1. Iterative Target Rotation with a Suboptimal Number of Factors Nicole Zelinsky - University of California, Merced -
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
0.039
0.096
0.921
Item 2
0.8413
0.015
0.219
0.746
Item 3
0.7741
0.103
0.275
0.579
Item 4
0.8225
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
-0.026
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
0.0034
Item 2
0.9267
−0.0858
0.7915
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
0.8550
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
0.0633
0.6712
Item 10
0.3105
0.2738
0.0063
0.7508
Item 11
0.4177
0.2051
0.0530
0.7469
Item 12
0.4529
0.0561
0.2438
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
n = 1000
0.2093
0.1779
0.0022
0.1097
0.1070
Average of 24 Cross Loadings
0.0138
0.1743
0.1424
0.0105
0.1739
0.1418
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. AAI ). Available from PsycINFO. ( ; ). Retrieved from
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),
Thompson, B. (2004). Exploratory and confirmatory factor analysis: Understanding concepts and applications. Washington, DC: American Psychological Association
References