, Frederik Aust [aut]
, Julia M. Haaf [aut]
, Joris Goosen [aut],
Quentin F. Gronau [aut], Maarten Marsman [aut]
| Title: | Evaluating Multinomial Order Restrictions with Bridge Sampling |
|---|---|
| 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>. |
| Authors: | Alexandra Sarafoglou [aut, cre]
, Frederik Aust [aut]
, Julia M. Haaf [aut]
, Joris Goosen [aut],
Quentin F. Gronau [aut], Maarten Marsman [aut]
|
| Maintainer: | Alexandra Sarafoglou <[email protected]> |
| License: | GPL-2 |
| Version: | 1.2.0 |
| Built: | 2024-10-31 22:28:43 UTC |
| Source: | CRAN |
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.
.adjustUpperBoundForFreeParameters( theta_mat, k, upper, nr_mult_equal, smaller_values, larger_values, hyp_direction ).adjustUpperBoundForFreeParameters( theta_mat, k, upper, nr_mult_equal, smaller_values, larger_values, hyp_direction )
theta_mat |
matrix with samples from truncated Dirichlet density |
k |
current parameter index |
upper |
current upper bound |
nr_mult_equal |
vector of multiplicative elements of collapsed parameters |
smaller_values |
index of parameters that are smaller than the current one |
larger_values |
index of parameters that are larger than the current one |
hyp_direction |
specifies whether the imposed inequality constrained imposes an increasing (i.e., 'smaller') or decreasing (i.e., 'larger') trend |
adjusted upper bound
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.
.computeLengthOfRemainingStick(theta_mat, k, hyp_direction).computeLengthOfRemainingStick(theta_mat, k, hyp_direction)
theta_mat |
matrix with samples from truncated Dirichlet density |
k |
current parameter index |
hyp_direction |
specifies whether the imposed inequality constrained imposes an increasing (i.e., 'smaller') or decreasing (i.e., 'larger') trend |
stick length
Extracts information about computed Bayes factors from object of class bmult
bayes_factor(x)bayes_factor(x)
x |
object of class |
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
In case the restricted hypothesis is tested against 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.
# 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)# 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)
bmult
Extracts information about computed Bayes factors from object of class bmult
## S3 method for class 'bmult' bayes_factor(x)## S3 method for class 'bmult' bayes_factor(x)
x |
object of class |
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
In case the restricted hypothesis is tested against 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.
# 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)# 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 factor for equality constrained binomial parameters.
Null hypothesis states that binomial proportions are exactly equal or
exactly equal and equal to p.
Alternative hypothesis states that binomial proportions are free to vary.
binom_bf_equality(x, n = NULL, a, b, p = NULL)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., ) are the observations of independent
binomial experiments, based on observations. Hence, the underlying likelihood is the product of the
individual binomial functions:
Furthermore, the model assigns a beta distribution as prior to each model parameter (i.e., underlying binomial proportions). That is:
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$nr_NHST a <- rep(1, nrow(journals)) b <- rep(1, nrow(journals)) binom_bf_equality(x=x, n=n, a=a, b=b)data(journals) x <- journals$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 factor for inequality constrained binomial parameters using a bridge sampling routine.
Restricted hypothesis states that binomial proportions follow a particular trend.
Alternative hypothesis states that binomial proportions are free to vary.
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 )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 )
samples |
matrix of dimension ( |
restrictions |
|
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 |
Hr |
string or character. Encodes the user specified informed hypothesis. Use either specified |
a |
numeric. Vector with alpha parameters. Must be the same length as |
b |
numeric. Vector with beta parameters. Must be the same length as |
factor_levels |
character. Vector with category names. Must be the same length as |
prior |
logical. If |
index |
numeric. Index of current restriction. Default is 1 |
maxiter |
numeric. Maximum number of iterations for the iterative updating scheme used in the bridge sampling routine. Default is 1,000 to avoid infinite loops |
seed |
numeric. Sets the seed for reproducible pseudo-random number generation |
niter |
numeric. Vector with number of samples to be drawn from truncated distribution |
nburnin |
numeric. A single value specifying the number of burn-in samples when drawing from the truncated distribution. Minimum number of burn-in samples is 10. Default is 5% of the number of samples. Burn-in samples are removed automatically after the sampling. |
The model assumes that the data in x (i.e., ) are the observations of independent
binomial experiments, based on observations. Hence, the underlying likelihood is the product of the
individual binomial functions:
Furthermore, the model assigns a beta distribution as prior to each model parameter (i.e., underlying binomial proportions). That is:
List consisting of the following elements:
$evalq11: 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
$niternumber of iterations of the iterative updating scheme
$logmlestimate of log marginal likelihood
$hypevaluated inequality constrained hypothesis
$error_measuresre2: 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
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
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.
Other functions to evaluate informed hypotheses:
binom_bf_equality(),
binom_bf_informed(),
mult_bf_equality(),
mult_bf_inequality(),
mult_bf_informed()
# 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))# 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.
These hypotheses can contain (a mixture of) inequality constraints, equality constraints, and free parameters.
Informed hypothesis states that binomial proportions obey a particular constraint.
can be tested against the encompassing hypothesis or the null hypothesis .
Encompassing hypothesis states that binomial proportions are free to vary.
Null hypothesis states that category proportions are exactly equal.
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 )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 )
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 |
Hr |
string or character. Encodes the user specified informed hypothesis. Use either specified |
a |
numeric. Vector with alpha parameters. Must be the same length as |
b |
numeric. Vector with beta parameters. Must be the same length as |
factor_levels |
character. Vector with category names. Must be the same length as |
cred_level |
numeric. Credible interval for the posterior point estimates. Must be a single number between 0 and 1 |
niter |
numeric. Vector with number of samples to be drawn from truncated distribution |
bf_type |
character. The Bayes factor type. When the informed hypothesis is compared to the encompassing hypothesis,
the Bayes factor type can be |
seed |
numeric. Sets the seed for reproducible pseudo-random number generation |
maxiter |
numeric. Maximum number of iterations for the iterative updating scheme used in the bridge sampling routine. Default is 1,000 to avoid infinite loops |
nburnin |
numeric. A single value specifying the number of burn-in samples when drawing from the truncated distribution. Minimum number of burn-in samples is 10. Default is 5% of the number of samples. Burn-in samples are removed automatically after the sampling. |
The model assumes that the data in x (i.e., ) are the observations of independent
binomial experiments, based on observations. Hence, the underlying likelihood is the product of the
individual binomial functions:
Furthermore, the model assigns a beta distribution as prior to each model parameter (i.e., underlying binomial proportions). That is:
List consisting of the following elements
$bf_listgives 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_levelnumeric. User specified credible interval
$restrictionslist 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_outputlist 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
$sampleslist 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.
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_equality(),
binom_bf_inequality(),
mult_bf_equality(),
mult_bf_inequality(),
mult_bf_informed()
# 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)# 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)
Based on specified inequality constraints, samples from truncated prior or posterior beta densities.
binom_tsampling( inequalities, index = 1, niter = 10000, prior = FALSE, nburnin = niter * 0.05, seed = NULL )binom_tsampling( inequalities, index = 1, niter = 10000, prior = FALSE, nburnin = niter * 0.05, seed = NULL )
inequalities |
list that contains inequality constraints for each independent inequality constrained hypotheses. The list
is created in the |
index |
numeric. If multiple independent inequality constraints are specified, this index determines for which inequality constraint samples should be drawn. Must be a single value. Default is 1 |
niter |
numeric. A single value specifying the number of samples. Default is set to |
prior |
logical. If |
nburnin |
numeric. A single value specifying the number of burn-in samples when drawing from the truncated distribution. Minimum number of burn-in samples is 10. Default is 5% of the number of samples. Burn-in samples are removed automatically after the sampling. |
seed |
numeric. Sets the seed for reproducible pseudo-random number generation |
The model assumes that the data in x (i.e., ) are the observations of independent
binomial experiments, based on observations. Hence, the underlying likelihood is the product of the
individual binomial functions:
Furthermore, the model assigns a beta distribution as prior to each model parameter (i.e., underlying binomial proportions). That is:
matrix of dimension niter * nsamples containing samples from truncated beta distributions.
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.
Damien P, Walker SG (2001). “Sampling truncated normal, beta, and gamma densities.” Journal of Computational and Graphical Statistics, 10, 206–215.
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)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)
bridge_output.bmult
Extracts bridge sampling output from object of class bmult
bridge_output(x)bridge_output(x)
x |
object of class |
Extracts output related to the bridge sampling routine. The output contains the following elements::
$evalq11: 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
$niternumber of iterations of the iterative updating scheme
$logmlestimate of log marginal likelihood
$hypevaluated inequality constrained hypothesis
$error_measuresre2: 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
# 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)# 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)
bmult
Extracts restriction list from an object of class bmult
## S3 method for class 'bmult' bridge_output(x)## S3 method for class 'bmult' bridge_output(x)
x |
object of class |
Extracts restriction list and associated hypothesis from an object of class bmult
# 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)# 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)
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.
generate_restriction_list(x = NULL, n = NULL, Hr, a, b = NULL, factor_levels)generate_restriction_list(x = NULL, n = NULL, Hr, a, b = NULL, factor_levels)
x |
a vector with data (for multinomial models) or a vector of counts of successes, or a two-dimensional table (or matrix) with 2 columns, giving the counts of successes and failures, respectively (for binomial models). |
n |
numeric. Vector of counts of trials. Must be the same length as |
Hr |
string or character. Encodes the user specified informed hypothesis. Use either specified |
a |
numeric. Vector with concentration parameters of Dirichlet distribution (for multinomial models) or alpha parameters for independent beta distributions (for binomial models). Default sets all parameters to 1 |
b |
numeric. Vector with beta parameters. Must be the same length as |
factor_levels |
character. Vector with category names. Must be the same length as |
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.
Restriction list containing the following elements:
$full_modelhyp: 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_constraintshyp: 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_constraintshyp: 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.
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
# 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)# 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)
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/).
data(journals)data(journals)
A data.frame with 8 rows and 14 variables:
| Variable Name | Description |
journal |
The journal name a research article was published in. |
articles_downloaded |
The number of articles downloaded per journal. |
articles_with_NHST |
The number of articles with NHST results. |
perc_articles_with_NHST |
The percentage of all downloaded articles that had NHST results. |
nr_NHST |
The total number of NHST results. |
mean_nr_NHST_per_article_with_NHST |
The mean number of NHST results per article that had at least one NHST result. |
mean_nr_NHST_per_article_all_included |
The mean number of NHST results in all downloaded articles. |
errors |
The total number of errors. |
dec_errors |
The total number of decision errors (i.e., an error that may have changed the statistical conclusion of the result). |
perc_errors |
The percentage of all results that was an error. |
perc_dec_errors |
The percentage of all results that was a decision error. |
perc_articles_with_errors |
The percentage of all articles that had at least one error. |
perc_articles_with_dec_errors |
The percentage of all articles that had at least one error. |
APAfactor |
APA factor: number of detected NHST results / total number of detected p values. |
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.
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)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$journal) # restricted hypothesis Hr1 <- c('JAP , PS , JCCP , PLOS , DP , FP , JEPG < JPSP') out <- binom_bf_informed(x=x, n=n, Hr=Hr1, a=a, b=b, factor_levels=factor_levels, niter = 2e3) summary(out)
This data set, "lifestresses", provides the number of reported life stresses (summed across participants) that occurred in specific months prior to an interview. This data set contains the subset of 147 participants who reported one negative life event over the time span of 18 months prior to an interview. Description taken from the JASP (2020) data library.
data(lifestresses)data(lifestresses)
A data.frame with 18 rows and 3 variables:
monthThe month in which participants reported a stressful life event.
stress.freqThe number of participants who reported a life stress in the particular month prior to an interview.
stress.percentageThe percentage of participants who reported a life stress in the particular month prior to an interview.
Haberman SJ (1978). Analysis of qualitative data: Introductory topics, volume 1. Academic Press.
JASP Team (2022). “JASP (Version 0.16.3.0) [Computer software].” https://jasp-stats.org/.
Sarafoglou A, Haaf JM, Ly A, Gronau QF, Wagenmakers EJ, Marsman M (2021). “Evaluating Multinomial Order Restrictions with Bridge Sampling.” Psychological Methods.
Uhlenhuth EH, Haberman SJ, Balter MD, Lipman RS (1977). “Remembering life events.” In The origins and course of psychopathology, 117–134. Springer Verlag.
data(lifestresses) # Prior specification # We assign a uniform Dirichlet distribution, that is, we set all # concentration parameters to 1 a <- rep(1, 18) x <- lifestresses$stress.freq factor_levels <- lifestresses$month # Test the following restricted Hypothesis: # Hr: month1 > month2 > ... > month18 Hr <- paste0(1:18, collapse=">"); Hr out <- mult_bf_informed(x=x, Hr=Hr, a=a, factor_levels=factor_levels, niter=1e3, bf_type = 'BFre', seed = 4) m1 <- summary(out) m1data(lifestresses) # Prior specification # We assign a uniform Dirichlet distribution, that is, we set all # concentration parameters to 1 a <- rep(1, 18) x <- lifestresses$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 factor for equality constrained multinomial parameters
using the standard Bayesian multinomial test.
Null hypothesis states that category proportions are exactly equal to those
specified in p.
Alternative hypothesis states that category proportions are free to vary.
mult_bf_equality(x, a, p = rep(1/length(a), length(a)))mult_bf_equality(x, a, p = rep(1/length(a), length(a)))
x |
numeric. Vector with data |
a |
numeric. Vector with concentration parameters of Dirichlet distribution. Must be the same length as |
p |
numeric. A vector of probabilities of the same length as |
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:
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_equality(),
binom_bf_inequality(),
binom_bf_informed(),
mult_bf_inequality(),
mult_bf_informed()
data(lifestresses) x <- lifestresses$stress.freq a <- rep(1, nrow(lifestresses)) mult_bf_equality(x=x, a=a)data(lifestresses) x <- lifestresses$stress.freq a <- rep(1, nrow(lifestresses)) mult_bf_equality(x=x, a=a)
Computes Bayes factor for inequality constrained multinomial parameters using a bridge sampling routine.
Restricted hypothesis states that category proportions follow a particular trend.
Alternative hypothesis states that category proportions are free to vary.
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 )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 )
samples |
matrix of dimension |
restrictions |
|
x |
numeric. Vector with data |
Hr |
string or character. Encodes the user specified informed hypothesis. Use either specified |
a |
numeric. Vector with concentration parameters of Dirichlet distribution. Must be the same length as |
factor_levels |
character. Vector with category names. Must be the same length as |
prior |
logical. If |
index |
numeric. Index of current restriction. Default is 1 |
maxiter |
numeric. Maximum number of iterations for the iterative updating scheme used in the bridge sampling routine. Default is 1,000 to avoid infinite loops |
seed |
numeric. Sets the seed for reproducible pseudo-random number generation |
niter |
numeric. Vector with number of samples to be drawn from truncated distribution |
nburnin |
numeric. A single value specifying the number of burn-in samples when drawing from the truncated distribution. Minimum number of burn-in samples is 10. Default is 5% of the number of samples. Burn-in samples are removed automatically after the sampling. |
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:
List consisting of the following elements:
$evalq11: 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
$niternumber of iterations of the iterative updating scheme
$logmlestimate of log marginal likelihood
$hypevaluated inequality constrained hypothesis
$error_measuresre2: 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
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_equality(),
binom_bf_inequality(),
binom_bf_informed(),
mult_bf_equality(),
mult_bf_informed()
# 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))# 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. These hypotheses can contain
(a mixture of) inequality constraints, equality constraints, and free parameters.
Informed hypothesis states that category proportions obey the particular constraint.
can be tested against the encompassing hypothesis or the null hypothesis .
Encompassing hypothesis states that category proportions are free to vary.
Null hypothesis states that category proportions are exactly equal.
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 )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 )
x |
numeric. Vector with data |
Hr |
string or character. Encodes the user specified informed hypothesis. Use either specified |
a |
numeric. Vector with concentration parameters of Dirichlet distribution. Must be the same length as |
factor_levels |
character. Vector with category names. Must be the same length as |
cred_level |
numeric. Credible interval for the posterior point estimates. Must be a single number between 0 and 1 |
niter |
numeric. Vector with number of samples to be drawn from truncated distribution |
bf_type |
character. The Bayes factor type. When the informed hypothesis is compared to the encompassing hypothesis,
the Bayes factor type can be |
seed |
numeric. Sets the seed for reproducible pseudo-random number generation |
maxiter |
numeric. Maximum number of iterations for the iterative updating scheme used in the bridge sampling routine. Default is 1,000 to avoid infinite loops |
nburnin |
numeric. A single value specifying the number of burn-in samples when drawing from the truncated distribution. Minimum number of burn-in samples is 10. Default is 5% of the number of samples. Burn-in samples are removed automatically after the sampling. |
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:
List consisting of the following elements
$bf_listgives 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_levelnumeric. User specified credible interval
$restrictionslist 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_outputlist 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
$sampleslist 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.
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_equality(),
binom_bf_inequality(),
binom_bf_informed(),
mult_bf_equality(),
mult_bf_inequality()
# 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)# 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)
Based on specified inequality constraints, samples from truncated prior or posterior Dirichlet density.
mult_tsampling( inequalities, index = 1, niter = 10000, prior = FALSE, nburnin = niter * 0.05, seed = NULL )mult_tsampling( inequalities, index = 1, niter = 10000, prior = FALSE, nburnin = niter * 0.05, seed = NULL )
inequalities |
list that contains inequality constraints for each independent inequality constrained hypotheses. The list
is created in the |
index |
numeric. If multiple independent inequality constraints are specified, this index determines for which inequality constraint samples should be drawn. Must be a single value. Default is 1 |
niter |
numeric. A single value specifying the number of samples. Default is set to |
prior |
logical. If |
nburnin |
numeric. A single value specifying the number of burn-in samples when drawing from the truncated distribution. Minimum number of burn-in samples is 10. Default is 5% of the number of samples. Burn-in samples are removed automatically after the sampling. |
seed |
numeric. Sets the seed for reproducible pseudo-random number generation |
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:
matrix of dimension niter * nsamples containing prior or posterior samples from truncated Dirichlet distribution.
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.
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.
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)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)$inequality_constraints # sample from prior distribution prior_samples <- mult_tsampling(inequalities, niter = 500, prior=TRUE) # sample from posterior distribution post_samples <- mult_tsampling(inequalities, niter = 500)
This data set, "peas", provides the categorization of crossbreeds between a plant variety that produced round yellow peas with a plant variety that produced wrinkled green peas. This data set contains the categorization of 556 plants that were categorized either as (1) round and yellow, (2) wrinkled and yellow, (3) round and green, or (4) wrinkled and green.
data(peas)data(peas)
A data.frame with 4 rows and 2 variables:
peasCrossbreeds that are categorized as 'roundYellow', 'wrinkledYellow', 'roundGreen', or 'wrinkledGreen'.
countsThe number of plants assigned to a one of the crossbreed categories.
Mulder J, Wagenmakers E, Marsman M (in press). “A Generalization of the Savage-Dickey Density Ratio for Testing Equality and Order Constrained Hypotheses.” The American Statistician.
Robertson T (1978). “Testing for and against an order restriction on multinomial parameters.” Journal of the American Statistical Association, 73, 197–202.
Sarafoglou A, Haaf JM, Ly A, Gronau QF, Wagenmakers EJ, Marsman M (2021). “Evaluating Multinomial Order Restrictions with Bridge Sampling.” Psychological Methods.
data("peas") # Prior specification # We assign a uniform Dirichlet distribution, that is, we set all # concentration parameters to 1 a <- c(1, 1, 1, 1) x <- peas$counts factor_levels <- levels(peas$peas) # Test the following mixed Hypothesis: # Hr: roundYellow > wrinkledYellow = roundGreen > wrinkledGreen # # Be careful: Factor levels are usually ordered alphabetically! # When specifying hypotheses using indexes, make sure they refer to the # correct factor levels. Hr <- c('1 > 2 = 3 > 4') # To avoid mistakes, write out factor levels explicitly: Hr <- c('roundYellow > wrinkledYellow = roundGreen > wrinkledGreen') out <- mult_bf_informed(x=x, Hr=Hr, a=a, factor_levels=factor_levels, niter=2e3, bf_type = 'BFre') summary(out)data("peas") # Prior specification # We assign a uniform Dirichlet distribution, that is, we set all # concentration parameters to 1 a <- c(1, 1, 1, 1) x <- peas$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)
Plots the posterior estimates from the unconstrained multi- or binomial model.
## S3 method for class 'summary.bmult' plot( x, main = NULL, xlab = NULL, ylab = NULL, xlim = NULL, ylim = NULL, panel.first = NULL, ... )## S3 method for class 'summary.bmult' plot( x, main = NULL, xlab = NULL, ylab = NULL, xlim = NULL, ylim = NULL, panel.first = NULL, ... )
x |
A |
main |
|
xlab |
|
ylab |
|
xlim, ylim
|
numeric vectors of length 2, giving the x and y coordinates ranges. |
panel.first |
|
... |
additional arguments passed to |
Invisibly returns a data.frame with the plotted estimates.
# 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))# 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))
bmult
Prints model specification
## S3 method for class 'bmult' print(x, ...)## S3 method for class 'bmult' print(x, ...)
x |
object of class |
... |
additional arguments, currently ignored |
The print methods print the model specifications and descriptives and return nothing
# 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# 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
bmult_bridge
Prints model specification
## S3 method for class 'bmult_bridge' print(x, ...)## S3 method for class 'bmult_bridge' print(x, ...)
x |
object of class |
... |
additional arguments, currently ignored |
The print methods print the results from the bridge sampling algorithm and return nothing
# 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# 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
summary.bmult
Prints the summary from Bayes factor analysis
## S3 method for class 'summary.bmult' print(x, ...)## S3 method for class 'summary.bmult' print(x, ...)
x |
object of class |
... |
additional arguments, currently ignored |
The print methods print the summary from the Bayes factor analysis and returns nothing
# 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)# 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.bmult_bridge
Prints the summary of bridge sampling output
## S3 method for class 'summary.bmult_bridge' print(x, ...)## S3 method for class 'summary.bmult_bridge' print(x, ...)
x |
object of class |
... |
additional arguments, currently ignored |
The print methods print the summary of the bridge sampling output
# 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)# 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)
restriction_list.bmult
Extracts restriction list from an object of class bmult
restriction_list(x, ...)restriction_list(x, ...)
x |
object of class |
... |
additional arguments passed to other methods |
Extracts restriction list and associated hypothesis from an object of class bmult
# 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)# 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)
bmult
Extracts restriction list from an object of class bmult
## S3 method for class 'bmult' restriction_list(x, restrictions = "inequalities", ...)## S3 method for class 'bmult' restriction_list(x, restrictions = "inequalities", ...)
x |
object of class |
restrictions |
specifies whether to extract restriction list for |
... |
additional arguments, currently ignored |
Extracts restriction list and associated hypothesis from an object of class bmult
# 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)# 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 prior and posterior samples (if applicable) from an object of class bmult
samples(x)samples(x)
x |
object of class |
Returns list with prior and posterior samples (if applicable) from an object of class bmult
# 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)# 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)
bmult
Extracts prior and posterior samples (if applicable) from an object of class bmult
## S3 method for class 'bmult' samples(x)## S3 method for class 'bmult' samples(x)
x |
object of class |
Returns list with prior and posterior samples (if applicable) from an object of class bmult
# 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)# 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)
bmult
Summarizes results from Bayes factor analysis
## S3 method for class 'bmult' summary(object, ...)## S3 method for class 'bmult' summary(object, ...)
object |
object of class |
... |
additional arguments, currently ignored |
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:
$hypVector containing the informed hypothesis as specified by the user
$bfContains Bayes factor
$logmlHeContains log marginal likelihood of the encompassing model
$logmlH0Contains log marginal likelihood of the null model
$logmlHrContains log marginal likelihood of the informed model
$re2Contains relative mean-square error for the Bayes factor
$bf_typeContains Bayes factor type as specified by the user
$cred_levelCredible interval for the posterior point estimates.
$priorList containing the prior parameters.
$dataList containing the data.
$nr_equalNumber of independent equality-constrained hypotheses.
$nr_inequalNumber of independent inequality-constrained hypotheses.
$estimatesParameter 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
# 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)# 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)
bmult_bridge
Summarizes bridge sampling results and associated error measures
## S3 method for class 'bmult_bridge' summary(object, ...)## S3 method for class 'bmult_bridge' summary(object, ...)
object |
object of class |
... |
additional arguments, currently ignored |
Invisibly returns a list which contains the log marginal likelihood for inequality constrained category proportions and associated error terms.
# 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)# 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)
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
tbinom_backtrans(xi_mat, boundaries, binom_equal, hyp_direction)tbinom_backtrans(xi_mat, boundaries, binom_equal, hyp_direction)
xi_mat |
matrix with samples from truncated beta density. These samples should be transformed, so they range over the entire real line |
boundaries |
list containing indices for upper and lower truncation boundaries |
binom_equal |
multiplicative elements for each lower and upper bound of each inequality constrained parameter. |
hyp_direction |
specifies whether the imposed inequality constrained imposes an increasing (i.e., 'smaller') or decreasing (i.e., 'larger') trend |
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 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
tbinom_trans(theta_mat, boundaries, binom_equal, hyp_direction)tbinom_trans(theta_mat, boundaries, binom_equal, hyp_direction)
theta_mat |
matrix with samples from truncated beta density |
boundaries |
list containing indices for upper and lower truncation boundaries |
binom_equal |
multiplicative elements for each lower and upper bound of each inequality constrained parameter. |
hyp_direction |
specifies whether the imposed inequality constrained imposes an increasing (i.e., 'smaller') or decreasing (i.e., 'larger') trend |
matrix with transformed samples
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
tdir_backtrans( xi_mat, boundaries, mult_equal, nr_mult_equal, nr_mult_free, hyp_direction )tdir_backtrans( xi_mat, boundaries, mult_equal, nr_mult_equal, nr_mult_free, hyp_direction )
xi_mat |
matrix with samples from truncated Dirichlet density. These samples should be transformed, so they range over the entire real line |
boundaries |
list containing indices for upper and lower truncation boundaries |
mult_equal |
multiplicative elements for each lower and upper bound of each inequality constrained parameter. |
nr_mult_equal |
vector of multiplicative elements of collapsed parameters |
nr_mult_free |
vector of multiplicative elements of free parameters |
hyp_direction |
specifies whether the imposed inequality constrained imposes an increasing (i.e., 'smaller') or decreasing (i.e., 'larger') trend |
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 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
tdir_trans( theta_mat, boundaries, mult_equal, nr_mult_equal, nr_mult_free, hyp_direction )tdir_trans( theta_mat, boundaries, mult_equal, nr_mult_equal, nr_mult_free, hyp_direction )
theta_mat |
matrix with samples from truncated Dirichlet density |
boundaries |
list containing indices for upper and lower truncation boundaries |
mult_equal |
multiplicative elements for each lower and upper bound of each inequality constrained parameter. |
nr_mult_equal |
vector of multiplicative elements of collapsed parameters |
nr_mult_free |
vector of multiplicative elements of free parameters |
hyp_direction |
specifies whether the imposed inequality constrained imposes an increasing (i.e., 'smaller') or decreasing (i.e., 'larger') trend |
matrix with transformed samples