The Rasch model is a one-parameter model. The model has one item parameter: the item difficulty parameter. In many tests the items differ not only with respect to difficulty but also with respect to discriminat- ing power. The two-parameter logistic model (Birnbaum, 1968), 2PL model for short, has a second item parameter, item discrimination.
The model is given by the following equation:
(9.2) where ai is the discrimination parameter of item i. The term logistic refers to the fact that the right-hand side of Equation 9.2 is equal to the cumulative logistic distribution function. The slope of the ICC at θ is equal to aiPi(θ)[1 −Pi(θ)]. At θ=bi the slope is equal to 0.25ai. So the slope at θ=bi is steeper for higher values of ai.
The person parameters in the 2PL model are defined on an interval scale. The probability of a correct response does not change if we transform θ into θ* = dθ + e under simultaneous transformations b*
= db+ e and a* = a/d. In order to fix the latent scale, we need two restrictions, for example, the mean θ can be set equal to 0.0 and the standard deviation of the θ’s can be set equal to 1.0.
ln ( ) ( ) P
P b
i
i i
θ
θ θ
1−
⎛
⎝⎜ ⎞
⎠⎟ = −
P a b
a b
i
i i
i i
( ) exp[ ( )]
exp[ ( )]
θ θ
= θ−
+ −
1
ITEM RESPONSE MODELS 137
In Figure 9.2 two item characteristic curves are displayed: one with ai= 1.0, the other with ai = 10.0. One can imagine what will happen to the ICC if the discrimination parameter of an item increases indef- initely. The item characteristic curve approximates a jump function with a value equal to 0 for θ smaller than bi, and a value equal to 1 for θ larger than bi. We then have a Guttman item with a perfect discrimination at the value θ= bi, and no discriminating power to the left and to the right of this point.
The item characteristic curves in Figure 9.2 cross. In the Rasch model, item characteristic curves do not cross, but run parallel. The Rasch model is not the only probabilistic model with nonintersecting item characteristic curves. Another model with this property is the nonparametric Mokken model of double monotonicity (Mokken, 1971).
With items of the multiple-choice type, guessing is possible and cannot be excluded. If an examinee does not know the answer on a four-choice item, he or she might correctly guess the answer with a probability equal to one fourth. With this kind of item, one better introduces a lower asymptote larger than 0 for the item characteristic curve. The three-parameter logistic model is obtained:
(9.3) Figure 9.2 Item characteristic curves for two items with different discrimi- nation parameters.
0 0.2 0.4 0.6 0.8 1
−4 −3 −2 −1 0 θ
1 2 3 4
P(θ)
P c c a b
a b
i i i
i i
i i
( ) ( ) exp[ ( )]
exp[ ( )]
θ θ
= + − θ−
+ −
1 1
where ci is the lower asymptote. The third parameter is called the pseudo-chance-level parameter. This parameter is not set equal to the inverse of the number of response alternatives, but it is estimated along with the other item parameters. Figure 9.3 displays two items:
one with ci equal to one fourth, and the other with ci equal to 0.0. The influence of the third item parameter at the lower level of θ is clear.
The 2PL model is a special case of the 3PL model with ci = 0.0 for all items. The 1PL model is obtained if all item discrimination param- eters are set equal. In the Rasch model (Equation 9.1), this common discrimination parameter is set equal to 1.0. The differences between the models seem to be very clear. Meredith and Kearns (1973), how- ever, demonstrated that a special case of the 3PL model can be refor- mulated in terms of the Rasch model.
In addition to the logistic model, we have the normal ogive model.
It has an ICC with the form of the cumulative normal distribution.
This model was the first to be used in test theory (see Lord, 1952).
The two-parameter normal ogive model is given by
(9.4) The normal ogive plays a role in some models with more dimensions or more than two response categories (Bock, Gibbons, and Muraki, 1988;
Figure 9.3 Item characteristic curves for two items with different pseudo- chance-level parameters.
0 0.2 0.4 0.6 0.8 1
−4 −3 −2 −1 0 θ
1 2 3 4
P(θ)
Pi ai bi t dt
ai bi
( ) [ ( )] ( ) exp
( )
θ θ ϕ
π
θ
= − = = −
−∞
−
∫
Φ 1
2
11 2
t dt2 ai bi
⎡
⎣⎢ ⎤
⎦⎥
−∞
−
∫
(θ )
Muraki and Carlson, 1995; Muthén, 1984). The application of the model to polytomous, multidimensional data will be discussed in Section 9.3.
The normal ogive model and the logistic model give practically the same probabilities if a scaling factor D = 1.7 is introduced in the logistic model:
For this reason, the factor D is frequently part of the logistic model as described in the literature. If the parameters of the logistic model are given by Equation 9.2 or Equation 9.3, the parameters are defined in the “logistic metric.” They can be transformed to the “normal metric”
through a division of the discrimination parameters by the scaling factor 1.7.