One of the distinctions between classical test theory and item response theory is that the former focuses on sum scores and their relationship to true scores, whereas the latter concerns item responses and their relationship to latent scores. Although item response theory is often viewed as the richer of the two theories, sum scores are still often used in practice. The issue addressed here is how to conduct item response modeling when only sum scores are available for some respondents; that is, their item responses are missing, but their sums scores are known. The author reviews the important role of sum scores in item response theory and shows how to estimate item response models using sum scores as data in lieu of item responses. The author also shows how this can be easily implemented in a Bayesian framework using the software package Just Another Gibbs Sampler (JAGS), and provides three examples for illustration.