Package 'TestDesign'

Title: Optimal Test Design Approach to Fixed and Adaptive Test Construction
Description: Uses the optimal test design approach by Birnbaum (1968, ISBN:9781593119348) and van der Linden (2018) <doi:10.1201/9781315117430> to construct fixed, adaptive, and parallel tests. Supports the following mixed-integer programming (MIP) solver packages: 'Rsymphony', 'highs', 'gurobi', 'lpSolve', and 'Rglpk'. The 'gurobi' package is not available from CRAN; see <https://www.gurobi.com/downloads/>.
Authors: Seung W. Choi [aut, cre] , Sangdon Lim [aut]
Maintainer: Seung W. Choi <[email protected]>
License: GPL (>= 2)
Version: 1.7.0
Built: 2024-11-21 06:54:35 UTC
Source: CRAN

Help Index

Calculate alpha angles from a-parameters

Description

a_to_alpha is a function for converting an a-parameter vector to an alpha angle vector. The returned values are in the radian metric.

Usage

a_to_alpha(a)

Arguments

a

the a-parameter vector.

Examples

a_to_alpha(c(1, 1))

Open TestDesign app

Description

app and OAT are aliases of TestDesign.

Usage

app()

OAT()

Details

TestDesign is a caller function for opening the Shiny interface of TestDesign package.

Examples

## Not run: 
if (interactive()) {
  TestDesign()
}

## End(Not run)

Build constraints (shortcut to other loading functions)

Description

buildConstraints is a data loading function for creating a constraints object. buildConstraints is a shortcut that calls other data loading functions. The constraints must be in the expected format; see the vignette in vignette("constraints").

Usage

buildConstraints(object, item_pool, item_attrib, st_attrib = NULL)

Arguments

object

constraint specifications. Can be a data.frame or the file path of a .csv file. See the vignette for the expected format.
item_pool

item parameters. Can be a item_pool object, a data.frame or the file path of a .csv file.
item_attrib

item attributes. Can be an item_attrib object, a data.frame or the file path of a .csv file.
st_attrib

(optional) stimulus attributes. Can be an st_attrib object, a data.frame or the file path of a .csv file.

Value

buildConstraints returns a constraints object. This object is used in Static and Shadow.

Examples

## Read from objects:
constraints_science <- buildConstraints(constraints_science_data,
  itempool_science, itemattrib_science)
constraints_reading <- buildConstraints(constraints_reading_data,
  itempool_reading, itemattrib_reading, stimattrib_reading)

## Read from data.frame:
constraints_science <- buildConstraints(constraints_science_data,
  itempool_science_data, itemattrib_science_data)
constraints_reading <- buildConstraints(constraints_reading_data,
  itempool_reading_data, itemattrib_reading_data, stimattrib_reading_data)

## Read from file: write to tempdir() for illustration and clean afterwards
f1 <- file.path(tempdir(), "constraints_science.csv")
f2 <- file.path(tempdir(), "itempool_science.csv")
f3 <- file.path(tempdir(), "itemattrib_science.csv")
write.csv(constraints_science_data, f1, row.names = FALSE)
write.csv(itempool_science_data   , f2, row.names = FALSE)
write.csv(itemattrib_science_data , f3, row.names = FALSE)
constraints_science <- buildConstraints(f1, f2, f3)
file.remove(f1)
file.remove(f2)
file.remove(f3)

(C++) For multiple items, calculate Fisher information

Description

calc_info() and calc_info_matrix() are functions for calculating Fisher information. These functions are designed for multiple items.

Usage

calc_info(x, item_parm, ncat, model)

calc_info_matrix(x, item_parm, ncat, model)

Arguments

x

the theta value. This must be a column vector in matrix form for calc_info_matrix().
item_parm

a matrix containing item parameters. Each row should represent an item.
ncat

a vector containing the number of response categories of each item.
model

a vector indicating item models of each item, using

  • 1: 1PL model

  • 2: 2PL model

  • 3: 3PL model

  • 4: PC model

  • 5: GPC model

  • 6: GR model

Details

calc_info() accepts a single theta value, and calc_info_matrix() accepts multiple theta values.

Currently supports unidimensional models.

References

Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: Danish Institute for Educational Research.

Lord, F. M. (1952). A theory of test scores (Psychometric Monograph No. 7). Richmond, VA: Psychometric Corporation.

Birnbaum, A. (1957). Efficient design and use of tests of mental ability for various decision-making problems (Series Report No. 58-16. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). On the estimation of mental ability (Series Report No. 15. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). Further considerations of efficiency in tests of a mental ability (Series Report No. 17. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In Lord, F. M., Novick, M. R. (eds.), Statistical Theories of Mental Test Scores, 397-479. Reading, MA: Addison-Wesley.

Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47(2), 149-174.

Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43(4), 561-573.

Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16(2), 159-176.

Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph, 17.

Examples

# item parameters
item_parm <- matrix(c(
  1, NA,   NA,
  1,  2,   NA,
  1,  2, 0.25,
  0,  1,   NA,
  2,  0,    1,
  2,  0,    2),
  nrow = 6,
  byrow = TRUE
)

ncat  <- c(2, 2, 2, 3, 3, 3)
model <- c(1, 2, 3, 4, 5, 6)

# single theta example
x <- 0.5
calc_info(x, item_parm, ncat, model)

# multiple thetas example
x <- matrix(seq(0.1, 0.5, 0.1)) # column vector in matrix form
calc_info_matrix(x, item_parm, ncat, model)

Calculate the Fisher information using empirical Bayes

Description

Calculate the Fisher information using empirical Bayes.

Usage

calc_info_EB(x, item_parm, ncat, model)

Arguments

x

A numeric vector of MCMC sampled theta values.
item_parm

A numeric matrix of item parameters.
ncat

a numeric vector specifying the number of response categories in each item.
model

a numeric vector indicating the IRT models of each item (1: 1PL, 2: 2PL, 3: 3PL, 4: PC, 5: GPC, 6: GR).

Calculate the Fisher information using full Bayesian

Description

Calculate the Fisher information using full Bayesian.

Usage

calc_info_FB(x, items_list, ncat, model, useEAP = FALSE)

Arguments

x

A numeric vector of MCMC sampled theta values.
items_list

A list of item parameter matrices.
ncat

a numeric vector specifying the number of response categories in each item.
model

a numeric vector indicating the IRT models of each item (1: 1PL, 2: 2PL, 3: 3PL, 4: PC, 5: GPC, 6: GR).
useEAP

TRUE to use the mean of MCMC theta draws.

(C++) For multiple items, calculate likelihoods

Description

calc_likelihood() and calc_likelihood_function() are functions for calculating likelihoods.

Usage

calc_likelihood(x, item_parm, resp, ncat, model)

calc_likelihood_function(theta_grid, item_parm, resp, ncat, model)

calc_log_likelihood(x, item_parm, resp, ncat, model, prior, prior_parm)

calc_log_likelihood_function(
  theta_grid,
  item_parm,
  resp,
  ncat,
  model,
  prior,
  prior_parm
)

Arguments

x, theta_grid

the theta value. This must be a column vector in matrix form for calc_*_function() functions.
item_parm

a matrix containing item parameters. Each row should represent an item.
resp

a vector containing responses on each item.
ncat

a vector containing the number of response categories of each item.
model

a vector indicating item models of each item, using

  • 1: 1PL model

  • 2: 2PL model

  • 3: 3PL model

  • 4: PC model

  • 5: GPC model

  • 6: GR model

prior

an integer indicating the type of prior distribution, using

  • 1: normal distribution

  • 2: uniform distribution

prior_parm

a vector containing parameters for the prior distribution.

Details

calc_log_likelihood() and calc_log_likelihood_function() are functions for calculating log likelihoods.

These functions are designed for multiple items.

calc_*() functions accept a single theta value, and calc_*_function() functions accept multiple theta values.

Currently supports unidimensional models.

References

Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: Danish Institute for Educational Research.

Lord, F. M. (1952). A theory of test scores (Psychometric Monograph No. 7). Richmond, VA: Psychometric Corporation.

Birnbaum, A. (1957). Efficient design and use of tests of mental ability for various decision-making problems (Series Report No. 58-16. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). On the estimation of mental ability (Series Report No. 15. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). Further considerations of efficiency in tests of a mental ability (Series Report No. 17. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In Lord, F. M., Novick, M. R. (eds.), Statistical Theories of Mental Test Scores, 397-479. Reading, MA: Addison-Wesley.

Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47(2), 149-174.

Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43(4), 561-573.

Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16(2), 159-176.

Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph, 17.

Examples

# item parameters
item_parm <- matrix(c(
  1, NA,   NA,
  1,  2,   NA,
  1,  2, 0.25,
  0,  1,   NA,
  2,  0,    1,
  2,  0,    2),
  nrow = 6,
  byrow = TRUE
)

ncat  <- c(2, 2, 2, 3, 3, 3)
model <- c(1, 2, 3, 4, 5, 6)
resp  <- c(0, 1, 0, 1, 0, 1)

x <- 3
l  <- calc_likelihood(x, item_parm, resp, ncat, model)
ll <- calc_log_likelihood(x, item_parm, resp, ncat, model, 2, NA)
log(l) == ll

x <- matrix(seq(-3, 3, .1))
l  <- calc_likelihood_function(x, item_parm, resp, ncat, model)
ll <- calc_log_likelihood_function(x, item_parm, resp, ncat, model, 2, NA)
all(log(l) == ll)

Calculate the mutual information using full Bayesian

Description

Calculate the mutual information using full Bayesian.

Usage

calc_MI_FB(x, items_list, ncat, model)

Arguments

x

A numeric vector of MCMC sampled theta values.
items_list

A list of item parameter matrices.
ncat

a numeric vector specifying the number of response categories in each item.
model

a numeric vector indicating the IRT models of each item (1: 1PL, 2: 2PL, 3: 3PL, 4: PC, 5: GPC, 6: GR).

Calculate a posterior value of theta

Description

Calculate a posterior value of theta.

Usage

calc_posterior(x, item_parm, resp, ncat, model, prior, prior_parm)

Arguments

x

A length-one numeric vector for a theta value.
item_parm

A numeric matrix of item parameters.
resp

a numeric vector containing item responses.
ncat

A numeric vector of the number of response categories by item.
model

A numeric vector indicating the IRT models of each item (1: 1PL, 2: 2PL, 3: 3PL, 4: PC, 5: GPC, 6: GR).
prior

The type of prior distribution (1: normal, 2: uniform).
prior_parm

A numeric vector of hyperparameters for the prior distribution, c(mu, sigma) or c(ll, ul).

Calculate a posterior distribution of theta

Description

Calculate a posterior distribution of theta.

Usage

calc_posterior_function(
  theta_grid,
  item_parm,
  resp,
  ncat,
  model,
  prior,
  prior_parm
)

Arguments

theta_grid

An equi-spaced grid of theta values.
item_parm

A numeric matrix of item parameters.
resp

a numeric vector containing item responses.
ncat

A numeric vector of the number of response categories by item.
model

A numeric vector indicating the IRT models of each item (1: 1PL, 2: 2PL, 3: 3PL, 4: PC, 5: GPC, 6: GR).
prior

The type of prior distribution (1: normal, 2: uniform).
prior_parm

A numeric vector of hyperparameters for the prior distribution, c(mu, sigma) or c(ll, ul).

Calculate a posterior value of theta for a single item

Description

Calculate a posterior value of theta for a single item.

Usage

calc_posterior_single(x, item_parm, resp, ncat, model, prior, prior_parm)

Arguments

x

A length-one numeric vector for a theta value.
item_parm

A numeric vector of item parameters (for one item).
resp

A length-one numeric vector of item responses.
ncat

A length-one numeric vector of the number of response categories by item.
model

A length-one numeric vector of the IRT model by item (1: 1PL, 2: 2PL, 3: 3PL, 4: PC, 5: GPC, 6: GR).
prior

The type of prior distribution (1: normal, 2: uniform).
prior_parm

A numeric vector of hyperparameters for the prior distribution, c(mu, sigma) or c(ll, ul).

Calculate expected scores

Description

calcEscore is a function for calculating expected scores.

Usage

calcEscore(object, theta)

## S4 method for signature 'item_1PL,numeric'
calcEscore(object, theta)

## S4 method for signature 'item_2PL,numeric'
calcEscore(object, theta)

## S4 method for signature 'item_3PL,numeric'
calcEscore(object, theta)

## S4 method for signature 'item_PC,numeric'
calcEscore(object, theta)

## S4 method for signature 'item_GPC,numeric'
calcEscore(object, theta)

## S4 method for signature 'item_GR,numeric'
calcEscore(object, theta)

## S4 method for signature 'item_pool,numeric'
calcEscore(object, theta)

## S4 method for signature 'item_1PL,matrix'
calcEscore(object, theta)

## S4 method for signature 'item_2PL,matrix'
calcEscore(object, theta)

## S4 method for signature 'item_3PL,matrix'
calcEscore(object, theta)

## S4 method for signature 'item_PC,matrix'
calcEscore(object, theta)

## S4 method for signature 'item_GPC,matrix'
calcEscore(object, theta)

## S4 method for signature 'item_GR,matrix'
calcEscore(object, theta)

## S4 method for signature 'item_pool,matrix'
calcEscore(object, theta)

## S4 method for signature 'item_pool_cluster,numeric'
calcEscore(object, theta)

Arguments

object

an item or an item_pool object.
theta

theta values to use.

Value

item object:

calcEscore a vector containing expected score of the item at the theta values.

item_pool object:

calcEscore returns a vector containing the pool-level expected score at the theta values.

References

Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: Danish Institute for Educational Research.

Lord, F. M. (1952). A theory of test scores (Psychometric Monograph No. 7). Richmond, VA: Psychometric Corporation.

Birnbaum, A. (1957). Efficient design and use of tests of mental ability for various decision-making problems (Series Report No. 58-16. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). On the estimation of mental ability (Series Report No. 15. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). Further considerations of efficiency in tests of a mental ability (Series Report No. 17. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In Lord, F. M., Novick, M. R. (eds.), Statistical Theories of Mental Test Scores, 397-479. Reading, MA: Addison-Wesley.

Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47(2), 149-174.

Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43(4), 561-573.

Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16(2), 159-176.

Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph, 17.

Examples

item_1     <- new("item_1PL", difficulty = 0.5)
item_2     <- new("item_2PL", slope = 1.0, difficulty = 0.5)
item_3     <- new("item_3PL", slope = 1.0, difficulty = 0.5, guessing = 0.2)
item_4     <- new("item_PC", threshold = c(-1, 0, 1), ncat = 4)
item_5     <- new("item_GPC", slope = 1.2, threshold = c(-0.8, -1.0, 0.5), ncat = 4)
item_6     <- new("item_GR", slope = 0.9, category = c(-1, 0, 1), ncat = 4)

ICC_item_1 <- calcEscore(item_1, seq(-3, 3, 1))
ICC_item_2 <- calcEscore(item_2, seq(-3, 3, 1))
ICC_item_3 <- calcEscore(item_3, seq(-3, 3, 1))
ICC_item_4 <- calcEscore(item_4, seq(-3, 3, 1))
ICC_item_5 <- calcEscore(item_5, seq(-3, 3, 1))
ICC_item_6 <- calcEscore(item_6, seq(-3, 3, 1))
TCC_pool   <- calcEscore(itempool_science, seq(-3, 3, 1))

Calculate Fisher information

Description

calcFisher is a function for calculating Fisher information.

Usage

calcFisher(object, theta)

## S4 method for signature 'item_1PL,numeric'
calcFisher(object, theta)

## S4 method for signature 'item_2PL,numeric'
calcFisher(object, theta)

## S4 method for signature 'item_3PL,numeric'
calcFisher(object, theta)

## S4 method for signature 'item_PC,numeric'
calcFisher(object, theta)

## S4 method for signature 'item_GPC,numeric'
calcFisher(object, theta)

## S4 method for signature 'item_GR,numeric'
calcFisher(object, theta)

## S4 method for signature 'item_pool,numeric'
calcFisher(object, theta)

## S4 method for signature 'item_1PL,matrix'
calcFisher(object, theta)

## S4 method for signature 'item_2PL,matrix'
calcFisher(object, theta)

## S4 method for signature 'item_3PL,matrix'
calcFisher(object, theta)

## S4 method for signature 'item_PC,matrix'
calcFisher(object, theta)

## S4 method for signature 'item_GPC,matrix'
calcFisher(object, theta)

## S4 method for signature 'item_GR,matrix'
calcFisher(object, theta)

## S4 method for signature 'item_pool,matrix'
calcFisher(object, theta)

## S4 method for signature 'item_pool_cluster,numeric'
calcFisher(object, theta)

Arguments

object

an item or an item_pool object.
theta

theta values to use.

Value

item object:

calcFisher returns a (nq, 1) matrix of information values.

item_pool object:

calcProb returns a (nq, ni) matrix of information values.

notations

  • nq denotes the number of theta values.

  • ni denotes the number of items in the item_pool object.

References

Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: Danish Institute for Educational Research.

Lord, F. M. (1952). A theory of test scores (Psychometric Monograph No. 7). Richmond, VA: Psychometric Corporation.

Birnbaum, A. (1957). Efficient design and use of tests of mental ability for various decision-making problems (Series Report No. 58-16. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). On the estimation of mental ability (Series Report No. 15. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). Further considerations of efficiency in tests of a mental ability (Series Report No. 17. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In Lord, F. M., Novick, M. R. (eds.), Statistical Theories of Mental Test Scores, 397-479. Reading, MA: Addison-Wesley.

Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47(2), 149-174.

Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43(4), 561-573.

Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16(2), 159-176.

Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph, 17.

Examples

item_1      <- new("item_1PL", difficulty = 0.5)
item_2      <- new("item_2PL", slope = 1.0, difficulty = 0.5)
item_3      <- new("item_3PL", slope = 1.0, difficulty = 0.5, guessing = 0.2)
item_4      <- new("item_PC", threshold = c(-1, 0, 1), ncat = 4)
item_5      <- new("item_GPC", slope = 1.2, threshold = c(-0.8, -1.0, 0.5), ncat = 4)
item_6      <- new("item_GR", slope = 0.9, category = c(-1, 0, 1), ncat = 4)

info_item_1 <- calcFisher(item_1, seq(-3, 3, 1))
info_item_2 <- calcFisher(item_2, seq(-3, 3, 1))
info_item_3 <- calcFisher(item_3, seq(-3, 3, 1))
info_item_4 <- calcFisher(item_4, seq(-3, 3, 1))
info_item_5 <- calcFisher(item_5, seq(-3, 3, 1))
info_item_6 <- calcFisher(item_6, seq(-3, 3, 1))
info_pool   <- calcFisher(itempool_science, seq(-3, 3, 1))

Calculate second derivative of log-likelihood

Description

calcHessian is a function for calculating the second derivative of the log-likelihood function.

Usage

calcHessian(object, theta, resp)

## S4 method for signature 'item_1PL,numeric,numeric'
calcHessian(object, theta, resp)

## S4 method for signature 'item_2PL,numeric,numeric'
calcHessian(object, theta, resp)

## S4 method for signature 'item_3PL,numeric,numeric'
calcHessian(object, theta, resp)

## S4 method for signature 'item_PC,numeric,numeric'
calcHessian(object, theta, resp)

## S4 method for signature 'item_GPC,numeric,numeric'
calcHessian(object, theta, resp)

## S4 method for signature 'item_GR,numeric,numeric'
calcHessian(object, theta, resp)

## S4 method for signature 'item_1PL,matrix,numeric'
calcHessian(object, theta, resp)

## S4 method for signature 'item_2PL,matrix,numeric'
calcHessian(object, theta, resp)

## S4 method for signature 'item_3PL,matrix,numeric'
calcHessian(object, theta, resp)

## S4 method for signature 'item_PC,matrix,numeric'
calcHessian(object, theta, resp)

## S4 method for signature 'item_GPC,matrix,numeric'
calcHessian(object, theta, resp)

## S4 method for signature 'item_GR,matrix,numeric'
calcHessian(object, theta, resp)

## S4 method for signature 'item_pool,numeric,numeric'
calcHessian(object, theta, resp)

## S4 method for signature 'item_pool_cluster,numeric,list'
calcHessian(object, theta, resp)

Arguments

object

an item or an item_pool object.
theta

theta values to use.
resp

the response data to use. This must be a single value for an item, or a length ni vector for an item_pool.

Details

notations

  • nq denotes the number of theta values.

  • ni denotes the number of items in the item_pool object.

Value

item object:

calcHessian returns a length nq vector containing the second derivative of the log-likelihood function, of observing the response at each theta.

item_pool object:

calcHessian returns a (nq, ni) matrix containing the second derivative of the log-likelihood function, of observing the response at each theta.

References

Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: Danish Institute for Educational Research.

Lord, F. M. (1952). A theory of test scores (Psychometric Monograph No. 7). Richmond, VA: Psychometric Corporation.

Birnbaum, A. (1957). Efficient design and use of tests of mental ability for various decision-making problems (Series Report No. 58-16. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). On the estimation of mental ability (Series Report No. 15. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). Further considerations of efficiency in tests of a mental ability (Series Report No. 17. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In Lord, F. M., Novick, M. R. (eds.), Statistical Theories of Mental Test Scores, 397-479. Reading, MA: Addison-Wesley.

Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47(2), 149-174.

Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43(4), 561-573.

Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16(2), 159-176.

Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph, 17.

Examples

item_1    <- new("item_1PL", difficulty = 0.5)
item_2    <- new("item_2PL", slope = 1.0, difficulty = 0.5)
item_3    <- new("item_3PL", slope = 1.0, difficulty = 0.5, guessing = 0.2)
item_4    <- new("item_PC", threshold = c(-1, 0, 1), ncat = 4)
item_5    <- new("item_GPC", slope = 1.2, threshold = c(-0.8, -1.0, 0.5), ncat = 4)
item_6    <- new("item_GR", slope = 0.9, category = c(-1, 0, 1), ncat = 4)

h_item_1 <- calcHessian(item_1, seq(-3, 3, 1), 0)
h_item_2 <- calcHessian(item_2, seq(-3, 3, 1), 0)
h_item_3 <- calcHessian(item_3, seq(-3, 3, 1), 0)
h_item_4 <- calcHessian(item_4, seq(-3, 3, 1), 0)
h_item_5 <- calcHessian(item_5, seq(-3, 3, 1), 0)
h_item_6 <- calcHessian(item_6, seq(-3, 3, 1), 0)
h_pool   <- calcHessian(
  itempool_science, seq(-3, 3, 1),
  rep(0, itempool_science@ni)
)

Calculate first derivative of log-likelihood

Description

calcJacobian is a function for calculating the first derivative of the log-likelihood function.

Usage

calcJacobian(object, theta, resp)

## S4 method for signature 'item_1PL,numeric,numeric'
calcJacobian(object, theta, resp)

## S4 method for signature 'item_2PL,numeric,numeric'
calcJacobian(object, theta, resp)

## S4 method for signature 'item_3PL,numeric,numeric'
calcJacobian(object, theta, resp)

## S4 method for signature 'item_PC,numeric,numeric'
calcJacobian(object, theta, resp)

## S4 method for signature 'item_GPC,numeric,numeric'
calcJacobian(object, theta, resp)

## S4 method for signature 'item_GR,numeric,numeric'
calcJacobian(object, theta, resp)

## S4 method for signature 'item_1PL,matrix,numeric'
calcJacobian(object, theta, resp)

## S4 method for signature 'item_2PL,matrix,numeric'
calcJacobian(object, theta, resp)

## S4 method for signature 'item_3PL,matrix,numeric'
calcJacobian(object, theta, resp)

## S4 method for signature 'item_PC,matrix,numeric'
calcJacobian(object, theta, resp)

## S4 method for signature 'item_GPC,matrix,numeric'
calcJacobian(object, theta, resp)

## S4 method for signature 'item_GR,matrix,numeric'
calcJacobian(object, theta, resp)

## S4 method for signature 'item_pool,numeric,numeric'
calcJacobian(object, theta, resp)

## S4 method for signature 'item_pool_cluster,numeric,list'
calcJacobian(object, theta, resp)

Arguments

object

an item or an item_pool object.
theta

theta values to use.
resp

the response value to use for each item.

Value

item object:

calcJacobian returns a length nq vector containing the first derivative of the log-likelihood function, of observing the response at each theta.

item_pool object:

calcJacobian returns a (nq, ni) matrix containing the first derivative of the log-likelihood function, of observing the response at each theta.

notations

  • nq denotes the number of theta values.

  • ni denotes the number of items in the item_pool object.

References

Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: Danish Institute for Educational Research.

Lord, F. M. (1952). A theory of test scores (Psychometric Monograph No. 7). Richmond, VA: Psychometric Corporation.

Birnbaum, A. (1957). Efficient design and use of tests of mental ability for various decision-making problems (Series Report No. 58-16. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). On the estimation of mental ability (Series Report No. 15. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). Further considerations of efficiency in tests of a mental ability (Series Report No. 17. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In Lord, F. M., Novick, M. R. (eds.), Statistical Theories of Mental Test Scores, 397-479. Reading, MA: Addison-Wesley.

Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47(2), 149-174.

Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43(4), 561-573.

Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16(2), 159-176.

Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph, 17.

Examples

item_1    <- new("item_1PL", difficulty = 0.5)
item_2    <- new("item_2PL", slope = 1.0, difficulty = 0.5)
item_3    <- new("item_3PL", slope = 1.0, difficulty = 0.5, guessing = 0.2)
item_4    <- new("item_PC", threshold = c(-1, 0, 1), ncat = 4)
item_5    <- new("item_GPC", slope = 1.2, threshold = c(-0.8, -1.0, 0.5), ncat = 4)
item_6    <- new("item_GR", slope = 0.9, category = c(-1, 0, 1), ncat = 4)

j_item_1 <- calcJacobian(item_1, seq(-3, 3, 1), 0)
j_item_2 <- calcJacobian(item_2, seq(-3, 3, 1), 0)
j_item_3 <- calcJacobian(item_3, seq(-3, 3, 1), 0)
j_item_4 <- calcJacobian(item_4, seq(-3, 3, 1), 0)
j_item_5 <- calcJacobian(item_5, seq(-3, 3, 1), 0)
j_item_6 <- calcJacobian(item_6, seq(-3, 3, 1), 0)
j_pool   <- calcJacobian(
  itempool_science, seq(-3, 3, 1),
  rep(0, itempool_science@ni)
)

Calculate central location (overall difficulty)

Description

calcLocation is a function for calculating the central location (overall difficulty) of items.

Usage

calcLocation(object)

## S4 method for signature 'item_1PL'
calcLocation(object)

## S4 method for signature 'item_2PL'
calcLocation(object)

## S4 method for signature 'item_3PL'
calcLocation(object)

## S4 method for signature 'item_PC'
calcLocation(object)

## S4 method for signature 'item_GPC'
calcLocation(object)

## S4 method for signature 'item_GR'
calcLocation(object)

## S4 method for signature 'item_pool'
calcLocation(object)

Arguments

object

an item or an item_pool object.

Value

item object:

calcLocation returns a theta value representing the central location.

item_pool object:

calcProb returns a length ni list, each containing the central location of the item.

notations

  • ni denotes the number of items in the item_pool object.

References

Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: Danish Institute for Educational Research.

Lord, F. M. (1952). A theory of test scores (Psychometric Monograph No. 7). Richmond, VA: Psychometric Corporation.

Birnbaum, A. (1957). Efficient design and use of tests of mental ability for various decision-making problems (Series Report No. 58-16. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). On the estimation of mental ability (Series Report No. 15. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). Further considerations of efficiency in tests of a mental ability (Series Report No. 17. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In Lord, F. M., Novick, M. R. (eds.), Statistical Theories of Mental Test Scores, 397-479. Reading, MA: Addison-Wesley.

Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47(2), 149-174.

Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43(4), 561-573.

Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16(2), 159-176.

Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph, 17.

Examples

item_1      <- new("item_1PL", difficulty = 0.5)
item_2      <- new("item_2PL", slope = 1.0, difficulty = 0.5)
item_3      <- new("item_3PL", slope = 1.0, difficulty = 0.5, guessing = 0.2)
item_4      <- new("item_PC", threshold = c(-1, 0, 1), ncat = 4)
item_5      <- new("item_GPC", slope = 1.2, threshold = c(-0.8, -1.0, 0.5), ncat = 4)
item_6      <- new("item_GR", slope = 0.9, category = c(-1, 0, 1), ncat = 4)

loc_item_1 <- calcLocation(item_1)
loc_item_2 <- calcLocation(item_2)
loc_item_3 <- calcLocation(item_3)
loc_item_4 <- calcLocation(item_4)
loc_item_5 <- calcLocation(item_5)
loc_item_6 <- calcLocation(item_6)
loc_pool   <- calcLocation(itempool_science)

Calculate log-likelihood

Description

calcLogLikelihood is a function for calculating log-likelihood values.

Usage

calcLogLikelihood(object, theta, resp)

## S4 method for signature 'item_pool,numeric,numeric'
calcLogLikelihood(object, theta, resp)

## S4 method for signature 'item_pool,numeric,matrix'
calcLogLikelihood(object, theta, resp)

## S4 method for signature 'item_pool,matrix,numeric'
calcLogLikelihood(object, theta, resp)

## S4 method for signature 'item_pool,matrix,matrix'
calcLogLikelihood(object, theta, resp)

Arguments

object

an item_pool object.
theta

theta values to use.
resp

the response data to use.

Value

calcLogLikelihood returns values of log-likelihoods.

References

Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: Danish Institute for Educational Research.

Lord, F. M. (1952). A theory of test scores (Psychometric Monograph No. 7). Richmond, VA: Psychometric Corporation.

Birnbaum, A. (1957). Efficient design and use of tests of mental ability for various decision-making problems (Series Report No. 58-16. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). On the estimation of mental ability (Series Report No. 15. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). Further considerations of efficiency in tests of a mental ability (Series Report No. 17. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In Lord, F. M., Novick, M. R. (eds.), Statistical Theories of Mental Test Scores, 397-479. Reading, MA: Addison-Wesley.

Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47(2), 149-174.

Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43(4), 561-573.

Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16(2), 159-176.

Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph, 17.

Examples

j_pool   <- calcLogLikelihood(itempool_science, seq(-3, 3, 1), 0)

Calculate item response probabilities

Description

calcProb is a function for calculating item response probabilities.

Usage

calcProb(object, theta)

## S4 method for signature 'item_1PL,numeric'
calcProb(object, theta)

## S4 method for signature 'item_2PL,numeric'
calcProb(object, theta)

## S4 method for signature 'item_3PL,numeric'
calcProb(object, theta)

## S4 method for signature 'item_PC,numeric'
calcProb(object, theta)

## S4 method for signature 'item_GPC,numeric'
calcProb(object, theta)

## S4 method for signature 'item_GR,numeric'
calcProb(object, theta)

## S4 method for signature 'item_pool,numeric'
calcProb(object, theta)

## S4 method for signature 'item_1PL,matrix'
calcProb(object, theta)

## S4 method for signature 'item_2PL,matrix'
calcProb(object, theta)

## S4 method for signature 'item_3PL,matrix'
calcProb(object, theta)

## S4 method for signature 'item_PC,matrix'
calcProb(object, theta)

## S4 method for signature 'item_GPC,matrix'
calcProb(object, theta)

## S4 method for signature 'item_GR,matrix'
calcProb(object, theta)

## S4 method for signature 'item_pool,matrix'
calcProb(object, theta)

## S4 method for signature 'item_pool_cluster,numeric'
calcProb(object, theta)

Arguments

object

an item or an item_pool object.
theta

theta values to use.

Value

item object:

calcProb returns a (nq, ncat) matrix of probability values.

item_pool object:

calcProb returns a length ni list, each containing a matrix of probability values.

notations

  • nq denotes the number of theta values.

  • ncat denotes the number of response categories.

  • ni denotes the number of items in the item_pool object.

References

Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: Danish Institute for Educational Research.

Lord, F. M. (1952). A theory of test scores (Psychometric Monograph No. 7). Richmond, VA: Psychometric Corporation.

Birnbaum, A. (1957). Efficient design and use of tests of mental ability for various decision-making problems (Series Report No. 58-16. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). On the estimation of mental ability (Series Report No. 15. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). Further considerations of efficiency in tests of a mental ability (Series Report No. 17. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In Lord, F. M., Novick, M. R. (eds.), Statistical Theories of Mental Test Scores, 397-479. Reading, MA: Addison-Wesley.

Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47(2), 149-174.

Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43(4), 561-573.

Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16(2), 159-176.

Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph, 17.

Examples

item_1      <- new("item_1PL", difficulty = 0.5)
item_2      <- new("item_2PL", slope = 1.0, difficulty = 0.5)
item_3      <- new("item_3PL", slope = 1.0, difficulty = 0.5, guessing = 0.2)
item_4      <- new("item_PC", threshold = c(-1, 0, 1), ncat = 4)
item_5      <- new("item_GPC", slope = 1.2, threshold = c(-0.8, -1.0, 0.5), ncat = 4)
item_6      <- new("item_GR", slope = 0.9, category = c(-1, 0, 1), ncat = 4)

prob_item_1 <- calcProb(item_1, seq(-3, 3, 1))
prob_item_2 <- calcProb(item_2, seq(-3, 3, 1))
prob_item_3 <- calcProb(item_3, seq(-3, 3, 1))
prob_item_4 <- calcProb(item_4, seq(-3, 3, 1))
prob_item_5 <- calcProb(item_5, seq(-3, 3, 1))
prob_item_6 <- calcProb(item_6, seq(-3, 3, 1))
prob_pool   <- calcProb(itempool_science, seq(-3, 3, 1))

Calculate Adaptivity Measures

Description

calculateAdaptivityMeasures is a function for calculating commonly used adaptivity measures.

Usage

calculateAdaptivityMeasures(x)

Arguments

x

an output_Shadow_all object.

Value

calculateAdaptivityMeasures returns a named list:

  • corr the correlation between final theta estimates and average test locations.

  • ratio the ratio of (1) standard deviation of average test locations, versus (2) standard deviation of final theta estimates.

  • PRV the proportion of variance reduced, from (1) the variance of item locations of all items in the pool, by (2) the average of test location variances.

  • info (1) average information of a test at final theta estimate, relative to (2) best average obtainable from item pool using same test length, adjusting for (3) average information from item pool using random selection.

Check the consistency of constraints and item usage

Description

Check the consistency of constraints and item usage.

Usage

checkConstraints(constraints, usage_matrix, true_theta = NULL)

Arguments

constraints

A constraints object generated by loadConstraints.
usage_matrix

A matrix of item usage data from Shadow.
true_theta

A vector of true theta values.

Create a config_Shadow object

Description

createShadowTestConfig is a config function for creating a config_Shadow object for shadowtest assembly. Default values are used for any unspecified parameters/slots.

Usage

createShadowTestConfig(
  item_selection = NULL,
  content_balancing = NULL,
  MIP = NULL,
  MCMC = NULL,
  exclude_policy = NULL,
  refresh_policy = NULL,
  exposure_control = NULL,
  overlap_control = NULL,
  stopping_criterion = NULL,
  interim_theta = NULL,
  final_theta = NULL,
  theta_grid = seq(-4, 4, 0.1)
)

Arguments

item_selection

a named list containing item selection criteria.

  • method the type of selection criteria. Accepts MFI, MPWI, FB, EB, GFI. (default = MFI)

  • info_type the type of information. Accepts FISHER. (default = FISHER)

  • initial_theta (optional) initial theta values to use.

  • fixed_theta (optional) fixed theta values to use throughout all item positions.

  • target_value (optional) the target value to use for method = 'GFI'.

content_balancing

a named list containing content balancing options.

  • method the type of balancing method. Accepts NONE, STA. (default = STA)

MIP

a named list containing solver options.

  • solver the type of solver. Accepts Rsymphony, highs, gurobi, lpSolve, Rglpk. (default = LPSOLVE)

  • verbosity verbosity level of the solver. (default = -2)

  • time_limit time limit in seconds. Used in solvers Rsymphony, gurobi, Rglpk. (default = 60)

  • gap_limit search termination criterion. Gap limit in relative scale passed onto the solver. Used in solver gurobi. (default = .05)

  • gap_limit_abs search termination criterion. Gap limit in absolute scale passed onto the solver. Used in solvers Rsymphony. (default = 0.05)

  • obj_tol search termination criterion. The lower bound to use on the minimax deviation variable. Used when item_selection$method is GFI, and ignored otherwise. (default = 0.05)

  • retry number of times to retry running the solver if the solver returns no solution. Some solvers incorrectly return no solution even when a solution exists. This is the number of attempts to verify that the problem is indeed infeasible in such cases. Set to 0 to not retry. (default = 5)

MCMC

a named list containing Markov-chain Monte Carlo configurations for obtaining posterior samples.

  • burn_in the number of chains from the start to discard. (default = 100)

  • post_burn_in the number of chains to use after discarding the first burn_in chains. (default = 500)

  • thin thinning interval to apply. 1 represents no thinning. (default = 1)

  • jump_factor the jump (scaling) factor for the proposal distribution. 1 represents no jumping. (default = 2.4)

exclude_policy

a named list containing the exclude policy for use with the exclude argument in Shadow.

  • method the type of policy. Accepts HARD, SOFT. (default = HARD)

  • M the Big M penalty to use on item information. Used in the SOFT method.

refresh_policy

a named list containing the refresh policy for when to obtain a new shadowtest.

  • method the type of policy. Accepts ALWAYS, POSITION, INTERVAL, THRESHOLD, INTERVAL-THRESHOLD, STIMULUS, SET, PASSAGE. (default = ALWAYS)

  • interval used in methods INTERVAL, INTERVAL-THRESHOLD. Set to 1 to refresh at each position, 2 to refresh at every two positions, and so on. (default = 1)

  • threshold used in methods THRESHOLD, INTERVAL-THRESHOLD. The absolute change in between interim theta estimates to trigger the refresh. (default = 0.1)

  • position used in methods POSITION. Item positions to trigger the refresh. (default = 1)

exposure_control

a named list containing exposure control settings.

  • method the type of exposure control method. Accepts NONE, ELIGIBILITY, BIGM, BIGM-BAYESIAN. (default = ELIGIBILITY)

  • M used in methods BIGM, BIGM-BAYESIAN. the Big M penalty to use on item information.

  • max_exposure_rate target exposure rates for each segment. (default = rep(0.25, 7))

  • acceleration_factor the acceleration factor to apply. (default = 1)

  • n_segment the number of theta segments to use. (default = 7)

  • first_segment (optional) the theta segment assumed at the beginning of test for all participants.

  • segment_cut theta segment cuts. (default = c(-Inf, seq(-2.5, 2.5, 1), Inf))

  • initial_eligibility_stats (optional) initial eligibility statistics to use.

  • fading_factor the fading factor to apply. (default = .999)

  • diagnostic_stats set to TRUE to generate segment-wise diagnostic statistics. (default = FALSE)

overlap_control

a named list containing overlap control settings.

  • method the type of overlap control method. Accepts NONE, ELIGIBILITY, BIGM, BIGM-BAYESIAN. (default = NONE)

  • M used in methods BIGM, BIGM-BAYESIAN. the Big M penalty to use on item information.

  • max_overlap_rate target overlap rate. (default = 0.20)

stopping_criterion

a named list containing stopping criterion.

  • method the type of stopping criterion. Accepts FIXED. (default = FIXED)

  • test_length test length.

  • min_ni the maximum number of items to administer.

  • max_ni the minimum number of items to administer.

  • se_threshold standard error threshold. Item administration is stopped when theta estimate standard error becomes lower than this value.

interim_theta

a named list containing interim theta estimation options.

  • method the type of estimation. Accepts EAP, MLE, MLEF, EB, FB, CARRYOVER. (default = EAP)

  • shrinkage_correction set TRUE to apply shrinkage correction. Used when method is EAP. (default = FALSE)

  • prior_dist the type of prior distribution. Accepts NORMAL, UNIFORM. (default = NORMAL)

  • prior_par distribution parameters for prior_dist. (default = c(0, 1))

  • bound_ML theta bound in c(lower_bound, upper_bound) format. Used when method is MLE. (default = -4, 4)

  • truncate_ML set TRUE to truncate ML estimate within bound_ML. (default = FALSE)

  • max_iter maximum number of Newton-Raphson iterations. Used when method is MLE. (default = 50)

  • crit convergence criterion. Used when method is MLE. (default = 1e-03)

  • max_change maximum change in ML estimates between iterations. Changes exceeding this value is clipped to this value. Used when method is MLE. (default = 1.0)

  • use_step_size set TRUE to use step_size. Used when method is MLE or MLEF. (default = FALSE)

  • step_size upper bound to impose on the absolute change in initial theta and estimated theta. Absolute changes exceeding this value will be capped to step_size. Used when method is MLE or MLEF. (default = 0.5)

  • do_Fisher set TRUE to use Fisher's method of scoring. Used when method is MLE. (default = TRUE)

  • fence_slope slope parameter to use for method = 'MLEF'. This must have two values in total, for the lower and upper bound item respectively. Use one value to use the same value for both bounds. (default = 5)

  • fence_difficulty difficulty parameters to use for method = 'MLEF'. This must have two values in total, for the lower and upper bound item respectively. (default = c(-5, 5))

  • hand_scored_attribute (optional) the item attribute name for whether each item is hand-scored or not. The attribute should have TRUE (hand-scored) and FALSE (machine-scored) values. If a hand-scored item is administered to an examinee, the previous interim theta (or the starting theta if this occurs for the first item) is reused without updating the estimate.

final_theta

a named list containing final theta estimation options.

  • method the type of estimation. Accepts EAP, MLE, MLEF, EB, FB, CARRYOVER. (default = EAP)

  • shrinkage_correction set TRUE to apply shrinkage correction. Used when method is EAP. (default = FALSE)

  • prior_dist the type of prior distribution. Accepts NORMAL, UNIFORM. (default = NORMAL)

  • prior_par distribution parameters for prior_dist. (default = c(0, 1))

  • bound_ML theta bound in c(lower_bound, upper_bound) format. Used when method is MLE. (default = -4, 4)

  • truncate_ML set TRUE to truncate ML estimate within bound_ML. (default = FALSE)

  • max_iter maximum number of Newton-Raphson iterations. Used when method is MLE. (default = 50)

  • crit convergence criterion. Used when method is MLE. (default = 1e-03)

  • max_change maximum change in ML estimates between iterations. Changes exceeding this value is clipped to this value. Used when method is MLE. (default = 1.0)

  • use_step_size set TRUE to use step_size. Used when method is MLE or MLEF. (default = FALSE)

  • step_size upper bound to impose on the absolute change in initial theta and estimated theta. Absolute changes exceeding this value will be capped to step_size. Used when method is MLE or MLEF. (default = 0.5)

  • do_Fisher set TRUE to use Fisher's method of scoring. Used when method is MLE. (default = TRUE)

  • fence_slope slope parameter to use for method = 'MLEF'. This must have two values in total, for the lower and upper bound item respectively. Use one value to use the same value for both bounds. (default = 5)

  • fence_difficulty difficulty parameters to use for method = 'MLEF'. This must have two values in total, for the lower and upper bound item respectively. (default = c(-5, 5))

theta_grid

the theta grid to use as quadrature points.

Examples

cfg1 <- createShadowTestConfig(refresh_policy = list(
  method = "STIMULUS"
))
cfg2 <- createShadowTestConfig(refresh_policy = list(
  method = "POSITION",
  position = c(1, 5, 9)
))

Create a config_Static object

Description

createStaticTestConfig is a config function for creating a config_Static object for Static (fixed-form) test assembly. Default values are used for any unspecified parameters/slots.

Usage

createStaticTestConfig(item_selection = NULL, MIP = NULL)

Arguments

item_selection

a named list containing item selection criteria.

  • method the type of selection criteria. Accepts MAXINFO, TIF, TCC. (default = MAXINFO)

  • info_type the type of information. Accepts FISHER. (default = FISHER)

  • target_location a numeric vector containing the locations of target theta points. (e.g. c(-1, 0, 1)) (default = c(-1.2, 0, 1.2))

  • target_value a numeric vector containing the target values at each theta location. This should have the same length with target_location. Ignored if method is MAXINFO. (default = NULL)

  • target_weight a numeric vector containing the weights for each theta location. This should have the same length with target_location. (default = rep(1, length(target_location))

MIP

a named list containing solver options.

  • solver the type of solver. Accepts Rsymphony, highs, gurobi, lpSolve, Rglpk. (default = LPSOLVE)

  • verbosity verbosity level of the solver. (default = -2)

  • time_limit time limit in seconds. Used in solvers Rsymphony, gurobi, Rglpk. (default = 60)

  • gap_limit search termination criterion. Gap limit in relative scale passed onto the solver. Used in solver gurobi. (default = .05)

  • gap_limit_abs search termination criterion. Gap limit in absolute scale passed onto the solver. Used in solvers Rsymphony. (default = 0.05)

  • obj_tol search termination criterion. The lower bound to use on the minimax deviation variable. Used when item_selection$method is TIF or TCC. (default = 0.05)

  • retry number of times to retry running the solver if the solver returns no solution. Some solvers incorrectly return no solution even when a solution exists. This is the number of attempts to verify that the problem is indeed infeasible in such cases. Set to 0 to not retry. (default = 5)

Value

createStaticTestConfig returns a config_Static object. This object is used in Static.

Examples

cfg1 <- createStaticTestConfig(
  list(
    method = "MAXINFO",
    info_type = "FISHER",
    target_location = c(-1, 0, 1),
    target_weight = c(1, 1, 1)
  )
)

cfg2 <- createStaticTestConfig(
  list(
    method = "TIF",
    info_type = "FISHER",
    target_location = c(-1, 0, 1),
    target_weight = c(1, 1, 1),
    target_value = c(8, 10, 12)
  )
)

cfg3 <- createStaticTestConfig(
  list(
    method = "TCC",
    info_type = "FISHER",
    target_location = c(-1, 0, 1),
    target_weight = c(1, 1, 1),
    target_value = c(10, 15, 20)
  )
)

Class 'constraint': a single constraint

Description

constraint is an S4 class for representing a single constraint.

Slots

constraint

the numeric index of the constraint.

constraint_id

the character ID of the constraint.

nc

the number of MIP-format constraints translated from this constraint.

mat,dir,rhs

these represent MIP-format constraints. A single MIP-format constraint is associated with a row in mat, a value in rhs, and a value in dir.

  • the i-th row of mat represents LHS coefficients to use on decision variables in the i-th MIP-format constraint.

  • the i-th value of rhs represents RHS values to use in the i-th MIP-format constraint.

  • the i-th value of dir represents the imposed constraint between LHS and RHS.

suspend

TRUE if the constraint is not to be imposed.

Class 'constraints': a set of constraints

Description

constraints is an S4 class for representing a set of constraints and its associated objects.

Details

See constraints-operators for object manipulation functions.

Slots

constraints

a data.frame containing the constraint specifications.

list_constraints

a list containing the constraint object representation of each constraint.

pool

the item_pool object associated with the constraints.

item_attrib

the item_attrib object associated with the constraints.

st_attrib

the st_attrib object associated with the constraints.

test_length

the test length specified in the constraints.

nv

the number of decision variables. Equals ni + ns.

ni

the number of items to search from.

ns

the number of stimulus to search from.

id

the item/stimulus ID string of each item/stimulus.

index,mat,dir,rhs

these represent MIP-format constraints. A single MIP-format constraint is associated with a value in index, a row in mat, a value in rhs, and a value in dir.

  • the i-th value of index represents which constraint specification in the constraints argument it was translated from.

  • the i-th row of mat represents LHS coefficients to use on decision variables in the i-th MIP-format constraint.

  • the i-th value of rhs represents RHS values to use in the i-th MIP-format constraint.

  • the i-th value of dir represents the imposed constraint between LHS and RHS.

set_based

TRUE if the constraint is set-based. FALSE otherwise.

item_order

the item attribute of each item to use in imposing an item order constraint, if any.

item_order_by

the name of the item attribute to use in imposing an item order constraint, if any.

stim_order

the stimulus attribute of each stimulus to use in imposing a stimulus order constraint, if any.

stim_order_by

the name of the stimulus attribute to use in imposing a stimulus order constraint, if any.

item_index_by_stimulus

a list containing item indices of each stimulus.

stimulus_index_by_item

the stimulus indices of each item.

Basic operators for constraints objects

Description

Create a subset of a constraints object:

  • constraints[i]

  • subsetConstraints(constraints, 1:10)

Combine two constraints objects:

  • c(constraints1, constraints2)

  • combineConstraints(constraints1, constraints2)

Usage

subsetConstraints(x, i = NULL)

combineConstraints(x1, x2)

## S4 method for signature 'constraints,numeric'
x[i, j, ..., drop = TRUE]

## S4 method for signature 'constraints'
c(x, ...)

Arguments

x, x1, x2

a constraints object.
i, j

indices to use in subsetting.
...

not used, exists for compatibility.
drop

not used, exists for compatibility.

Examples

c1 <- constraints_science
c2 <- c1[1:10]
c3 <- c1[c(1, 11:36)] # keep constraint 1 for test length
c4 <- c(c2, c3)

Bayes dataset

Description

Item-based example item pool with standard errors (320 items).

Details

This pool is associated with the following objects:

  • itempool_bayes an item_pool object containing 320 items.

  • itemattrib_bayes a item_attrib object containing 5 item-level attributes.

  • constraints_bayes a constraints object containing 14 constraints.

Also, the following objects are intended for illustrating expected data structures.

  • itempool_bayes_data a data.frame containing item parameters.

  • itempool_se_bayes_data a data.frame containing item parameter standard errors.

  • itemattrib_bayes_data a data.frame containing item attributes.

  • constraints_bayes_data a data.frame containing constraint specifications.

Examples

itempool_bayes    <- loadItemPool(itempool_bayes_data, itempool_se_bayes_data)
itemattrib_bayes  <- loadItemAttrib(itemattrib_bayes_data, itempool_bayes)
constraints_bayes <- loadConstraints(constraints_bayes_data,
  itempool_bayes, itemattrib_bayes)

Fatigue dataset

Description

Item-based example pool with item contents (95 items).

Details

This pool is associated with the following objects:

  • itempool_fatigue an item_pool object containing 95 items.

  • itemattrib_fatigue an item_attrib object containing 7 item-level attributes.

  • constraints_fatigue a constraints object containing 111 constraints.

Also, the following objects are intended for illustrating expected data structures.

  • itempool_fatigue_data a data.frame containing item parameters.

  • itemattrib_fatigue_data a data.frame containing item attributes.

  • itemtext_fatigue_data a data.frame containing item texts.

  • constraints_fatigue_data a data.frame containing constraint specifications.

  • resp_fatigue_data a data.frame containing raw response data.

Examples

itempool_fatigue   <- loadItemPool(itempool_fatigue_data)
itemattrib_fatigue <- loadItemAttrib(itemattrib_fatigue_data, itempool_fatigue)
constraints_fatigue <- loadConstraints(constraints_fatigue_data,
  itempool_fatigue, itemattrib_fatigue)

Reading dataset

Description

Stimulus-based example item pool (303 items, 35 stimuli).

Details

This pool is associated with the following objects:

  • itempool_reading an item_pool object containing 303 items.

  • itemattrib_reading an item_attrib object containing 12 item-level attributes.

  • stimattrib_reading a st_attrib object containing 4 stimulus-level attributes.

  • constraints_reading a constraints object containing 18 constraints.

Also, the following objects are intended for illustrating expected data structures.

  • itempool_reading_data a data.frame containing item parameters.

  • itemattrib_reading_data a data.frame containing item attributes.

  • stimattrib_reading_data a data.frame containing stimulus attributes.

  • constraints_reading_data a data.frame containing constraint specifications.

Examples

itempool_reading    <- loadItemPool(itempool_reading_data)
itemattrib_reading  <- loadItemAttrib(itemattrib_reading_data, itempool_reading)
stimattrib_reading  <- loadStAttrib(stimattrib_reading_data, itemattrib_reading)
constraints_reading <- loadConstraints(constraints_reading_data,
  itempool_reading, itemattrib_reading, stimattrib_reading)

Science dataset

Description

Item-based example item pool (1000 items).

Details

This pool is associated with the following objects:

  • itempool_science an item_pool object containing 1000 items.

  • itemattrib_science an item_attrib object containing 9 item-level attributes.

  • constraints_science a constraints object containing 36 constraints.

Also, the following objects are intended for illustrating expected data structures.

  • itempool_science_data a data.frame containing item parameters.

  • itemattrib_science_data a data.frame containing item attributes.

  • constraints_science_data a data.frame containing constraint specifications.

Examples

itempool_science    <- loadItemPool(itempool_science_data)
itemattrib_science  <- loadItemAttrib(itemattrib_science_data, itempool_science)
constraints_science <- loadConstraints(constraints_science_data,
  itempool_science, itemattrib_science)

Detect best solver

Description

Detect best solver

Usage

detectBestSolver()

Value

the package name of the best available solver on the system.

Examples

solver <- detectBestSolver()
cfg <- createStaticTestConfig(MIP = list(solver = solver))
cfg <- createShadowTestConfig(MIP = list(solver = solver))

(C++) Calculate expected scores

Description

e_*() and array_e_*() are C++ functions for calculating expected scores.

Usage

e_1pl(x, b)

e_2pl(x, a, b)

e_m_2pl(x, a, d)

e_3pl(x, a, b, c)

e_m_3pl(x, a, d, c)

e_pc(x, b)

e_gpc(x, a, b)

e_m_gpc(x, a, d)

e_gr(x, a, b)

e_m_gr(x, a, d)

array_e_1pl(x, b)

array_e_2pl(x, a, b)

array_e_3pl(x, a, b, c)

array_e_pc(x, b)

array_e_gpc(x, a, b)

array_e_gr(x, a, b)

Arguments

x

the theta value. The number of columns should correspond to the number of dimensions. For array_*() functions, the number of theta values must correspond to the number of rows.
b, d

the difficulty parameter. b is used for unidimensional items, and d is used for multidimensional items.
a

the a-parameter.
c

the c-parameter.

Details

e_*() functions accept a single theta value, and array_p_*() functions accept multiple theta values.

Supports unidimensional and multidimensional models.

  • e_1pl(), array_e_1pl(): 1PL models

  • e_2pl(), array_e_2pl(): 2PL models

  • e_3pl(), array_e_3pl(): 3PL models

  • e_pc(), array_e_pc(): PC (partial credit) models

  • e_gpc(), array_e_gpc(): GPC (generalized partial credit) models

  • e_gr(), array_e_gr(): GR (graded response) models

  • e_m_2pl(), array_e_m_2pl(): multidimensional 2PL models

  • e_m_3pl(), array_e_m_3pl(): multidimensional 3PL models

  • e_m_gpc(), array_e_m_gpc(): multidimensional GPC models

  • e_m_gr(), array_e_m_gr(): multidimensional GR models

References

Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: Danish Institute for Educational Research.

Lord, F. M. (1952). A theory of test scores (Psychometric Monograph No. 7). Richmond, VA: Psychometric Corporation.

Birnbaum, A. (1957). Efficient design and use of tests of mental ability for various decision-making problems (Series Report No. 58-16. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). On the estimation of mental ability (Series Report No. 15. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). Further considerations of efficiency in tests of a mental ability (Series Report No. 17. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In Lord, F. M., Novick, M. R. (eds.), Statistical Theories of Mental Test Scores, 397-479. Reading, MA: Addison-Wesley.

Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47(2), 149-174.

Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43(4), 561-573.

Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16(2), 159-176.

Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph, 17.

Examples

x <- 0.5

e_1pl(x, 1)
e_2pl(x, 1, 2)
e_3pl(x, 1, 2, 0.25)
e_pc(x, c(0, 1))
e_gpc(x, 2, c(0, 1))
e_gr(x, 2, c(0, 2))

x <- matrix(seq(-3, 3, 1)) # three theta values, unidimensional

array_e_1pl(x, 1)
array_e_2pl(x, 1, 2)
array_e_3pl(x, 1, 2, 0.25)
array_e_pc(x, c(0, 1))
array_e_gpc(x, 2, c(0, 1))
array_e_gr(x, 2, c(0, 2))

Compute expected a posteriori estimates of theta

Description

eap is a function for computing expected a posteriori estimates of theta.

Usage

eap(
  object,
  select = NULL,
  resp,
  theta_grid = seq(-4, 4, 0.1),
  prior = rep(1/81, 81)
)

## S4 method for signature 'item_pool'
eap(
  object,
  select = NULL,
  resp,
  theta_grid = seq(-4, 4, 0.1),
  prior = rep(1/81, 81)
)

EAP(object, select = NULL, prior, reset_prior = FALSE)

## S4 method for signature 'test'
EAP(object, select = NULL, prior, reset_prior = FALSE)

## S4 method for signature 'test_cluster'
EAP(object, select = NULL, prior, reset_prior = FALSE)

Arguments

object

an item_pool object.
select

(optional) if item indices are supplied, only the specified items are used.
resp

item response on all (or selected) items in the object argument. Can be a vector, a matrix, or a data frame. length(resp) or ncol(resp) must be equal to the number of all (or selected) items.
theta_grid

the theta grid to use as quadrature points. (default = seq(-4, 4, .1))
prior

a prior distribution, a numeric vector for a common prior or a matrix for individualized priors. (default = rep(1 / 81, 81))
reset_prior

used for test_cluster objects. If TRUE, reset the prior distribution for each test object.

Value

eap returns a list containing estimated values.

  • th theta value.

  • se standard error.

Examples

eap(itempool_fatigue, resp = resp_fatigue_data[10, ])
eap(itempool_fatigue, select = 1:20, resp = resp_fatigue_data[10, 1:20])

(C++) Classify theta values into segments using cutpoints

Description

find_segment() is a function for classifying theta values into segments based on supplied cutpoints.

Usage

find_segment(x, segment)

Arguments

x

the theta value. This can be a vector.
segment

segment cutpoints. Values of -Inf, Inf are not implied and must be explicitly supplied if intended.

Examples

cuts <- c(-Inf, -2, 0, 2, Inf)

find_segment(-3, cuts)
find_segment(-1, cuts)
find_segment(1, cuts)
find_segment(3, cuts)
find_segment(seq(-3, 3, 2), cuts)

Retrieve constraints-related scores from solution

Description

getScoreAttributes is a helper function for retrieving constraints-related scores from a solution.

Usage

getScoreAttributes(constraints, item_idx, item_resp, item_ncat)

Arguments

constraints

a constraints object.
item_idx

item indices from a solution.
item_resp

item scores for item_idx.
item_ncat

number of score categories for item_idx.

Examples

item_idx <-
  c( 29,  33,  26,  36,  34,
    295, 289, 296, 291, 126,
    133, 124, 134, 129,  38,
     47,  39,  41,  46,  45,
    167, 166, 170, 168, 113,
    116, 119, 117, 118, 114)

item_resp <-
  c( 1, 0, 1, 1, 0,
     0, 1, 1, 0, 0,
     1, 0, 1, 0, 1,
     1, 1, 1, 0, 1,
     0, 1, 1, 1, 1,
     1, 0, 1, 0, 1)

item_ncat <-
  c( 2, 2, 2, 2, 2,
     2, 2, 2, 2, 2,
     2, 2, 2, 2, 2,
     2, 2, 2, 2, 2,
     2, 2, 2, 2, 2,
     2, 2, 2, 2, 2)

getScoreAttributes(constraints_reading, item_idx, item_resp, item_ncat)

Print solution items

Description

Print solution items

Usage

getSolution(object, examinee = NA, position = NA, index_only = TRUE)

## S4 method for signature 'list'
getSolution(object, examinee = NA, position = NA, index_only = TRUE)

## S4 method for signature 'output_Static'
getSolution(object, examinee = NA, position = NA, index_only = TRUE)

Arguments

object

an output_Static object or an output_Shadow object.
examinee

(optional) the examinee index to display the solution. Used when the 'object' argument is an output_Shadow object.
position

(optional) if supplied, display the item attributes of the assembled test at that item position. If not supplied, display the item attributes of the administered items. Used when the 'object' argument is an output_Shadow object.
index_only

if TRUE, only print item indices. if FALSE, print all item attributes. (default = TRUE)

Value

Item attributes of solution items.

Retrieve constraints-related attributes from solution

Description

getSolutionAttributes is a helper function for retrieving constraints-related attributes from a solution.

Usage

getSolutionAttributes(constraints, item_idx, all_values = FALSE)

Arguments

constraints

a constraints object.
item_idx

item indices from a solution.
all_values

if TRUE, return all values as-is without taking the mean when there are multiple values. If FALSE, return the mean when there are multiple values. This has an effect when there is a constraint on items per stimulus, where there are multiple values of number of items per stimulus. In this case, if TRUE, the number of items for every stimuli are returned as-is. If FALSE, the average number of items across stimuli is returned. (default = FALSE)

Value

Examples

item_idx <-
  c( 29,  33,  26,  36,  34,
    295, 289, 296, 291, 126,
    133, 124, 134, 129,  38,
     47,  39,  41,  46,  45,
    167, 166, 170, 168, 113,
    116, 119, 117, 118, 114)

getSolutionAttributes(constraints_reading, item_idx, FALSE)
getSolutionAttributes(constraints_reading, item_idx, TRUE)

(C++) Calculate second derivative of log-likelihood

Description

h_*() and array_h_*() are C++ functions for calculating the second derivative of the log-likelihood function.

Usage

h_1pl(x, b, u)

h_2pl(x, a, b, u)

h_m_2pl(x, a, d, u)

h_3pl(x, a, b, c, u)

h_m_3pl(x, a, d, c, u)

h_pc(x, b, u)

h_gpc(x, a, b, u)

h_m_gpc(x, a, d, u)

h_gr(x, a, b, u)

h_m_gr(x, a, d, u)

array_h_1pl(x, b, u)

array_h_2pl(x, a, b, u)

array_h_3pl(x, a, b, c, u)

array_h_pc(x, b, u)

array_h_gpc(x, a, b, u)

array_h_gr(x, a, b, u)

Arguments

x

the theta value. The number of columns should correspond to the number of dimensions. For array_*() functions, the number of theta values must correspond to the number of rows.
b, d

the difficulty parameter. b is used for unidimensional items, and d is used for multidimensional items.
u

the response value.
a

the a-parameter.
c

the c-parameter.

Details

h_*() functions accept a single theta value, and array_h_*() functions accept multiple theta values.

Supports unidimensional and multidimensional models.

  • h_1pl(), array_h_1pl(): 1PL models

  • h_2pl(), array_h_2pl(): 2PL models

  • h_3pl(), array_h_3pl(): 3PL models

  • h_pc(), array_h_pc(): PC (partial credit) models

  • h_gpc(), array_h_gpc(): GPC (generalized partial credit) models

  • h_gr(), array_h_gr(): GR (graded response) models

  • h_m_2pl(), array_h_m_2pl(): multidimensional 2PL models

  • h_m_3pl(), array_h_m_3pl(): multidimensional 3PL models

  • h_m_gpc(), array_h_m_gpc(): multidimensional GPC models

  • h_m_gr(), array_h_m_gr(): multidimensional GR models

References

Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: Danish Institute for Educational Research.

Lord, F. M. (1952). A theory of test scores (Psychometric Monograph No. 7). Richmond, VA: Psychometric Corporation.

Birnbaum, A. (1957). Efficient design and use of tests of mental ability for various decision-making problems (Series Report No. 58-16. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). On the estimation of mental ability (Series Report No. 15. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). Further considerations of efficiency in tests of a mental ability (Series Report No. 17. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In Lord, F. M., Novick, M. R. (eds.), Statistical Theories of Mental Test Scores, 397-479. Reading, MA: Addison-Wesley.

Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47(2), 149-174.

Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43(4), 561-573.

Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16(2), 159-176.

Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph, 17.

Examples

u <- 1

x <- 0.5
h_1pl(x, 1, u)
h_2pl(x, 1, 2, u)
h_3pl(x, 1, 2, 0.25, u)
h_pc(x, c(0, 1), u)
h_gpc(x, 2, c(0, 1), u)
h_gr(x, 2, c(0, 2), u)

x <- matrix(seq(-3, 3, 1)) # three theta values, unidimensional
array_h_1pl(x, 1, u)
array_h_2pl(x, 1, 2, u)
array_h_3pl(x, 1, 2, 0.25, u)
array_h_pc(x, c(0, 1), u)
array_h_gpc(x, 2, c(0, 1), u)
array_h_gr(x, 2, c(0, 2), u)

(C++) Calculate Fisher information

Description

info_*() and array_info_*() are functions for calculating Fisher information.

Usage

info_1pl(x, b)

info_2pl(x, a, b)

info_m_2pl(x, a, d)

dirinfo_m_2pl(x, a, d)

thisdirinfo_m_2pl(x, alpha_vec, a, d)

info_3pl(x, a, b, c)

info_m_3pl(x, a, d, c)

dirinfo_m_3pl(x, a, d, c)

thisdirinfo_m_3pl(x, alpha_vec, a, d, c)

info_pc(x, b)

info_gpc(x, a, b)

info_m_gpc(x, a, d)

dirinfo_m_gpc(x, a, d)

thisdirinfo_m_gpc(x, alpha_vec, a, d)

info_gr(x, a, b)

info_m_gr(x, a, d)

dirinfo_m_gr(x, a, d)

thisdirinfo_m_gr(x, alpha_vec, a, d)

array_info_1pl(x, b)

array_info_2pl(x, a, b)

array_info_m_2pl(x, a, d)

array_dirinfo_m_2pl(x, a, d)

array_thisdirinfo_m_2pl(x, alpha_vec, a, d)

array_info_3pl(x, a, b, c)

array_info_m_3pl(x, a, d, c)

array_dirinfo_m_3pl(x, a, d, c)

array_thisdirinfo_m_3pl(x, alpha_vec, a, d, c)

array_info_pc(x, b)

array_info_gpc(x, a, b)

array_info_m_gpc(x, a, d)

array_dirinfo_m_gpc(x, a, d)

array_thisdirinfo_m_gpc(x, alpha_vec, a, d)

array_info_gr(x, a, b)

array_info_m_gr(x, a, d)

array_dirinfo_m_gr(x, a, d)

array_thisdirinfo_m_gr(x, alpha_vec, a, d)

Arguments

x

the theta value. The number of columns should correspond to the number of dimensions. For array_*() functions, the number of theta values must correspond to the number of rows.
b, d

the difficulty parameter. b is used for unidimensional items, and d is used for multidimensional items.
a

the a-parameter.
alpha_vec

the alpha angle vector. Used for directional information in thisdirinfo_*() and array_thisdirinfo_*().
c

the c-parameter.

Details

info_*() functions accept a single theta value, and array_info_* functions accept multiple theta values.

Supports unidimensional and multidimensional models.

  • info_1pl(), array_info_1pl(): 1PL models

  • info_2pl(), array_info_2pl(): 2PL models

  • info_3pl(), array_info_3pl(): 3PL models

  • info_pc(), array_info_pc(): PC (partial credit) models

  • info_gpc(), array_info_gpc(): GPC (generalized partial credit) models

  • info_gr(), array_info_gr(): GR (graded response) models

  • info_m_2pl(), array_info_m_2pl(): multidimensional 2PL models

  • info_m_3pl(), array_info_m_3pl(): multidimensional 3PL models

  • info_m_gpc(), array_info_m_gpc(): multidimensional GPC models

  • info_m_gr(), array_info_m_gr(): multidimensional GR models

  • Directional information for a specific angle

    • thisdirinfo_m_2pl(), array_thisdirinfo_m_2pl(): multidimensional 2PL models

    • thisdirinfo_m_3pl(), array_thisdirinfo_m_3pl(): multidimensional 3PL models

    • thisdirinfo_m_gpc(), array_thisdirinfo_m_gpc(): multidimensional GPC models

    • thisdirinfo_m_gr(), array_thisdirinfo_m_gr(): multidimensional GR models

References

Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: Danish Institute for Educational Research.

Lord, F. M. (1952). A theory of test scores (Psychometric Monograph No. 7). Richmond, VA: Psychometric Corporation.

Birnbaum, A. (1957). Efficient design and use of tests of mental ability for various decision-making problems (Series Report No. 58-16. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). On the estimation of mental ability (Series Report No. 15. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). Further considerations of efficiency in tests of a mental ability (Series Report No. 17. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In Lord, F. M., Novick, M. R. (eds.), Statistical Theories of Mental Test Scores, 397-479. Reading, MA: Addison-Wesley.

Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47(2), 149-174.

Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43(4), 561-573.

Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16(2), 159-176.

Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph, 17.

Examples

x <- 0.5

info_1pl(x, 1)
info_2pl(x, 1, 2)
info_3pl(x, 1, 2, 0.25)
info_pc(x, c(0, 1))
info_gpc(x, 2, c(0, 1))
info_gr(x, 2, c(0, 2))

x <- matrix(seq(0.1, 0.5, 0.1)) # three theta values, unidimensional

array_info_1pl(x, 1)
array_info_2pl(x, 1, 2)
array_info_3pl(x, 1, 2, 0.25)
array_info_pc(x, c(0, 1))
array_info_gpc(x, 2, c(0, 1))
array_info_gr(x, 2, c(0, 2))

Generate item parameter samples for Bayesian purposes

Description

iparPosteriorSample is a function for generating item parameter samples. Used for the FB (full-Bayesian) estimation method.

Usage

iparPosteriorSample(pool, n_sample = 500)

Arguments

pool

an item_pool object.
n_sample

the number of samples to draw.

Value

iparPosteriorSample returns a length-ni list of item parameter matrices, with each matrix having n_sample rows.

Examples

ipar <- iparPosteriorSample(itempool_bayes, 5)
ipar <- iparPosteriorSample(itempool_science, 5) # no variation

Load item attributes

Description

loadItemAttrib is a data loading function for creating an item_attrib object. loadItemAttrib can read item attributes from a data.frame or a .csv file.

Usage

loadItemAttrib(object, pool)

Arguments

object

item attributes. Can be a data.frame or the file path of a .csv file. The content should at least include an 'ID' column that matches with item IDs (the 'ID' column) of the item_pool object.
pool

an item_pool object. Use loadItemPool for this.

Value

loadItemAttrib returns an item_attrib object.

See Also

dataset_science, dataset_reading, dataset_fatigue, dataset_bayes for examples.

Examples

## Read from data.frame:
itempool_science   <- loadItemPool(itempool_science_data)
itemattrib_science <- loadItemAttrib(itemattrib_science_data, itempool_science)

## Read from file: write to tempdir() for illustration and clean afterwards
f <- file.path(tempdir(), "itemattrib_science.csv")
write.csv(itemattrib_science_data, f, row.names = FALSE)
itemattrib_science <- loadItemAttrib(f, itempool_science)
file.remove(f)

Basic functions for item attribute objects

Description

Basic functions for item attribute objects

Usage

## S4 method for signature 'item_attrib,numeric'
x[i, j, ..., drop = TRUE]

## S4 method for signature 'item_attrib'
dim(x)

## S4 method for signature 'item_attrib'
colnames(x)

## S4 method for signature 'item_attrib'
rownames(x)

## S4 method for signature 'item_attrib'
names(x)

## S4 method for signature 'item_attrib'
as.data.frame(x, row.names = NULL, optional = FALSE, ...)

Arguments

x

an item_attrib object.
i, j

indices to use in subsetting.
...

not used, exists for compatibility.
drop

not used, exists for compatibility.
row.names

not used, exists for compatibility.
optional

not used, exists for compatibility.

Examples

x <- itemattrib_science
x[1:10]
dim(x)
ncol(x)
nrow(x)
colnames(x)
rownames(x)
names(x)
as.data.frame(x)

Class 'item_pool_cluster': an item pool

Description

item_pool_cluster is an S4 class for representing a group of item pools.

Slots

np

the number of item pools.

pools

a list of item_pool objects.

names

a vector containing item pool names.

Class 'item_pool': an item pool

Description

item_pool is an S4 class for representing an item pool.

Details

See item_pool-operators for object manipulation functions.

Slots

ni

the number of items in the pool.

max_cat

the maximum number of response categories across the pool.

index

the numeric index of each item.

id

the ID string of each item.

model

the item class name of each item. See item-classes.

NCAT

the number of response categories of each item.

parms

a list containing item class objects. See item-classes.

ipar

a matrix containing item parameters.

se

a matrix containing item parameter standard errors.

raw

the raw input data.frame used in loadItemPool to create this object.

raw_se

the raw input data.frame used in loadItemPool to create this object.

unique

whether item IDs must be unique for this object to be a valid object.

Basic operators for item pool objects

Description

Create a subset of an item_pool object:

  • pool[i]

  • subsetItemPool(pool, i)

Combine two item_pool objects:

  • c(pool1, pool2)

  • combineItemPool(pool1, pool2)

  • pool1 + pool2

pool1 - pool2 excludes items in pool2 from pool1.

pool1 == pool2 tests whether two item_pool objects are identical.

Usage

subsetItemPool(x, i = NULL)

combineItemPool(x1, x2, unique = TRUE, verbose = TRUE)

## S4 method for signature 'item_pool,numeric'
x[i, j, ..., drop = TRUE]

## S4 method for signature 'item_pool'
c(x, ...)

## S3 method for class 'item_pool'
x1 + x2

## S3 method for class 'item_pool'
x1 - x2

## S3 method for class 'item_pool'
x1 == x2

Arguments

x, x1, x2

an item_pool object.
i

item indices to use in subsetting.
unique

if TRUE, remove items with duplicate IDs after combining. (default = TRUE)
verbose

if TRUE, raise a warning if duplicate IDs are found after combining. (default = TRUE)
j, drop, ...

not used, exists for compatibility.

Examples

p1 <- itempool_science[1:100]
p2 <- c(itempool_science, itempool_reading)
p3 <- p2 - p1

p1 <- itempool_science[1:500]
p2 <- itempool_science - p1
p3 <- itempool_science[501:1000]
identical(p2, p3)  ## TRUE

p <- p1 + p3
p == itempool_science ## TRUE

Item classes

Description

  • item_1PL class represents a 1PL item.

  • item_2PL class represents a 2PL item.

  • item_3PL class represents a 3PL item.

  • item_PC class represents a partial credit item.

  • item_GPC class represents a generalized partial credit item.

  • item_GR class represents a graded response item.

Slots

slope

a slope parameter value

difficulty

a difficulty parameter value

guessing

a guessing parameter value

threshold

a vector of threshold parameter values

category

a vector of category boundary values

ncat

the number of response categories

References

Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: Danish Institute for Educational Research.

Lord, F. M. (1952). A theory of test scores (Psychometric Monograph No. 7). Richmond, VA: Psychometric Corporation.

Birnbaum, A. (1957). Efficient design and use of tests of mental ability for various decision-making problems (Series Report No. 58-16. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). On the estimation of mental ability (Series Report No. 15. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). Further considerations of efficiency in tests of a mental ability (Series Report No. 17. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In Lord, F. M., Novick, M. R. (eds.), Statistical Theories of Mental Test Scores, 397-479. Reading, MA: Addison-Wesley.

Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47(2), 149-174.

Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43(4), 561-573.

Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16(2), 159-176.

Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph, 17.

Examples

item_1 <- new("item_1PL", difficulty = 0.5)
item_2 <- new("item_2PL", slope = 1.0, difficulty = 0.5)
item_3 <- new("item_3PL", slope = 1.0, difficulty = 0.5, guessing = 0.2)
item_4 <- new("item_PC", threshold = c(-0.5, 0.5), ncat = 3)
item_5 <- new("item_GPC", slope = 1.0, threshold = c(-0.5, 0.0, 0.5), ncat = 4)
item_6 <- new("item_GR", slope = 1.0, category = c(-2.0, -1.0, 0, 1.0, 2.0), ncat = 6)

(C++) Calculate first derivative of log-likelihood

Description

j_*() and array_j_*() are C++ functions for calculating the first derivative of the log-likelihood function.

Usage

j_1pl(x, b, u)

j_2pl(x, a, b, u)

j_m_2pl(x, a, d, u)

j_3pl(x, a, b, c, u)

j_m_3pl(x, a, d, c, u)

j_pc(x, b, u)

j_gpc(x, a, b, u)

j_m_gpc(x, a, d, u)

j_gr(x, a, b, u)

j_m_gr(x, a, d, u)

array_j_1pl(x, b, u)

array_j_2pl(x, a, b, u)

array_j_3pl(x, a, b, c, u)

array_j_pc(x, b, u)

array_j_gpc(x, a, b, u)

array_j_gr(x, a, b, u)

Arguments

x

the theta value. The number of columns should correspond to the number of dimensions. For array_*() functions, the number of theta values must correspond to the number of rows.
b, d

the difficulty parameter. b is used for unidimensional items, and d is used for multidimensional items.
u

the response value.
a

the a-parameter.
c

the c-parameter.

Details

j_*() functions accept a single theta value, and array_j_*() functions accept multiple theta values.

Supports unidimensional and multidimensional models.

  • j_1pl(), array_j_1pl(): 1PL models

  • j_2pl(), array_j_2pl(): 2PL models

  • j_3pl(), array_j_3pl(): 3PL models

  • j_pc(), array_j_pc(): PC (partial credit) models

  • j_gpc(), array_j_gpc(): GPC (generalized partial credit) models

  • j_gr(), array_j_gr(): GR (graded response) models

  • j_m_2pl(), array_j_m_2pl(): multidimensional 2PL models

  • j_m_3pl(), array_j_m_3pl(): multidimensional 3PL models

  • j_m_gpc(), array_j_m_gpc(): multidimensional GPC models

  • j_m_gr(), array_j_m_gr(): multidimensional GR models

References

Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: Danish Institute for Educational Research.

Lord, F. M. (1952). A theory of test scores (Psychometric Monograph No. 7). Richmond, VA: Psychometric Corporation.

Birnbaum, A. (1957). Efficient design and use of tests of mental ability for various decision-making problems (Series Report No. 58-16. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). On the estimation of mental ability (Series Report No. 15. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). Further considerations of efficiency in tests of a mental ability (Series Report No. 17. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In Lord, F. M., Novick, M. R. (eds.), Statistical Theories of Mental Test Scores, 397-479. Reading, MA: Addison-Wesley.

Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47(2), 149-174.

Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43(4), 561-573.

Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16(2), 159-176.

Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph, 17.

Examples

u <- 1

x <- 0.5
j_1pl(x, 1, u)
j_2pl(x, 1, 2, u)
j_3pl(x, 1, 2, 0.25, u)
j_pc(x, c(0, 1), u)
j_gpc(x, 2, c(0, 1), u)
j_gr(x, 2, c(0, 2), u)

x <- matrix(seq(-3, 3, 1)) # three theta values, unidimensional
array_j_1pl(x, 1, u)
array_j_2pl(x, 1, 2, u)
array_j_3pl(x, 1, 2, 0.25, u)
array_j_pc(x, c(0, 1), u)
array_j_gpc(x, 2, c(0, 1), u)
array_j_gr(x, 2, c(0, 2), u)

Convert mean and standard deviation into log-normal distribution parameters

Description

lnHyperPars is a function for calculating parameters for a log-normal distribution, such that the distribution yields desired mean and standard deviation. Used for sampling the a-parameter.

Usage

lnHyperPars(mean, sd)

Arguments

mean

the desired mean.
sd

the desired standard deviation.

Value

lnHyperPars returns two values. These can be directly supplied to rlnorm.

Examples

pars <- lnHyperPars(2, 4)
x <- rlnorm(1000000, pars[1], pars[2])
mean(x) # close to 2
sd(x)   # close to 4

Load constraints

Description

loadConstraints is a data loading function for creating a constraints object. loadConstraints can read constraints from a data.frame or a .csv file. The contents must be in the expected format; see the vignette in vignette("constraints") for a documentation.

Usage

loadConstraints(object, pool, item_attrib, st_attrib = NULL)

Arguments

object

constraint specifications. Can be a data.frame or the file path of a .csv file. See the vignette for a description of the expected format.
pool

an item_pool object. Use loadItemPool for this.
item_attrib

an item_attrib object. Use loadItemAttrib for this.
st_attrib

(optional) an st_attrib object. Use loadStAttrib for this.

Value

loadConstraints returns a constraints object. This object is used in Static and Shadow.

See Also

dataset_science, dataset_reading, dataset_fatigue, dataset_bayes for examples.

Examples

## Read from data.frame:
itempool_science    <- loadItemPool(itempool_science_data)
itemattrib_science  <- loadItemAttrib(itemattrib_science_data, itempool_science)
constraints_science <- loadConstraints(constraints_science_data,
  itempool_science, itemattrib_science)

## Read from file: write to tempdir() for illustration and clean afterwards
f <- file.path(tempdir(), "constraints_science.csv")
write.csv(constraints_science_data, f, row.names = FALSE)
constraints_science <- loadConstraints(f,
  itempool_science, itemattrib_science)
file.remove(f)

Load item pool

Description

loadItemPool is a data loading function for creating an item_pool object. loadItemPool can read item parameters and standard errors from a data.frame or a .csv file.

Usage

loadItemPool(ipar, ipar_se = NULL, unique = FALSE)

Arguments

ipar

item parameters. Can be a data.frame or the file path of a .csv file. The content should at least include columns 'ID' and 'MODEL'.
ipar_se

(optional) standard errors. Can be a data.frame or the file path of a .csv file.
unique

if TRUE, item IDs must be unique to create a valid item_pool object. (default = FALSE)

Value

loadItemPool returns an item_pool object.

  • ni the number of items in the pool.

  • max_cat the maximum number of response categories across all items in the pool.

  • index the numeric item index of each item.

  • id the item ID string of each item.

  • model the object class names of each item representing an item model type. Can be item_1PL, item_2PL, item_3PL, item_PC, item_GPC, or item_GR.

  • NCAT the number of response categories of each item.

  • parms a list containing the item object of each item.

  • ipar a matrix containing all item parameters.

  • se a matrix containing all item parameter standard errors. The values will be 0 if the argument ipar_se was not supplied.

  • raw the original input ipar argument used to create this object.

  • raw_se the original input ipar_se argument used to create this object. If the argument was not supplied, this will be in the same structure with the ipar argument but the item parameter values will be filled with 0s.

  • unique the original input unique argument used to create this object.

See Also

dataset_science, dataset_reading, dataset_fatigue, dataset_bayes for examples.

Examples

## Read from data.frame:
itempool_science <- loadItemPool(itempool_science_data)

## Read from file: write to tempdir() for illustration and clean afterwards
f <- file.path(tempdir(), "itempool_science.csv")
write.csv(itempool_science_data, f, row.names = FALSE)
itempool_science <- loadItemPool(f)
file.remove(f)

Convert mean and standard deviation into logit-normal distribution parameters

Description

logitHyperPars is a function for calculating parameters for a logit-normal distribution, such that the distribution yields desired mean and standard deviation. Used for sampling the c-parameter.

Usage

logitHyperPars(mean, sd)

Arguments

mean

the desired mean.
sd

the desired standard deviation.

Value

logitHyperPars returns two values. These can be directly supplied to rlogitnorm.

Examples

pars <- logitHyperPars(0.2, 0.1)
x <- logitnorm::rlogitnorm(1000000, pars[1], pars[2])
mean(x) # close to 0.2
sd(x)   # close to 0.1

make constraints objects from Split() solution indices

Description

makeConstraintsByEachPartition is a helper function for making constraints objects from Split solution indices.

Usage

makeConstraintsByEachPartition(constraints, solution_per_bin)

Arguments

constraints

a constraints object representing test specifications. Use loadConstraints for this.
solution_per_bin

a list containing item/stimulus indices for each partition. This accepts a list stored in the output slot of an output_Split object.

Value

makeConstraintsByEachPartition returns a list of constraints objects.

Create an item pool cluster object

Description

Create a item_pool_cluster object.

item_pool_cluster1 == item_pool_cluster2 tests equality of two item_pool_cluster objects.

Usage

makeItemPoolCluster(x, ..., names = NULL)

## S4 method for signature 'item_pool'
makeItemPoolCluster(x, ..., names = NULL)

## S3 method for class 'item_pool_cluster'
item_pool_cluster1 == item_pool_cluster2

Arguments

x, ...

item_pool objects.
names

(optional) names to use for item_pool.
item_pool_cluster1

an item_pool_cluster object.
item_pool_cluster2

an item_pool_cluster object.

Examples

cluster <- makeItemPoolCluster(itempool_science, itempool_reading)
cluster1 <- makeItemPoolCluster(itempool_science, itempool_reading)
cluster2 <- makeItemPoolCluster(cluster1@pools[[1]], cluster1@pools[[2]])
cluster1 == cluster2  ## TRUE

Create a simulation data cache object

Description

makeSimulationDataCache is a function for creating a simulation_data_cache object. This is used in Shadow to make all necessary data (e.g., item information, response data) prior to the main simulation.

Usage

makeSimulationDataCache(
  item_pool,
  info_type = "FISHER",
  theta_grid = seq(-4, 4, 0.1),
  seed = NULL,
  true_theta = NULL,
  response_data = NULL
)

## S4 method for signature 'item_pool'
makeSimulationDataCache(
  item_pool,
  info_type = "FISHER",
  theta_grid = seq(-4, 4, 0.1),
  seed = NULL,
  true_theta = NULL,
  response_data = NULL
)

Arguments

item_pool

an item_pool object.
info_type

the type of information.
theta_grid

a grid of theta values.
seed

(optional) seed to use for generating response data if needed.
true_theta

(optional) true theta values of all simulees.
response_data

(optional) response data on all items for all simulees.

Create a test object

Description

makeTest is a function for creating a test object. This is used to make all necessary data (e.g., item information, response data) prior to the main simulation. This function is only kept for backwards compatibility. The functionality of this function is superseded by makeSimulationDataCache.

Usage

makeTest(
  object,
  theta = seq(-4, 4, 0.1),
  info_type = "FISHER",
  true_theta = NULL
)

## S4 method for signature 'item_pool'
makeTest(
  object,
  theta = seq(-4, 4, 0.1),
  info_type = "FISHER",
  true_theta = NULL
)

Arguments

object

an item_pool object.
theta

a grid of theta values.
info_type

the type of information.
true_theta

(optional) true theta values to simulate response data.

Examples

test <- makeTest(itempool_science, seq(-3, 3, 1))

Create a test cluster object

Description

makeTestCluster is a function for creating a test_cluster object. This is used to make all necessary data (e.g., item information, response data) prior to the main simulation. This function is only kept for backwards compatibility.

Usage

makeTestCluster(object, theta, true_theta)

## S4 method for signature 'item_pool_cluster,numeric,numeric'
makeTestCluster(object, theta, true_theta)

## S4 method for signature 'item_pool_cluster,numeric,list'
makeTestCluster(object, theta, true_theta)

Arguments

object

an item_pool_cluster object.
theta

a grid of theta values.
true_theta

an optional vector of true theta values to simulate response data.

Compute maximum likelihood estimates of theta

Description

mle is a function for computing maximum likelihood estimates of theta.

Usage

mle(
  object,
  select = NULL,
  resp,
  start_theta = NULL,
  max_iter = 100,
  crit = 0.001,
  truncate = FALSE,
  theta_range = c(-4, 4),
  max_change = 1,
  use_step_size = FALSE,
  step_size = 0.5,
  do_Fisher = TRUE
)

## S4 method for signature 'item_pool'
mle(
  object,
  select = NULL,
  resp,
  start_theta = NULL,
  max_iter = 50,
  crit = 0.005,
  truncate = FALSE,
  theta_range = c(-4, 4),
  max_change = 1,
  use_step_size = FALSE,
  step_size = 0.5,
  do_Fisher = TRUE
)

MLE(
  object,
  select = NULL,
  start_theta = NULL,
  max_iter = 100,
  crit = 0.001,
  theta_range = c(-4, 4),
  truncate = FALSE,
  max_change = 1,
  do_Fisher = TRUE
)

## S4 method for signature 'test'
MLE(
  object,
  select = NULL,
  start_theta = NULL,
  max_iter = 100,
  crit = 0.001,
  theta_range = c(-4, 4),
  truncate = FALSE,
  max_change = 1,
  do_Fisher = TRUE
)

## S4 method for signature 'test_cluster'
MLE(object, select = NULL, start_theta = NULL, max_iter = 100, crit = 0.001)

Arguments

object

an item_pool object.
select

(optional) if item indices are supplied, only the specified items are used.
resp

item response on all (or selected) items in the object argument. Can be a vector, a matrix, or a data frame. length(resp) or ncol(resp) must be equal to the number of all (or selected) items.
start_theta

(optional) initial theta values. If not supplied, EAP estimates using uniform priors are used as initial values. Uniform priors are computed using the theta_range argument below, with increments of .1.
max_iter

maximum number of iterations. (default = 100)
crit

convergence criterion to use. (default = 0.001)
truncate

set TRUE to impose a bound using theta_range on the estimate. (default = FALSE)
theta_range

a range of theta values to bound the estimate. Only effective when truncate is TRUE. (default = c(-4, 4))
max_change

upper bound to impose on the absolute change in theta between iterations. Absolute changes exceeding this value will be capped to max_change. (default = 1.0)
use_step_size

set TRUE to use step_size. (default = FALSE)
step_size

upper bound to impose on the absolute change in initial theta and estimated theta. Absolute changes exceeding this value will be capped to step_size. (default = 0.5)
do_Fisher

set TRUE to use Fisher scoring instead of Newton-Raphson method. (default = TRUE)

Value

mle returns a list containing estimated values.

  • th theta value.

  • se standard error.

  • conv TRUE if estimation converged.

  • trunc TRUE if truncation was applied on th.

Examples

mle(itempool_fatigue, resp = resp_fatigue_data[10, ])
mle(itempool_fatigue, select = 1:20, resp = resp_fatigue_data[10, 1:20])

Compute maximum likelihood estimates of theta using fence items

Description

mlef is a function for computing maximum likelihood estimates of theta using fence items.

Usage

mlef(
  object,
  select = NULL,
  resp,
  fence_slope = 5,
  fence_difficulty = c(-5, 5),
  start_theta = NULL,
  max_iter = 100,
  crit = 0.001,
  truncate = FALSE,
  theta_range = c(-4, 4),
  max_change = 1,
  use_step_size = FALSE,
  step_size = 0.5,
  do_Fisher = TRUE
)

## S4 method for signature 'item_pool'
mlef(
  object,
  select = NULL,
  resp,
  fence_slope = 5,
  fence_difficulty = c(-5, 5),
  start_theta = NULL,
  max_iter = 50,
  crit = 0.005,
  truncate = FALSE,
  theta_range = c(-4, 4),
  max_change = 1,
  use_step_size = FALSE,
  step_size = 0.5,
  do_Fisher = TRUE
)

Arguments

object

an item_pool object.
select

(optional) if item indices are supplied, only the specified items are used.
resp

item response on all (or selected) items in the object argument. Can be a vector, a matrix, or a data frame. length(resp) or ncol(resp) must be equal to the number of all (or selected) items.
fence_slope

the slope parameter to use on fence items. Can be one value, or two values for the lower and the upper fence respectively. (default = 5)
fence_difficulty

the difficulty parameter to use on fence items. Must have two values for the lower and the upper fence respectively. (default = c(-5, 5))
start_theta

(optional) initial theta values. If not supplied, EAP estimates using uniform priors are used as initial values. Uniform priors are computed using the theta_range argument below, with increments of .1.
max_iter

maximum number of iterations. (default = 100)
crit

convergence criterion to use. (default = 0.001)
truncate

set TRUE to impose a bound using theta_range on the estimate. (default = FALSE)
theta_range

a range of theta values to bound the estimate. Only effective when truncate is TRUE. (default = c(-4, 4))
max_change

upper bound to impose on the absolute change in theta between iterations. Absolute changes exceeding this value will be capped to max_change. (default = 1.0)
use_step_size

set TRUE to use step_size. (default = FALSE)
step_size

upper bound to impose on the absolute change in initial theta and estimated theta. Absolute changes exceeding this value will be capped to step_size. (default = 0.5)
do_Fisher

set TRUE to use Fisher scoring instead of Newton-Raphson method. (default = TRUE)

Value

mlef returns a list containing estimated values.

  • th theta value.

  • se standard error.

  • conv TRUE if estimation converged.

  • trunc TRUE if truncation was applied on th.

References

Han, K. T. (2016). Maximum likelihood score estimation method with fences for short-length tests and computerized adaptive tests. Applied Psychological Measurement, 40(4), 289-301.

Examples

mlef(itempool_fatigue, resp = resp_fatigue_data[10, ])
mlef(itempool_fatigue, select = 1:20, resp = resp_fatigue_data[10, 1:20])

Class 'output_Shadow_all': a set of adaptive assembly solutions

Description

output_Shadow_all is an S4 class for representing a set of adaptive assembly solutions.

Details

notations

  • ni denotes the number of items in the item_pool object.

  • ns denotes the number of stimuli.

  • nj denotes the number of participants.

Slots

call

the function call used for obtaining this object.

output

a length-*nj* list of output_Shadow objects, containing the assembly results for each participant.

final_theta_est

a length-*nj* vector containing final theta estimates for each participant.

final_se_est

a length-*nj* vector standard errors of the final theta estimates for each participant.

exposure_rate

a matrix containing item-level exposure rates of all items in the pool. Also contains stimulus-level exposure rates if the assembly was set-based.

usage_matrix

a *nj* by (*ni* + *ns*) matrix representing whether the item/stimulus was administered to each participant. Stimuli representations are appended to the right side of the matrix.

cumulative_usage_matrix

a *nj* by (*ni* + *ns*) matrix representing the number of times the item/stimulus was administered to each participant over multiple administrations.

true_segment_count

a length-*nj* vector containing the how many examinees are now in their segment based on the true theta. This will tend to increase. This can be reproduced with true theta values alone.

est_segment_count

a length-*nj* vector containing the how many examinees are now in their segment based on the estimated theta. This will tend to increase. This can be reproduced with estimated theta values alone.

eligibility_stats

exposure record for diagnostics.

check_eligibility_stats

detailed segment-wise exposure record for diagnostics. available when config_Shadow@exposure_control$diagnostic_stats is TRUE.

no_fading_eligibility_stats

detailed segment-wise exposure record without fading for diagnostics. available when config_Shadow@exposure_control$diagnostic_stats is TRUE.

freq_infeasible

a table representing the number of times the assembly was initially infeasible.

pool

the item_pool used in the assembly.

config

the config_Shadow used in the assembly.

constraints

the constraints used in the assembly.

true_theta

the true_theta argument used in the assembly.

data

the data argument used in the assembly.

prior

the prior argument used in the assembly.

prior_par

the prior_par argument used in the assembly.

adaptivity

a list of adaptivity indices.

simulation_constants

a list containing simulation constants parsed from input.

Class 'output_Shadow': adaptive assembly solution for one simulee

Description

output_Shadow is an S4 class for representing the adaptive assembly solution for one simulee.

Slots

simulee_id

the numeric ID of the simulee.

true_theta

the true theta of the simulee, if was specified.

true_theta_segment

the segment number of the true theta.

final_theta_est

final theta estimate.

final_se_est

the standard error of final_theta_est.

administered_item_index

item IDs administered at each position.

administered_item_resp

item responses from the simulee at each position.

administered_item_ncat

the number of categories of each administered item.

administered_stimulus_index

stimulus IDs administered at each position.

shadow_test_refreshed

TRUE indicates the shadowtest was refreshed for the position.

shadow_test_feasible

TRUE indicates the MIP was feasible with all constraints.

solve_time

elapsed time in running the solver at each position.

initial_theta_est

initial theta estimate.

interim_theta_est

interim theta estimates at each position.

interim_se_est

the standard error of the interim estimate at each position.

theta_segment_index

segment numbers of interim theta estimates.

prior

prior distribution, if was specified.

prior_par

prior parameters, if were specified.

posterior

the posterior distribution after completing test.

posterior_sample

posterior samples of interim theta before the estimation of final theta. mean(posterior_sample) == interim_theta_est[test_length] holds.

likelihood

the likelihood distribution after completing test.

shadow_test

the list containing the item IDs within the shadowtest used in each position.

max_cat_pool

the maximum number of response categories the item pool had.

ni_pool

the total number of items the item pool had.

ns_pool

the total number of stimuli the item pool had.

test_length_constraints

the test length constraint used in assembly.

set_based

whether the item pool was set-based.

item_index_by_stimulus

the list of items by each stimulus the item pool had.

Class 'output_Split': partitioning solution

Description

output_Split is an S4 class for representing the partitioning solution of an item pool.

Slots

call

the function call used for obtaining this object.

output

a list containing item/set indices of each partition.

feasible

for partitioning into sub-pools, TRUE indicates the complete assignment problem was feasible.

solve_time

elapsed time in running the solver.

set_based

whether the item pool is set-based.

config

the config_Static used in the assembly.

constraints

the constraints used in the assembly.

partition_size_range

the partition size range for splitting into sub-pools.

partition_type

the partition type. Can be a test or a pool.

constraints_by_each_partition

a list of constraints objects that represent each partition.

Class 'output_Static': fixed-form assembly solution

Description

output_Static is an S4 class for representing a fixed-form assembly solution.

Slots

call

the function call used for obtaining this object.

MIP

a list containing the result from MIP solver.

selected

a data.frame containing the selected items and their attributes.

obj_value

the objective value of the solution.

solve_time

the elapsed time in running the solver.

achieved

a data.frame containing attributes of the assembled test, by each constraint.

pool

the item_pool used in the assembly.

config

the config_Static used in the assembly.

constraints

the constraints used in the assembly.

(C++) Calculate item response probability

Description

p_*() and array_p_*() are C++ functions for calculating item response probability.

Usage

p_1pl(x, b)

p_2pl(x, a, b)

p_m_2pl(x, a, d)

p_3pl(x, a, b, c)

p_m_3pl(x, a, d, c)

p_pc(x, b)

p_gpc(x, a, b)

p_m_gpc(x, a, d)

p_gr(x, a, b)

p_m_gr(x, a, d)

array_p_1pl(x, b)

array_p_2pl(x, a, b)

array_p_m_2pl(x, a, d)

array_p_3pl(x, a, b, c)

array_p_m_3pl(x, a, d, c)

array_p_pc(x, b)

array_p_gpc(x, a, b)

array_p_m_gpc(x, a, d)

array_p_gr(x, a, b)

array_p_m_gr(x, a, d)

Arguments

x

the theta value. The number of columns should correspond to the number of dimensions. For array_*() functions, the number of theta values must correspond to the number of rows.
b, d

the difficulty parameter. b is used for unidimensional items, and d is used for multidimensional items.
a

the a-parameter.
c

the c-parameter.

Details

p_*() functions accept a single theta value, and array_p_*() functions accept multiple theta values.

Supports unidimensional and multidimensional models.

  • p_1pl(), array_p_1pl(): 1PL models

  • p_2pl(), array_p_2pl(): 2PL models

  • p_3pl(), array_p_3pl(): 3PL models

  • p_pc(), array_p_pc(): PC (partial credit) models

  • p_gpc(), array_p_gpc(): GPC (generalized partial credit) models

  • p_gr(), array_p_gr(): GR (graded response) models

  • p_m_2pl(), array_p_m_2pl(): multidimensional 2PL models

  • p_m_3pl(), array_p_m_3pl(): multidimensional 3PL models

  • p_m_gpc(), array_p_m_gpc(): multidimensional GPC models

  • p_m_gr(), array_p_m_gr(): multidimensional GR models

References

Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: Danish Institute for Educational Research.

Lord, F. M. (1952). A theory of test scores (Psychometric Monograph No. 7). Richmond, VA: Psychometric Corporation.

Birnbaum, A. (1957). Efficient design and use of tests of mental ability for various decision-making problems (Series Report No. 58-16. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). On the estimation of mental ability (Series Report No. 15. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). Further considerations of efficiency in tests of a mental ability (Series Report No. 17. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In Lord, F. M., Novick, M. R. (eds.), Statistical Theories of Mental Test Scores, 397-479. Reading, MA: Addison-Wesley.

Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47(2), 149-174.

Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43(4), 561-573.

Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16(2), 159-176.

Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph, 17.

Examples

x <- 0.5

p_1pl(x, 1)
p_2pl(x, 1, 2)
p_3pl(x, 1, 2, 0.25)
p_pc(x, c(0, 1))
p_gpc(x, 2, c(0, 1))
p_gr(x, 2, c(0, 2))

x <- matrix(seq(0.1, 0.5, 0.1)) # three theta values, unidimensional

array_p_1pl(x, 1)
array_p_2pl(x, 1, 2)
array_p_3pl(x, 1, 2, 0.25)
array_p_pc(x, c(0, 1))
array_p_gpc(x, 2, c(0, 1))
array_p_gr(x, 2, c(0, 2))

Extension of plot() for objects in TestDesign package

Description

Extension of plot() for objects in TestDesign package

Usage

## S4 method for signature 'item_pool'
plot(
  x,
  y,
  type = "info",
  theta = seq(-3, 3, 0.1),
  info_type = "FISHER",
  plot_sum = TRUE,
  select = NULL,
  examinee_id = 1,
  position = NULL,
  theta_range = c(-5, 5),
  ylim = NULL,
  color = "blue",
  z_ci = 1.96,
  simple = TRUE,
  theta_type = "Estimated",
  color_final = "blue",
  color_stim = "red",
  segment = NULL,
  rmse = FALSE,
  use_segment_label = TRUE,
  use_par = TRUE,
  ...
)

## S4 method for signature 'output_Static'
plot(
  x,
  y,
  type = NULL,
  theta = seq(-3, 3, 0.1),
  info_type = "FISHER",
  plot_sum = TRUE,
  select = NULL,
  examinee_id = 1,
  position = NULL,
  theta_range = c(-5, 5),
  ylim = NULL,
  color = "blue",
  z_ci = 1.96,
  simple = TRUE,
  use_par = TRUE,
  ...
)

## S4 method for signature 'constraints'
plot(
  x,
  y,
  type = "info",
  theta = seq(-3, 3, 0.1),
  info_type = "FISHER",
  plot_sum = TRUE,
  select = NULL,
  examinee_id = 1,
  position = NULL,
  theta_range = c(-5, 5),
  ylim = NULL,
  color = "blue",
  z_ci = 1.96,
  simple = TRUE,
  use_par = TRUE,
  ...
)

## S4 method for signature 'output_Shadow'
plot(
  x,
  y,
  type = "audit",
  theta = seq(-3, 3, 0.1),
  info_type = "FISHER",
  plot_sum = TRUE,
  select = NULL,
  examinee_id = 1,
  theta_range = c(-5, 5),
  ylim = NULL,
  color = "blue",
  z_ci = 1.96,
  simple = FALSE,
  theta_type = "Estimated",
  use_par = TRUE,
  ...
)

## S4 method for signature 'output_Shadow_all'
plot(
  x,
  y,
  type = "audit",
  theta = seq(-3, 3, 0.1),
  info_type = "FISHER",
  plot_sum = TRUE,
  select = NULL,
  examinee_id = 1,
  position = NULL,
  theta_range = c(-5, 5),
  ylim = NULL,
  color = "blue",
  z_ci = 1.96,
  simple = FALSE,
  theta_type = "Estimated",
  color_final = "blue",
  color_stim = "red",
  segment = NULL,
  rmse = FALSE,
  use_segment_label = TRUE,
  use_par = TRUE,
  ...
)

## S4 method for signature 'output_Split'
plot(
  x,
  y,
  type = NULL,
  theta = seq(-3, 3, 0.1),
  info_type = "FISHER",
  plot_sum = TRUE,
  select = NULL,
  examinee_id = 1,
  position = NULL,
  theta_range = c(-5, 5),
  ylim = NULL,
  color = "blue",
  z_ci = 1.96,
  simple = TRUE,
  use_par = TRUE,
  ...
)

Arguments

x

accepts the following signatures:

  • item_pool: plot information and expected scores.

  • constraints: plot information range based on the test length constraint.

  • output_Static: plot information and expected scores based on the fixed assembly solution.

  • output_Shadow_all: plot audit trail, shadowtest chart, exposure rates, and item overlap data from the adaptive assembly solution.

  • output_Shadow: plot audit trail and shadowtest chart from the adaptive assembly solution.

y

not used, exists for compatibility with plot in the base R package.
type

the type of plot.

theta

the theta grid to use in plotting. (default = seq(-3, 3, .1))
info_type

the type of information. Currently accepts FISHER. (default = FISHER)
plot_sum

used in item_pool objects.

  • if TRUE then plot pool-level values.

  • if FALSE then plot item-level values, and repeat for all items in the pool.

  • (default = TRUE)

select

used in item_pool objects. Item indices to subset.
examinee_id

used in output_Shadow and output_Shadow_all with type = 'audit' and type = 'shadow'. The examinee numeric ID to draw the plot.
position

used in output_Shadow_all with type = 'info'. The item position to draw the plot.
theta_range

used in output_Shadow and output_Shadow_all with type = 'audit'. The theta range to plot. (default = c(-5, 5))
ylim

(optional) the y-axis plot range. Used in most plot types.
color

the color of the curve.
z_ci

used in output_Shadow and output_Shadow_all with type = 'audit'. The range to use for confidence intervals. (default = 1.96)
simple

used in output_Shadow and output_Shadow_all with type = 'shadow'. If TRUE, simplify the chart by hiding unused items.
theta_type

used in output_Shadow_all with type = 'exposure'. The type of theta to determine exposure segments. Accepts Estimated or True. (default = Estimated)
color_final

used in output_Shadow_all with type = 'exposure'. The color of item-wise exposure rates, only counting the items administered in the final theta segment as exposed.
color_stim

used in output_Shadow_all with type = 'exposure' or type = 'overlap'. The color of stimulus exposure rates or stimulus overlap data.
segment

used in output_Shadow_all with type = 'exposure'. (optional) The segment index to draw the plot. Leave empty to use all segments.
rmse

used in output_Shadow_all with type = 'exposure'. If TRUE, display the RMSE value for each segment. (default = FALSE)
use_segment_label

used in output_Shadow_all with type = 'exposure'. If TRUE, display the segment label for each segment. (default = TRUE)
use_par

if FALSE, graphical parameters are not overridden inside the function. (default = TRUE)
...

arguments to pass onto plot.

Examples

subitempool <- itempool_science[1:8]

## Plot item information of a pool
plot(subitempool)
plot(itempool_science, select = 1:8)

## Plot expected score of a pool
plot(subitempool, type = "score")
plot(itempool_science, type = "score", select = 1:8)

## Plot assembly results from Static()
cfg <- createStaticTestConfig()
solution <- Static(cfg, constraints_science)
plot(solution)                 # defaults to the objective type
plot(solution, type = "score") # plot expected scores

## Plot attainable information range from constraints
plot(constraints_science)

## Plot assembly results from Shadow()
cfg <- createShadowTestConfig()
set.seed(1)
solution <- Shadow(cfg, constraints_science, true_theta = rnorm(1))
plot(solution, type = 'audit' , examinee_id = 1)
plot(solution, type = 'shadow', examinee_id = 1, simple = TRUE)

## plot(solution, type = 'exposure')
## plot(solution, type = 'overlap')

Extension of print() for objects in TestDesign package

Description

Extension of print() for objects in TestDesign package

Usage

## S4 method for signature 'item_1PL'
print(x)

## S4 method for signature 'item_2PL'
print(x)

## S4 method for signature 'item_3PL'
print(x)

## S4 method for signature 'item_PC'
print(x)

## S4 method for signature 'item_GPC'
print(x)

## S4 method for signature 'item_GR'
print(x)

## S4 method for signature 'item_pool'
print(x)

## S4 method for signature 'item_attrib'
print(x)

## S4 method for signature 'st_attrib'
print(x)

## S4 method for signature 'summary_item_attrib'
print(x)

## S4 method for signature 'summary_st_attrib'
print(x)

## S4 method for signature 'constraints'
print(x)

## S4 method for signature 'config_Static'
print(x)

## S4 method for signature 'config_Shadow'
print(x)

## S4 method for signature 'output_Static'
print(x, index_only = TRUE)

## S4 method for signature 'output_Shadow'
print(x)

## S4 method for signature 'output_Shadow_all'
print(x)

## S4 method for signature 'exposure_rate_plot'
print(x)

## S4 method for signature 'summary_item_pool'
print(x)

## S4 method for signature 'summary_constraints'
print(x)

## S4 method for signature 'summary_output_Static'
print(x, digits = 3)

## S4 method for signature 'summary_output_Shadow_all'
print(x, digits = 3)

Arguments

x

an object to print.
index_only

if TRUE then only print item indices. If FALSE then print all item attributes. (default = TRUE)
digits

minimal number of *significant* digits. See print.default.

Calculate Relative Errors

Description

Calculate Relative Errors.

Usage

RE(RMSE_foc, RMSE_ref)

Arguments

RMSE_foc

A vector of RMSE values for the focal group.
RMSE_ref

A vector of RMSE values for the reference group.

Calculate Root Mean Squared Error

Description

Calculate Root Mean Squared Error.

Usage

RMSE(x, y, conditional = TRUE)

Arguments

x

A vector of values.
y

A vector of values.
conditional

If TRUE, calculate RMSE conditional on x.

Run adaptive test assembly

Description

Shadow is a test assembly function for performing adaptive test assembly based on the generalized shadow-test framework.

Usage

Shadow(
  config,
  constraints = NULL,
  true_theta = NULL,
  data = NULL,
  prior = NULL,
  prior_par = NULL,
  exclude = NULL,
  include_items_for_estimation = NULL,
  force_solver = FALSE,
  session = NULL,
  seed = NULL,
  cumulative_usage_matrix = NULL
)

## S4 method for signature 'config_Shadow'
Shadow(
  config,
  constraints = NULL,
  true_theta = NULL,
  data = NULL,
  prior = NULL,
  prior_par = NULL,
  exclude = NULL,
  include_items_for_estimation = NULL,
  force_solver = FALSE,
  session = NULL,
  seed = NULL,
  cumulative_usage_matrix = NULL
)

Arguments

config

a config_Shadow object. Use createShadowTestConfig for this.
constraints

a constraints object representing test specifications. Use loadConstraints for this.
true_theta

(optional) true theta values to use in simulation. Either true_theta or data must be supplied.
data

(optional) a matrix containing item response data to use in simulation. Either true_theta or data must be supplied.
prior

(optional) density at each config@theta_grid to use as prior. Must be a length-nq vector or a nj * nq matrix. This overrides prior_dist and prior_par in the config. prior and prior_par cannot be used simultaneously.
prior_par

(optional) normal distribution parameters c(mean, sd) to use as prior. Must be a length-nq vector or a nj * nq matrix. This overrides prior_dist and prior_par in the config. prior and prior_par cannot be used simultaneously.
exclude

(optional) a list containing item names in $i and set names in $s to exclude from selection for each participant. The length of the list must be equal to the number of participants.
include_items_for_estimation

(optional) an examinee-wise list containing:

  • administered_item_pool items to include in theta estimation as item_pool object.

  • administered_item_resp item responses to include in theta estimation.

force_solver

if TRUE, do not check whether the solver is one of recommended solvers for complex problems (set-based assembly, partitioning). (default = FALSE)
session

(optional) used to communicate with Shiny app TestDesign.
seed

(optional) used to perform data generation internally.
cumulative_usage_matrix

(optional) a *nj* by (*ni* + *ns*) matrix containing the number of times the item/stimulus was administered previously to each participant. Stimuli representations are appended to the right side of the matrix.

Value

Shadow returns an output_Shadow_all object containing assembly results.

References

van der Linden, W. J., Reese, L. M. (1998). A model for optimal constrained adaptive testing. Applied Psychological Measurement, 22, 259-270.

van der Linden, W. J. (1998). Optimal assembly of psychological and educational tests. Applied Psychological Measurement, 22, 195-211.

van der Linden, W. J. (2000). Optimal assembly of tests with item sets. Applied Psychological Measurement, 24, 225-240.

van der Linden, W. J. (2005). Linear models for optimal test design. Springer Science & Business Media.

Examples

config <- createShadowTestConfig()
true_theta <- rnorm(1)
solution <- Shadow(config, constraints_science, true_theta)
solution@output

Extension of show() for objects in TestDesign package

Description

Extension of show() for objects in TestDesign package

Usage

## S4 method for signature 'item_1PL'
show(object)

## S4 method for signature 'item_2PL'
show(object)

## S4 method for signature 'item_3PL'
show(object)

## S4 method for signature 'item_PC'
show(object)

## S4 method for signature 'item_GPC'
show(object)

## S4 method for signature 'item_GR'
show(object)

## S4 method for signature 'item_pool'
show(object)

## S4 method for signature 'item_pool_cluster'
show(object)

## S4 method for signature 'item_attrib'
show(object)

## S4 method for signature 'st_attrib'
show(object)

## S4 method for signature 'constraints'
show(object)

## S4 method for signature 'summary_item_pool'
show(object)

## S4 method for signature 'summary_item_attrib'
show(object)

## S4 method for signature 'summary_st_attrib'
show(object)

## S4 method for signature 'summary_constraints'
show(object)

## S4 method for signature 'config_Static'
show(object)

## S4 method for signature 'config_Shadow'
show(object)

## S4 method for signature 'output_Static'
show(object)

## S4 method for signature 'output_Shadow'
show(object)

## S4 method for signature 'output_Shadow_all'
show(object)

## S4 method for signature 'summary_output_Static'
show(object)

## S4 method for signature 'summary_output_Shadow_all'
show(object)

## S4 method for signature 'exposure_rate_plot'
show(object)

Arguments

object

an object to display.

Simulate item response data

Description

simResp is a function for simulating item response data.

Usage

simResp(object, theta)

## S4 method for signature 'item_1PL,numeric'
simResp(object, theta)

## S4 method for signature 'item_1PL,matrix'
simResp(object, theta)

## S4 method for signature 'item_2PL,numeric'
simResp(object, theta)

## S4 method for signature 'item_2PL,matrix'
simResp(object, theta)

## S4 method for signature 'item_3PL,numeric'
simResp(object, theta)

## S4 method for signature 'item_3PL,matrix'
simResp(object, theta)

## S4 method for signature 'item_PC,numeric'
simResp(object, theta)

## S4 method for signature 'item_PC,matrix'
simResp(object, theta)

## S4 method for signature 'item_GPC,numeric'
simResp(object, theta)

## S4 method for signature 'item_GPC,matrix'
simResp(object, theta)

## S4 method for signature 'item_GR,numeric'
simResp(object, theta)

## S4 method for signature 'item_GR,matrix'
simResp(object, theta)

## S4 method for signature 'item_pool,numeric'
simResp(object, theta)

## S4 method for signature 'item_pool,matrix'
simResp(object, theta)

## S4 method for signature 'item_pool_cluster,numeric'
simResp(object, theta)

## S4 method for signature 'item_pool_cluster,list'
simResp(object, theta)

Arguments

object

an item or an item_pool object.
theta

theta values to use.

Details

notations

  • nq denotes the number of theta values.

  • ni denotes the number of items in the item_pool object.

Value

item object:

simResp returns a length nq vector containing simulated item response data.

item_pool object:

simResp returns a (nq, ni) matrix containing simulated item response data.

References

Rasch, G. (1960). Probabilistic models for some intelligence and attainment tests. Copenhagen: Danish Institute for Educational Research.

Lord, F. M. (1952). A theory of test scores (Psychometric Monograph No. 7). Richmond, VA: Psychometric Corporation.

Birnbaum, A. (1957). Efficient design and use of tests of mental ability for various decision-making problems (Series Report No. 58-16. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). On the estimation of mental ability (Series Report No. 15. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1958). Further considerations of efficiency in tests of a mental ability (Series Report No. 17. Project No. 7755-23). Randolph Air Force Base, TX: USAF School of Aviation Medicine.

Birnbaum, A. (1968). Some latent trait models and their use in inferring an examinee's ability. In Lord, F. M., Novick, M. R. (eds.), Statistical Theories of Mental Test Scores, 397-479. Reading, MA: Addison-Wesley.

Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47(2), 149-174.

Andrich, D. (1978). A rating formulation for ordered response categories. Psychometrika, 43(4), 561-573.

Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16(2), 159-176.

Samejima, F. (1969). Estimation of latent ability using a response pattern of graded scores. Psychometrika Monograph, 17.

Examples

item_1    <- new("item_1PL", difficulty = 0.5)
item_2    <- new("item_2PL", slope = 1.0, difficulty = 0.5)
item_3    <- new("item_3PL", slope = 1.0, difficulty = 0.5, guessing = 0.2)
item_4    <- new("item_PC", threshold = c(-1, 0, 1), ncat = 4)
item_5    <- new("item_GPC", slope = 1.2, threshold = c(-0.8, -1.0, 0.5), ncat = 4)
item_6    <- new("item_GR", slope = 0.9, category = c(-1, 0, 1), ncat = 4)

sim_item_1 <- simResp(item_1, seq(-3, 3, 1))
sim_item_2 <- simResp(item_2, seq(-3, 3, 1))
sim_item_3 <- simResp(item_3, seq(-3, 3, 1))
sim_item_4 <- simResp(item_4, seq(-3, 3, 1))
sim_item_5 <- simResp(item_5, seq(-3, 3, 1))
sim_item_6 <- simResp(item_6, seq(-3, 3, 1))
sim_pool   <- simResp(itempool_science, seq(-3, 3, 1))

Class 'simulation_data_cache': data cache for Shadow()

Description

simulation_data_cache is an S4 class for representing data cache for Shadow().

Slots

item_pool

the item_pool object.

theta_grid

the theta grid to use as quadrature points.

prob_grid

the list containing item response probabilities at theta quadratures.

info_grid

the matrix containing item information values at theta quadratures.

max_info

the maximum value of info_grid.

true_theta

(optional) the true theta values.

response_data

(optional) the matrix containing item responses.

Split an item pool into partitions

Description

Split is a function for splitting a pool into multiple parallel tests or pools. When constructing parallel tests, each test is constructed to satisfy all constraints. When constructing parallel pools, each pool is constructed so that it contains a test that satisfies all constraints.

Usage

Split(
  config,
  constraints,
  n_partition,
  partition_type,
  partition_size_range = NULL,
  n_maximum_partitions_per_item = 1,
  force_solver = FALSE
)

## S4 method for signature 'config_Static'
Split(
  config,
  constraints,
  n_partition,
  partition_type,
  partition_size_range = NULL,
  n_maximum_partitions_per_item = 1,
  force_solver = FALSE
)

Arguments

config

a config_Static object. Use createStaticTestConfig for this.
constraints

a constraints object representing test specifications. Use loadConstraints for this.
n_partition

the number of partitions to create.
partition_type

test to create tests, or pool to create pools.
partition_size_range

(optional) two integer values for the desired range for the size of a partition. Has no effect when partition_type is test. For discrete item pools, the default partition size is (pool size / number of partitions). For set-based item pools, the default partition size is (pool size / number of partitions) +/- smallest set size.
n_maximum_partitions_per_item

(optional) the number of times an item can be assigned to a partition. Setting this to 1 is equivalent to requiring all partitions to be mutually exclusive. A caveat is that when this is equal to n_partition, the assembled partitions will be identical to each other, because Split aims to minimize the test information difference between all partitions. (default = 1)
force_solver

if TRUE, do not check whether the solver is one of recommended solvers for complex problems (set-based assembly, partitioning). (default = FALSE)

Value

Split returns an output_Split object containing item/set indices of created tests/pools.

Examples

## Not run: 
config <- createStaticTestConfig(MIP = list(solver = "RSYMPHONY"))
constraints <- constraints_science[1:10]

solution <- Split(config, constraints, n_partition = 4, partition_type = "test"))
plot(solution)
solution <- Split(config, constraints, n_partition = 4, partition_type = "pool"))
plot(solution)

## End(Not run)

Load set/stimulus/passage attributes

Description

loadStAttrib is a data loading function for creating an st_attrib object. loadStAttrib can read itemset-level attributes from a data.frame or a .csv file.

Usage

loadStAttrib(object, item_attrib)

Arguments

object

itemset-level attributes. Can be a data.frame or the file path of a .csv file. The content should at least include an 'STID' column that matches with itemset IDs (the 'STID' column) of the item_attrib object.
item_attrib

an item_attrib object. Use loadItemAttrib for this.

Value

loadStAttrib returns a st_attrib object.

  • data a data.frame containing itemset-level attributes.

See Also

dataset_reading for examples.

Examples

## Read from data.frame:
itempool_reading   <- loadItemPool(itempool_reading_data)
itemattrib_reading <- loadItemAttrib(itemattrib_reading_data, itempool_reading)
stimattrib_reading <- loadStAttrib(stimattrib_reading_data, itemattrib_reading)

## Read from file: write to tempdir() for illustration and clean afterwards
f <- file.path(tempdir(), "stimattrib_reading.csv")
write.csv(stimattrib_reading_data, f, row.names = FALSE)
stimattrib_reading <- loadStAttrib(f, itemattrib_reading)
file.remove(f)

Basic functions for stimulus attribute objects

Description

Basic functions for stimulus attribute objects

Usage

## S4 method for signature 'st_attrib,numeric'
x[i, j, ..., drop = TRUE]

## S4 method for signature 'st_attrib'
dim(x)

## S4 method for signature 'st_attrib'
colnames(x)

## S4 method for signature 'st_attrib'
rownames(x)

## S4 method for signature 'st_attrib'
names(x)

## S4 method for signature 'st_attrib'
as.data.frame(x, row.names = NULL, optional = FALSE, ...)

Arguments

x

a st_attrib object.
i, j

indices to use in subsetting.
...

not used, exists for compatibility.
drop

not used, exists for compatibility.
row.names

not used, exists for compatibility.
optional

not used, exists for compatibility.

Examples

x <- stimattrib_reading
x[1:10]
dim(x)
ncol(x)
nrow(x)
colnames(x)
rownames(x)
names(x)
as.data.frame(x)

Run fixed-form test assembly

Description

Static is a test assembly function for performing fixed-form test assembly based on the generalized shadow-test framework.

Usage

Static(config, constraints, force_solver = FALSE)

## S4 method for signature 'config_Static'
Static(config, constraints, force_solver = FALSE)

Arguments

config

a config_Static object. Use createStaticTestConfig for this.
constraints

a constraints object representing test specifications. Use loadConstraints for this.
force_solver

if TRUE, do not check whether the solver is one of recommended solvers for complex problems (set-based assembly, partitioning). (default = FALSE)

Value

Static returns a output_Static object containing the selected items.

References

van der Linden, W. J. (2005). Linear models for optimal test design. Springer Science & Business Media.

Examples

config_science <- createStaticTestConfig(
  list(
    method = "MAXINFO",
    target_location = c(-1, 1)
  )
)
solution <- Static(config_science, constraints_science)

Extension of summary() for objects in TestDesign package

Description

Extension of summary() for objects in TestDesign package

Usage

## S4 method for signature 'item_pool'
summary(object)

## S4 method for signature 'item_attrib'
summary(object)

## S4 method for signature 'st_attrib'
summary(object)

## S4 method for signature 'constraints'
summary(object)

## S4 method for signature 'output_Static'
summary(object, simple = FALSE)

## S4 method for signature 'output_Shadow_all'
summary(object, simple = FALSE)

Arguments

object

an object to summarize.
simple

if TRUE, do not print constraints. (default = FALSE)

Examples

summary(itempool_science)
summary(itemattrib_science)

cfg <- createStaticTestConfig()
solution <- Static(cfg, constraints_science)
summary(solution)
summary(solution, simple = TRUE)

cfg <- createShadowTestConfig()
solution <- Shadow(cfg, constraints_science, true_theta = seq(-1, 1, 1))
summary(solution)
summary(solution, simple = TRUE)

Summary classes

Description

Summary classes

Class 'test_cluster': data cache for simulations

Description

test_cluster is an S4 class for representing data cache for running simulations. Despite the name, this class does not represent a series of tests and is not related to a series of tests. That is, test length is not stored in this class. This class is only kept for backwards compatibility.

Slots

nt

the number of test objects in this cluster.

tests

the list containing test objects.

names

test ID strings for each test object.

Basic operators for test objects

Description

Create a subset of a test object.

Usage

subsetTest(x, i = NULL)

## S4 method for signature 'test,ANY'
x[i, j, ..., drop = TRUE]

Arguments

x

a test object.
i

item indices to use in subsetting.
j, drop, ...

not used, exists for compatibility.

Class 'test': data cache for simulations

Description

test is an S4 class for representing data cache for running simulations. Despite the name, this class does not represent a test and is not related to a test. That is, test length is not stored in this class. This class is only kept for backwards compatibility. The functionality of this class is superseded by simulation_data_cache.

Slots

pool

the item_pool object.

theta

the theta grid to use as quadrature points.

prob

the list containing item response probabilities.

info

the matrix containing item information values.

true_theta

(optional) the true theta values.

data

(optional) the matrix containing item responses.

Open TestDesign app

Description

TestDesign is a caller function for opening the Shiny interface of TestDesign package.

Usage

TestDesign()

Examples

## Not run: 
if (interactive()) {
  TestDesign()
}

## End(Not run)

Test solver

Description

Test solver

Usage

testSolver(solver)

Arguments

solver

a solver package name. Accepts lpSolve, Rsymphony, highs, gurobi, Rglpk.

Value

empty string "" if solver works. A string containing error messages otherwise.

(C++) Calculate a theta estimate using EAP (expected a posteriori) method

Description

theta_EAP() and theta_EAP_matrix() are functions for calculating a theta estimate using EAP (expected a posteriori) method.

Usage

theta_EAP(theta_grid, item_parm, resp, ncat, model, prior, prior_parm)

theta_EAP_matrix(theta_grid, item_parm, resp, ncat, model, prior, prior_parm)

Arguments

theta_grid

theta quadrature points.
item_parm

a matrix containing item parameters.
resp

responses on each item. Must be a vector for theta_EAP(), and a matrix for theta_EAP_matrix(). Each row should represent an examinee.
ncat

a vector containing the number of response categories of each item.
model

a vector indicating item models of each item, using

  • 1: 1PL model

  • 2: 2PL model

  • 3: 3PL model

  • 4: PC model

  • 5: GPC model

  • 6: GR model

prior

an integer indicating the type of prior distribution, using

  • 1: normal distribution

  • 2: uniform distribution

prior_parm

a vector containing parameters for the prior distribution.

Details

theta_EAP() and theta_EAP_matrix() are designed for multiple items.

theta_EAP() is designed for one examinee, and theta_EAP_matrix() is designed for multiple examinees.

Currently supports unidimensional models.

Examples

# item parameters
item_parm <- matrix(c(
  1, NA,   NA,
  1,  2,   NA,
  1,  2, 0.25,
  0,  1,   NA,
  2,  0,    1,
  2,  0,    2),
  nrow = 6,
  byrow = TRUE
)

ncat  <- c(2, 2, 2, 3, 3, 3)
model <- c(1, 2, 3, 4, 5, 6)

# simulate response
item_parm <- as.data.frame(item_parm)
item_parm <- cbind(101:106, 1:6, item_parm)
pool <- loadItemPool(item_parm)
true_theta <- seq(-3, 3, 1)
resp <- simResp(pool, true_theta)

theta_grid <- matrix(seq(-3, 3, .1), , 1)

theta_EAP(theta_grid, pool@ipar, resp[1, ], ncat, model, 1, c(1, 2))
theta_EAP_matrix(theta_grid, pool@ipar, resp, ncat, model, 1, c(1, 2))

(C++) Calculate a theta estimate using EB (Empirical Bayes) method

Description

theta_EB_single() and theta_EB() are functions for calculating a theta estimate using EB (Empirical Bayes) method.

Usage

theta_EB(
  nx,
  theta_init,
  theta_prop,
  item_parm,
  resp,
  ncat,
  model,
  prior,
  prior_parm
)

theta_EB_single(
  nx,
  theta_init,
  theta_prop,
  item_parm,
  resp,
  ncat,
  model,
  prior,
  prior_parm
)

Arguments

nx

the number of MCMC draws.
theta_init

the initial estimate to use.
theta_prop

the SD of the proposal distribution.
item_parm

a matrix containing item parameters. Each row should represent an item.
resp

a vector containing responses on each item.
ncat

a vector containing the number of response categories of each item.
model

a vector indicating item models of each item, using

  • 1: 1PL model

  • 2: 2PL model

  • 3: 3PL model

  • 4: PC model

  • 5: GPC model

  • 6: GR model

prior

an integer indicating the type of prior distribution, using

  • 1: normal distribution

  • 2: uniform distribution

prior_parm

a vector containing parameters for the prior distribution.

Details

theta_EB_single() is designed for one item, and theta_EB() is designed for multiple items.

Currently supports unidimensional models.

Examples

# item parameters
item_parm <- matrix(c(
  1, NA,   NA,
  1,  2,   NA,
  1,  2, 0.25,
  0,  1,   NA,
  2,  0,    1,
  2,  0,    2),
  nrow = 6,
  byrow = TRUE
)

ncat  <- c(2, 2, 2, 3, 3, 3)
model <- c(1, 2, 3, 4, 5, 6)
resp  <- c(0, 1, 0, 1, 0, 1)

nx <- 100
theta_init <- 0
theta_prop <- 1.0
set.seed(1)
theta_EB_single(nx, theta_init, theta_prop, item_parm[1, ], resp[1], ncat[1], model[1], 1, c(0, 1))
theta_EB(nx, theta_init, theta_prop, item_parm, resp, ncat, model, 1, c(0, 1))

(C++) Calculate a theta estimate using FB (Full Bayes) method

Description

theta_FB_single() and theta_FB() are functions for calculating a theta estimate using FB (Full Bayes) method.

Usage

theta_FB(
  nx,
  theta_init,
  theta_prop,
  items_list,
  item_init,
  resp,
  ncat,
  model,
  prior,
  prior_parm
)

theta_FB_single(
  nx,
  theta_init,
  theta_prop,
  item_mcmc,
  item_init,
  resp,
  ncat,
  model,
  prior,
  prior_parm
)

Arguments

nx

the number of MCMC draws.
theta_init

the initial estimate to use.
theta_prop

the SD of the proposal distribution.
item_init

item parameter estimates. Must be a vector for theta_FB_single(), and a matrix for theta_FB().
resp

a vector containing responses on each item.
ncat

a vector containing the number of response categories of each item.
model

a vector indicating item models of each item, using

  • 1: 1PL model

  • 2: 2PL model

  • 3: 3PL model

  • 4: PC model

  • 5: GPC model

  • 6: GR model

prior

an integer indicating the type of prior distribution, using

  • 1: normal distribution

  • 2: uniform distribution

prior_parm

a vector containing parameters for the prior distribution.
item_mcmc, items_list

sampled item parameters. Must be a matrix for theta_FB_single(), and a list of matrices for theta_FB().

Details

theta_FB_single() is designed for one item, and theta_FB() is designed for multiple items.

Currently supports unidimensional models.

Toggle constraints

Description

toggleConstraints is a function for toggling individual constraints in a constraints object.

Usage

toggleConstraints(object, on = NULL, off = NULL)

Arguments

object

a constraints object from loadConstraints.
on

constraint indices to mark as active. Also accepts character IDs.
off

constraint indices to mark as inactive. Also accepts character IDs.

Value

toggleConstraints returns the updated constraints object.

Examples

constraints_science2 <- toggleConstraints(constraints_science, off = 32:36)
constraints_science3 <- toggleConstraints(constraints_science2, on = 32:36)
constraints_science4 <- toggleConstraints(constraints_science, off = "C32")