Formulas for computing the exact signed and unsigned areas between two item characteristic curves (ICCs) are presented. It is further shown that when thec parameters are unequal, the area between two ICCs is infinite. The significance of the exact area measures for item bias research is discussed.

In Item Response Theory (IRT), item characteristic curves (ICCs) are illustrated through logistic models or normal ogive models, and the probability that examinees give the correct answer is usually a monotonically increasing function of their ability parameters. However, since only limited patterns of shapes can be obtained from logistic models or normal ogive models, there is a possibility that the model applied does not fit the data. As a result, the existing method can be rejected because it cannot deal with various item response patterns.

To overcome these problems, we propose a new semiparametric IRT model using a Dirichlet process mixture logistic distribution. Our method does not rely on assumptions but only requires that the ICCs be a monotonically nondecreasing function; that is, our method can deal with more types of item response patterns than the existing methods, such as the one-parameter normal ogive models or the two- or three-parameter logistic models.

We conducted two simulation studies whose results indicate that the proposed method can express more patterns of shapes for ICCs and can estimate the ability parameters more accurately than the existing parametric and nonparametric methods. The proposed method has also been applied to Facial Expression Recognition data with noteworthy results.

The paper addresses and discusses whether the tradition of accepting point-symmetric item characteristic curves is justified by uncovering the inconsistent relationship between the difficulties of items and the order of maximum likelihood estimates of ability. This inconsistency is intrinsic in models that provide point-symmetric item characteristic curves, and in this paper focus is put on the normal ogive model for observation. It is also questioned if in the logistic model the sufficient statistic has forfeited the rationale that is appropriate to the psychological reality. It is observed that the logistic model can be interpreted as the case in which the inconsistency in ordering the maximum likelihood estimates is degenerated.

The paper proposes a family of models, called the logistic positive exponent family, which provides asymmetric item chacteristic curves. A model in this family has a consistent principle in ordering the maximum likelihood estimates of ability. The family is divided into two subsets each of which has its own principle, and includes the logistic model as a transition from one principle to the other. Rationale and some illustrative examples are given.