Package 'simcdm' reference manual

Title:	Simulate Cognitive Diagnostic Model ('CDM') Data
Description:	Provides efficient R and 'C++' routines to simulate cognitive diagnostic model data for Deterministic Input, Noisy "And" Gate ('DINA') and reduced Reparameterized Unified Model ('rRUM') from Culpepper and Hudson (2017) <doi: 10.1177/0146621617707511>, Culpepper (2015) <doi:10.3102/1076998615595403>, and de la Torre (2009) <doi:10.3102/1076998607309474>.
Authors:	James Joseph Balamuta [aut, cre, cph] , Steven Andrew Culpepper [aut, cph] , Aaron Hudson [ctb, cph]
Maintainer:	James Joseph Balamuta <[email protected]>
License:	GPL (>= 2)
Version:	0.1.2
Built:	2024-11-06 04:43:03 UTC
Source:	https://github.com/tmsalab/simcdm

simcdm: Simulate Cognitive Diagnostic Model ('CDM') Data

Description

Provides efficient R and 'C++' routines to simulate cognitive diagnostic model data for Deterministic Input, Noisy "And" Gate ('DINA') and reduced Reparameterized Unified Model ('rRUM') from Culpepper and Hudson (2017) doi: 10.1177/0146621617707511, Culpepper (2015) doi:10.3102/1076998615595403, and de la Torre (2009) doi:10.3102/1076998607309474.

Author(s)

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

Authors:

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

Other contributors:

Aaron Hudson [email protected] (ORCID) [contributor, copyright holder]

Constructs Unique Attribute Pattern Map

Description

Computes the powers of 2 from $0$ up to $K - 1$ for $K$ -dimensional attribute pattern.

Usage

attribute_bijection(K)
attribute_bijection(K)

Arguments

`K`	Number of Attributes.

Value

A vec with length $K$ detailing the power's of 2.

Author(s)

Steven Andrew Culpepper and James Joseph Balamuta

Examples

## Construct an attribute bijection ----
biject = attribute_bijection(3)
## Construct an attribute bijection ----
biject = attribute_bijection(3)

Simulate all the Latent Attribute Profile $\mathbf{\alpha}_c$ in Matrix form

Description

Generate the $\mathbf{\alpha}_c = (\alpha_{c1}, \ldots, \alpha_{cK})'$ attribute profile matrix for members of class $c$ such that $\alpha_{ck}$ ' is 1 if members of class $c$ possess skill $k$ and zero otherwise.

Usage

attribute_classes(K)
attribute_classes(K)

Arguments

`K`	Number of Attributes

Value

A $2^K$ by $K$ matrix of latent classes corresponding to entry $c$ of $pi$ based upon mastery and nonmastery of the $K$ skills.

Author(s)

James Joseph Balamuta and Steven Andrew Culpepper

Examples

## Simulate Attribute Class Matrix ----

# Define number of attributes
K = 3

# Generate an Latent Attribute Profile (Alpha) Matrix
alphas = attribute_classes(K)
## Simulate Attribute Class Matrix ----

# Define number of attributes
K = 3

# Generate an Latent Attribute Profile (Alpha) Matrix
alphas = attribute_classes(K)

Perform an Inverse Bijection of an Integer to Attribute Pattern

Description

Convert an integer between $0$ and $2^{K-1}$ to $K$ -dimensional attribute pattern.

Usage

attribute_inv_bijection(K, CL)
attribute_inv_bijection(K, CL)

Arguments

`K`	Number of Attributes.
`CL`	An `integer` between $0$ and $2^{K-1}$

Value

A $K$ -dimensional vector with an attribute pattern corresponding to CL.

Author(s)

Steven Andrew Culpepper and James Joseph Balamuta

Examples

## Construct an attribute inversion bijection ----
inv_biject1 = attribute_inv_bijection(5, 1)
inv_biject2 = attribute_inv_bijection(5, 2)
## Construct an attribute inversion bijection ----
inv_biject1 = attribute_inv_bijection(5, 1)
inv_biject2 = attribute_inv_bijection(5, 2)

Simulate a DINA Model's $\eta$ Matrix

Description

Generates a DINA model's $\eta$ matrix based on alphas and the $\mathbf{Q}$ matrix.

Usage

sim_dina_attributes(alphas, Q)
sim_dina_attributes(alphas, Q)

Arguments

`alphas`	A $N$ by $K$ `matrix` of latent attributes.
`Q`	A $J$ by $K$ `matrix` indicating which skills are required for which items.

Value

The $\eta$ matrix with dimensions $N \times J$ under the DINA model.

Author(s)

Steven Andrew Culpepper and James Joseph Balamuta

Examples

N = 200
K = 5
J = 30
delta0 = rep(1, 2 ^ K)

# Creating Q matrix
Q = matrix(rep(diag(K), 2), 2 * K, K, byrow = TRUE)
for (mm in 2:K) {
  temp = combn(seq_len(K), m = mm)
  tempmat = matrix(0, ncol(temp), K)
  for (j in seq_len(ncol(temp)))
    tempmat[j, temp[, j]] = 1
  Q = rbind(Q, tempmat)
}
Q = Q[seq_len(J), ]

# Setting item parameters and generating attribute profiles
ss = gs = rep(.2, J)
PIs = rep(1 / (2 ^ K), 2 ^ K)
CLs = c((1:(2 ^ K)) %*% rmultinom(n = N, size = 1, prob = PIs))

# Defining matrix of possible attribute profiles
As = rep(0, K)
for (j in seq_len(K)) {
  temp = combn(1:K, m = j)
  tempmat = matrix(0, ncol(temp), K)
  for (j in seq_len(ncol(temp)))
    tempmat[j, temp[, j]] = 1
  As = rbind(As, tempmat)
}
As = as.matrix(As)

# Sample true attribute profiles
Alphas = As[CLs, ]

# Simulate item data under DINA model 
dina_items = sim_dina_items(Alphas, Q, ss, gs)

# Simulate attribute data under DINA model 
dina_attributes = sim_dina_attributes(Alphas, Q)
N = 200
K = 5
J = 30
delta0 = rep(1, 2 ^ K)

# Creating Q matrix
Q = matrix(rep(diag(K), 2), 2 * K, K, byrow = TRUE)
for (mm in 2:K) {
  temp = combn(seq_len(K), m = mm)
  tempmat = matrix(0, ncol(temp), K)
  for (j in seq_len(ncol(temp)))
    tempmat[j, temp[, j]] = 1
  Q = rbind(Q, tempmat)
}
Q = Q[seq_len(J), ]

# Setting item parameters and generating attribute profiles
ss = gs = rep(.2, J)
PIs = rep(1 / (2 ^ K), 2 ^ K)
CLs = c((1:(2 ^ K)) %*% rmultinom(n = N, size = 1, prob = PIs))

# Defining matrix of possible attribute profiles
As = rep(0, K)
for (j in seq_len(K)) {
  temp = combn(1:K, m = j)
  tempmat = matrix(0, ncol(temp), K)
  for (j in seq_len(ncol(temp)))
    tempmat[j, temp[, j]] = 1
  As = rbind(As, tempmat)
}
As = as.matrix(As)

# Sample true attribute profiles
Alphas = As[CLs, ]

# Simulate item data under DINA model 
dina_items = sim_dina_items(Alphas, Q, ss, gs)

# Simulate attribute data under DINA model 
dina_attributes = sim_dina_attributes(Alphas, Q)

Simulate Binary Responses for a DINA Model

Description

Generate the dichotomous item matrix for a DINA Model.

Usage

sim_dina_class(N, J, CLASS, ETA, gs, ss)
sim_dina_class(N, J, CLASS, ETA, gs, ss)

Arguments

`N`	Number of Observations
`J`	Number of Assessment Items
`CLASS`	Does the individual possess all the necessary attributes?
`ETA`	$\eta$ Matrix containing indicators.
`gs`	A `vec` describing the probability of guessing or the probability subject correctly answers item $j$ when at least one attribute is lacking.
`ss`	A `vec` describing the probability of slipping or the probability of an incorrect response for individuals with all of the required attributes

Value

A dichotomous item matrix with dimensions $N \times J$ .

Author(s)

Steven Andrew Culpepper and James Joseph Balamuta

Examples

# Set 
N       = 100
rho     = 0
K       = 3

# Fixed Number of Assessment Items for Q
J = 18

# Specify Q
qbj = c(4, 2, 1, 4, 2, 1, 4, 2, 1, 6, 5, 3, 6, 5, 3, 7, 7, 7)

# Fill Q Matrix
Q = matrix(, J, K)
for (j in seq_len(J)) {
  Q[j,] = attribute_inv_bijection(K, qbj[j])
}

# Item parm vals
ss = gs = rep(.2, J)

# Generating attribute classes depending on correlation
if (rho == 0) {
  PIs = rep(1 / (2 ^ K), 2 ^ K)
  CLs = c(seq_len(2 ^ K) %*% rmultinom(n = N, size = 1, prob = PIs)) - 1
}

if (rho > 0) {
  Z = matrix(rnorm(N * K), N, K)
  Sig = matrix(rho, K, K)
  diag(Sig) = 1
  X = Z %*% chol(Sig)
  thvals = matrix(rep(0, K), N, K, byrow = T)
  Alphas = 1 * (X > thvals)
  CLs = Alphas %*% attribute_bijection(K)
}

# Simulate data under DINA model
ETA = sim_eta_matrix(K, J, Q)
Y_sim = sim_dina_class(N, J, CLs, ETA, gs, ss)
# Set 
N       = 100
rho     = 0
K       = 3

# Fixed Number of Assessment Items for Q
J = 18

# Specify Q
qbj = c(4, 2, 1, 4, 2, 1, 4, 2, 1, 6, 5, 3, 6, 5, 3, 7, 7, 7)

# Fill Q Matrix
Q = matrix(, J, K)
for (j in seq_len(J)) {
  Q[j,] = attribute_inv_bijection(K, qbj[j])
}

# Item parm vals
ss = gs = rep(.2, J)

# Generating attribute classes depending on correlation
if (rho == 0) {
  PIs = rep(1 / (2 ^ K), 2 ^ K)
  CLs = c(seq_len(2 ^ K) %*% rmultinom(n = N, size = 1, prob = PIs)) - 1
}

if (rho > 0) {
  Z = matrix(rnorm(N * K), N, K)
  Sig = matrix(rho, K, K)
  diag(Sig) = 1
  X = Z %*% chol(Sig)
  thvals = matrix(rep(0, K), N, K, byrow = T)
  Alphas = 1 * (X > thvals)
  CLs = Alphas %*% attribute_bijection(K)
}

# Simulate data under DINA model
ETA = sim_eta_matrix(K, J, Q)
Y_sim = sim_dina_class(N, J, CLs, ETA, gs, ss)

Simulation Responses from the DINA model

Description

Sample responses from the DINA model for given attribute profiles, Q matrix, and item parmeters. Returns a matrix of dichotomous responses generated under DINA model.

Usage

sim_dina_items(alphas, Q, ss, gs)
sim_dina_items(alphas, Q, ss, gs)

Arguments

`alphas`	A $N$ by $K$ `matrix` of latent attributes.
`Q`	A $J$ by $K$ `matrix` indicating which skills are required for which items.
`ss`	A $J$ `vector` of item slipping parameters.
`gs`	A $J$ `vector` of item guessing parameters.

Value

A $N$ by $J$ matrix of responses from the DINA model.

Author(s)

Steven Andrew Culpepper and James Joseph Balamuta

Examples

N = 200
K = 5
J = 30
delta0 = rep(1, 2 ^ K)

# Creating Q matrix
Q = matrix(rep(diag(K), 2), 2 * K, K, byrow = TRUE)
for (mm in 2:K) {
  temp = combn(seq_len(K), m = mm)
  tempmat = matrix(0, ncol(temp), K)
  for (j in seq_len(ncol(temp)))
    tempmat[j, temp[, j]] = 1
  Q = rbind(Q, tempmat)
}
Q = Q[seq_len(J), ]

# Setting item parameters and generating attribute profiles
ss = gs = rep(.2, J)
PIs = rep(1 / (2 ^ K), 2 ^ K)
CLs = c((1:(2 ^ K)) %*% rmultinom(n = N, size = 1, prob = PIs))

# Defining matrix of possible attribute profiles
As = rep(0, K)
for (j in seq_len(K)) {
  temp = combn(1:K, m = j)
  tempmat = matrix(0, ncol(temp), K)
  for (j in seq_len(ncol(temp)))
    tempmat[j, temp[, j]] = 1
  As = rbind(As, tempmat)
}
As = as.matrix(As)

# Sample true attribute profiles
Alphas = As[CLs, ]

# Simulate item data under DINA model 
dina_items = sim_dina_items(Alphas, Q, ss, gs)

# Simulate attribute data under DINA model 
dina_attributes = sim_dina_attributes(Alphas, Q)
N = 200
K = 5
J = 30
delta0 = rep(1, 2 ^ K)

# Creating Q matrix
Q = matrix(rep(diag(K), 2), 2 * K, K, byrow = TRUE)
for (mm in 2:K) {
  temp = combn(seq_len(K), m = mm)
  tempmat = matrix(0, ncol(temp), K)
  for (j in seq_len(ncol(temp)))
    tempmat[j, temp[, j]] = 1
  Q = rbind(Q, tempmat)
}
Q = Q[seq_len(J), ]

# Setting item parameters and generating attribute profiles
ss = gs = rep(.2, J)
PIs = rep(1 / (2 ^ K), 2 ^ K)
CLs = c((1:(2 ^ K)) %*% rmultinom(n = N, size = 1, prob = PIs))

# Defining matrix of possible attribute profiles
As = rep(0, K)
for (j in seq_len(K)) {
  temp = combn(1:K, m = j)
  tempmat = matrix(0, ncol(temp), K)
  for (j in seq_len(ncol(temp)))
    tempmat[j, temp[, j]] = 1
  As = rbind(As, tempmat)
}
As = as.matrix(As)

# Sample true attribute profiles
Alphas = As[CLs, ]

# Simulate item data under DINA model 
dina_items = sim_dina_items(Alphas, Q, ss, gs)

# Simulate attribute data under DINA model 
dina_attributes = sim_dina_attributes(Alphas, Q)

Generate ideal response $\eta$ Matrix

Description

Creates the ideal response matrix for each trait

Usage

sim_eta_matrix(K, J, Q)
sim_eta_matrix(K, J, Q)

Arguments

`K`	Number of Attribute Levels
`J`	Number of Assessment Items
`Q`	Q Matrix with dimensions $K \times J$ .

Value

A mat with dimensions $J \times 2^K$ .

Author(s)

Steven Andrew Culpepper and James Joseph Balamuta

Examples

## Simulation Settings ----

# Fixed Number of Assessment Items for Q
J = 18

# Fixed Number of Attributes for Q
K = 3

## Pre-specified configuration ----

# Specify Q
qbj = c(4, 2, 1, 4, 2, 1, 4, 2, 1, 6, 5, 3, 6, 5, 3, 7, 7, 7)

# Fill Q Matrix
Q = matrix(, J, K)
for (j in seq_len(J)) {
  Q[j,] = attribute_inv_bijection(K, qbj[j])
}

# Create an eta matrix
ETA = sim_eta_matrix(K, J, Q)

## Random generation of Q matrix with ETA matrix ----

# Construct a random q matrix
Q_sim = sim_q_matrix(J, K)

# Generate the eta matrix
ETA_gen = sim_eta_matrix(K, J, Q_sim)
## Simulation Settings ----

# Fixed Number of Assessment Items for Q
J = 18

# Fixed Number of Attributes for Q
K = 3

## Pre-specified configuration ----

# Specify Q
qbj = c(4, 2, 1, 4, 2, 1, 4, 2, 1, 6, 5, 3, 6, 5, 3, 7, 7, 7)

# Fill Q Matrix
Q = matrix(, J, K)
for (j in seq_len(J)) {
  Q[j,] = attribute_inv_bijection(K, qbj[j])
}

# Create an eta matrix
ETA = sim_eta_matrix(K, J, Q)

## Random generation of Q matrix with ETA matrix ----

# Construct a random q matrix
Q_sim = sim_q_matrix(J, K)

# Generate the eta matrix
ETA_gen = sim_eta_matrix(K, J, Q_sim)

Generate a Random Identifiable Q Matrix

Description

Simulates a Q matrix containing three identity matrices after a row permutation that is identifiable.

Usage

sim_q_matrix(J, K)
sim_q_matrix(J, K)

Arguments

`J`	Number of Items
`K`	Number of Attributes

Value

A dichotomous matrix for Q.

Author(s)

Steven Andrew Culpepper and James Joseph Balamuta

Examples

## Simulate identifiable Q matrices ----

# 7 items and 2 attributes
q_matrix_j7_k2 = sim_q_matrix(7, 2)

# 10 items and 3 attributes
q_matrix_j10_k3 = sim_q_matrix(10, 3)
## Simulate identifiable Q matrices ----

# 7 items and 2 attributes
q_matrix_j7_k2 = sim_q_matrix(7, 2)

# 10 items and 3 attributes
q_matrix_j10_k3 = sim_q_matrix(10, 3)

Generate data from the rRUM

Description

Randomly generate response data according to the reduced Reparameterized Unified Model (rRUM).

Usage

sim_rrum_items(Q, rstar, pistar, alpha)
sim_rrum_items(Q, rstar, pistar, alpha)

Arguments

`Q`	A `matrix` with $J$ rows and $K$ columns indicating which attributes are required to answer each of the items, where $J$ represents the number of items and $K$ the number of attributes. An entry of 1 indicates attribute $k$ is required to answer item $j$ . An entry of one indicates attribute $k$ is not required.
`rstar`	A `matrix` a matrix with $J$ rows and $K$ columns indicating the penalties for failing to have each of the required attributes, where $J$ represents the number of items and $K$ the number of attributes. `rstar` and `Q` must share the same 0 entries.
`pistar`	A `vector` of length $J$ indicating the probabiliies of answering each item correctly for individuals who do not lack any required attribute, where $J$ represents the number of items.
`alpha`	A `matrix` with $N$ rows and $K$ columns indicating the subjects attribute acquisition, where $K$ represents the number of attributes. An entry of 1 indicates individual $i$ has attained attribute $k$ . An entry of 0 indicates the attribute has not been attained.

Value

Y A matrix with $N$ rows and $J$ columns indicating the indviduals' responses to each of the items, where $J$ represents the number of items.

Author(s)

Steven Andrew Culpepper, Aaron Hudson, and James Joseph Balamuta

References

Culpepper, S. A. & Hudson, A. (In Press). An improved strategy for Bayesian estimation of the reduced reparameterized unified model. Applied Psychological Measurement.

Hudson, A., Culpepper, S. A., & Douglas, J. (2016, July). Bayesian estimation of the generalized NIDA model with Gibbs sampling. Paper presented at the annual International Meeting of the Psychometric Society, Asheville, North Carolina.

Examples

# Set seed for reproducibility
set.seed(217)

# Define Simulation Parameters
N = 1000 # number of individuals
J = 6 # number of items
K = 2 # number of attributes

# Matrix where rows represent attribute classes
As = attribute_classes(K) 

# Latent Class probabilities
pis = c(.1, .2, .3, .4) 

# Q Matrix
Q = rbind(c(1, 0),
          c(0, 1),
          c(1, 0),
          c(0, 1),
          c(1, 1),
          c(1, 1)
    )
    
# The probabiliies of answering each item correctly for individuals 
# who do not lack any required attribute
pistar = rep(.9, J)

# Penalties for failing to have each of the required attributes
rstar  = .5 * Q

# Randomized alpha profiles
alpha  = As[sample(1:(K ^ 2), N, replace = TRUE, pis),]

# Simulate data
rrum_items = sim_rrum_items(Q, rstar, pistar, alpha)
# Set seed for reproducibility
set.seed(217)

# Define Simulation Parameters
N = 1000 # number of individuals
J = 6 # number of items
K = 2 # number of attributes

# Matrix where rows represent attribute classes
As = attribute_classes(K) 

# Latent Class probabilities
pis = c(.1, .2, .3, .4) 

# Q Matrix
Q = rbind(c(1, 0),
          c(0, 1),
          c(1, 0),
          c(0, 1),
          c(1, 1),
          c(1, 1)
    )
    
# The probabiliies of answering each item correctly for individuals 
# who do not lack any required attribute
pistar = rep(.9, J)

# Penalties for failing to have each of the required attributes
rstar  = .5 * Q

# Randomized alpha profiles
alpha  = As[sample(1:(K ^ 2), N, replace = TRUE, pis),]

# Simulate data
rrum_items = sim_rrum_items(Q, rstar, pistar, alpha)

Simulate Subject Latent Attribute Profiles $\mathbf{\alpha}_c$

Description

Generate a sample from the $\mathbf{\alpha}_c = (\alpha_{c1}, \ldots, \alpha_{cK})'$ attribute profile matrix for members of class $c$ such that $\alpha_{ck}$ ' is 1 if members of class $c$ possess skill $k$ and zero otherwise.

Usage

sim_subject_attributes(N, K, probs = NULL)
sim_subject_attributes(N, K, probs = NULL)

Arguments

`N`	Number of Observations
`K`	Number of Skills
`probs`	A `vector` of probabilities that sum to 1.

Value

A $N$ by $K$ matrix of latent classes corresponding to entry $c$ of $pi$ based upon mastery and nonmastery of the $K$ skills.

Author(s)

James Joseph Balamuta and Steven Andrew Culpepper

Examples

# Define number of subjects and attributes
N = 100
K = 3

# Generate a sample from the Latent Attribute Profile (Alpha) Matrix
# By default, we sample from a uniform distribution weighting of classes.
alphas_builtin = sim_subject_attributes(N, K)

# Generate a sample using custom probabilities from the
# Latent Attribute Profile (Alpha) Matrix
probs = rep(1 / (2 ^ K), 2 ^ K)
alphas_custom = sim_subject_attributes(N, K, probs)
# Define number of subjects and attributes
N = 100
K = 3

# Generate a sample from the Latent Attribute Profile (Alpha) Matrix
# By default, we sample from a uniform distribution weighting of classes.
alphas_builtin = sim_subject_attributes(N, K)

# Generate a sample using custom probabilities from the
# Latent Attribute Profile (Alpha) Matrix
probs = rep(1 / (2 ^ K), 2 ^ K)
alphas_custom = sim_subject_attributes(N, K, probs)

Package 'simcdm'

Help Index

simcdm: Simulate Cognitive Diagnostic Model ('CDM') Data

Description

Author(s)

See Also

Constructs Unique Attribute Pattern Map

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Simulate all the Latent Attribute Profile αc\mathbf{\alpha}_cαc​ in Matrix form

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Perform an Inverse Bijection of an Integer to Attribute Pattern

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Simulate a DINA Model's η\etaη Matrix

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Simulate Binary Responses for a DINA Model

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Simulation Responses from the DINA model

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Generate ideal response η\etaη Matrix

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Generate a Random Identifiable Q Matrix

Description

Usage

Arguments

Value

Author(s)

See Also

Examples

Generate data from the rRUM

Description

Usage

Arguments

Value

Author(s)

References

Examples

Simulate Subject Latent Attribute Profiles αc\mathbf{\alpha}_cαc​

Description

Simulate all the Latent Attribute Profile $\mathbf{\alpha}_c$ in Matrix form

Simulate a DINA Model's $\eta$ Matrix

Generate ideal response $\eta$ Matrix

Simulate Subject Latent Attribute Profiles $\mathbf{\alpha}_c$