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*'''Root mean square error of approximation (RMSEA) fit index:''' RMSEA is an estimate of the discrepancy between the model and the data per degree of freedom for the model. Values less that .05 constitute good fit, values between 0.05 and 0.08 constitute acceptable fit, a values between 0.08 and 0.10 constitute marginal fit and values greater than 0.10 indicate poor fit .<ref name =Browne/><ref>Steiger, J. H. (1989). EzPATH: A supplementary module for SYSTAT andsygraph. Evanston, IL: SYSTAT</ref> An advantage of the RMSEA fit index is that it provides confidence intervals which allow researchers to compare a series of models with varying numbers of factors.
*'''Information Criteria:''' Information criteria such as Akaike Information Criterion (AIC) or the Bayesian Information Criterion (BIC) <ref>Neath, A. A., & Cavanaugh, J. E. (2012). The Bayesian information criterion: background, derivation, and applications. Wiley Interdisciplinary Reviews: Computational Statistics, 4(2), 199-203.</ref> can be used to trade-off model fit with model complexity and select an optimal number of factors.
*'''Out-of-sample Prediction Errors (PE):''' Using the connection between model-implied covariance matrices and standardized regression weights, the number of factors can be selected using out-of-sample prediction errors.<ref name=":0" /> In other words, the PE approach tests the ability of a factor model with ''k'' factors to predict scores on ''p'' items in held-out respondents, using the model-implied covariance structure to derive item-level regressions (e.g., predicting item ''i'' as a linear combination of all other items, with coefficients given by the inverse covariance matrix), selecting the value of ''k'' that best predicts out-of-sample item scores. In an extensive 2022 simulation study, Haslbeck and van Bork<ref name=":0" /> found that the PE method compares favorably with the best-performing existing methods (e.g., parallel analysis, exploratory graph analysis, AIC).
===Optimal Coordinate and Acceleration Factor===
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