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*'''Likelihood ratio statistic:'''<ref>{{Lawley, D. N. (1940). The estimation of factor loadings by the method of maximumlikelihood. Proceedings of the Royal Society ofedinborough, 60A, 64-82.}}</ref> Used to test the null hypothesis that a model has perfect model fit. It should be applied to models with an increasing number of factors until the result is nonsignificant, indicating that the model is not rejected as good model fit of the population. This statistic should be used with a large sample size and normally distributed data. There are some drawbacks to the likelihood ratio test. First, when there is a large sample size, even small discrepancies between the model and the data result in model rejection.<ref>{{Hakstian, A. R., Rogers, W. T., & Cattell, R. B. (1982). The behavior of number-offactors rules with simulated data. Multivariate Behavioral Research, 17(2), 193-219}}</ref> <ref>{{Humphreys, L. G. & Montanelli, R. G., Jr. 1975. An investigation of the parallel analysis criterion for determining the number of common factors. Multivariate Behavioral Research, 10(2): 193-205.}}</ref> <ref>{{cite journal|last=Harris|first=M. L.|coauthors=Harris, C. W.|title=A Factor Analytic Interpretation Strategy|journal=Educational and Psychological Measurement|date=1 October 1971|volume=31|issue=3|pages=589–606|doi=10.1177/001316447103100301}}</ref> When there is a small sample size, even large discrepancies between the model and data may not be significant, which leads to underfactoring <ref>{{Humphreys, L. G. & Montanelli, R. G., Jr. 1975. An investigation of the parallel analysis criterion for determining the number of common factors. Multivariate Behavioral Research, 10(2): 193-205.}}</ref> . Another disadvantage of the likelihood ratio test is that the null hypothesis of perfect fit is an unrealistic standard.<ref>{{MacCallum, R. C. (1990). The need for alternative measures of fit in covariance structure modeling. Multivariate Behavioral Research, 25, 157-162.}}</ref> <ref>{{Browne, M. W., & Cudeck, R. (1992). Alternative ways of assessing model fit. Sociological Methods and Research, 21, 230-258.}}</ref> <ref>{{Cudeck, R., & Henly, S. J. (1991). Model selection in covariance structures analysis and the "problem" of sample size: A clarification. Psychological Bulletin, 109 (3), 512-519.1991-20270-00110.1037//0033-2909.109.3.512}}</ref>
*'''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.{{cn|date=April 2012}} 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>{{Browne, M. W., & Cudeck, R. (1992). Alternative ways of assessing model fit. Sociological Methods and Research, 21, 230-258.}}</ref> <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.
==Factor rotation==
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