samples from proposal

• q22: log proposal evaluations for samples from proposal

$niter

number of iterations of the iterative updating scheme

$logml

estimate of log marginal likelihood

$hyp

evaluated inequality constrained hypothesis

$error_measures

• re2: the approximate relative mean-squared error for the marginal likelihood estimate

• cv: the approximate coefficient of variation for the marginal likelihood estimate (assumes that bridge estimate is unbiased)

• percentage: the approximate percentage error of the marginal likelihood estimate

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

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.

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

See Also

generate_restriction_list

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

Examples

# priors
a <- c(1, 1, 1, 1)
b <- c(1, 1, 1, 1)

# informed hypothesis
factor_levels <- c('theta1', 'theta2', 'theta3', 'theta4')
Hr            <- c('theta1', '<',  'theta2', '<', 'theta3', '<', 'theta4')

results_prior  <- binom_bf_inequality(Hr=Hr, a=a, b=b, 
factor_levels=factor_levels, prior=TRUE, seed = 2020)
# corresponds to
cbind(exp(results_prior$logml), 1/factorial(4))

# alternative - if you have samples and a restriction list
inequalities  <- generate_restriction_list(Hr=Hr, a=a,b=b,
factor_levels=factor_levels)$inequality_constraints
prior_samples <- binom_tsampling(inequalities, niter = 2e3, 
prior=TRUE, seed = 2020)
results_prior <- binom_bf_inequality(prior_samples, inequalities, seed=2020)
cbind(exp(results_prior$logml), 1/factorial(4))

Evaluates Informed Hypotheses on Multiple Binomial Parameters

Description

Evaluates informed hypotheses on multiple binomial parameters. These hypotheses can contain (a mixture of) inequality constraints, equality constraints, and free parameters. Informed hypothesis H_r states that binomial proportions obey a particular constraint. H_r can be tested against the encompassing hypothesis H_e or the null hypothesis H_0. Encompassing hypothesis H_e states that binomial proportions are free to vary. Null hypothesis H_0 states that category proportions are exactly equal.

Usage

binom_bf_informed(
  x,
  n = NULL,
  Hr,
  a,
  b,
  factor_levels = NULL,
  cred_level = 0.95,
  niter = 5000,
  bf_type = "LogBFer",
  seed = NULL,
  maxiter = 1000,
  nburnin = niter * 0.05
)

Arguments

Details

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)

Value

List consisting of the following elements

$bf_list

gives an overview of the Bayes factor analysis:

• bf_type: string. Contains Bayes factor type as specified by the user

• bf: data.frame. Contains Bayes factors for all Bayes factor types

• error_measures: data.frame. Contains for the overall Bayes factor the approximate relative mean-squared error re2, the approximate coefficient of variation cv, and the approximate percentage error percentage

• logBFe_equalities: data.frame. Lists the log Bayes factors for all independent equality constrained hypotheses

• logBFe_inequalities: data.frame. Lists the log Bayes factor for all independent inequality constrained hypotheses

$cred_level

numeric. User specified credible interval

$restrictions

list that encodes informed hypothesis for each independent restriction:

• full_model: list containing the hypothesis, parameter names, data and prior specifications for the full model.

• equality_constraints: list containing the hypothesis, parameter names, data and prior specifications for each equality constrained hypothesis.

• inequality_constraints: list containing the hypothesis, parameter names, data and prior specifications for each inequality constrained hypothesis. In addition, in nr_mult_equal and nr_mult_free encodes which and how many parameters are equality constraint or free, in boundaries includes the boundaries of each parameter, in nineq_per_hyp states the number of inequality constraint parameters per independent inequality constrained hypothesis, and in direction states the direction of the inequality constraint.

$bridge_output

list containing output from bridge sampling function:

• eval: list containing the log prior or posterior evaluations (q11) and the log proposal evaluations (q12) for the prior or posterior samples, as well as the log prior or posterior evaluations (q21) and the log proposal evaluations (q22) for the samples from the proposal distribution

• niter: number of iterations of the iterative updating scheme

• logml: estimate of log marginal likelihood

• hyp: evaluated inequality constrained hypothesis

• error_measures: list containing in re2 the approximate relative mean-squared error for the marginal likelihood estimate, in cv the approximate coefficient of variation for the marginal likelihood estimate (assumes that bridge estimate is unbiased), and in percentage the approximate percentage error of the marginal likelihood estimate

$samples

list containing a list for prior samples and a list of posterior samples from truncated distributions which were used to evaluate inequality constraints. Prior and posterior samples of independent inequality constraints are again saved in separate lists. Samples are stored as matrix of dimension nsamples x nparams.

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(), mult_bf_equality(), mult_bf_inequality(), mult_bf_informed()

Examples

# data
x <- c(3, 4, 10, 11)
n <- c(15, 12, 12, 12)
# priors
a <- c(1, 1, 1, 1)
b <- c(1, 1, 1, 1)
# informed hypothesis
factor_levels <- c('binom1', 'binom2', 'binom3', 'binom4')
Hr            <- c('binom1', '<',  'binom2', '<', 'binom3', '<', 'binom4')
output_total  <- binom_bf_informed(x, n, Hr, a, b, niter=2e3, factor_levels, seed=2020)

Samples From Truncated Beta Densities

Description

Based on specified inequality constraints, samples from truncated prior or posterior beta densities.

Usage

binom_tsampling(
  inequalities,
  index = 1,
  niter = 10000,
  prior = FALSE,
  nburnin = niter * 0.05,
  seed = NULL
)

Arguments

Details

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)

Value

matrix of dimension niter * nsamples containing samples from truncated beta distributions.

Note

When equality constraints are specified in the restricted hypothesis, this function samples from the conditional Beta distributions given that the equality constraints hold.

Only inequality constrained parameters are sampled. Free parameters or parameters that are exclusively equality constrained will be ignored.

References

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

See Also

generate_restriction_list

Examples

x <- c(200, 130, 40, 10)
n <- c(200, 200, 200, 200)
a <- c(1, 1, 1, 1)
b <- c(1, 1, 1, 1)
factor_levels <- c('binom1', 'binom2', 'binom3', 'binom4')
Hr <- c('binom1 > binom2 > binom3 > binom4')

# generate restriction list
inequalities <- generate_restriction_list(x=x, n=n, Hr=Hr, a=a, b=b, 
factor_levels=factor_levels)$inequality_constraints

# sample from prior distribution
prior_samples <- binom_tsampling(inequalities, niter = 500, 
prior=TRUE)
# sample from posterior distribution
post_samples <- binom_tsampling(inequalities, niter = 500)

S3 method for class bridge_output.bmult

Description

Extracts bridge sampling output from object of class bmult

Usage

bridge_output(x)

Arguments

Value

Extracts output related to the bridge sampling routine. The output contains the following elements::

$eval

• q11: log prior or posterior evaluations for prior or posterior samples

• q12: log proposal evaluations for prior or posterior samples

• q21: log prior or posterior evaluations for samples from proposal

• q22: log proposal evaluations for samples from proposal

$niter

number of iterations of the iterative updating scheme

$logml

estimate of log marginal likelihood

$hyp

evaluated inequality constrained hypothesis

$error_measures

• re2: the approximate relative mean-squared error for the marginal likelihood estimate

• cv: the approximate coefficient of variation for the marginal likelihood estimate (assumes that bridge estimate is unbiased)

• percentage: the approximate percentage error of the marginal likelihood estimate

Examples

# data
x <- c(3, 4, 10, 11)
n <- c(15, 12, 12, 12)
# priors
a <- c(1, 1, 1, 1)
b <- c(1, 1, 1, 1)
# informed hypothesis
factor_levels <- c('theta1', 'theta2', 'theta3', 'theta4')
Hr            <- c('theta1', '<',  'theta2', '<', 'theta3', '<', 'theta4')

## Multinomial Case
out_mult  <- mult_bf_informed(x=x, Hr=Hr, a=a, factor_levels=factor_levels,
niter=1e3, seed=2020)
bridge_output <- bridge_output(out_mult)

Extracts bridge sampling output from object of class bmult

Description

Extracts restriction list from an object of class bmult

Usage

## S3 method for class 'bmult'
bridge_output(x)

Arguments

Value

Extracts restriction list and associated hypothesis from an object of class bmult

Examples

# data
x <- c(3, 4, 10, 11)
n <- c(15, 12, 12, 12)
# priors
a <- c(1, 1, 1, 1)
b <- c(1, 1, 1, 1)
# informed hypothesis
factor_levels <- c('theta1', 'theta2', 'theta3', 'theta4')
Hr            <- c('theta1', '<',  'theta2', '<', 'theta3', '<', 'theta4')

## Multinomial Case
out_mult  <- mult_bf_informed(x=x, Hr=Hr, a=a, factor_levels=factor_levels,
niter=1e3, seed=2020)
restriction_list <- restriction_list(out_mult)

Creates Restriction List Based On User Specified Informed Hypothesis

Description

Encodes the user specified informed hypothesis. It creates a separate restriction list for the full model, and all independent equality and inequality constraints. The returned list features relevant information for the transformation and sampling of the model parameters, such as information about the upper and lower bound for each parameter, and the indexes of equality constrained and free parameters.

Usage

generate_restriction_list(x = NULL, n = NULL, Hr, a, b = NULL, factor_levels)

Arguments

Details

The restriction list can be created for both binomial and multinomial models. If multinomial models are specified, the arguments b and n should be left empty and x should not be a table or matrix.

Value

Restriction list containing the following elements:

$full_model

• hyp: character. Vector containing the informed hypothesis as specified by the user

• parameters_full: character. Vector containing the names for each constrained parameter

• alpha_full: numeric. Vector containing the concentration parameters of the Dirichlet distribution (when evaluating ordered multinomial parameters) or alpha parameters of the beta distribution (when evaluating ordered binomial parameters)

• beta_full: numeric. Vector containing the values of beta parameters of the beta distribution (when evaluating ordered binomial parameters)

• counts_full: numeric. Vector containing data values (when evaluating multinomial parameters), or number of successes (when evaluating ordered binomial parameters)

• total_full: numeric. Vector containing the number of observations (when evaluating ordered binomial parameters, that is, number of successes and failures)

$equality_constraints

• hyp: list. Contains all independent equality constrained hypotheses

• parameters_equality: character. Vector containing the names for each equality constrained parameter.

• equality_hypotheses: list. Contains the indexes of each equality constrained parameter. Note that these indices are based on the vector of all factor levels

• alpha_equalities: list. Contains the concentration parameters for equality constrained hypotheses (when evaluating multinomial parameters) or alpha parameters of the beta distribution (when evaluating ordered binomial parameters).

• beta_equalities: list. Contains the values of beta parameters of the beta distribution (when evaluating ordered binomial parameters)

• counts_equalities: list. Contains data values (when evaluating multinomial parameters), or number of successes (when evaluating ordered binomial parameters) of each equality constrained parameter

• total_equalitiesl: list. Contains the number of observations of each equality constrained parameter (when evaluating ordered binomial parameters, that is, number of successes and failures)

$inequality_constraints

• hyp: list. Contains all independent inequality constrained hypotheses

• parameters_inequality: list. Contains the names for each inequality constrained parameter

• inequality_hypotheses: list. Contains the indices of each inequality constrained parameter

• alpha_inequalities: list. Contains for inequality constrained hypotheses the concentration parameters of the Dirichlet distribution (when evaluating ordered multinomial parameters) or alpha parameters of the beta distribution (when evaluating ordered binomial parameters).

• beta_inequalities: list. Contains for inequality constrained hypotheses the values of beta parameters of the beta distribution (when evaluating ordered binomial parameters).

• counts_inequalities: list. Contains for inequality constrained parameter data values (when evaluating multinomial parameters), or number of successes (when evaluating ordered binomial parameters).

• total_inequalities: list. Contains for each inequality constrained parameter the number of observations (when evaluating ordered binomial parameters, that is, number of successes and failures).

• boundaries: list that lists for each inequality constrained parameter the index of parameters that serve as its upper and lower bounds. Note that these indices refer to the collapsed categories (i.e., categories after conditioning for equality constraints). If a lower or upper bound is missing, for instance because the current parameter is set to be the smallest or the largest, the bounds take the value int(0).

• nr_mult_equal: list. Contains multiplicative elements of collapsed categories

• nr_mult_free: list. Contains multiplicative elements of free parameters

• mult_equal: list. Contains for each lower and upper bound of each inequality constrained parameter necessary multiplicative elements to recreate the implied order restriction, even for collapsed parameter values. If there is no upper or lower bound, the multiplicative element will be 0.

• nineq_per_hyp: numeric. Vector containing the total number of inequality constrained parameters for each independent inequality constrained hypotheses.

• direction: character. Vector containing the direction for each independent inequality constrained hypothesis. Takes the values smaller or larger.

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

Examples

# Restriction list for ordered multinomial
x <- c(1, 4, 1, 10)
a <- c(1, 1, 1, 1)
factor_levels <- c('mult1', 'mult2', 'mult3', 'mult4')
Hr <- c('mult2 > mult1 , mult3 = mult4')
restrictions <- generate_restriction_list(x=x, Hr=Hr, a=a, 
factor_levels=factor_levels)

Prevalence of Statistical Reporting Errors

Description

This data set, "journals" provides a summary of statistical reporting errors (i.e., inconsistencies between reported test statistic and reported p-value) of 16,695 research articles reporting results from null hypothesis significance testing (NHST). The selected articles were published in eight major journals in psychology between 1985 to 2013:

• Developmental Psychology (DP)

• Frontiers in Psychology (FP)

• Journal of Applied Psychology (JAP)

• Journal of Consulting and Clinical Psychology (JCCP)

• Journal of Experimental Psychology: General (JEPG)

• Journal of Personality and Social Psychology (JPSP)

• Public Library of Science (PLoS)

• Psychological Science (PS)

In total, Nuijten et al. (2016) recomputed 258,105 p-values with the R software package statcheck which extracts statistics from articles and recomputes the p-values. The anonymized dataset and the data documentation was openly available on the Open Science Framework (https://osf.io/d3ukb/; https://osf.io/c6ap2/).

Usage

data(journals)

Format

A data.frame with 8 rows and 14 variables:

References

Nuijten MB, Hartgerink CH, van Assen MA, Epskamp S, Wicherts JM (2016). “The prevalence of statistical reporting errors in psychology (1985–2013).” Behavior Research Methods, 48, 1205–1226.

Examples

data(journals)
# Prior specification 
# We assign a uniform Beta distribution on each binomial probability
a <- rep(1, 8)  
b <- rep(1, 8)  

x <- journals$errors 
n  <- journals$nr_NHST
factor_levels <- levels(journals$journal)

# restricted hypothesis
Hr1 <- c('JAP , PS , JCCP , PLOS , DP , FP , JEPG < JPSP')
out <- binom_bf_informed(x=x, n=n, Hr=Hr1, a=a, b=b, 
factor_levels=factor_levels, niter = 2e3)

summary(out)

Memory of Life Stresses

Description

This data set, "lifestresses", provides the number of reported life stresses (summed across participants) that occurred in specific months prior to an interview. This data set contains the subset of 147 participants who reported one negative life event over the time span of 18 months prior to an interview. Description taken from the JASP (2020) data library.

Usage

data(lifestresses)

Format

A data.frame with 18 rows and 3 variables:

month

The month in which participants reported a stressful life event.

stress.freq

The number of participants who reported a life stress in the particular month prior to an interview.

stress.percentage

The percentage of participants who reported a life stress in the particular month prior to an interview.

References

Haberman SJ (1978). Analysis of qualitative data: Introductory topics, volume 1. Academic Press.

JASP Team (2022). “JASP (Version 0.16.3.0) [Computer software].” https://jasp-stats.org/.

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

Uhlenhuth EH, Haberman SJ, Balter MD, Lipman RS (1977). “Remembering life events.” In The origins and course of psychopathology, 117–134. Springer Verlag.

Examples

data(lifestresses)
# Prior specification 
# We assign a uniform Dirichlet distribution, that is, we set all 
# concentration parameters to 1
a             <- rep(1, 18)
x             <- lifestresses$stress.freq
factor_levels <- lifestresses$month
# Test the following restricted Hypothesis:
# Hr: month1 > month2 > ... > month18 
Hr            <- paste0(1:18, collapse=">"); Hr
out  <- mult_bf_informed(x=x, Hr=Hr, a=a, factor_levels=factor_levels,
niter=1e3, bf_type = 'BFre', seed = 4)
m1 <- summary(out)
m1

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

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)

Computes Bayes Factors For Inequality Constrained Multinomial Parameters

Description

Computes Bayes factor for inequality constrained multinomial parameters using a bridge sampling routine. Restricted hypothesis H_r states that category proportions follow a particular trend. Alternative hypothesis H_e states that category proportions are free to vary.

Usage

mult_bf_inequality(
  samples = NULL,
  restrictions = NULL,
  x = NULL,
  Hr = NULL,
  a = rep(1, ncol(samples)),
  factor_levels = NULL,
  prior = FALSE,
  index = 1,
  maxiter = 1000,
  seed = NULL,
  niter = 5000,
  nburnin = niter * 0.05
)

Arguments

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

List consisting of the following elements:

$eval

• q11: log prior or posterior evaluations for prior or posterior samples

• q12: log proposal evaluations for prior or posterior samples

• q21: log prior or posterior evaluations for samples from proposal

• q22: log proposal evaluations for samples from proposal

$niter

number of iterations of the iterative updating scheme

$logml

estimate of log marginal likelihood

$hyp

evaluated inequality constrained hypothesis

$error_measures

• re2: the approximate relative mean-squared error for the marginal likelihood estimate

• cv: the approximate coefficient of variation for the marginal likelihood estimate (assumes that bridge estimate is unbiased)

• percentage: the approximate percentage error of the marginal likelihood estimate

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

generate_restriction_list

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

Examples

# priors
a <- c(1, 1, 1, 1)

# informed hypothesis
factor_levels <- c('theta1', 'theta2', 'theta3', 'theta4')
Hr            <- c('theta1', '<',  'theta2', '<', 'theta3', '<', 'theta4')

results_prior  <- mult_bf_inequality(Hr=Hr, a=a, factor_levels=factor_levels, 
prior=TRUE, seed = 2020)
# corresponds to
cbind(exp(results_prior$logml), 1/factorial(4))

# alternative - if you have samples and a restriction list
inequalities  <- generate_restriction_list(Hr=Hr, a=a,
factor_levels=factor_levels)$inequality_constraints
prior_samples <- mult_tsampling(inequalities, niter = 2e3, 
prior=TRUE, seed = 2020)
results_prior <- mult_bf_inequality(prior_samples, inequalities, seed=2020)
cbind(exp(results_prior$logml), 1/factorial(4))

Evaluates Informed Hypotheses on Multinomial Parameters

Description

Evaluates informed hypotheses on multinomial parameters. These hypotheses can contain (a mixture of) inequality constraints, equality constraints, and free parameters. Informed hypothesis H_r states that category proportions obey the particular constraint. H_r can be tested against the encompassing hypothesis H_e or the null hypothesis H_0. Encompassing hypothesis H_e states that category proportions are free to vary. Null hypothesis H_0 states that category proportions are exactly equal.

Usage

mult_bf_informed(
  x,
  Hr,
  a = rep(1, length(x)),
  factor_levels = NULL,
  cred_level = 0.95,
  niter = 5000,
  bf_type = "LogBFer",
  seed = NULL,
  maxiter = 1000,
  nburnin = niter * 0.05
)

Arguments

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

List consisting of the following elements

$bf_list

gives an overview of the Bayes factor analysis:

• bf_type: string. Contains Bayes factor type as specified by the user

• bf: data.frame. Contains Bayes factors for all Bayes factor types

• error_measures: data.frame. Contains for the overall Bayes factor the approximate relative mean-squared error re2, the approximate coefficient of variation cv, and the approximate percentage error percentage

• logBFe_equalities: data.frame. Lists the log Bayes factors for all independent equality constrained hypotheses

• logBFe_inequalities: data.frame. Lists the log Bayes factor for all independent inequality constrained hypotheses

$cred_level

numeric. User specified credible interval

$restrictions

list that encodes informed hypothesis for each independent restriction:

• full_model: list containing the hypothesis, parameter names, data and prior specifications for the full model.

• equality_constraints: list containing the hypothesis, parameter names, data and prior specifications for each equality constrained hypothesis.

• inequality_constraints: list containing the hypothesis, parameter names, data and prior specifications for each inequality constrained hypothesis. In addition, in nr_mult_equal and nr_mult_free encodes which and how many parameters are equality constraint or free, in boundaries includes the boundaries of each parameter, in nineq_per_hyp states the number of inequality constraint parameters per independent inequality constrained hypothesis, and in direction states the direction of the inequality constraint.

$bridge_output

list containing output from bridge sampling function:

• eval: list containing the log prior or posterior evaluations (q11) and the log proposal evaluations (q12) for the prior or posterior samples, as well as the log prior or posterior evaluations (q21) and the log proposal evaluations (q22) for the samples from the proposal distribution

• niter: number of iterations of the iterative updating scheme

• logml: estimate of log marginal likelihood

• hyp: evaluated inequality constrained hypothesis

• error_measures: list containing in re2 the approximate relative mean-squared error for the marginal likelihood estimate, in cv the approximate coefficient of variation for the marginal likelihood estimate (assumes that bridge estimate is unbiased), and in percentage the approximate percentage error of the marginal likelihood estimate

$samples

list containing a list for prior samples and a list of posterior samples from truncated distributions which were used to evaluate inequality constraints. Prior and posterior samples of independent inequality constraints are again saved in separate lists. Samples are stored as matrix of dimension nsamples x nparams.

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_equality(), mult_bf_inequality()

Examples

# data
x <- c(3, 4, 10, 11, 7, 30)
# priors
a <- c(1, 1, 1, 1, 1, 1)
# restricted hypothesis
factor_levels <- c('theta1', 'theta2', 'theta3', 'theta4', 'theta5', 
'theta6')
Hr            <- c('theta1', '<',  'theta2', '&', 'theta3', '=', 'theta4', 
',', 'theta5', '<', 'theta6')
output_total  <- mult_bf_informed(x, Hr, a, factor_levels, seed=2020, niter=2e3)

Samples From Truncated Dirichlet Density

Description

Based on specified inequality constraints, samples from truncated prior or posterior Dirichlet density.

Usage

mult_tsampling(
  inequalities,
  index = 1,
  niter = 10000,
  prior = FALSE,
  nburnin = niter * 0.05,
  seed = NULL
)

Arguments

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

matrix of dimension niter * nsamples containing prior or posterior samples from truncated Dirichlet distribution.

Note

When equality constraints are specified in the restricted hypothesis, this function samples from the conditional Dirichlet distribution given that the equality constraints hold.

Only inequality constrained parameters are sampled. Free parameters or parameters that are exclusively equality constrained will be ignored.

References

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

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

See Also

generate_restriction_list

Examples

x <- c(200, 130, 40, 10)
a <- c(1, 1, 1, 1)
factor_levels <- c('mult1', 'mult2', 'mult3', 'mult4')
Hr <- c('mult1 > mult2 > mult3 > mult4')

# generate restriction list
inequalities <- generate_restriction_list(x=x, Hr=Hr, a=a, 
factor_levels=factor_levels)$inequality_constraints

# sample from prior distribution
prior_samples <- mult_tsampling(inequalities, niter = 500, prior=TRUE)
# sample from posterior distribution
post_samples <- mult_tsampling(inequalities, niter = 500)

Mendelian Laws of Inheritance

Description

This data set, "peas", provides the categorization of crossbreeds between a plant variety that produced round yellow peas with a plant variety that produced wrinkled green peas. This data set contains the categorization of 556 plants that were categorized either as (1) round and yellow, (2) wrinkled and yellow, (3) round and green, or (4) wrinkled and green.

Usage

data(peas)

Format

A data.frame with 4 rows and 2 variables:

peas

Crossbreeds that are categorized as 'roundYellow', 'wrinkledYellow', 'roundGreen', or 'wrinkledGreen'.

counts

The number of plants assigned to a one of the crossbreed categories.

References

Mulder J, Wagenmakers E, Marsman M (in press). “A Generalization of the Savage-Dickey Density Ratio for Testing Equality and Order Constrained Hypotheses.” The American Statistician.

Robertson T (1978). “Testing for and against an order restriction on multinomial parameters.” Journal of the American Statistical Association, 73, 197–202.

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

Examples

data("peas")
# Prior specification 
# We assign a uniform Dirichlet distribution, that is, we set all 
# concentration parameters to 1
a <- c(1, 1, 1, 1)     

x <- peas$counts
factor_levels <- levels(peas$peas)
# Test the following mixed Hypothesis:
# Hr: roundYellow > wrinkledYellow = roundGreen > wrinkledGreen 
#
# Be careful: Factor levels are usually ordered alphabetically!
# When specifying hypotheses using indexes, make sure they refer to the 
# correct factor levels.
Hr <- c('1 > 2 = 3 > 4') 
# To avoid mistakes, write out factor levels explicitly:
Hr <- c('roundYellow > wrinkledYellow = roundGreen > wrinkledGreen')

out <- mult_bf_informed(x=x, Hr=Hr, a=a, factor_levels=factor_levels, niter=2e3,
bf_type = 'BFre')
summary(out)

Plot estimates

Description

Plots the posterior estimates from the unconstrained multi- or binomial model.

Usage

## S3 method for class 'summary.bmult'
plot(
  x,
  main = NULL,
  xlab = NULL,
  ylab = NULL,
  xlim = NULL,
  ylim = NULL,
  panel.first = NULL,
  ...
)

Arguments

Value

Invisibly returns a data.frame with the plotted estimates.

Examples

# data
x <- c(3, 4, 10, 11, 7, 30)
# priors
a <- c(1, 1, 1, 1, 1, 1)
# restricted hypothesis
factor_levels <- c('theta1', 'theta2', 'theta3', 'theta4', 'theta5', 
                   'theta6')
Hr <- c('theta1', '<',  'theta2', '&', 'theta3', '=', 
'theta4', ',', 'theta5', '<', 'theta6')
output_total  <- mult_bf_informed(x, Hr, a, factor_levels, seed=2020, 
niter=1e3, bf_type = "BFer")
plot(summary(output_total))

# data for a big Bayes factor
x <- c(3, 4, 10, 11, 7, 30) * 1000
output_total  <- mult_bf_informed(x, Hr, a, factor_levels, seed=2020, 
niter=1e3, bf_type = "BFre")
plot(summary(output_total))

print method for class bmult

Description

Prints model specification

Usage

## S3 method for class 'bmult'
print(x, ...)

Arguments

Value

The print methods print the model specifications and descriptives and return nothing

Examples

# data
x <- c(3, 4, 10, 11)
n <- c(15, 12, 12, 12)
# priors
a <- c(1, 1, 1, 1)
b <- c(1, 1, 1, 1)
# informed hypothesis
factor_levels <- c('theta1', 'theta2', 'theta3', 'theta4')
Hr            <- c('theta1', '<',  'theta2', '<', 'theta3', '<', 'theta4')

## Binomial Case
out_binom  <- binom_bf_informed(x=x, n=n, Hr=Hr, a=a, b=b, niter=1e3,factor_levels, seed=2020)
out_binom
## Multinomial Case
out_mult  <- mult_bf_informed(x=x, Hr=Hr, a=a, niter=1e3,factor_levels, seed=2020)
out_mult

Print method for class bmult_bridge

Description

Prints model specification

Usage

## S3 method for class 'bmult_bridge'
print(x, ...)

Arguments

Value

The print methods print the results from the bridge sampling algorithm and return nothing

Examples

# data
x <- c(3, 4, 10, 11)
n <- c(15, 12, 12, 12)
# priors
a <- c(1, 1, 1, 1)
b <- c(1, 1, 1, 1)
# informed hypothesis
factor_levels <- c('theta1', 'theta2', 'theta3', 'theta4')
Hr            <- c('theta1', '<',  'theta2', '<', 'theta3', '<', 'theta4')

## Multinomial Case
out_mult  <- mult_bf_inequality(x=x, Hr=Hr, a=a, factor_levels=factor_levels,
niter=1e3, seed=2020)
out_mult

print method for class summary.bmult

Description

Prints the summary from Bayes factor analysis

Usage

## S3 method for class 'summary.bmult'
print(x, ...)

Arguments

Value

The print methods print the summary from the Bayes factor analysis and returns nothing

Examples

# data
x <- c(3, 4, 10, 11)
n <- c(15, 12, 12, 12)
# priors
a <- c(1, 1, 1, 1)
b <- c(1, 1, 1, 1)
# informed hypothesis
factor_levels <- c('theta1', 'theta2', 'theta3', 'theta4')
Hr            <- c('theta1', '<',  'theta2', '<', 'theta3', '<', 'theta4')

## Binomial Case
out_binom  <- binom_bf_informed(x=x, n=n, Hr=Hr, a=a, b=b, niter=1e3,factor_levels, seed=2020)
summary(out_binom)
## Multinomial Case
out_mult  <- mult_bf_informed(x=x, Hr=Hr, a=a, niter=1e3,factor_levels, seed=2020)
summary(out_mult)

print method for class summary.bmult_bridge

Description

Prints the summary of bridge sampling output

Usage

## S3 method for class 'summary.bmult_bridge'
print(x, ...)

Arguments

Value

The print methods print the summary of the bridge sampling output

Examples

# data
x <- c(3, 4, 10, 11)
n <- c(15, 12, 12, 12)
# priors
a <- c(1, 1, 1, 1)
b <- c(1, 1, 1, 1)
# informed hypothesis
factor_levels <- c('theta1', 'theta2', 'theta3', 'theta4')
Hr            <- c('theta1', '<',  'theta2', '<', 'theta3', '<', 'theta4')

## Multinomial Case
out_mult  <- mult_bf_inequality(x=x, Hr=Hr, a=a, factor_levels=factor_levels,
niter=1e3, seed=2020)
summary(out_mult)

S3 method for class restriction_list.bmult

Description

Extracts restriction list from an object of class bmult

Usage

restriction_list(x, ...)

Arguments

Value

Extracts restriction list and associated hypothesis from an object of class bmult

Examples

# data
x <- c(3, 4, 10, 11)
n <- c(15, 12, 12, 12)
# priors
a <- c(1, 1, 1, 1)
b <- c(1, 1, 1, 1)
# informed hypothesis
factor_levels <- c('theta1', 'theta2', 'theta3', 'theta4')
Hr            <- c('theta1', '<',  'theta2', '<', 'theta3', '<', 'theta4')

## Multinomial Case
out_mult  <- mult_bf_informed(x=x, Hr=Hr, a=a, factor_levels=factor_levels,
niter=1e3, seed=2020)
restriction_list <- restriction_list(out_mult)

Extracts restriction list from an object of class bmult

Description

Extracts restriction list from an object of class bmult

Usage

## S3 method for class 'bmult'
restriction_list(x, restrictions = "inequalities", ...)

Arguments

Value

Extracts restriction list and associated hypothesis from an object of class bmult

Examples

# data
x <- c(3, 4, 10, 11)
n <- c(15, 12, 12, 12)
# priors
a <- c(1, 1, 1, 1)
b <- c(1, 1, 1, 1)
# informed hypothesis
factor_levels <- c('theta1', 'theta2', 'theta3', 'theta4')
Hr            <- c('theta1', '<',  'theta2', '<', 'theta3', '<', 'theta4')

## Multinomial Case
out_mult  <- mult_bf_informed(x=x, Hr=Hr, a=a, factor_levels=factor_levels,
niter=1e3, seed=2020)
restriction_list <- restriction_list(out_mult)

S3 method for class 'samples.bmult'

Description

Extracts prior and posterior samples (if applicable) from an object of class bmult

Usage

samples(x)

Arguments

Value

Returns list with prior and posterior samples (if applicable) from an object of class bmult

Examples

# data
x <- c(3, 4, 10, 11)
n <- c(15, 12, 12, 12)
# priors
a <- c(1, 1, 1, 1)
b <- c(1, 1, 1, 1)
# informed hypothesis
factor_levels <- c('theta1', 'theta2', 'theta3', 'theta4')
Hr            <- c('theta1', '<',  'theta2', '<', 'theta3', '<', 'theta4')

## Multinomial Case
out_mult  <- mult_bf_informed(x=x, Hr=Hr, a=a, factor_levels=factor_levels,
niter=1e3, seed=2020)
sample_list <- samples(out_mult)

Extracts prior and posterior samples (if applicable) from an object of class bmult

Description

Extracts prior and posterior samples (if applicable) from an object of class bmult

Usage

## S3 method for class 'bmult'
samples(x)

Arguments

Value

Returns list with prior and posterior samples (if applicable) from an object of class bmult

Examples

# data
x <- c(3, 4, 10, 11)
n <- c(15, 12, 12, 12)
# priors
a <- c(1, 1, 1, 1)
b <- c(1, 1, 1, 1)
# informed hypothesis
factor_levels <- c('theta1', 'theta2', 'theta3', 'theta4')
Hr            <- c('theta1', '<',  'theta2', '<', 'theta3', '<', 'theta4')

## Multinomial Case
out_mult  <- mult_bf_informed(x=x, Hr=Hr, a=a, factor_levels=factor_levels,
niter=1e3, seed=2020)
sample_list <- samples(out_mult)

summary method for class bmult

Description

Summarizes results from Bayes factor analysis

Usage

## S3 method for class 'bmult'
summary(object, ...)

Arguments

Value

Invisibly returns a list which contains the Bayes factor and associated hypotheses for the full model, but also the separate for the independent equality and inequality constraints.

The summary method returns a list with the following elements:

$hyp

Vector containing the informed hypothesis as specified by the user

$bf

Contains Bayes factor

$logmlHe

Contains log marginal likelihood of the encompassing model

$logmlH0

Contains log marginal likelihood of the null model

$logmlHr

Contains log marginal likelihood of the informed model

$re2

Contains relative mean-square error for the Bayes factor

$bf_type

Contains Bayes factor type as specified by the user

$cred_level

Credible interval for the posterior point estimates.

$prior

List containing the prior parameters.

$data

List containing the data.

$nr_equal

Number of independent equality-constrained hypotheses.

$nr_inequal

Number of independent inequality-constrained hypotheses.

$estimates

Parameter estimates for the encompassing model

• factor_level: Vector with category names

• alpha: Vector with posterior concentration parameters of Dirichlet distribution (for multinomial models) or alpha parameters for independent beta distributions (for binomial models)

• beta: Vector with beta parameters for independent beta distributions (for binomial models)

• lower: Lower value of credible intervals of marginal beta distributions

• median: Posterior median of marginal beta distributions

• upper: Upper value of credible intervals of marginal beta distributions

Examples

# data
x <- c(3, 4, 10, 11)
n <- c(15, 12, 12, 12)
# priors
a <- c(1, 1, 1, 1)
b <- c(1, 1, 1, 1)
# informed hypothesis
factor_levels <- c('theta1', 'theta2', 'theta3', 'theta4')
Hr            <- c('theta1', '<',  'theta2', '<', 'theta3', '<', 'theta4')

## Binomial Case
out_binom  <- binom_bf_informed(x=x, n=n, Hr=Hr, a=a, b=b, niter=1e3,factor_levels, seed=2020)
summary(out_binom)
## Multinomial Case
out_mult  <- mult_bf_informed(x=x, Hr=Hr, a=a, niter=1e3,factor_levels, seed=2020)
summary(out_mult)

summary method for class bmult_bridge

Description

Summarizes bridge sampling results and associated error measures

Usage

## S3 method for class 'bmult_bridge'
summary(object, ...)

Arguments

Value

Invisibly returns a list which contains the log marginal likelihood for inequality constrained category proportions and associated error terms.

Examples

# data
x <- c(3, 4, 10, 11)
n <- c(15, 12, 12, 12)
# priors
a <- c(1, 1, 1, 1)
b <- c(1, 1, 1, 1)
# informed hypothesis
factor_levels <- c('theta1', 'theta2', 'theta3', 'theta4')
Hr            <- c('theta1', '<',  'theta2', '<', 'theta3', '<', 'theta4')

## Multinomial Case
out_mult  <- mult_bf_inequality(x=x, Hr=Hr, a=a, factor_levels=factor_levels,
niter=1e3, seed=2020)
summary(out_mult)

Backtransforms Samples From Real Line To Beta Parameters

Description

Transforms samples from the real line to samples from a truncated beta density using a stick-breaking algorithm. This algorithm is suitable for mixtures of equality constrained parameters, inequality constrained parameters, and free parameters

Usage

tbinom_backtrans(xi_mat, boundaries, binom_equal, hyp_direction)

Arguments

Value

list consisting of the following elements: (1) theta_mat: matrix with transformed samples (2) lower_mat: matrix containing the lower bound for each parameter (3) upper_mat: matrix containing the upper bound for each parameter

Transforms Truncated Beta Samples To Real Line

Description

Transforms samples from a truncated beta density to the real line using a stick-breaking algorithm. This algorithm is suitable for mixtures of equality constrained parameters, inequality constrained parameters, and free parameters

Usage

tbinom_trans(theta_mat, boundaries, binom_equal, hyp_direction)

Arguments

Value

matrix with transformed samples

Backtransforms Samples From Real Line To Dirichlet Parameters

Description

Transforms samples from the real line to samples from a truncated Dirichlet density using a stick-breaking algorithm. This algorithm is suitable for mixtures of equality constrained parameters, inequality constrained parameters, and free parameters

Usage

tdir_backtrans(
  xi_mat,
  boundaries,
  mult_equal,
  nr_mult_equal,
  nr_mult_free,
  hyp_direction
)

Arguments

Value

list consisting of the following elements: (1) theta_mat: matrix with transformed samples (2) lower_mat: matrix containing the lower bound for each parameter (3) upper_mat: matrix containing the upper bound for each parameter

Transforms Truncated Dirichlet Samples To Real Line

Description

Transforms samples from a truncated Dirichlet density to the real line using a stick-breaking algorithm. This algorithm is suitable for mixtures of equality constrained parameters, inequality constrained parameters, and free parameters

Usage

tdir_trans(
  theta_mat,
  boundaries,
  mult_equal,
  nr_mult_equal,
  nr_mult_free,
  hyp_direction
)

Arguments

Value

matrix with transformed samples