Grant: Discovery Grants Program - Individual

Latent variable (LV) models have gained much popularity in the natural sciences due to their flexibility in assessing complex multivariable relations, and ability to account for measurement error. Like all statistical models, LV models cannot fully represent complex real world phenomena and at best provide a good approximation (Box, 1979). Such model imperfection raises the need to quantify and address uncertainties inherent in results because strong scientific conclusions can be drawn only when all aspects of uncertainly are quantified (MacCallum, Lee & Browne, 2013). This research program focuses on three aspects of model uncertainty: (a) sensitivity of results due to influential cases, (b) uncertainty of parameter estimates and (c) modeling latent functions of unknown form. The proposed program of research is designed to specifically address existing gaps in the literature of uncertainty quantification in LV models, extend the quantification of uncertainty to a broader class of LV models, and develop novel graphical approaches of uncertainty quantification so as to promote their dissemination and use by bridging theory and practice. The presence of influential cases can strongly impact model results, making the identification of such cases an important diagnostic endeavor. As there has been little applied use in LV models, graphical methods are to be developed and implemented in software so as to better disseminate case influence and their use. Additionally, the analytical relationship between exact and approximate measures of case influence is to be established and evaluated under different conditions such that these measures would be applied with better discernment. Several measures of case influence on model selection exist (Sterba & Pek, 2012), and the most popular model selection criterion of model modification (Sorbom, 1989) is to be developed so as to promote the application of case influence diagnostics in the context of selecting among competing LV models. Finally, to bridge the gap between two related nonlinear LV models (item factor analysis and item response theory), it is proposed that exact measures stemming from these two distinct modeling traditions be unified in a more general framework of case influence. Parameter estimates reflect sampling variability that is quantified by confidence intervals (CIs), and likelihood-based CIs perform better than Wald-type CIs in LV models especially when constructed about a function of parameters. It is proposed that profile likelihood-based CIs are extended from linear LV models and developed for a general class of nonlinear LV models, including CIs for a function of parameters, so as to provide better devices for inference and quantifying estimate precision. Finally, structural equation mixture models (SEMMs) can reasonably recover latent relationships of unknown functional form (Bauer, 2005). Approximate CIs for this indirect application of SEMMs (Pek, Losardo & Bauer, 2011) only allow for point-wise inference. It is proposed that delta method and bootstrap simultaneous confidence bands or confidence envelopes (CEs) are developed to allow for inference about the latent function as a whole. A line searching algorithm would also be developed to implement a general test of linearity that makes use of the CEs such that researchers can rule out linearity in the presence of unspecified nonlinearity. These proposed studies represent steps toward acknowledging and communicating that all statistical models carry uncertainty in their results, which should be fully considered before strong conclusions are drawn. The development and implementation of these new and existing methods promises to contribute to the better use of LV models.