5.11 Unrotated Factor Solution
According to the eigenvalue, scree plot and parallel analysis, we are to retain 4, 3 and 6 factors respectively. We will explore all these factors and then look out for best factor structure with good model data fit.
First, let’s explore the unrotated factor solution. Since the data is larger than 350, we can set a minimum factor loading of 0.3. In the code, we set rotate = "none" because we want results for unrotated solution; nfactors = 3 indicate the number of factor to extract, cut = 0.30 to get results with minimum loading of 0.30. Finally, we set sort = T so that we get results from the highest factor to the least.
# three factor solution
unrotated_three_factors <- fa(data.us.only, nfactors = 3, rotate = "none")
print(unrotated_three_factors, sort = T, cut = .30)## Factor Analysis using method = minres
## Call: fa(r = data.us.only, nfactors = 3, rotate = "none")
## Standardized loadings (pattern matrix) based upon correlation matrix
## item MR1 MR2 MR3 h2 u2 com
## COMPICT2 8 0.68 0.47 0.53 1.1
## AUICT4 15 0.67 -0.46 0.66 0.34 1.8
## COMPICT5 11 0.67 0.48 0.52 1.2
## COMPICT4 10 0.65 0.47 0.53 1.3
## AUICT3 14 0.63 0.48 0.52 1.4
## INTICT4 4 0.63 0.49 0.51 1.5
## AUICT5 16 0.61 0.47 0.53 1.5
## AUICT2 13 0.61 -0.31 0.53 0.47 1.9
## SOIAICT4 20 0.60 -0.55 0.72 0.28 2.3
## INTICT6 6 0.60 0.39 0.57 0.43 2.1
## COMPICT3 9 0.59 0.37 0.49 0.51 1.7
## AUICT1 12 0.57 -0.37 0.50 0.50 1.9
## SOIAICT2 18 0.57 -0.48 0.60 0.40 2.3
## SOIAICT5 21 0.56 -0.46 0.59 0.41 2.4
## COMPICT1 7 0.52 0.30 0.70 1.2
## SOIAICT3 19 0.52 -0.37 0.43 0.57 2.0
## SOIAICT1 17 0.50 -0.34 0.43 0.57 2.3
## INTICT3 3 0.50 0.42 0.50 0.50 2.6
## INTICT2 2 0.42 0.40 0.36 0.64 2.3
## INTICT5 5 0.38 0.23 0.77 2.1
## INTICT1 1 0.36 0.25 0.75 2.6
##
## MR1 MR2 MR3
## SS loadings 6.84 1.91 1.28
## Proportion Var 0.33 0.09 0.06
## Cumulative Var 0.33 0.42 0.48
## Proportion Explained 0.68 0.19 0.13
## Cumulative Proportion 0.68 0.87 1.00
##
## Mean item complexity = 1.9
## Test of the hypothesis that 3 factors are sufficient.
##
## df null model = 210 with the objective function = 9.5 with Chi Square = 51537.58
## df of the model are 150 and the objective function was 1.04
##
## The root mean square of the residuals (RMSR) is 0.04
## The df corrected root mean square of the residuals is 0.05
##
## The harmonic n.obs is 5433 with the empirical chi square 3868.82 with prob < 0
## The total n.obs was 5433 with Likelihood Chi Square = 5633.66 with prob < 0
##
## Tucker Lewis Index of factoring reliability = 0.85
## RMSEA index = 0.082 and the 90 % confidence intervals are 0.08 0.084
## BIC = 4343.62
## Fit based upon off diagonal values = 0.99
## Measures of factor score adequacy
## MR1 MR2 MR3
## Correlation of (regression) scores with factors 0.97 0.90 0.86
## Multiple R square of scores with factors 0.93 0.81 0.74
## Minimum correlation of possible factor scores 0.87 0.62 0.47For the unrotated three factor solution, it is evident that all the items load on a single factor. This doesn’t align with our expectation.
Next, we can explore the unrotated four factors and six factor solution.
# four factor solution
unrotated_four_factors <- fa(data.us.only, nfactors = 4, rotate = "none")
print(unrotated_four_factors, sort = T, cut = .30)## Factor Analysis using method = minres
## Call: fa(r = data.us.only, nfactors = 4, rotate = "none")
## Standardized loadings (pattern matrix) based upon correlation matrix
## item MR1 MR2 MR3 MR4 h2 u2 com
## COMPICT2 8 0.69 -0.31 0.59 0.41 1.5
## COMPICT5 11 0.68 -0.39 0.67 0.33 1.9
## AUICT4 15 0.67 -0.45 0.68 0.32 1.9
## COMPICT4 10 0.66 -0.35 0.62 0.38 1.8
## AUICT3 14 0.64 0.56 0.44 1.8
## INTICT4 4 0.62 0.49 0.51 1.6
## AUICT5 16 0.62 0.54 0.46 1.8
## AUICT2 13 0.61 -0.30 0.54 0.46 2.0
## INTICT6 6 0.60 0.38 0.57 0.43 2.1
## SOIAICT4 20 0.60 -0.55 0.72 0.28 2.3
## COMPICT3 9 0.59 0.37 0.50 0.50 1.8
## AUICT1 12 0.57 -0.36 0.51 0.49 2.1
## SOIAICT2 18 0.56 -0.48 0.59 0.41 2.3
## SOIAICT5 21 0.56 -0.46 0.59 0.41 2.4
## COMPICT1 7 0.52 0.34 0.66 1.5
## SOIAICT3 19 0.51 -0.37 0.43 0.57 2.0
## SOIAICT1 17 0.50 -0.34 0.43 0.57 2.3
## INTICT3 3 0.50 0.42 0.54 0.46 2.9
## INTICT2 2 0.42 0.40 0.40 0.60 2.6
## INTICT5 5 0.38 0.23 0.77 2.1
## INTICT1 1 0.36 0.26 0.74 2.7
##
## MR1 MR2 MR3 MR4
## SS loadings 6.89 1.92 1.31 0.70
## Proportion Var 0.33 0.09 0.06 0.03
## Cumulative Var 0.33 0.42 0.48 0.51
## Proportion Explained 0.64 0.18 0.12 0.06
## Cumulative Proportion 0.64 0.81 0.94 1.00
##
## Mean item complexity = 2.1
## Test of the hypothesis that 4 factors are sufficient.
##
## df null model = 210 with the objective function = 9.5 with Chi Square = 51537.58
## df of the model are 132 and the objective function was 0.52
##
## The root mean square of the residuals (RMSR) is 0.03
## The df corrected root mean square of the residuals is 0.03
##
## The harmonic n.obs is 5433 with the empirical chi square 1598.07 with prob < 5.9e-250
## The total n.obs was 5433 with Likelihood Chi Square = 2829.87 with prob < 0
##
## Tucker Lewis Index of factoring reliability = 0.916
## RMSEA index = 0.061 and the 90 % confidence intervals are 0.059 0.063
## BIC = 1694.64
## Fit based upon off diagonal values = 0.99
## Measures of factor score adequacy
## MR1 MR2 MR3 MR4
## Correlation of (regression) scores with factors 0.97 0.90 0.87 0.79
## Multiple R square of scores with factors 0.94 0.82 0.75 0.63
## Minimum correlation of possible factor scores 0.88 0.63 0.50 0.26# six factor solution
unrotated_six_factors <- fa(data.us.only, nfactors = 6, rotate = "none")
print(unrotated_six_factors, sort = T, cut = .30)## Factor Analysis using method = minres
## Call: fa(r = data.us.only, nfactors = 6, rotate = "none")
## Standardized loadings (pattern matrix) based upon correlation matrix
## item MR1 MR2 MR3 MR4 MR5 MR6 h2 u2 com
## COMPICT2 8 0.68 -0.32 0.58 0.42 1.5
## COMPICT5 11 0.68 -0.40 0.67 0.33 1.9
## AUICT4 15 0.67 -0.43 0.65 0.35 1.8
## COMPICT4 10 0.66 -0.36 0.63 0.37 1.9
## AUICT3 14 0.65 0.30 0.69 0.31 2.4
## INTICT4 4 0.62 0.52 0.48 1.7
## AUICT5 16 0.62 0.60 0.40 2.2
## AUICT2 13 0.61 -0.31 0.64 0.36 2.6
## INTICT6 6 0.60 0.38 0.57 0.43 2.1
## SOIAICT4 20 0.60 -0.54 0.72 0.28 2.4
## COMPICT3 9 0.59 0.37 0.54 0.46 2.0
## AUICT1 12 0.58 -0.38 0.63 0.37 2.8
## SOIAICT2 18 0.56 -0.47 0.59 0.41 2.3
## SOIAICT5 21 0.55 -0.45 0.60 0.40 2.5
## COMPICT1 7 0.52 0.35 0.65 1.6
## SOIAICT3 19 0.51 -0.36 0.43 0.57 2.1
## SOIAICT1 17 0.50 -0.34 0.46 0.54 2.6
## INTICT3 3 0.50 0.43 0.56 0.44 3.1
## INTICT5 5 0.40 0.38 0.46 0.54 3.8
## INTICT1 1 0.36 0.27 0.73 3.1
## INTICT2 2 0.44 0.45 0.30 0.59 0.41 4.0
##
## MR1 MR2 MR3 MR4 MR5 MR6
## SS loadings 6.92 1.96 1.35 0.74 0.42 0.37
## Proportion Var 0.33 0.09 0.06 0.04 0.02 0.02
## Cumulative Var 0.33 0.42 0.49 0.52 0.54 0.56
## Proportion Explained 0.59 0.17 0.12 0.06 0.04 0.03
## Cumulative Proportion 0.59 0.75 0.87 0.93 0.97 1.00
##
## Mean item complexity = 2.4
## Test of the hypothesis that 6 factors are sufficient.
##
## df null model = 210 with the objective function = 9.5 with Chi Square = 51537.58
## df of the model are 99 and the objective function was 0.21
##
## The root mean square of the residuals (RMSR) is 0.01
## The df corrected root mean square of the residuals is 0.02
##
## The harmonic n.obs is 5433 with the empirical chi square 478.16 with prob < 4.9e-51
## The total n.obs was 5433 with Likelihood Chi Square = 1121.08 with prob < 9.3e-173
##
## Tucker Lewis Index of factoring reliability = 0.958
## RMSEA index = 0.044 and the 90 % confidence intervals are 0.041 0.046
## BIC = 269.65
## Fit based upon off diagonal values = 1
## Measures of factor score adequacy
## MR1 MR2 MR3 MR4 MR5 MR6
## Correlation of (regression) scores with factors 0.97 0.91 0.88 0.81 0.70 0.69
## Multiple R square of scores with factors 0.94 0.83 0.77 0.66 0.48 0.48
## Minimum correlation of possible factor scores 0.89 0.65 0.54 0.32 -0.03 -0.05Results from the unrotated four and six factor solution also show that almost all the items load on one factor only.
As you can see, it is quite challenging to identify a clear pattern for the unrotated factor solution. To overcome this challenge, we need to rotate the factors to get a finer solution.