Item is a class to represent an item. An object in Item class should
have a model name and parameters.
idItem id. Default value is NULL.
modelThe 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.
parametersA list containing numeric vectors that represent item parameters. Depending on the model these can change.
se_parametersStandard error of the item parameters. This should be
a list of standard error values. For example, for "2PL", if the parameters
are list(a = 1.2, b = -0.22), the standard error values of
parameters can be either NULL (which is the default value) or
list(a = 0.24, b = 0.42). None of the standard error values can
be smaller than 0. Individual SE values can be NA. For example,
list(a = 0.24, b = NA) is acceptable, whereas
list(a = 0.24, b = NULL) is not acceptable.
For models like polytomous items, the SE values should match the parameter
values in length. For example, if the parameter values of a "GPCM"
is parameters = list(a = 1.4, b = c(-1, 0.42, 2.1), D = 1.7), then
the SE values should be like
se_parameters = list(a = .2, b = c(.32, 0.34, .3)). Since the
scaling parameter D is constant, it does not have a standard error.
contentContent information for the Item object.
miscThis 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).
Masters, G. N. (1982). A Rasch model for partial credit scoring. Psychometrika, 47, 149<U+2013>174.
Muraki, E. (1992). A generalized partial credit model: Application of an EM algorithm. Applied Psychological Measurement, 16, 159<U+2013>176.