library(irt)
This vignettes covers the psychometric models that has been implemented in
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 | Reparametrized 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:
itm_rasch <- item(b = -1.29)
itm_rasch
#> A 'Rasch' item.
#> Model: Rasch (Rasch Model)
#> Model Parameters:
#> b = -1.29
#>
#> --------------------------
The probability of correct response at \(\theta = -.65\) is:
prob(ip = itm_rasch, theta = -0.65)
#> 0 1
#> [1,] 0.3452465 0.6547535
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 1).
User needs to specify the following parameters:
itm_1pl <- item(b = 0.83, D = 1)
itm_1pl
#> A '1PL' item.
#> Model: 1PL (One-Parameter Logistic Model)
#> Model Parameters:
#> b = 0.83
#> D = 1
#>
#> --------------------------
The probability of correct response at \(\theta = .73\) is:
prob(ip = itm_1pl, theta = 0.73)
#> 0 1
#> [1,] 0.5249792 0.4750208
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 1).
User needs to specify all of the parameter values:
itm_2pl <- item(a = .94, b = -1.302, D = 1)
itm_2pl
#> A '2PL' item.
#> Model: 2PL (Two-Parameter Logistic Model)
#> Model Parameters:
#> a = 0.94
#> b = -1.302
#> D = 1
#>
#> --------------------------
The probability of correct response at \(\theta = -0.53\) is:
prob(ip = itm_2pl, theta = -0.53)
#> 0 1
#> [1,] 0.3261434 0.6738566
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 1).
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
#> A '3PL' item.
#> Model: 3PL (Three-Parameter Logistic Model)
#> Model Parameters:
#> a = 1.51
#> b = 2.04
#> c = 0.16
#> D = 1.7
#>
#> --------------------------
The probability of correct response at \(\theta = 1.5\) is:
prob(ip = itm_3pl, theta = 1.5)
#> 0 1
#> [1,] 0.6719846 0.3280154
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 1).
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
#> A '4PL' item.
#> Model: 4PL (Four-Parameter Logistic Model)
#> Model Parameters:
#> a = 1.2
#> b = -0.74
#> c = 0.22
#> d = 0.99
#> D = 1.7
#>
#> --------------------------
The probability of correct response at \(\theta = 1.2\) is:
prob(ip = itm_4pl, theta = 1.2)
#> 0 1
#> [1,] 0.02443797 0.975562
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
#> A 'GRM' item.
#> Model: GRM (Graded Response Model)
#> Model Parameters:
#> a = 0.84
#> b = -1; -0.2; 0.75; 1.78
#> D = 1.7
#>
#> --------------------------
The probability of each response option at \(\theta = 1.13\) is:
prob(ip = itm_grm, theta = 1.13)
#> 0 1 2 3 4
#> [1,] 0.04557977 0.08461474 0.2373791 0.3491363 0.28329
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 1). \(\sum_{v = 0}^0 Da_j(\theta_i - b_{jv}) = 0\).
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
#> A 'GPCM' item.
#> Model: GPCM (Generalized Partial Credit Model)
#> Model Parameters:
#> a = 1.1
#> b = -0.74; 0.3; 0.91; 2.19
#> D = 1.7
#>
#> --------------------------
The probability of each response option at \(\theta = -0.53\) is:
prob(ip = itm_gpcm, theta = -0.53)
#> 0 1 2 3 4
#> [1,] 0.3551121 0.5259117 0.1113895 0.007540084 4.659945e-05
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
#> A 'PCM' item.
#> Model: PCM (Partial Credit Model)
#> Model Parameters:
#> b = -1.38; -0.18; 1.1
#>
#> --------------------------
The probability of each response option at \(\theta = -1.09\) is:
prob(ip = itm_pcm, theta = -1.09)
#> 0 1 2 3
#> [1,] 0.3407646 0.4554072 0.1833124 0.02051573
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 1).
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
#> A 'GPCM2' item.
#> Model: GPCM2 (Reparametrized Generalized Partial Credit Model)
#> Model Parameters:
#> a = 0.71
#> b = 0.37
#> d = -0.18; 0.11; 1.29
#> D = 1
#>
#> --------------------------
The probability of each response option at \(\theta 1.3\) is:
prob(ip = itm_gpcm2, theta = 1.3)
#> 0 1 2 3
#> [1,] 0.04254421 0.07246065 0.15163 0.7333651
The option characteristic curves of this item is:
plot(itm_gpcm2)
