Package 'cIRT' reference manual

Title:	Choice Item Response Theory
Description:	Jointly model the accuracy of cognitive responses and item choices within a Bayesian hierarchical framework as described by Culpepper and Balamuta (2015) <doi:10.1007/s11336-015-9484-7>. In addition, the package contains the datasets used within the analysis of the paper.
Authors:	Steven Andrew Culpepper [aut, cph] , James Joseph Balamuta [aut, cph, cre]
Maintainer:	James Joseph Balamuta <[email protected]>
License:	GPL (>= 2)
Version:	1.3.2
Built:	2024-11-19 04:14:26 UTC
Source:	https://github.com/tmsalab/cirt

cIRT: Choice Item Response Theory

Description

Jointly model the accuracy of cognitive responses and item choices within a Bayesian hierarchical framework as described by Culpepper and Balamuta (2015) <doi:10.1007/s11336-015-9484-7>. In addition, the package contains the datasets used within the analysis of the paper.

Author(s)

Maintainer: James Joseph Balamuta [email protected] (ORCID) [copyright holder]

Authors:

Steven Andrew Culpepper [email protected] (ORCID) [copyright holder]

Center a Matrix

Description

Obtains the mean of each column of the matrix and subtracts it from the given matrix in a centering operation.

Usage

center_matrix(x)
center_matrix(x)

Arguments

`x`	A `matrix` with any dimensions

Details

The application of this function to a matrix mimics the use of a centering matrix given by:

${C_n} = {I_n} - \frac{1}{n}{11^T}$

Value

A matrix with the same dimensions of X that has been centered.

Author(s)

James Joseph Balamuta

Examples

nobs = 500
nvars = 20
x = matrix(rnorm(nobs * nvars), nrow = nobs, ncol = nvars) 
r_centered = scale(x) 
arma_centered1 = center_matrix(x)
nobs = 500
nvars = 20
x = matrix(rnorm(nobs * nvars), nrow = nobs, ncol = nvars) 
r_centered = scale(x) 
arma_centered1 = center_matrix(x)

Choice Matrix Data

Description

This data set contains the subject's choices and point values for the difficult questions.

Usage

choice_matrix
choice_matrix

Format

A data frame with 3780 observations on the following 5 variables.

subject_id: Research Participant Subject ID. There are 102 IDs and each ID has 15 observations.
hard_q_id: The item ID of the hard question assigned to the student (16-30)
easy_q_id: The item ID of the easy question assigned to the student (1-15)
choose_hard_q: Selected either: Difficult Question (1) or Easy Question (0)
high_value: Range of values associated with Difficult Question that span from 12 to 16, repeated three times per subject
low_value: Range of values associated with Easy Question that span from 4 to 6, repeated five times per subject
is_correct_choice: Did the user select an item that was answered correctly?

Author(s)

Steven Andrew Culpepper and James Joseph Balamuta

Source

Choice38 Experiment at UIUC during Spring 2014 - Fall 2014

Generic Implementation of Choice IRT MCMC

Description

Builds a model using MCMC

Usage

cIRT(
  subject_ids,
  fixed_effects,
  B_elem_plus1,
  rv_effects,
  trial_matrix,
  choices_nk,
  burnit,
  chain_length = 10000L
)
cIRT(
  subject_ids,
  fixed_effects,
  B_elem_plus1,
  rv_effects,
  trial_matrix,
  choices_nk,
  burnit,
  chain_length = 10000L
)

Arguments

`subject_ids`	A `vector` that contains subject IDs for each line of data in the choice vector (e.g. For 1 subject that made 5 choices, we would have the number 1 appear five times consecutively.)
`fixed_effects`	A `matrix` with NK x P1 dimensions that acts as the design matrix for terms WITHOUT theta.
`B_elem_plus1`	A `V[[1]]` dimensional column `vector` indicating which zeta_i relate to theta_i.
`rv_effects`	A `matrix` with NK x V dimensions for random effects design matrix.
`trial_matrix`	A `matrix` with N x J dimensions, where J denotes the number of items presented. The matrix MUST contain only 1's and 0's.
`choices_nk`	A `vector` with NK length that contains the choice value e.g. 0 or 1.
`burnit`	An `int` that describes how many MCMC draws should be discarded.
`chain_length`	An `int` that controls how many MCMC draws there are. (> 0)

Value

A list that contains:

as: A matrix of dimension chain_length x J
bs: A matrix of dimension chain_length x J
gs: A matrix of dimension chain_length x P_1
Sigma_zeta_inv: An array of dimension V x V x chain_length
betas: A matrix of dimension chain_length x P_2

Author(s)

Steven Andrew Culpepper and James Joseph Balamuta

Examples

## Not run: 
# Variables
# Y = trial matix
# C = KN vector of binary choices
# N = #of subjects
# J = # of items
# K= # of choices
# atrue = true item discriminations
# btrue = true item locations
# thetatrue = true thetas/latent performance
# gamma = fixed effects coefficients
# Sig = random-effects variance-covariance
# subid = id variable for subjects

# Load the Package
library(cIRT)

# Load the Data
data(trial_matrix)
data(choice_matrix)

# Thurstone design matrices
all_nopractice = subset(all_data_trials, experiment_loop.thisN > -1)
hard_items = choice_matrix$hard_q_id
easy_items = choice_matrix$easy_q_id

D_easy = model.matrix( ~ -1 + factor(easy_items))
D_hard = -1 * model.matrix( ~ -1 + factor(hard_items))[, -c(5, 10, 15)]

# Defining effect-coded contrasts
high_contrasts = rbind(-1, diag(4))
rownames(high_contrasts) = 12:16
low_contrasts = rbind(-1, diag(2))
rownames(low_contrasts) = 4:6

# Creating high & low factors
high = factor(choice_matrix[, 'high_value'])
low = factor(choice_matrix[, 'low_value'])
contrasts(high) = high_contrasts
contrasts(low) = low_contrasts

fixed_effects = model.matrix( ~ high + low)
fixed_effects_base = fixed_effects[, 1]
fixed_effects_int = model.matrix( ~ high * low)


# Model with Thurstone D Matrix
system.time({
 out_model_thurstone = cIRT(
   choice_matrix[, 'subject_id'],
   cbind(fixed_effects[, -1], D_easy, D_hard),
   c(1:ncol(fixed_effects)),
   as.matrix(fixed_effects),
   as.matrix(trial_matrix),
   choice_matrix[, 'choose_hard_q'],
   20000,
   25000
 )
})


vlabels_thurstone = colnames(cbind(fixed_effects[, -1], D_easy, D_hard))
G_thurstone = t(apply(
 out_model_thurstone$gs0,
 2,
 FUN = quantile,
 probs = c(.5, .025, .975)
))

rownames(G_thurstone) = vlabels_thurstone
B_thurstone = t(apply(
 out_model_thurstone$beta,
 2,
 FUN = quantile,
 probs = c(.5, 0.025, .975)
))

rownames(B_thurstone) = colnames(fixed_effects)

S_thurstone = solve(
  apply(out_model_thurstone$Sigma_zeta_inv, c(1, 2), FUN = mean)
)

inv_sd = diag(1 / sqrt(diag(solve(
 apply(out_model_thurstone$Sigma_zeta_inv, c(1, 2),
       FUN = mean)
))))

inv_sd %*% S_thurstone %*% inv_sd
apply(out_model_thurstone$as, 2, FUN = mean)
apply(out_model_thurstone$bs, 2, FUN = mean)

## End(Not run)
## Not run: 
# Variables
# Y = trial matix
# C = KN vector of binary choices
# N = #of subjects
# J = # of items
# K= # of choices
# atrue = true item discriminations
# btrue = true item locations
# thetatrue = true thetas/latent performance
# gamma = fixed effects coefficients
# Sig = random-effects variance-covariance
# subid = id variable for subjects

# Load the Package
library(cIRT)

# Load the Data
data(trial_matrix)
data(choice_matrix)

# Thurstone design matrices
all_nopractice = subset(all_data_trials, experiment_loop.thisN > -1)
hard_items = choice_matrix$hard_q_id
easy_items = choice_matrix$easy_q_id

D_easy = model.matrix( ~ -1 + factor(easy_items))
D_hard = -1 * model.matrix( ~ -1 + factor(hard_items))[, -c(5, 10, 15)]

# Defining effect-coded contrasts
high_contrasts = rbind(-1, diag(4))
rownames(high_contrasts) = 12:16
low_contrasts = rbind(-1, diag(2))
rownames(low_contrasts) = 4:6

# Creating high & low factors
high = factor(choice_matrix[, 'high_value'])
low = factor(choice_matrix[, 'low_value'])
contrasts(high) = high_contrasts
contrasts(low) = low_contrasts

fixed_effects = model.matrix( ~ high + low)
fixed_effects_base = fixed_effects[, 1]
fixed_effects_int = model.matrix( ~ high * low)


# Model with Thurstone D Matrix
system.time({
 out_model_thurstone = cIRT(
   choice_matrix[, 'subject_id'],
   cbind(fixed_effects[, -1], D_easy, D_hard),
   c(1:ncol(fixed_effects)),
   as.matrix(fixed_effects),
   as.matrix(trial_matrix),
   choice_matrix[, 'choose_hard_q'],
   20000,
   25000
 )
})


vlabels_thurstone = colnames(cbind(fixed_effects[, -1], D_easy, D_hard))
G_thurstone = t(apply(
 out_model_thurstone$gs0,
 2,
 FUN = quantile,
 probs = c(.5, .025, .975)
))

rownames(G_thurstone) = vlabels_thurstone
B_thurstone = t(apply(
 out_model_thurstone$beta,
 2,
 FUN = quantile,
 probs = c(.5, 0.025, .975)
))

rownames(B_thurstone) = colnames(fixed_effects)

S_thurstone = solve(
  apply(out_model_thurstone$Sigma_zeta_inv, c(1, 2), FUN = mean)
)

inv_sd = diag(1 / sqrt(diag(solve(
 apply(out_model_thurstone$Sigma_zeta_inv, c(1, 2),
       FUN = mean)
))))

inv_sd %*% S_thurstone %*% inv_sd
apply(out_model_thurstone$as, 2, FUN = mean)
apply(out_model_thurstone$bs, 2, FUN = mean)

## End(Not run)

Direct Sum of Matrices

Description

Computes the direct sum of all matrices passed in via the list.

Usage

direct_sum(x)
direct_sum(x)

Arguments

`x`	A `⁠field<matrix>⁠` or `list` containing matrices

Details

Consider matrix $A$ ( $M \times N$ ) and $B$ ( $K \times P$ ). A direct sum is a diagonal matrix $A (+) B$ with dimensions $(m + k) x (n + p)$ .

Value

Matrix containing the direct sum of all matrices in the list.

Author(s)

James Joseph Balamuta

Examples


x = list(matrix(0, nrow = 5, ncol = 3),
         matrix(1, nrow = 5, ncol = 3))
direct_sum(x)

x = list(matrix(rnorm(15), nrow = 5, ncol = 3),
         matrix(rnorm(30), nrow = 5, ncol = 6),
         matrix(rnorm(18), nrow = 2, ncol = 9))
direct_sum(x)
x = list(matrix(0, nrow = 5, ncol = 3),
         matrix(1, nrow = 5, ncol = 3))
direct_sum(x)

x = list(matrix(rnorm(15), nrow = 5, ncol = 3),
         matrix(rnorm(30), nrow = 5, ncol = 6),
         matrix(rnorm(18), nrow = 2, ncol = 9))
direct_sum(x)

Generate Observed Data from choice model

Description

Generates observed cognitive and choice data from the IRT-Thurstone model.

Usage

Generate_Choice(
  N,
  J,
  K,
  theta,
  as,
  bs,
  zeta,
  gamma,
  X,
  W,
  subject_ids,
  unique_subject_ids
)
Generate_Choice(
  N,
  J,
  K,
  theta,
  as,
  bs,
  zeta,
  gamma,
  X,
  W,
  subject_ids,
  unique_subject_ids
)

Arguments

`N`	An `integer` for the number of observations.
`J`	An `integer` for the number of items.
`K`	An `integer` for the number of paired comparisons.
`theta`	A `vector` of latent cognitive variables.
`as`	A `vector` of length J with item discriminations.
`bs`	A `vector` of length J with item locations.
`zeta`	A `matrix` with dimensions N x V containing random parameter estimates.
`gamma`	A `vector` with dimensions P x 1 containing fixed parameter estimates, where $P = P_1 + P_2$
`X`	A `matrix` with dimensions N*K x P_1 containing fixed effect design matrix without theta.
`W`	A `matrix` with dimensions N*K x V containing random effect variables.
`subject_ids`	A `vector` with length NK x 1 containing subject-choice IDs.
`unique_subject_ids`	A `vector` with length N x 1 containing unique subject IDs.

Value

A list that contains:

Y: A matrix of dimension N by J
C: A vector of length NK

Author(s)

Steven Andrew Culpepper and James Joseph Balamuta

Payout Matrix Data

Description

This data set contains the payout information for each subject.

Usage

payout_matrix
payout_matrix

Format

A data frame with 252 observations on the following 4 variables.

Participant: Subject ID
cum_sum: Sum of all payouts
num_correct_choices: Total number of correct choices (out of 15)
num_correct_trials: Total number of correct trials (out of 30)

Author(s)

Steven Andrew Culpepper and James Joseph Balamuta

Source

Choice38 Experiment at UIUC during Spring 2014 - Fall 2014

Probit Hierarchical Level Model

Description

Performs modeling procedure for a Probit Hierarchical Level Model.

Usage

probitHLM(
  unique_subject_ids,
  subject_ids,
  choices_nk,
  fixed_effects_design,
  rv_effects_design,
  B_elem_plus1,
  gamma,
  beta,
  theta,
  zeta_rv,
  WtW,
  Z_c,
  Wzeta_0,
  inv_Sigma_gamma,
  mu_gamma,
  Sigma_zeta_inv,
  S0,
  mu_beta,
  sigma_beta_inv
)
probitHLM(
  unique_subject_ids,
  subject_ids,
  choices_nk,
  fixed_effects_design,
  rv_effects_design,
  B_elem_plus1,
  gamma,
  beta,
  theta,
  zeta_rv,
  WtW,
  Z_c,
  Wzeta_0,
  inv_Sigma_gamma,
  mu_gamma,
  Sigma_zeta_inv,
  S0,
  mu_beta,
  sigma_beta_inv
)

Arguments

`unique_subject_ids`	A `vector` with length N x 1 containing unique subject IDs.
`subject_ids`	A `vector` with length N*K x 1 containing subject IDs.
`choices_nk`	A `vector` with length N*K x 1 containing subject choices.
`fixed_effects_design`	A `matrix` with dimensions N*K x P containing fixed effect variables.
`rv_effects_design`	A `matrix` with dimensions N*K x V containing random effect variables.
`B_elem_plus1`	A `V[[1]]` dimensional column `vector` indicating which zeta_i relate to theta_i.
`gamma`	A `vector` with dimensions P_1 x 1 containing fixed parameter estimates.
`beta`	A `vector` with dimensions P_2 x 1 containing random parameter estimates.
`theta`	A `vector` with dimensions N x 1 containing subject understanding estimates.
`zeta_rv`	A `matrix` with dimensions N x V containing random parameter estimates.
`WtW`	A `⁠field<matrix>⁠` P x P x N contains the caching for direct sum.
`Z_c`	A `vector` with dimensions N*K x 1
`Wzeta_0`	A `vector` with dimensions N*K x 1
`inv_Sigma_gamma`	A `matrix` with dimensions P x P that is the prior inverse sigma matrix for gamma.
`mu_gamma`	A `vector` with length P x 1 that is the prior mean vector for gamma.
`Sigma_zeta_inv`	A `matrix` with dimensions V x V that is the prior inverse sigma matrix for zeta.
`S0`	A `matrix` with dimensions V x V that is the prior sigma matrix for zeta.
`mu_beta`	A `vector` with dimensions P_2 x 1, that is the mean of beta.
`sigma_beta_inv`	A `matrix` with dimensions P_2 x P_2, that is the inverse sigma matrix of beta.

Details

The function is implemented to decrease the amount of vectorizations necessary.

Value

A list that contains:

zeta_1: A vector of length N
sigma_zeta_inv_1: A matrix of dimensions V x V
gamma_1: A vector of length P
beta_1: A vector of length V
B: A matrix of length V

Author(s)

Steven Andrew Culpepper and James Joseph Balamuta

Generate Random Inverse Wishart Distribution

Description

Creates a random inverse wishart distribution when given degrees of freedom and a sigma matrix.

Usage

riwishart(df, S)
riwishart(df, S)

Arguments

`df`	An `integer` that represents the degrees of freedom. (> 0)
`S`	A `matrix` with dimensions m x m that provides Sigma, the covariance matrix.

Value

A matrix that is an inverse wishart distribution.

Author(s)

James Joseph Balamuta

Examples

#Call with the following data:
riwishart(3, diag(2))
#Call with the following data:
riwishart(3, diag(2))

Generate Random Multivariate Normal Distribution

Description

Creates a random Multivariate Normal when given number of obs, mean, and sigma.

Usage

rmvnorm(n, mu, S)
rmvnorm(n, mu, S)

Arguments

`n`	An `integer`, which gives the number of observations. (> 0)
`mu`	A `vector` length m that represents the means of the normals.
`S`	A `matrix` with dimensions m x m that provides Sigma, the covariance matrix.

Value

A matrix that is a Multivariate Normal distribution.

Author(s)

James Joseph Balamuta

Examples

# Call with the following data: 
rmvnorm(2, c(0,0), diag(2))
# Call with the following data: 
rmvnorm(2, c(0,0), diag(2))

Generate Random Wishart Distribution

Description

Creates a random wishart distribution when given degrees of freedom and a sigma matrix.

Usage

rwishart(df, S)
rwishart(df, S)

Arguments

`df`	An `integer`, which gives the degrees of freedom of the Wishart. (> 0)
`S`	A `matrix` with dimensions m x m that provides Sigma, the covariance matrix.

Value

A matrix that is a Wishart distribution, aka the sample covariance matrix of a Multivariate Normal Distribution

Author(s)

James Joseph Balamuta

Examples

# Call with the following data:
rwishart(3, diag(2))

# Validation
set.seed(1337)
S = toeplitz((10:1)/10)
n = 10000
o = array(dim = c(10,10,n))
for(i in 1:n){
o[,,i] = rwishart(20, S)
}
mR = apply(o, 1:2, mean)
Va = 20*(S^2 + tcrossprod(diag(S)))
vR = apply(o, 1:2, var)
stopifnot(all.equal(vR, Va, tolerance = 1/16))
# Call with the following data:
rwishart(3, diag(2))

# Validation
set.seed(1337)
S = toeplitz((10:1)/10)
n = 10000
o = array(dim = c(10,10,n))
for(i in 1:n){
o[,,i] = rwishart(20, S)
}
mR = apply(o, 1:2, mean)
Va = 20*(S^2 + tcrossprod(diag(S)))
vR = apply(o, 1:2, var)
stopifnot(all.equal(vR, Va, tolerance = 1/16))

Survey Data

Description

This data set contains the subject's responses survey questions administered using Choice38.

Usage

survey_data
survey_data

Format

A data frame with 102 observations on the following 2 variables.

id

Subject's Assigned Research ID

sex

Subject's sex:

Male
Female

Author(s)

Steven Andrew Culpepper and James Joseph Balamuta

Source

Choice38 Experiment at UIUC during Spring 2014 - Fall 2014

Calculate Tabulated Total Scores

Description

Internal function to -2LL

Usage

Total_Tabulate(N, J, Y)
Total_Tabulate(N, J, Y)

Arguments

`N`	An `integer`, which gives the number of observations. (> 0)
`J`	An `integer`, which gives the number of items. (> 0)
`Y`	A N by J `matrix` of item responses.

Value

A vector of tabulated total scores.

Author(s)

Steven Andrew Culpepper

Trial Matrix Data

Description

This data set contains the subject's responses to items. Correct answers are denoted by 1 and incorrect answers are denoted by 0.

Usage

trial_matrix
trial_matrix

Format

A data frame with 252 observations on the following 30 variables.

t1: Subject's Response to Item 1.
t2: Subject's Response to Item 2.
t3: Subject's Response to Item 3.
t4: Subject's Response to Item 4.
t5: Subject's Response to Item 5.
t6: Subject's Response to Item 6.
t7: Subject's Response to Item 7.
t8: Subject's Response to Item 8.
t9: Subject's Response to Item 9.
t10: Subject's Response to Item 10.
t11: Subject's Response to Item 11.
t12: Subject's Response to Item 12.
t13: Subject's Response to Item 13.
t14: Subject's Response to Item 14.
t15: Subject's Response to Item 15.
t16: Subject's Response to Item 16.
t17: Subject's Response to Item 17.
t18: Subject's Response to Item 18.
t19: Subject's Response to Item 19.
t20: Subject's Response to Item 20.
t21: Subject's Response to Item 21.
t22: Subject's Response to Item 22.
t23: Subject's Response to Item 23.
t24: Subject's Response to Item 24.
t25: Subject's Response to Item 25.
t26: Subject's Response to Item 26.
t27: Subject's Response to Item 27.
t28: Subject's Response to Item 28.
t29: Subject's Response to Item 29.
t30: Subject's Response to Item 30.

Author(s)

Steven Andrew Culpepper and James Joseph Balamuta

Source

Choice38 Experiment at UIUC during Spring 2014 - Fall 2014

Two Parameter Choice IRT Model MCMC

Description

Performs an MCMC routine for a two parameter IRT Model using Choice Data

Usage

TwoPLChoicemcmc(
  unique_subject_ids,
  subject_ids,
  choices_nk,
  fixed_effects,
  B,
  rv_effects_design,
  gamma,
  beta,
  zeta_rv,
  Sigma_zeta_inv,
  Y,
  theta0,
  a0,
  b0,
  mu_xi0,
  Sig_xi0
)
TwoPLChoicemcmc(
  unique_subject_ids,
  subject_ids,
  choices_nk,
  fixed_effects,
  B,
  rv_effects_design,
  gamma,
  beta,
  zeta_rv,
  Sigma_zeta_inv,
  Y,
  theta0,
  a0,
  b0,
  mu_xi0,
  Sig_xi0
)

Arguments

`unique_subject_ids`	A `vector` with length $N \times 1$ containing unique subject IDs.
`subject_ids`	A `vector` with length $NK \times 1$ containing subject IDs.
`choices_nk`	A `vector` with length $NK \times 1$ containing subject choices.
`fixed_effects`	A `matrix` with dimensions $NK \times P_1$ containing fixed effect design matrix without theta.
`B`	A $V$ dimensional column `vector` relating $\theta_i$ and $\zeta_i$ .
`rv_effects_design`	A `matrix` with dimensions $NK \times V$ containing random effect variables.
`gamma`	A `vector` with dimensions $P \times 1$ containing fixed parameter estimates, where $P = P_1 + P_2$
`beta`	A `vector` with dimensions $P_2$ containing random parameter estimates.
`zeta_rv`	A `matrix` with dimensions $N \times V$ containing random parameter estimates.
`Sigma_zeta_inv`	A `matrix` with dimensions $P_2 \times P_2$ .
`Y`	A `matrix` of dimensions $N \times J$ for Dichotomous item responses
`theta0`	A `vector` of length $N \times 1$ for latent theta.
`a0`	A `vector` of length $J$ for item discriminations.
`b0`	A `vector` of length $J$ for item locations.
`mu_xi0`	A `vector` of dimension 2 (i.e. c(0,1)) that is a prior for item parameter means.
`Sig_xi0`	A `matrix` of dimension 2x2 (i.e. diag(2)) that is a prior for item parameter vc matrix.

Value

A list that contains:

ai1: A vector of length J
bi1: A vector of length J
theta1: A vector of length N
Z_c: A matrix of length NK
Wzeta_0: A matrix of length NK

Author(s)

Steven Andrew Culpepper and James Joseph Balamuta

Examples

## Not run: 
# Call with the following data:
TwoPLChoicemcmc(cogDAT, theta0, a0, b0, mu_xi0, Sig_xi0)

## End(Not run)
## Not run: 
# Call with the following data:
TwoPLChoicemcmc(cogDAT, theta0, a0, b0, mu_xi0, Sig_xi0)

## End(Not run)

Package 'cIRT'

Help Index

cIRT: Choice Item Response Theory

Description

Author(s)

See Also

Center a Matrix

Description

Usage

Arguments

Details

Value

Author(s)

See Also

Examples

Choice Matrix Data

Description

Usage

Format

Author(s)

Source

Generic Implementation of Choice IRT MCMC

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Direct Sum of Matrices

Description

Usage

Arguments

Details

Value

Author(s)

Examples

Generate Observed Data from choice model

Description

Usage

Arguments

Value

Author(s)

Payout Matrix Data

Description

Usage

Format

Author(s)

Source

Probit Hierarchical Level Model

Description

Usage

Arguments

Details

Value

Author(s)

See Also

Generate Random Inverse Wishart Distribution

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Generate Random Multivariate Normal Distribution

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Generate Random Wishart Distribution

Description

Usage

Arguments

Value

Author(s)

See Also