Psychometric Models in irt Package" In irt: Item Response Theory and Computerized Adaptive Testing Functions

knitr::opts_chunk$set( fig.width = 5, collapse = TRUE, comment = "#>" ) library(irt) This vignettes covers the psychometric models that has been implemented in irt package. Item Models | Name | Description | Parameters | |-------|-------------------------------------------------|---------------| | Rasch | Rasch Model | b | | 1PL | One-Parameter Logistic Model | b, D | | 2PL | Two-Parameter Logistic Model | a, b, D | | 3PL | Three-Parameter Logistic Model | a, b, c, D | | 4PL | Four-Parameter Logistic Model | a, b, c, d, D | | GRM | Graded Response Model | a, b, D | | PCM | Partial Credit Model | b | | GPCM | Generalized Partial Credit Model | a, b, D | | GPCM2 | Reparametrized Generalized Partial Credit Model | a, b, d, D | Rasch Model For an examinee$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: itm_rasch <- item(b = -1.29) itm_rasch The probability of correct response 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 correct response 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 correct response 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 correct response 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 correct response 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) Re-parametrized 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)

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irt documentation built on Nov. 9, 2021, 9:07 a.m.