generate_restriction_list {multibridge}

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

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 x. Ignored if x is a matrix or a table

Hr

string or character. Encodes the user specified informed hypothesis. Use either specified factor_levels or indices to refer to parameters. See “Note” section for details on how to formulate informed hypotheses

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 x. Default sets all beta parameters to 1

factor_levels

character. Vector with category names. Must be the same length as x

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)