A multicomponent latent trait model for diagnosis.

This paper presents a noncompensatory latent trait model, the multicomponent latent trait model for diagnosis (MLTM-D), for cognitive diagnosis. In MLTM-D, a hierarchical relationship between components and attributes is specified to be applicable to permit diagnosis at two levels. MLTM-D is a generalization of the multicomponent latent trait model (MLTM; Whitely in Psychometrika, 45:479-494, 1980; Embretson in Psychometrika, 49:175-186, 1984) to be applicable to measures of broad traits, such as achievement tests, in which component structure varies between items. Conditions for model identification are described and marginal maximum likelihood estimators are presented, along with simulation data to demonstrate parameter recovery. To illustrate how MLTM-D can be used for diagnosis, an application to a large-scale test of mathematics achievement is presented. An advantage of MLTM-D for diagnosis is that it may be more applicable to large-scale assessments with more heterogeneous items than are latent class models.