Overview of simcdm
Author: James Joseph Balamuta
Date: 2019-03-09

# Overview of `simcdm`

#### James Joseph Balamuta

#### 2019-03-09

# Overview

Within this document, we highlight the different features of the `simcdm` package as it relates to simulating cognitive diagnostic modeling data.

## Notation

For consistency, we aim to use the following notation.

Denoting individuals:

- \(N\) is the total number of individuals taking the assessment.
- \(i\) is the current individual.

Denoting items:

- \(J\) is the total number of items on the assessment.
- \(j\) is the current item
- \(Y_{ij}\) is the observed binary response for individual \(i\) (\(1\leq i \leq N\)) to item \(j\) (\(1\leq j\leq J\)).
- \(s_j\) is the probability of slipping on item \(j\).
- \(g_j\) is the probability of guessing on item \(j\).

Denoting attributes:

- \(K\) is the total number of attributes for the assessment item.
- \(k\) is the current attribute.
- \(\boldsymbol\alpha_i=\left(\alpha_{i1},\dots,\alpha_{iK}\right)^\prime\) where \(\boldsymbol\alpha_i\in \left\{0,1\right\}^K\) and \(\alpha_{ik}\) is the latent binary attribute for individual \(i\) on attribute \(k\) (\(1\leq k\leq K\)).

Denoting the skill/attribute “Q” matrix:

- \(\boldsymbol q_{j}=\left(q_{j1},\dots,q_{jK}\right)^\prime\) be the \(j\)th row of \(\boldsymbol Q\) such that \(q_{jk}=1\) if attribute \(k\) is required for item \(j\) and zero otherwise.

# Usage

To use `simcdm`, please load the package.

    library(simcdm)

## Matrix Simulation

Simulations within this section are done underneath the following settings.

    # Set a seed for reproducibility
    set.seed(888)

    # Setup Parameters
    N = 15   # Number of Examinees / Subjects
    J = 10   # Number of Items
    K = 2    # Number of Skills / Attributes

### Identifiable Q Matrix Simulation

Simulate an identifiable \(Q\) matrix (\(J\) items by \(K\) skills).

    # Set a seed for reproducibility
    set.seed(1512)

    # Simulate an identifiable Q matrix
    Q = sim_q_matrix(J, K)
    Q

    ##       [,1] [,2]
    ##  [1,]    1    0
    ##  [2,]    1    0
    ##  [3,]    0    1
    ##  [4,]    0    1
    ##  [5,]    0    1
    ##  [6,]    0    1
    ##  [7,]    1    1
    ##  [8,]    1    0
    ##  [9,]    1    0
    ## [10,]    0    1

### \(\eta\) Matrix Simulation

Create the ideal response matrix for each trait (\(J\) items by \(2^K\) latent classes).

    # Set a seed for reproducibility
    set.seed(4421)

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

    ##       [,1] [,2] [,3] [,4]
    ##  [1,]    0    0    1    1
    ##  [2,]    0    0    1    1
    ##  [3,]    0    1    0    1
    ##  [4,]    0    1    0    1
    ##  [5,]    0    1    0    1
    ##  [6,]    0    1    0    1
    ##  [7,]    0    0    0    1
    ##  [8,]    0    0    1    1
    ##  [9,]    0    0    1    1
    ## [10,]    0    1    0    1

### Attribute profile simulation

Generate latent attribute profile classes (\(2^K\) latent classes by \(K\) skills).

    # Create a listing of all attribute classes
    class_alphas = attribute_classes(K)
    class_alphas

    ##      [,1] [,2]
    ## [1,]    0    0
    ## [2,]    0    1
    ## [3,]    1    0
    ## [4,]    1    1

Generate latent attribute profile class for each subject (\(N\) subjects by \(K\) skills).

    # Set a seed for reproducibility
    set.seed(5126)

    # Create attributes for a subject
    subject_alphas = sim_subject_attributes(N, K)
    subject_alphas

    ##       [,1] [,2]
    ##  [1,]    0    0
    ##  [2,]    1    1
    ##  [3,]    1    0
    ##  [4,]    1    1
    ##  [5,]    0    0
    ##  [6,]    0    1
    ##  [7,]    0    1
    ##  [8,]    1    0
    ##  [9,]    0    1
    ## [10,]    0    1
    ## [11,]    0    1
    ## [12,]    0    1
    ## [13,]    1    0
    ## [14,]    0    0
    ## [15,]    1    0

    # Equivalent to:
    # subject_alphas = class_alphas[sample(2 ^ K, N, replace = TRUE),]

### DINA Simulation

Simulations within this section are done underneath the following settings.

    # Set a seed for reproducibility
    set.seed(888)

    # Setup Parameters
    N = 15   # Number of Examinees / Subjects
    J = 10   # Number of Items
    K = 2    # Number of Skills / Attributes

    # Assign slipping and guessing values for each item
    ss = gs = rep(.2, J)

    # Simulate identifiable Q matrix
    Q = sim_q_matrix(J, K)

    # Simulate subject attributes
    subject_alphas = sim_subject_attributes(N, K)

### DINA Item Simulation

Simulate item data, \(Y\), under DINA model (\(N\) by \(J\))

    # Set a seed for reproducibility
    set.seed(2019)

    # Simulate items under the DINA model
    items_dina = sim_dina_items(subject_alphas, Q, ss, gs)
    items_dina

    ##       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
    ##  [1,]    0    0    0    1    1    0    0    0    1     1
    ##  [2,]    1    1    1    0    1    1    1    1    0     1
    ##  [3,]    0    0    0    0    1    0    0    0    1     1
    ##  [4,]    1    1    1    0    0    0    1    1    1     0
    ##  [5,]    1    1    1    1    1    1    1    1    1     1
    ##  [6,]    1    1    1    1    0    1    0    1    0     1
    ##  [7,]    0    0    0    1    1    0    0    0    0     1
    ##  [8,]    1    0    1    1    1    0    0    1    0     1
    ##  [9,]    1    1    1    1    0    1    1    1    1     1
    ## [10,]    0    1    0    1    1    0    0    0    1     1
    ## [11,]    1    1    1    1    1    1    1    0    1     1
    ## [12,]    0    0    0    0    1    0    0    0    1     0
    ## [13,]    0    0    0    0    1    0    0    0    1     1
    ## [14,]    1    0    0    1    1    0    0    0    0     0
    ## [15,]    1    1    0    1    0    0    1    1    0     0

### DINA Attribute Simulation

Simulate attribute data under DINA model (\(N\) by \(J\))

    # Set a seed for reproducibility
    set.seed(51823)

    # Simulate attributes under the DINA model
    attributes = sim_dina_attributes(subject_alphas, Q)
    attributes

    ##       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
    ##  [1,]    0    0    0    1    1    0    0    0    1     1
    ##  [2,]    1    1    1    1    1    1    1    1    1     1
    ##  [3,]    0    0    0    1    1    0    0    0    1     1
    ##  [4,]    1    1    1    0    0    0    1    1    0     0
    ##  [5,]    1    1    1    1    1    1    1    1    1     1
    ##  [6,]    1    1    1    1    1    1    1    1    1     1
    ##  [7,]    0    0    0    1    1    0    0    0    1     1
    ##  [8,]    1    1    1    1    1    1    1    1    1     1
    ##  [9,]    1    1    1    1    1    1    1    1    1     1
    ## [10,]    0    0    0    1    1    0    0    0    1     1
    ## [11,]    1    1    1    1    1    1    1    1    1     1
    ## [12,]    0    0    0    1    1    0    0    0    1     1
    ## [13,]    0    0    0    1    1    0    0    0    1     1
    ## [14,]    0    0    0    0    0    0    0    0    0     0
    ## [15,]    1    1    1    0    0    0    1    1    0     0

## rRUM Simulation

The rRUM simulations are done using the following settings.

    # Set a seed for reproducibility
    set.seed(888)

    # Setup Parameters
    N = 15   # Number of Examinees / Subjects
    J = 10   # Number of Items
    K = 2    # Number of Skills / Attributes

    # The probabilities of answering each item correctly for individuals
    # who do not lack any required attribute
    pistar = rep(.9, J)

    # Simulate an identifiable Q matrix
    Q = sim_q_matrix(J, K)

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

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

    # Generate latent attribute profile with custom probability (N subjects by K skills)
    subject_alphas = sim_subject_attributes(N, K, prob = pis)

    # Equivalent to:
    # class_alphas = attribute_classes(K)
    # subject_alphas = class_alphas[sample(2 ^ K, N, replace = TRUE, prob = pis),]

### Simulate rRUM items

Simulate rRUM item data \(Y\) (\(N\) by \(J\))

    # Set a seed for reproducibility
    set.seed(912)

    # Generate rRUM items
    rrum_items = sim_rrum_items(Q, rstar, pistar, subject_alphas)
    rrum_items

    ##       [,1] [,2] [,3] [,4] [,5] [,6] [,7] [,8] [,9] [,10]
    ##  [1,]    1    1    1    0    1    0    1    1    1     0
    ##  [2,]    1    1    0    1    1    0    1    0    0     0
    ##  [3,]    0    1    1    1    0    1    1    1    0     0
    ##  [4,]    0    1    1    1    0    1    1    1    1     0
    ##  [5,]    1    1    0    0    1    0    1    1    1     0
    ##  [6,]    1    1    1    1    0    0    1    1    0     1
    ##  [7,]    1    0    1    0    1    0    0    0    0     1
    ##  [8,]    0    0    1    0    0    0    1    0    0     1
    ##  [9,]    0    1    0    1    1    1    0    1    1     0
    ## [10,]    1    1    1    1    0    0    0    1    1     0
    ## [11,]    1    1    1    0    0    0    1    1    0     1
    ## [12,]    0    1    0    1    1    0    1    1    1     1
    ## [13,]    1    0    1    1    1    0    1    1    1     1
    ## [14,]    1    0    1    1    1    1    1    1    1     1
    ## [15,]    0    0    1    0    1    1    1    1    1     1