The generalized partial credit model (GPCM) is an IRT model developed to analyze partial credit data, where responses are scored 0,1,…, k, where k is the highest score category for the item. Masters (1982) developed the partial credit model (PCM) by formulating a polytomous rating response model based on the Rasch (1960) dichotomous model. In Masters’ formulation, the probability of choosing the kth category over the k  1th category of item i is governed by the dichotomous response model. Let P_{jk}(q) denote the specific probability of selecting the kth category from m_{j} possible categories of item j. For each adjacent categories, the probability of the specific categorical response k over k  1 is given by the conditional probability: where k = 1,2,..., m_{j}. Note that in this model, all items are assumed to have uniform discriminating power. Muraki (1992) extented Masters’ PCM by relaxing the assumption of uniform discriminating power of test items based on the twoparameter (2PL) logistic response model. In Muraki’s formulation, the probability of choosing category k over category k  1 is given by the conditional probability: where k = 1,2,...,m_{j}. The above equation can be written as After normalizing each P_{jk}(q) so that S P_{jk}(q) = 1, the GPCM is written as where D, is a scaling constant set to 1.7 to approximate the normal ogive model, a_{j} is a slope parameter , b_{j} is an item location parameter , and d_{j},v is a category parameter . The slope parameter indicates the degree to which categorical responses vary among items as q level changes. With m_{j} categories, only m_{j}  1 category parameters can be identified. Indeterminacies in the parameters of the GPCM are resolved by setting d_{j},0 = 0 and setting . Muraki (1992) points out that b_{j}  d_{j},k is the point on the q scale at which the plots of P_{j},k1(q) and P_{jk}(q) intersect and so characterizes the point on the q scale at which the response to item j has equal probability of falling in response category k 1 and falling in response category k. The figure below illustrates the item category response functions (what is known as item characteristic curves for dichotomous items) for a GPCM item with parameters a=1 and b=(0,2,4). It is clear from this graph that the point of intersection of P_{j},k1(q) and P_{jk}(q) corresponds to an equal probability (i.e., P = .5) of being in category k 1 or category k.
