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==Fitting procedures==
Fitting procedures are used to estimate the factor loadings and unique variances of the model (''Factor loadings'' are the regression coefficients between items and factors and measure the influence of a common factor on a measured variable). There are several factor analysis fitting methods to choose from, however there is little information on all of their strengths and weaknesses and many don’tdon't even have an exact name that is used consistently. Principal axis factoring (PAF) and [[maximum likelihood]] (ML) are two extraction methods that are generally recommended. In general, ML or PAF give the best results, depending on whether data are normally-distributed or if the assumption of normality has been violated.<ref name =Fabrigar/>
===Maximum likelihood (ML)===
===Revelle and Rocklin (1979) very simple structure===
Revelle and Rocklin’sRocklin's (1979) VSS criterion operationalizes this tendency by assessing the extent to which the original correlation matrix is reproduced by a simplified pattern matrix, in which only the highest loading for each item is retained, all other loadings being set to zero. The VSS criterion for assessing the extent of replication can take values between 0 and 1, and is a measure of the goodness-of-fit of the factor solution. The VSS criterion is gathered from factor solutions that involve one factor (k = 1) to a user-specified theoretical maximum number of factors. Thereafter, the factor solution that provides the highest VSS criterion determines the optimal number of interpretable factors in the matrix. In an attempt to accommodate datasets where items covary with more than one factor (i.e., more factorially complex data), the criterion can also be carried out with simplified pattern matrices in which the highest two loadings are retained, with the rest set to zero (Max VSS complexity 2). Courtney also does not recommend VSS because of lack of robust simulation research concerning the performance of the VSS criterion.<ref name="pareonline.net"/>
===Model comparison techniques===
===Optimal Coordinate and Acceleration Factor===
In an attempt to overcome the subjective weakness of Cattell’sCattell's (1966) scree test,<ref name="Cattell, R. B. 1966"/><ref name =Raiche>Raiche, Roipel, and Blais (2006)</ref> presented two families of non-graphical solutions. The first method, coined the optimal coordinate (OC), attempts to determine the ___location of the scree by measuring the gradients associated with eigenvalues and their preceding coordinates. The second method, coined the acceleration factor (AF), pertains to a numerical solution for determining the coordinate where the slope of the curve changes most abruptly. Both of these methods have out-performed the K1 method in simulation.<ref name =Ruscio/> In the Ruscio and Roche study (2012),<ref name =Ruscio/> the OC method was correct 74.03% of the time rivaling the PA technique (76.42%). The AF method was correct 45.91% of the time with a tendency toward under-estimation. Both the OC and AF methods, generated with the use of Pearson correlation coefficients, were reviewed in Ruscio and Roche’sRoche's (2012) simulation study. Results suggested that both techniques performed quite well under ordinal response categories of two to seven (C = 2-7) and quasi-continuous (C = 10 or 20) data situations. Given the accuracy of these procedures under simulation, they are highly recommended{{by whom|date=February 2014}} for determining the number of factors to retain in EFA. It is one of Courtney's 5 recommended modern procedures.<ref name="pareonline.net"/>
===Velicer's Minimum Average Partial test (MAP)===
Velicer’sVelicer's (1976) MAP test<ref name=Velicer/> “involves a complete principal components analysis followed by the examination of a series of matrices of partial correlations” (p. 397). The squared correlation for Step “0” (see Figure 4) is the average squared off-diagonal correlation for the unpartialed correlation matrix. On Step 1, the first principal component and its associated items are partialed out. Thereafter, the average squared off-diagonal correlation for the subsequent correlation matrix is computed for Step 1. On Step 2, the first two principal components are partialed out and the resultant average squared off-diagonal correlation is again computed. The computations are carried out for k minus one steps (k representing the total number of variables in the matrix). Finally, the average squared correlations for all steps are lined up and the step number that resulted in the lowest average squared partial correlation determines the number of components or factors to retain (Velicer, 1976). By this method, components are maintained as long as the variance in the correlation matrix represents systematic variance, as opposed to residual or error variance. Although methodologically akin to principal components analysis, the MAP technique has been shown to perform quite well in determining the number of factors to retain in multiple simulation studies.<ref name =Ruscio/><ref name=Garrido>Garrido, L. E., & Abad, F. J., & Ponsoda, V. (2012). A new look at Horn's parallel analysis with ordinal variables. Psychological Methods. Advance online publication. doi:10.1037/a0030005</ref> However, in a very small minority of cases MAP may grossly overestimate the number of factors in a dataset for unknown reasons.<ref>{{cite journal | last1 = Warne | first1 = R. T. | last2 = Larsen | first2 = R. | year = 2014 | title = Evaluating a proposed modification of the Guttman rule for determinig the number of factors in an exploratory factor analysis. P | url = | journal = Sychological Test and Assessment Modeling | volume = 56 | issue = | pages = 104–123 }}</ref> This procedure is made available through SPSS's user interface. See Courtney (2013)<ref name="pareonline.net"/> for guidance. This is one of his five recommended modern procedures.
===Parallel analysis===
===Convergence of multiple tests===
A review of 60 journal articles by Henson and Roberts (2006) found that none used multiple modern techniques in an attempt to find convergence, such as PA and Velicer’sVelicer's (1976) minimum average partial (MAP) procedures. Ruscio and Roche (2012) simulation study demonstrated the empirical advantage of seeking convergence. When the CD and PA procedures agreed, the accuracy of the estimated number of factors was correct 92.2% of the time. Ruscio and Roche (2012) demonstrated that when further tests were in agreement, the accuracy of the estimation could be increased even further.<ref name="pareonline.net"/>
===Tailoring Courtney's recommended procedures for ordinal and continuous data===
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