Items that are clustered according to shared content may violate the principle of conditional independence commonly used in item response theory. This paper investigates the capabilities of a logistic item response model in relation to locally dependent item responses. The model includes main effect and interaction parameters that are computed as linear functions of the latent trait. The paper explains the interpretation of the parameters, the maximum likelihood estimation algorithm, the information matrix and some results concerning parameter identifiability. The problem of over-fitting the data is addressed in a simulation study, and two real data examples are described to illustrate the approach, one from the context of a sample survey and the other from ability testing using testlets.