Item-class: An S4 class to represent an Item
In irt: Item Response Theory and Computerized Adaptive Testing Functions

Item-class

R Documentation

An S4 class to represent an Item

Description

Item is a class to represent an item. An object in Item class should have a model name and parameters.

The model that item parameters represents. Currently, following models are available:

"Rasch"

Rasch Model.

Required parameters:

"b": Item difficulty parameter.

Probability of correct response at ability estimate θ:

P(θ) = \frac{e^{(θ - b)}}{1+e^{(θ - b)}}

Model family: Unidimensional Item Response Theory (UIRT) Models

"1PL"

Unidimensional One-Parameter Logistic Model.

Required parameters:

"b": Item difficulty parameter.
"D": Scaling constant. Default value is 1.

Probability of correct response at ability estimate θ:

P(θ) = \frac{e^{D(θ - b)}}{1+e^{D(θ - b)}}

Model family: Unidimensional Item Response Theory (UIRT) Models

"2PL"

Unidimensional Two-Parameter Logistic Model.

Required parameters:

"a": Item discrimination parameter.
"b": Item difficulty parameter.
"D": Scaling constant. Default value is 1.

Probability of correct response at ability estimate θ:

P(θ) = \frac{e^{Da(θ - b)}}{1+e^{Da(θ - b)}}

Model family: Unidimensional Item Response Theory (UIRT) Models

"3PL"

Unidimensional Three-Parameter Logistic Model.

Required parameters:

"a": Item discrimination parameter.
"b": Item difficulty parameter.
"c": Pseudo-guessing parameter (lower asymptote).
"D": Scaling constant. Default value is 1.

Probability of correct response at ability estimate θ:

P(θ) = c + (1-c) \frac{e^{Da(θ - b)}}{1+e^{Da(θ - b)}}

Model family: Unidimensional Item Response Theory (UIRT) Models

"4PL"

Unidimensional Four-Parameter Logistic Model.

Required parameters:

"a": Item discrimination parameter.
"b": Item difficulty parameter.
"c": Pseudo-guessing parameter (lower asymptote).
"d": Upper asymptote parameter.
"D": Scaling constant. Default value is 1.

Probability of correct response at ability estimate θ:

P(θ) = c + (d-c) \frac{e^{Da(θ - b)}}{1+e^{Da(θ - b)}}

Model family: Unidimensional Item Response Theory (UIRT) Models

"GRM"

Graded Response Model

Required parameters:

"a": Item discrimination parameter.
"b": Item threshold parameters (a vector of values). Each value refers to the ability level for which the probability of responding at or above that category is equal to 0.5.
"D": Scaling constant. Default value is 1.

Probability of scoring at or above the category k:

P^*_k(θ) = \frac{e^{Da(θ - b_k)}}{1+e^{Da(θ - b_k)}}

Probability of responding at category k where the possible scores are 0, …, m:

P_0(θ) = 1 - P^*_1(θ)

P_1(θ) = P^*_1(θ) - P^*_2(θ)

\cdots

P_k(θ) = P^*_{k}(θ) - P^*_{k+1}(θ)

\cdots

P_m(θ) = P^*_{m}(θ)

Model family: Polytomous Item Response Theory (PIRT) Models

"GPCM"

Generalized Partial Credit Model

Required parameters:

"a": Item discrimination parameter.
"b": Item step difficulty parameters (a vector of values).
"D": Scaling constant. Default value is 1.

Probability of scoring at category k:

P_k(θ) = \frac{exp[∑_{v = 0}^{k} Da(θ - b_v)]} {∑_{c = 0}^{m-1}exp[∑_{v = 0}^{c}Da(θ - b_v)]}

Model family: Polytomous Item Response Theory (PIRT) Models

"PCM"

Partial Credit Model (Masters, 1982)

Required parameters:

"b": Item step difficulty parameters (a vector of values).

Probability of scoring at category k:

P_k(θ) = \frac{exp[∑_{v = 0}^{k} (θ - b_v)]}{∑_{c = 0}^{m-1}exp[∑_{v = 0}^{c}(θ - b_v)]}

Model family: Polytomous Item Response Theory (PIRT) Models

"GPCM2"

An alternative parametrization of Generalized Partial Credit Model "GPCM" where b_k = b - d_k. See Muraki (1997), Equation 15 on page 164.

Required parameters:

"a": Item discrimination parameter.
"b": Location parameter.
"d": A vector of threshold parameters.
"D": Scaling constant. Default value is 1.

Probability of scoring at category k:

P_k(θ) = \frac{exp[∑_{v = 0}^{k} Da(θ - b + d_v)]}{∑_{c = 0}^{m-1}exp[∑_{v = 0}^{c}Da(θ - b + d_v)]}

Model family: Polytomous Item Response Theory (PIRT) Models

A model must be specified for the construction of an Item object.

Slots

item_id: Item ID. Default value is NULL.
content: Content information for the Item object.
misc: This slot is a list where one can put any information about the Item object. For example, one can enter the ID's of the enemies of the current Item as misc = list(enemies = c("i1", i2)). Or, one can enter Sympson-Hetter exposure control parameter K: misc = list(sympson_hetter_k = .75).