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Type:

Package

Title:

Bayesian 4 Parameter Item Response Model

Version:

1.1.0

Description:

Estimate Barton & Lord's (1981) <doi:10.1002/j.2333-8504.1981.tb01255.x> four parameter IRT model with lower and upper asymptotes using Bayesian formulation described by Culpepper (2016) <doi:10.1007/s11336-015-9477-6>.

URL:

https://github.com/tmsalab/fourPNO

BugReports:

https://github.com/tmsalab/fourPNO/issues

License:

GPL-2 | GPL-3 [expanded from: GPL (≥ 2)]

Depends:

R (≥ 3.5.0)

Imports:

Rcpp (≥ 1.0.0)

LinkingTo:

Rcpp, RcppArmadillo (≥ 0.9.200)

RoxygenNote:

6.1.1

Encoding:

UTF-8

NeedsCompilation:

yes

Packaged:

2019-09-23 22:19:50 UTC; jjb

Author:

Steven Andrew Culpepper

[aut, cre, cph]

Maintainer:

Steven Andrew Culpepper <sculpepp@illinois.edu>

Repository:

CRAN

Date/Publication:

2019-09-24 04:40:02 UTC

fourPNO: Bayesian 4 Parameter Item Response Model

Description

Author(s)

Maintainer: Steven Andrew Culpepper sculpepp@illinois.edu (0000-0003-4226-6176) [copyright holder]

Gibbs Implementation of 2PNO

Description

Implement Gibbs 2PNO Sampler

Usage

Gibbs_2PNO(Y, mu_xi, Sigma_xi_inv, mu_theta, Sigma_theta_inv, burnin,
  chain_length = 10000L)

Arguments

Y

A N by J matrix of item responses.

mu_xi

A two dimensional vector of prior item parameter means.

Sigma_xi_inv

A two dimensional identity matrix of prior item parameter VC matrix.

mu_theta

The prior mean for theta.

Sigma_theta_inv

The prior inverse variance for theta.

burnin

The number of MCMC samples to discard.

chain_length

The number of MCMC samples.

Value

Samples from posterior.

Author(s)

Steven Andrew Culpepper

Examples

# simulate small 2PNO dataset to demonstrate function
J = 5
N = 100

# Population item parameters
as_t = rnorm(J,mean=2,sd=.5)
bs_t = rnorm(J,mean=0,sd=.5)

# Sampling gs and ss with truncation
gs_t = rbeta(J,1,8)
ps_g = pbeta(1-gs_t,1,8)
ss_t = qbeta(runif(J)*ps_g,1,8)
theta_t = rnorm(N)
Y_t = Y_4pno_simulate(N,J,as=as_t,bs=bs_t,gs=gs_t,ss=ss_t,theta=theta_t)

# Setting prior parameters
mu_theta = 0
Sigma_theta_inv = 1
mu_xi = c(0,0)
alpha_c = alpha_s = beta_c = beta_s = 1
Sigma_xi_inv = solve(2*matrix(c(1,0,0,1), 2, 2))
burnin = 1000

# Execute Gibbs sampler. This should take about 15.5 minutes
out_t = Gibbs_4PNO(Y_t,mu_xi,Sigma_xi_inv,mu_theta,Sigma_theta_inv,
                    alpha_c,beta_c,alpha_s, beta_s,burnin,
                    rep(1,J),rep(1,J),gwg_reps=5,chain_length=burnin*2)

# Summarizing posterior distribution
OUT = cbind(
    apply(out_t$AS[, -c(1:burnin)], 1, mean),
    apply(out_t$BS[, -c(1:burnin)], 1, mean),
    apply(out_t$GS[, -c(1:burnin)], 1, mean),
    apply(out_t$SS[, -c(1:burnin)], 1, mean),
    apply(out_t$AS[, -c(1:burnin)], 1, sd),
    apply(out_t$BS[, -c(1:burnin)], 1, sd),
    apply(out_t$GS[, -c(1:burnin)], 1, sd),
    apply(out_t$SS[, -c(1:burnin)], 1, sd)
)
OUT = cbind(1:J, OUT)
colnames(OUT) = c('Item','as','bs','gs','ss','as_sd','bs_sd',
                  'gs_sd','ss_sd')
print(OUT, digits = 3)

Gibbs Implementation of 4PNO

Description

Internal function to -2LL

Usage

Gibbs_4PNO(Y, mu_xi, Sigma_xi_inv, mu_theta, Sigma_theta_inv, alpha_c,
  beta_c, alpha_s, beta_s, burnin, cTF, sTF, gwg_reps,
  chain_length = 10000L)

Arguments

Y

A N by J matrix of item responses.

mu_xi

A two dimensional vector of prior item parameter means.

Sigma_xi_inv

A two dimensional identity matrix of prior item parameter VC matrix.

mu_theta

The prior mean for theta.

Sigma_theta_inv

The prior inverse variance for theta.

alpha_c

The lower asymptote prior 'a' parameter.

beta_c

The lower asymptote prior 'b' parameter.

alpha_s

The upper asymptote prior 'a' parameter.

beta_s

The upper asymptote prior 'b' parameter.

burnin

The number of MCMC samples to discard.

cTF

A J dimensional vector indicating which lower asymptotes to estimate. 0 = exclude lower asymptote and 1 = include lower asymptote.

sTF

A J dimensional vector indicating which upper asymptotes to estimate. 0 = exclude upper asymptote and 1 = include upper asymptote.

gwg_reps

The number of Gibbs within Gibbs MCMC samples for marginal distribution of gamma. Values between 5 to 10 are adequate.

chain_length

The number of MCMC samples.

Value

Samples from posterior.

Author(s)

Steven Andrew Culpepper

Examples

# Simulate small 4PNO dataset to demonstrate function
J = 5
N = 100

# Population item parameters
as_t = rnorm(J,mean=2,sd=.5)
bs_t = rnorm(J,mean=0,sd=.5)

# Sampling gs and ss with truncation
gs_t = rbeta(J,1,8)
ps_g = pbeta(1-gs_t,1,8)
ss_t = qbeta(runif(J)*ps_g,1,8)
theta_t <- rnorm(N)
Y_t = Y_4pno_simulate(N,J,as=as_t,bs=bs_t,gs=gs_t,ss=ss_t,theta=theta_t)

# Setting prior parameters
mu_theta=0
Sigma_theta_inv=1
mu_xi = c(0,0)
alpha_c=alpha_s=beta_c=beta_s=1
Sigma_xi_inv = solve(2*matrix(c(1,0,0,1),2,2))
burnin = 1000

# Execute Gibbs sampler
out_t = Gibbs_4PNO(Y_t,mu_xi,Sigma_xi_inv,mu_theta,
                   Sigma_theta_inv,alpha_c,beta_c,alpha_s,
                   beta_s,burnin,rep(1,J),rep(1,J),
                   gwg_reps=5,chain_length=burnin*2)

# Summarizing posterior distribution
OUT = cbind(apply(out_t$AS[,-c(1:burnin)],1,mean),
            apply(out_t$BS[,-c(1:burnin)],1,mean),
            apply(out_t$GS[,-c(1:burnin)],1,mean),
            apply(out_t$SS[,-c(1:burnin)],1,mean),
            apply(out_t$AS[,-c(1:burnin)],1,sd),
            apply(out_t$BS[,-c(1:burnin)],1,sd),
            apply(out_t$GS[,-c(1:burnin)],1,sd),
            apply(out_t$SS[,-c(1:burnin)],1,sd) )

OUT = cbind(1:J,OUT)
colnames(OUT) = c('Item', 'as', 'bs', 'gs', 'ss', 'as_sd', 'bs_sd',
                  'gs_sd', 'ss_sd')
print(OUT, digits = 3)

Calculate Tabulated Total Scores

Description

Internal function to -2LL

Usage

Total_Tabulate(N, J, Y)

Arguments

N

An int, which gives the number of observations. (> 0)

J

An int, 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

Simulate from 4PNO Model

Description

Generate item responses under the 4PNO

Usage

Y_4pno_simulate(N, J, as, bs, gs, ss, theta)

Arguments

N

An int, which gives the number of observations. (> 0)

J

An int, which gives the number of items. (> 0)

as

A vector of item discrimination parameters.

bs

A vector of item threshold parameters.

gs

A vector of item lower asymptote parameters.

ss

A vector of item upper asymptote parameters.

theta

A vector of prior thetas.

Value

A N by J matrix of dichotomous item responses.

Author(s)

Steven Andrew Culpepper

Compute 4PNO Deviance

Description

Internal function to -2LL

Usage

min2LL_4pno(N, J, Y, as, bs, gs, ss, theta)

Arguments

N

An int, which gives the number of observations. (> 0)

J

An int, which gives the number of items. (> 0)

Y

A N by J matrix of item responses.

as

A vector of item discrimination parameters.

bs

A vector of item threshold parameters.

gs

A vector of item lower asymptote parameters.

ss

A vector of item upper asymptote parameters.

theta

A vector of prior thetas.

Value

-2LL.

Author(s)

Steven Andrew Culpepper

Generate Random Multivariate Normal Distribution

Description

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

Usage

rmvnorm(n, mu, sigma)

Arguments

n

An int, which gives the number of observations. (> 0)

mu

A vector length m that represents the means of the normals.

sigma

A matrix with dimensions m x m that provides the covariance matrix.

Value

A matrix that is a Multivariate Normal distribution

Author(s)

James J Balamuta

Examples

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

fourPNO: Bayesian 4 Parameter Item Response Model

Description

Author(s)

See Also

Gibbs Implementation of 2PNO

Description

Usage

Arguments

Value

Author(s)

Examples

Gibbs Implementation of 4PNO

Description

Usage

Arguments

Value

Author(s)

Examples

Calculate Tabulated Total Scores

Description

Usage

Arguments

Value

Author(s)

See Also

Simulate from 4PNO Model

Description

Usage

Arguments

Value

Author(s)

See Also

Compute 4PNO Deviance

Description

Usage

Arguments

Value

Author(s)

See Also

Generate Random Multivariate Normal Distribution

Description

Usage

Arguments

Value

Author(s)

Examples