Observable data in empirical social and behavioral science studies are often categorical (i.e., binary, ordinal, or nominal). When categorical data are outcomes, they fail to maintain the scale and distributional properties of linear regression and factor analysis. Attempting to estimate model parameters for categorical outcome data with the linear factor analysis model will result in biased and non-interpretable parameter estimates. A solution to this problem is to conduct estimation using either an underlying variable approach or item response theory. However, due to the assumptions of these methods, models with many latent factors or many items become computationally demanding and limit their application with large models (i.e., many factors, many items, Wirth & Edwards, 2007). This difficulty exists because parameter estimation requires intractable high-dimensional numerical integration of the marginal likelihood that can be computationally demanding. Several estimation methods have been developed to approximate such integration. However, there is a need for research that comprehensively evaluates the comparative precision and computational efficiency of these existing estimation methods. This dissertation attempts to add to the literature on comparative estimator performance by comparing conventional and modern methods for estimating factor analysis models with ordinal categorical outcomes in a large model context. [The dissertation citations contained here are published with the permission of ProQuest LLC. Further reproduction is prohibited without permission. Copies of dissertations may be obtained by Telephone (800) 1-800-521-0600. Web page: http://www.proquest.com.bibliotheek.ehb.be/en-US/products/dissertations/individuals.shtml.]