faRotate.RdA dirty little secret of factor rotation algorithms is the problem of local minima (Nguyen and Waller,2022). Following ideas in that article, we allow for multiple random restarts and then return the global optimal solution. Used as part of the fa function or available as a stand alone function.
faRotations(loadings, r = NULL, rotate = "oblimin", hyper = 0.15, n.rotations = 10,...)Factor loadings matrix from fa or pca or any N x k loadings matrix
The correlation matrix used to find the factors. (Used to find the factor indeterminancy of the solution)
"none", "varimax", "quartimax", "bentlerT", "equamax", "varimin", "geominT" and "bifactor" are orthogonal rotations. "Promax", "promax", "oblimin", "simplimax", "bentlerQ, "geominQ" and "biquartimin" and "cluster" are possible oblique transformations of the solution. Defaults to oblimin.
The value defining when a loading is in the “hyperplane".
The number of random restarts to use.
additional parameters, specifically, keys may be passed if using the target rotation, or delta if using geominQ, or whether to normalize if using Varimax
Nguyen and Waller review the problem of local minima in factor analysis. This is a problem for all rotation algorithms, but is more so for some. faRotate generates n.rotations different starting values and then applies the specified rotation to the original loadings using multiple start values. Hyperplane counts and complexity indices are reported for each starting matrix, and the one with the highest hyoerplane count and the lowest complexity is returned.
The best rotated solution
Factor correlations
Hyperplane count, complexity.
The rotation matrix used.
Nguyen, H. V., & Waller, N. G. (2022, January 6). Local Minima and Factor Rotations in Exploratory Factor Analysis. Psychological Methods. Advance online publication. doi 10.1037/met0000467
Adapted from the fungible package by Waller
f5 <- fa(bfi[,1:25],5,rotate="none")
faRotations(f5,n.rotations=10) #note that the factor analysis needs to not do the rotation
#> Factor Analysis using method =
#> Call: faRotations(loadings = f5, n.rotations = 10)
#> Standardized loadings (pattern matrix) based upon correlation matrix
#> MR1 MR2 MR3 MR4 MR5 h2 u2
#> A1 -0.07 -0.21 0.17 -0.06 -0.41 0.19 0.81
#> A2 -0.08 0.02 0.00 0.03 0.64 0.45 0.55
#> A3 -0.02 0.03 0.12 0.03 0.66 0.52 0.48
#> A4 -0.19 0.06 0.06 -0.15 0.43 0.28 0.72
#> A5 -0.01 0.11 0.23 0.04 0.53 0.46 0.54
#> C1 -0.55 -0.07 -0.03 0.15 -0.02 0.33 0.67
#> C2 -0.67 -0.15 -0.09 0.04 0.08 0.45 0.55
#> C3 -0.57 -0.03 -0.06 -0.07 0.09 0.32 0.68
#> C4 0.61 -0.17 0.00 -0.05 0.04 0.45 0.55
#> C5 0.55 -0.19 -0.14 0.09 0.02 0.43 0.57
#> E1 -0.11 0.06 -0.56 -0.10 -0.08 0.35 0.65
#> E2 0.02 -0.10 -0.68 -0.06 -0.05 0.54 0.46
#> E3 0.00 -0.08 0.42 0.28 0.25 0.44 0.56
#> E4 -0.02 -0.01 0.59 -0.08 0.29 0.53 0.47
#> E5 -0.27 -0.15 0.42 0.21 0.05 0.40 0.60
#> N1 0.00 -0.81 0.10 -0.05 -0.11 0.65 0.35
#> N2 -0.01 -0.78 0.04 0.01 -0.09 0.60 0.40
#> N3 0.04 -0.71 -0.10 0.02 0.08 0.55 0.45
#> N4 0.14 -0.47 -0.39 0.08 0.09 0.49 0.51
#> N5 0.00 -0.49 -0.20 -0.15 0.21 0.35 0.65
#> O1 -0.07 -0.02 0.10 0.51 0.02 0.31 0.69
#> O2 0.08 -0.19 0.06 -0.46 0.16 0.26 0.74
#> O3 -0.02 -0.03 0.15 0.61 0.08 0.46 0.54
#> O4 0.02 -0.13 -0.32 0.37 0.17 0.25 0.75
#> O5 0.03 -0.13 0.10 -0.54 0.04 0.30 0.70
#>
#> MR1 MR2 MR3 MR4 MR5
#> SS loadings 2.03 2.57 2.20 1.59 1.99
#> Proportion Var 0.08 0.10 0.09 0.06 0.08
#> Cumulative Var 0.08 0.18 0.27 0.34 0.41
#> Proportion Explained 0.20 0.25 0.21 0.15 0.19
#> Cumulative Proportion 0.20 0.44 0.66 0.81 1.00
#>
#> With factor correlations of
#> MR1 MR2 MR3 MR4 MR5
#> MR1 1.00 -0.19 -0.23 -0.19 -0.20
#> MR2 -0.19 1.00 0.21 0.01 0.04
#> MR3 -0.23 0.21 1.00 0.17 0.33
#> MR4 -0.19 0.01 0.17 1.00 0.19
#> MR5 -0.20 0.04 0.33 0.19 1.00
faRotations(f5$loadings) #matrix input
#>
#> Call: faRotations(loadings = f5$loadings)
#> Standardized loadings (pattern matrix) based upon correlation matrix
#> MR1 MR2 MR3 MR4 MR5 h2 u2
#> A1 0.21 -0.17 0.41 -0.06 -0.07 0.19 0.81
#> A2 -0.02 0.00 -0.64 0.03 -0.08 0.45 0.55
#> A3 -0.03 -0.12 -0.66 0.03 -0.02 0.52 0.48
#> A4 -0.06 -0.06 -0.43 -0.15 -0.19 0.28 0.72
#> A5 -0.11 -0.23 -0.53 0.04 -0.01 0.46 0.54
#> C1 0.07 0.03 0.02 0.15 -0.55 0.33 0.67
#> C2 0.15 0.09 -0.08 0.04 -0.67 0.45 0.55
#> C3 0.03 0.06 -0.09 -0.07 -0.57 0.32 0.68
#> C4 0.17 0.00 -0.04 -0.05 0.61 0.45 0.55
#> C5 0.19 0.14 -0.02 0.09 0.55 0.43 0.57
#> E1 -0.06 0.56 0.08 -0.10 -0.11 0.35 0.65
#> E2 0.10 0.68 0.05 -0.06 0.02 0.54 0.46
#> E3 0.08 -0.42 -0.25 0.28 0.00 0.44 0.56
#> E4 0.01 -0.59 -0.29 -0.08 -0.02 0.53 0.47
#> E5 0.15 -0.42 -0.05 0.21 -0.27 0.40 0.60
#> N1 0.81 -0.10 0.11 -0.05 0.00 0.65 0.35
#> N2 0.78 -0.04 0.09 0.01 -0.01 0.60 0.40
#> N3 0.71 0.10 -0.08 0.02 0.04 0.55 0.45
#> N4 0.47 0.39 -0.09 0.08 0.14 0.49 0.51
#> N5 0.49 0.20 -0.21 -0.15 0.00 0.35 0.65
#> O1 0.02 -0.10 -0.02 0.51 -0.07 0.31 0.69
#> O2 0.19 -0.06 -0.16 -0.46 0.08 0.26 0.74
#> O3 0.03 -0.15 -0.08 0.61 -0.02 0.46 0.54
#> O4 0.13 0.32 -0.17 0.37 0.02 0.25 0.75
#> O5 0.13 -0.10 -0.04 -0.54 0.03 0.30 0.70
#>
#> MR1 MR2 MR3 MR4 MR5
#> SS loadings 2.57 2.20 1.99 1.59 2.03
#> Proportion Var 0.10 0.09 0.08 0.06 0.08
#> Cumulative Var 0.10 0.19 0.27 0.33 0.41
#> Proportion Explained 0.25 0.21 0.19 0.15 0.20
#> Cumulative Proportion 0.25 0.46 0.65 0.80 1.00
#> MR1 MR2 MR3 MR4 MR5
#> MR1 1.00 0.21 0.04 -0.01 0.19
#> MR2 0.21 1.00 0.33 -0.17 0.23
#> MR3 0.04 0.33 1.00 -0.19 0.20
#> MR4 -0.01 -0.17 -0.19 1.00 -0.19
#> MR5 0.19 0.23 0.20 -0.19 1.00
geo <- faRotations(f5,rotate="geominQ",n.rotation=10)
# a popular alternative, but more sensitive to local minima
describe(geo$rotation.stats[,1:3])
#> vars n mean sd median trimmed mad min max range skew kurtosis se
#> hyperplane 1 10 0.61 0 0.61 0.61 0 0.61 0.61 0 NaN NaN 0
#> fit 2 10 0.10 0 0.10 0.10 0 0.10 0.10 0 0 -2.16 0
#> complexity 3 10 1.45 0 1.45 1.45 0 1.45 1.45 0 0 -2.18 0