Iterative Target Rotation with a Suboptimal Number of Factors - PowerPoint Presentation

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Iterative Target Rotation with a Suboptimal Number of FactorsNicole Zelinsky - University of California, Merced - nzelinsky@ucmerced.edu

Introduction and Motivation

Exploratory Factor AnalysisAnalytic tool which helps researchers develop scales, generate theory, and inform structure for a confirmatory factor analysisTarget RotationPartially specify what the final solution might look like Determining what the partially specified matrix contains can be subjectiveIterative Target Rotation (ITR)Requires less user input than regular target rotationResearcher specifies the target matrix based on information from previous rotationMoore (2013) and Moore et al. (2015) tested rotation method through simulation Found ITR had better factor loadings estimates than the traditional single rotationsFocused estimates based on the optimal number of factorsProvided little detail on how ITR estimates based on suboptimal factorsPresent ResearchImpossible to tell whether the number of factors in a simulation is optimal or notImportant to know if ITR estimates are also more accurate when a suboptimal number of factors are extractedIllustrates exploratory factor analysis using data with ITRDifferences between true parameters and parameter estimates from a single factor and ITR.

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 methodChoose a cutoff value for a factor loading that is ‘too small’Form a new target matrixPerform a target rotation using the target matrixReturn to step 3 and form another target matrix until factor loadings remain the same between iterations

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=14515Moore, 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/10.1080/00273171.2014.973990Thompson, B. (2004). Exploratory and confirmatory factor analysis: Understanding concepts and applications. Washington, DC: American Psychological Association

Simulation Cells

References

Initial (Traditional

) Rotation Output

1230.520.04-0.060.300.310.26-0.050.55-0.080.180.530.340.44-0.020.450.28-0.070.2

Average differences in factor loading estimates between the initial rotation and ITRs

Data Generating Model

Illustration of Iterative Target Rotation

1

st

Target Matrix123?00???0?0????0??0?

Output from Target Matrix1230.510.000.010.520.270.260.000.380.020.280.530.240.220.080.530.01-0.010.51

2nd Target Matrix123?00???0?0????0?00?

Single Crawford Ferguson Rotation at k = 0.4Factor 1Factor 2Factor 1Factor 2Factor 3Item 10.9321-0.29340.0390.0960.921Item 20.8413-0.22180.0150.2190.746Item 30.7741-0.07700.1030.2750.579Item 40.8225-0.14470.0900.2030.693Item 50.59810.70120.8200.2170.004Item 60.54760.62600.7900.0810.066Item 70.65090.63560.7980.1710.108Item 80.62660.52030.6660.1910.145Item 90.48740.19910.0760.665-0.004Item 100.49140.17730.0230.737-0.021Item 110.54100.1136-0.0200.7370.059Item 120.4783-0.0235-0.0260.4120.233

Average of 4 Cross LoadingsSample Size2 Factors3 Factors4 Factors5 Factorsn = 250QuartiminParsimaxk = 0.4Facparsim0.00170.21140.18140.00000.003360.004620.118520.115970.002490.095610.092570.000250.006140.096860.094650.00099n = 1000QuartiminParsimaxk = 0.4Facparsim0.00000.20930.17790.00000.000000.00268 0.123640.121340.00220.10970.10700.00000.004390.117510.118540.00369Average of 24 Cross Loadingsn = 250QuartiminParsimaxk = 0.4Facparsim0.01380.17430.14240.00000.016280.004130.088040.086280.013830.078760.071840.000000.014060.076530.074180.00375n = 1000QuartiminParsimaxk = 0.4Facparsim0.01050.17390.14180.00000.017290.008620.103130.096650.021630.088720.082470.000480.014590.092710.091030.00059

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.

Iterative Target RotationFactor 1Factor 2Factor 1Factor 2Factor 3Item 11.0057−0.11260.9791-0.00480.0034Item 20.9267−0.08580.7915-0.02480.1483Item 30.74470.07000.60960.07450.2139Item 40.84630.00130.73220.05740.13Item 5−0.06760.9604-0.02950.85500.1489Item 6−0.05670.86290.04020.82420.007Item 70.02720.89490.08300.82880.0934Item 80.10930.75150.12690.68820.1223Item 90.28800.3016-0.01600.06330.6712Item 100.31050.2738-0.03290.00630.7508Item 110.41770.20510.0530-0.04210.7469Item 120.45290.05610.2438-0.04890.3993

LibQUAL+ Subsection: Twelve Library Quality ItemsData from Thompson (2004)Scale with 9-points from “Low” to “High”Rotation OptionsSingle rotationIterative target rotationTwo or Three Factors2 FactorsUtilitarian Environment3 FactorsResourcesHelpful Staff EnvironmentNumber of FactorsWays to determine numberWork under different circumstances

Average of 4 Cross LoadingsSample Size2 Factors3 Factors4 Factors5 Factorsn = 250Q/P/0.4/FQ/P/0.4/FQ/P/0.4/FQ/P/0.4/Fn = 1000Q/P/0.4/FQ/P/0.4/FQ/P/0.4/FQ/P/0.4/FAverage of 24 Cross Loadingsn = 250Q/P/0.4/FQ/P/0.4/FQ/P/0.4/FQ/P/0.4/Fn = 1000Q/P/0.4/FQ/P/0.4/FQ/P/0.4/FQ/P/0.4/F

Simulation Results

LibQUAL+ Main factor loadings between 0.5 and 0.6; factor cross loadings .25Factor correlations between 0.1 and 0.4Cutoff to form the target matrix at 0.1

Four different types of initial rotation were used: Q = Quartimin, P = Parsimax, 0.4 = Crawford Family k = .04, F = Facparsim

All factor loadings below a cut off are specified as

0

All values above the cut off are freely estimated (specified as

?)