The graded response model (GRM)

5.3 The graded response model (GRM)

This model (Samejima 1969, 1972, 1973) allows for collapsing of adjacent categories in a double sense: the parameters are invariant, and if the model is valid before collapsing, it remains valid after collapsing.

An easy way to see the structure of the model is to use cumulative logits (Agresti 1990: 321), which are logits defined on cumulative probabilities: the probability of obtaining category j or higher is compared to its complement, the probability of obtaining a category lower than j. The GRM assumes that these cumulative logits are linear in ␪:

Notice that the right-hand sides of equations (22) – the PCM – and (24) – the GRM – have an identical structure, but the interpretation is quite different. Equation (24) is equivalent with

exp( θβ − ij ) PX ( i ≥ j |) θ =

,( j = 1…m ,, m i ),

1 + exp( θβ − ij ) where one sees again the same structure as in the Rasch model. However, (25)

is not an expression for the category response functions, because it expresses the probability that the response is observed in category j or higher. The category response functions then are given by

178 Different methodological orientations P(X i = 0|␪) = 1 – P(X i ⱖ 1|␪),

(26a) P(X i = j|␪) = P(X i ⱖ j|␪) – P(X i ⱖ j + 1|␪), (j = 1, . . ., m i – 1), (26b) P(X i =m i |␪) = P(X i ⱖm i |␪).

(26c) If the items are binary, that is, for all items it holds that m i = 1, then only (26a)

and (26c) apply, and the model is identical to the Rasch model. If m i > 1 for one or more items, then (26b) applies for the middle categories, and the category response function contains a difference of two logistic functions, and this means that the model is not an exponential family model.

Figure 8.2 is a graph of the functions (25) for a four-category item – that is, m i = 3. However, where in this figure – which was meant as an illustration of the Rasch model – each curve represented a separate item, now the three curves have to be interpreted as belonging to the same four-category item. The labels for the parameters in the figure (␤ i ,␤ j and ␤ m ) have to be replaced by ␤ i1 ,␤ i2 and ␤ i3 respectively, where i is the index for the item. In the GRM the category parameters are necessarily ordered in increasing order of the category numbers, otherwise equation (26b) would yield a negative probability.

In Figure 8.9, the category response curves, corresponding to the three cumulative curves given in Figure 8.1 are displayed. Notice that, similar to the PCM, the curve corresponding to the zero category is decreasing, the curve corresponding to the highest category is increasing and curves corresponding to all other categories are single-peaked. In this figure, the location of the category parameters do not have an elegant interpretation, unlike in the PCM; the obvious interpretation is associated with Figure 8.1.

The collapsibility of adjacent categories is immediately clear from equation (25): the parameter ␤ ij is the location where the probability of obtaining category

obab Pr

Figure 8.9 Category response functions in the GRM

Using Item Response Theory 179 j or higher is 50 per cent, and this not dependent on how lower or higher

categories are defined. In this sense, the GRM has a nice interpretation in terms of Thurstonian thresholds, which the PCM has not.