The partial credit model (PCM)

5.2 The partial credit model (PCM)

This model was introduced formally by Andersen (1977), but the name is due to Masters (1982). It can be understood in several ways, and two of them will

be discussed here. In one approach, it will be considered as a special case of the NRM, in the other, it will be characterized as a linear logit model. As the PCM is a model for ordered categories, this implies that the order of the categories is known, unlike in the NRM. Situations where this can arise are items that can be partially correct, and a partial credit is given for such an answer. Formally, the PCM has the same category response functions as the NRM, but the ␣-parameters are given a fixed value. It is customary in this model to number the categories starting from zero. If the highest category is m, then there are m + 1 response categories. The PCM is equivalent to the NRM with the parameter ␣ ij fixed at j for all items. The zero category is automatically the reference category.

In the parameterization proposed by Andersen (1977), the category response functions are given by

P(X i = j) ⬀ 冦 exp(j␪ – ␩

Masters uses another parameterization, where the ␩-parameters in (20) are cumulative sums:

η j ij = β ig . (21)

Such a reparameterization does not change the model, and if one knows the ␩-parameters, the ␤-parameters are immediately available:

␤ i1 =␩ i1 , ␤ ij =␩ ij –␩ i, j – 1 , (j > 1).

Using Item Response Theory 175 The PCM is an exponential family model. The sufficient statistic for the latent

variable ␪ is the raw score: the sum of the partial item scores obtained, and the sufficient statistic for the category parameters, is just the number of times the category has been obtained.

The ␤-parameters in the Masters parameterization have a nice interpretation: the parameter ␤ ij is the location on the ␪-scale where the categories j and j – 1 have the same probability, which is shown in Figure 8.8 for two items with three response categories, 0, 1 and 2: in the left-hand panel of the figure it holds that the ␤-parameters are ordered in the same order as the categories, that is, ␤ i1 < ␤ i2 ; for the right-hand panel, the order is reversed, ␤ i2 < ␤ i1 . If the parameter values are ordered in the same order as the categories, then for each category there is an interval on the ␪-scale where that category is the modal one, that is, the category is the most probable one. In the left-hand panel of Figure 8.8, it is seen that category 1 is the most probable one for ␤ i1 < ␪< ␤ i2 . In the right-hand panel, category 1 is never the modal one. Sometimes it is claimed that the ␤-parameters must be ordered in the same way as the categories, but there is nothing in the model definition that prescribes such a rule.

The second derivation of the PCM is one where it is conceived as a possible generalization of the Rasch model. In Section 1 it was shown that the Rasch

obab Pr

β i1

β i2

obab Pr

β i2

β i1

Figure 8.8 Two examples of the partial credit model

176 Different methodological orientations model can be viewed as a logit-linear model. For binary outcomes, the logit

function is well defined: it is the logarithm of p/(1 – p), where p is the probability of a success. With variables with more than two outcomes, however, the logit function is not uniquely determined, although it is used in these contexts, with

a suitable modification. The PCM can be characterized by the adjacent category logits (Agresti 1990) as follows:

PX ( j |) θ ln

(22) PX ( i = j − 1 |) θ

= θβ − ,( j > 0 ).

ij

Together with the requirement that P(X i = 0|␪) ⬀ 1, this defines the PCM in the Masters parameterization. Equation (22) is also equivalent with the following conditional probability:

exp( θβ − ij )

PX ( i = j |, θ X i = j or X i = j − 1 ) =

1 + exp( θ − ββ ij ) that is, the probability that the highest category is obtained, given that the choice

is between this category and the preceding one. The right-hand side of (23) has exactly the same structure as the IRF for the Rasch model, but for the inter- pretation, it is important to look carefully at the left-hand side. It is not easy to imagine what could be a meaningful interpretation of this conditional probability in an educational context. Moreover, the model is not invariant under a collapsing of categories, as will be shown in a simplified example.

Suppose an item i has been scored into four categories, and furthermore that it is applied in a population where the ability of all the test takers is constant, say ␪ = 0. The probabilities of responding to each category are given in the second row of Table 8.1. Now assume that the PCM is valid for the original categorization. The value of P(X i = 2 | ␪ = 0, X i = 2 or X i = 1) is 0.35/ (0.35 + 0.20) = 0.636, and solving the right-hand side of (21) for ␤ i2 gives ␤ i2 = –0.56. Now, suppose one wants to apply the PCM but with one category less: the original categories 0 and 1 are collapsed into a single category, which is the new reference category; the other categories are renumbered from 2 and

3 to 1 and 2 respectively. But applying the PCM rationale to this collapsed table will give P(X i = 2|␪ = 0, X i = 2 or X i = 1) = 0.35/(0.35 + 0.35) = 0.5 and as

Table 8.1 Collapsing categories Original categories

0 1 2 3 Probabilities

0.15 0.20 0.35 0.30 Collapsed categories

Probabilities

Using Item Response Theory 177

a result ␤ i1 = 0. This means that the ␤-parameters in the PCM cannot be interpreted as a kind of lower bound of the categories, or as steps or thresholds, because in the example, the definition of the original category 2 (category 1 after collapsing) has not changed by the collapsing of the two lower categories, but the value of the associated ␤-parameter has changed substantially.

Moreover, if the PCM is the exact model for the original categorization, it cannot be exact for the data after collapsing.

A slight generalization of the PCM yields the generalized partial credit model (GPCM) (Muraki 1992). The generalization implies that a different discrimination per item is added to the PCM. The weight or score parameters (the coefficient of ␪) for category j in the two models is:

PCM: j, GPCM: j × ␣ i .

The parameter ␣ i is a discrimination parameter for item i. The GPCM relates to the PCM as the 2PLM relates to the Rasch model.