The article proposes a family of item-response models that allow the separate and independent specification of three orthogonal components: item attribute, person covariate, and local item dependence. Special interest lies in extending the linear logistic test model, which is commonly used to measure item attributes, to tests with embedded item clusters. The problem of local item dependence arises in item clusters. Existing methods for handling such dependence, however, often fail to satisfy the property of invariant marginal interpretation of the item attribute parameters. Although such a property may not be necessary for applications that focus on predictive analysis, it is critical for linear logistic test models. To achieve the marginal property, we implement an iterative estimation method, which is illustrated using data collected from an inventory on verbal aggressiveness.