The two-parameter logistic model

3.1 The two-parameter logistic model

In the Rasch model, each item has one parameter ␤ i , commonly referred to as the difficulty parameter. From elementary techniques in Classical Test Theory it is well known that items do not differ only in difficulty but also in discrimi- nation. In the Rasch model, there is no possibility to make the items differ in discrimination. In Figure 8.3, two response curves are displayed for items i and j, both having the same difficulty, but they differ in discrimination in the following sense: imagine two students with a latent ability in a small neighborhood of the common difficulty parameter of the two items – that is, where the two curves cross, one being a bit lower, the other a bit higher. With item i the difference in probability of success for the two students is quite small (this difference is displayed as the distance between the two dashed lines on the left vertical axis), while with item j the corresponding difference is much larger (refer to the right vertical axis), meaning that item j discriminates better than item i. This difference in probabilities is associated with the steepness of the two curves in the neighborhood of the difficulty parameter.

To grasp this difference in a mathematical expression, one needs an extra parameter. The IRF for this generalized model is given by

exp[ ( αθβ − )] PX ( i = 1 |) θ =

1 + exp[ ( αθβ − )]

Using Item Response Theory 165

0.5 obab Pr

0.0 θ Figure 8.3 Two items with equal difficulty but different discrimination

The larger the parameter ␣, the steeper the curve is. This parameter is commonly referred to as the discrimination parameter. The function in (9) is also a logistic function; its argument is ␣ i (␪ – ␤ i ). The model with IRFs given by (9) is known as the two-parameter logistic model (2PLM).

It may be interesting to see what happens if the discrimination parameter becomes very large. Taking limits of (9) for ␣ i → ⬁ gives different results, depending on the sign of the difference ␪ – ␤ i :

lim ␣i→⬁ P(X i = 1|␪) = 1 if ␪ > ␤ i , lim ␣i→⬁ P(X i = 1|␪) = 0 if ␪ < ␤ i .

This means that with a very large discrimination parameter the item behaves as

a Guttman item. The log-likelihood function for this model is given by

ln P ( X = x | θ 1 ,, … θ n )

= k ∑ w v θ v + ∑ t i ( − αβ i i ) − ∑ ∑ lln 1 ⎡⎣ + exp[ ( αθ i v − β i )] ,

nk

where the sufficient statistic for the difficulty parameters is t i =⌺ v x vi , as in the Rasch model, and

w v = ∑ α i x vi ,

the weighted score – that is, the sum of the discrimination parameters of the correctly answered items by student v. However, this weighted sum is not a

166 Different methodological orientations mere statistic, that is, a function of the observed data; it also depends on the

unknown discrimination parameters, and therefore the 2PLM is not an expo- nential family model. If one knows the value of these parameters, or treats them as known constants (by hypothesis, for example), then the model becomes an exponential family model. More on this will be said in the section on parameter estimation in the next chapter.