Many scientific research programs aim to learn the causal structure of real world phenomena. This learning problem is made more difficult when the target of study cannot be directly observed. One strategy commonly used by social scientists is to create measurable "indicator" variables that covary with the latent variables of interest. Before leveraging the indicator variables to learn about the latent variables, however, one needs a measurement model of the causal relations between the indicators and their corresponding latents. These measurement models are a special class of Bayesian networks. This paper addresses the problem of reliably inferring measurement models from measured indicators, without prior knowledge of the causal relations or the number of latent variables. We present a provably correct novel algorithm, FindOneFactorClusters (FOFC), for solving this inference problem. Compared to other state of the art algorithms, FOFC is faster, scales to larger sets of indicators, and is more reliable at small sample sizes. We also present the first correctness proofs for this problem that do not assume linearity or acyclicity among the latent variables.
Several studies have indicated that bi–factor models fit a broad range of psychometric data better than alternative multidimensional models such as second–order models, e.g Rodriguez, Reise and Haviland (2016), Gignac (2016), and Carnivez (2016). Murray and Johnson (2013) and Gignac (2016) argue that this phenomenon is partially due to un–modeled complexities (e.g. un–modeled cross-factor loadings) that induce a bias in standard statistical measures that favors bi–factor models over second–order models. We extend the Murray and Johnson simulation studies to show how the ability to distinguish second–order and bi–factor models diminishes as the amount of un–modeled complexity increases. By using theorems about rank constraints on the covariance matrix to find sub–models of measurement models that have less un–modeled complexity, we are able to reduce the statistical bias in favor of bi–factor models; this allows researchers to reliably distinguish between bi-factor and second-order models.
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