knitr::opts_chunk$set( fig.width = 5, collapse = TRUE, comment = "#>" )
This vignettes covers the psychometric models that are implemented in
the irt
package.
| 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 |
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)
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)
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)
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)
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)
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)
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)
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)
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)
Any scripts or data that you put into this service are public.
Add the following code to your website.
For more information on customizing the embed code, read Embedding Snippets.