Iterative Target Rotation with a Suboptimal Number of Factors Nicole Zelinsky - University of California, Merced - nzelinsky@ucmerced.edu Introduction and Motivation Illustration of Iterative Target Rotation Simulation Cells Exploratory Factor Analysis Analytic tool which helps researchers develop scales, generate theory, and inform structure for a confirmatory factor analysis Target Rotation Partially specify what the final solution might look like Determining what the partially specified matrix contains can be subjective Iterative Target Rotation (ITR) Requires less user input than regular target rotation Researcher specifies the target matrix based on information from previous rotation Moore (2013) and Moore et al. (2015) tested rotation method through simulation Found ITR had better factor loadings estimates than the traditional single rotations Focused estimates based on the optimal number of factors Provided little detail on how ITR estimates based on suboptimal factors Present Research Impossible to tell whether the number of factors in a simulation is optimal or not Important to know if ITR estimates are also more accurate when a suboptimal number of factors are extracted Illustrates exploratory factor analysis using data with ITR Differences between true parameters and parameter estimates from a single factor and ITR. Four different types of initial rotation were used: Q = Quartimin, P = Parsimax, 0.4 = Crawford Family k = .04, F = Facparsim Single Crawford Ferguson Rotation at k = 0.4 Factor 1 Factor 2 Factor 3 Item 1 0.9321 -0.2934 0.039 0.096 0.921 Item 2 0.8413 -0.2218 0.015 0.219 0.746 Item 3 0.7741 -0.0770 0.103 0.275 0.579 Item 4 0.8225 -0.1447 0.090 0.203 0.693 Item 5 0.5981 0.7012 0.820 0.217 0.004 Item 6 0.5476 0.6260 0.790 0.081 0.066 Item 7 0.6509 0.6356 0.798 0.171 0.108 Item 8 0.6266 0.5203 0.666 0.191 0.145 Item 9 0.4874 0.1991 0.076 0.665 -0.004 Item 10 0.4914 0.1773 0.023 0.737 -0.021 Item 11 0.5410 0.1136 -0.020 0.059 Item 12 0.4783 -0.0235 -0.026 0.412 0.233 LibQUAL+ Subsection: Twelve Library Quality Items Data from Thompson (2004) Scale with 9-points from “Low” to “High” Rotation Options Single rotation Iterative target rotation Two or Three Factors 2 Factors Utilitarian Environment 3 Factors Resources Helpful Staff Number of Factors Ways to determine number Work under different circumstances Average of 4 Cross Loadings Sample Size 2 Factors 3 Factors 4 Factors 5 Factors n = 250 Q/P/0.4/F n = 1000 Average of 24 Cross Loadings Simulation Results Iterative Target Rotation Factor 1 Factor 2 Factor 3 Item 1 1.0057 −0.1126 0.9791 -0.0048 0.0034 Item 2 0.9267 −0.0858 0.7915 -0.0248 0.1483 Item 3 0.7447 0.0700 0.6096 0.0745 0.2139 Item 4 0.8463 0.0013 0.7322 0.0574 0.13 Item 5 −0.0676 0.9604 -0.0295 0.8550 0.1489 Item 6 −0.0567 0.8629 0.0402 0.8242 0.007 Item 7 0.0272 0.8949 0.0830 0.8288 0.0934 Item 8 0.1093 0.7515 0.1269 0.6882 0.1223 Item 9 0.2880 0.3016 -0.0160 0.0633 0.6712 Item 10 0.3105 0.2738 -0.0329 0.0063 0.7508 Item 11 0.4177 0.2051 0.0530 -0.0421 0.7469 Item 12 0.4529 0.0561 0.2438 -0.0489 0.3993 Average differences in factor loading estimates between the initial rotation and ITRs Average of 4 Cross Loadings Sample Size 2 Factors 3 Factors 4 Factors 5 Factors n = 250 Quartimin Parsimax k = 0.4 Facparsim 0.0017 0.2114 0.1814 0.0000 0.00336 0.00462 0.11852 0.11597 0.00249 0.09561 0.09257 0.00025 0.00614 0.09686 0.09465 0.00099 n = 1000 0.2093 0.1779 0.00000 0.00268 0.12364 0.12134 0.0022 0.1097 0.1070 0.00439 0.11751 0.11854 0.00369 Average of 24 Cross Loadings 0.0138 0.1743 0.1424 0.01628 0.00413 0.08804 0.08628 0.01383 0.07876 0.07184 0.01406 0.07653 0.07418 0.00375 0.0105 0.1739 0.1418 0.01729 0.00862 0.10313 0.09665 0.02163 0.08872 0.08247 0.00048 0.01459 0.09271 0.09103 0.00059 Model Description Iterative Target Rotation (Moore et al., 2015) Extraction for exploratory factor analysis remains the same, only rotation changes Start with a standard analytic rotation method Choose a cutoff value for a factor loading that is ‘too small’ Form a new target matrix Perform a target rotation using the target matrix Return to step 3 and form another target matrix until factor loadings remain the same between iterations Data Generating Model LibQUAL+ Main factor loadings between 0.5 and 0.6; factor cross loadings .25 Factor correlations between 0.1 and 0.4 Cutoff to form the target matrix at 0.1 All factor loadings below a cut off are specified as 0 All values above the cut off are freely estimated (specified as ?) Conclusions ITR works just as well or better than single rotation for correctly specified and mis-specified factors across the range of Crawford Family rotations. ITR worked slightly better (~0.161) for cells with an average of 4 than 24 cross loadings and slightly better (~0.068) for cells with the 1,000 sample size than 250. Initial (Traditional) Rotation Output 1 2 3 0.52 0.04 -0.06 0.30 0.31 0.26 -0.05 0.55 -0.08 0.18 0.53 0.34 0.44 -0.02 0.45 0.28 -0.07 0.2 Output from Target Matrix 1 2 3 0.51 0.00 0.01 0.52 0.27 0.26 0.38 0.02 0.28 0.53 0.24 0.22 0.08 -0.01 2nd Target Matrix 1 2 3 ? 1st Target Matrix 1 2 3 ? Moore, T. M. (2013). Iteration of target matrices in exploratory factor analysis (Order No. AAI3551433). Available from PsycINFO. (1499086987; 2013-99240-026). Retrieved from http://search.proquest.com/docview/1499086987?accountid=14515 Moore, T. M., Reise, S. P., Depaoli, S., & Haviland, M. G. (2015). Iteration of partially specified target matrices: Applications in exploratory and bayesian confirmatory factor analysis. Multivariate Behavioral Research, 50(2), 149-161. http://dx.doi.org/10.1080/00273171.2014.973990 Thompson, B. (2004). Exploratory and confirmatory factor analysis: Understanding concepts and applications. Washington, DC: American Psychological Association References