Latent variable models are ubiquitous in psychometrics and educational measurement. These models are used in the context of assessment to provide standardized measures of achievement and in the context of learning environments to track and diagnose mastery of skills. The simplest statistical model utilized in assessment is the Rasch model. Treating the student ability as a nuisance parameter, a conditional maximum likelihood approach can be utilized to estimate item difficulty. This involves evaluating a computational expensive function. A linear-time approximation is presented and impact of the approximation is explored. Due to the simplicity of the Rasch model many extensions have been proposed in the literature. Most of these extensions are no longer exponential families and many have received criticism for difficulty in their estimation. An extension which remains in the exponential family is discussed and shown to fit to real data as well as other models in the literature. Finally, a model is introduced which is purpose built to help diagnose and provide personalized recommendations for an assessment.