Item-class {irt}

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 \theta:

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

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

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

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

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

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

P_0(\theta) = 1 - P^*_1(\theta)

P_1(\theta) = P^*_1(\theta) - P^*_2(\theta)

\cdots

P_k(\theta) = P^*_{k}(\theta) - P^*_{k+1}(\theta)

\cdots

P_m(\theta) = P^*_{m}(\theta)

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(\theta) = \frac{exp[\sum_{v = 0}^{k} Da(\theta - b_v)]} {\sum_{c = 0}^{m-1}exp[\sum_{v = 0}^{c}Da(\theta - 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(\theta) = \frac{exp[\sum_{v = 0}^{k} (\theta - b_v)]}{\sum_{c = 0}^{m-1}exp[\sum_{v = 0}^{c}(\theta - 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(\theta) = \frac{exp[\sum_{v = 0}^{k} Da(\theta - b + d_v)]}{\sum_{c = 0}^{m-1}exp[\sum_{v = 0}^{c}Da(\theta - 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).

Author(s)

Emre Gonulates

References

Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149–174.

Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16, 159–176.