The present dissertation project investigates different approaches to adding covariates and the impact in fitting mixture item response theory (IRT) models. Mixture IRT models serve as an important methodology for tackling several important psychometric issues in test development, including detecting latent differential item functioning (DIF). A Monte Carlo simulation study is conducted in which data generated according to a two-class mixture Rasch model (MRM) with both dichotomous and continuous covariates are fitted to several MRMs with misspecified covariates to examine the effects of covariate inclusion on model parameter estimation. In addition, both complete response data and incomplete response data with different types of missingness are considered in the present study in order to simulate practical assessment settings. Parameter estimation is carried out within a Bayesian framework vis-à-vis Markov chain Monte Carlo (MCMC) algorithms. Two empirical examples using the Programme for International Student Assessment (PISA) 2009 U.S. reading assessment data are presented to demonstrate the impact of different specifications of covariate effects for an MRM in real applications.