Psychometric Models in `irt` Package"

In irt: Item Response Theory and Computerized Adaptive Testing Functions

knitr::opts_chunk$set(
  fig.width = 5,
  collapse = TRUE,
  comment = "#>"
)

This vignettes covers the psychometric models that are implemented in the irt package.

Item Models

| Name | Description | Parameters | |-------|-------------------------------------------------|---------------| | Rasch | Rasch Model | b | | 1PL | One-Parameter Logistic Model | b, D | | 2PL | Two-Parameter Logistic Model | a, b, D | | 3PL | Three-Parameter Logistic Model | a, b, c, D | | 4PL | Four-Parameter Logistic Model | a, b, c, d, D | | GRM | Graded Response Model | a, b, D | | PCM | Partial Credit Model | b | | GPCM | Generalized Partial Credit Model | a, b, D | | GPCM2 | Reparameterized Generalized Partial Credit Model | a, b, d, D |

Rasch Model

For an examinee $i$ with ability $\theta_i$, the probability of correct response to an item $j$ is:

$$P\left(X_{ij} = 1 | \theta_i;b_j\right) = \frac{e^{(\theta_i - b_j)}}{1 + e^{(\theta_i - b_j)}} $$

where , $b_j$ is the item difficulty (or threshold) of item $j$.

User needs to specify only the item difficulty parameter:

library(irt)

itm_rasch <- item(b = -1.29)
itm_rasch

The probability of each response option at $\theta = -.65$ is:

prob(ip = itm_rasch, theta = -0.65)

The item characteristic curve of this item is:

plot(itm_rasch)

One-Parameter Logistic Model

For an examinee $i$ with ability $\theta_i$, the probability of correct response to an item $j$ is:

$$P\left(X_{ij} = 1 | \theta_i;b_j\right) = \frac{e^{D(\theta_i - b_j)}}{1 + e^{D(\theta_i - b_j)}} $$

where , $b_j$ is the item difficulty (or threshold) of item $j$. D is the scaling constant (the default value is r irt:::DEFAULT_D_SCALING).

User needs to specify the following parameters:

itm_1pl <- item(b = 0.83, D = 1)
itm_1pl

The probability of each response option at $\theta = .73$ is:

prob(ip = itm_1pl, theta = 0.73)

The item characteristic curve of this item is:

plot(itm_1pl)

Two-Parameter Logistic Model

For an examinee $i$ with ability $\theta_i$, the probability of correct response to an item $j$ is:

$$P\left(X_{ij} = 1 | \theta_i;a_j, b_j\right) = \frac{e^{Da_j(\theta_i - b_j)}}{1 + e^{Da_j(\theta_i - b_j)}} $$

where $a_j$ is the item discrimination (or slope) of item $j$, $b_j$ is the item difficulty (or threshold). D is the scaling constant (the default value is r irt:::DEFAULT_D_SCALING).

User needs to specify all of the parameter values:

itm_2pl <- item(a = .94, b = -1.302, D = 1)
itm_2pl

The probability of each response option at $\theta = -0.53$ is:

prob(ip = itm_2pl, theta = -0.53)

The item characteristic curve of this item is:

plot(itm_2pl)

Three-Parameter Logistic Model

For an examinee $i$ with ability $\theta_i$, the probability of correct response to an item $j$ is:

$$P\left(X_{ij} = 1 | \theta_i;a_j, b_j, c_j\right) = c_j + (1 - c_j)\frac{e^{Da_j(\theta_i - b_j)}}{1 + e^{Da_j(\theta_i - b_j)}} $$

where $a_j$ is the item discrimination (or slope) of item $j$, $b_j$ is the item difficulty (or threshold), and $c_j$ is the lower asymptote (or pseudo-guessing) value. D is the scaling constant (the default value is r irt:::DEFAULT_D_SCALING).

As can be seen from the equation above, the user needs to specify all of the parameter values:

itm_3pl <- item(a = 1.51, b = 2.04, c = .16, D = 1.7)
itm_3pl

The probability of each response option at $\theta = 1.5$ is:

prob(ip = itm_3pl, theta = 1.5)

The item characteristic curve of this item:

plot(itm_3pl)

Four-Parameter Logistic Model

For an examinee $i$ with ability $\theta_i$, the probability of correct response to an item $j$ is:

$$P\left(X_{ij} = 1 | \theta_i;a_j, b_j, c_j, d_j\right) = c_j + (d_j - c_j)\frac{e^{Da_j(\theta_i - b_j)}}{1 + e^{Da_j(\theta_i - b_j)}} $$

where $a_j$ is the item discrimination (or slope) of item $j$, $b_j$ is the item difficulty (or threshold), $c_j$ is the lower asymptote (or pseudo-guessing) value and $d_j$ is the upper asymptote. D is the scaling constant (the default value is r irt:::DEFAULT_D_SCALING).

As can be seen from the equation above, the user needs to specify all of the parameter values:

itm_4pl <- item(a = 1.2, b = -.74, c = .22, d = .99, D = 1.7)
itm_4pl

The probability of each response option at $\theta = 1.2$ is:

prob(ip = itm_4pl, theta = 1.2)

The item characteristic curve of this item is:

plot(itm_4pl)

Graded Response Model (GRM)

For an examinee $i$ with ability $\theta_i$, the probability of responding at or above the category $k$ to an item $j$ with possible scores $k = 0, 1, \ldots, m_j$:

$$P^*\left(X_{ij} = k | \theta_i;a_j, b_j\right) = \frac{e^{Da_j(\theta_i - b_{jk})} }{1 + e^{Da_j(\theta_i - b_{jk})}} $$

where $a_j$ is the item discrimination (or slope) of item $j$, $b_{jk}$ is the threshold parameter. Note that the probability of responding at or above the lowest category is $P^*\left(X_{ij} = 0\right) = 1$. Responding at a category $k$ can be calculated as:

$$P(X_{ij}=k|\theta_i) = P^(X_{ij} = k) - P^(X_{ij} = k+1)$$

The user needs to specify the following parameter values:

itm_grm <- item(a = 0.84, b = c(-1, -.2, .75, 1.78), D = 1.7, model = "GRM")
itm_grm

The probability of each response option at $\theta = 1.13$ is:

prob(ip = itm_grm, theta = 1.13)

The option characteristic curves of this item is:

plot(itm_grm)

Generalized Partial Credit Model (GPCM)

For an examinee $i$ with ability $\theta_i$, the probability of a response $k$ to an item $j$ with possible scores $k = 0, 1, \ldots, m_j$:

$$P\left(X_{ij} = k | \theta_i;a_j, b_j\right) = \frac{\text{exp} \left( \sum_{v = 0}^{k}Da_j(\theta_i - b_{jv}) \right) } {{\sum_{h = 0}^{m_j} \text{exp} \left[ \sum_{v = 0}^{h} Da_j(\theta_i - b_{jv}) \right] }} $$

where $a_j$ is the item discrimination (or slope) of item $j$, $b_{jv}$ are the step difficulty parameters. Note that $b_{jv}$ values are not necessarily ordered from smallest to the largest. D is the scaling constant (the default value is r irt:::DEFAULT_D_SCALING). $\sum_{v = 0}^0 Da_j(\theta_i - b_{jv}) = 0$.

The user needs to specify the following parameter values:

itm_gpcm <- item(a = 1.1, b = c(-.74, .3, .91, 2.19), D = 1.7, model = "GPCM")
itm_gpcm

The probability of each response option at $\theta = -0.53$ is:

prob(ip = itm_gpcm, theta = -0.53)

The option characteristic curves of this item is:

plot(itm_gpcm)

Partial Credit Model (PCM)

For an examinee $i$ with ability $\theta_i$, the probability of a
response $k$ to an item $j$ with possible scores $k = 0, 1, \ldots, m_j$:

$$P\left(X_{ij} = k | \theta_i;b_j\right) = \frac{\text{exp} \left( \sum_{v = 0}^{k}(\theta_i - b_{jv}) \right) } {{\sum_{h = 0}^{m_j} \text{exp} \left[ \sum_{v = 0}^{h} (\theta_i - b_{jv}) \right] }} $$

where $b_{jv}$ are the step difficulty parameters. $\sum_{v = 0}^0 (\theta_i - b_{jv}) = 0$.

The user needs to specify the following parameter values:

itm_pcm <- item(b = c(-1.38, -.18, 1.1), model = "PCM")
itm_pcm

The probability of each response option at $\theta = -1.09$ is:

prob(ip = itm_pcm, theta = -1.09)

The option characteristic curves of this item is:

plot(itm_pcm)

Reparameterized Generalized Partial Credit Model (GPCM2)

For an examinee $i$ with ability $\theta_i$, the probability of a response $k$ to an item $j$ with possible scores $k = 0, 1, \ldots, m_j$:

$$P\left(X_{ij} = k | \theta_i;a_j, b_j, d_j\right) = \frac{\text{exp} \left( \sum_{v = 0}^{k}Da_j(\theta_i - b_j + d_{jv}) \right) } {{\sum_{h = 0}^{m_j} \text{exp} \left[ \sum_{v = 0}^{h} Da_j(\theta_i - b_j + d_{jv}) \right] }} $$

where $a_j$ is the item discrimination (or slope) of item $j$, $b_j$ is the overall location parameter and $d_{jv}$ are the threshold parameters. D is the scaling constant (the default value is r irt:::DEFAULT_D_SCALING).

The user needs to specify the following parameter values:

itm_gpcm2 <- item(a = .71, b = .37, d = c(-.18, .11, 1.29), D = 1, 
                  model = "GPCM2")
itm_gpcm2

The probability of each response option at $\theta 1.3$ is:

prob(ip = itm_gpcm2, theta = 1.3)

The option characteristic curves of this item is:

plot(itm_gpcm2)

irt documentation built on Nov. 10, 2022, 5:50 p.m.